LMFDB - Embedded newform 378.4.k.c.215.4

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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 178 x^{14} + 23185 x^{12} - 1395488 x^{10} + 61706754 x^{8} - 468877357 x^{6} + \cdots + 6975757441 $ x^16 - 178x^14 + 23185x^12 - 1395488x^10 + 61706754x^8 - 468877357x^6 + 2731971910x^4 - 4899926507*x^2 + 6975757441
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			215.4
Root			$-8.42688 - 4.86526i$ of defining polynomial
Character	$\chi$	$=$	378.215
Dual form			378.4.k.c.269.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} +(6.91387 - 11.9752i) q^{5} +(2.37938 - 18.3668i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} +(6.91387 - 11.9752i) q^{5} +(2.37938 - 18.3668i) q^{7} -8.00000i q^{8} +(-23.9503 + 13.8277i) q^{10} +(45.0525 - 26.0111i) q^{11} -29.5683i q^{13} +(-22.4880 + 29.4328i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-4.22592 - 7.31950i) q^{17} +(-38.0663 - 21.9776i) q^{19} +55.3109 q^{20} -104.044 q^{22} +(103.107 + 59.5286i) q^{23} +(-33.1031 - 57.3362i) q^{25} +(-29.5683 + 51.2137i) q^{26} +(68.3831 - 28.4911i) q^{28} +100.748i q^{29} +(-174.866 + 100.959i) q^{31} +(27.7128 - 16.0000i) q^{32} +16.9037i q^{34} +(-203.495 - 155.479i) q^{35} +(175.193 - 303.443i) q^{37} +(43.9552 + 76.1326i) q^{38} +(-95.8013 - 55.3109i) q^{40} +334.098 q^{41} +18.3440 q^{43} +(180.210 + 104.044i) q^{44} +(-119.057 - 206.213i) q^{46} +(-122.646 + 212.429i) q^{47} +(-331.677 - 87.4031i) q^{49} +132.412i q^{50} +(102.427 - 59.1365i) q^{52} +(-6.61069 + 3.81669i) q^{53} -719.348i q^{55} +(-146.934 - 19.0350i) q^{56} +(100.748 - 174.500i) q^{58} +(-208.579 - 361.270i) q^{59} +(-537.277 - 310.197i) q^{61} +403.835 q^{62} -64.0000 q^{64} +(-354.085 - 204.431i) q^{65} +(25.5858 + 44.3159i) q^{67} +(16.9037 - 29.2780i) q^{68} +(196.984 + 472.792i) q^{70} -1151.73i q^{71} +(-877.473 + 506.609i) q^{73} +(-606.885 + 350.385i) q^{74} -175.821i q^{76} +(-370.543 - 889.360i) q^{77} +(-213.929 + 370.536i) q^{79} +(110.622 + 191.603i) q^{80} +(-578.675 - 334.098i) q^{82} +307.968 q^{83} -116.870 q^{85} +(-31.7728 - 18.3440i) q^{86} +(-208.089 - 360.420i) q^{88} +(-322.016 + 557.749i) q^{89} +(-543.074 - 70.3541i) q^{91} +476.229i q^{92} +(424.858 - 245.292i) q^{94} +(-526.371 + 303.900i) q^{95} +474.008i q^{97} +(487.078 + 483.064i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} + 50 q^{7}+O(q^{10}) $ 16 * q + 32 * q^4 + 50 * q^7	$ 16 q + 32 q^{4} + 50 q^{7} + 60 q^{10} - 128 q^{16} - 498 q^{19} - 240 q^{22} - 470 q^{25} + 160 q^{28} - 582 q^{31} + 40 q^{37} + 240 q^{40} + 1900 q^{43} - 456 q^{46} - 1634 q^{49} - 720 q^{52} + 1200 q^{58} - 1302 q^{61} - 1024 q^{64} - 100 q^{67} - 1620 q^{70} - 5280 q^{73} + 590 q^{79} - 480 q^{82} + 2580 q^{85} - 480 q^{88} - 2382 q^{91} - 3396 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 + 50 * q^7 + 60 * q^10 - 128 * q^16 - 498 * q^19 - 240 * q^22 - 470 * q^25 + 160 * q^28 - 582 * q^31 + 40 * q^37 + 240 * q^40 + 1900 * q^43 - 456 * q^46 - 1634 * q^49 - 720 * q^52 + 1200 * q^58 - 1302 * q^61 - 1024 * q^64 - 100 * q^67 - 1620 * q^70 - 5280 * q^73 + 590 * q^79 - 480 * q^82 + 2580 * q^85 - 480 * q^88 - 2382 * q^91 - 3396 * q^94

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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
$5$	6.91387	−	11.9752i	0.618395	−	1.07109i	−0.371384	−	0.928479i	$-0.621117\pi$
$5$	6.91387	−	11.9752i	0.618395	−	1.07109i	0.989779	−	0.142612i	$-0.0455501\pi$
$6$		0			0
$7$	2.37938	−	18.3668i	0.128474	−	0.991713i
$7$	2.37938	−	18.3668i	0.128474	−	0.991713i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−23.9503	+	13.8277i	−0.757376	+	0.437271i
$11$	45.0525	−	26.0111i	1.23489	−	0.712967i	0.266848	−	0.963738i	$-0.414018\pi$
$11$	45.0525	−	26.0111i	1.23489	−	0.712967i	0.968046	+	0.250772i	$0.0806843\pi$
$12$		0			0
$13$		−	29.5683i		−	0.630828i	−0.948954	−	0.315414i	$-0.897857\pi$
$13$		−	29.5683i		−	0.630828i	0.948954	−	0.315414i	$-0.102143\pi$
$14$	−22.4880	+	29.4328i	−0.429298	+	0.561875i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−4.22592	−	7.31950i	−0.0602903	−	0.104426i	0.834305	−	0.551303i	$-0.185869\pi$
$17$	−4.22592	−	7.31950i	−0.0602903	−	0.104426i	−0.894595	+	0.446877i	$0.852536\pi$
$18$		0			0
$19$	−38.0663	−	21.9776i	−0.459632	−	0.265369i	0.252257	−	0.967660i	$-0.418827\pi$
$19$	−38.0663	−	21.9776i	−0.459632	−	0.265369i	−0.711890	+	0.702291i	$0.752160\pi$
$20$	55.3109			0.618395
$21$		0			0
$22$	−104.044			−1.00829
$23$	103.107	+	59.5286i	0.934748	+	0.539677i	0.888310	−	0.459244i	$-0.151880\pi$
$23$	103.107	+	59.5286i	0.934748	+	0.539677i	0.0464379	+	0.998921i	$0.485213\pi$
$24$		0			0
$25$	−33.1031	−	57.3362i	−0.264825	−	0.458690i
$26$	−29.5683	+	51.2137i	−0.223031	+	0.386301i
$27$		0			0
$28$	68.3831	−	28.4911i	0.461543	−	0.192297i
$29$			100.748i			0.645116i	0.946550	+	0.322558i	$0.104543\pi$
$29$			100.748i			0.645116i	−0.946550	+	0.322558i	$0.895457\pi$
$30$		0			0
$31$	−174.866	+	100.959i	−1.01312	+	0.584927i	−0.912105	−	0.409957i	$-0.865544\pi$
$31$	−174.866	+	100.959i	−1.01312	+	0.584927i	−0.101019	+	0.994885i	$0.532210\pi$
$32$	27.7128	−	16.0000i	0.153093	−	0.0883883i
$33$		0			0
$34$			16.9037i			0.0852634i
$35$	−203.495	−	155.479i	−0.982767	−	0.750878i
$36$		0			0
$37$	175.193	−	303.443i	0.778419	−	1.34826i	−0.154434	−	0.988003i	$-0.549355\pi$
$37$	175.193	−	303.443i	0.778419	−	1.34826i	0.932853	−	0.360258i	$-0.117311\pi$
$38$	43.9552	+	76.1326i	0.187644	+	0.325009i
$39$		0			0
$40$	−95.8013	−	55.3109i	−0.378688	−	0.218636i
$41$	334.098			1.27262			0.636309	−	0.771434i	$-0.280460\pi$
$41$	334.098			1.27262			0.636309	+	0.771434i	$0.280460\pi$
$42$		0			0
$43$	18.3440			0.0650567			0.0325283	−	0.999471i	$-0.489644\pi$
$43$	18.3440			0.0650567			0.0325283	+	0.999471i	$0.489644\pi$
$44$	180.210	+	104.044i	0.617447	+	0.356483i
$45$		0			0
$46$	−119.057	−	206.213i	−0.381609	−	0.660967i
$47$	−122.646	+	212.429i	−0.380633	+	0.659275i	−0.991153	−	0.132726i	$-0.957627\pi$
$47$	−122.646	+	212.429i	−0.380633	+	0.659275i	0.610520	+	0.792001i	$0.290960\pi$
$48$		0			0
$49$	−331.677	−	87.4031i	−0.966989	−	0.254820i
$50$			132.412i			0.374519i
$51$		0			0
$52$	102.427	−	59.1365i	0.273156	−	0.157707i
$53$	−6.61069	+	3.81669i	−0.0171330	+	0.00989174i	−0.508542	−	0.861037i	$-0.669815\pi$
$53$	−6.61069	+	3.81669i	−0.0171330	+	0.00989174i	0.491409	+	0.870929i	$0.336482\pi$
$54$		0			0
$55$		−	719.348i		−	1.76358i
$56$	−146.934	−	19.0350i	−0.350623	−	0.0454226i
$57$		0			0
$58$	100.748	−	174.500i	0.228083	−	0.395051i
$59$	−208.579	−	361.270i	−0.460249	−	0.797175i	0.538724	−	0.842482i	$-0.318907\pi$
$59$	−208.579	−	361.270i	−0.460249	−	0.797175i	−0.998973	+	0.0453074i	$0.985573\pi$
$60$		0			0
$61$	−537.277	−	310.197i	−1.12773	−	0.651093i	−0.184363	−	0.982858i	$-0.559022\pi$
$61$	−537.277	−	310.197i	−1.12773	−	0.651093i	−0.943362	+	0.331766i	$0.892356\pi$
$62$	403.835			0.827212
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−354.085	−	204.431i	−0.675674	−	0.390101i
$66$		0			0
$67$	25.5858	+	44.3159i	0.0466538	+	0.0808067i	0.888409	−	0.459052i	$-0.151811\pi$
$67$	25.5858	+	44.3159i	0.0466538	+	0.0808067i	−0.841756	+	0.539859i	$0.818478\pi$
$68$	16.9037	−	29.2780i	0.0301452	−	0.0522129i
$69$		0			0
$70$	196.984	+	472.792i	0.336344	+	0.807278i
$71$		−	1151.73i		−	1.92514i	−0.271027	−	0.962572i	$-0.587363\pi$
$71$		−	1151.73i		−	1.92514i	0.271027	−	0.962572i	$-0.412637\pi$
$72$		0			0
$73$	−877.473	+	506.609i	−1.40686	+	0.812248i	−0.995084	−	0.0990390i	$-0.968423\pi$
$73$	−877.473	+	506.609i	−1.40686	+	0.812248i	−0.411771	+	0.911287i	$0.635090\pi$
$74$	−606.885	+	350.385i	−0.953364	+	0.550425i
$75$		0			0
$76$		−	175.821i		−	0.265369i
$77$	−370.543	−	889.360i	−0.548406	−	1.31626i
$78$		0			0
$79$	−213.929	+	370.536i	−0.304670	+	0.527704i	−0.977188	−	0.212377i	$-0.931880\pi$
$79$	−213.929	+	370.536i	−0.304670	+	0.527704i	0.672518	+	0.740081i	$0.265213\pi$
$80$	110.622	+	191.603i	0.154599	+	0.267773i
$81$		0			0
$82$	−578.675	−	334.098i	−0.779317	−	0.449939i
$83$	307.968			0.407275			0.203638	−	0.979046i	$-0.434724\pi$
$83$	307.968			0.407275			0.203638	+	0.979046i	$0.434724\pi$
$84$		0			0
$85$	−116.870			−0.149133
$86$	−31.7728	−	18.3440i	−0.0398389	−	0.0230010i
$87$		0			0
$88$	−208.089	−	360.420i	−0.252072	−	0.436601i
$89$	−322.016	+	557.749i	−0.383524	+	0.664284i	−0.991563	−	0.129623i	$-0.958623\pi$
$89$	−322.016	+	557.749i	−0.383524	+	0.664284i	0.608039	+	0.793907i	$0.291956\pi$
$90$		0			0
$91$	−543.074	−	70.3541i	−0.625600	−	0.0810453i
$92$			476.229i			0.539677i
$93$		0			0
$94$	424.858	−	245.292i	0.466178	−	0.269148i
$95$	−526.371	+	303.900i	−0.568469	+	0.328206i
$96$		0			0
$97$			474.008i			0.496167i	0.968739	+	0.248083i	$0.0798007\pi$
$97$			474.008i			0.496167i	−0.968739	+	0.248083i	$0.920199\pi$
$98$	487.078	+	483.064i	0.502065	+	0.497927i
$99$		0			0
$100$	132.412	−	229.345i	0.132412	−	0.229345i
$101$	487.904	+	845.075i	0.480676	+	0.832555i	0.999754	−	0.0221714i	$-0.00705796\pi$
$101$	487.904	+	845.075i	0.480676	+	0.832555i	−0.519078	+	0.854727i	$0.673725\pi$
$102$		0			0
$103$	946.832	+	546.654i	0.905769	+	0.522946i	0.879067	−	0.476698i	$-0.158166\pi$
$103$	946.832	+	546.654i	0.905769	+	0.522946i	0.0267014	+	0.999643i	$0.491500\pi$
$104$	−236.546			−0.223031
$105$		0			0
$106$	15.2667			0.0139890
$107$	−979.376	−	565.443i	−0.884858	−	0.510873i	−0.0126010	−	0.999921i	$-0.504011\pi$
$107$	−979.376	−	565.443i	−0.884858	−	0.510873i	−0.872257	+	0.489047i	$0.837344\pi$
$108$		0			0
$109$	−766.895	−	1328.30i	−0.673901	−	1.16723i	−0.976789	−	0.214205i	$-0.931284\pi$
$109$	−766.895	−	1328.30i	−0.673901	−	1.16723i	0.302888	−	0.953026i	$-0.402049\pi$
$110$	−719.348	+	1245.95i	−0.623520	+	1.07997i
$111$		0			0
$112$	235.462	+	179.904i	0.198653	+	0.151780i
$113$		−	982.606i		−	0.818016i	−0.912531	−	0.409008i	$-0.865875\pi$
$113$		−	982.606i		−	0.818016i	0.912531	−	0.409008i	$-0.134125\pi$
$114$		0			0
$115$	1425.73	−	823.146i	1.15609	−	0.667467i
$116$	−349.000	+	201.495i	−0.279343	+	0.161279i
$117$		0			0
$118$			834.317i			0.650891i
$119$	−144.491	+	60.2006i	−0.111306	+	0.0463746i
$120$		0			0
$121$	687.652	−	1191.05i	0.516643	−	0.894852i
$122$	620.394	+	1074.55i	0.460392	+	0.797422i
$123$		0			0
$124$	−699.463	−	403.835i	−0.506562	−	0.292464i
$125$	812.985			0.581725
$126$		0			0
$127$	2082.23			1.45486			0.727432	−	0.686180i	$-0.240714\pi$
$127$	2082.23			1.45486			0.727432	+	0.686180i	$0.240714\pi$
$128$	110.851	+	64.0000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	408.862	+	708.170i	0.275843	+	0.477774i
$131$	1340.04	−	2321.01i	0.893737	−	1.54800i	0.0583778	−	0.998295i	$-0.481407\pi$
$131$	1340.04	−	2321.01i	0.893737	−	1.54800i	0.835360	−	0.549704i	$-0.185259\pi$
$132$		0			0
$133$	−494.232	+	646.863i	−0.322221	+	0.421730i
$134$		−	102.343i		−	0.0659784i
$135$		0			0
$136$	−58.5560	+	33.8073i	−0.0369201	+	0.0213158i
$137$	−1089.35	+	628.936i	−0.679338	+	0.392216i	−0.799606	−	0.600525i	$-0.794958\pi$
$137$	−1089.35	+	628.936i	−0.679338	+	0.392216i	0.120267	+	0.992742i	$0.461625\pi$
$138$		0			0
$139$			1549.78i			0.945685i	0.881147	+	0.472843i	$0.156772\pi$
$139$			1549.78i			0.945685i	−0.881147	+	0.472843i	$0.843228\pi$
$140$	131.606	−	1015.88i	0.0794480	−	0.613270i
$141$		0			0
$142$	−1151.73	+	1994.85i	−0.680641	+	1.17890i
$143$	−769.102	−	1332.12i	−0.449759	−	0.779006i
$144$		0			0
$145$	1206.47	+	696.555i	0.690978	+	0.398936i
$146$	2026.44			1.14869
$147$		0			0
$148$	1401.54			0.778419
$149$	−1191.70	−	688.030i	−0.655222	−	0.378293i	0.135232	−	0.990814i	$-0.456822\pi$
$149$	−1191.70	−	688.030i	−0.655222	−	0.378293i	−0.790454	+	0.612521i	$0.790155\pi$
$150$		0			0
$151$	1590.77	+	2755.30i	0.857319	+	1.48492i	0.874477	+	0.485066i	$0.161205\pi$
$151$	1590.77	+	2755.30i	0.857319	+	1.48492i	−0.0171588	+	0.999853i	$0.505462\pi$
$152$	−175.821	+	304.531i	−0.0938220	+	0.162505i
$153$		0			0
$154$	−247.561	+	1910.96i	−0.129539	+	0.999931i
$155$			2792.06i			1.44686i
$156$		0			0
$157$	−1749.07	+	1009.83i	−0.889117	+	0.513332i	−0.873653	−	0.486549i	$-0.838256\pi$
$157$	−1749.07	+	1009.83i	−0.889117	+	0.513332i	−0.0154633	+	0.999880i	$0.504922\pi$
$158$	741.073	−	427.859i	0.373143	−	0.215434i
$159$		0			0
$160$		−	442.487i		−	0.218636i
$161$	1338.68	−	1752.09i	0.655296	−	0.857667i
$162$		0			0
$163$	1893.98	−	3280.47i	0.910110	−	1.57636i	0.0962022	−	0.995362i	$-0.469330\pi$
$163$	1893.98	−	3280.47i	0.910110	−	1.57636i	0.813908	−	0.580994i	$-0.197336\pi$
$164$	668.196	+	1157.35i	0.318155	+	0.551060i
$165$		0			0
$166$	−533.416	−	307.968i	−0.249404	−	0.143994i
$167$	3182.97			1.47489			0.737443	−	0.675410i	$-0.236033\pi$
$167$	3182.97			1.47489			0.737443	+	0.675410i	$0.236033\pi$
$168$		0			0
$169$	1322.72			0.602056
$170$	202.424	+	116.870i	0.0913249	+	0.0527265i
$171$		0			0
$172$	36.6880	+	63.5456i	0.0162642	+	0.0281704i
$173$	−1451.52	+	2514.11i	−0.637902	+	1.10488i	0.347991	+	0.937498i	$0.386864\pi$
$173$	−1451.52	+	2514.11i	−0.637902	+	1.10488i	−0.985892	+	0.167380i	$0.946469\pi$
$174$		0			0
$175$	−1131.85	+	471.572i	−0.488912	+	0.203700i
$176$			832.354i			0.356483i
$177$		0			0
$178$	1115.50	−	644.033i	0.469720	−	0.271193i
$179$	3390.78	−	1957.67i	1.41586	−	0.817446i	0.419927	−	0.907558i	$-0.362056\pi$
$179$	3390.78	−	1957.67i	1.41586	−	0.817446i	0.995932	+	0.0901114i	$0.0287223\pi$
$180$		0			0
$181$			2963.94i			1.21717i	0.793488	+	0.608586i	$0.208263\pi$
$181$			2963.94i			1.21717i	−0.793488	+	0.608586i	$0.791737\pi$
$182$	870.277	+	664.931i	0.354446	+	0.270813i
$183$		0			0
$184$	476.229	−	824.852i	0.190805	−	0.330483i
$185$	−2422.52	−	4195.92i	−0.962741	−	1.66752i
$186$		0			0
$187$	−380.776	−	219.841i	−0.148904	−	0.0859700i
$188$	−981.167			−0.380633
$189$		0			0
$190$	1215.60			0.464153
$191$	−389.717	−	225.003i	−0.147639	−	0.0852391i	0.424361	−	0.905493i	$-0.360499\pi$
$191$	−389.717	−	225.003i	−0.147639	−	0.0852391i	−0.572000	+	0.820254i	$0.693832\pi$
$192$		0			0
$193$	2553.12	+	4422.13i	0.952215	+	1.64928i	0.740617	+	0.671927i	$0.234533\pi$
$193$	2553.12	+	4422.13i	0.952215	+	1.64928i	0.211597	+	0.977357i	$0.432133\pi$
$194$	474.008	−	821.006i	0.175422	−	0.303839i
$195$		0			0
$196$	−360.581	−	1323.77i	−0.131407	−	0.482423i
$197$			3951.86i			1.42923i	0.699518	+	0.714615i	$0.253398\pi$
$197$			3951.86i			1.42923i	−0.699518	+	0.714615i	$0.746602\pi$
$198$		0			0
$199$	−1362.36	+	786.559i	−0.485302	+	0.280189i	−0.722623	−	0.691242i	$-0.757064\pi$
$199$	−1362.36	+	786.559i	−0.485302	+	0.280189i	0.237321	+	0.971431i	$0.423731\pi$
$200$	−458.690	+	264.825i	−0.162171	+	0.0936297i
$201$		0			0
$202$		−	1951.62i		−	0.679779i
$203$	1850.41	+	239.717i	0.639770	+	0.0828809i
$204$		0			0
$205$	2309.91	−	4000.88i	0.786981	−	1.36309i
$206$	−1093.31	−	1893.66i	−0.369779	−	0.640475i
$207$		0			0
$208$	409.710	+	236.546i	0.136578	+	0.0788535i
$209$	−2286.64			−0.756797
$210$		0			0
$211$	−5485.14			−1.78963			−0.894817	−	0.446434i	$-0.852694\pi$
$211$	−5485.14			−1.78963			−0.894817	+	0.446434i	$0.852694\pi$
$212$	−26.4428	−	15.2667i	−0.00856650	−	0.00494587i
$213$		0			0
$214$	1130.89	+	1958.75i	0.361242	+	0.625689i
$215$	126.828	−	219.673i	0.0402307	−	0.0696817i
$216$		0			0
$217$	1438.22	+	3451.94i	0.449919	+	1.07988i
$218$			3067.58i			0.953040i
$219$		0			0
$220$	2491.90	−	1438.70i	0.763653	−	0.440895i
$221$	−216.425	+	124.953i	−0.0658747	+	0.0380328i
$222$		0			0
$223$			2397.01i			0.719802i	0.932991	+	0.359901i	$0.117189\pi$
$223$			2397.01i			0.719802i	−0.932991	+	0.359901i	$0.882811\pi$
$224$	−227.929	−	547.065i	−0.0679873	−	0.163180i
$225$		0			0
$226$	−982.606	+	1701.92i	−0.289212	+	0.500930i
$227$	−199.618	−	345.748i	−0.0583661	−	0.101093i	0.835366	−	0.549694i	$-0.185256\pi$
$227$	−199.618	−	345.748i	−0.0583661	−	0.101093i	−0.893732	+	0.448601i	$0.851922\pi$
$228$		0			0
$229$	−3660.44	−	2113.36i	−1.05628	−	0.609845i	−0.131881	−	0.991266i	$-0.542102\pi$
$229$	−3660.44	−	2113.36i	−1.05628	−	0.609845i	−0.924401	+	0.381421i	$0.875435\pi$
$230$	−3292.58			−0.943941
$231$		0			0
$232$	805.981			0.228083
$233$	1538.77	+	888.408i	0.432653	+	0.249792i	0.700476	−	0.713676i	$-0.252971\pi$
$233$	1538.77	+	888.408i	0.432653	+	0.249792i	−0.267823	+	0.963468i	$0.586304\pi$
$234$		0			0
$235$	1695.91	+	2937.41i	0.470763	+	0.815385i
$236$	834.317	−	1445.08i	0.230125	−	0.398588i
$237$		0			0
$238$	310.466	+	40.2203i	0.0845568	+	0.0109542i
$239$			5575.18i			1.50890i	0.656355	+	0.754452i	$0.272097\pi$
$239$			5575.18i			1.50890i	−0.656355	+	0.754452i	$0.727903\pi$
$240$		0			0
$241$	2897.07	−	1672.62i	0.774343	−	0.447067i	−0.0600787	−	0.998194i	$-0.519135\pi$
$241$	2897.07	−	1672.62i	0.774343	−	0.447067i	0.834422	+	0.551127i	$0.185802\pi$
$242$	−2382.10	+	1375.30i	−0.632756	+	0.365322i
$243$		0			0
$244$		−	2481.58i		−	0.651093i
$245$	−3339.84	+	3367.60i	−0.870916	+	0.878154i
$246$		0			0
$247$	−649.839	+	1125.55i	−0.167402	+	0.289949i
$248$	807.671	+	1398.93i	0.206803	+	0.358193i
$249$		0			0
$250$	−1408.13	−	812.985i	−0.356232	−	0.205671i
$251$	907.432			0.228194			0.114097	−	0.993470i	$-0.463603\pi$
$251$	907.432			0.228194			0.114097	+	0.993470i	$0.463603\pi$
$252$		0			0
$253$	6193.61			1.53909
$254$	−3606.52	−	2082.23i	−0.890919	−	0.514372i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1219.47	−	2112.19i	0.295987	−	0.512665i	−0.679227	−	0.733928i	$-0.737685\pi$
$257$	1219.47	−	2112.19i	0.295987	−	0.512665i	0.975214	+	0.221263i	$0.0710181\pi$
$258$		0			0
$259$	−5156.41	−	3939.73i	−1.23708	−	0.945185i
$260$		−	1635.45i		−	0.390101i
$261$		0			0
$262$	−4642.03	+	2680.07i	−1.09460	+	0.631968i
$263$	1174.49	−	678.091i	0.275369	−	0.158984i	−0.355956	−	0.934503i	$-0.615845\pi$
$263$	1174.49	−	678.091i	0.275369	−	0.158984i	0.631325	+	0.775518i	$0.282511\pi$
$264$		0			0
$265$			105.552i			0.0244680i
$266$	1502.90	−	626.167i	0.346423	−	0.144334i
$267$		0			0
$268$	−102.343	+	177.264i	−0.0233269	+	0.0404034i
$269$	−1270.70	−	2200.92i	−0.288015	−	0.498857i	0.685321	−	0.728241i	$-0.259662\pi$
$269$	−1270.70	−	2200.92i	−0.288015	−	0.498857i	−0.973336	+	0.229384i	$0.926329\pi$
$270$		0			0
$271$	−4735.98	−	2734.32i	−1.06159	−	0.612908i	−0.135716	−	0.990748i	$-0.543334\pi$
$271$	−4735.98	−	2734.32i	−1.06159	−	0.612908i	−0.925871	+	0.377840i	$0.876667\pi$
$272$	135.229			0.0301452
$273$		0			0
$274$	2515.74			0.554678
$275$	−2982.75	−	1722.09i	−0.654061	−	0.377623i
$276$		0			0
$277$	−1411.68	−	2445.10i	−0.306208	−	0.530368i	0.671322	−	0.741166i	$-0.265727\pi$
$277$	−1411.68	−	2445.10i	−0.306208	−	0.530368i	−0.977530	+	0.210798i	$0.932394\pi$
$278$	1549.78	−	2684.29i	0.334350	−	0.579112i
$279$		0			0
$280$	−1243.83	+	1627.96i	−0.265476	+	0.347461i
$281$			3007.63i			0.638505i	0.947670	+	0.319252i	$0.103432\pi$
$281$			3007.63i			0.638505i	−0.947670	+	0.319252i	$0.896568\pi$
$282$		0			0
$283$	3277.62	−	1892.33i	0.688460	−	0.397482i	−0.114575	−	0.993415i	$-0.536551\pi$
$283$	3277.62	−	1892.33i	0.688460	−	0.397482i	0.803035	+	0.595932i	$0.203217\pi$
$284$	3989.71	−	2303.46i	0.833612	−	0.481286i
$285$		0			0
$286$			3076.41i			0.636056i
$287$	794.947	−	6136.31i	0.163499	−	1.26207i
$288$		0			0
$289$	2420.78	−	4192.92i	0.492730	−	0.853434i
$290$	−1393.11	−	2412.94i	−0.282091	−	0.488595i
$291$		0			0
$292$	−3509.89	−	2026.44i	−0.703428	−	0.406124i
$293$	−3289.54			−0.655894			−0.327947	−	0.944696i	$-0.606357\pi$
$293$	−3289.54			−0.655894			−0.327947	+	0.944696i	$0.606357\pi$
$294$		0			0
$295$	−5768.36			−1.13846
$296$	−2427.54	−	1401.54i	−0.476682	−	0.275213i
$297$		0			0
$298$	1376.06	+	2383.41i	0.267493	+	0.463312i
$299$	1760.16	−	3048.68i	0.340443	−	0.589665i
$300$		0			0
$301$	43.6474	−	336.921i	0.00835812	−	0.0645175i
$302$		−	6363.08i		−	1.21243i
$303$		0			0
$304$	609.061	−	351.642i	0.114908	−	0.0663422i
$305$	−7429.32	+	4289.32i	−1.39476	+	0.805265i
$306$		0			0
$307$		−	6283.43i		−	1.16813i	−0.811709	−	0.584063i	$-0.801462\pi$
$307$		−	6283.43i		−	1.16813i	0.811709	−	0.584063i	$-0.198538\pi$
$308$	2339.75	−	3062.32i	0.432855	−	0.566531i
$309$		0			0
$310$	2792.06	−	4836.00i	0.511544	−	0.886020i
$311$	87.8805	+	152.213i	0.0160233	+	0.0277532i	0.873926	−	0.486059i	$-0.161566\pi$
$311$	87.8805	+	152.213i	0.0160233	+	0.0277532i	−0.857903	+	0.513812i	$0.828233\pi$
$312$		0			0
$313$	−5600.66	−	3233.54i	−1.01140	−	0.583932i	−0.0997989	−	0.995008i	$-0.531820\pi$
$313$	−5600.66	−	3233.54i	−1.01140	−	0.583932i	−0.911601	+	0.411075i	$0.865153\pi$
$314$	4039.31			0.725961
$315$		0			0
$316$	−1711.43			−0.304670
$317$	2796.51	+	1614.56i	0.495481	+	0.286066i	0.726845	−	0.686801i	$-0.240986\pi$
$317$	2796.51	+	1614.56i	0.495481	+	0.286066i	−0.231365	+	0.972867i	$0.574319\pi$
$318$		0			0
$319$	2620.55	+	4538.93i	0.459946	+	0.796650i
$320$	−442.487	+	766.411i	−0.0772994	+	0.133886i
$321$		0			0
$322$	−4070.75	+	1696.04i	−0.704516	+	0.293529i
$323$			371.502i			0.0639967i
$324$		0			0
$325$	−1695.33	+	978.801i	−0.289354	+	0.167059i
$326$	−6560.94	+	3787.96i	−1.11465	+	0.643545i
$327$		0			0
$328$		−	2672.79i		−	0.449939i
$329$	3609.81	+	2758.06i	0.604910	+	0.462178i
$330$		0			0
$331$	4064.21	−	7039.41i	0.674891	−	1.16895i	−0.301610	−	0.953432i	$-0.597524\pi$
$331$	4064.21	−	7039.41i	0.674891	−	1.16895i	0.976501	−	0.215514i	$-0.0691427\pi$
$332$	615.935	+	1066.83i	0.101819	+	0.176355i
$333$		0			0
$334$	−5513.07	−	3182.97i	−0.903179	−	0.521451i
$335$	707.587			0.115402
$336$		0			0
$337$	4654.58			0.752378			0.376189	−	0.926543i	$-0.377234\pi$
$337$	4654.58			0.752378			0.376189	+	0.926543i	$0.377234\pi$
$338$	−2291.01	−	1322.72i	−0.368683	−	0.212859i
$339$		0			0
$340$	−233.739	−	404.849i	−0.0372832	−	0.0645765i
$341$	−5252.10	+	9096.90i	−0.834067	+	1.44465i
$342$		0			0
$343$	−2394.50	+	5883.87i	−0.376941	+	0.926237i
$344$		−	146.752i		−	0.0230010i
$345$		0			0
$346$	5028.21	−	2903.04i	0.781267	−	0.451065i
$347$	3660.07	−	2113.14i	0.566232	−	0.326914i	−0.189411	−	0.981898i	$-0.560658\pi$
$347$	3660.07	−	2113.14i	0.566232	−	0.326914i	0.755643	+	0.654983i	$0.227324\pi$
$348$		0			0
$349$			5382.10i			0.825493i	0.910846	+	0.412747i	$0.135430\pi$
$349$			5382.10i			0.825493i	−0.910846	+	0.412747i	$0.864570\pi$
$350$	2431.99	+	315.059i	0.371415	+	0.0481161i
$351$		0			0
$352$	832.354	−	1441.68i	0.126036	−	0.218301i
$353$	5622.19	+	9737.92i	0.847702	+	1.46826i	0.883253	+	0.468896i	$0.155348\pi$
$353$	5622.19	+	9737.92i	0.847702	+	1.46826i	−0.0355511	+	0.999368i	$0.511319\pi$
$354$		0			0
$355$	−13792.2	−	7962.91i	−2.06200	−	1.19050i
$356$	−2576.13			−0.383524
$357$		0			0
$358$	−7830.67			−1.15604
$359$	2743.69	+	1584.07i	0.403360	+	0.232880i	0.687933	−	0.725774i	$-0.258518\pi$
$359$	2743.69	+	1584.07i	0.403360	+	0.232880i	−0.284573	+	0.958654i	$0.591852\pi$
$360$		0			0
$361$	−2463.47	−	4266.86i	−0.359159	−	0.622081i
$362$	2963.94	−	5133.70i	0.430335	−	0.745362i
$363$		0			0
$364$	−842.433	−	2021.97i	−0.121306	−	0.291154i
$365$			14010.5i			2.00916i
$366$		0			0
$367$	9175.52	−	5297.49i	1.30506	−	0.753479i	0.323796	−	0.946127i	$-0.395041\pi$
$367$	9175.52	−	5297.49i	1.30506	−	0.753479i	0.981268	+	0.192648i	$0.0617076\pi$
$368$	−1649.70	+	952.458i	−0.233687	+	0.134919i
$369$		0			0
$370$			9690.07i			1.36152i
$371$	54.3709	+	130.499i	0.00760861	+	0.0182618i
$372$		0			0
$373$	−1060.86	+	1837.46i	−0.147263	+	0.255068i	−0.930215	−	0.367015i	$-0.880380\pi$
$373$	−1060.86	+	1837.46i	−0.147263	+	0.255068i	0.782952	+	0.622083i	$0.213713\pi$
$374$	439.683	+	761.553i	0.0607900	+	0.105291i
$375$		0			0
$376$	1699.43	+	981.167i	0.233089	+	0.134574i
$377$	2978.93			0.406957
$378$		0			0
$379$	−2725.81			−0.369434			−0.184717	−	0.982792i	$-0.559137\pi$
$379$	−2725.81			−0.369434			−0.184717	+	0.982792i	$0.559137\pi$
$380$	−2105.48	−	1215.60i	−0.284234	−	0.164103i
$381$		0			0
$382$	450.007	+	779.435i	0.0602732	+	0.104396i
$383$	−1159.76	+	2008.76i	−0.154728	+	0.267997i	−0.932960	−	0.359980i	$-0.882784\pi$
$383$	−1159.76	+	2008.76i	−0.154728	+	0.267997i	0.778232	+	0.627977i	$0.216117\pi$
$384$		0			0
$385$	−13212.1	−	1711.60i	−1.74897	−	0.226575i
$386$		−	10212.5i		−	1.34663i
$387$		0			0
$388$	−1642.01	+	948.015i	−0.214847	+	0.124042i
$389$	5808.13	−	3353.32i	0.757027	−	0.437070i	−0.0712000	−	0.997462i	$-0.522683\pi$
$389$	5808.13	−	3353.32i	0.757027	−	0.437070i	0.828227	+	0.560392i	$0.189350\pi$
$390$		0			0
$391$		−	1006.25i		−	0.130149i
$392$	−699.225	+	2653.42i	−0.0900923	+	0.341882i
$393$		0			0
$394$	3951.86	−	6844.82i	0.505309	−	0.875221i
$395$	2958.16	+	5123.68i	0.376813	+	0.652659i
$396$		0			0
$397$	−6052.95	−	3494.67i	−0.765211	−	0.441795i	0.0659525	−	0.997823i	$-0.478991\pi$
$397$	−6052.95	−	3494.67i	−0.765211	−	0.441795i	−0.831164	+	0.556028i	$0.812325\pi$
$398$	3146.23			0.396247
$399$		0			0
$400$	1059.30			0.132412
$401$	90.5710	+	52.2912i	0.0112791	+	0.00651197i	0.505629	−	0.862751i	$-0.331260\pi$
$401$	90.5710	+	52.2912i	0.0112791	+	0.00651197i	−0.494350	+	0.869263i	$0.664594\pi$
$402$		0			0
$403$	2985.18	+	5170.48i	0.368988	+	0.639106i
$404$	−1951.62	+	3380.30i	−0.240338	+	0.416278i
$405$		0			0
$406$	−2965.29	−	2265.61i	−0.362474	−	0.276947i
$407$		−	18227.8i		−	2.21995i
$408$		0			0
$409$	−10037.2	+	5795.00i	−1.21347	+	0.700598i	−0.963514	−	0.267659i	$-0.913750\pi$
$409$	−10037.2	+	5795.00i	−1.21347	+	0.700598i	−0.249957	+	0.968257i	$0.580417\pi$
$410$	−8001.76	+	4619.82i	−0.963851	+	0.556480i
$411$		0			0
$412$			4373.23i			0.522946i
$413$	−7131.65	+	2971.33i	−0.849699	+	0.354018i
$414$		0			0
$415$	2129.25	−	3687.96i	0.251857	−	0.436229i
$416$	−473.092	−	819.420i	−0.0557578	−	0.0965754i
$417$		0			0
$418$	3960.58	+	2286.64i	0.463441	+	0.267568i
$419$	9004.98			1.04993			0.524967	−	0.851123i	$-0.324078\pi$
$419$	9004.98			1.04993			0.524967	+	0.851123i	$0.324078\pi$
$420$		0			0
$421$	12230.3			1.41584			0.707919	−	0.706293i	$-0.249634\pi$
$421$	12230.3			1.41584			0.707919	+	0.706293i	$0.249634\pi$
$422$	9500.54	+	5485.14i	1.09592	+	0.632731i
$423$		0			0
$424$	30.5335	+	52.8856i	0.00349726	+	0.00605743i
$425$	−279.782	+	484.596i	−0.0319327	+	0.0553091i
$426$		0			0
$427$	−6975.70	+	9129.97i	−0.790581	+	1.03473i
$428$		−	4523.54i		−	0.510873i
$429$		0			0
$430$	−439.345	+	253.656i	−0.0492724	+	0.0284474i
$431$	6629.06	−	3827.29i	0.740860	−	0.427736i	−0.0815219	−	0.996672i	$-0.525978\pi$
$431$	6629.06	−	3827.29i	0.740860	−	0.427736i	0.822382	+	0.568936i	$0.192645\pi$
$432$		0			0
$433$		−	6569.44i		−	0.729116i	−0.931181	−	0.364558i	$-0.881220\pi$
$433$		−	6569.44i		−	0.729116i	0.931181	−	0.364558i	$-0.118780\pi$
$434$	960.878	−	7417.16i	0.106276	−	0.820357i
$435$		0			0
$436$	3067.58	−	5313.20i	0.336951	−	0.583615i
$437$	−2616.59	−	4532.07i	−0.286427	−	0.496106i
$438$		0			0
$439$	8700.97	+	5023.51i	0.945956	+	0.546148i	0.891822	−	0.452386i	$-0.149427\pi$
$439$	8700.97	+	5023.51i	0.945956	+	0.546148i	0.0541338	+	0.998534i	$0.482760\pi$
$440$	−5754.79			−0.623520
$441$		0			0
$442$	499.812			0.0537865
$443$	8890.50	+	5132.93i	0.953499	+	0.550503i	0.894166	−	0.447735i	$-0.147769\pi$
$443$	8890.50	+	5132.93i	0.953499	+	0.550503i	0.0593331	+	0.998238i	$0.481103\pi$
$444$		0			0
$445$	4452.76	+	7712.40i	0.474339	+	0.821580i
$446$	2397.01	−	4151.75i	0.254488	−	0.440787i
$447$		0			0
$448$	−152.280	+	1175.47i	−0.0160593	+	0.123964i
$449$		−	995.326i		−	0.104615i	−0.998631	−	0.0523077i	$-0.983342\pi$
$449$		−	995.326i		−	0.104615i	0.998631	−	0.0523077i	$-0.0166577\pi$
$450$		0			0
$451$	15052.0	−	8690.25i	1.57155	−	0.907335i
$452$	3403.85	−	1965.21i	0.354211	−	0.204504i
$453$		0			0
$454$			798.471i			0.0825421i
$455$	−4597.24	+	6016.98i	−0.473675	+	0.619957i
$456$		0			0
$457$	−3339.22	+	5783.70i	−0.341799	+	0.592013i	−0.984767	−	0.173880i	$-0.944369\pi$
$457$	−3339.22	+	5783.70i	−0.341799	+	0.592013i	0.642968	+	0.765893i	$0.277703\pi$
$458$	4226.71	+	7320.88i	0.431225	+	0.746904i
$459$		0			0
$460$	5702.92	+	3292.58i	0.578043	+	0.333734i
$461$	4459.61			0.450553			0.225276	−	0.974295i	$-0.427672\pi$
$461$	4459.61			0.450553			0.225276	+	0.974295i	$0.427672\pi$
$462$		0			0
$463$	3163.17			0.317505			0.158753	−	0.987318i	$-0.449253\pi$
$463$	3163.17			0.317505			0.158753	+	0.987318i	$0.449253\pi$
$464$	−1396.00	−	805.981i	−0.139672	−	0.0806395i
$465$		0			0
$466$	−1776.82	−	3077.54i	−0.176630	−	0.305932i
$467$	−1325.74	+	2296.25i	−0.131366	+	0.227533i	−0.924203	−	0.381901i	$-0.875270\pi$
$467$	−1325.74	+	2296.25i	−0.131366	+	0.227533i	0.792837	+	0.609433i	$0.208603\pi$
$468$		0			0
$469$	874.819	−	364.484i	0.0861309	−	0.0358855i
$470$		−	6783.66i		−	0.665759i
$471$		0			0
$472$	−2890.16	+	1668.63i	−0.281844	+	0.162723i
$473$	826.444	−	477.148i	0.0803381	−	0.0463833i
$474$		0			0
$475$			2910.11i			0.281105i
$476$	−497.522	−	380.129i	−0.0479074	−	0.0366034i
$477$		0			0
$478$	5575.18	−	9656.49i	0.533478	−	0.924012i
$479$	−8833.32	−	15299.8i	−0.842598	−	1.45942i	−0.887691	−	0.460440i	$-0.847692\pi$
$479$	−8833.32	−	15299.8i	−0.842598	−	1.45942i	0.0450928	−	0.998983i	$-0.485642\pi$
$480$		0			0
$481$	−8972.27	−	5180.14i	−0.850520	−	0.491048i
$482$	−6690.50			−0.632248
$483$		0			0
$484$	5501.22			0.516643
$485$	5676.32	+	3277.23i	0.531440	+	0.306827i
$486$		0			0
$487$	7037.64	+	12189.6i	0.654838	+	1.13421i	0.981934	+	0.189221i	$0.0605964\pi$
$487$	7037.64	+	12189.6i	0.654838	+	1.13421i	−0.327097	+	0.944991i	$0.606070\pi$
$488$	−2481.58	+	4298.21i	−0.230196	+	0.398711i
$489$		0			0
$490$	9152.36	−	2493.01i	0.843799	−	0.229842i
$491$			909.632i			0.0836071i	0.999126	+	0.0418036i	$0.0133104\pi$
$491$			909.632i			0.0836071i	−0.999126	+	0.0418036i	$0.986690\pi$
$492$		0			0
$493$	737.422	−	425.751i	0.0673668	−	0.0388942i
$494$	2251.11	−	1299.68i	0.205025	−	0.118371i
$495$		0			0
$496$		−	3230.68i		−	0.292464i
$497$	−21153.6	−	2740.40i	−1.90919	−	0.247332i
$498$		0			0
$499$	2421.32	−	4193.86i	0.217221	−	0.376238i	−0.736736	−	0.676180i	$-0.763634\pi$
$499$	2421.32	−	4193.86i	0.217221	−	0.376238i	0.953957	+	0.299942i	$0.0969674\pi$
$500$	1625.97	+	2816.26i	0.145431	+	0.251894i
$501$		0			0
$502$	−1571.72	−	907.432i	−0.139739	−	0.0806786i
$503$	7463.74			0.661614			0.330807	−	0.943698i	$-0.392679\pi$
$503$	7463.74			0.661614			0.330807	+	0.943698i	$0.392679\pi$
$504$		0			0
$505$	13493.2			1.18899
$506$	−10727.7	−	6193.61i	−0.942495	−	0.544149i
$507$		0			0
$508$	4164.45	+	7213.05i	0.363716	+	0.629975i
$509$	−9633.66	+	16686.0i	−0.838908	+	1.45303i	0.0519000	+	0.998652i	$0.483472\pi$
$509$	−9633.66	+	16686.0i	−0.838908	+	1.45303i	−0.890808	+	0.454379i	$0.849861\pi$
$510$		0			0
$511$	7216.94	+	17321.8i	0.624772	+	1.49955i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−4224.38	+	2438.95i	−0.362509	+	0.209295i
$515$	13092.5	−	7558.98i	1.12025	−	0.646774i
$516$		0			0
$517$			12760.6i			1.08551i
$518$	4991.44	+	11980.2i	0.423381	+	1.01618i
$519$		0			0
$520$	−1635.45	+	2832.68i	−0.137921	+	0.238887i
$521$	−3011.26	−	5215.65i	−0.253216	−	0.438583i	0.711193	−	0.702996i	$-0.248155\pi$
$521$	−3011.26	−	5215.65i	−0.253216	−	0.438583i	−0.964409	+	0.264413i	$0.914822\pi$
$522$		0			0
$523$	−8892.03	−	5133.82i	−0.743444	−	0.429228i	0.0798760	−	0.996805i	$-0.474548\pi$
$523$	−8892.03	−	5133.82i	−0.743444	−	0.429228i	−0.823320	+	0.567577i	$0.807881\pi$
$524$	10720.3			0.893737
$525$		0			0
$526$	−2712.36			−0.224838
$527$	1477.94	+	853.287i	0.122163	+	0.0705309i
$528$		0			0
$529$	1003.81	+	1738.65i	0.0825025	+	0.142899i
$530$	105.552	−	182.822i	0.00865075	−	0.0149835i
$531$		0			0
$532$	−3229.26	−	418.345i	−0.263170	−	0.0340931i
$533$		−	9878.70i		−	0.802803i
$534$		0			0
$535$	−13542.5	+	7818.79i	−1.09438	+	0.631843i
$536$	354.527	−	204.686i	0.0285695	−	0.0164946i
$537$		0			0
$538$			5082.81i			0.407315i
$539$	−17216.3	+	4689.55i	−1.37581	+	0.374755i
$540$		0			0
$541$	5472.84	−	9479.24i	0.434928	−	0.753317i	−0.562362	−	0.826891i	$-0.690107\pi$
$541$	5472.84	−	9479.24i	0.434928	−	0.753317i	0.997290	+	0.0735742i	$0.0234406\pi$
$542$	5468.63	+	9471.95i	0.433391	+	0.750655i
$543$		0			0
$544$	−234.224	−	135.229i	−0.0184601	−	0.0106579i
$545$	−21208.8			−1.66695
$546$		0			0
$547$	14007.6			1.09492			0.547460	−	0.836832i	$-0.315595\pi$
$547$	14007.6			1.09492			0.547460	+	0.836832i	$0.315595\pi$
$548$	−4357.40	−	2515.74i	−0.339669	−	0.196108i
$549$		0			0
$550$	3444.19	+	5965.51i	0.267019	+	0.462491i
$551$	2214.19	−	3835.09i	0.171194	−	0.296516i
$552$		0			0
$553$	6296.54	+	4810.84i	0.484188	+	0.369942i
$554$			5646.72i			0.433043i
$555$		0			0
$556$	−5368.58	+	3099.55i	−0.409494	+	0.236421i
$557$	−9582.43	+	5532.42i	−0.728942	+	0.420855i	−0.818035	−	0.575169i	$-0.804936\pi$
$557$	−9582.43	+	5532.42i	−0.728942	+	0.420855i	0.0890931	+	0.996023i	$0.471603\pi$
$558$		0			0
$559$		−	542.401i		−	0.0410396i
$560$	3782.34	−	1575.87i	0.285416	−	0.118916i
$561$		0			0
$562$	3007.63	−	5209.36i	0.225746	−	0.391003i
$563$	1242.01	+	2151.22i	0.0929741	+	0.161036i	0.908761	−	0.417316i	$-0.137029\pi$
$563$	1242.01	+	2151.22i	0.0929741	+	0.161036i	−0.815787	+	0.578352i	$0.803696\pi$
$564$		0			0
$565$	−11766.9	−	6793.61i	−0.876170	−	0.505857i
$566$	−7569.33			−0.562125
$567$		0			0
$568$	−9213.84			−0.680641
$569$	−4226.29	−	2440.05i	−0.311380	−	0.179775i	0.336164	−	0.941804i	$-0.390870\pi$
$569$	−4226.29	−	2440.05i	−0.311380	−	0.179775i	−0.647544	+	0.762028i	$0.724204\pi$
$570$		0			0
$571$	6767.06	+	11720.9i	0.495959	+	0.859026i	0.999989	−	0.00466002i	$-0.00148334\pi$
$571$	6767.06	+	11720.9i	0.495959	+	0.859026i	−0.504030	+	0.863686i	$0.668150\pi$
$572$	3076.41	−	5328.50i	0.224880	−	0.389503i
$573$		0			0
$574$	−7513.19	+	9833.45i	−0.546332	+	0.715053i
$575$		−	7882.32i		−	0.571679i
$576$		0			0
$577$	1312.90	−	758.004i	0.0947259	−	0.0546900i	−0.451889	−	0.892074i	$-0.649250\pi$
$577$	1312.90	−	758.004i	0.0947259	−	0.0546900i	0.546615	+	0.837384i	$0.315916\pi$
$578$	−8385.84	+	4841.57i	−0.603469	+	0.348413i
$579$		0			0
$580$			5572.44i			0.398936i
$581$	732.772	−	5656.37i	0.0523245	−	0.403900i
$582$		0			0
$583$	−198.552	+	343.903i	−0.0141050	+	0.0244305i
$584$	4052.87	+	7019.78i	0.287173	+	0.497398i
$585$		0			0
$586$	5697.65	+	3289.54i	0.401651	+	0.231893i
$587$	−866.107			−0.0608996			−0.0304498	−	0.999536i	$-0.509694\pi$
$587$	−866.107			−0.0608996			−0.0304498	+	0.999536i	$0.509694\pi$
$588$		0			0
$589$	8875.33			0.620886
$590$	9991.09	+	5768.36i	0.697164	+	0.402508i
$591$		0			0
$592$	2803.08	+	4855.08i	0.194605	+	0.337065i
$593$	−2626.36	+	4549.00i	−0.181875	+	0.315017i	−0.942519	−	0.334153i	$-0.891550\pi$
$593$	−2626.36	+	4549.00i	−0.181875	+	0.315017i	0.760644	+	0.649169i	$0.224883\pi$
$594$		0			0
$595$	−278.078	+	2146.52i	−0.0191598	+	0.147897i
$596$		−	5504.24i		−	0.378293i
$597$		0			0
$598$	−6097.36	+	3520.31i	−0.416956	+	0.240730i
$599$	9664.14	−	5579.59i	0.659209	−	0.380594i	−0.132767	−	0.991147i	$-0.542386\pi$
$599$	9664.14	−	5579.59i	0.659209	−	0.380594i	0.791975	+	0.610553i	$0.209053\pi$
$600$		0			0
$601$			5087.23i			0.345279i	0.984985	+	0.172639i	$0.0552295\pi$
$601$			5087.23i			0.345279i	−0.984985	+	0.172639i	$0.944771\pi$
$602$	−412.520	+	539.916i	−0.0279287	+	0.0365537i
$603$		0			0
$604$	−6363.08	+	11021.2i	−0.428659	+	0.742460i
$605$	−9508.67	−	16469.5i	−0.638979	−	1.10674i
$606$		0			0
$607$	15728.0	+	9080.56i	1.05170	+	0.607197i	0.923124	−	0.384503i	$-0.125627\pi$
$607$	15728.0	+	9080.56i	1.05170	+	0.607197i	0.128572	+	0.991700i	$0.458961\pi$
$608$	−1406.57			−0.0938220
$609$		0			0
$610$	17157.3			1.13882
$611$	6281.15	+	3626.42i	0.415889	+	0.240114i
$612$		0			0
$613$	7101.83	+	12300.7i	0.467928	+	0.810476i	0.999328	−	0.0366454i	$-0.0116672\pi$
$613$	7101.83	+	12300.7i	0.467928	+	0.810476i	−0.531400	+	0.847121i	$0.678334\pi$
$614$	−6283.43	+	10883.2i	−0.412995	+	0.715328i
$615$		0			0
$616$	−7114.88	+	2964.34i	−0.465368	+	0.193891i
$617$			27974.5i			1.82530i	0.408739	+	0.912651i	$0.365969\pi$
$617$			27974.5i			1.82530i	−0.408739	+	0.912651i	$0.634031\pi$
$618$		0			0
$619$	12853.8	−	7421.13i	0.834631	−	0.481875i	−0.0208044	−	0.999784i	$-0.506623\pi$
$619$	12853.8	−	7421.13i	0.834631	−	0.481875i	0.855436	+	0.517909i	$0.173289\pi$
$620$	−9671.99	+	5584.13i	−0.626511	+	0.361716i
$621$		0			0
$622$		−	351.522i		−	0.0226604i
$623$	9477.85	+	7241.50i	0.609506	+	0.465690i
$624$		0			0
$625$	9758.76	−	16902.7i	0.624560	−	1.08177i
$626$	6467.09	+	11201.3i	0.412902	+	0.715168i
$627$		0			0
$628$	−6996.30	−	4039.31i	−0.444558	−	0.256666i
$629$	−2961.40			−0.187724
$630$		0			0
$631$	−10234.4			−0.645679			−0.322840	−	0.946454i	$-0.604637\pi$
$631$	−10234.4			−0.645679			−0.322840	+	0.946454i	$0.604637\pi$
$632$	2964.29	+	1711.43i	0.186572	+	0.107717i
$633$		0			0
$634$	−3229.13	−	5593.01i	−0.202279	−	0.350358i
$635$	14396.2	−	24935.0i	0.899681	−	1.55829i
$636$		0			0
$637$	−2584.36	+	9807.11i	−0.160747	+	0.610003i
$638$		−	10482.2i		−	0.650462i
$639$		0			0
$640$	1532.82	−	884.975i	0.0946720	−	0.0546589i
$641$	9726.02	−	5615.32i	0.599306	−	0.346009i	−0.169463	−	0.985537i	$-0.554203\pi$
$641$	9726.02	−	5615.32i	0.599306	−	0.346009i	0.768768	+	0.639527i	$0.220870\pi$
$642$		0			0
$643$		−	19549.3i		−	1.19899i	−0.800379	−	0.599494i	$-0.795369\pi$
$643$		−	19549.3i		−	1.19899i	0.800379	−	0.599494i	$-0.204631\pi$
$644$	8746.79	+	1133.13i	0.535205	+	0.0693347i
$645$		0			0
$646$	371.502	−	643.460i	0.0226262	−	0.0391898i
$647$	−9943.48	−	17222.6i	−0.604201	−	1.04651i	−0.992177	−	0.124838i	$-0.960159\pi$
$647$	−9943.48	−	17222.6i	−0.604201	−	1.04651i	0.387976	−	0.921670i	$-0.373174\pi$
$648$		0			0
$649$	−18794.0	−	10850.7i	−1.13672	−	0.656285i
$650$	3915.20			0.236257
$651$		0			0
$652$	15151.8			0.910110
$653$	3126.25	+	1804.94i	0.187350	+	0.108167i	0.590741	−	0.806861i	$-0.298835\pi$
$653$	3126.25	+	1804.94i	0.187350	+	0.108167i	−0.403391	+	0.915028i	$0.632169\pi$
$654$		0			0
$655$	−18529.7	−	32094.3i	−1.10537	−	1.91455i
$656$	−2672.79	+	4629.40i	−0.159077	+	0.275530i
$657$		0			0
$658$	−3494.32	−	8386.91i	−0.207026	−	0.496893i
$659$		−	27554.0i		−	1.62876i	−0.580333	−	0.814379i	$-0.697078\pi$
$659$		−	27554.0i		−	1.62876i	0.580333	−	0.814379i	$-0.302922\pi$
$660$		0			0
$661$	12341.2	−	7125.17i	0.726196	−	0.419269i	−0.0908331	−	0.995866i	$-0.528953\pi$
$661$	12341.2	−	7125.17i	0.726196	−	0.419269i	0.817029	+	0.576597i	$0.195620\pi$
$662$	−14078.8	+	8128.41i	−0.826569	+	0.477220i
$663$		0			0
$664$		−	2463.74i		−	0.143994i
$665$	4329.23	+	10390.8i	0.252452	+	0.605924i
$666$		0			0
$667$	−5997.36	+	10387.7i	−0.348154	+	0.603021i
$668$	6365.95	+	11026.1i	0.368721	+	0.638644i
$669$		0			0
$670$	−1225.58	−	707.587i	−0.0706689	−	0.0408007i
$671$	−32274.2			−1.85683
$672$		0			0
$673$	−4469.53			−0.256000			−0.128000	−	0.991774i	$-0.540856\pi$
$673$	−4469.53			−0.256000			−0.128000	+	0.991774i	$0.540856\pi$
$674$	−8061.98	−	4654.58i	−0.460735	−	0.266006i
$675$		0			0
$676$	2645.44	+	4582.03i	0.150514	+	0.260698i
$677$	15414.2	−	26698.1i	0.875059	−	1.51565i	0.0183596	−	0.999831i	$-0.494156\pi$
$677$	15414.2	−	26698.1i	0.875059	−	1.51565i	0.856700	−	0.515816i	$-0.172511\pi$
$678$		0			0
$679$	8706.00	+	1127.84i	0.492055	+	0.0637448i
$680$			934.958i			0.0527265i
$681$		0			0
$682$	18193.8	−	10504.2i	1.02152	−	0.589775i
$683$	18545.3	−	10707.1i	1.03897	−	0.599849i	0.119428	−	0.992843i	$-0.461894\pi$
$683$	18545.3	−	10707.1i	1.03897	−	0.599849i	0.919541	+	0.392994i	$0.128561\pi$
$684$		0			0
$685$			17393.5i			0.970178i
$686$	10031.3	−	7796.67i	0.558303	−	0.433933i
$687$		0			0
$688$	−146.752	+	254.182i	−0.00813208	+	0.0140852i
$689$	112.853	+	195.467i	0.00623998	+	0.0108080i
$690$		0			0
$691$	11527.9	+	6655.63i	0.634648	+	0.366414i	0.782550	−	0.622588i	$-0.213919\pi$
$691$	11527.9	+	6655.63i	0.634648	+	0.366414i	−0.147902	+	0.989002i	$0.547252\pi$
$692$	−11612.2			−0.637902
$693$		0			0
$694$	−8452.56			−0.462327
$695$	18558.8	+	10714.9i	1.01292	+	0.584807i
$696$		0			0
$697$	−1411.87	−	2445.43i	−0.0767266	−	0.132894i
$698$	5382.10	−	9322.07i	0.291856	−	0.505509i
$699$		0			0
$700$	−3897.27	−	2977.69i	−0.210433	−	0.160780i
$701$		−	26877.2i		−	1.44813i	−0.689732	−	0.724065i	$-0.742271\pi$
$701$		−	26877.2i		−	1.44813i	0.689732	−	0.724065i	$-0.257729\pi$
$702$		0			0
$703$	−13337.9	+	7700.63i	−0.715573	+	0.413136i
$704$	−2883.36	+	1664.71i	−0.154362	+	0.0891209i
$705$		0			0
$706$		−	22488.8i		−	1.19883i
$707$	16682.2	−	6950.47i	0.887410	−	0.369731i
$708$		0			0
$709$	−6390.45	+	11068.6i	−0.338502	+	0.586304i	−0.984151	−	0.177331i	$-0.943254\pi$
$709$	−6390.45	+	11068.6i	−0.338502	+	0.586304i	0.645649	+	0.763635i	$0.276587\pi$
$710$	15925.8	+	27584.3i	0.841810	+	1.45806i
$711$		0			0
$712$	4461.99	+	2576.13i	0.234860	+	0.135596i
$713$	−24039.8			−1.26269
$714$		0			0
$715$	−21269.9			−1.11252
$716$	13563.1	+	7830.67i	0.707929	+	0.408723i
$717$		0			0
$718$	−3168.14	−	5487.37i	−0.164671	−	0.285219i
$719$	−7113.47	+	12320.9i	−0.368968	+	0.639070i	−0.989404	−	0.145186i	$-0.953622\pi$
$719$	−7113.47	+	12320.9i	−0.368968	+	0.639070i	0.620437	+	0.784256i	$0.286955\pi$
$720$		0			0
$721$	12293.1	−	16089.6i	0.634980	−	0.831077i
$722$			9853.88i			0.507927i
$723$		0			0
$724$	−10267.4	+	5927.88i	−0.527051	+	0.304293i
$725$	5776.49	−	3335.06i	0.295908	−	0.170843i
$726$		0			0
$727$			1891.93i			0.0965171i	0.998835	+	0.0482585i	$0.0153671\pi$
$727$			1891.93i			0.0965171i	−0.998835	+	0.0482585i	$0.984633\pi$
$728$	−562.833	+	4344.59i	−0.0286538	+	0.221183i
$729$		0			0
$730$	14010.5	−	24266.9i	0.710346	−	1.23035i
$731$	−77.5203	−	134.269i	−0.00392229	−	0.00679360i
$732$		0			0
$733$	3112.50	+	1797.00i	0.156839	+	0.0905508i	0.576365	−	0.817192i	$-0.304471\pi$
$733$	3112.50	+	1797.00i	0.156839	+	0.0905508i	−0.419527	+	0.907743i	$0.637804\pi$
$734$	−21190.0			−1.06558
$735$		0			0
$736$	3809.83			0.190805
$737$	2305.41	+	1331.03i	0.115225	+	0.0665252i
$738$		0			0
$739$	5252.85	+	9098.20i	0.261474	+	0.452886i	0.966634	−	0.256162i	$-0.0824582\pi$
$739$	5252.85	+	9098.20i	0.261474	+	0.452886i	−0.705160	+	0.709048i	$0.749125\pi$
$740$	9690.07	−	16783.7i	0.481370	−	0.833758i
$741$		0			0
$742$	36.3254	−	280.401i	0.00179723	−	0.0138731i
$743$			31704.8i			1.56546i	0.622363	+	0.782729i	$0.286173\pi$
$743$			31704.8i			1.56546i	−0.622363	+	0.782729i	$0.713827\pi$
$744$		0			0
$745$	−16478.6	+	9513.90i	−0.810372	+	0.467869i
$746$	3674.93	−	2121.72i	0.180360	−	0.104131i
$747$		0			0
$748$		−	1758.73i		−	0.0859700i
$749$	−12715.7	+	16642.6i	−0.620321	+	0.811891i
$750$		0			0
$751$	12195.5	−	21123.2i	0.592571	−	1.02636i	−0.401314	−	0.915940i	$-0.631446\pi$
$751$	12195.5	−	21123.2i	0.592571	−	1.02636i	0.993885	−	0.110422i	$-0.0352202\pi$
$752$	−1962.33	−	3398.86i	−0.0951582	−	0.164819i
$753$		0			0
$754$	−5159.66	−	2978.93i	−0.249209	−	0.143881i
$755$	43993.5			2.12065
$756$		0			0
$757$	−28508.8			−1.36878			−0.684391	−	0.729115i	$-0.739932\pi$
$757$	−28508.8			−1.36878			−0.684391	+	0.729115i	$0.739932\pi$
$758$	4721.25	+	2725.81i	0.226231	+	0.130615i
$759$		0			0
$760$	2431.20	+	4210.97i	0.116038	+	0.200984i
$761$	14929.1	−	25858.0i	0.711145	−	1.23174i	−0.253283	−	0.967392i	$-0.581510\pi$
$761$	14929.1	−	25858.0i	0.711145	−	1.23174i	0.964428	−	0.264347i	$-0.0851564\pi$
$762$		0			0
$763$	−26221.3	+	10924.9i	−1.24414	+	0.518357i
$764$		−	1800.03i		−	0.0852391i
$765$		0			0
$766$	4017.52	−	2319.52i	0.189503	−	0.109409i
$767$	−10682.1	+	6167.33i	−0.502880	+	0.290338i
$768$		0			0
$769$			18804.7i			0.881812i	0.897553	+	0.440906i	$0.145343\pi$
$769$			18804.7i			0.881812i	−0.897553	+	0.440906i	$0.854657\pi$
$770$	21172.4	+	16176.7i	0.990912	+	0.757101i
$771$		0			0
$772$	−10212.5	+	17688.5i	−0.476107	+	0.824642i
$773$	13746.1	+	23809.0i	0.639604	+	1.10783i	0.985520	+	0.169561i	$0.0542350\pi$
$773$	13746.1	+	23809.0i	0.639604	+	1.10783i	−0.345916	+	0.938266i	$0.612432\pi$
$774$		0			0
$775$	11577.2	+	6684.10i	0.536600	+	0.309806i
$776$	3792.06			0.175422
$777$		0			0
$778$	−13413.3			−0.618110
$779$	−12717.9	−	7342.68i	−0.584937	−	0.337713i
$780$		0			0
$781$	−29957.7	−	51888.3i	−1.37256	−	2.37735i
$782$	−1006.25	+	1742.88i	−0.0460147	+	0.0796998i
$783$		0			0
$784$	3864.51	−	3896.63i	0.176044	−	0.177507i
$785$			27927.3i			1.26977i
$786$		0			0
$787$	−28710.4	+	16575.9i	−1.30040	+	0.750785i	−0.980472	−	0.196657i	$-0.936991\pi$
$787$	−28710.4	+	16575.9i	−1.30040	+	0.750785i	−0.319926	+	0.947442i	$0.603658\pi$
$788$	−13689.6	+	7903.72i	−0.618875	+	0.357307i
$789$		0			0
$790$		−	11832.6i		−	0.532894i
$791$	−18047.3	−	2337.99i	−0.811237	−	0.105094i
$792$		0			0
$793$	−9171.98	+	15886.3i	−0.410727	+	0.711400i
$794$	6989.34	+	12105.9i	0.312396	+	0.541086i
$795$		0			0
$796$	−5449.44	−	3146.23i	−0.242651	−	0.140095i
$797$	−21513.9			−0.956162			−0.478081	−	0.878316i	$-0.658667\pi$
$797$	−21513.9			−0.956162			−0.478081	+	0.878316i	$0.658667\pi$
$798$		0			0
$799$	2073.16			0.0917939
$800$	−1834.76	−	1059.30i	−0.0810857	−	0.0468148i
$801$		0			0
$802$	−104.582	−	181.142i	−0.00460466	−	0.00797550i
$803$	−26354.9	+	45648.0i	−1.15821	+	2.00608i
$804$		0			0
$805$	−11726.2	−	28144.6i	−0.513408	−	1.23226i
$806$		−	11940.7i		−	0.521828i
$807$		0			0
$808$	6760.60	−	3903.23i	0.294353	−	0.169945i
$809$	7573.93	−	4372.81i	0.329154	−	0.190037i	−0.326312	−	0.945262i	$-0.605806\pi$
$809$	7573.93	−	4372.81i	0.329154	−	0.190037i	0.655465	+	0.755225i	$0.272473\pi$
$810$		0			0
$811$		−	15053.4i		−	0.651783i	−0.945407	−	0.325892i	$-0.894336\pi$
$811$		−	15053.4i		−	0.651783i	0.945407	−	0.325892i	$-0.105664\pi$
$812$	2870.41	+	6889.44i	0.124054	+	0.297749i
$813$		0			0
$814$	−18227.8	+	31571.5i	−0.784870	+	1.35943i
$815$	−26189.4	−	45361.4i	−1.12561	−	1.94962i
$816$		0			0
$817$	−698.289	−	403.158i	−0.0299021	−	0.0172640i
$818$	23180.0			0.990795
$819$		0			0
$820$	18479.3			0.786981
$821$	−11315.3	−	6532.87i	−0.481005	−	0.277708i	0.239830	−	0.970815i	$-0.422908\pi$
$821$	−11315.3	−	6532.87i	−0.481005	−	0.277708i	−0.720835	+	0.693106i	$0.756242\pi$
$822$		0			0
$823$	−18054.9	−	31272.0i	−0.764708	−	1.32451i	−0.940401	−	0.340069i	$-0.889550\pi$
$823$	−18054.9	−	31272.0i	−0.764708	−	1.32451i	0.175692	−	0.984445i	$-0.443784\pi$
$824$	4373.23	−	7574.66i	0.184889	−	0.320238i
$825$		0			0
$826$	15323.7	+	1985.16i	0.645497	+	0.0836228i
$827$			25074.7i			1.05433i	0.849762	+	0.527167i	$0.176746\pi$
$827$			25074.7i			1.05433i	−0.849762	+	0.527167i	$0.823254\pi$
$828$		0			0
$829$	33362.1	−	19261.6i	1.39772	−	0.806976i	0.403570	−	0.914949i	$-0.367769\pi$
$829$	33362.1	−	19261.6i	1.39772	−	0.806976i	0.994154	+	0.107972i	$0.0344358\pi$
$830$	−7375.93	+	4258.49i	−0.308461	+	0.178090i
$831$		0			0
$832$			1892.37i			0.0788535i
$833$	761.892	+	2797.07i	0.0316903	+	0.116342i
$834$		0			0
$835$	22006.7	−	38116.6i	0.912062	−	1.57974i
$836$	−4573.29	−	7921.17i	−0.189199	−	0.327703i
$837$		0			0
$838$	−15597.1	−	9004.98i	−0.642950	−	0.371208i
$839$	−6763.95			−0.278328			−0.139164	−	0.990269i	$-0.544442\pi$
$839$	−6763.95			−0.278328			−0.139164	+	0.990269i	$0.544442\pi$
$840$		0			0
$841$	14238.9			0.583826
$842$	−21183.5	−	12230.3i	−0.867021	−	0.500575i
$843$		0			0
$844$	−10970.3	−	19001.1i	−0.447408	−	0.774934i
$845$	9145.10	−	15839.8i	0.372309	−	0.644858i
$846$		0			0
$847$	−20239.5	−	15463.9i	−0.821061	−	0.627328i
$848$		−	122.134i		−	0.00494587i
$849$		0			0
$850$	969.193	−	559.564i	0.0391095	−	0.0225799i
$851$	36127.0	−	20857.9i	1.45525	−	0.840189i
$852$		0			0
$853$			9040.05i			0.362867i	0.983403	+	0.181434i	$0.0580737\pi$
$853$			9040.05i			0.362867i	−0.983403	+	0.181434i	$0.941926\pi$
$854$	21212.2	−	8837.86i	0.849962	−	0.354128i
$855$		0			0
$856$	−4523.54	+	7835.01i	−0.180621	+	0.312845i
$857$	−914.312	−	1583.64i	−0.0364438	−	0.0631225i	0.847228	−	0.531229i	$-0.178270\pi$
$857$	−914.312	−	1583.64i	−0.0364438	−	0.0631225i	−0.883672	+	0.468107i	$0.844936\pi$
$858$		0			0
$859$	−19278.7	−	11130.6i	−0.765753	−	0.442108i	0.0656043	−	0.997846i	$-0.479102\pi$
$859$	−19278.7	−	11130.6i	−0.765753	−	0.442108i	−0.831358	+	0.555738i	$0.812436\pi$
$860$	1014.62			0.0402307
$861$		0			0
$862$	−15309.2			−0.604910
$863$	−30131.1	−	17396.2i	−1.18850	−	0.686180i	−0.230534	−	0.973064i	$-0.574047\pi$
$863$	−30131.1	−	17396.2i	−1.18850	−	0.686180i	−0.957965	+	0.286884i	$0.907381\pi$
$864$		0			0
$865$	20071.2	+	34764.4i	0.788950	+	1.36650i
$866$	−6569.44	+	11378.6i	−0.257781	+	0.446490i
$867$		0			0
$868$	−9081.44	+	11886.0i	−0.355120	+	0.464790i
$869$			22258.1i			0.868878i
$870$		0			0
$871$	1310.34	−	756.528i	0.0509751	−	0.0294305i
$872$	−10626.4	+	6135.16i	−0.412678	+	0.238260i
$873$		0			0
$874$			10466.4i			0.405069i
$875$	1934.40	−	14931.9i	0.0747368	−	0.576904i
$876$		0			0
$877$	−4333.24	+	7505.40i	−0.166845	+	0.288984i	−0.937309	−	0.348499i	$-0.886691\pi$
$877$	−4333.24	+	7505.40i	−0.166845	+	0.288984i	0.770464	+	0.637484i	$0.220025\pi$
$878$	−10047.0	−	17401.9i	−0.386185	−	0.668892i
$879$		0			0
$880$	9967.58	+	5754.79i	0.381826	+	0.220448i
$881$	10439.5			0.399224			0.199612	−	0.979875i	$-0.436032\pi$
$881$	10439.5			0.399224			0.199612	+	0.979875i	$0.436032\pi$
$882$		0			0
$883$	−5933.13			−0.226122			−0.113061	−	0.993588i	$-0.536066\pi$
$883$	−5933.13			−0.226122			−0.113061	+	0.993588i	$0.536066\pi$
$884$	−865.700	−	499.812i	−0.0329374	−	0.0190164i
$885$		0			0
$886$	−10265.9	−	17781.0i	−0.389264	−	0.674226i
$887$	18422.1	−	31908.0i	0.697355	−	1.20785i	−0.272025	−	0.962290i	$-0.587693\pi$
$887$	18422.1	−	31908.0i	0.697355	−	1.20785i	0.969380	−	0.245564i	$-0.0789732\pi$
$888$		0			0
$889$	4954.41	−	38243.8i	0.186913	−	1.44281i
$890$		−	17811.0i		−	0.670817i
$891$		0			0
$892$	−8303.49	+	4794.02i	−0.311683	+	0.179950i
$893$	9337.35	−	5390.92i	0.349902	−	0.202016i
$894$		0			0
$895$		−	54140.2i		−	2.02202i
$896$	1439.23	−	1883.70i	0.0536622	−	0.0702344i
$897$		0			0
$898$	−995.326	+	1723.95i	−0.0369871	+	0.0640636i
$899$	−10171.4	−	17617.3i	−0.377346	−	0.653582i
$900$		0			0
$901$	55.8725	+	32.2580i	0.00206591	+	0.00119275i
$902$	−34761.0			−1.28317
$903$		0			0
$904$	−7860.85			−0.289212
$905$	35493.7	+	20492.3i	1.30370	+	0.752693i
$906$		0			0
$907$	20245.6	+	35066.5i	0.741175	+	1.28375i	0.951961	+	0.306220i	$0.0990644\pi$
$907$	20245.6	+	35066.5i	0.741175	+	1.28375i	−0.210786	+	0.977532i	$0.567602\pi$
$908$	798.471	−	1382.99i	0.0291830	−	0.0505465i
$909$		0			0
$910$	13979.6	−	5824.47i	0.509253	−	0.212175i
$911$			13793.4i			0.501643i	0.968033	+	0.250822i	$0.0807008\pi$
$911$			13793.4i			0.501643i	−0.968033	+	0.250822i	$0.919299\pi$
$912$		0			0
$913$	13874.7	−	8010.57i	0.502942	−	0.290374i
$914$	11567.4	−	6678.44i	0.418616	−	0.241688i
$915$		0			0
$916$		−	16906.8i		−	0.609845i
$917$	−39441.1	−	30134.7i	−1.42035	−	1.08521i
$918$		0			0
$919$	−27234.1	+	47170.8i	−0.977551	+	1.69317i	−0.306303	+	0.951934i	$0.599092\pi$
$919$	−27234.1	+	47170.8i	−0.977551	+	1.69317i	−0.671247	+	0.741233i	$0.734241\pi$
$920$	−6585.16	−	11405.8i	−0.235985	−	0.408738i
$921$		0			0
$922$	−7724.27	−	4459.61i	−0.275906	−	0.159294i
$923$	−34054.6			−1.21443
$924$		0			0
$925$	−23197.7			−0.824578
$926$	−5478.77	−	3163.17i	−0.194431	−	0.112255i
$927$		0			0
$928$	1611.96	+	2792.00i	0.0570207	+	0.0987628i
$929$	−21791.6	+	37744.1i	−0.769601	+	1.33299i	0.168179	+	0.985756i	$0.446211\pi$
$929$	−21791.6	+	37744.1i	−0.769601	+	1.33299i	−0.937780	+	0.347231i	$0.887122\pi$
$930$		0			0
$931$	10704.8	+	10616.6i	0.376838	+	0.373732i
$932$			7107.26i			0.249792i
$933$		0			0
$934$	4592.50	−	2651.48i	0.160890	−	0.0928899i
$935$	−5265.27	+	3039.91i	−0.184163	+	0.106327i
$936$		0			0
$937$			13915.0i			0.485148i	0.970133	+	0.242574i	$0.0779918\pi$
$937$			13915.0i			0.485148i	−0.970133	+	0.242574i	$0.922008\pi$
$938$	−1879.71	−	243.513i	−0.0654316	−	0.00847654i
$939$		0			0
$940$	−6783.66	+	11749.6i	−0.235381	+	0.407692i
$941$	−17696.6	−	30651.5i	−0.613065	−	1.06186i	−0.990721	−	0.135914i	$-0.956603\pi$
$941$	−17696.6	−	30651.5i	−0.613065	−	1.06186i	0.377656	−	0.925946i	$-0.376730\pi$
$942$		0			0
$943$	34447.7	+	19888.4i	1.18958	+	0.686803i
$944$	6674.54			0.230125
$945$		0			0
$946$	−1908.59			−0.0655958
$947$	38055.6	+	21971.4i	1.30585	+	0.753933i	0.981401	−	0.191970i	$-0.0614877\pi$
$947$	38055.6	+	21971.4i	1.30585	+	0.753933i	0.324449	+	0.945903i	$0.394821\pi$
$948$		0			0
$949$	14979.6	+	25945.3i	0.512389	+	0.887483i
$950$	2910.11	−	5040.45i	0.0993856	−	0.172141i
$951$		0			0
$952$	481.605	+	1155.93i	0.0163959	+	0.0393527i
$953$		−	8634.71i		−	0.293500i	−0.989174	−	0.146750i	$-0.953119\pi$
$953$		−	8634.71i		−	0.293500i	0.989174	−	0.146750i	$-0.0468813\pi$
$954$		0			0
$955$	−5388.91	+	3111.29i	−0.182598	+	0.105423i
$956$	−19313.0	+	11150.4i	−0.653375	+	0.377226i
$957$		0			0
$958$			35333.3i			1.19161i
$959$	8959.55	+	21504.3i	0.301688	+	0.724098i
$960$		0			0
$961$	5489.88	−	9508.75i	0.184280	−	0.319182i
$962$	10360.3	+	17944.5i	0.347223	+	0.601409i
$963$		0			0
$964$	11588.3	+	6690.50i	0.387171	+	0.223534i
$965$	70607.7			2.35538
$966$		0			0
$967$	−21086.3			−0.701231			−0.350616	−	0.936519i	$-0.614028\pi$
$967$	−21086.3			−0.701231			−0.350616	+	0.936519i	$0.614028\pi$
$968$	−9528.39	−	5501.22i	−0.316378	−	0.182661i
$969$		0			0
$970$	−6554.45	−	11352.6i	−0.216960	−	0.375785i
$971$	−13104.4	+	22697.5i	−0.433100	+	0.750152i	−0.997138	−	0.0755969i	$-0.975914\pi$
$971$	−13104.4	+	22697.5i	−0.433100	+	0.750152i	0.564038	+	0.825749i	$0.309247\pi$
$972$		0			0
$973$	28464.4	+	3687.51i	0.937848	+	0.121496i
$974$		−	28150.6i		−	0.926081i
$975$		0			0
$976$	8596.43	−	4963.15i	0.281931	−	0.162773i
$977$	27246.6	−	15730.9i	0.892218	−	0.515122i	0.0175507	−	0.999846i	$-0.494413\pi$
$977$	27246.6	−	15730.9i	0.892218	−	0.515122i	0.874667	+	0.484724i	$0.161080\pi$
$978$		0			0
$979$			33504.0i			1.09376i
$980$	−18345.4	−	4834.35i	−0.597981	−	0.157579i
$981$		0			0
$982$	909.632	−	1575.53i	0.0295596	−	0.0511987i
$983$	9690.42	+	16784.3i	0.314422	+	0.544594i	0.979314	−	0.202345i	$-0.0648561\pi$
$983$	9690.42	+	16784.3i	0.314422	+	0.544594i	−0.664893	+	0.746939i	$0.731523\pi$
$984$		0			0
$985$	47324.2	+	27322.6i	1.53084	+	0.883829i
$986$	−1703.00			−0.0550048
$987$		0			0
$988$	−5198.72			−0.167402
$989$	1891.39	+	1091.99i	0.0608116	+	0.0351096i
$990$		0			0
$991$	−27458.1	−	47558.7i	−0.880155	−	1.52447i	−0.851168	−	0.524894i	$-0.824105\pi$
$991$	−27458.1	−	47558.7i	−0.880155	−	1.52447i	−0.0289876	−	0.999580i	$-0.509228\pi$
$992$	−3230.68	+	5595.71i	−0.103402	+	0.179097i
$993$		0			0
$994$	33898.6	+	25900.1i	1.08169	+	0.826460i
$995$			21752.6i			0.693071i
$996$		0			0
$997$	−35254.5	+	20354.2i	−1.11988	+	0.646563i	−0.941370	−	0.337375i	$-0.890461\pi$
$997$	−35254.5	+	20354.2i	−1.11988	+	0.646563i	−0.178510	+	0.983938i	$0.557128\pi$
$998$	−8387.71	+	4842.65i	−0.266040	+	0.153599i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.c.215.4	✓	16
3.2	odd	2	inner	378.4.k.c.215.5	yes	16
7.3	odd	6	inner	378.4.k.c.269.5	yes	16
21.17	even	6	inner	378.4.k.c.269.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.c.215.4	✓	16	1.1	even	1	trivial
378.4.k.c.215.5	yes	16	3.2	odd	2	inner
378.4.k.c.269.4	yes	16	21.17	even	6	inner
378.4.k.c.269.5	yes	16	7.3	odd	6	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)

\(f(q)\)	\(=\)	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} +(6.91387 - 11.9752i) q^{5} +(2.37938 - 18.3668i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} +(6.91387 - 11.9752i) q^{5} +(2.37938 - 18.3668i) q^{7} -8.00000i q^{8} +(-23.9503 + 13.8277i) q^{10} +(45.0525 - 26.0111i) q^{11} -29.5683i q^{13} +(-22.4880 + 29.4328i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-4.22592 - 7.31950i) q^{17} +(-38.0663 - 21.9776i) q^{19} +55.3109 q^{20} -104.044 q^{22} +(103.107 + 59.5286i) q^{23} +(-33.1031 - 57.3362i) q^{25} +(-29.5683 + 51.2137i) q^{26} +(68.3831 - 28.4911i) q^{28} +100.748i q^{29} +(-174.866 + 100.959i) q^{31} +(27.7128 - 16.0000i) q^{32} +16.9037i q^{34} +(-203.495 - 155.479i) q^{35} +(175.193 - 303.443i) q^{37} +(43.9552 + 76.1326i) q^{38} +(-95.8013 - 55.3109i) q^{40} +334.098 q^{41} +18.3440 q^{43} +(180.210 + 104.044i) q^{44} +(-119.057 - 206.213i) q^{46} +(-122.646 + 212.429i) q^{47} +(-331.677 - 87.4031i) q^{49} +132.412i q^{50} +(102.427 - 59.1365i) q^{52} +(-6.61069 + 3.81669i) q^{53} -719.348i q^{55} +(-146.934 - 19.0350i) q^{56} +(100.748 - 174.500i) q^{58} +(-208.579 - 361.270i) q^{59} +(-537.277 - 310.197i) q^{61} +403.835 q^{62} -64.0000 q^{64} +(-354.085 - 204.431i) q^{65} +(25.5858 + 44.3159i) q^{67} +(16.9037 - 29.2780i) q^{68} +(196.984 + 472.792i) q^{70} -1151.73i q^{71} +(-877.473 + 506.609i) q^{73} +(-606.885 + 350.385i) q^{74} -175.821i q^{76} +(-370.543 - 889.360i) q^{77} +(-213.929 + 370.536i) q^{79} +(110.622 + 191.603i) q^{80} +(-578.675 - 334.098i) q^{82} +307.968 q^{83} -116.870 q^{85} +(-31.7728 - 18.3440i) q^{86} +(-208.089 - 360.420i) q^{88} +(-322.016 + 557.749i) q^{89} +(-543.074 - 70.3541i) q^{91} +476.229i q^{92} +(424.858 - 245.292i) q^{94} +(-526.371 + 303.900i) q^{95} +474.008i q^{97} +(487.078 + 483.064i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 50 q^{7}+O(q^{10}) \) 16 * q + 32 * q^4 + 50 * q^7	\( 16 q + 32 q^{4} + 50 q^{7} + 60 q^{10} - 128 q^{16} - 498 q^{19} - 240 q^{22} - 470 q^{25} + 160 q^{28} - 582 q^{31} + 40 q^{37} + 240 q^{40} + 1900 q^{43} - 456 q^{46} - 1634 q^{49} - 720 q^{52} + 1200 q^{58} - 1302 q^{61} - 1024 q^{64} - 100 q^{67} - 1620 q^{70} - 5280 q^{73} + 590 q^{79} - 480 q^{82} + 2580 q^{85} - 480 q^{88} - 2382 q^{91} - 3396 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 + 50 * q^7 + 60 * q^10 - 128 * q^16 - 498 * q^19 - 240 * q^22 - 470 * q^25 + 160 * q^28 - 582 * q^31 + 40 * q^37 + 240 * q^40 + 1900 * q^43 - 456 * q^46 - 1634 * q^49 - 720 * q^52 + 1200 * q^58 - 1302 * q^61 - 1024 * q^64 - 100 * q^67 - 1620 * q^70 - 5280 * q^73 + 590 * q^79 - 480 * q^82 + 2580 * q^85 - 480 * q^88 - 2382 * q^91 - 3396 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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