LMFDB - Embedded newform 1764.3.z.m.325.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (of order $6$, degree $2$, not 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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			325.2
Root			$-0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1764.325
Dual form			1764.3.z.m.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.416265 + 0.240331i) q^{5} +O(q^{10})$	$q+(-0.416265 + 0.240331i) q^{5} +(2.88158 - 4.99104i) q^{11} -1.41991i q^{13} +(19.9767 + 11.5336i) q^{17} +(9.24384 - 5.33693i) q^{19} +(-3.29990 - 5.71559i) q^{23} +(-12.3845 + 21.4506i) q^{25} +6.20258 q^{29} +(-36.3142 - 20.9660i) q^{31} +(30.0464 + 52.0419i) q^{37} -48.8250i q^{41} -51.5603 q^{43} +(16.6641 - 9.62104i) q^{47} +(41.0672 - 71.1306i) q^{53} +2.77013i q^{55} +(80.1766 + 46.2900i) q^{59} +(4.32309 - 2.49594i) q^{61} +(0.341248 + 0.591060i) q^{65} +(1.10350 - 1.91131i) q^{67} +80.5899 q^{71} +(12.0454 + 6.95439i) q^{73} +(32.4442 + 56.1951i) q^{79} -118.005i q^{83} -11.0875 q^{85} +(90.2959 - 52.1324i) q^{89} +(-2.56526 + 4.44316i) q^{95} -31.7875i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) $ 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.416265	+	0.240331i	−0.0832530	+	0.0480662i	−0.541049	−	0.840991i	$-0.681972\pi$
$5$	−0.416265	+	0.240331i	−0.0832530	+	0.0480662i	0.457796	+	0.889057i	$0.348639\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.88158	−	4.99104i	0.261962	−	0.453731i	−0.704801	−	0.709405i	$-0.748964\pi$
$11$	2.88158	−	4.99104i	0.261962	−	0.453731i	0.966763	+	0.255673i	$0.0822972\pi$
$12$		0			0
$13$		−	1.41991i		−	0.109224i	−0.998508	−	0.0546120i	$-0.982608\pi$
$13$		−	1.41991i		−	0.109224i	0.998508	−	0.0546120i	$-0.0173922\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.9767	+	11.5336i	1.17510	+	0.678445i	0.954876	−	0.297004i	$-0.0959876\pi$
$17$	19.9767	+	11.5336i	1.17510	+	0.678445i	0.220225	+	0.975449i	$0.429321\pi$
$18$		0			0
$19$	9.24384	−	5.33693i	0.486518	−	0.280891i	−0.236611	−	0.971605i	$-0.576037\pi$
$19$	9.24384	−	5.33693i	0.486518	−	0.280891i	0.723129	+	0.690713i	$0.242703\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.29990	−	5.71559i	−0.143474	−	0.248504i	0.785329	−	0.619079i	$-0.212494\pi$
$23$	−3.29990	−	5.71559i	−0.143474	−	0.248504i	−0.928803	+	0.370575i	$0.879161\pi$
$24$		0			0
$25$	−12.3845	+	21.4506i	−0.495379	+	0.858022i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.20258			0.213882			0.106941	−	0.994265i	$-0.465894\pi$
$29$	6.20258			0.213882			0.106941	+	0.994265i	$0.465894\pi$
$30$		0			0
$31$	−36.3142	−	20.9660i	−1.17143	−	0.676323i	−0.217410	−	0.976080i	$-0.569761\pi$
$31$	−36.3142	−	20.9660i	−1.17143	−	0.676323i	−0.954016	+	0.299757i	$0.903094\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	30.0464	+	52.0419i	0.812066	+	1.40654i	0.911416	+	0.411486i	$0.134990\pi$
$37$	30.0464	+	52.0419i	0.812066	+	1.40654i	−0.0993502	+	0.995053i	$0.531676\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	48.8250i		−	1.19085i	−0.803409	−	0.595427i	$-0.796983\pi$
$41$		−	48.8250i		−	1.19085i	0.803409	−	0.595427i	$-0.203017\pi$
$42$		0			0
$43$	−51.5603			−1.19908			−0.599539	−	0.800346i	$-0.704649\pi$
$43$	−51.5603			−1.19908			−0.599539	+	0.800346i	$0.704649\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	16.6641	−	9.62104i	0.354556	−	0.204703i	−0.312134	−	0.950038i	$-0.601044\pi$
$47$	16.6641	−	9.62104i	0.354556	−	0.204703i	0.666690	+	0.745335i	$0.267710\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	41.0672	−	71.1306i	0.774854	−	1.34209i	−0.160023	−	0.987113i	$-0.551157\pi$
$53$	41.0672	−	71.1306i	0.774854	−	1.34209i	0.934877	−	0.354973i	$-0.115510\pi$
$54$		0			0
$55$			2.77013i			0.0503660i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	80.1766	+	46.2900i	1.35893	+	0.784576i	0.989479	−	0.144676i	$-0.0462138\pi$
$59$	80.1766	+	46.2900i	1.35893	+	0.784576i	0.369447	+	0.929252i	$0.379547\pi$
$60$		0			0
$61$	4.32309	−	2.49594i	0.0708703	−	0.0409170i	−0.464146	−	0.885759i	$-0.653639\pi$
$61$	4.32309	−	2.49594i	0.0708703	−	0.0409170i	0.535016	+	0.844842i	$0.320305\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.341248	+	0.591060i	0.00524998	+	0.00909323i
$66$		0			0
$67$	1.10350	−	1.91131i	0.0164701	−	0.0285270i	−0.857673	−	0.514196i	$-0.828091\pi$
$67$	1.10350	−	1.91131i	0.0164701	−	0.0285270i	0.874143	+	0.485669i	$0.161424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	80.5899			1.13507			0.567535	−	0.823349i	$-0.307897\pi$
$71$	80.5899			1.13507			0.567535	+	0.823349i	$0.307897\pi$
$72$		0			0
$73$	12.0454	+	6.95439i	0.165005	+	0.0952657i	0.580228	−	0.814454i	$-0.302963\pi$
$73$	12.0454	+	6.95439i	0.165005	+	0.0952657i	−0.415223	+	0.909719i	$0.636297\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	32.4442	+	56.1951i	0.410687	+	0.711330i	0.994965	−	0.100223i	$-0.0319557\pi$
$79$	32.4442	+	56.1951i	0.410687	+	0.711330i	−0.584278	+	0.811553i	$0.698622\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	118.005i		−	1.42174i	−0.703322	−	0.710872i	$-0.748301\pi$
$83$		−	118.005i		−	1.42174i	0.703322	−	0.710872i	$-0.251699\pi$
$84$		0			0
$85$	−11.0875			−0.130441
$86$		0			0
$87$		0			0
$88$		0			0
$89$	90.2959	−	52.1324i	1.01456	−	0.585757i	0.102037	−	0.994781i	$-0.467464\pi$
$89$	90.2959	−	52.1324i	1.01456	−	0.585757i	0.912524	+	0.409024i	$0.134131\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.56526	+	4.44316i	−0.0270027	+	0.0467701i
$96$		0			0
$97$		−	31.7875i		−	0.327706i	−0.986485	−	0.163853i	$-0.947608\pi$
$97$		−	31.7875i		−	0.327706i	0.986485	−	0.163853i	$-0.0523923\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	62.7901	+	36.2519i	0.621684	+	0.358929i	0.777524	−	0.628853i	$-0.216475\pi$
$101$	62.7901	+	36.2519i	0.621684	+	0.358929i	−0.155840	+	0.987782i	$0.549809\pi$
$102$		0			0
$103$	92.6323	−	53.4813i	0.899343	−	0.519236i	0.0223558	−	0.999750i	$-0.492883\pi$
$103$	92.6323	−	53.4813i	0.899343	−	0.519236i	0.876987	+	0.480514i	$0.159550\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	98.3097	+	170.277i	0.918782	+	1.59138i	0.801268	+	0.598306i	$0.204159\pi$
$107$	98.3097	+	170.277i	0.918782	+	1.59138i	0.117515	+	0.993071i	$0.462507\pi$
$108$		0			0
$109$	−21.3461	+	36.9726i	−0.195836	+	0.339198i	−0.947174	−	0.320719i	$-0.896075\pi$
$109$	−21.3461	+	36.9726i	−0.195836	+	0.339198i	0.751338	+	0.659917i	$0.229409\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−175.501			−1.55310			−0.776552	−	0.630053i	$-0.783033\pi$
$113$	−175.501			−1.55310			−0.776552	+	0.630053i	$0.783033\pi$
$114$		0			0
$115$	2.74726	+	1.58613i	0.0238893	+	0.0137925i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	43.8930	+	76.0249i	0.362752	+	0.628305i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	23.9220i		−	0.191376i
$126$		0			0
$127$	31.0434			0.244436			0.122218	−	0.992503i	$-0.460999\pi$
$127$	31.0434			0.244436			0.122218	+	0.992503i	$0.460999\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	40.2784	−	23.2547i	0.307469	−	0.177517i	−0.338325	−	0.941029i	$-0.609860\pi$
$131$	40.2784	−	23.2547i	0.307469	−	0.177517i	0.645793	+	0.763512i	$0.276527\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	22.7327	−	39.3742i	0.165932	−	0.287403i	−0.771054	−	0.636770i	$-0.780270\pi$
$137$	22.7327	−	39.3742i	0.165932	−	0.287403i	0.936986	+	0.349367i	$0.113603\pi$
$138$		0			0
$139$		−	138.075i		−	0.993343i	−0.867939	−	0.496672i	$-0.834555\pi$
$139$		−	138.075i		−	0.993343i	0.867939	−	0.496672i	$-0.165445\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.08684	−	4.09159i	−0.0495583	−	0.0286125i
$144$		0			0
$145$	−2.58192	+	1.49067i	−0.0178063	+	0.0102805i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	119.614	+	207.178i	0.802780	+	1.39045i	0.917780	+	0.397089i	$0.129980\pi$
$149$	119.614	+	207.178i	0.802780	+	1.39045i	−0.115000	+	0.993365i	$0.536687\pi$
$150$		0			0
$151$	94.0732	−	162.940i	0.623001	−	1.07907i	−0.365922	−	0.930645i	$-0.619246\pi$
$151$	94.0732	−	162.940i	0.623001	−	1.07907i	0.988924	−	0.148425i	$-0.0474202\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.1551			0.130033
$156$		0			0
$157$	186.402	+	107.619i	1.18728	+	0.685474i	0.957686	−	0.287814i	$-0.0929284\pi$
$157$	186.402	+	107.619i	1.18728	+	0.685474i	0.229589	+	0.973288i	$0.426262\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−74.9053	−	129.740i	−0.459542	−	0.795949i	0.539395	−	0.842053i	$-0.318653\pi$
$163$	−74.9053	−	129.740i	−0.459542	−	0.795949i	−0.998937	+	0.0461035i	$0.985320\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	137.195i		−	0.821528i	−0.911742	−	0.410764i	$-0.865262\pi$
$167$		−	137.195i		−	0.821528i	0.911742	−	0.410764i	$-0.134738\pi$
$168$		0			0
$169$	166.984			0.988070
$170$		0			0
$171$		0			0
$172$		0			0
$173$	217.529	−	125.591i	1.25739	−	0.725957i	0.284828	−	0.958579i	$-0.408064\pi$
$173$	217.529	−	125.591i	1.25739	−	0.725957i	0.972567	+	0.232621i	$0.0747303\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−58.3967	+	101.146i	−0.326239	+	0.565062i	−0.981762	−	0.190113i	$-0.939115\pi$
$179$	−58.3967	+	101.146i	−0.326239	+	0.565062i	0.655524	+	0.755175i	$0.272448\pi$
$180$		0			0
$181$		−	117.148i		−	0.647228i	−0.946189	−	0.323614i	$-0.895102\pi$
$181$		−	117.148i		−	0.647228i	0.946189	−	0.323614i	$-0.104898\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−25.0146	−	14.4422i	−0.135214	−	0.0780657i
$186$		0			0
$187$	115.129	−	66.4698i	0.615663	−	0.355453i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	170.909	+	296.023i	0.894812	+	1.54986i	0.834037	+	0.551709i	$0.186024\pi$
$191$	170.909	+	296.023i	0.894812	+	1.54986i	0.0607755	+	0.998151i	$0.480643\pi$
$192$		0			0
$193$	112.275	−	194.467i	0.581738	−	1.00760i	−0.413535	−	0.910488i	$-0.635706\pi$
$193$	112.275	−	194.467i	0.581738	−	1.00760i	0.995273	−	0.0971119i	$-0.0309605\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−257.109			−1.30512			−0.652560	−	0.757737i	$-0.726305\pi$
$197$	−257.109			−1.30512			−0.652560	+	0.757737i	$0.726305\pi$
$198$		0			0
$199$	212.591	+	122.740i	1.06830	+	0.616782i	0.927716	−	0.373287i	$-0.121769\pi$
$199$	212.591	+	122.740i	1.06830	+	0.616782i	0.140582	+	0.990069i	$0.455103\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	11.7342	+	20.3241i	0.0572398	+	0.0991422i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	61.5152i		−	0.294331i
$210$		0			0
$211$	−95.8210			−0.454128			−0.227064	−	0.973880i	$-0.572913\pi$
$211$	−95.8210			−0.454128			−0.227064	+	0.973880i	$0.572913\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	21.4628	−	12.3915i	0.0998268	−	0.0576350i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	16.3766	−	28.3652i	0.0741024	−	0.128349i
$222$		0			0
$223$		−	94.2091i		−	0.422462i	−0.977436	−	0.211231i	$-0.932253\pi$
$223$		−	94.2091i		−	0.422462i	0.977436	−	0.211231i	$-0.0677473\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−194.209	−	112.127i	−0.855545	−	0.493949i	0.00697266	−	0.999976i	$-0.497781\pi$
$227$	−194.209	−	112.127i	−0.855545	−	0.493949i	−0.862518	+	0.506026i	$0.831114\pi$
$228$		0			0
$229$	−317.592	+	183.362i	−1.38687	+	0.800708i	−0.992961	−	0.118443i	$-0.962210\pi$
$229$	−317.592	+	183.362i	−1.38687	+	0.800708i	−0.393906	+	0.919151i	$0.628876\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	34.8370	+	60.3395i	0.149515	+	0.258968i	0.931048	−	0.364896i	$-0.118895\pi$
$233$	34.8370	+	60.3395i	0.149515	+	0.258968i	−0.781533	+	0.623864i	$0.785562\pi$
$234$		0			0
$235$	−4.62447	+	8.00981i	−0.0196786	+	0.0340843i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	214.544			0.897674			0.448837	−	0.893614i	$-0.351838\pi$
$239$	214.544			0.897674			0.448837	+	0.893614i	$0.351838\pi$
$240$		0			0
$241$	141.504	+	81.6976i	0.587155	+	0.338994i	0.763972	−	0.645250i	$-0.223247\pi$
$241$	141.504	+	81.6976i	0.587155	+	0.338994i	−0.176817	+	0.984244i	$0.556580\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.57798	−	13.1254i	−0.0306801	−	0.0531394i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	330.546i		−	1.31692i	−0.752617	−	0.658458i	$-0.771209\pi$
$251$		−	330.546i		−	1.31692i	0.752617	−	0.658458i	$-0.228791\pi$
$252$		0			0
$253$	−38.0357			−0.150339
$254$		0			0
$255$		0			0
$256$		0			0
$257$	139.022	−	80.2644i	0.540942	−	0.312313i	−0.204519	−	0.978863i	$-0.565563\pi$
$257$	139.022	−	80.2644i	0.540942	−	0.312313i	0.745461	+	0.666550i	$0.232230\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−53.2577	+	92.2451i	−0.202501	+	0.350742i	−0.949334	−	0.314270i	$-0.898240\pi$
$263$	−53.2577	+	92.2451i	−0.202501	+	0.350742i	0.746833	+	0.665012i	$0.231574\pi$
$264$		0			0
$265$			39.4789i			0.148977i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	134.627	+	77.7268i	0.500471	+	0.288947i	0.728908	−	0.684611i	$-0.240028\pi$
$269$	134.627	+	77.7268i	0.500471	+	0.288947i	−0.228437	+	0.973559i	$0.573361\pi$
$270$		0			0
$271$	446.938	−	258.040i	1.64922	−	0.952176i	0.671834	−	0.740702i	$-0.265507\pi$
$271$	446.938	−	258.040i	1.64922	−	0.952176i	0.977383	−	0.211475i	$-0.0678266\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	71.3738	+	123.623i	0.259541	+	0.449538i
$276$		0			0
$277$	−100.101	+	173.380i	−0.361375	+	0.625919i	−0.988187	−	0.153251i	$-0.951026\pi$
$277$	−100.101	+	173.380i	−0.361375	+	0.625919i	0.626813	+	0.779170i	$0.284359\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−228.093			−0.811720			−0.405860	−	0.913935i	$-0.633028\pi$
$281$	−228.093			−0.811720			−0.405860	+	0.913935i	$0.633028\pi$
$282$		0			0
$283$	−225.781	−	130.355i	−0.797814	−	0.460618i	0.0448922	−	0.998992i	$-0.485706\pi$
$283$	−225.781	−	130.355i	−0.797814	−	0.460618i	−0.842706	+	0.538374i	$0.819039\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	121.546	+	210.524i	0.420575	+	0.728457i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			349.885i			1.19415i	0.802186	+	0.597074i	$0.203670\pi$
$293$			349.885i			1.19415i	−0.802186	+	0.597074i	$0.796330\pi$
$294$		0			0
$295$	−44.4996			−0.150846
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.11563	+	4.68556i	−0.0271426	+	0.0156708i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.19970	+	2.07794i	−0.00393345	+	0.00681293i
$306$		0			0
$307$			146.898i			0.478495i	0.970959	+	0.239247i	$0.0769007\pi$
$307$			146.898i			0.478495i	−0.970959	+	0.239247i	$0.923099\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	69.9177	+	40.3670i	0.224816	+	0.129797i	0.608178	−	0.793801i	$-0.291901\pi$
$311$	69.9177	+	40.3670i	0.224816	+	0.129797i	−0.383362	+	0.923598i	$0.625234\pi$
$312$		0			0
$313$	−133.661	+	77.1695i	−0.427033	+	0.246548i	−0.698082	−	0.716018i	$-0.745963\pi$
$313$	−133.661	+	77.1695i	−0.427033	+	0.246548i	0.271049	+	0.962566i	$0.412630\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−94.2726	−	163.285i	−0.297390	−	0.515094i	0.678148	−	0.734925i	$-0.262783\pi$
$317$	−94.2726	−	163.285i	−0.297390	−	0.515094i	−0.975538	+	0.219831i	$0.929449\pi$
$318$		0			0
$319$	17.8732	−	30.9574i	0.0560290	−	0.0970450i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	246.216			0.762277
$324$		0			0
$325$	30.4579	+	17.5849i	0.0937166	+	0.0541073i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−73.3936	−	127.121i	−0.221733	−	0.384053i	0.733601	−	0.679580i	$-0.237838\pi$
$331$	−73.3936	−	127.121i	−0.221733	−	0.384053i	−0.955334	+	0.295527i	$0.904505\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.06082i			0.00316662i
$336$		0			0
$337$	101.231			0.300388			0.150194	−	0.988657i	$-0.452010\pi$
$337$	101.231			0.300388			0.150194	+	0.988657i	$0.452010\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−209.285	+	120.831i	−0.613738	+	0.354342i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	175.844	−	304.571i	0.506756	−	0.877727i	−0.493213	−	0.869908i	$-0.664178\pi$
$347$	175.844	−	304.571i	0.506756	−	0.877727i	0.999969	−	0.00781897i	$-0.00248888\pi$
$348$		0			0
$349$		−	88.3780i		−	0.253232i	−0.991952	−	0.126616i	$-0.959588\pi$
$349$		−	88.3780i		−	0.253232i	0.991952	−	0.126616i	$-0.0404116\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	450.803	+	260.271i	1.27706	+	0.737312i	0.976307	−	0.216390i	$-0.0694281\pi$
$353$	450.803	+	260.271i	1.27706	+	0.737312i	0.300755	+	0.953702i	$0.402761\pi$
$354$		0			0
$355$	−33.5468	+	19.3682i	−0.0944980	+	0.0545584i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−257.132	−	445.366i	−0.716246	−	1.24058i	−0.962477	−	0.271364i	$-0.912525\pi$
$359$	−257.132	−	445.366i	−0.716246	−	1.24058i	0.246231	−	0.969211i	$-0.420808\pi$
$360$		0			0
$361$	−123.534	+	213.968i	−0.342200	+	0.592708i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.68542			−0.0183162
$366$		0			0
$367$	−361.726	−	208.843i	−0.985630	−	0.569054i	−0.0816651	−	0.996660i	$-0.526024\pi$
$367$	−361.726	−	208.843i	−0.985630	−	0.569054i	−0.903965	+	0.427606i	$0.859357\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	70.1568	+	121.515i	0.188088	+	0.325778i	0.944613	−	0.328187i	$-0.106438\pi$
$373$	70.1568	+	121.515i	0.188088	+	0.325778i	−0.756525	+	0.653965i	$0.773104\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.80712i		−	0.0233611i
$378$		0			0
$379$	153.298			0.404480			0.202240	−	0.979336i	$-0.435178\pi$
$379$	153.298			0.404480			0.202240	+	0.979336i	$0.435178\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−105.410	+	60.8586i	−0.275222	+	0.158900i	−0.631258	−	0.775573i	$-0.717461\pi$
$383$	−105.410	+	60.8586i	−0.275222	+	0.158900i	0.356036	+	0.934472i	$0.384128\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	200.636	−	347.511i	0.515773	−	0.893345i	−0.484060	−	0.875035i	$-0.660838\pi$
$389$	200.636	−	347.511i	0.515773	−	0.893345i	0.999832	−	0.0183096i	$-0.00582845\pi$
$390$		0			0
$391$		−	152.238i		−	0.389356i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−27.0108	−	15.5947i	−0.0683818	−	0.0394802i
$396$		0			0
$397$	52.1564	−	30.1125i	0.131376	−	0.0758502i	−0.432871	−	0.901456i	$-0.642500\pi$
$397$	52.1564	−	30.1125i	0.131376	−	0.0758502i	0.564248	+	0.825605i	$0.309166\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−187.154	−	324.160i	−0.466718	−	0.808380i	0.532559	−	0.846393i	$-0.321230\pi$
$401$	−187.154	−	324.160i	−0.466718	−	0.808380i	−0.999277	+	0.0380132i	$0.987897\pi$
$402$		0			0
$403$	−29.7699	+	51.5630i	−0.0738707	+	0.127948i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	346.325			0.850921
$408$		0			0
$409$	−532.267	−	307.305i	−1.30139	−	0.751356i	−0.320745	−	0.947166i	$-0.603933\pi$
$409$	−532.267	−	307.305i	−1.30139	−	0.751356i	−0.980642	+	0.195810i	$0.937267\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	28.3602	+	49.1212i	0.0683377	+	0.118364i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	129.067i		−	0.308035i	−0.988068	−	0.154017i	$-0.950779\pi$
$419$		−	129.067i		−	0.308035i	0.988068	−	0.154017i	$-0.0492212\pi$
$420$		0			0
$421$	−697.880			−1.65767			−0.828836	−	0.559492i	$-0.810996\pi$
$421$	−697.880			−1.65767			−0.828836	+	0.559492i	$0.810996\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−494.803	+	285.674i	−1.16424	+	0.672175i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−138.546	+	239.969i	−0.321453	+	0.556773i	−0.980788	−	0.195076i	$-0.937505\pi$
$431$	−138.546	+	239.969i	−0.321453	+	0.556773i	0.659335	+	0.751849i	$0.270838\pi$
$432$		0			0
$433$			822.794i			1.90022i	0.311919	+	0.950109i	$0.399028\pi$
$433$			822.794i			1.90022i	−0.311919	+	0.950109i	$0.600972\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−61.0075	−	35.2227i	−0.139605	−	0.0806011i
$438$		0			0
$439$	−316.816	+	182.914i	−0.721676	+	0.416660i	−0.815369	−	0.578941i	$-0.803466\pi$
$439$	−316.816	+	182.914i	−0.721676	+	0.416660i	0.0936932	+	0.995601i	$0.470133\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−286.696	−	496.572i	−0.647169	−	1.12093i	−0.983796	−	0.179292i	$-0.942619\pi$
$443$	−286.696	−	496.572i	−0.647169	−	1.12093i	0.336626	−	0.941638i	$-0.390714\pi$
$444$		0			0
$445$	−25.0580	+	43.4018i	−0.0563102	+	0.0975321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−227.961			−0.507708			−0.253854	−	0.967243i	$-0.581698\pi$
$449$	−227.961			−0.507708			−0.253854	+	0.967243i	$0.581698\pi$
$450$		0			0
$451$	−243.688	−	140.693i	−0.540328	−	0.311958i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	171.562	+	297.154i	0.375409	+	0.650228i	0.990388	−	0.138316i	$-0.0441688\pi$
$457$	171.562	+	297.154i	0.375409	+	0.650228i	−0.614979	+	0.788544i	$0.710836\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			611.314i			1.32606i	0.748593	+	0.663030i	$0.230730\pi$
$461$			611.314i			1.32606i	−0.748593	+	0.663030i	$0.769270\pi$
$462$		0			0
$463$	−67.2682			−0.145288			−0.0726439	−	0.997358i	$-0.523144\pi$
$463$	−67.2682			−0.145288			−0.0726439	+	0.997358i	$0.523144\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	81.1897	−	46.8749i	0.173854	−	0.100375i	−0.410548	−	0.911839i	$-0.634663\pi$
$467$	81.1897	−	46.8749i	0.173854	−	0.100375i	0.584402	+	0.811464i	$0.301329\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−148.575	+	257.340i	−0.314113	+	0.544059i
$474$		0			0
$475$			264.381i			0.556591i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−306.396	−	176.898i	−0.639658	−	0.369306i	0.144825	−	0.989457i	$-0.453738\pi$
$479$	−306.396	−	176.898i	−0.639658	−	0.369306i	−0.784483	+	0.620151i	$0.787071\pi$
$480$		0			0
$481$	73.8950	−	42.6633i	0.153628	−	0.0886970i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.63952	+	13.2320i	0.0157516	+	0.0272825i
$486$		0			0
$487$	−476.338	+	825.042i	−0.978107	+	1.69413i	−0.308833	+	0.951116i	$0.599939\pi$
$487$	−476.338	+	825.042i	−0.978107	+	1.69413i	−0.669274	+	0.743016i	$0.733395\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	818.649			1.66731			0.833655	−	0.552286i	$-0.186244\pi$
$491$	818.649			1.66731			0.833655	+	0.552286i	$0.186244\pi$
$492$		0			0
$493$	123.907	+	71.5379i	0.251333	+	0.145107i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.0411	+	20.8557i	0.0241304	+	0.0417950i	0.877838	−	0.478957i	$-0.158985\pi$
$499$	12.0411	+	20.8557i	0.0241304	+	0.0417950i	−0.853708	+	0.520752i	$0.825652\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			420.445i			0.835874i	0.908476	+	0.417937i	$0.137247\pi$
$503$			420.445i			0.835874i	−0.908476	+	0.417937i	$0.862753\pi$
$504$		0			0
$505$	−34.8498			−0.0690094
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−207.732	+	119.934i	−0.408119	+	0.235627i	−0.689981	−	0.723827i	$-0.742381\pi$
$509$	−207.732	+	119.934i	−0.408119	+	0.235627i	0.281862	+	0.959455i	$0.409048\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.7064	+	44.5248i	−0.0499153	+	0.0864559i
$516$		0			0
$517$		−	110.895i		−	0.214498i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−23.2584	−	13.4282i	−0.0446418	−	0.0257739i	0.477513	−	0.878625i	$-0.341538\pi$
$521$	−23.2584	−	13.4282i	−0.0446418	−	0.0257739i	−0.522155	+	0.852851i	$0.674872\pi$
$522$		0			0
$523$	−193.762	+	111.869i	−0.370482	+	0.213898i	−0.673669	−	0.739033i	$-0.735283\pi$
$523$	−193.762	+	111.869i	−0.370482	+	0.213898i	0.303187	+	0.952931i	$0.401949\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−483.626	−	837.664i	−0.917696	−	1.58950i
$528$		0			0
$529$	242.721	−	420.406i	0.458831	−	0.794718i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−69.3272			−0.130070
$534$		0			0
$535$	−81.8458	−	47.2537i	−0.152983	−	0.0883246i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−493.095	−	854.065i	−0.911451	−	1.57868i	−0.812016	−	0.583635i	$-0.801630\pi$
$541$	−493.095	−	854.065i	−0.911451	−	1.57868i	−0.0994342	−	0.995044i	$-0.531703\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	20.5205i		−	0.0376523i
$546$		0			0
$547$	735.369			1.34437			0.672183	−	0.740385i	$-0.265357\pi$
$547$	735.369			1.34437			0.672183	+	0.740385i	$0.265357\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	57.3357	−	33.1028i	0.104058	−	0.0600776i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	209.284	−	362.491i	0.375735	−	0.650791i	−0.614702	−	0.788759i	$-0.710724\pi$
$557$	209.284	−	362.491i	0.375735	−	0.650791i	0.990437	+	0.137968i	$0.0440571\pi$
$558$		0			0
$559$			73.2111i			0.130968i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−646.694	−	373.369i	−1.14866	−	0.663177i	−0.200097	−	0.979776i	$-0.564126\pi$
$563$	−646.694	−	373.369i	−1.14866	−	0.663177i	−0.948560	+	0.316599i	$0.897459\pi$
$564$		0			0
$565$	73.0549	−	42.1782i	0.129301	−	0.0746518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−144.355	−	250.030i	−0.253699	−	0.439420i	0.710842	−	0.703351i	$-0.248314\pi$
$569$	−144.355	−	250.030i	−0.253699	−	0.439420i	−0.964541	+	0.263932i	$0.914981\pi$
$570$		0			0
$571$	458.104	−	793.459i	0.802283	−	1.38960i	−0.115827	−	0.993269i	$-0.536952\pi$
$571$	458.104	−	793.459i	0.802283	−	1.38960i	0.918110	−	0.396326i	$-0.129715\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	163.470			0.284296
$576$		0			0
$577$	859.156	+	496.034i	1.48901	+	0.859678i	0.999921	−	0.0125584i	$-0.00399756\pi$
$577$	859.156	+	496.034i	1.48901	+	0.859678i	0.489085	+	0.872236i	$0.337331\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−236.677	−	409.937i	−0.405964	−	0.703151i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			154.965i			0.263996i	0.991250	+	0.131998i	$0.0421392\pi$
$587$			154.965i			0.263996i	−0.991250	+	0.131998i	$0.957861\pi$
$588$		0			0
$589$	−447.577			−0.759893
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−869.636	+	502.084i	−1.46650	+	0.846685i	−0.999298	−	0.0374640i	$-0.988072\pi$
$593$	−869.636	+	502.084i	−1.46650	+	0.846685i	−0.467204	+	0.884149i	$0.654739\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−398.459	+	690.150i	−0.665206	+	1.15217i	0.314023	+	0.949415i	$0.398323\pi$
$599$	−398.459	+	690.150i	−0.665206	+	1.15217i	−0.979229	+	0.202756i	$0.935010\pi$
$600$		0			0
$601$			467.002i			0.777041i	0.921440	+	0.388521i	$0.127014\pi$
$601$			467.002i			0.777041i	−0.921440	+	0.388521i	$0.872986\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−36.5422	−	21.0977i	−0.0604004	−	0.0348722i
$606$		0			0
$607$	−788.270	+	455.108i	−1.29863	+	0.749766i	−0.980168	−	0.198169i	$-0.936501\pi$
$607$	−788.270	+	455.108i	−1.29863	+	0.749766i	−0.318465	+	0.947935i	$0.603167\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.6610	−	23.6616i	−0.0223585	−	0.0387260i
$612$		0			0
$613$	−387.900	+	671.863i	−0.632790	+	1.09602i	0.354189	+	0.935174i	$0.384757\pi$
$613$	−387.900	+	671.863i	−0.632790	+	1.09602i	−0.986979	+	0.160850i	$0.948576\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−974.803			−1.57991			−0.789954	−	0.613166i	$-0.789896\pi$
$617$	−974.803			−1.57991			−0.789954	+	0.613166i	$0.789896\pi$
$618$		0			0
$619$	−172.564	−	99.6300i	−0.278779	−	0.160953i	0.354092	−	0.935211i	$-0.384790\pi$
$619$	−172.564	−	99.6300i	−0.278779	−	0.160953i	−0.632870	+	0.774258i	$0.718123\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−303.863	−	526.306i	−0.486181	−	0.842089i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1386.17i			2.20377i
$630$		0			0
$631$	751.062			1.19027			0.595136	−	0.803625i	$-0.297098\pi$
$631$	751.062			1.19027			0.595136	+	0.803625i	$0.297098\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.9223	+	7.46068i	−0.0203500	+	0.0117491i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−569.138	+	985.776i	−0.887891	+	1.53787i	−0.0455278	+	0.998963i	$0.514497\pi$
$641$	−569.138	+	985.776i	−0.887891	+	1.53787i	−0.842364	+	0.538910i	$0.818836\pi$
$642$		0			0
$643$		−	647.823i		−	1.00750i	−0.863849	−	0.503751i	$-0.831953\pi$
$643$		−	647.823i		−	1.00750i	0.863849	−	0.503751i	$-0.168047\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−764.651	−	441.471i	−1.18184	−	0.682336i	−0.225401	−	0.974266i	$-0.572369\pi$
$647$	−764.651	−	441.471i	−1.18184	−	0.682336i	−0.956440	+	0.291930i	$0.905703\pi$
$648$		0			0
$649$	462.071	−	266.777i	0.711974	−	0.411058i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	272.558	+	472.085i	0.417394	+	0.722948i	0.995676	−	0.0928890i	$-0.0296102\pi$
$653$	272.558	+	472.085i	0.417394	+	0.722948i	−0.578282	+	0.815837i	$0.696277\pi$
$654$		0			0
$655$	−11.1777	+	19.3603i	−0.0170651	+	0.0295577i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−698.290			−1.05962			−0.529810	−	0.848116i	$-0.677737\pi$
$659$	−698.290			−1.05962			−0.529810	+	0.848116i	$0.677737\pi$
$660$		0			0
$661$	−495.128	−	285.863i	−0.749060	−	0.432470i	0.0762943	−	0.997085i	$-0.475691\pi$
$661$	−495.128	−	285.863i	−0.749060	−	0.432470i	−0.825354	+	0.564616i	$0.809024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−20.4679	−	35.4514i	−0.0306865	−	0.0531505i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	28.7690i		−	0.0428748i
$672$		0			0
$673$	221.015			0.328403			0.164202	−	0.986427i	$-0.447495\pi$
$673$	221.015			0.328403			0.164202	+	0.986427i	$0.447495\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−407.608	+	235.333i	−0.602080	+	0.347611i	−0.769860	−	0.638213i	$-0.779674\pi$
$677$	−407.608	+	235.333i	−0.602080	+	0.347611i	0.167779	+	0.985825i	$0.446340\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	107.980	−	187.027i	0.158097	−	0.273831i	−0.776086	−	0.630628i	$-0.782798\pi$
$683$	107.980	−	187.027i	0.158097	−	0.273831i	0.934182	+	0.356796i	$0.116131\pi$
$684$		0			0
$685$			21.8535i			0.0319029i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−100.999	−	58.3119i	−0.146588	−	0.0846326i
$690$		0			0
$691$	69.3808	−	40.0570i	0.100406	−	0.0579697i	−0.448956	−	0.893554i	$-0.648204\pi$
$691$	69.3808	−	40.0570i	0.100406	−	0.0579697i	0.549362	+	0.835584i	$0.314871\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.1836	+	57.4757i	0.0477462	+	0.0826988i
$696$		0			0
$697$	563.126	−	975.363i	0.807929	−	1.39937i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	821.973			1.17257			0.586286	−	0.810104i	$-0.300589\pi$
$701$	821.973			1.17257			0.586286	+	0.810104i	$0.300589\pi$
$702$		0			0
$703$	555.489	+	320.712i	0.790169	+	0.456204i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−287.728	−	498.359i	−0.405822	−	0.702904i	0.588595	−	0.808428i	$-0.299681\pi$
$709$	−287.728	−	498.359i	−0.405822	−	0.702904i	−0.994417	+	0.105524i	$0.966348\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			276.743i			0.388139i
$714$		0			0
$715$	3.93334			0.00550117
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1051.29	−	606.964i	1.46216	−	0.844178i	0.463048	−	0.886333i	$-0.346756\pi$
$719$	1051.29	−	606.964i	1.46216	−	0.844178i	0.999111	+	0.0421556i	$0.0134225\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−76.8158	+	133.049i	−0.105953	+	0.183516i
$726$		0			0
$727$		−	379.498i		−	0.522005i	−0.965338	−	0.261003i	$-0.915947\pi$
$727$		−	379.498i		−	0.522005i	0.965338	−	0.261003i	$-0.0840532\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1030.01	−	594.674i	−1.40904	−	0.813508i
$732$		0			0
$733$	−1082.93	+	625.230i	−1.47739	+	0.852973i	−0.999674	−	0.0255391i	$-0.991870\pi$
$733$	−1082.93	+	625.230i	−1.47739	+	0.852973i	−0.477719	+	0.878512i	$0.658536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.35963	−	11.0152i	−0.00862908	−	0.0149460i
$738$		0			0
$739$	−250.617	+	434.081i	−0.339130	+	0.587390i	−0.984269	−	0.176675i	$-0.943466\pi$
$739$	−250.617	+	434.081i	−0.339130	+	0.587390i	0.645139	+	0.764065i	$0.276799\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−444.584			−0.598363			−0.299181	−	0.954196i	$-0.596714\pi$
$743$	−444.584			−0.598363			−0.299181	+	0.954196i	$0.596714\pi$
$744$		0			0
$745$	−99.5824	−	57.4939i	−0.133668	−	0.0771730i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−119.667	−	207.270i	−0.159344	−	0.275992i	0.775288	−	0.631608i	$-0.217605\pi$
$751$	−119.667	−	207.270i	−0.159344	−	0.275992i	−0.934632	+	0.355616i	$0.884271\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			90.4347i			0.119781i
$756$		0			0
$757$	249.486			0.329572			0.164786	−	0.986329i	$-0.447307\pi$
$757$	249.486			0.329572			0.164786	+	0.986329i	$0.447307\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1191.37	+	687.840i	−1.56554	+	0.903864i	−0.568858	+	0.822436i	$0.692615\pi$
$761$	−1191.37	+	687.840i	−1.56554	+	0.903864i	−0.996679	+	0.0814280i	$0.974052\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	65.7277	−	113.844i	0.0856945	−	0.148427i
$768$		0			0
$769$			528.594i			0.687379i	0.939083	+	0.343689i	$0.111677\pi$
$769$			528.594i			0.687379i	−0.939083	+	0.343689i	$0.888323\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	53.6879	+	30.9967i	0.0694539	+	0.0400992i	0.534325	−	0.845279i	$-0.320566\pi$
$773$	53.6879	+	30.9967i	0.0694539	+	0.0400992i	−0.464871	+	0.885378i	$0.653899\pi$
$774$		0			0
$775$	899.465	−	519.307i	1.16060	−	0.670073i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−260.576	−	451.331i	−0.334501	−	0.579372i
$780$		0			0
$781$	232.226	−	402.228i	0.297345	−	0.515017i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−103.457			−0.131792
$786$		0			0
$787$	141.654	+	81.7839i	0.179992	+	0.103919i	0.587289	−	0.809377i	$-0.300195\pi$
$787$	141.654	+	81.7839i	0.179992	+	0.103919i	−0.407297	+	0.913296i	$0.633529\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.54401	−	6.13841i	−0.00446912	−	0.00774074i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			533.394i			0.669253i	0.942351	+	0.334626i	$0.108610\pi$
$797$			533.394i			0.669253i	−0.942351	+	0.334626i	$0.891390\pi$
$798$		0			0
$799$	443.860			0.555519
$800$		0			0
$801$		0			0
$802$		0			0
$803$	69.4194	−	40.0793i	0.0864500	−	0.0499119i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−275.137	+	476.551i	−0.340095	+	0.589062i	−0.984450	−	0.175664i	$-0.943793\pi$
$809$	−275.137	+	476.551i	−0.340095	+	0.589062i	0.644355	+	0.764727i	$0.277126\pi$
$810$		0			0
$811$			415.532i			0.512370i	0.966628	+	0.256185i	$0.0824656\pi$
$811$			415.532i			0.512370i	−0.966628	+	0.256185i	$0.917534\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	62.3609	+	36.0041i	0.0765164	+	0.0441768i
$816$		0			0
$817$	−476.616	+	275.174i	−0.583373	+	0.336810i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	434.596	+	752.742i	0.529349	+	0.916860i	0.999414	+	0.0342278i	$0.0108972\pi$
$821$	434.596	+	752.742i	0.529349	+	0.916860i	−0.470065	+	0.882632i	$0.655769\pi$
$822$		0			0
$823$	210.891	−	365.275i	0.256247	−	0.443833i	−0.708986	−	0.705222i	$-0.750847\pi$
$823$	210.891	−	365.275i	0.256247	−	0.443833i	0.965233	+	0.261389i	$0.0841806\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−70.7290			−0.0855248			−0.0427624	−	0.999085i	$-0.513616\pi$
$827$	−70.7290			−0.0855248			−0.0427624	+	0.999085i	$0.513616\pi$
$828$		0			0
$829$	1098.86	+	634.426i	1.32552	+	0.765290i	0.984603	−	0.174803i	$-0.0559288\pi$
$829$	1098.86	+	634.426i	1.32552	+	0.765290i	0.340918	+	0.940093i	$0.389262\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	32.9722	+	57.1096i	0.0394877	+	0.0683947i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	613.254i		−	0.730935i	−0.930824	−	0.365467i	$-0.880909\pi$
$839$		−	613.254i		−	0.730935i	0.930824	−	0.365467i	$-0.119091\pi$
$840$		0			0
$841$	−802.528			−0.954254
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−69.5095	+	40.1314i	−0.0822598	+	0.0474927i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	198.300	−	343.466i	0.233020	−	0.403603i
$852$		0			0
$853$		−	668.244i		−	0.783404i	−0.920092	−	0.391702i	$-0.871886\pi$
$853$		−	668.244i		−	0.783404i	0.920092	−	0.391702i	$-0.128114\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	450.083	+	259.856i	0.525185	+	0.303216i	0.739053	−	0.673647i	$-0.235273\pi$
$857$	450.083	+	259.856i	0.525185	+	0.303216i	−0.213869	+	0.976862i	$0.568606\pi$
$858$		0			0
$859$	−619.687	+	357.777i	−0.721405	+	0.416504i	−0.815270	−	0.579081i	$-0.803411\pi$
$859$	−619.687	+	357.777i	−0.721405	+	0.416504i	0.0938643	+	0.995585i	$0.470078\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−171.358	−	296.801i	−0.198561	−	0.343917i	0.749501	−	0.662003i	$-0.230293\pi$
$863$	−171.358	−	296.801i	−0.198561	−	0.343917i	−0.948062	+	0.318086i	$0.896960\pi$
$864$		0			0
$865$	−60.3666	+	104.558i	−0.0697879	+	0.120876i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	373.963			0.430337
$870$		0			0
$871$	−2.71389	−	1.56687i	−0.00311584	−	0.00179893i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	79.2750	+	137.308i	0.0903934	+	0.156566i	0.907677	−	0.419670i	$-0.137854\pi$
$877$	79.2750	+	137.308i	0.0903934	+	0.156566i	−0.817283	+	0.576236i	$0.804521\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1100.63i		−	1.24930i	−0.780906	−	0.624648i	$-0.785242\pi$
$881$		−	1100.63i		−	1.24930i	0.780906	−	0.624648i	$-0.214758\pi$
$882$		0			0
$883$	255.888			0.289794			0.144897	−	0.989447i	$-0.453715\pi$
$883$	255.888			0.289794			0.144897	+	0.989447i	$0.453715\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1359.00	−	784.617i	1.53213	−	0.884574i	0.532864	−	0.846201i	$-0.321116\pi$
$887$	1359.00	−	784.617i	1.53213	−	0.884574i	0.999263	−	0.0383728i	$-0.0122175\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	102.694	−	177.871i	0.114999	−	0.199183i
$894$		0			0
$895$		−	56.1381i		−	0.0627241i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−225.242	−	130.043i	−0.250547	−	0.144653i
$900$		0			0
$901$	1640.78	−	947.303i	1.82106	−	1.05139i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	28.1544	+	48.7648i	0.0311098	+	0.0538837i
$906$		0			0
$907$	−438.230	+	759.037i	−0.483164	+	0.836865i	−0.999813	−	0.0193324i	$-0.993846\pi$
$907$	−438.230	+	759.037i	−0.483164	+	0.836865i	0.516649	+	0.856197i	$0.327179\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1778.49			1.95224			0.976118	−	0.217241i	$-0.0697057\pi$
$911$	1778.49			1.95224			0.976118	+	0.217241i	$0.0697057\pi$
$912$		0			0
$913$	−588.967	−	340.040i	−0.645089	−	0.372443i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	132.431	+	229.378i	0.144104	+	0.249595i	0.929038	−	0.369984i	$-0.120637\pi$
$919$	132.431	+	229.378i	0.144104	+	0.249595i	−0.784934	+	0.619579i	$0.787303\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	114.431i		−	0.123977i
$924$		0			0
$925$	−1488.44			−1.60912
$926$		0			0
$927$		0			0
$928$		0			0
$929$	683.631	−	394.695i	0.735879	−	0.424860i	−0.0846901	−	0.996407i	$-0.526990\pi$
$929$	683.631	−	394.695i	0.735879	−	0.424860i	0.820569	+	0.571547i	$0.193657\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.9495	+	55.3381i	−0.0341706	+	0.0591851i
$936$		0			0
$937$			1446.06i			1.54329i	0.636053	+	0.771645i	$0.280566\pi$
$937$			1446.06i			1.54329i	−0.636053	+	0.771645i	$0.719434\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1040.18	+	600.548i	1.10540	+	0.638202i	0.937634	−	0.347625i	$-0.113012\pi$
$941$	1040.18	+	600.548i	1.10540	+	0.638202i	0.167765	+	0.985827i	$0.446345\pi$
$942$		0			0
$943$	−279.064	+	161.118i	−0.295932	+	0.170856i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−419.360	−	726.353i	−0.442830	−	0.767004i	0.555068	−	0.831805i	$-0.312692\pi$
$947$	−419.360	−	726.353i	−0.442830	−	0.767004i	−0.997898	+	0.0648007i	$0.979359\pi$
$948$		0			0
$949$	9.87462	−	17.1033i	0.0104053	−	0.0180225i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1377.68			−1.44562			−0.722812	−	0.691045i	$-0.757151\pi$
$953$	−1377.68			−1.44562			−0.722812	+	0.691045i	$0.757151\pi$
$954$		0			0
$955$	−142.287	−	82.1495i	−0.148992	−	0.0860204i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	398.648	+	690.478i	0.414826	+	0.718500i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			107.933i			0.111848i
$966$		0			0
$967$	848.834			0.877802			0.438901	−	0.898536i	$-0.355368\pi$
$967$	848.834			0.877802			0.438901	+	0.898536i	$0.355368\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1092.99	+	631.040i	−1.12564	+	0.649887i	−0.942834	−	0.333262i	$-0.891851\pi$
$971$	−1092.99	+	631.040i	−1.12564	+	0.649887i	−0.182803	+	0.983149i	$0.558517\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	322.084	−	557.866i	0.329666	−	0.570999i	−0.652779	−	0.757548i	$-0.726397\pi$
$977$	322.084	−	557.866i	0.329666	−	0.570999i	0.982446	+	0.186549i	$0.0597305\pi$
$978$		0			0
$979$		−	600.895i		−	0.613784i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1051.15	+	606.883i	1.06933	+	0.617378i	0.927998	−	0.372584i	$-0.121528\pi$
$983$	1051.15	+	606.883i	1.06933	+	0.617378i	0.141332	+	0.989962i	$0.454862\pi$
$984$		0			0
$985$	107.025	−	61.7911i	0.108655	−	0.0627321i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	170.144	+	294.698i	0.172036	+	0.297975i
$990$		0			0
$991$	−845.562	+	1464.56i	−0.853241	+	1.47786i	0.0250262	+	0.999687i	$0.492033\pi$
$991$	−845.562	+	1464.56i	−0.853241	+	1.47786i	−0.878267	+	0.478170i	$0.841300\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−117.992			−0.118585
$996$		0			0
$997$	−978.241	−	564.788i	−0.981185	−	0.566487i	−0.0785570	−	0.996910i	$-0.525031\pi$
$997$	−978.241	−	564.788i	−0.981185	−	0.566487i	−0.902628	+	0.430422i	$0.858365\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.m.325.2		8
3.2	odd	2		588.3.m.e.325.3		8
7.2	even	3		1764.3.z.l.901.3		8
7.3	odd	6		1764.3.d.h.685.5		8
7.4	even	3		1764.3.d.h.685.4		8
7.5	odd	6	inner	1764.3.z.m.901.2		8
7.6	odd	2		1764.3.z.l.325.3		8
21.2	odd	6		588.3.m.f.313.2		8
21.5	even	6		588.3.m.e.313.3		8
21.11	odd	6		588.3.d.c.97.7	yes	8
21.17	even	6		588.3.d.c.97.2	✓	8
21.20	even	2		588.3.m.f.325.2		8
84.11	even	6		2352.3.f.j.97.3		8
84.59	odd	6		2352.3.f.j.97.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.2	✓	8	21.17	even	6
588.3.d.c.97.7	yes	8	21.11	odd	6
588.3.m.e.313.3		8	21.5	even	6
588.3.m.e.325.3		8	3.2	odd	2
588.3.m.f.313.2		8	21.2	odd	6
588.3.m.f.325.2		8	21.20	even	2
1764.3.d.h.685.4		8	7.4	even	3
1764.3.d.h.685.5		8	7.3	odd	6
1764.3.z.l.325.3		8	7.6	odd	2
1764.3.z.l.901.3		8	7.2	even	3
1764.3.z.m.325.2		8	1.1	even	1	trivial
1764.3.z.m.901.2		8	7.5	odd	6	inner
2352.3.f.j.97.3		8	84.11	even	6
2352.3.f.j.97.6		8	84.59	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.416265 + 0.240331i) q^{5} +O(q^{10})\)	\(q+(-0.416265 + 0.240331i) q^{5} +(2.88158 - 4.99104i) q^{11} -1.41991i q^{13} +(19.9767 + 11.5336i) q^{17} +(9.24384 - 5.33693i) q^{19} +(-3.29990 - 5.71559i) q^{23} +(-12.3845 + 21.4506i) q^{25} +6.20258 q^{29} +(-36.3142 - 20.9660i) q^{31} +(30.0464 + 52.0419i) q^{37} -48.8250i q^{41} -51.5603 q^{43} +(16.6641 - 9.62104i) q^{47} +(41.0672 - 71.1306i) q^{53} +2.77013i q^{55} +(80.1766 + 46.2900i) q^{59} +(4.32309 - 2.49594i) q^{61} +(0.341248 + 0.591060i) q^{65} +(1.10350 - 1.91131i) q^{67} +80.5899 q^{71} +(12.0454 + 6.95439i) q^{73} +(32.4442 + 56.1951i) q^{79} -118.005i q^{83} -11.0875 q^{85} +(90.2959 - 52.1324i) q^{89} +(-2.56526 + 4.44316i) q^{95} -31.7875i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) \) 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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