LMFDB - Embedded newform 1764.4.t.b.521.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

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, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (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:	$104.079369250$
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} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.3
Root			$-1.35249 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.b.1097.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.27106 - 7.39769i) q^{5} +O(q^{10})$	$q+(-4.27106 - 7.39769i) q^{5} +(-27.4679 - 15.8586i) q^{11} +9.92568i q^{13} +(-63.7751 + 110.462i) q^{17} +(100.923 - 58.2677i) q^{19} +(55.8271 - 32.2318i) q^{23} +(26.0161 - 45.0612i) q^{25} -113.016i q^{29} +(-6.33498 - 3.65751i) q^{31} +(-184.736 - 319.972i) q^{37} -211.959 q^{41} -432.263 q^{43} +(200.021 + 346.446i) q^{47} +(-121.869 - 70.3612i) q^{53} +270.933i q^{55} +(-259.447 + 449.375i) q^{59} +(-23.5627 + 13.6039i) q^{61} +(73.4271 - 42.3932i) q^{65} +(68.3597 - 118.403i) q^{67} +604.779i q^{71} +(41.9780 + 24.2360i) q^{73} +(415.618 + 719.872i) q^{79} +37.2350 q^{83} +1089.55 q^{85} +(-235.356 - 407.649i) q^{89} +(-862.093 - 497.730i) q^{95} +522.691i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

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
$5$	−4.27106	−	7.39769i	−0.382015	−	0.661670i	0.609335	−	0.792913i	$-0.291436\pi$
$5$	−4.27106	−	7.39769i	−0.382015	−	0.661670i	−0.991350	+	0.131243i	$0.958103\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−27.4679	−	15.8586i	−0.752900	−	0.434687i	0.0738410	−	0.997270i	$-0.476474\pi$
$11$	−27.4679	−	15.8586i	−0.752900	−	0.434687i	−0.826741	+	0.562583i	$0.809808\pi$
$12$		0			0
$13$			9.92568i			0.211761i	0.994379	+	0.105880i	$0.0337660\pi$
$13$			9.92568i			0.211761i	−0.994379	+	0.105880i	$0.966234\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−63.7751	+	110.462i	−0.909866	+	1.57593i	−0.0956180	+	0.995418i	$0.530483\pi$
$17$	−63.7751	+	110.462i	−0.909866	+	1.57593i	−0.814248	+	0.580517i	$0.802851\pi$
$18$		0			0
$19$	100.923	−	58.2677i	1.21859	−	0.703554i	0.253974	−	0.967211i	$-0.418262\pi$
$19$	100.923	−	58.2677i	1.21859	−	0.703554i	0.964616	+	0.263657i	$0.0849288\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	55.8271	−	32.2318i	0.506120	−	0.292208i	−0.225118	−	0.974332i	$-0.572277\pi$
$23$	55.8271	−	32.2318i	0.506120	−	0.292208i	0.731237	+	0.682123i	$0.238943\pi$
$24$		0			0
$25$	26.0161	−	45.0612i	0.208129	−	0.360489i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	113.016i		−	0.723671i	−0.932242	−	0.361836i	$-0.882150\pi$
$29$		−	113.016i		−	0.723671i	0.932242	−	0.361836i	$-0.117850\pi$
$30$		0			0
$31$	−6.33498	−	3.65751i	−0.0367031	−	0.0211906i	0.481536	−	0.876426i	$-0.340079\pi$
$31$	−6.33498	−	3.65751i	−0.0367031	−	0.0211906i	−0.518239	+	0.855236i	$0.673412\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−184.736	−	319.972i	−0.820822	−	1.42171i	−0.905071	−	0.425261i	$-0.860182\pi$
$37$	−184.736	−	319.972i	−0.820822	−	1.42171i	0.0842481	−	0.996445i	$-0.473151\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−211.959			−0.807376			−0.403688	−	0.914897i	$-0.632272\pi$
$41$	−211.959			−0.807376			−0.403688	+	0.914897i	$0.632272\pi$
$42$		0			0
$43$	−432.263			−1.53301			−0.766506	−	0.642237i	$-0.778007\pi$
$43$	−432.263			−1.53301			−0.766506	+	0.642237i	$0.778007\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	200.021	+	346.446i	0.620766	+	1.07520i	0.989343	+	0.145601i	$0.0465117\pi$
$47$	200.021	+	346.446i	0.620766	+	1.07520i	−0.368577	+	0.929597i	$0.620155\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−121.869	−	70.3612i	−0.315849	−	0.182356i	0.333692	−	0.942682i	$-0.391706\pi$
$53$	−121.869	−	70.3612i	−0.315849	−	0.182356i	−0.649541	+	0.760327i	$0.725039\pi$
$54$		0			0
$55$			270.933i			0.664228i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−259.447	+	449.375i	−0.572493	+	0.991587i	0.423816	+	0.905748i	$0.360690\pi$
$59$	−259.447	+	449.375i	−0.572493	+	0.991587i	−0.996309	+	0.0858389i	$0.972643\pi$
$60$		0			0
$61$	−23.5627	+	13.6039i	−0.0494573	+	0.0285542i	−0.524525	−	0.851395i	$-0.675757\pi$
$61$	−23.5627	+	13.6039i	−0.0494573	+	0.0285542i	0.475067	+	0.879949i	$0.342424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	73.4271	−	42.3932i	0.140116	−	0.0808958i
$66$		0			0
$67$	68.3597	−	118.403i	0.124649	−	0.215898i	−0.796947	−	0.604050i	$-0.793553\pi$
$67$	68.3597	−	118.403i	0.124649	−	0.215898i	0.921596	+	0.388151i	$0.126886\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			604.779i			1.01090i	0.862855	+	0.505451i	$0.168674\pi$
$71$			604.779i			1.01090i	−0.862855	+	0.505451i	$0.831326\pi$
$72$		0			0
$73$	41.9780	+	24.2360i	0.0673035	+	0.0388577i	0.533274	−	0.845942i	$-0.320961\pi$
$73$	41.9780	+	24.2360i	0.0673035	+	0.0388577i	−0.465971	+	0.884800i	$0.654295\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	415.618	+	719.872i	0.591907	+	1.02521i	0.993975	+	0.109604i	$0.0349582\pi$
$79$	415.618	+	719.872i	0.591907	+	1.02521i	−0.402068	+	0.915610i	$0.631708\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	37.2350			0.0492418			0.0246209	−	0.999697i	$-0.492162\pi$
$83$	37.2350			0.0492418			0.0246209	+	0.999697i	$0.492162\pi$
$84$		0			0
$85$	1089.55			1.39033
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−235.356	−	407.649i	−0.280311	−	0.485513i	0.691150	−	0.722711i	$-0.257104\pi$
$89$	−235.356	−	407.649i	−0.280311	−	0.485513i	−0.971461	+	0.237198i	$0.923771\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−862.093	−	497.730i	−0.931041	−	0.537537i
$96$		0			0
$97$			522.691i			0.547126i	0.961854	+	0.273563i	$0.0882022\pi$
$97$			522.691i			0.547126i	−0.961854	+	0.273563i	$0.911798\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	527.361	−	913.417i	0.519549	−	0.899885i	−0.480193	−	0.877163i	$-0.659433\pi$
$101$	527.361	−	913.417i	0.519549	−	0.899885i	0.999742	−	0.0227221i	$-0.00723329\pi$
$102$		0			0
$103$	643.991	−	371.808i	0.616061	−	0.355683i	−0.159273	−	0.987235i	$-0.550915\pi$
$103$	643.991	−	371.808i	0.616061	−	0.355683i	0.775334	+	0.631552i	$0.217582\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−204.383	+	118.001i	−0.184659	+	0.106613i	−0.589480	−	0.807783i	$-0.700667\pi$
$107$	−204.383	+	118.001i	−0.184659	+	0.106613i	0.404821	+	0.914396i	$0.367334\pi$
$108$		0			0
$109$	116.269	−	201.385i	0.102171	−	0.176965i	−0.810408	−	0.585866i	$-0.800755\pi$
$109$	116.269	−	201.385i	0.102171	−	0.176965i	0.912579	+	0.408901i	$0.134088\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2266.43i			1.88680i	0.331661	+	0.943398i	$0.392391\pi$
$113$			2266.43i			1.88680i	−0.331661	+	0.943398i	$0.607609\pi$
$114$		0			0
$115$	−476.882	−	275.328i	−0.386691	−	0.223256i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−162.508	−	281.472i	−0.122095	−	0.211474i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1512.23			−1.08206
$126$		0			0
$127$	1675.71			1.17083			0.585414	−	0.810735i	$-0.300932\pi$
$127$	1675.71			1.17083			0.585414	+	0.810735i	$0.300932\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	593.103	+	1027.28i	0.395570	+	0.685147i	0.993174	−	0.116645i	$-0.0372138\pi$
$131$	593.103	+	1027.28i	0.395570	+	0.685147i	−0.597604	+	0.801791i	$0.703881\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2743.44	+	1583.92i	1.71086	+	0.987764i	0.933406	+	0.358821i	$0.116821\pi$
$137$	2743.44	+	1583.92i	1.71086	+	0.987764i	0.777451	+	0.628943i	$0.216512\pi$
$138$		0			0
$139$		−	607.821i		−	0.370897i	−0.982654	−	0.185448i	$-0.940626\pi$
$139$		−	607.821i		−	0.370897i	0.982654	−	0.185448i	$-0.0593738\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	157.408	−	272.638i	0.0920496	−	0.159435i
$144$		0			0
$145$	−836.055	+	482.696i	−0.478832	+	0.276454i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1117.57	+	645.230i	−0.614463	+	0.354760i	−0.774710	−	0.632317i	$-0.782104\pi$
$149$	−1117.57	+	645.230i	−0.614463	+	0.354760i	0.160247	+	0.987077i	$0.448771\pi$
$150$		0			0
$151$	−1452.49	+	2515.79i	−0.782796	+	1.35584i	0.147510	+	0.989060i	$0.452874\pi$
$151$	−1452.49	+	2515.79i	−0.782796	+	1.35584i	−0.930307	+	0.366782i	$0.880459\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			62.4857i			0.0323805i
$156$		0			0
$157$	1022.35	+	590.254i	0.519697	+	0.300047i	0.736811	−	0.676099i	$-0.236331\pi$
$157$	1022.35	+	590.254i	0.519697	+	0.300047i	−0.217114	+	0.976146i	$0.569664\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−803.791	−	1392.21i	−0.386244	−	0.668994i	0.605697	−	0.795695i	$-0.292894\pi$
$163$	−803.791	−	1392.21i	−0.386244	−	0.668994i	−0.991941	+	0.126701i	$0.959561\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2621.57			−1.21475			−0.607374	−	0.794416i	$-0.707777\pi$
$167$	−2621.57			−1.21475			−0.607374	+	0.794416i	$0.707777\pi$
$168$		0			0
$169$	2098.48			0.955157
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1321.78	+	2289.40i	0.580887	+	1.00613i	0.995375	+	0.0960706i	$0.0306275\pi$
$173$	1321.78	+	2289.40i	0.580887	+	1.00613i	−0.414488	+	0.910055i	$0.636039\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3594.15	−	2075.08i	−1.50078	−	0.866475i	−1.00000		0.000899820i	$-0.999714\pi$
$179$	−3594.15	−	2075.08i	−1.50078	−	0.866475i	−0.500779	−	0.865575i	$-0.666953\pi$
$180$		0			0
$181$			4040.50i			1.65927i	0.558304	+	0.829636i	$0.311452\pi$
$181$			4040.50i			1.65927i	−0.558304	+	0.829636i	$0.688548\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1578.04	+	2733.24i	−0.627133	+	1.08623i
$186$		0			0
$187$	3503.54	−	2022.77i	1.37008	−	0.791014i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3227.37	−	1863.32i	1.22264	−	0.705892i	0.257161	−	0.966369i	$-0.417213\pi$
$191$	3227.37	−	1863.32i	1.22264	−	0.705892i	0.965480	+	0.260477i	$0.0838797\pi$
$192$		0			0
$193$	−668.472	+	1157.83i	−0.249314	+	0.431825i	−0.963336	−	0.268299i	$-0.913539\pi$
$193$	−668.472	+	1157.83i	−0.249314	+	0.431825i	0.714022	+	0.700124i	$0.246872\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3451.92i			1.24842i	0.781256	+	0.624210i	$0.214579\pi$
$197$			3451.92i			1.24842i	−0.781256	+	0.624210i	$0.785421\pi$
$198$		0			0
$199$	4621.37	+	2668.15i	1.64623	+	0.950453i	0.978551	+	0.206006i	$0.0660467\pi$
$199$	4621.37	+	2668.15i	1.64623	+	0.950453i	0.667682	+	0.744447i	$0.267287\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	905.289	+	1568.01i	0.308430	+	0.534216i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3696.18			−1.22330
$210$		0			0
$211$	1800.90			0.587578			0.293789	−	0.955870i	$-0.405084\pi$
$211$	1800.90			0.587578			0.293789	+	0.955870i	$0.405084\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1846.22	+	3197.75i	0.585634	+	1.01435i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1096.41	−	633.011i	−0.333721	−	0.192674i
$222$		0			0
$223$			2815.72i			0.845535i	0.906238	+	0.422768i	$0.138941\pi$
$223$			2815.72i			0.845535i	−0.906238	+	0.422768i	$0.861059\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1976.56	+	3423.50i	−0.577925	+	1.00100i	0.417792	+	0.908543i	$0.362804\pi$
$227$	−1976.56	+	3423.50i	−0.577925	+	1.00100i	−0.995717	+	0.0924528i	$0.970529\pi$
$228$		0			0
$229$	2132.66	−	1231.29i	0.615416	−	0.355311i	−0.159666	−	0.987171i	$-0.551042\pi$
$229$	2132.66	−	1231.29i	0.615416	−	0.355311i	0.775082	+	0.631860i	$0.217708\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4505.33	−	2601.16i	1.26676	−	0.731362i	0.292384	−	0.956301i	$-0.405552\pi$
$233$	4505.33	−	2601.16i	1.26676	−	0.731362i	0.974373	+	0.224939i	$0.0722182\pi$
$234$		0			0
$235$	1708.60	−	2959.38i	0.474284	−	0.821485i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1510.80i		−	0.408894i	−0.978878	−	0.204447i	$-0.934460\pi$
$239$		−	1510.80i		−	0.408894i	0.978878	−	0.204447i	$-0.0655396\pi$
$240$		0			0
$241$	1360.97	+	785.759i	0.363768	+	0.210022i	0.670732	−	0.741699i	$-0.265980\pi$
$241$	1360.97	+	785.759i	0.363768	+	0.210022i	−0.306964	+	0.951721i	$0.599313\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	578.346	+	1001.73i	0.148985	+	0.258050i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3863.31			−0.971514			−0.485757	−	0.874094i	$-0.661456\pi$
$251$	−3863.31			−0.971514			−0.485757	+	0.874094i	$0.661456\pi$
$252$		0			0
$253$	−2044.61			−0.508077
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2911.09	+	5042.16i	0.706572	+	1.22382i	0.966121	+	0.258089i	$0.0830929\pi$
$257$	2911.09	+	5042.16i	0.706572	+	1.22382i	−0.259549	+	0.965730i	$0.583574\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−935.656	−	540.201i	−0.219373	−	0.126655i	0.386287	−	0.922379i	$-0.373757\pi$
$263$	−935.656	−	540.201i	−0.219373	−	0.126655i	−0.605660	+	0.795724i	$0.707091\pi$
$264$		0			0
$265$			1202.07i			0.278651i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1142.76	+	1979.32i	−0.259016	+	0.448630i	−0.965979	−	0.258622i	$-0.916732\pi$
$269$	−1142.76	+	1979.32i	−0.259016	+	0.448630i	0.706962	+	0.707251i	$0.250065\pi$
$270$		0			0
$271$	1966.85	−	1135.56i	0.440877	−	0.254540i	−0.263093	−	0.964771i	$-0.584743\pi$
$271$	1966.85	−	1135.56i	0.440877	−	0.254540i	0.703969	+	0.710230i	$0.251409\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1429.22	+	825.159i	−0.313400	+	0.180942i
$276$		0			0
$277$	−824.549	+	1428.16i	−0.178853	+	0.309783i	−0.941488	−	0.337046i	$-0.890572\pi$
$277$	−824.549	+	1428.16i	−0.178853	+	0.309783i	0.762635	+	0.646829i	$0.223905\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7573.33i			1.60778i	0.594776	+	0.803892i	$0.297241\pi$
$281$			7573.33i			1.60778i	−0.594776	+	0.803892i	$0.702759\pi$
$282$		0			0
$283$	−4526.00	−	2613.09i	−0.950682	−	0.548877i	−0.0573894	−	0.998352i	$-0.518278\pi$
$283$	−4526.00	−	2613.09i	−0.950682	−	0.548877i	−0.893293	+	0.449475i	$0.851611\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5678.02	−	9834.62i	−1.15571	−	2.00175i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4924.11			−0.981808			−0.490904	−	0.871214i	$-0.663333\pi$
$293$	−4924.11			−0.981808			−0.490904	+	0.871214i	$0.663333\pi$
$294$		0			0
$295$	4432.45			0.874805
$296$		0			0
$297$		0			0
$298$		0			0
$299$	319.922	+	554.122i	0.0618782	+	0.107176i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	201.275	+	116.206i	0.0377869	+	0.0218163i
$306$		0			0
$307$			10064.5i			1.87105i	0.353256	+	0.935527i	$0.385075\pi$
$307$			10064.5i			1.87105i	−0.353256	+	0.935527i	$0.614925\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1150.12	−	1992.06i	0.209701	−	0.363214i	−0.741919	−	0.670490i	$-0.766084\pi$
$311$	1150.12	−	1992.06i	0.209701	−	0.363214i	0.951620	+	0.307276i	$0.0994175\pi$
$312$		0			0
$313$	−898.271	+	518.617i	−0.162215	+	0.0936548i	−0.578910	−	0.815392i	$-0.696522\pi$
$313$	−898.271	+	518.617i	−0.162215	+	0.0936548i	0.416695	+	0.909046i	$0.363188\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7214.93	+	4165.54i	−1.27833	+	0.738044i	−0.976541	−	0.215330i	$-0.930917\pi$
$317$	−7214.93	+	4165.54i	−1.27833	+	0.738044i	−0.301789	+	0.953375i	$0.597584\pi$
$318$		0			0
$319$	−1792.27	+	3104.31i	−0.314570	+	0.544852i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			14864.1i			2.56056i
$324$		0			0
$325$	447.263	+	258.227i	0.0763375	+	0.0440735i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1397.99	+	2421.39i	0.232146	+	0.402089i	0.958440	−	0.285296i	$-0.0920918\pi$
$331$	1397.99	+	2421.39i	0.232146	+	0.402089i	−0.726293	+	0.687385i	$0.758759\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1167.87			−0.190471
$336$		0			0
$337$	4675.27			0.755721			0.377861	−	0.925863i	$-0.376660\pi$
$337$	4675.27			0.755721			0.377861	+	0.925863i	$0.376660\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	116.006	+	200.928i	0.0184225	+	0.0319087i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2303.50	−	1329.93i	−0.356364	−	0.205747i	0.311120	−	0.950370i	$-0.399296\pi$
$347$	−2303.50	−	1329.93i	−0.356364	−	0.205747i	−0.667485	+	0.744623i	$0.732629\pi$
$348$		0			0
$349$		−	12826.3i		−	1.96727i	−0.180181	−	0.983633i	$-0.557668\pi$
$349$		−	12826.3i		−	1.96727i	0.180181	−	0.983633i	$-0.442332\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4113.76	+	7125.25i	−0.620265	+	1.07433i	0.369171	+	0.929361i	$0.379642\pi$
$353$	−4113.76	+	7125.25i	−0.620265	+	1.07433i	−0.989436	+	0.144969i	$0.953692\pi$
$354$		0			0
$355$	4473.97	−	2583.05i	0.668883	−	0.386180i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.38244	+	4.26226i	−0.00108532	+	0.000626611i	−0.500543	−	0.865712i	$-0.666866\pi$
$359$	−7.38244	+	4.26226i	−0.00108532	+	0.000626611i	0.499457	+	0.866339i	$0.333533\pi$
$360$		0			0
$361$	3360.74	−	5820.98i	0.489976	−	0.848663i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	414.054i		−	0.0593769i
$366$		0			0
$367$	−6325.97	−	3652.30i	−0.899763	−	0.519478i	−0.0226398	−	0.999744i	$-0.507207\pi$
$367$	−6325.97	−	3652.30i	−0.899763	−	0.519478i	−0.877123	+	0.480265i	$0.840540\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4253.40	+	7367.10i	0.590436	+	1.02266i	0.994174	+	0.107790i	$0.0343774\pi$
$373$	4253.40	+	7367.10i	0.590436	+	1.02266i	−0.403738	+	0.914875i	$0.632289\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1121.76			0.153245
$378$		0			0
$379$	7735.02			1.04834			0.524171	−	0.851613i	$-0.324375\pi$
$379$	7735.02			1.04834			0.524171	+	0.851613i	$0.324375\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3963.43	−	6864.86i	−0.528777	−	0.915869i	−0.999437	−	0.0335545i	$-0.989317\pi$
$383$	−3963.43	−	6864.86i	−0.528777	−	0.915869i	0.470659	−	0.882315i	$-0.344016\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9283.67	−	5359.93i	−1.21003	−	0.698609i	−0.247262	−	0.968949i	$-0.579531\pi$
$389$	−9283.67	−	5359.93i	−1.21003	−	0.698609i	−0.962765	+	0.270339i	$0.912864\pi$
$390$		0			0
$391$			8222.34i			1.06348i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3550.26	−	6149.23i	0.452235	−	0.783295i
$396$		0			0
$397$	4886.86	−	2821.43i	0.617795	−	0.356684i	−0.158215	−	0.987405i	$-0.550574\pi$
$397$	4886.86	−	2821.43i	0.617795	−	0.356684i	0.776010	+	0.630721i	$0.217241\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−159.937	+	92.3395i	−0.0199174	+	0.0114993i	−0.509926	−	0.860218i	$-0.670327\pi$
$401$	−159.937	+	92.3395i	−0.0199174	+	0.0114993i	0.490008	+	0.871718i	$0.336994\pi$
$402$		0			0
$403$	36.3032	−	62.8790i	0.00448733	−	0.00777228i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			11718.6i			1.42720i
$408$		0			0
$409$	2321.40	+	1340.26i	0.280651	+	0.162034i	0.633718	−	0.773564i	$-0.281528\pi$
$409$	2321.40	+	1340.26i	0.280651	+	0.162034i	−0.353067	+	0.935598i	$0.614861\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−159.033	−	275.453i	−0.0188111	−	0.0325818i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12735.7			−1.48492			−0.742460	−	0.669890i	$-0.766341\pi$
$419$	−12735.7			−1.48492			−0.742460	+	0.669890i	$0.766341\pi$
$420$		0			0
$421$	3317.68			0.384070			0.192035	−	0.981388i	$-0.438491\pi$
$421$	3317.68			0.384070			0.192035	+	0.981388i	$0.438491\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3318.36	+	5747.56i	0.378739	+	0.655994i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7748.89	+	4473.82i	0.866011	+	0.499992i	0.866021	−	0.500008i	$-0.166670\pi$
$431$	7748.89	+	4473.82i	0.866011	+	0.499992i	−9.69024e−6		1.00000i	$0.500003\pi$
$432$		0			0
$433$			11833.0i			1.31330i	0.754194	+	0.656651i	$0.228028\pi$
$433$			11833.0i			1.31330i	−0.754194	+	0.656651i	$0.771972\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3756.14	−	6505.83i	0.411169	−	0.712165i
$438$		0			0
$439$	−7376.74	+	4258.96i	−0.801988	+	0.463028i	−0.844166	−	0.536082i	$-0.819904\pi$
$439$	−7376.74	+	4258.96i	−0.801988	+	0.463028i	0.0421781	+	0.999110i	$0.486570\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7190.84	−	4151.64i	0.771213	−	0.445260i	−0.0620942	−	0.998070i	$-0.519778\pi$
$443$	7190.84	−	4151.64i	0.771213	−	0.445260i	0.833307	+	0.552810i	$0.186445\pi$
$444$		0			0
$445$	−2010.44	+	3482.19i	−0.214166	+	0.370947i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14235.2i		−	1.49622i	−0.663575	−	0.748110i	$-0.730961\pi$
$449$		−	14235.2i		−	1.49622i	0.663575	−	0.748110i	$-0.269039\pi$
$450$		0			0
$451$	5822.08	+	3361.38i	0.607873	+	0.350956i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3046.98	+	5277.52i	0.311885	+	0.540201i	0.978770	−	0.204959i	$-0.0657062\pi$
$457$	3046.98	+	5277.52i	0.311885	+	0.540201i	−0.666885	+	0.745160i	$0.732373\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13335.2			1.34725			0.673625	−	0.739073i	$-0.264736\pi$
$461$	13335.2			1.34725			0.673625	+	0.739073i	$0.264736\pi$
$462$		0			0
$463$	−6596.99			−0.662178			−0.331089	−	0.943600i	$-0.607416\pi$
$463$	−6596.99			−0.662178			−0.331089	+	0.943600i	$0.607416\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2176.35	+	3769.54i	0.215652	+	0.373519i	0.953474	−	0.301475i	$-0.0974791\pi$
$467$	2176.35	+	3769.54i	0.215652	+	0.373519i	−0.737822	+	0.674995i	$0.764146\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11873.4	+	6855.10i	1.15420	+	0.666380i
$474$		0			0
$475$		−	6063.59i		−	0.585719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	159.875	−	276.912i	0.0152503	−	0.0264143i	−0.858300	−	0.513149i	$-0.828479\pi$
$479$	159.875	−	276.912i	0.0152503	−	0.0264143i	0.873550	+	0.486735i	$0.161812\pi$
$480$		0			0
$481$	3175.94	−	1833.63i	0.301061	−	0.173818i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3866.71	−	2232.45i	0.362017	−	0.209011i
$486$		0			0
$487$	−2919.49	+	5056.70i	−0.271652	+	0.470515i	−0.969285	−	0.245940i	$-0.920903\pi$
$487$	−2919.49	+	5056.70i	−0.271652	+	0.470515i	0.697633	+	0.716455i	$0.254237\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12442.2i		−	1.14360i	−0.820392	−	0.571801i	$-0.806245\pi$
$491$		−	12442.2i		−	1.14360i	0.820392	−	0.571801i	$-0.193755\pi$
$492$		0			0
$493$	12483.9	+	7207.58i	1.14046	+	0.658444i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8348.40	−	14459.9i	−0.748949	−	1.29722i	−0.948327	−	0.317295i	$-0.897225\pi$
$499$	−8348.40	−	14459.9i	−0.748949	−	1.29722i	0.199378	−	0.979923i	$-0.436108\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4435.36			−0.393167			−0.196583	−	0.980487i	$-0.562985\pi$
$503$	−4435.36			−0.393167			−0.196583	+	0.980487i	$0.562985\pi$
$504$		0			0
$505$	−9009.57			−0.793902
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11207.4	+	19411.8i	0.975951	+	1.69040i	0.676761	+	0.736203i	$0.263383\pi$
$509$	11207.4	+	19411.8i	0.975951	+	1.69040i	0.299190	+	0.954193i	$0.403283\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5501.05	−	3176.03i	−0.470689	−	0.271753i
$516$		0			0
$517$		−	12688.2i		−	1.07936i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9561.15	+	16560.4i	−0.803996	+	1.39256i	0.112971	+	0.993598i	$0.463963\pi$
$521$	−9561.15	+	16560.4i	−0.803996	+	1.39256i	−0.916967	+	0.398963i	$0.869370\pi$
$522$		0			0
$523$	−18873.4	+	10896.5i	−1.57796	+	0.911037i	−0.582819	+	0.812602i	$0.698050\pi$
$523$	−18873.4	+	10896.5i	−1.57796	+	0.911037i	−0.995143	+	0.0984349i	$0.968616\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	808.028	−	466.515i	0.0667899	−	0.0385612i
$528$		0			0
$529$	−4005.72	+	6938.12i	−0.329229	+	0.570241i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2103.84i		−	0.170970i
$534$		0			0
$535$	1745.87	+	1007.98i	0.141085	+	0.0814554i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	101.107	+	175.122i	0.00803498	+	0.0139170i	0.870015	−	0.493025i	$-0.164109\pi$
$541$	101.107	+	175.122i	0.00803498	+	0.0139170i	−0.861980	+	0.506942i	$0.830776\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1986.38			−0.156123
$546$		0			0
$547$	1510.85			0.118097			0.0590486	−	0.998255i	$-0.481193\pi$
$547$	1510.85			0.118097			0.0590486	+	0.998255i	$0.481193\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6585.16	−	11405.8i	−0.509142	−	0.881859i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3266.16	+	1885.72i	0.248459	+	0.143448i	0.619058	−	0.785345i	$-0.287514\pi$
$557$	3266.16	+	1885.72i	0.248459	+	0.143448i	−0.370599	+	0.928793i	$0.620848\pi$
$558$		0			0
$559$		−	4290.51i		−	0.324632i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3916.56	−	6783.67i	0.293185	−	0.507811i	−0.681376	−	0.731933i	$-0.738618\pi$
$563$	3916.56	−	6783.67i	0.293185	−	0.507811i	0.974561	+	0.224122i	$0.0719515\pi$
$564$		0			0
$565$	16766.4	−	9680.07i	1.24844	−	0.720785i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4941.75	−	2853.12i	0.364093	−	0.210209i	−0.306782	−	0.951780i	$-0.599252\pi$
$569$	4941.75	−	2853.12i	0.364093	−	0.210209i	0.670875	+	0.741571i	$0.265919\pi$
$570$		0			0
$571$	−3568.83	+	6181.40i	−0.261560	+	0.453036i	−0.966657	−	0.256076i	$-0.917570\pi$
$571$	−3568.83	+	6181.40i	−0.261560	+	0.453036i	0.705096	+	0.709111i	$0.250904\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3354.18i		−	0.243268i
$576$		0			0
$577$	6579.24	+	3798.52i	0.474692	+	0.274064i	0.718202	−	0.695835i	$-0.244965\pi$
$577$	6579.24	+	3798.52i	0.474692	+	0.274064i	−0.243510	+	0.969898i	$0.578299\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2231.66	+	3865.36i	0.158535	+	0.274591i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8212.13			−0.577429			−0.288714	−	0.957415i	$-0.593228\pi$
$587$	−8212.13			−0.577429			−0.288714	+	0.957415i	$0.593228\pi$
$588$		0			0
$589$	−852.457			−0.0596348
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9490.08	−	16437.3i	−0.657186	−	1.13828i	−0.981341	−	0.192275i	$-0.938413\pi$
$593$	−9490.08	−	16437.3i	−0.657186	−	1.13828i	0.324155	−	0.946004i	$-0.394920\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17794.3	−	10273.5i	−1.21378	−	0.700776i	−0.250199	−	0.968195i	$-0.580496\pi$
$599$	−17794.3	−	10273.5i	−1.21378	−	0.700776i	−0.963580	+	0.267419i	$0.913829\pi$
$600$		0			0
$601$			15444.1i			1.04821i	0.851652	+	0.524107i	$0.175601\pi$
$601$			15444.1i			1.04821i	−0.851652	+	0.524107i	$0.824399\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1388.16	+	2404.37i	−0.0932840	+	0.161573i
$606$		0			0
$607$	−1320.67	+	762.488i	−0.0883102	+	0.0509859i	−0.543505	−	0.839406i	$-0.682903\pi$
$607$	−1320.67	+	762.488i	−0.0883102	+	0.0509859i	0.455195	+	0.890392i	$0.349570\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3438.71	+	1985.34i	−0.227685	+	0.131454i
$612$		0			0
$613$	8883.48	−	15386.6i	0.585319	−	1.01380i	−0.409517	−	0.912302i	$-0.634303\pi$
$613$	8883.48	−	15386.6i	0.585319	−	1.01380i	0.994836	−	0.101499i	$-0.0323639\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	21575.8i		−	1.40779i	−0.710302	−	0.703897i	$-0.751442\pi$
$617$		−	21575.8i		−	1.40779i	0.710302	−	0.703897i	$-0.248558\pi$
$618$		0			0
$619$	−7448.71	−	4300.51i	−0.483665	−	0.279244i	0.238277	−	0.971197i	$-0.423417\pi$
$619$	−7448.71	−	4300.51i	−0.483665	−	0.279244i	−0.721943	+	0.691953i	$0.756751\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3206.82	+	5554.37i	0.205236	+	0.355480i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	47126.2			2.98736
$630$		0			0
$631$	−5922.32			−0.373635			−0.186817	−	0.982395i	$-0.559817\pi$
$631$	−5922.32			−0.373635			−0.186817	+	0.982395i	$0.559817\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7157.05	−	12396.4i	−0.447274	−	0.774701i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19398.9	+	11199.9i	1.19533	+	0.690127i	0.959512	−	0.281669i	$-0.0908881\pi$
$641$	19398.9	+	11199.9i	1.19533	+	0.690127i	0.235823	+	0.971796i	$0.424221\pi$
$642$		0			0
$643$		−	19715.7i		−	1.20919i	−0.796533	−	0.604595i	$-0.793335\pi$
$643$		−	19715.7i		−	1.20919i	0.796533	−	0.604595i	$-0.206665\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3763.68	−	6518.89i	0.228695	−	0.396111i	−0.728727	−	0.684805i	$-0.759887\pi$
$647$	3763.68	−	6518.89i	0.228695	−	0.396111i	0.957422	+	0.288693i	$0.0932208\pi$
$648$		0			0
$649$	14252.9	−	8228.94i	0.862060	−	0.497711i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14524.4	+	8385.65i	−0.870417	+	0.502536i	−0.867487	−	0.497460i	$-0.834266\pi$
$653$	−14524.4	+	8385.65i	−0.870417	+	0.502536i	−0.00293044	+	0.999996i	$0.500933\pi$
$654$		0			0
$655$	5066.36	−	8775.19i	0.302227	−	0.523473i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	20834.0i		−	1.23153i	−0.787930	−	0.615765i	$-0.788847\pi$
$659$		−	20834.0i		−	1.23153i	0.787930	−	0.615765i	$-0.211153\pi$
$660$		0			0
$661$	−22838.3	−	13185.7i	−1.34388	−	0.775892i	−0.356509	−	0.934292i	$-0.616033\pi$
$661$	−22838.3	−	13185.7i	−1.34388	−	0.775892i	−0.987375	+	0.158400i	$0.949367\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3642.69	−	6309.33i	−0.211463	−	0.366264i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	862.959			0.0496485
$672$		0			0
$673$	−18150.5			−1.03960			−0.519800	−	0.854288i	$-0.673993\pi$
$673$	−18150.5			−1.03960			−0.519800	+	0.854288i	$0.673993\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11357.7	−	19672.2i	−0.644776	−	1.11679i	−0.984353	−	0.176207i	$-0.943617\pi$
$677$	−11357.7	−	19672.2i	−0.644776	−	1.11679i	0.339577	−	0.940578i	$-0.389716\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11788.2	+	6805.92i	0.660414	+	0.381290i	0.792435	−	0.609957i	$-0.208813\pi$
$683$	11788.2	+	6805.92i	0.660414	+	0.381290i	−0.132021	+	0.991247i	$0.542146\pi$
$684$		0			0
$685$		−	27060.1i		−	1.50936i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	698.383	−	1209.63i	0.0386158	−	0.0668845i
$690$		0			0
$691$	−24565.6	+	14182.9i	−1.35241	+	0.780817i	−0.988587	−	0.150650i	$-0.951863\pi$
$691$	−24565.6	+	14182.9i	−1.35241	+	0.780817i	−0.363827	+	0.931467i	$0.618530\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4496.47	+	2596.04i	−0.245411	+	0.141688i
$696$		0			0
$697$	13517.7	−	23413.3i	0.734604	−	1.27237i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			9203.01i			0.495853i	0.968779	+	0.247927i	$0.0797492\pi$
$701$			9203.01i			0.495853i	−0.968779	+	0.247927i	$0.920251\pi$
$702$		0			0
$703$	−37288.1	−	21528.3i	−2.00049	−	1.15499i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10975.7	+	19010.4i	0.581383	+	1.00698i	0.995316	+	0.0966770i	$0.0308214\pi$
$709$	10975.7	+	19010.4i	0.581383	+	1.00698i	−0.413933	+	0.910307i	$0.635845\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−471.552			−0.0247682
$714$		0			0
$715$	−2689.19			−0.140657
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5090.07	+	8816.26i	0.264016	+	0.457289i	0.967305	−	0.253615i	$-0.0816195\pi$
$719$	5090.07	+	8816.26i	0.264016	+	0.457289i	−0.703289	+	0.710904i	$0.748286\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5092.62	−	2940.22i	−0.260876	−	0.150617i
$726$		0			0
$727$			19192.2i			0.979093i	0.871977	+	0.489546i	$0.162838\pi$
$727$			19192.2i			0.979093i	−0.871977	+	0.489546i	$0.837162\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27567.6	−	47748.5i	1.39484	−	2.41593i
$732$		0			0
$733$	−30916.5	+	17849.6i	−1.55788	+	0.899442i	−0.560419	+	0.828209i	$0.689360\pi$
$733$	−30916.5	+	17849.6i	−1.55788	+	0.899442i	−0.997460	+	0.0712328i	$0.977307\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3755.40	+	2168.18i	−0.187696	+	0.108366i
$738$		0			0
$739$	−13871.9	+	24026.9i	−0.690511	+	1.19600i	0.281160	+	0.959661i	$0.409281\pi$
$739$	−13871.9	+	24026.9i	−0.690511	+	1.19600i	−0.971671	+	0.236339i	$0.924052\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			25212.0i			1.24487i	0.782671	+	0.622435i	$0.213857\pi$
$743$			25212.0i			1.24487i	−0.782671	+	0.622435i	$0.786143\pi$
$744$		0			0
$745$	9546.43	+	5511.63i	0.469469	+	0.271048i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6479.18	−	11222.3i	−0.314818	−	0.545282i	0.664580	−	0.747217i	$-0.268610\pi$
$751$	−6479.18	−	11222.3i	−0.314818	−	0.545282i	−0.979399	+	0.201935i	$0.935277\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24814.7			1.19616
$756$		0			0
$757$	−1671.41			−0.0802488			−0.0401244	−	0.999195i	$-0.512775\pi$
$757$	−1671.41			−0.0802488			−0.0401244	+	0.999195i	$0.512775\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13983.9	−	24220.9i	−0.666120	−	1.15375i	−0.978980	−	0.203954i	$-0.934621\pi$
$761$	−13983.9	−	24220.9i	−0.666120	−	1.15375i	0.312861	−	0.949799i	$-0.398713\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4460.35	−	2575.19i	−0.209979	−	0.121231i
$768$		0			0
$769$		−	22787.5i		−	1.06858i	−0.845301	−	0.534290i	$-0.820579\pi$
$769$		−	22787.5i		−	1.06858i	0.845301	−	0.534290i	$-0.179421\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−754.803	+	1307.36i	−0.0351208	+	0.0608310i	−0.883052	−	0.469276i	$-0.844515\pi$
$773$	−754.803	+	1307.36i	−0.0351208	+	0.0608310i	0.847931	+	0.530107i	$0.177848\pi$
$774$		0			0
$775$	−329.623	+	190.308i	−0.0152779	+	0.00882072i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−21391.4	+	12350.4i	−0.983861	+	0.568032i
$780$		0			0
$781$	9590.96	−	16612.0i	0.439426	−	0.761108i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	10084.0i		−	0.458490i
$786$		0			0
$787$	4781.43	+	2760.56i	0.216569	+	0.125036i	0.604360	−	0.796711i	$-0.293429\pi$
$787$	4781.43	+	2760.56i	0.216569	+	0.125036i	−0.387792	+	0.921747i	$0.626762\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−135.028	−	233.876i	−0.00604665	−	0.0104731i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	32608.8			1.44926			0.724631	−	0.689137i	$-0.242010\pi$
$797$	32608.8			1.44926			0.724631	+	0.689137i	$0.242010\pi$
$798$		0			0
$799$	−51025.3			−2.25926
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−768.700	−	1331.43i	−0.0337819	−	0.0585119i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8150.32	+	4705.59i	0.354203	+	0.204499i	0.666535	−	0.745474i	$-0.267777\pi$
$809$	8150.32	+	4705.59i	0.354203	+	0.204499i	−0.312332	+	0.949973i	$0.601110\pi$
$810$		0			0
$811$		−	26945.2i		−	1.16668i	−0.812230	−	0.583338i	$-0.801746\pi$
$811$		−	26945.2i		−	1.16668i	0.812230	−	0.583338i	$-0.198254\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6866.08	+	11892.4i	−0.295102	+	0.511132i
$816$		0			0
$817$	−43625.1	+	25187.0i	−1.86812	+	1.07856i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8629.02	+	4981.97i	−0.366815	+	0.211781i	−0.672066	−	0.740491i	$-0.734593\pi$
$821$	−8629.02	+	4981.97i	−0.366815	+	0.211781i	0.305251	+	0.952272i	$0.401260\pi$
$822$		0			0
$823$	−38.5282	+	66.7327i	−0.00163184	+	0.00282644i	−0.866840	−	0.498586i	$-0.833853\pi$
$823$	−38.5282	+	66.7327i	−0.00163184	+	0.00282644i	0.865208	+	0.501413i	$0.167186\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	17261.3i		−	0.725797i	−0.931829	−	0.362898i	$-0.881787\pi$
$827$		−	17261.3i		−	0.725797i	0.931829	−	0.362898i	$-0.118213\pi$
$828$		0			0
$829$	1435.89	+	829.014i	0.0601576	+	0.0347320i	0.529777	−	0.848137i	$-0.322276\pi$
$829$	1435.89	+	829.014i	0.0601576	+	0.0347320i	−0.469620	+	0.882869i	$0.655609\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	11196.9	+	19393.6i	0.464053	+	0.803763i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31632.6			1.30164			0.650821	−	0.759231i	$-0.274425\pi$
$839$	31632.6			1.30164			0.650821	+	0.759231i	$0.274425\pi$
$840$		0			0
$841$	11616.5			0.476300
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8962.74	−	15523.9i	−0.364885	−	0.631999i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20626.6	−	11908.8i	−0.830869	−	0.479702i
$852$		0			0
$853$		−	7848.64i		−	0.315044i	−0.987516	−	0.157522i	$-0.949650\pi$
$853$		−	7848.64i		−	0.315044i	0.987516	−	0.157522i	$-0.0503505\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4290.84	+	7431.96i	−0.171030	+	0.296232i	−0.938780	−	0.344517i	$-0.888043\pi$
$857$	−4290.84	+	7431.96i	−0.171030	+	0.296232i	0.767750	+	0.640749i	$0.221376\pi$
$858$		0			0
$859$	35896.5	−	20724.8i	1.42581	−	0.823192i	0.429024	−	0.903293i	$-0.358858\pi$
$859$	35896.5	−	20724.8i	1.42581	−	0.823192i	0.996787	+	0.0801010i	$0.0255243\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5665.14	−	3270.77i	0.223457	−	0.129013i	−0.384093	−	0.923294i	$-0.625486\pi$
$863$	5665.14	−	3270.77i	0.223457	−	0.129013i	0.607550	+	0.794281i	$0.292152\pi$
$864$		0			0
$865$	11290.8	−	19556.3i	0.443815	−	0.768711i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	26364.5i		−	1.02918i
$870$		0			0
$871$	1175.23	+	678.517i	0.0457187	+	0.0263957i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7853.26	−	13602.3i	−0.302378	−	0.523735i	0.674296	−	0.738461i	$-0.264447\pi$
$877$	−7853.26	−	13602.3i	−0.302378	−	0.523735i	−0.976674	+	0.214727i	$0.931114\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15432.4			−0.590161			−0.295080	−	0.955472i	$-0.595346\pi$
$881$	−15432.4			−0.590161			−0.295080	+	0.955472i	$0.595346\pi$
$882$		0			0
$883$	17372.6			0.662101			0.331050	−	0.943613i	$-0.392597\pi$
$883$	17372.6			0.662101			0.331050	+	0.943613i	$0.392597\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15942.5	−	27613.3i	−0.603493	−	1.04528i	−0.992288	−	0.123956i	$-0.960442\pi$
$887$	−15942.5	−	27613.3i	−0.603493	−	1.04528i	0.388795	−	0.921324i	$-0.372891\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	40373.2	+	23309.5i	1.51292	+	0.873485i
$894$		0			0
$895$			35451.2i			1.32403i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−413.355	+	715.952i	−0.0153350	+	0.0265610i
$900$		0			0
$901$	15544.4	−	8974.58i	0.574762	−	0.331839i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29890.4	−	17257.2i	1.09789	−	0.633867i
$906$		0			0
$907$	−4017.29	+	6958.16i	−0.147070	+	0.254732i	−0.930143	−	0.367197i	$-0.880317\pi$
$907$	−4017.29	+	6958.16i	−0.147070	+	0.254732i	0.783074	+	0.621929i	$0.213651\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	51249.1i		−	1.86384i	−0.362662	−	0.931921i	$-0.618132\pi$
$911$		−	51249.1i		−	1.86384i	0.362662	−	0.931921i	$-0.381868\pi$
$912$		0			0
$913$	−1022.77	−	590.496i	−0.0370742	−	0.0214048i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	18992.6	+	32896.2i	0.681730	+	1.18079i	0.974453	+	0.224593i	$0.0721053\pi$
$919$	18992.6	+	32896.2i	0.681730	+	1.18079i	−0.292723	+	0.956197i	$0.594561\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6002.84			−0.214069
$924$		0			0
$925$	−19224.4			−0.683347
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13009.3	+	22532.8i	0.459443	+	0.795778i	0.998932	−	0.0462144i	$-0.0147158\pi$
$929$	13009.3	+	22532.8i	0.459443	+	0.795778i	−0.539489	+	0.841993i	$0.681382\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−29927.7	−	17278.7i	−1.04678	−	0.604359i
$936$		0			0
$937$		−	44218.3i		−	1.54168i	−0.637031	−	0.770838i	$-0.719838\pi$
$937$		−	44218.3i		−	1.54168i	0.637031	−	0.770838i	$-0.280162\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3103.46	+	5375.35i	−0.107513	+	0.186218i	−0.914762	−	0.403993i	$-0.867622\pi$
$941$	−3103.46	+	5375.35i	−0.107513	+	0.186218i	0.807249	+	0.590211i	$0.200955\pi$
$942$		0			0
$943$	−11833.0	+	6831.81i	−0.408629	+	0.235922i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34277.2	−	19790.0i	1.17620	−	0.679079i	0.221067	−	0.975259i	$-0.429046\pi$
$947$	34277.2	−	19790.0i	1.17620	−	0.679079i	0.955132	+	0.296180i	$0.0957127\pi$
$948$		0			0
$949$	−240.559	+	416.660i	−0.00822853	+	0.0142522i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	23083.3i		−	0.784619i	−0.919833	−	0.392310i	$-0.871676\pi$
$953$		−	23083.3i		−	0.784619i	0.919833	−	0.392310i	$-0.128324\pi$
$954$		0			0
$955$	−27568.6	−	15916.7i	−0.934135	−	0.539323i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14868.7	−	25753.4i	−0.499102	−	0.864470i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11420.3			0.380967
$966$		0			0
$967$	−34489.9			−1.14697			−0.573486	−	0.819215i	$-0.694409\pi$
$967$	−34489.9			−1.14697			−0.573486	+	0.819215i	$0.694409\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2501.31	−	4332.39i	−0.0826681	−	0.143185i	0.821727	−	0.569881i	$-0.193011\pi$
$971$	−2501.31	−	4332.39i	−0.0826681	−	0.143185i	−0.904395	+	0.426696i	$0.859677\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41676.7	+	24062.0i	1.36474	+	0.787936i	0.990251	−	0.139294i	$-0.0444833\pi$
$977$	41676.7	+	24062.0i	1.36474	+	0.787936i	0.374493	+	0.927230i	$0.377817\pi$
$978$		0			0
$979$			14929.7i			0.487391i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10394.8	−	18004.3i	0.337276	−	0.584179i	−0.646643	−	0.762793i	$-0.723828\pi$
$983$	10394.8	−	18004.3i	0.337276	−	0.584179i	0.983919	+	0.178613i	$0.0571611\pi$
$984$		0			0
$985$	25536.2	−	14743.3i	0.826042	−	0.476916i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24132.0	+	13932.6i	−0.775888	+	0.447959i
$990$		0			0
$991$	−24701.8	+	42784.7i	−0.791803	+	1.37144i	0.133046	+	0.991110i	$0.457524\pi$
$991$	−24701.8	+	42784.7i	−0.791803	+	1.37144i	−0.924849	+	0.380334i	$0.875809\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	45583.3i		−	1.45235i
$996$		0			0
$997$	36971.1	+	21345.3i	1.17441	+	0.678047i	0.954715	−	0.297522i	$-0.0961601\pi$
$997$	36971.1	+	21345.3i	1.17441	+	0.678047i	0.219696	+	0.975568i	$0.429493\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.b.521.3		16
3.2	odd	2	inner	1764.4.t.b.521.6		16
7.2	even	3		252.4.t.a.89.3	yes	16
7.3	odd	6		1764.4.f.a.881.6		16
7.4	even	3		1764.4.f.a.881.12		16
7.5	odd	6	inner	1764.4.t.b.1097.6		16
7.6	odd	2		252.4.t.a.17.6	yes	16
21.2	odd	6		252.4.t.a.89.6	yes	16
21.5	even	6	inner	1764.4.t.b.1097.3		16
21.11	odd	6		1764.4.f.a.881.5		16
21.17	even	6		1764.4.f.a.881.11		16
21.20	even	2		252.4.t.a.17.3	✓	16
28.23	odd	6		1008.4.bt.b.593.3		16
28.27	even	2		1008.4.bt.b.17.6		16
84.23	even	6		1008.4.bt.b.593.6		16
84.83	odd	2		1008.4.bt.b.17.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.3	✓	16	21.20	even	2
252.4.t.a.17.6	yes	16	7.6	odd	2
252.4.t.a.89.3	yes	16	7.2	even	3
252.4.t.a.89.6	yes	16	21.2	odd	6
1008.4.bt.b.17.3		16	84.83	odd	2
1008.4.bt.b.17.6		16	28.27	even	2
1008.4.bt.b.593.3		16	28.23	odd	6
1008.4.bt.b.593.6		16	84.23	even	6
1764.4.f.a.881.5		16	21.11	odd	6
1764.4.f.a.881.6		16	7.3	odd	6
1764.4.f.a.881.11		16	21.17	even	6
1764.4.f.a.881.12		16	7.4	even	3
1764.4.t.b.521.3		16	1.1	even	1	trivial
1764.4.t.b.521.6		16	3.2	odd	2	inner
1764.4.t.b.1097.3		16	21.5	even	6	inner
1764.4.t.b.1097.6		16	7.5	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.27106 - 7.39769i) q^{5} +O(q^{10})\)	\(q+(-4.27106 - 7.39769i) q^{5} +(-27.4679 - 15.8586i) q^{11} +9.92568i q^{13} +(-63.7751 + 110.462i) q^{17} +(100.923 - 58.2677i) q^{19} +(55.8271 - 32.2318i) q^{23} +(26.0161 - 45.0612i) q^{25} -113.016i q^{29} +(-6.33498 - 3.65751i) q^{31} +(-184.736 - 319.972i) q^{37} -211.959 q^{41} -432.263 q^{43} +(200.021 + 346.446i) q^{47} +(-121.869 - 70.3612i) q^{53} +270.933i q^{55} +(-259.447 + 449.375i) q^{59} +(-23.5627 + 13.6039i) q^{61} +(73.4271 - 42.3932i) q^{65} +(68.3597 - 118.403i) q^{67} +604.779i q^{71} +(41.9780 + 24.2360i) q^{73} +(415.618 + 719.872i) q^{79} +37.2350 q^{83} +1089.55 q^{85} +(-235.356 - 407.649i) q^{89} +(-862.093 - 497.730i) q^{95} +522.691i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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