LMFDB - Embedded newform 378.4.k.a.269.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 11x^{10} + 88x^{8} - 331x^{6} + 913x^{4} - 528x^{2} + 256 $ x^12 - 11x^10 + 88x^8 - 331x^6 + 913x^4 - 528*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.3
Root			$-2.07100 - 1.19569i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.a.215.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(4.70894 + 8.15613i) q^{5} +(8.82127 - 16.2845i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(4.70894 + 8.15613i) q^{5} +(8.82127 - 16.2845i) q^{7} +8.00000i q^{8} +(-16.3123 - 9.41789i) q^{10} +(23.3903 + 13.5044i) q^{11} -2.58331i q^{13} +(1.00563 + 37.0269i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(29.4673 - 51.0388i) q^{17} +(-17.7852 + 10.2683i) q^{19} +37.6715 q^{20} -54.0177 q^{22} +(42.2502 - 24.3932i) q^{23} +(18.1517 - 31.4397i) q^{25} +(2.58331 + 4.47442i) q^{26} +(-38.7687 - 63.1268i) q^{28} -0.0600052i q^{29} +(-151.031 - 87.1976i) q^{31} +(27.7128 + 16.0000i) q^{32} +117.869i q^{34} +(174.357 - 4.73544i) q^{35} +(22.7628 + 39.4263i) q^{37} +(20.5366 - 35.5705i) q^{38} +(-65.2490 + 37.6715i) q^{40} +166.852 q^{41} +28.8256 q^{43} +(93.5614 - 54.0177i) q^{44} +(-48.7864 + 84.5004i) q^{46} +(92.4868 + 160.192i) q^{47} +(-187.370 - 287.300i) q^{49} +72.6068i q^{50} +(-8.94884 - 5.16662i) q^{52} +(564.433 + 325.875i) q^{53} +254.366i q^{55} +(130.276 + 70.5701i) q^{56} +(0.0600052 + 0.103932i) q^{58} +(234.651 - 406.428i) q^{59} +(643.906 - 371.759i) q^{61} +348.790 q^{62} -64.0000 q^{64} +(21.0698 - 12.1647i) q^{65} +(231.332 - 400.679i) q^{67} +(-117.869 - 204.155i) q^{68} +(-297.260 + 182.559i) q^{70} +519.129i q^{71} +(892.655 + 515.375i) q^{73} +(-78.8526 - 45.5256i) q^{74} +82.1465i q^{76} +(426.245 - 261.774i) q^{77} +(92.9573 + 161.007i) q^{79} +(75.3431 - 130.498i) q^{80} +(-288.996 + 166.852i) q^{82} -120.345 q^{83} +555.039 q^{85} +(-49.9274 + 28.8256i) q^{86} +(-108.035 + 187.123i) q^{88} +(503.915 + 872.807i) q^{89} +(-42.0679 - 22.7881i) q^{91} -195.145i q^{92} +(-320.384 - 184.974i) q^{94} +(-167.499 - 96.7058i) q^{95} +1244.91i q^{97} +(611.835 + 310.248i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{4} - 42 q^{7}+O(q^{10}) $ 12 * q + 24 * q^4 - 42 * q^7	$ 12 q + 24 q^{4} - 42 q^{7} - 48 q^{10} - 96 q^{16} - 66 q^{19} + 240 q^{22} + 366 q^{25} - 168 q^{28} + 630 q^{31} + 348 q^{37} - 192 q^{40} - 252 q^{43} + 600 q^{46} - 1218 q^{49} + 192 q^{52} - 144 q^{58} + 114 q^{61} - 768 q^{64} + 108 q^{67} - 1344 q^{70} - 978 q^{73} - 744 q^{79} + 240 q^{82} + 7248 q^{85} + 480 q^{88} - 5544 q^{91} - 1920 q^{94}+O(q^{100}) $ 12 * q + 24 * q^4 - 42 * q^7 - 48 * q^10 - 96 * q^16 - 66 * q^19 + 240 * q^22 + 366 * q^25 - 168 * q^28 + 630 * q^31 + 348 * q^37 - 192 * q^40 - 252 * q^43 + 600 * q^46 - 1218 * q^49 + 192 * q^52 - 144 * q^58 + 114 * q^61 - 768 * q^64 + 108 * q^67 - 1344 * q^70 - 978 * q^73 - 744 * q^79 + 240 * q^82 + 7248 * q^85 + 480 * q^88 - 5544 * q^91 - 1920 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	4.70894	+	8.15613i	0.421181	+	0.729506i	0.996055	−	0.0887352i	$-0.0282825\pi$
$5$	4.70894	+	8.15613i	0.421181	+	0.729506i	−0.574875	+	0.818242i	$0.694949\pi$
$6$		0			0
$7$	8.82127	−	16.2845i	0.476304	−	0.879281i
$7$	8.82127	−	16.2845i	0.476304	−	0.879281i
$8$			8.00000i			0.353553i
$9$		0			0
$10$	−16.3123	−	9.41789i	−0.515839	−	0.297820i
$11$	23.3903	+	13.5044i	0.641132	+	0.370158i	0.785051	−	0.619432i	$-0.212637\pi$
$11$	23.3903	+	13.5044i	0.641132	+	0.370158i	−0.143918	+	0.989590i	$0.545970\pi$
$12$		0			0
$13$		−	2.58331i		−	0.0551139i	−0.999620	−	0.0275570i	$-0.991227\pi$
$13$		−	2.58331i		−	0.0551139i	0.999620	−	0.0275570i	$-0.00877276\pi$
$14$	1.00563	+	37.0269i	0.0191975	+	0.706846i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	29.4673	−	51.0388i	0.420404	−	0.728161i	−0.575575	−	0.817749i	$-0.695222\pi$
$17$	29.4673	−	51.0388i	0.420404	−	0.728161i	0.995979	+	0.0895879i	$0.0285550\pi$
$18$		0			0
$19$	−17.7852	+	10.2683i	−0.214748	+	0.123985i	−0.603516	−	0.797351i	$-0.706234\pi$
$19$	−17.7852	+	10.2683i	−0.214748	+	0.123985i	0.388768	+	0.921336i	$0.372901\pi$
$20$	37.6715			0.421181
$21$		0			0
$22$	−54.0177			−0.523482
$23$	42.2502	−	24.3932i	0.383034	−	0.221145i	−0.296104	−	0.955156i	$-0.595687\pi$
$23$	42.2502	−	24.3932i	0.383034	−	0.221145i	0.679137	+	0.734011i	$0.262354\pi$
$24$		0			0
$25$	18.1517	−	31.4397i	0.145214	−	0.251517i
$26$	2.58331	+	4.47442i	0.0194857	+	0.0337502i
$27$		0			0
$28$	−38.7687	−	63.1268i	−0.261664	−	0.426066i
$29$		−	0.0600052i		−	0.000384230i	−1.00000		0.000192115i	$-0.999939\pi$
$29$		−	0.0600052i		−	0.000384230i	1.00000		0.000192115i	$-6.11521e-5\pi$
$30$		0			0
$31$	−151.031	−	87.1976i	−0.875029	−	0.505198i	−0.00601293	−	0.999982i	$-0.501914\pi$
$31$	−151.031	−	87.1976i	−0.875029	−	0.505198i	−0.869016	+	0.494784i	$0.835247\pi$
$32$	27.7128	+	16.0000i	0.153093	+	0.0883883i
$33$		0			0
$34$			117.869i			0.594541i
$35$	174.357	−	4.73544i	0.842051	−	0.0228696i
$36$		0			0
$37$	22.7628	+	39.4263i	0.101140	+	0.175180i	0.912155	−	0.409846i	$-0.134418\pi$
$37$	22.7628	+	39.4263i	0.101140	+	0.175180i	−0.811015	+	0.585026i	$0.801084\pi$
$38$	20.5366	−	35.5705i	0.0876705	−	0.151850i
$39$		0			0
$40$	−65.2490	+	37.6715i	−0.257919	+	0.148910i
$41$	166.852			0.635558			0.317779	−	0.948165i	$-0.397063\pi$
$41$	166.852			0.635558			0.317779	+	0.948165i	$0.397063\pi$
$42$		0			0
$43$	28.8256			0.102229			0.0511147	−	0.998693i	$-0.483723\pi$
$43$	28.8256			0.102229			0.0511147	+	0.998693i	$0.483723\pi$
$44$	93.5614	−	54.0177i	0.320566	−	0.185079i
$45$		0			0
$46$	−48.7864	+	84.5004i	−0.156373	+	0.270846i
$47$	92.4868	+	160.192i	0.287034	+	0.497157i	0.973100	−	0.230382i	$-0.0739975\pi$
$47$	92.4868	+	160.192i	0.287034	+	0.497157i	−0.686067	+	0.727539i	$0.740664\pi$
$48$		0			0
$49$	−187.370	−	287.300i	−0.546270	−	0.837609i
$50$			72.6068i			0.205363i
$51$		0			0
$52$	−8.94884	−	5.16662i	−0.0238650	−	0.0137785i
$53$	564.433	+	325.875i	1.46285	+	0.844574i	0.999142	−	0.0414161i	$-0.0131869\pi$
$53$	564.433	+	325.875i	1.46285	+	0.844574i	0.463704	+	0.885990i	$0.346520\pi$
$54$		0			0
$55$			254.366i			0.623613i
$56$	130.276	+	70.5701i	0.310873	+	0.168399i
$57$		0			0
$58$	0.0600052	+	0.103932i	0.000135846	+	0.000235292i
$59$	234.651	−	406.428i	0.517779	−	0.896820i	−0.482007	−	0.876167i	$-0.660092\pi$
$59$	234.651	−	406.428i	0.517779	−	0.896820i	0.999787	−	0.0206531i	$-0.00657454\pi$
$60$		0			0
$61$	643.906	−	371.759i	1.35154	−	0.780309i	0.363071	−	0.931762i	$-0.381728\pi$
$61$	643.906	−	371.759i	1.35154	−	0.780309i	0.988465	+	0.151452i	$0.0483949\pi$
$62$	348.790			0.714458
$63$		0			0
$64$	−64.0000			−0.125000
$65$	21.0698	−	12.1647i	0.0402059	−	0.0232129i
$66$		0			0
$67$	231.332	−	400.679i	0.421817	−	0.730609i	−0.574300	−	0.818645i	$-0.694726\pi$
$67$	231.332	−	400.679i	0.421817	−	0.730609i	0.996117	+	0.0880361i	$0.0280591\pi$
$68$	−117.869	−	204.155i	−0.210202	−	0.364081i
$69$		0			0
$70$	−297.260	+	182.559i	−0.507563	+	0.311715i
$71$			519.129i			0.867736i	0.900977	+	0.433868i	$0.142851\pi$
$71$			519.129i			0.867736i	−0.900977	+	0.433868i	$0.857149\pi$
$72$		0			0
$73$	892.655	+	515.375i	1.43120	+	0.826302i	0.997212	−	0.0746186i	$-0.0237739\pi$
$73$	892.655	+	515.375i	1.43120	+	0.826302i	0.433984	+	0.900920i	$0.357107\pi$
$74$	−78.8526	−	45.5256i	−0.123871	−	0.0715168i
$75$		0			0
$76$			82.1465i			0.123985i
$77$	426.245	−	261.774i	0.630846	−	0.387428i
$78$		0			0
$79$	92.9573	+	161.007i	0.132386	+	0.229300i	0.924596	−	0.380949i	$-0.124403\pi$
$79$	92.9573	+	161.007i	0.132386	+	0.229300i	−0.792210	+	0.610249i	$0.791069\pi$
$80$	75.3431	−	130.498i	0.105295	−	0.182377i
$81$		0			0
$82$	−288.996	+	166.852i	−0.389198	+	0.224704i
$83$	−120.345			−0.159151			−0.0795755	−	0.996829i	$-0.525356\pi$
$83$	−120.345			−0.159151			−0.0795755	+	0.996829i	$0.525356\pi$
$84$		0			0
$85$	555.039			0.708264
$86$	−49.9274	+	28.8256i	−0.0626025	+	0.0361436i
$87$		0			0
$88$	−108.035	+	187.123i	−0.130871	+	0.226674i
$89$	503.915	+	872.807i	0.600167	+	1.03952i	0.992795	+	0.119823i	$0.0382327\pi$
$89$	503.915	+	872.807i	0.600167	+	1.03952i	−0.392628	+	0.919697i	$0.628434\pi$
$90$		0			0
$91$	−42.0679	−	22.7881i	−0.0484606	−	0.0262510i
$92$		−	195.145i		−	0.221145i
$93$		0			0
$94$	−320.384	−	184.974i	−0.351543	−	0.202964i
$95$	−167.499	−	96.7058i	−0.180895	−	0.104440i
$96$		0			0
$97$			1244.91i			1.30311i	0.758600	+	0.651556i	$0.225884\pi$
$97$			1244.91i			1.30311i	−0.758600	+	0.651556i	$0.774116\pi$
$98$	611.835	+	310.248i	0.630660	+	0.319793i
$99$		0			0
$100$	−72.6068	−	125.759i	−0.0726068	−	0.125759i
$101$	160.925	−	278.730i	0.158541	−	0.274601i	−0.775802	−	0.630977i	$-0.782654\pi$
$101$	160.925	−	278.730i	0.158541	−	0.274601i	0.934343	+	0.356376i	$0.115988\pi$
$102$		0			0
$103$	777.847	−	449.090i	0.744112	−	0.429613i	−0.0794503	−	0.996839i	$-0.525317\pi$
$103$	777.847	−	449.090i	0.744112	−	0.429613i	0.823563	+	0.567225i	$0.191983\pi$
$104$	20.6665			0.0194857
$105$		0			0
$106$	−1303.50			−1.19441
$107$	−462.122	+	266.806i	−0.417524	+	0.241057i	−0.694017	−	0.719958i	$-0.744161\pi$
$107$	−462.122	+	266.806i	−0.417524	+	0.241057i	0.276494	+	0.961016i	$0.410828\pi$
$108$		0			0
$109$	311.414	−	539.385i	0.273652	−	0.473979i	−0.696142	−	0.717904i	$-0.745102\pi$
$109$	311.414	−	539.385i	0.273652	−	0.473979i	0.969794	+	0.243925i	$0.0784350\pi$
$110$	−254.366	−	440.575i	−0.220481	−	0.381884i
$111$		0			0
$112$	−296.215	+	8.04501i	−0.249908	+	0.00678734i
$113$			853.663i			0.710671i	0.934739	+	0.355336i	$0.115633\pi$
$113$			853.663i			0.710671i	−0.934739	+	0.355336i	$0.884367\pi$
$114$		0			0
$115$	397.908	+	229.732i	0.322653	+	0.186284i
$116$	−0.207864	−	0.120010i	−0.000166377	−	9.60576e-5i
$117$		0			0
$118$			938.605i			0.732251i
$119$	−571.204	−	930.088i	−0.440018	−	0.716479i
$120$		0			0
$121$	−300.761	−	520.934i	−0.225966	−	0.391385i
$122$	−743.518	+	1287.81i	−0.551762	+	0.955680i
$123$		0			0
$124$	−604.123	+	348.790i	−0.437515	+	0.252599i
$125$	1519.14			1.08701
$126$		0			0
$127$	732.202			0.511594			0.255797	−	0.966731i	$-0.417662\pi$
$127$	732.202			0.511594			0.255797	+	0.966731i	$0.417662\pi$
$128$	110.851	−	64.0000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−24.3293	+	42.1396i	−0.0164140	+	0.0284299i
$131$	−1102.72	−	1909.97i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$131$	−1102.72	−	1909.97i	−0.735460	−	1.27385i	0.219062	−	0.975711i	$-0.429700\pi$
$132$		0			0
$133$	10.3261	+	380.203i	0.00673222	+	0.247878i
$134$			925.329i			0.596539i
$135$		0			0
$136$	408.311	+	235.738i	0.257444	+	0.148635i
$137$	−2395.25	−	1382.90i	−1.49372	−	0.862400i	−0.493747	−	0.869605i	$-0.664373\pi$
$137$	−2395.25	−	1382.90i	−1.49372	−	0.862400i	−0.999974	+	0.00720514i	$0.997707\pi$
$138$		0			0
$139$			198.429i			0.121083i	0.998166	+	0.0605413i	$0.0192827\pi$
$139$			198.429i			0.121083i	−0.998166	+	0.0605413i	$0.980717\pi$
$140$	332.311	−	613.463i	0.200610	−	0.370336i
$141$		0			0
$142$	−519.129	−	899.157i	−0.306791	−	0.531377i
$143$	34.8861	−	60.4245i	0.0204008	−	0.0353353i
$144$		0			0
$145$	0.489410	−	0.282561i	0.000280298	−	0.000161830i
$146$	−2061.50			−1.16857
$147$		0			0
$148$	182.102			0.101140
$149$	−2670.63	+	1541.89i	−1.46837	+	0.847763i	−0.999372	−	0.0354370i	$-0.988718\pi$
$149$	−2670.63	+	1541.89i	−1.46837	+	0.847763i	−0.468997	+	0.883200i	$0.655384\pi$
$150$		0			0
$151$	1222.57	−	2117.55i	0.658883	−	1.14122i	−0.322022	−	0.946732i	$-0.604362\pi$
$151$	1222.57	−	2117.55i	0.658883	−	1.14122i	0.980905	−	0.194487i	$-0.0623043\pi$
$152$	−82.1465	−	142.282i	−0.0438353	−	0.0759249i
$153$		0			0
$154$	−476.505	+	879.652i	−0.249337	+	0.460288i
$155$		−	1642.43i		−	0.851119i
$156$		0			0
$157$	−2503.31	−	1445.29i	−1.27252	−	0.734691i	−0.297060	−	0.954859i	$-0.596006\pi$
$157$	−2503.31	−	1445.29i	−1.27252	−	0.734691i	−0.975462	+	0.220168i	$0.929339\pi$
$158$	−322.013	−	185.915i	−0.162139	−	0.0936112i
$159$		0			0
$160$			301.372i			0.148910i
$161$	−24.5304	−	903.203i	−0.0120079	−	0.442126i
$162$		0			0
$163$	−1632.25	−	2827.13i	−0.784340	−	1.35852i	−0.929393	−	0.369092i	$-0.879669\pi$
$163$	−1632.25	−	2827.13i	−0.784340	−	1.35852i	0.145053	−	0.989424i	$-0.453665\pi$
$164$	333.703	−	577.991i	0.158889	−	0.275204i
$165$		0			0
$166$	208.443	−	120.345i	0.0974597	−	0.0562684i
$167$	−2000.73			−0.927075			−0.463537	−	0.886077i	$-0.653420\pi$
$167$	−2000.73			−0.927075			−0.463537	+	0.886077i	$0.653420\pi$
$168$		0			0
$169$	2190.33			0.996962
$170$	−961.356	+	555.039i	−0.433721	+	0.250409i
$171$		0			0
$172$	57.6512	−	99.8548i	0.0255574	−	0.0442666i
$173$	−1272.74	−	2204.45i	−0.559334	−	0.968795i	−0.997552	−	0.0699262i	$-0.977724\pi$
$173$	−1272.74	−	2204.45i	−0.559334	−	0.968795i	0.438218	−	0.898869i	$-0.355610\pi$
$174$		0			0
$175$	−351.859	−	572.929i	−0.151989	−	0.247482i
$176$		−	432.142i		−	0.185079i
$177$		0			0
$178$	−1745.61	−	1007.83i	−0.735052	−	0.424382i
$179$	3613.10	+	2086.02i	1.50869	+	0.871043i	0.999949	+	0.0101238i	$0.00322255\pi$
$179$	3613.10	+	2086.02i	1.50869	+	0.871043i	0.508742	+	0.860919i	$0.330111\pi$
$180$		0			0
$181$			13.4947i			0.00554174i	0.999996	+	0.00277087i	$0.000881997\pi$
$181$			13.4947i			0.00554174i	−0.999996	+	0.00277087i	$0.999118\pi$
$182$	95.6518	−	2.59784i	0.0389570	−	0.00105805i
$183$		0			0
$184$	195.145	+	338.002i	0.0781865	+	0.135423i
$185$	−214.377	+	371.313i	−0.0851964	+	0.147565i
$186$		0			0
$187$	1378.50	−	795.878i	0.539069	−	0.311232i
$188$	739.894			0.287034
$189$		0			0
$190$	386.823			0.147700
$191$	−4512.07	+	2605.04i	−1.70933	+	0.986881i	−0.773941	+	0.633258i	$0.781717\pi$
$191$	−4512.07	+	2605.04i	−1.70933	+	0.986881i	−0.935388	+	0.353624i	$0.884949\pi$
$192$		0			0
$193$	−2166.32	+	3752.17i	−0.807953	+	1.39942i	0.106326	+	0.994331i	$0.466091\pi$
$193$	−2166.32	+	3752.17i	−0.807953	+	1.39942i	−0.914279	+	0.405085i	$0.867242\pi$
$194$	−1244.91	−	2156.25i	−0.460720	−	0.797990i
$195$		0			0
$196$	−1369.98	+	74.4704i	−0.499263	+	0.0271394i
$197$			2929.41i			1.05945i	0.848169	+	0.529725i	$0.177705\pi$
$197$			2929.41i			1.05945i	−0.848169	+	0.529725i	$0.822295\pi$
$198$		0			0
$199$	−1205.78	−	696.156i	−0.429524	−	0.247986i	0.269620	−	0.962967i	$-0.413102\pi$
$199$	−1205.78	−	696.156i	−0.429524	−	0.247986i	−0.699144	+	0.714981i	$0.746435\pi$
$200$	251.517	+	145.214i	0.0889248	+	0.0513408i
$201$		0			0
$202$			643.699i			0.224211i
$203$	−0.977154	−	0.529321i	−0.000337846	−	0.000183010i
$204$		0			0
$205$	785.695	+	1360.86i	0.267685	+	0.463643i
$206$	−898.181	+	1555.69i	−0.303783	+	0.526167i
$207$		0			0
$208$	−35.7954	+	20.6665i	−0.0119325	+	0.00688924i
$209$	−554.670			−0.183576
$210$		0			0
$211$	4687.57			1.52941			0.764706	−	0.644379i	$-0.222884\pi$
$211$	4687.57			1.52941			0.764706	+	0.644379i	$0.222884\pi$
$212$	2257.73	−	1303.50i	0.731423	−	0.422287i
$213$		0			0
$214$	533.613	−	924.244i	0.170453	−	0.295234i
$215$	135.738	+	235.105i	0.0430571	+	0.0745770i
$216$		0			0
$217$	−2752.25	+	1690.27i	−0.860991	+	0.528769i
$218$			1245.66i			0.387002i
$219$		0			0
$220$	881.151	+	508.733i	0.270033	+	0.155903i
$221$	−131.849	−	76.1231i	−0.0401318	−	0.0231701i
$222$		0			0
$223$		−	523.586i		−	0.157228i	−0.996905	−	0.0786142i	$-0.974950\pi$
$223$		−	523.586i		−	0.157228i	0.996905	−	0.0786142i	$-0.0250495\pi$
$224$	505.014	−	310.149i	0.150637	−	0.0925121i
$225$		0			0
$226$	−853.663	−	1478.59i	−0.251260	−	0.435195i
$227$	−1094.73	+	1896.12i	−0.320087	+	0.554406i	−0.980506	−	0.196491i	$-0.937045\pi$
$227$	−1094.73	+	1896.12i	−0.320087	+	0.554406i	0.660419	+	0.750897i	$0.270379\pi$
$228$		0			0
$229$	−2147.90	+	1240.09i	−0.619813	+	0.357849i	−0.776796	−	0.629752i	$-0.783156\pi$
$229$	−2147.90	+	1240.09i	−0.619813	+	0.357849i	0.156983	+	0.987601i	$0.449823\pi$
$230$	−918.929			−0.263445
$231$		0			0
$232$	0.480041			0.000135846
$233$	−2285.26	+	1319.40i	−0.642543	+	0.370972i	−0.785593	−	0.618743i	$-0.787642\pi$
$233$	−2285.26	+	1319.40i	−0.642543	+	0.370972i	0.143050	+	0.989715i	$0.454309\pi$
$234$		0			0
$235$	−871.030	+	1508.67i	−0.241786	+	0.418786i
$236$	−938.605	−	1625.71i	−0.258890	−	0.448410i
$237$		0			0
$238$	1919.44	+	1039.76i	0.522769	+	0.283182i
$239$		−	1177.28i		−	0.318628i	−0.987228	−	0.159314i	$-0.949072\pi$
$239$		−	1177.28i		−	0.318628i	0.987228	−	0.159314i	$-0.0509283\pi$
$240$		0			0
$241$	1155.79	+	667.295i	0.308925	+	0.178358i	0.646445	−	0.762960i	$-0.276255\pi$
$241$	1155.79	+	667.295i	0.308925	+	0.178358i	−0.337520	+	0.941318i	$0.609588\pi$
$242$	1041.87	+	601.522i	0.276751	+	0.159782i
$243$		0			0
$244$		−	2974.07i		−	0.780309i
$245$	1460.94	−	2881.10i	0.380963	−	0.751292i
$246$		0			0
$247$	26.5262	+	45.9447i	0.00683329	+	0.0118356i
$248$	697.581	−	1208.25i	0.178615	−	0.309370i
$249$		0			0
$250$	−2631.22	+	1519.14i	−0.665653	+	0.384315i
$251$	2541.89			0.639214			0.319607	−	0.947550i	$-0.396449\pi$
$251$	2541.89			0.639214			0.319607	+	0.947550i	$0.396449\pi$
$252$		0			0
$253$	1317.66			0.327434
$254$	−1268.21	+	732.202i	−0.313286	+	0.180876i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−2967.84	−	5140.45i	−0.720346	−	1.24768i	−0.960861	−	0.277030i	$-0.910650\pi$
$257$	−2967.84	−	5140.45i	−0.720346	−	1.24768i	0.240516	−	0.970645i	$-0.422683\pi$
$258$		0			0
$259$	842.835	−	22.8909i	0.202205	−	0.00549177i
$260$		−	97.3172i		−	0.0232129i
$261$		0			0
$262$	3819.94	+	2205.44i	0.900750	+	0.520049i
$263$	5501.25	+	3176.15i	1.28981	+	0.744675i	0.978621	−	0.205671i	$-0.0659378\pi$
$263$	5501.25	+	3176.15i	1.28981	+	0.744675i	0.311194	+	0.950346i	$0.399271\pi$
$264$		0			0
$265$			6138.12i			1.42287i
$266$	−398.089	−	648.205i	−0.0917608	−	0.149414i
$267$		0			0
$268$	−925.329	−	1602.72i	−0.210909	−	0.365304i
$269$	918.605	−	1591.07i	0.208209	−	0.360629i	−0.742941	−	0.669357i	$-0.766570\pi$
$269$	918.605	−	1591.07i	0.208209	−	0.360629i	0.951151	+	0.308727i	$0.0999031\pi$
$270$		0			0
$271$	812.837	−	469.292i	0.182201	−	0.105194i	−0.406126	−	0.913817i	$-0.633120\pi$
$271$	812.837	−	469.292i	0.182201	−	0.105194i	0.588326	+	0.808624i	$0.299787\pi$
$272$	−942.953			−0.210202
$273$		0			0
$274$	5531.59			1.21962
$275$	849.149	−	490.257i	0.186202	−	0.107504i
$276$		0			0
$277$	843.429	−	1460.86i	0.182949	−	0.316876i	−0.759935	−	0.649999i	$-0.774769\pi$
$277$	843.429	−	1460.86i	0.182949	−	0.316876i	0.942883	+	0.333123i	$0.108102\pi$
$278$	−198.429	−	343.688i	−0.0428092	−	0.0741477i
$279$		0			0
$280$	37.8835	+	1394.86i	0.00808562	+	0.297710i
$281$		−	2696.01i		−	0.572351i	−0.958177	−	0.286175i	$-0.907616\pi$
$281$		−	2696.01i		−	0.572351i	0.958177	−	0.286175i	$-0.0923840\pi$
$282$		0			0
$283$	−5433.02	−	3136.76i	−1.14120	−	0.658872i	−0.194473	−	0.980908i	$-0.562300\pi$
$283$	−5433.02	−	3136.76i	−1.14120	−	0.658872i	−0.946728	+	0.322036i	$0.895633\pi$
$284$	1798.31	+	1038.26i	0.375741	+	0.216934i
$285$		0			0
$286$			139.544i			0.0288512i
$287$	1471.84	−	2717.10i	0.302718	−	0.558834i
$288$		0			0
$289$	719.857	+	1246.83i	0.146521	+	0.253782i
$290$	−0.565122	+	0.978820i	−0.000114431	+	0.000198201i
$291$		0			0
$292$	3570.62	−	2061.50i	0.715598	−	0.413151i
$293$	6270.97			1.25035			0.625177	−	0.780483i	$-0.285027\pi$
$293$	6270.97			1.25035			0.625177	+	0.780483i	$0.285027\pi$
$294$		0			0
$295$	4419.84			0.872315
$296$	−315.410	+	182.102i	−0.0619353	+	0.0357584i
$297$		0			0
$298$	3083.78	−	5341.27i	0.599459	−	1.03829i
$299$	−63.0151	−	109.145i	−0.0121881	−	0.0211105i
$300$		0			0
$301$	254.278	−	469.411i	0.0486922	−	0.0898884i
$302$			4890.28i			0.931802i
$303$		0			0
$304$	284.564	+	164.293i	0.0536870	+	0.0309962i
$305$	6064.23	+	3501.18i	1.13848	+	0.657303i
$306$		0			0
$307$		−	5129.14i		−	0.953535i	−0.879029	−	0.476768i	$-0.841808\pi$
$307$		−	5129.14i		−	0.953535i	0.879029	−	0.476768i	$-0.158192\pi$
$308$	−54.3216	−	2000.11i	−0.0100496	−	0.370021i
$309$		0			0
$310$	1642.43	+	2844.78i	0.300916	+	0.521202i
$311$	−1710.97	+	2963.48i	−0.311961	+	0.540333i	−0.978787	−	0.204881i	$-0.934319\pi$
$311$	−1710.97	+	2963.48i	−0.311961	+	0.540333i	0.666825	+	0.745214i	$0.267653\pi$
$312$		0			0
$313$	−5236.79	+	3023.46i	−0.945690	+	0.545994i	−0.891739	−	0.452549i	$-0.850515\pi$
$313$	−5236.79	+	3023.46i	−0.945690	+	0.545994i	−0.0539507	+	0.998544i	$0.517181\pi$
$314$	5781.15			1.03901
$315$		0			0
$316$	743.658			0.132386
$317$	−2545.70	+	1469.76i	−0.451043	+	0.260410i	−0.708271	−	0.705941i	$-0.750524\pi$
$317$	−2545.70	+	1469.76i	−0.451043	+	0.260410i	0.257228	+	0.966351i	$0.417191\pi$
$318$		0			0
$319$	0.810335	−	1.40354i	0.000142226	−	0.000246342i
$320$	−301.372	−	521.992i	−0.0526476	−	0.0911883i
$321$		0			0
$322$	945.691	+	1539.86i	0.163669	+	0.266501i
$323$			1210.32i			0.208495i
$324$		0			0
$325$	−81.2184	−	46.8914i	−0.0138621	−	0.00800329i
$326$	5654.27	+	3264.49i	0.960616	+	0.554612i
$327$		0			0
$328$			1334.81i			0.224704i
$329$	3424.50	−	93.0072i	0.573856	−	0.0155856i
$330$		0			0
$331$	−2258.94	−	3912.60i	−0.375114	−	0.649716i	0.615230	−	0.788348i	$-0.289063\pi$
$331$	−2258.94	−	3912.60i	−0.375114	−	0.649716i	−0.990344	+	0.138631i	$0.955730\pi$
$332$	−240.689	+	416.886i	−0.0397878	+	0.0689144i
$333$		0			0
$334$	3465.37	−	2000.73i	0.567715	−	0.327770i
$335$	4357.32			0.710645
$336$		0			0
$337$	−3729.56			−0.602855			−0.301427	−	0.953489i	$-0.597463\pi$
$337$	−3729.56			−0.602855			−0.301427	+	0.953489i	$0.597463\pi$
$338$	−3793.76	+	2190.33i	−0.610512	+	0.352479i
$339$		0			0
$340$	1110.08	−	1922.71i	0.177066	−	0.306687i
$341$	−2355.11	−	4079.16i	−0.374006	−	0.647798i
$342$		0			0
$343$	−6331.38	+	516.886i	−0.996684	+	0.0813680i
$344$			230.605i			0.0361436i
$345$		0			0
$346$	4408.91	+	2545.48i	0.685041	+	0.395509i
$347$	8243.34	+	4759.29i	1.27529	+	0.736289i	0.975978	−	0.217867i	$-0.0699099\pi$
$347$	8243.34	+	4759.29i	1.27529	+	0.736289i	0.299311	+	0.954156i	$0.403243\pi$
$348$		0			0
$349$			7997.82i			1.22669i	0.789817	+	0.613343i	$0.210176\pi$
$349$			7997.82i			1.22669i	−0.789817	+	0.613343i	$0.789824\pi$
$350$	1182.37	+	640.484i	0.180572	+	0.0978152i
$351$		0			0
$352$	432.142	+	748.491i	0.0654353	+	0.113337i
$353$	−1450.59	+	2512.50i	−0.218718	+	0.378830i	−0.954416	−	0.298479i	$-0.903521\pi$
$353$	−1450.59	+	2512.50i	−0.218718	+	0.378830i	0.735699	+	0.677309i	$0.236854\pi$
$354$		0			0
$355$	−4234.08	+	2444.55i	−0.633019	+	0.365473i
$356$	4031.32			0.600167
$357$		0			0
$358$	−8344.09			−1.23184
$359$	−581.616	+	335.796i	−0.0855056	+	0.0493667i	−0.542143	−	0.840286i	$-0.682387\pi$
$359$	−581.616	+	335.796i	−0.0855056	+	0.0493667i	0.456638	+	0.889653i	$0.349054\pi$
$360$		0			0
$361$	−3218.62	+	5574.82i	−0.469256	+	0.812774i
$362$	−13.4947	−	23.3736i	−0.00195930	−	0.00339361i
$363$		0			0
$364$	−163.076	+	100.151i	−0.0234821	+	0.0144213i
$365$			9707.48i			1.39209i
$366$		0			0
$367$	−4519.72	−	2609.46i	−0.642855	−	0.371152i	0.142859	−	0.989743i	$-0.454371\pi$
$367$	−4519.72	−	2609.46i	−0.642855	−	0.371152i	−0.785713	+	0.618591i	$0.787704\pi$
$368$	−676.004	−	390.291i	−0.0957585	−	0.0552862i
$369$		0			0
$370$		−	857.509i		−	0.120486i
$371$	10285.7	−	6316.88i	1.43938	−	0.883978i
$372$		0			0
$373$	761.779	+	1319.44i	0.105746	+	0.183158i	0.914043	−	0.405617i	$-0.132943\pi$
$373$	761.779	+	1319.44i	0.105746	+	0.183158i	−0.808296	+	0.588776i	$0.799610\pi$
$374$	−1591.76	+	2757.00i	−0.220074	+	0.381179i
$375$		0			0
$376$	−1281.53	+	739.894i	−0.175772	+	0.101482i
$377$	−0.155012			−2.11764e−5
$378$		0			0
$379$	−8144.22			−1.10380			−0.551900	−	0.833910i	$-0.686097\pi$
$379$	−8144.22			−1.10380			−0.551900	+	0.833910i	$0.686097\pi$
$380$	−669.997	+	386.823i	−0.0904477	+	0.0522200i
$381$		0			0
$382$	5210.09	−	9024.14i	0.697831	−	1.20868i
$383$	−3818.61	−	6614.02i	−0.509456	−	0.882404i	−0.999940	−	0.0109534i	$-0.996513\pi$
$383$	−3818.61	−	6614.02i	−0.509456	−	0.882404i	0.490484	−	0.871450i	$-0.336820\pi$
$384$		0			0
$385$	4142.23	+	2243.83i	0.548331	+	0.297029i
$386$		−	8665.27i		−	1.14262i
$387$		0			0
$388$	4312.51	+	2489.83i	0.564264	+	0.325778i
$389$	6135.65	+	3542.42i	0.799717	+	0.461717i	0.843372	−	0.537330i	$-0.180567\pi$
$389$	6135.65	+	3542.42i	0.799717	+	0.461717i	−0.0436552	+	0.999047i	$0.513900\pi$
$390$		0			0
$391$		−	2875.20i		−	0.371881i
$392$	2298.40	−	1498.96i	0.296140	−	0.193135i
$393$		0			0
$394$	−2929.41	−	5073.89i	−0.374572	−	0.648778i
$395$	−875.461	+	1516.34i	−0.111517	+	0.193153i
$396$		0			0
$397$	490.109	−	282.965i	0.0619594	−	0.0357722i	−0.468700	−	0.883357i	$-0.655278\pi$
$397$	490.109	−	282.965i	0.0619594	−	0.0357722i	0.530660	+	0.847585i	$0.321944\pi$
$398$	2784.62			0.350705
$399$		0			0
$400$	−580.855			−0.0726068
$401$	3039.76	−	1755.00i	0.378549	−	0.218555i	−0.298638	−	0.954367i	$-0.596532\pi$
$401$	3039.76	−	1755.00i	0.378549	−	0.218555i	0.677187	+	0.735811i	$0.263199\pi$
$402$		0			0
$403$	−225.258	+	390.159i	−0.0278435	+	0.0482263i
$404$	−643.699	−	1114.92i	−0.0792704	−	0.137300i
$405$		0			0
$406$	2.22180	−	0.0603428i	0.000271592	−	7.37626e-6i
$407$			1229.59i			0.149751i
$408$		0			0
$409$	−5312.45	−	3067.14i	−0.642258	−	0.370808i	0.143226	−	0.989690i	$-0.454253\pi$
$409$	−5312.45	−	3067.14i	−0.642258	−	0.370808i	−0.785484	+	0.618882i	$0.787586\pi$
$410$	−2721.73	−	1571.39i	−0.327845	−	0.189282i
$411$		0			0
$412$		−	3592.72i		−	0.429613i
$413$	−4548.56	−	7406.39i	−0.541937	−	0.882432i
$414$		0			0
$415$	−566.696	−	981.546i	−0.0670314	−	0.116102i
$416$	41.3329	−	71.5907i	0.00487143	−	0.00843756i
$417$		0			0
$418$	960.717	−	554.670i	0.112417	−	0.0649039i
$419$	4434.51			0.517040			0.258520	−	0.966006i	$-0.416765\pi$
$419$	4434.51			0.517040			0.258520	+	0.966006i	$0.416765\pi$
$420$		0			0
$421$	384.764			0.0445421			0.0222711	−	0.999752i	$-0.492910\pi$
$421$	384.764			0.0445421			0.0222711	+	0.999752i	$0.492910\pi$
$422$	−8119.12	+	4687.57i	−0.936570	+	0.540729i
$423$		0			0
$424$	−2607.00	+	4515.46i	−0.298602	+	0.517194i
$425$	−1069.76	−	1852.88i	−0.122097	−	0.211478i
$426$		0			0
$427$	−373.851	−	13765.1i	−0.0423698	−	1.56004i
$428$			2134.45i			0.241057i
$429$		0			0
$430$	−470.211	−	271.476i	−0.0527339	−	0.0304459i
$431$	−7054.76	−	4073.07i	−0.788435	−	0.455203i	0.0509760	−	0.998700i	$-0.483767\pi$
$431$	−7054.76	−	4073.07i	−0.788435	−	0.455203i	−0.839411	+	0.543496i	$0.817100\pi$
$432$		0			0
$433$		−	1421.98i		−	0.157820i	−0.996882	−	0.0789098i	$-0.974856\pi$
$433$		−	1421.98i		−	0.157820i	0.996882	−	0.0789098i	$-0.0251439\pi$
$434$	3076.77	−	5679.88i	0.340299	−	0.628210i
$435$		0			0
$436$	−1245.66	−	2157.54i	−0.136826	−	0.236990i
$437$	−500.953	+	867.677i	−0.0548372	+	0.0949808i
$438$		0			0
$439$	13655.9	−	7884.25i	1.48465	−	0.857163i	0.484802	−	0.874624i	$-0.338891\pi$
$439$	13655.9	−	7884.25i	1.48465	−	0.857163i	0.999848	+	0.0174606i	$0.00555816\pi$
$440$	−2034.93			−0.220481
$441$		0			0
$442$	304.492			0.0327675
$443$	−8399.99	+	4849.74i	−0.900893	+	0.520131i	−0.877490	−	0.479595i	$-0.840783\pi$
$443$	−8399.99	+	4849.74i	−0.900893	+	0.520131i	−0.0234030	+	0.999726i	$0.507450\pi$
$444$		0			0
$445$	−4745.82	+	8219.99i	−0.505558	+	0.875652i
$446$	523.586	+	906.878i	0.0555886	+	0.0962823i
$447$		0			0
$448$	−564.561	+	1042.21i	−0.0595380	+	0.109910i
$449$			13556.2i			1.42485i	0.701750	+	0.712423i	$0.252402\pi$
$449$			13556.2i			1.42485i	−0.701750	+	0.712423i	$0.747598\pi$
$450$		0			0
$451$	3902.72	+	2253.24i	0.407476	+	0.235257i
$452$	2957.17	+	1707.33i	0.307730	+	0.177668i
$453$		0			0
$454$		−	4378.91i		−	0.452671i
$455$	−12.2331	−	450.419i	−0.00126043	−	0.0464087i
$456$		0			0
$457$	7570.78	+	13113.0i	0.774937	+	1.34223i	0.934830	+	0.355096i	$0.115552\pi$
$457$	7570.78	+	13113.0i	0.774937	+	1.34223i	−0.159893	+	0.987134i	$0.551115\pi$
$458$	2480.18	−	4295.80i	0.253038	−	0.438274i
$459$		0			0
$460$	1591.63	−	918.929i	0.161326	−	0.0931419i
$461$	73.0400			0.00737920			0.00368960	−	0.999993i	$-0.498826\pi$
$461$	73.0400			0.00737920			0.00368960	+	0.999993i	$0.498826\pi$
$462$		0			0
$463$	−9972.52			−1.00100			−0.500499	−	0.865737i	$-0.666850\pi$
$463$	−9972.52			−1.00100			−0.500499	+	0.865737i	$0.666850\pi$
$464$	−0.831456	+	0.480041i	−8.31883e−5	+	4.80288e-5i
$465$		0			0
$466$	2638.79	−	4570.53i	0.262317	−	0.454347i
$467$	−8316.76	−	14405.0i	−0.824098	−	1.42738i	−0.902607	−	0.430466i	$-0.858349\pi$
$467$	−8316.76	−	14405.0i	−0.824098	−	1.42738i	0.0785089	−	0.996913i	$-0.474984\pi$
$468$		0			0
$469$	−4484.22	−	7301.63i	−0.441497	−	0.718887i
$470$		−	3484.12i		−	0.341937i
$471$		0			0
$472$	3251.42	+	1877.21i	0.317074	+	0.183063i
$473$	674.241	+	389.273i	0.0655426	+	0.0378410i
$474$		0			0
$475$			745.549i			0.0720171i
$476$	−4364.33	+	118.532i	−0.420249	+	0.0114137i
$477$		0			0
$478$	1177.28	+	2039.11i	0.112652	+	0.195119i
$479$	−5145.03	+	8911.45i	−0.490778	+	0.850052i	−0.999944	−	0.0106167i	$-0.996621\pi$
$479$	−5145.03	+	8911.45i	−0.490778	+	0.850052i	0.509166	+	0.860668i	$0.329954\pi$
$480$		0			0
$481$	101.850	−	58.8033i	0.00965483	−	0.00557422i
$482$	−2669.18			−0.252236
$483$		0			0
$484$	−2406.09			−0.225966
$485$	−10153.7	+	5862.23i	−0.950629	+	0.548846i
$486$		0			0
$487$	−4937.47	+	8551.94i	−0.459421	+	0.795740i	−0.998930	−	0.0462393i	$-0.985276\pi$
$487$	−4937.47	+	8551.94i	−0.459421	+	0.795740i	0.539510	+	0.841979i	$0.318610\pi$
$488$	2974.07	+	5151.24i	0.275881	+	0.477840i
$489$		0			0
$490$	350.677	+	6451.15i	0.0323305	+	0.594761i
$491$		−	640.761i		−	0.0588944i	−0.999566	−	0.0294472i	$-0.990625\pi$
$491$		−	640.761i		−	0.0588944i	0.999566	−	0.0294472i	$-0.00937469\pi$
$492$		0			0
$493$	−3.06259	−	1.76819i	−0.000279782	−	0.000161532i
$494$	−91.8895	−	53.0524i	−0.00836903	−	0.00483186i
$495$		0			0
$496$			2790.32i			0.252599i
$497$	8453.75	+	4579.37i	0.762983	+	0.413306i
$498$		0			0
$499$	−3110.23	−	5387.08i	−0.279024	−	0.483284i	0.692118	−	0.721784i	$-0.256678\pi$
$499$	−3110.23	−	5387.08i	−0.279024	−	0.483284i	−0.971142	+	0.238500i	$0.923344\pi$
$500$	3038.27	−	5262.45i	0.271752	−	0.470687i
$501$		0			0
$502$	−4402.68	+	2541.89i	−0.391437	+	0.225996i
$503$	−20335.9			−1.80265			−0.901326	−	0.433142i	$-0.857405\pi$
$503$	−20335.9			−1.80265			−0.901326	+	0.433142i	$0.857405\pi$
$504$		0			0
$505$	3031.14			0.267097
$506$	−2282.26	+	1317.66i	−0.200511	+	0.115765i
$507$		0			0
$508$	1464.40	−	2536.42i	0.127898	−	0.221527i
$509$	−7723.80	−	13378.0i	−0.672596	−	1.16497i	−0.977165	−	0.212481i	$-0.931846\pi$
$509$	−7723.80	−	13378.0i	−0.672596	−	1.16497i	0.304569	−	0.952490i	$-0.401488\pi$
$510$		0			0
$511$	16267.0	−	9990.19i	1.40824	−	0.864853i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	10280.9	+	5935.68i	0.882240	+	0.509361i
$515$	7325.68	+	4229.48i	0.626811	+	0.361890i
$516$		0			0
$517$			4995.92i			0.424991i
$518$	−1436.94	+	882.483i	−0.121883	+	0.0748534i
$519$		0			0
$520$	97.3172	+	168.558i	0.00820700	+	0.0142149i
$521$	5440.52	−	9423.25i	0.457492	−	0.792400i	−0.541336	−	0.840807i	$-0.682081\pi$
$521$	5440.52	−	9423.25i	0.457492	−	0.792400i	0.998828	+	0.0484070i	$0.0154145\pi$
$522$		0			0
$523$	7788.70	−	4496.81i	0.651197	−	0.375969i	−0.137718	−	0.990472i	$-0.543977\pi$
$523$	7788.70	−	4496.81i	0.651197	−	0.375969i	0.788915	+	0.614503i	$0.210643\pi$
$524$	−8821.77			−0.735460
$525$		0			0
$526$	−12704.6			−1.05313
$527$	−8900.93	+	5138.95i	−0.735732	+	0.424775i
$528$		0			0
$529$	−4893.45	+	8475.70i	−0.402190	+	0.696614i
$530$	−6138.12	−	10631.5i	−0.503062	−	0.871329i
$531$		0			0
$532$	1337.71	+	724.636i	0.109017	+	0.0590544i
$533$		−	431.029i		−	0.0350281i
$534$		0			0
$535$	−4352.21	−	2512.75i	−0.351706	−	0.203057i
$536$	3205.44	+	1850.66i	0.258309	+	0.149135i
$537$		0			0
$538$			3674.42i			0.294453i
$539$	−502.840	−	9250.38i	−0.0401834	−	0.739224i
$540$		0			0
$541$	5332.27	+	9235.77i	0.423757	+	0.733968i	0.996303	−	0.0859041i	$-0.0273779\pi$
$541$	5332.27	+	9235.77i	0.423757	+	0.733968i	−0.572547	+	0.819872i	$0.694045\pi$
$542$	−938.583	+	1625.67i	−0.0743830	+	0.128835i
$543$		0			0
$544$	1633.24	−	942.953i	0.128722	−	0.0743176i
$545$	5865.73			0.461028
$546$		0			0
$547$	7676.68			0.600057			0.300028	−	0.953930i	$-0.403004\pi$
$547$	7676.68			0.600057			0.300028	+	0.953930i	$0.403004\pi$
$548$	−9580.99	+	5531.59i	−0.746861	+	0.431200i
$549$		0			0
$550$	−980.513	+	1698.30i	−0.0760168	+	0.131665i
$551$	0.616151	+	1.06721i	4.76387e−5	+	8.25127e-5i
$552$		0			0
$553$	3441.92	−	93.4803i	0.264675	−	0.00718840i
$554$			3373.72i			0.258728i
$555$		0			0
$556$	687.377	+	396.857i	0.0524303	+	0.0302707i
$557$	−8751.95	−	5052.94i	−0.665766	−	0.384380i	0.128704	−	0.991683i	$-0.458918\pi$
$557$	−8751.95	−	5052.94i	−0.665766	−	0.384380i	−0.794471	+	0.607303i	$0.792252\pi$
$558$		0			0
$559$		−	74.4654i		−	0.00563426i
$560$	−1460.48	−	2378.08i	−0.110208	−	0.179451i
$561$		0			0
$562$	2696.01	+	4669.63i	0.202356	+	0.350492i
$563$	−2089.62	+	3619.33i	−0.156424	+	0.270935i	−0.933577	−	0.358377i	$-0.883330\pi$
$563$	−2089.62	+	3619.33i	−0.156424	+	0.270935i	0.777152	+	0.629312i	$0.216663\pi$
$564$		0			0
$565$	−6962.58	+	4019.85i	−0.518439	+	0.299321i
$566$	12547.0			0.931786
$567$		0			0
$568$	−4153.03			−0.306791
$569$	−7686.89	+	4438.03i	−0.566347	+	0.326980i	−0.755689	−	0.654931i	$-0.772698\pi$
$569$	−7686.89	+	4438.03i	−0.566347	+	0.326980i	0.189342	+	0.981911i	$0.439364\pi$
$570$		0			0
$571$	6929.43	−	12002.1i	0.507859	−	0.879638i	−0.492099	−	0.870539i	$-0.663770\pi$
$571$	6929.43	−	12002.1i	0.507859	−	0.879638i	0.999959	−	0.00909921i	$-0.00289641\pi$
$572$	−139.544	−	241.698i	−0.0102004	−	0.0176677i
$573$		0			0
$574$	167.790	+	6178.00i	0.0122011	+	0.449241i
$575$		−	1771.11i		−	0.128453i
$576$		0			0
$577$	3478.55	+	2008.34i	0.250977	+	0.144902i	0.620212	−	0.784435i	$-0.287047\pi$
$577$	3478.55	+	2008.34i	0.250977	+	0.144902i	−0.369234	+	0.929336i	$0.620380\pi$
$578$	−2493.66	−	1439.71i	−0.179451	−	0.103606i
$579$		0			0
$580$		−	2.26049i		−	0.000161830i
$581$	−1061.59	+	1959.75i	−0.0758042	+	0.139939i
$582$		0			0
$583$	8801.52	+	15244.7i	0.625252	+	1.08297i
$584$	−4123.00	+	7141.24i	−0.292142	+	0.506004i
$585$		0			0
$586$	−10861.6	+	6270.97i	−0.765683	+	0.442067i
$587$	−17062.7			−1.19975			−0.599876	−	0.800093i	$-0.704783\pi$
$587$	−17062.7			−1.19975			−0.599876	+	0.800093i	$0.704783\pi$
$588$		0			0
$589$	3581.49			0.250548
$590$	−7655.38	+	4419.84i	−0.534182	+	0.308410i
$591$		0			0
$592$	364.205	−	630.821i	0.0252850	−	0.0437949i
$593$	−13932.2	−	24131.2i	−0.964800	−	1.67108i	−0.710152	−	0.704048i	$-0.751374\pi$
$593$	−13932.2	−	24131.2i	−0.964800	−	1.67108i	−0.254647	−	0.967034i	$-0.581959\pi$
$594$		0			0
$595$	4896.15	−	9038.54i	0.337349	−	0.622763i
$596$			12335.1i			0.847763i
$597$		0			0
$598$	218.291	+	126.030i	0.0149274	+	0.00861832i
$599$	−7546.23	−	4356.82i	−0.514742	−	0.297187i	0.220039	−	0.975491i	$-0.429382\pi$
$599$	−7546.23	−	4356.82i	−0.514742	−	0.297187i	−0.734781	+	0.678305i	$0.762715\pi$
$600$		0			0
$601$		−	1567.31i		−	0.106376i	−0.998585	−	0.0531881i	$-0.983062\pi$
$601$		−	1567.31i		−	0.106376i	0.998585	−	0.0531881i	$-0.0169383\pi$
$602$	28.9878	+	1067.32i	0.00196255	+	0.0722605i
$603$		0			0
$604$	−4890.28	−	8470.22i	−0.329442	−	0.570610i
$605$	2832.53	−	4906.09i	0.190345	−	0.329688i
$606$		0			0
$607$	8066.22	−	4657.03i	0.539370	−	0.311406i	−0.205453	−	0.978667i	$-0.565867\pi$
$607$	8066.22	−	4657.03i	0.539370	−	0.311406i	0.744824	+	0.667261i	$0.232534\pi$
$608$	−657.172			−0.0438353
$609$		0			0
$610$	−14004.7			−0.929566
$611$	413.825	−	238.922i	0.0274003	−	0.0158196i
$612$		0			0
$613$	−12480.5	+	21616.9i	−0.822322	+	1.42430i	0.0816265	+	0.996663i	$0.473989\pi$
$613$	−12480.5	+	21616.9i	−0.822322	+	1.42430i	−0.903949	+	0.427641i	$0.859345\pi$
$614$	5129.14	+	8883.93i	0.337126	+	0.583919i
$615$		0			0
$616$	2094.19	+	3409.96i	0.136976	+	0.223038i
$617$			21801.6i			1.42252i	0.702927	+	0.711262i	$0.251876\pi$
$617$			21801.6i			1.42252i	−0.702927	+	0.711262i	$0.748124\pi$
$618$		0			0
$619$	−17775.9	−	10262.9i	−1.15424	−	0.666401i	−0.204324	−	0.978903i	$-0.565500\pi$
$619$	−17775.9	−	10262.9i	−1.15424	−	0.666401i	−0.949917	+	0.312502i	$0.898833\pi$
$620$	−5689.56	−	3284.87i	−0.368545	−	0.212780i
$621$		0			0
$622$		−	6843.87i		−	0.441180i
$623$	18658.4	−	506.750i	1.19989	−	0.0325883i
$624$		0			0
$625$	4884.57	+	8460.32i	0.312612	+	0.541460i
$626$	6046.93	−	10473.6i	0.386076	−	0.668704i
$627$		0			0
$628$	−10013.2	+	5781.15i	−0.636261	+	0.367345i
$629$	2683.03			0.170079
$630$		0			0
$631$	19654.0			1.23996			0.619978	−	0.784619i	$-0.287141\pi$
$631$	19654.0			1.23996			0.619978	+	0.784619i	$0.287141\pi$
$632$	−1288.05	+	743.658i	−0.0810697	+	0.0468056i
$633$		0			0
$634$	2939.52	−	5091.40i	0.184138	−	0.318936i
$635$	3447.90	+	5971.93i	0.215473	+	0.373211i
$636$		0			0
$637$	−742.184	+	484.036i	−0.0461639	+	0.0301071i
$638$			3.24134i			0.000201138i
$639$		0			0
$640$	1043.98	+	602.745i	0.0644799	+	0.0372275i
$641$	8694.99	+	5020.05i	0.535774	+	0.309329i	0.743365	−	0.668886i	$-0.233229\pi$
$641$	8694.99	+	5020.05i	0.535774	+	0.309329i	−0.207590	+	0.978216i	$0.566562\pi$
$642$		0			0
$643$		−	17764.7i		−	1.08954i	−0.838587	−	0.544768i	$-0.816618\pi$
$643$		−	17764.7i		−	1.08954i	0.838587	−	0.544768i	$-0.183382\pi$
$644$	−3177.85	−	1721.43i	−0.194448	−	0.105332i
$645$		0			0
$646$	−1210.32	−	2096.33i	−0.0737141	−	0.127677i
$647$	−2191.61	+	3795.98i	−0.133170	+	0.230657i	−0.924897	−	0.380218i	$-0.875849\pi$
$647$	−2191.61	+	3795.98i	−0.133170	+	0.230657i	0.791727	+	0.610875i	$0.209182\pi$
$648$		0			0
$649$	10977.1	−	6337.66i	0.663930	−	0.383320i
$650$	187.566			0.0113184
$651$		0			0
$652$	−13058.0			−0.784340
$653$	−6755.82	+	3900.47i	−0.404863	+	0.233748i	−0.688580	−	0.725160i	$-0.741766\pi$
$653$	−6755.82	+	3900.47i	−0.404863	+	0.233748i	0.283717	+	0.958908i	$0.408432\pi$
$654$		0			0
$655$	10385.3	−	17987.9i	0.619523	−	1.07305i
$656$	−1334.81	−	2311.97i	−0.0794447	−	0.137602i
$657$		0			0
$658$	−5838.39	+	3585.59i	−0.345903	+	0.212433i
$659$		−	24968.4i		−	1.47592i	−0.674844	−	0.737960i	$-0.735789\pi$
$659$		−	24968.4i		−	1.47592i	0.674844	−	0.737960i	$-0.264211\pi$
$660$		0			0
$661$	14745.5	+	8513.33i	0.867676	+	0.500953i	0.866575	−	0.499046i	$-0.166316\pi$
$661$	14745.5	+	8513.33i	0.867676	+	0.500953i	0.00110096	+	0.999999i	$0.499650\pi$
$662$	7825.21	+	4517.89i	0.459419	+	0.265246i
$663$		0			0
$664$		−	962.757i		−	0.0562684i
$665$	−3052.36	+	1874.58i	−0.177993	+	0.109313i
$666$		0			0
$667$	−1.46372	−	2.53523i	−8.49705e−5	−	0.000147173i
$668$	−4001.47	+	6930.75i	−0.231769	+	0.401435i
$669$		0			0
$670$	−7547.11	+	4357.32i	−0.435179	+	0.251251i
$671$	20081.6			1.15535
$672$		0			0
$673$	−8289.60			−0.474800			−0.237400	−	0.971412i	$-0.576295\pi$
$673$	−8289.60			−0.474800			−0.237400	+	0.971412i	$0.576295\pi$
$674$	6459.79	−	3729.56i	0.369172	−	0.213141i
$675$		0			0
$676$	4380.65	−	7587.51i	0.249241	−	0.431697i
$677$	13312.5	+	23058.0i	0.755749	+	1.30900i	0.945001	+	0.327067i	$0.106060\pi$
$677$	13312.5	+	23058.0i	0.755749	+	1.30900i	−0.189252	+	0.981928i	$0.560606\pi$
$678$		0			0
$679$	20272.8	+	10981.7i	1.14580	+	0.620677i
$680$			4440.31i			0.250409i
$681$		0			0
$682$	8158.33	+	4710.21i	0.458062	+	0.264462i
$683$	−2342.28	−	1352.32i	−0.131223	−	0.0757614i	0.432952	−	0.901417i	$-0.357472\pi$
$683$	−2342.28	−	1352.32i	−0.131223	−	0.0757614i	−0.564174	+	0.825656i	$0.690805\pi$
$684$		0			0
$685$		−	26047.9i		−	1.45291i
$686$	10449.4	−	7226.66i	0.581574	−	0.402209i
$687$		0			0
$688$	−230.605	−	399.419i	−0.0127787	−	0.0221333i
$689$	841.837	−	1458.10i	0.0465478	−	0.0806231i
$690$		0			0
$691$	407.582	−	235.318i	0.0224387	−	0.0129550i	−0.488739	−	0.872430i	$-0.662543\pi$
$691$	407.582	−	235.318i	0.0224387	−	0.0129550i	0.511177	+	0.859475i	$0.329210\pi$
$692$	−10181.9			−0.559334
$693$		0			0
$694$	−19037.2			−1.04127
$695$	−1618.41	+	934.389i	−0.0883306	+	0.0509977i
$696$		0			0
$697$	4916.67	−	8515.92i	0.267191	−	0.462788i
$698$	−7997.82	−	13852.6i	−0.433699	−	0.751189i
$699$		0			0
$700$	−2688.40	+	73.0153i	−0.145160	+	0.00394246i
$701$		−	9201.23i		−	0.495757i	−0.968791	−	0.247879i	$-0.920267\pi$
$701$		−	9201.23i		−	0.495757i	0.968791	−	0.247879i	$-0.0797334\pi$
$702$		0			0
$703$	−809.683	−	467.471i	−0.0434392	−	0.0250796i
$704$	−1496.98	−	864.283i	−0.0801415	−	0.0462697i
$705$		0			0
$706$		−	5802.38i		−	0.309313i
$707$	−3119.42	−	5079.33i	−0.165938	−	0.270195i
$708$		0			0
$709$	4308.44	+	7462.44i	0.228219	+	0.395286i	0.957280	−	0.289162i	$-0.0933766\pi$
$709$	4308.44	+	7462.44i	0.228219	+	0.395286i	−0.729062	+	0.684448i	$0.760043\pi$
$710$	4889.09	−	8468.16i	0.258429	−	0.447612i
$711$		0			0
$712$	−6982.45	+	4031.32i	−0.367526	+	0.212191i
$713$	−8508.10			−0.446888
$714$		0			0
$715$	657.106			0.0343698
$716$	14452.4	−	8344.09i	0.754345	−	0.435521i
$717$		0			0
$718$	671.592	−	1163.23i	0.0349075	−	0.0604616i
$719$	9135.32	+	15822.8i	0.473839	+	0.820713i	0.999551	−	0.0299495i	$-0.00953465\pi$
$719$	9135.32	+	15822.8i	0.473839	+	0.820713i	−0.525713	+	0.850662i	$0.676201\pi$
$720$		0			0
$721$	−451.617	−	16628.4i	−0.0233275	−	0.858910i
$722$		−	12874.5i		−	0.663628i
$723$		0			0
$724$	46.7471	+	26.9895i	0.00239965	+	0.00138544i
$725$	−1.88654	−	1.08920i	−9.66406e−5	−	5.57955e-5i
$726$		0			0
$727$		−	20012.8i		−	1.02096i	−0.859891	−	0.510478i	$-0.829469\pi$
$727$		−	20012.8i		−	1.02096i	0.859891	−	0.510478i	$-0.170531\pi$
$728$	182.304	−	336.543i	0.00928111	−	0.0171334i
$729$		0			0
$730$	−9707.48	−	16813.8i	−0.492178	−	0.852477i
$731$	849.413	−	1471.23i	0.0429777	−	0.0744395i
$732$		0			0
$733$	−22142.2	+	12783.8i	−1.11575	+	0.644176i	−0.940311	−	0.340315i	$-0.889466\pi$
$733$	−22142.2	+	12783.8i	−1.11575	+	0.644176i	−0.175434	+	0.984491i	$0.556133\pi$
$734$	10437.9			0.524889
$735$		0			0
$736$	1561.16			0.0781865
$737$	10821.9	−	6248.02i	0.540881	−	0.312278i
$738$		0			0
$739$	−17297.1	+	29959.4i	−0.861007	+	1.49131i	0.00995183	+	0.999950i	$0.496832\pi$
$739$	−17297.1	+	29959.4i	−0.861007	+	1.49131i	−0.870958	+	0.491357i	$0.836501\pi$
$740$	857.509	+	1485.25i	0.0425982	+	0.0737823i
$741$		0			0
$742$	−11498.5	+	21226.9i	−0.568901	+	1.05022i
$743$			12083.7i			0.596646i	0.954465	+	0.298323i	$0.0964272\pi$
$743$			12083.7i			0.596646i	−0.954465	+	0.298323i	$0.903573\pi$
$744$		0			0
$745$	−25151.7	−	14521.4i	−1.23690	−	0.714123i
$746$	−2638.88	−	1523.56i	−0.129512	−	0.0747741i
$747$		0			0
$748$		−	6367.02i		−	0.311232i
$749$	268.307	+	9879.00i	0.0130891	+	0.481937i
$750$		0			0
$751$	−5919.69	−	10253.2i	−0.287633	−	0.498195i	0.685611	−	0.727968i	$-0.259535\pi$
$751$	−5919.69	−	10253.2i	−0.287633	−	0.498195i	−0.973244	+	0.229773i	$0.926202\pi$
$752$	1479.79	−	2563.07i	0.0717584	−	0.124289i
$753$		0			0
$754$	0.268488	−	0.155012i	1.29679e−5	−	7.48700e-6i
$755$	23028.1			1.11004
$756$		0			0
$757$	22220.2			1.06685			0.533425	−	0.845847i	$-0.320904\pi$
$757$	22220.2			1.06685			0.533425	+	0.845847i	$0.320904\pi$
$758$	14106.2	−	8144.22i	0.675937	−	0.390253i
$759$		0			0
$760$	773.646	−	1339.99i	0.0369251	−	0.0639562i
$761$	−8508.86	−	14737.8i	−0.405317	−	0.702029i	0.589042	−	0.808103i	$-0.299505\pi$
$761$	−8508.86	−	14737.8i	−0.405317	−	0.702029i	−0.994358	+	0.106074i	$0.966172\pi$
$762$		0			0
$763$	−6036.55	−	9829.29i	−0.286419	−	0.466375i
$764$			20840.4i			0.986881i
$765$		0			0
$766$	13228.0	+	7637.21i	0.623954	+	0.360240i
$767$	−1049.93	−	606.176i	−0.0494273	−	0.0285368i
$768$		0			0
$769$			6690.03i			0.313717i	0.987621	+	0.156859i	$0.0501367\pi$
$769$			6690.03i			0.313717i	−0.987621	+	0.156859i	$0.949863\pi$
$770$	−9418.38	+	255.797i	−0.440799	+	0.0119718i
$771$		0			0
$772$	8665.27	+	15008.7i	0.403977	+	0.699708i
$773$	−4705.19	+	8149.63i	−0.218931	+	0.379200i	−0.954482	−	0.298270i	$-0.903590\pi$
$773$	−4705.19	+	8149.63i	−0.218931	+	0.379200i	0.735550	+	0.677470i	$0.236924\pi$
$774$		0			0
$775$	−5482.93	+	3165.57i	−0.254132	+	0.146723i
$776$	−9959.31			−0.460720
$777$		0			0
$778$	−14169.7			−0.652966
$779$	−2967.50	+	1713.28i	−0.136485	+	0.0787995i
$780$		0			0
$781$	−7010.53	+	12142.6i	−0.321199	+	0.556333i
$782$	2875.20	+	4980.00i	0.131480	+	0.227729i
$783$		0			0
$784$	−2481.98	+	4894.68i	−0.113064	+	0.222972i
$785$		−	27223.1i		−	1.23775i
$786$		0			0
$787$	3783.96	+	2184.67i	0.171389	+	0.0989517i	0.583241	−	0.812299i	$-0.301784\pi$
$787$	3783.96	+	2184.67i	0.171389	+	0.0989517i	−0.411851	+	0.911251i	$0.635118\pi$
$788$	10147.8	+	5858.82i	0.458756	+	0.264863i
$789$		0			0
$790$		−	3501.84i		−	0.157709i
$791$	13901.5	+	7530.39i	0.624879	+	0.338495i
$792$		0			0
$793$	−960.368	−	1663.41i	−0.0430059	−	0.0744884i
$794$	−565.929	+	980.218i	−0.0252948	+	0.0438119i
$795$		0			0
$796$	−4823.11	+	2784.62i	−0.214762	+	0.123993i
$797$	33765.4			1.50067			0.750334	−	0.661059i	$-0.229893\pi$
$797$	33765.4			1.50067			0.750334	+	0.661059i	$0.229893\pi$
$798$		0			0
$799$	10901.3			0.482681
$800$	1006.07	−	580.855i	0.0444624	−	0.0256704i
$801$		0			0
$802$	−3510.01	+	6079.51i	−0.154542	+	0.267675i
$803$	13919.7	+	24109.6i	0.611724	+	1.05954i
$804$		0			0
$805$	7251.13	−	4453.20i	0.317477	−	0.194975i
$806$		−	901.033i		−	0.0393766i
$807$		0			0
$808$	2229.84	+	1287.40i	0.0970860	+	0.0560526i
$809$	20962.6	+	12102.8i	0.911008	+	0.525970i	0.880755	−	0.473572i	$-0.157036\pi$
$809$	20962.6	+	12102.8i	0.911008	+	0.525970i	0.0302523	+	0.999542i	$0.490369\pi$
$810$		0			0
$811$		−	13589.1i		−	0.588380i	−0.955747	−	0.294190i	$-0.904950\pi$
$811$		−	13589.1i		−	0.588380i	0.955747	−	0.294190i	$-0.0950498\pi$
$812$	−3.78793	+	2.32632i	−0.000163707	+	0.000100539i
$813$		0			0
$814$	−1229.59	−	2129.72i	−0.0529450	−	0.0917034i
$815$	15372.3	−	26625.6i	0.660698	−	1.14436i
$816$		0			0
$817$	−512.670	+	295.990i	−0.0219536	+	0.0126749i
$818$	12268.6			0.524402
$819$		0			0
$820$	6285.56			0.267685
$821$	31438.8	−	18151.2i	1.33645	−	0.771598i	0.350169	−	0.936687i	$-0.386124\pi$
$821$	31438.8	−	18151.2i	1.33645	−	0.771598i	0.986279	+	0.165089i	$0.0527910\pi$
$822$		0			0
$823$	−11909.9	+	20628.6i	−0.504441	+	0.873717i	0.495546	+	0.868582i	$0.334968\pi$
$823$	−11909.9	+	20628.6i	−0.504441	+	0.873717i	−0.999987	+	0.00513537i	$0.998365\pi$
$824$	3592.72	+	6222.78i	0.151891	+	0.263083i
$825$		0			0
$826$	15284.7	+	8279.68i	0.643854	+	0.348774i
$827$			43595.8i			1.83310i	0.399917	+	0.916551i	$0.369039\pi$
$827$			43595.8i			1.83310i	−0.399917	+	0.916551i	$0.630961\pi$
$828$		0			0
$829$	−9921.91	−	5728.42i	−0.415684	−	0.239995i	0.277545	−	0.960713i	$-0.410479\pi$
$829$	−9921.91	−	5728.42i	−0.415684	−	0.239995i	−0.693229	+	0.720717i	$0.743813\pi$
$830$	1963.09	+	1133.39i	0.0820963	+	0.0473983i
$831$		0			0
$832$			165.332i			0.00688924i
$833$	−20184.8	+	1097.22i	−0.839569	+	0.0456380i
$834$		0			0
$835$	−9421.34	−	16318.2i	−0.390466	−	0.676307i
$836$	−1109.34	+	1921.43i	−0.0458940	+	0.0794907i
$837$		0			0
$838$	−7680.79	+	4434.51i	−0.316621	+	0.182801i
$839$	−29369.9			−1.20853			−0.604267	−	0.796782i	$-0.706534\pi$
$839$	−29369.9			−1.20853			−0.604267	+	0.796782i	$0.706534\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$	−666.430	+	384.764i	−0.0272764	+	0.0157480i
$843$		0			0
$844$	9375.15	−	16238.2i	0.382353	−	0.662255i
$845$	10314.1	+	17864.6i	0.419901	+	0.727290i
$846$		0			0
$847$	−11136.2	+	302.453i	−0.451766	+	0.0122697i
$848$		−	10428.0i		−	0.422287i
$849$		0			0
$850$	3705.77	+	2139.53i	0.149537	+	0.0863355i
$851$	1923.47	+	1110.51i	0.0774801	+	0.0447332i
$852$		0			0
$853$			24640.8i			0.989081i	0.869154	+	0.494541i	$0.164664\pi$
$853$			24640.8i			0.989081i	−0.869154	+	0.494541i	$0.835336\pi$
$854$	14412.6	+	23468.0i	0.577505	+	0.940348i
$855$		0			0
$856$	−2134.45	−	3696.98i	−0.0852267	−	0.147617i
$857$	−18228.7	+	31573.0i	−0.726580	+	1.25847i	0.231740	+	0.972778i	$0.425558\pi$
$857$	−18228.7	+	31573.0i	−0.726580	+	1.25847i	−0.958320	+	0.285696i	$0.907775\pi$
$858$		0			0
$859$	9867.48	−	5697.00i	0.391937	−	0.226285i	−0.291062	−	0.956704i	$-0.594009\pi$
$859$	9867.48	−	5697.00i	0.391937	−	0.226285i	0.682999	+	0.730419i	$0.260675\pi$
$860$	1085.91			0.0430571
$861$		0			0
$862$	16292.3			0.643755
$863$	29928.4	−	17279.2i	1.18050	−	0.681564i	0.224372	−	0.974503i	$-0.427967\pi$
$863$	29928.4	−	17279.2i	1.18050	−	0.681564i	0.956131	+	0.292940i	$0.0946335\pi$
$864$		0			0
$865$	11986.5	−	20761.3i	0.471161	−	0.816075i
$866$	1421.98	+	2462.94i	0.0557976	+	0.0966443i
$867$		0			0
$868$	350.753	+	12914.6i	0.0137158	+	0.505012i
$869$			5021.34i			0.196015i
$870$		0			0
$871$	−1035.08	−	597.603i	−0.0402667	−	0.0232480i
$872$	4315.08	+	2491.31i	0.167577	+	0.0967506i
$873$		0			0
$874$		−	2003.81i		−	0.0775515i
$875$	13400.7	−	24738.4i	0.517745	−	0.955784i
$876$		0			0
$877$	14127.6	+	24469.7i	0.543962	+	0.942170i	0.998671	+	0.0515300i	$0.0164098\pi$
$877$	14127.6	+	24469.7i	0.543962	+	0.942170i	−0.454709	+	0.890640i	$0.650257\pi$
$878$	−15768.5	+	27311.8i	−0.606106	+	1.04981i
$879$		0			0
$880$	3524.60	−	2034.93i	0.135016	−	0.0779517i
$881$	35990.0			1.37632			0.688158	−	0.725561i	$-0.258420\pi$
$881$	35990.0			1.37632			0.688158	+	0.725561i	$0.258420\pi$
$882$		0			0
$883$	−12624.3			−0.481133			−0.240567	−	0.970633i	$-0.577333\pi$
$883$	−12624.3			−0.481133			−0.240567	+	0.970633i	$0.577333\pi$
$884$	−527.396	+	304.492i	−0.0200659	+	0.0115851i
$885$		0			0
$886$	9699.47	−	16800.0i	0.367788	−	0.637027i
$887$	10569.1	+	18306.2i	0.400085	+	0.692967i	0.993736	−	0.111756i	$-0.0356474\pi$
$887$	10569.1	+	18306.2i	0.400085	+	0.692967i	−0.593651	+	0.804723i	$0.702314\pi$
$888$		0			0
$889$	6458.95	−	11923.5i	0.243674	−	0.449835i
$890$		−	18983.3i		−	0.714967i
$891$		0			0
$892$	−1813.76	−	1047.17i	−0.0680819	−	0.0393071i
$893$	−3289.80	−	1899.37i	−0.123280	−	0.0711757i
$894$		0			0
$895$			39291.9i			1.46747i
$896$	−64.3601	−	2369.72i	−0.00239969	−	0.0883558i
$897$		0			0
$898$	−13556.2	−	23480.0i	−0.503759	−	0.872537i
$899$	−5.23230	+	9.06262i	−0.000194112	+	0.000336213i
$900$		0			0
$901$	33264.6	−	19205.3i	1.22997	−	0.710125i
$902$	−9012.94			−0.332703
$903$		0			0
$904$	−6829.30			−0.251260
$905$	−110.065	+	63.5459i	−0.00404274	+	0.00233408i
$906$		0			0
$907$	−7584.68	+	13137.0i	−0.277668	+	0.480936i	−0.970805	−	0.239871i	$-0.922895\pi$
$907$	−7584.68	+	13137.0i	−0.277668	+	0.480936i	0.693137	+	0.720806i	$0.256228\pi$
$908$	4378.91	+	7584.50i	0.160043	+	0.277203i
$909$		0			0
$910$	471.607	+	767.915i	0.0171798	+	0.0279738i
$911$		−	48167.6i		−	1.75177i	−0.482518	−	0.875886i	$-0.660278\pi$
$911$		−	48167.6i		−	1.75177i	0.482518	−	0.875886i	$-0.339722\pi$
$912$		0			0
$913$	−2814.90	−	1625.19i	−0.102037	−	0.0589110i
$914$	−26225.9	−	15141.6i	−0.949100	−	0.547963i
$915$		0			0
$916$			9920.72i			0.357849i
$917$	−40830.3	+	1108.93i	−1.47038	+	0.0399345i
$918$		0			0
$919$	18156.0	+	31447.2i	0.651700	+	1.12878i	0.982710	+	0.185150i	$0.0592772\pi$
$919$	18156.0	+	31447.2i	0.651700	+	1.12878i	−0.331010	+	0.943627i	$0.607390\pi$
$920$	−1837.86	+	3183.26i	−0.0658613	+	0.114075i
$921$		0			0
$922$	−126.509	+	73.0400i	−0.00451882	+	0.00260894i
$923$	1341.07			0.0478243
$924$		0			0
$925$	1652.73			0.0587476
$926$	17272.9	−	9972.52i	0.612984	−	0.353906i
$927$		0			0
$928$	0.960082	−	1.66291i	3.39615e−5	−	5.88230e-5i
$929$	−6858.43	−	11879.1i	−0.242215	−	0.419529i	0.719130	−	0.694876i	$-0.244541\pi$
$929$	−6858.43	−	11879.1i	−0.242215	−	0.419529i	−0.961345	+	0.275347i	$0.911207\pi$
$930$		0			0
$931$	6282.51	+	3185.72i	0.221161	+	0.112146i
$932$			10555.2i			0.370972i
$933$		0			0
$934$	28810.1	+	16633.5i	1.00931	+	0.582725i
$935$	12982.6	+	7495.49i	0.454091	+	0.262170i
$936$		0			0
$937$		−	13181.3i		−	0.459568i	−0.973242	−	0.229784i	$-0.926198\pi$
$937$		−	13181.3i		−	0.459568i	0.973242	−	0.229784i	$-0.0738020\pi$
$938$	15068.5	+	8162.58i	0.524526	+	0.284134i
$939$		0			0
$940$	3484.12	+	6034.67i	0.120893	+	0.209393i
$941$	24331.9	−	42144.2i	0.842932	−	1.46000i	−0.0444738	−	0.999011i	$-0.514161\pi$
$941$	24331.9	−	42144.2i	0.842932	−	1.46000i	0.887405	−	0.460990i	$-0.152506\pi$
$942$		0			0
$943$	7049.52	−	4070.04i	0.243440	−	0.140550i
$944$	−7508.84			−0.258890
$945$		0			0
$946$	−1557.09			−0.0535153
$947$	−30733.7	+	17744.1i	−1.05461	+	0.608877i	−0.923935	−	0.382549i	$-0.875046\pi$
$947$	−30733.7	+	17744.1i	−1.05461	+	0.608877i	−0.130671	+	0.991426i	$0.541713\pi$
$948$		0			0
$949$	1331.37	−	2306.00i	0.0455407	−	0.0788788i
$950$	−745.549	−	1291.33i	−0.0254619	−	0.0441013i
$951$		0			0
$952$	7440.70	−	4569.63i	0.253314	−	0.155570i
$953$			2282.25i			0.0775753i	0.999247	+	0.0387877i	$0.0123496\pi$
$953$			2282.25i			0.0775753i	−0.999247	+	0.0387877i	$0.987650\pi$
$954$		0			0
$955$	−42494.1	−	24534.0i	−1.43987	−	0.831311i
$956$	−4078.23	−	2354.57i	−0.137970	−	0.0796570i
$957$		0			0
$958$		−	20580.1i		−	0.694064i
$959$	−43648.9	+	26806.5i	−1.46976	+	0.902636i
$960$		0			0
$961$	311.336	+	539.250i	0.0104507	+	0.0181011i
$962$	−117.607	+	203.701i	−0.00394157	+	0.00682700i
$963$		0			0
$964$	4623.16	−	2669.18i	0.154463	−	0.0891790i
$965$	−40804.3			−1.36118
$966$		0			0
$967$	38287.9			1.27328			0.636638	−	0.771163i	$-0.280325\pi$
$967$	38287.9			1.27328			0.636638	+	0.771163i	$0.280325\pi$
$968$	4167.47	−	2406.09i	0.138376	−	0.0798911i
$969$		0			0
$970$	11724.5	−	20307.4i	0.388093	−	0.672196i
$971$	−4769.32	−	8260.70i	−0.157626	−	0.273016i	0.776386	−	0.630257i	$-0.217051\pi$
$971$	−4769.32	−	8260.70i	−0.157626	−	0.273016i	−0.934012	+	0.357241i	$0.883717\pi$
$972$		0			0
$973$	3231.31	+	1750.39i	0.106466	+	0.0576721i
$974$		−	19749.9i		−	0.649719i
$975$		0			0
$976$	−10302.5	−	5948.15i	−0.337884	−	0.195077i
$977$	−41164.6	−	23766.4i	−1.34798	−	0.778255i	−0.360014	−	0.932947i	$-0.617228\pi$
$977$	−41164.6	−	23766.4i	−1.34798	−	0.778255i	−0.987963	+	0.154692i	$0.950561\pi$
$978$		0			0
$979$			27220.3i			0.888627i
$980$	−7058.54	−	10823.0i	−0.230078	−	0.352785i
$981$		0			0
$982$	640.761	+	1109.83i	0.0208223	+	0.0360653i
$983$	12801.9	−	22173.6i	0.415379	−	0.719458i	−0.580089	−	0.814553i	$-0.696982\pi$
$983$	12801.9	−	22173.6i	0.415379	−	0.719458i	0.995468	+	0.0950950i	$0.0303155\pi$
$984$		0			0
$985$	−23892.6	+	13794.4i	−0.772876	+	0.446220i
$986$	7.07276			0.000228441
$987$		0			0
$988$	212.210			0.00683329
$989$	1217.89	−	703.148i	0.0391573	−	0.0226075i
$990$		0			0
$991$	−24464.4	+	42373.6i	−0.784195	+	1.35827i	0.145283	+	0.989390i	$0.453591\pi$
$991$	−24464.4	+	42373.6i	−0.784195	+	1.35827i	−0.929479	+	0.368876i	$0.879743\pi$
$992$	−2790.32	−	4832.98i	−0.0893073	−	0.154685i
$993$		0			0
$994$	−19221.7	+	522.049i	−0.613356	+	0.0166584i
$995$		−	13112.6i		−	0.417787i
$996$		0			0
$997$	−37064.9	−	21399.4i	−1.17739	−	0.679767i	−0.221981	−	0.975051i	$-0.571252\pi$
$997$	−37064.9	−	21399.4i	−1.17739	−	0.679767i	−0.955409	+	0.295285i	$0.904586\pi$
$998$	10774.2	+	6220.46i	0.341733	+	0.197300i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.a.269.3	yes	12
3.2	odd	2	inner	378.4.k.a.269.4	yes	12
7.5	odd	6	inner	378.4.k.a.215.4	yes	12
21.5	even	6	inner	378.4.k.a.215.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.a.215.3	✓	12	21.5	even	6	inner
378.4.k.a.215.4	yes	12	7.5	odd	6	inner
378.4.k.a.269.3	yes	12	1.1	even	1	trivial
378.4.k.a.269.4	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(4.70894 + 8.15613i) q^{5} +(8.82127 - 16.2845i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(4.70894 + 8.15613i) q^{5} +(8.82127 - 16.2845i) q^{7} +8.00000i q^{8} +(-16.3123 - 9.41789i) q^{10} +(23.3903 + 13.5044i) q^{11} -2.58331i q^{13} +(1.00563 + 37.0269i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(29.4673 - 51.0388i) q^{17} +(-17.7852 + 10.2683i) q^{19} +37.6715 q^{20} -54.0177 q^{22} +(42.2502 - 24.3932i) q^{23} +(18.1517 - 31.4397i) q^{25} +(2.58331 + 4.47442i) q^{26} +(-38.7687 - 63.1268i) q^{28} -0.0600052i q^{29} +(-151.031 - 87.1976i) q^{31} +(27.7128 + 16.0000i) q^{32} +117.869i q^{34} +(174.357 - 4.73544i) q^{35} +(22.7628 + 39.4263i) q^{37} +(20.5366 - 35.5705i) q^{38} +(-65.2490 + 37.6715i) q^{40} +166.852 q^{41} +28.8256 q^{43} +(93.5614 - 54.0177i) q^{44} +(-48.7864 + 84.5004i) q^{46} +(92.4868 + 160.192i) q^{47} +(-187.370 - 287.300i) q^{49} +72.6068i q^{50} +(-8.94884 - 5.16662i) q^{52} +(564.433 + 325.875i) q^{53} +254.366i q^{55} +(130.276 + 70.5701i) q^{56} +(0.0600052 + 0.103932i) q^{58} +(234.651 - 406.428i) q^{59} +(643.906 - 371.759i) q^{61} +348.790 q^{62} -64.0000 q^{64} +(21.0698 - 12.1647i) q^{65} +(231.332 - 400.679i) q^{67} +(-117.869 - 204.155i) q^{68} +(-297.260 + 182.559i) q^{70} +519.129i q^{71} +(892.655 + 515.375i) q^{73} +(-78.8526 - 45.5256i) q^{74} +82.1465i q^{76} +(426.245 - 261.774i) q^{77} +(92.9573 + 161.007i) q^{79} +(75.3431 - 130.498i) q^{80} +(-288.996 + 166.852i) q^{82} -120.345 q^{83} +555.039 q^{85} +(-49.9274 + 28.8256i) q^{86} +(-108.035 + 187.123i) q^{88} +(503.915 + 872.807i) q^{89} +(-42.0679 - 22.7881i) q^{91} -195.145i q^{92} +(-320.384 - 184.974i) q^{94} +(-167.499 - 96.7058i) q^{95} +1244.91i q^{97} +(611.835 + 310.248i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{4} - 42 q^{7}+O(q^{10}) \) 12 * q + 24 * q^4 - 42 * q^7	\( 12 q + 24 q^{4} - 42 q^{7} - 48 q^{10} - 96 q^{16} - 66 q^{19} + 240 q^{22} + 366 q^{25} - 168 q^{28} + 630 q^{31} + 348 q^{37} - 192 q^{40} - 252 q^{43} + 600 q^{46} - 1218 q^{49} + 192 q^{52} - 144 q^{58} + 114 q^{61} - 768 q^{64} + 108 q^{67} - 1344 q^{70} - 978 q^{73} - 744 q^{79} + 240 q^{82} + 7248 q^{85} + 480 q^{88} - 5544 q^{91} - 1920 q^{94}+O(q^{100}) \) 12 * q + 24 * q^4 - 42 * q^7 - 48 * q^10 - 96 * q^16 - 66 * q^19 + 240 * q^22 + 366 * q^25 - 168 * q^28 + 630 * q^31 + 348 * q^37 - 192 * q^40 - 252 * q^43 + 600 * q^46 - 1218 * q^49 + 192 * q^52 - 144 * q^58 + 114 * q^61 - 768 * q^64 + 108 * q^67 - 1344 * q^70 - 978 * q^73 - 744 * q^79 + 240 * q^82 + 7248 * q^85 + 480 * q^88 - 5544 * q^91 - 1920 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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