LMFDB - Embedded newform 5292.2.w.c.521.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(521,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5292.521");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.w (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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.12
Character	$\chi$	$=$	5292.521
Dual form			5292.2.w.c.1097.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0565510 + 0.0979493i) q^{5} +O(q^{10})$	$q+(-0.0565510 + 0.0979493i) q^{5} +(4.86969 - 2.81151i) q^{11} +(2.71286 - 1.56627i) q^{13} +(0.615632 - 1.06631i) q^{17} +(3.38980 - 1.95710i) q^{19} +(4.06316 + 2.34587i) q^{23} +(2.49360 + 4.31905i) q^{25} +(0.117222 + 0.0676783i) q^{29} -6.82790i q^{31} +(-3.39182 - 5.87480i) q^{37} +(1.27192 + 2.20303i) q^{41} +(-2.88806 + 5.00226i) q^{43} -5.32646 q^{47} +(-2.48935 - 1.43723i) q^{53} +0.635976i q^{55} +11.4458 q^{59} +0.0386207i q^{61} +0.354297i q^{65} -10.4899 q^{67} +12.4663i q^{71} +(-11.3768 - 6.56842i) q^{73} +11.2236 q^{79} +(0.344701 - 0.597040i) q^{83} +(0.0696293 + 0.120601i) q^{85} +(-5.37075 - 9.30241i) q^{89} +0.442705i q^{95} +(-5.89327 - 3.40248i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 24 q^{11} + 48 q^{23} - 24 q^{25} + 48 q^{53} + 48 q^{79} - 24 q^{85}+O(q^{100}) $ 48 * q - 24 * q^11 + 48 * q^23 - 24 * q^25 + 48 * q^53 + 48 * q^79 - 24 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	−0.0565510	+	0.0979493i	−0.0252904	+	0.0438043i	−0.878394	−	0.477938i	$-0.841384\pi$
$5$	−0.0565510	+	0.0979493i	−0.0252904	+	0.0438043i	0.853103	+	0.521742i	$0.174718\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.86969	−	2.81151i	1.46827	−	0.847703i	0.468897	−	0.883253i	$-0.344651\pi$
$11$	4.86969	−	2.81151i	1.46827	−	0.847703i	0.999368	+	0.0355493i	$0.0113181\pi$
$12$		0			0
$13$	2.71286	−	1.56627i	0.752412	−	0.434405i	−0.0741527	−	0.997247i	$-0.523625\pi$
$13$	2.71286	−	1.56627i	0.752412	−	0.434405i	0.826565	+	0.562842i	$0.190292\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.615632	−	1.06631i	0.149313	−	0.258617i	−0.781661	−	0.623704i	$-0.785627\pi$
$17$	0.615632	−	1.06631i	0.149313	−	0.258617i	0.930974	+	0.365086i	$0.118961\pi$
$18$		0			0
$19$	3.38980	−	1.95710i	0.777673	−	0.448990i	−0.0579318	−	0.998321i	$-0.518451\pi$
$19$	3.38980	−	1.95710i	0.777673	−	0.448990i	0.835605	+	0.549331i	$0.185117\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.06316	+	2.34587i	0.847227	+	0.489147i	0.859714	−	0.510775i	$-0.170642\pi$
$23$	4.06316	+	2.34587i	0.847227	+	0.489147i	−0.0124873	+	0.999922i	$0.503975\pi$
$24$		0			0
$25$	2.49360	+	4.31905i	0.498721	+	0.863810i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.117222	+	0.0676783i	0.0217676	+	0.0125675i	0.510844	−	0.859673i	$-0.329333\pi$
$29$	0.117222	+	0.0676783i	0.0217676	+	0.0125675i	−0.489077	+	0.872241i	$0.662666\pi$
$30$		0			0
$31$		−	6.82790i		−	1.22633i	−0.789956	−	0.613164i	$-0.789897\pi$
$31$		−	6.82790i		−	1.22633i	0.789956	−	0.613164i	$-0.210103\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.39182	−	5.87480i	−0.557611	−	0.965811i	−0.997695	−	0.0678544i	$-0.978385\pi$
$37$	−3.39182	−	5.87480i	−0.557611	−	0.965811i	0.440084	−	0.897957i	$-0.354949\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.27192	+	2.20303i	0.198640	+	0.344055i	0.948088	−	0.318009i	$-0.103014\pi$
$41$	1.27192	+	2.20303i	0.198640	+	0.344055i	−0.749448	+	0.662064i	$0.769681\pi$
$42$		0			0
$43$	−2.88806	+	5.00226i	−0.440425	+	0.762838i	−0.997721	−	0.0674759i	$-0.978505\pi$
$43$	−2.88806	+	5.00226i	−0.440425	+	0.762838i	0.557296	+	0.830314i	$0.311839\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.32646			−0.776944			−0.388472	−	0.921461i	$-0.626997\pi$
$47$	−5.32646			−0.776944			−0.388472	+	0.921461i	$0.626997\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.48935	−	1.43723i	−0.341939	−	0.197419i	0.319190	−	0.947691i	$-0.396589\pi$
$53$	−2.48935	−	1.43723i	−0.341939	−	0.197419i	−0.661129	+	0.750272i	$0.729922\pi$
$54$		0			0
$55$			0.635976i			0.0857550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11.4458			1.49012			0.745058	−	0.666999i	$-0.232422\pi$
$59$	11.4458			1.49012			0.745058	+	0.666999i	$0.232422\pi$
$60$		0			0
$61$			0.0386207i			0.00494488i	0.999997	+	0.00247244i	$0.000787002\pi$
$61$			0.0386207i			0.00494488i	−0.999997	+	0.00247244i	$0.999213\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.354297i			0.0439451i
$66$		0			0
$67$	−10.4899			−1.28155			−0.640774	−	0.767730i	$-0.721386\pi$
$67$	−10.4899			−1.28155			−0.640774	+	0.767730i	$0.721386\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.4663i			1.47948i	0.672894	+	0.739739i	$0.265051\pi$
$71$			12.4663i			1.47948i	−0.672894	+	0.739739i	$0.734949\pi$
$72$		0			0
$73$	−11.3768	−	6.56842i	−1.33156	−	0.768775i	−0.346019	−	0.938227i	$-0.612467\pi$
$73$	−11.3768	−	6.56842i	−1.33156	−	0.768775i	−0.985538	+	0.169452i	$0.945800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	11.2236			1.26275			0.631377	−	0.775476i	$-0.282490\pi$
$79$	11.2236			1.26275			0.631377	+	0.775476i	$0.282490\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.344701	−	0.597040i	0.0378359	−	0.0655337i	−0.846487	−	0.532409i	$-0.821287\pi$
$83$	0.344701	−	0.597040i	0.0378359	−	0.0655337i	0.884323	+	0.466875i	$0.154620\pi$
$84$		0			0
$85$	0.0696293	+	0.120601i	0.00755235	+	0.0130811i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.37075	−	9.30241i	−0.569298	−	0.986053i	−0.996636	−	0.0819609i	$-0.973882\pi$
$89$	−5.37075	−	9.30241i	−0.569298	−	0.986053i	0.427338	−	0.904092i	$-0.359452\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.442705i			0.0454205i
$96$		0			0
$97$	−5.89327	−	3.40248i	−0.598371	−	0.345470i	0.170029	−	0.985439i	$-0.445614\pi$
$97$	−5.89327	−	3.40248i	−0.598371	−	0.345470i	−0.768401	+	0.639969i	$0.778947\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.14757	+	14.1120i	0.810713	+	1.40420i	0.912365	+	0.409377i	$0.134254\pi$
$101$	8.14757	+	14.1120i	0.810713	+	1.40420i	−0.101652	+	0.994820i	$0.532413\pi$
$102$		0			0
$103$	−15.3680	−	8.87273i	−1.51426	−	0.874256i	−0.999860	−	0.0167032i	$-0.994683\pi$
$103$	−15.3680	−	8.87273i	−1.51426	−	0.874256i	−0.514396	−	0.857553i	$-0.671984\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.2674	−	6.50525i	1.08926	−	0.628887i	0.155883	−	0.987775i	$-0.450178\pi$
$107$	11.2674	−	6.50525i	1.08926	−	0.628887i	0.933380	+	0.358889i	$0.116844\pi$
$108$		0			0
$109$	−0.685898	+	1.18801i	−0.0656971	+	0.113791i	−0.897003	−	0.442024i	$-0.854260\pi$
$109$	−0.685898	+	1.18801i	−0.0656971	+	0.113791i	0.831306	+	0.555815i	$0.187594\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.75422	−	4.47690i	0.729455	−	0.421151i	−0.0887675	−	0.996052i	$-0.528293\pi$
$113$	7.75422	−	4.47690i	0.729455	−	0.421151i	0.818223	+	0.574901i	$0.194959\pi$
$114$		0			0
$115$	−0.459552	+	0.265322i	−0.0428534	+	0.0247414i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.3092	−	17.8561i	0.937202	−	1.62328i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.12957			−0.101032
$126$		0			0
$127$	−7.21996			−0.640668			−0.320334	−	0.947305i	$-0.603795\pi$
$127$	−7.21996			−0.640668			−0.320334	+	0.947305i	$0.603795\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.16622	−	7.21611i	0.364005	−	0.630474i	−0.624611	−	0.780936i	$-0.714743\pi$
$131$	4.16622	−	7.21611i	0.364005	−	0.630474i	0.988616	+	0.150461i	$0.0480759\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.10942	+	4.10463i	−0.607399	+	0.350682i	−0.771947	−	0.635687i	$-0.780717\pi$
$137$	−7.10942	+	4.10463i	−0.607399	+	0.350682i	0.164548	+	0.986369i	$0.447384\pi$
$138$		0			0
$139$	3.19880	−	1.84683i	0.271318	−	0.156646i	−0.358168	−	0.933657i	$-0.616599\pi$
$139$	3.19880	−	1.84683i	0.271318	−	0.156646i	0.629487	+	0.777011i	$0.283265\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.80718	−	15.2545i	0.736494	−	1.27564i
$144$		0			0
$145$	−0.0132581	+	0.00765456i	−0.00110102	+	0.000635677i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	20.4407	+	11.8014i	1.67456	+	0.966810i	0.965031	+	0.262137i	$0.0844273\pi$
$149$	20.4407	+	11.8014i	1.67456	+	0.966810i	0.709533	+	0.704672i	$0.248906\pi$
$150$		0			0
$151$	−2.19194	−	3.79656i	−0.178378	−	0.308959i	0.762947	−	0.646461i	$-0.223752\pi$
$151$	−2.19194	−	3.79656i	−0.178378	−	0.308959i	−0.941325	+	0.337501i	$0.890418\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.668788	+	0.386125i	0.0537184	+	0.0310143i
$156$		0			0
$157$		−	14.1053i		−	1.12572i	−0.826551	−	0.562861i	$-0.809700\pi$
$157$		−	14.1053i		−	1.12572i	0.826551	−	0.562861i	$-0.190300\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.80340	−	11.7838i	−0.532884	−	0.922982i	−0.999263	−	0.0383965i	$-0.987775\pi$
$163$	−6.80340	−	11.7838i	−0.532884	−	0.922982i	0.466379	−	0.884585i	$-0.345558\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.49355	+	14.7113i	0.657251	+	1.13839i	0.981324	+	0.192360i	$0.0616141\pi$
$167$	8.49355	+	14.7113i	0.657251	+	1.13839i	−0.324074	+	0.946032i	$0.605053\pi$
$168$		0			0
$169$	−1.59359	+	2.76018i	−0.122584	+	0.212322i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.77136			0.134674			0.0673371	−	0.997730i	$-0.478550\pi$
$173$	1.77136			0.134674			0.0673371	+	0.997730i	$0.478550\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.13609	+	4.12002i	0.533376	+	0.307945i	0.742390	−	0.669968i	$-0.233692\pi$
$179$	7.13609	+	4.12002i	0.533376	+	0.307945i	−0.209014	+	0.977913i	$0.567025\pi$
$180$		0			0
$181$			0.934986i			0.0694970i	0.999396	+	0.0347485i	$0.0110630\pi$
$181$			0.934986i			0.0694970i	−0.999396	+	0.0347485i	$0.988937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.767243			0.0564088
$186$		0			0
$187$		−	6.92343i		−	0.506291i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.2825i		−	0.816377i	−0.912898	−	0.408188i	$-0.866161\pi$
$191$		−	11.2825i		−	0.816377i	0.912898	−	0.408188i	$-0.133839\pi$
$192$		0			0
$193$	20.9886			1.51079			0.755396	−	0.655268i	$-0.227444\pi$
$193$	20.9886			1.51079			0.755396	+	0.655268i	$0.227444\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.2865i			1.23161i	0.787898	+	0.615805i	$0.211169\pi$
$197$			17.2865i			1.23161i	−0.787898	+	0.615805i	$0.788831\pi$
$198$		0			0
$199$	−4.50187	−	2.59915i	−0.319129	−	0.184249i	0.331875	−	0.943323i	$-0.392319\pi$
$199$	−4.50187	−	2.59915i	−0.319129	−	0.184249i	−0.651004	+	0.759074i	$0.725652\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.287713			−0.0200948
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.0048	−	19.0609i	0.761220	−	1.31847i
$210$		0			0
$211$	12.6664	+	21.9388i	0.871990	+	1.51033i	0.859935	+	0.510404i	$0.170504\pi$
$211$	12.6664	+	21.9388i	0.871990	+	1.51033i	0.0120556	+	0.999927i	$0.496162\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.326645	−	0.565766i	−0.0222770	−	0.0385849i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3.85698i		−	0.259449i
$222$		0			0
$223$	23.0004	+	13.2793i	1.54022	+	0.889247i	0.998824	+	0.0484802i	$0.0154378\pi$
$223$	23.0004	+	13.2793i	1.54022	+	0.889247i	0.541397	+	0.840767i	$0.317896\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.15231	−	1.99585i	−0.0764813	−	0.132469i	0.825248	−	0.564770i	$-0.191035\pi$
$227$	−1.15231	−	1.99585i	−0.0764813	−	0.132469i	−0.901729	+	0.432301i	$0.857702\pi$
$228$		0			0
$229$	1.32361	+	0.764188i	0.0874668	+	0.0504990i	0.543095	−	0.839671i	$-0.317252\pi$
$229$	1.32361	+	0.764188i	0.0874668	+	0.0504990i	−0.455629	+	0.890170i	$0.650586\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.60738	+	0.928021i	−0.105303	+	0.0607966i	−0.551727	−	0.834025i	$-0.686031\pi$
$233$	−1.60738	+	0.928021i	−0.105303	+	0.0607966i	0.446424	+	0.894822i	$0.352697\pi$
$234$		0			0
$235$	0.301217	−	0.521723i	0.0196492	−	0.0340335i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.5014	+	15.3006i	−1.71423	+	0.989712i	−0.785578	+	0.618763i	$0.787634\pi$
$239$	−26.5014	+	15.3006i	−1.71423	+	0.989712i	−0.928653	+	0.370949i	$0.879032\pi$
$240$		0			0
$241$	20.5380	−	11.8576i	1.32297	−	0.763817i	0.338769	−	0.940870i	$-0.389989\pi$
$241$	20.5380	−	11.8576i	1.32297	−	0.763817i	0.984201	+	0.177052i	$0.0566561\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.13070	−	10.6187i	0.390087	−	0.675651i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.5555			−0.918736			−0.459368	−	0.888246i	$-0.651924\pi$
$251$	−14.5555			−0.918736			−0.459368	+	0.888246i	$0.651924\pi$
$252$		0			0
$253$	26.3817			1.65861
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.23554	−	15.9964i	0.576097	−	0.997829i	−0.419824	−	0.907605i	$-0.637908\pi$
$257$	9.23554	−	15.9964i	0.576097	−	0.997829i	0.995921	−	0.0902241i	$-0.0287583\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.3889	−	11.7716i	1.25724	−	0.725865i	0.284699	−	0.958617i	$-0.408106\pi$
$263$	20.3889	−	11.7716i	1.25724	−	0.725865i	0.972536	+	0.232751i	$0.0747729\pi$
$264$		0			0
$265$	0.281551	−	0.162554i	0.0172955	−	0.00998559i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.1399	−	26.2232i	0.923099	−	1.59885i	0.128509	−	0.991708i	$-0.458981\pi$
$269$	15.1399	−	26.2232i	0.923099	−	1.59885i	0.794590	−	0.607146i	$-0.207686\pi$
$270$		0			0
$271$	−24.9128	+	14.3834i	−1.51334	+	0.873730i	−0.513467	+	0.858110i	$0.671639\pi$
$271$	−24.9128	+	14.3834i	−1.51334	+	0.873730i	−0.999878	+	0.0156202i	$0.995028\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24.2861	+	14.0216i	1.46451	+	0.845535i
$276$		0			0
$277$	2.50103	+	4.33191i	0.150272	+	0.260279i	0.931328	−	0.364183i	$-0.118652\pi$
$277$	2.50103	+	4.33191i	0.150272	+	0.260279i	−0.781055	+	0.624462i	$0.785318\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.4362	−	6.60270i	−0.682227	−	0.393884i	0.118467	−	0.992958i	$-0.462202\pi$
$281$	−11.4362	−	6.60270i	−0.682227	−	0.393884i	−0.800694	+	0.599074i	$0.795535\pi$
$282$		0			0
$283$		−	11.3492i		−	0.674641i	−0.941390	−	0.337321i	$-0.890479\pi$
$283$		−	11.3492i		−	0.674641i	0.941390	−	0.337321i	$-0.109521\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.74199	+	13.4095i	0.455411	+	0.788796i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.62606	+	8.01258i	0.270258	+	0.468100i	0.968928	−	0.247344i	$-0.0795578\pi$
$293$	4.62606	+	8.01258i	0.270258	+	0.468100i	−0.698670	+	0.715444i	$0.746224\pi$
$294$		0			0
$295$	−0.647272	+	1.12111i	−0.0376857	+	0.0652735i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.6970			0.849952
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.00378287	−	0.00218404i	−0.000216607	−	0.000125058i
$306$		0			0
$307$		−	25.1926i		−	1.43782i	−0.695104	−	0.718909i	$-0.744642\pi$
$307$		−	25.1926i		−	1.43782i	0.695104	−	0.718909i	$-0.255358\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−23.1865			−1.31478			−0.657392	−	0.753548i	$-0.728341\pi$
$311$	−23.1865			−1.31478			−0.657392	+	0.753548i	$0.728341\pi$
$312$		0			0
$313$		−	23.8472i		−	1.34793i	−0.738765	−	0.673963i	$-0.764591\pi$
$313$		−	23.8472i		−	1.34793i	0.738765	−	0.673963i	$-0.235409\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10.3450i		−	0.581036i	−0.956870	−	0.290518i	$-0.906172\pi$
$317$		−	10.3450i		−	0.581036i	0.956870	−	0.290518i	$-0.0938275\pi$
$318$		0			0
$319$	0.761114			0.0426142
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.81942i		−	0.268159i
$324$		0			0
$325$	13.5296	+	7.81132i	0.750487	+	0.433294i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.28956			−0.400671			−0.200335	−	0.979727i	$-0.564203\pi$
$331$	−7.28956			−0.400671			−0.200335	+	0.979727i	$0.564203\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.593216	−	1.02748i	0.0324109	−	0.0561373i
$336$		0			0
$337$	7.67533	+	13.2941i	0.418102	+	0.724173i	0.995749	−	0.0921135i	$-0.0293623\pi$
$337$	7.67533	+	13.2941i	0.418102	+	0.724173i	−0.577647	+	0.816287i	$0.696029\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−19.1967	−	33.2497i	−1.03956	−	1.80057i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			18.3604i			0.985639i	0.870132	+	0.492819i	$0.164034\pi$
$347$			18.3604i			0.985639i	−0.870132	+	0.492819i	$0.835966\pi$
$348$		0			0
$349$	−14.6555	−	8.46135i	−0.784490	−	0.452925i	0.0535293	−	0.998566i	$-0.482953\pi$
$349$	−14.6555	−	8.46135i	−0.784490	−	0.452925i	−0.838019	+	0.545641i	$0.816286\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.1058	−	19.2358i	−0.591101	−	1.02382i	−0.994084	−	0.108610i	$-0.965360\pi$
$353$	−11.1058	−	19.2358i	−0.591101	−	1.02382i	0.402983	−	0.915207i	$-0.367973\pi$
$354$		0			0
$355$	−1.22107	−	0.704982i	−0.0648074	−	0.0374166i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.26853	−	2.46444i	0.225285	−	0.130068i	−0.383110	−	0.923703i	$-0.625147\pi$
$359$	4.26853	−	2.46444i	0.225285	−	0.130068i	0.608395	+	0.793635i	$0.291814\pi$
$360$		0			0
$361$	−1.83951	+	3.18612i	−0.0968162	+	0.167691i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.28674	−	0.742902i	0.0673512	−	0.0388853i
$366$		0			0
$367$	−6.85336	+	3.95679i	−0.357743	+	0.206543i	−0.668090	−	0.744080i	$-0.732888\pi$
$367$	−6.85336	+	3.95679i	−0.357743	+	0.206543i	0.310347	+	0.950623i	$0.399555\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.1651	−	27.9989i	0.837000	−	1.44973i	−0.0553923	−	0.998465i	$-0.517641\pi$
$373$	16.1651	−	27.9989i	0.837000	−	1.44973i	0.892392	−	0.451261i	$-0.149026\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.424010			0.0218376
$378$		0			0
$379$	9.46110			0.485984			0.242992	−	0.970028i	$-0.421871\pi$
$379$	9.46110			0.485984			0.242992	+	0.970028i	$0.421871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.71048	+	13.3549i	−0.393987	+	0.682406i	−0.992971	−	0.118355i	$-0.962238\pi$
$383$	−7.71048	+	13.3549i	−0.393987	+	0.682406i	0.598984	+	0.800761i	$0.295571\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.7483	+	17.7525i	−1.55900	+	0.900089i	−0.561647	+	0.827377i	$0.689832\pi$
$389$	−30.7483	+	17.7525i	−1.55900	+	0.900089i	−0.997353	+	0.0727122i	$0.976835\pi$
$390$		0			0
$391$	5.00282	−	2.88838i	0.253003	−	0.146072i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.634707	+	1.09934i	−0.0319356	+	0.0553140i
$396$		0			0
$397$	−14.5897	+	8.42339i	−0.732238	+	0.422758i	−0.819240	−	0.573450i	$-0.805605\pi$
$397$	−14.5897	+	8.42339i	−0.732238	+	0.422758i	0.0870024	+	0.996208i	$0.472271\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.48987	+	3.16958i	0.274151	+	0.158281i	0.630772	−	0.775968i	$-0.282738\pi$
$401$	5.48987	+	3.16958i	0.274151	+	0.158281i	−0.356622	+	0.934249i	$0.616071\pi$
$402$		0			0
$403$	−10.6943	−	18.5231i	−0.532723	−	0.922704i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−33.0342	−	19.0723i	−1.63744	−	0.945378i
$408$		0			0
$409$			17.3680i			0.858792i	0.903116	+	0.429396i	$0.141274\pi$
$409$			17.3680i			0.858792i	−0.903116	+	0.429396i	$0.858726\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.0389864	+	0.0675265i	0.00191377	+	0.00331474i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.0242	+	31.2189i	0.880542	+	1.52514i	0.850740	+	0.525587i	$0.176154\pi$
$419$	18.0242	+	31.2189i	0.880542	+	1.52514i	0.0298018	+	0.999556i	$0.490512\pi$
$420$		0			0
$421$	4.22463	−	7.31727i	0.205896	−	0.356622i	−0.744522	−	0.667598i	$-0.767323\pi$
$421$	4.22463	−	7.31727i	0.205896	−	0.356622i	0.950418	+	0.310976i	$0.100656\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.14057			0.297861
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−26.6040	−	15.3598i	−1.28147	−	0.739856i	−0.304351	−	0.952560i	$-0.598440\pi$
$431$	−26.6040	−	15.3598i	−1.28147	−	0.739856i	−0.977117	+	0.212704i	$0.931773\pi$
$432$		0			0
$433$			9.85040i			0.473380i	0.971585	+	0.236690i	$0.0760626\pi$
$433$			9.85040i			0.473380i	−0.971585	+	0.236690i	$0.923937\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.3644			0.878488
$438$		0			0
$439$			9.51745i			0.454243i	0.973866	+	0.227121i	$0.0729314\pi$
$439$			9.51745i			0.454243i	−0.973866	+	0.227121i	$0.927069\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	19.1399i		−	0.909364i	−0.890654	−	0.454682i	$-0.849753\pi$
$443$		−	19.1399i		−	0.909364i	0.890654	−	0.454682i	$-0.150247\pi$
$444$		0			0
$445$	1.21489			0.0575911
$446$		0			0
$447$		0			0
$448$		0			0
$449$			34.8874i			1.64644i	0.567724	+	0.823219i	$0.307824\pi$
$449$			34.8874i			1.64644i	−0.567724	+	0.823219i	$0.692176\pi$
$450$		0			0
$451$	12.3877	+	7.15203i	0.583313	+	0.336776i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	32.4540			1.51813			0.759067	−	0.651013i	$-0.225656\pi$
$457$	32.4540			1.51813			0.759067	+	0.651013i	$0.225656\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.1696	−	24.5425i	0.659946	−	1.14306i	−0.320683	−	0.947186i	$-0.603913\pi$
$461$	14.1696	−	24.5425i	0.659946	−	1.14306i	0.980629	−	0.195873i	$-0.0627541\pi$
$462$		0			0
$463$	2.11604	+	3.66508i	0.0983405	+	0.170331i	0.910998	−	0.412411i	$-0.135313\pi$
$463$	2.11604	+	3.66508i	0.0983405	+	0.170331i	−0.812657	+	0.582742i	$0.801980\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.61042	+	9.71752i	0.259619	+	0.449673i	0.966140	−	0.258019i	$-0.0830697\pi$
$467$	5.61042	+	9.71752i	0.259619	+	0.449673i	−0.706521	+	0.707692i	$0.749736\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			32.4793i			1.49340i
$474$		0			0
$475$	16.9056	+	9.76047i	0.775684	+	0.447841i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.8043	+	30.8379i	0.813498	+	1.40902i	0.910401	+	0.413726i	$0.135773\pi$
$479$	17.8043	+	30.8379i	0.813498	+	1.40902i	−0.0969034	+	0.995294i	$0.530894\pi$
$480$		0			0
$481$	−18.4030	−	10.6250i	−0.839107	−	0.484459i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.666542	−	0.384828i	0.0302661	−	0.0174741i
$486$		0			0
$487$	12.3856	−	21.4525i	0.561244	−	0.972104i	−0.436144	−	0.899877i	$-0.643656\pi$
$487$	12.3856	−	21.4525i	0.561244	−	0.972104i	0.997388	−	0.0722269i	$-0.0230106\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.4482	−	8.34166i	0.652037	−	0.376454i	−0.137199	−	0.990543i	$-0.543810\pi$
$491$	14.4482	−	8.34166i	0.652037	−	0.376454i	0.789236	+	0.614090i	$0.210477\pi$
$492$		0			0
$493$	0.144332	−	0.0833299i	0.00650037	−	0.00375299i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8.58644	+	14.8722i	−0.384382	+	0.665769i	−0.991683	−	0.128702i	$-0.958919\pi$
$499$	−8.58644	+	14.8722i	−0.384382	+	0.665769i	0.607301	+	0.794472i	$0.292252\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−38.7122			−1.72609			−0.863045	−	0.505127i	$-0.831446\pi$
$503$	−38.7122			−1.72609			−0.863045	+	0.505127i	$0.831446\pi$
$504$		0			0
$505$	−1.84301			−0.0820131
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.4933	−	25.1032i	0.642406	−	1.11268i	−0.342488	−	0.939522i	$-0.611270\pi$
$509$	14.4933	−	25.1032i	0.642406	−	1.11268i	0.984894	−	0.173158i	$-0.0553970\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.73816	−	1.00352i	0.0765923	−	0.0442206i
$516$		0			0
$517$	−25.9382	+	14.9754i	−1.14076	+	0.658618i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.43678	−	5.95267i	0.150568	−	0.260791i	−0.780868	−	0.624696i	$-0.785223\pi$
$521$	3.43678	−	5.95267i	0.150568	−	0.260791i	0.931436	+	0.363904i	$0.118556\pi$
$522$		0			0
$523$	−16.8198	+	9.71091i	−0.735478	+	0.424628i	−0.820423	−	0.571757i	$-0.806262\pi$
$523$	−16.8198	+	9.71091i	−0.735478	+	0.424628i	0.0849449	+	0.996386i	$0.472929\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.28063	−	4.20348i	−0.317149	−	0.183106i
$528$		0			0
$529$	−0.493834	−	0.855346i	−0.0214711	−	0.0371890i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.90107	+	3.98434i	0.298919	+	0.172581i
$534$		0			0
$535$			1.47152i			0.0636192i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.62684	−	16.6742i	−0.413890	−	0.716879i	0.581421	−	0.813603i	$-0.302497\pi$
$541$	−9.62684	−	16.6742i	−0.413890	−	0.716879i	−0.995311	+	0.0967242i	$0.969164\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.0775765	−	0.134366i	−0.00332301	−	0.00575563i
$546$		0			0
$547$	−9.03080	+	15.6418i	−0.386129	+	0.668795i	−0.991925	−	0.126825i	$-0.959521\pi$
$547$	−9.03080	+	15.6418i	−0.386129	+	0.668795i	0.605796	+	0.795620i	$0.292855\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.529813			0.0225708
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.79413	+	2.19054i	0.160763	+	0.0928163i	0.578223	−	0.815879i	$-0.303746\pi$
$557$	3.79413	+	2.19054i	0.160763	+	0.0928163i	−0.417460	+	0.908695i	$0.637080\pi$
$558$		0			0
$559$			18.0939i			0.765291i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−43.8972			−1.85005			−0.925024	−	0.379909i	$-0.875955\pi$
$563$	−43.8972			−1.85005			−0.925024	+	0.379909i	$0.875955\pi$
$564$		0			0
$565$			1.01269i			0.0426043i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4.58113i		−	0.192051i	−0.995379	−	0.0960254i	$-0.969387\pi$
$569$		−	4.58113i		−	0.192051i	0.995379	−	0.0960254i	$-0.0306130\pi$
$570$		0			0
$571$	3.16869			0.132606			0.0663029	−	0.997800i	$-0.478880\pi$
$571$	3.16869			0.132606			0.0663029	+	0.997800i	$0.478880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			23.3986i			0.975790i
$576$		0			0
$577$	−10.4992	−	6.06171i	−0.437087	−	0.252352i	0.265274	−	0.964173i	$-0.414538\pi$
$577$	−10.4992	−	6.06171i	−0.437087	−	0.252352i	−0.702361	+	0.711821i	$0.747871\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−16.1632			−0.669409
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.40737	−	5.90173i	0.140637	−	0.243591i	−0.787100	−	0.616826i	$-0.788418\pi$
$587$	3.40737	−	5.90173i	0.140637	−	0.243591i	0.927737	+	0.373235i	$0.121752\pi$
$588$		0			0
$589$	−13.3629	−	23.1452i	−0.550609	−	0.953682i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.0992	+	26.1526i	0.620050	+	1.07396i	0.989476	+	0.144697i	$0.0462209\pi$
$593$	15.0992	+	26.1526i	0.620050	+	1.07396i	−0.369426	+	0.929260i	$0.620446\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	17.6120i		−	0.719607i	−0.933028	−	0.359803i	$-0.882844\pi$
$599$		−	17.6120i		−	0.719607i	0.933028	−	0.359803i	$-0.117156\pi$
$600$		0			0
$601$	−27.0004	−	15.5887i	−1.10137	−	0.635875i	−0.164788	−	0.986329i	$-0.552694\pi$
$601$	−27.0004	−	15.5887i	−1.10137	−	0.635875i	−0.936580	+	0.350454i	$0.886027\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.16599	+	2.01956i	0.0474044	+	0.0821069i
$606$		0			0
$607$	−0.765446	−	0.441931i	−0.0310685	−	0.0179374i	0.484385	−	0.874855i	$-0.339043\pi$
$607$	−0.765446	−	0.441931i	−0.0310685	−	0.0179374i	−0.515454	+	0.856917i	$0.672377\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.4499	+	8.34268i	−0.584582	+	0.337509i
$612$		0			0
$613$	0.112197	−	0.194331i	0.00453159	−	0.00784895i	−0.863751	−	0.503919i	$-0.831891\pi$
$613$	0.112197	−	0.194331i	0.00453159	−	0.00784895i	0.868282	+	0.496070i	$0.165224\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	41.3032	−	23.8464i	1.66280	−	0.960020i	0.691436	−	0.722438i	$-0.256978\pi$
$617$	41.3032	−	23.8464i	1.66280	−	0.960020i	0.971367	−	0.237583i	$-0.0763550\pi$
$618$		0			0
$619$	10.1722	−	5.87292i	0.408855	−	0.236053i	−0.281442	−	0.959578i	$-0.590813\pi$
$619$	10.1722	−	5.87292i	0.408855	−	0.236053i	0.690298	+	0.723525i	$0.257480\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.4041	+	21.4846i	−0.496166	+	0.859384i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.35244			−0.333034
$630$		0			0
$631$	−17.4415			−0.694335			−0.347168	−	0.937803i	$-0.612857\pi$
$631$	−17.4415			−0.694335			−0.347168	+	0.937803i	$0.612857\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.408297	−	0.707190i	0.0162028	−	0.0280640i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	27.9439	−	16.1334i	1.10372	−	0.637231i	0.166522	−	0.986038i	$-0.446746\pi$
$641$	27.9439	−	16.1334i	1.10372	−	0.637231i	0.937195	+	0.348806i	$0.113413\pi$
$642$		0			0
$643$	23.4858	−	13.5595i	0.926191	−	0.534736i	0.0405858	−	0.999176i	$-0.487078\pi$
$643$	23.4858	−	13.5595i	0.926191	−	0.534736i	0.885605	+	0.464440i	$0.153744\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.2588	−	35.0893i	0.796456	−	1.37950i	−0.125454	−	0.992099i	$-0.540039\pi$
$647$	20.2588	−	35.0893i	0.796456	−	1.37950i	0.921910	−	0.387403i	$-0.126628\pi$
$648$		0			0
$649$	55.7375	−	32.1800i	2.18789	−	1.26318i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.9550	+	10.3663i	0.702633	+	0.405665i	0.808327	−	0.588734i	$-0.200373\pi$
$653$	17.9550	+	10.3663i	0.702633	+	0.405665i	−0.105695	+	0.994399i	$0.533707\pi$
$654$		0			0
$655$	0.471209	+	0.816157i	0.0184116	+	0.0318899i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.2785	−	8.82104i	−0.595165	−	0.343619i	0.171972	−	0.985102i	$-0.444986\pi$
$659$	−15.2785	−	8.82104i	−0.595165	−	0.343619i	−0.767137	+	0.641483i	$0.778319\pi$
$660$		0			0
$661$		−	4.63072i		−	0.180114i	−0.995937	−	0.0900571i	$-0.971295\pi$
$661$		−	4.63072i		−	0.180114i	0.995937	−	0.0900571i	$-0.0287049\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.317528	+	0.549975i	0.0122947	+	0.0212951i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.108583	+	0.188071i	0.00419179	+	0.00726039i
$672$		0			0
$673$	7.90990	−	13.7004i	0.304904	−	0.528110i	−0.672336	−	0.740246i	$-0.734709\pi$
$673$	7.90990	−	13.7004i	0.304904	−	0.528110i	0.977240	+	0.212137i	$0.0680422\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.2672			0.471468			0.235734	−	0.971818i	$-0.424251\pi$
$677$	12.2672			0.471468			0.235734	+	0.971818i	$0.424251\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.56676	+	3.79132i	0.251270	+	0.145071i	0.620346	−	0.784329i	$-0.286992\pi$
$683$	6.56676	+	3.79132i	0.251270	+	0.145071i	−0.369076	+	0.929399i	$0.620326\pi$
$684$		0			0
$685$		−	0.928484i		−	0.0354756i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.00435			−0.343039
$690$		0			0
$691$			28.9241i			1.10032i	0.835058	+	0.550162i	$0.185434\pi$
$691$			28.9241i			1.10032i	−0.835058	+	0.550162i	$0.814566\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.417760i			0.0158465i
$696$		0			0
$697$	3.13213			0.118638
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	14.8034i		−	0.559118i	−0.960128	−	0.279559i	$-0.909812\pi$
$701$		−	14.8034i		−	0.559118i	0.960128	−	0.279559i	$-0.0901883\pi$
$702$		0			0
$703$	−22.9952	−	13.2763i	−0.867279	−	0.500724i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.85894			−0.332704			−0.166352	−	0.986066i	$-0.553199\pi$
$709$	−8.85894			−0.332704			−0.166352	+	0.986066i	$0.553199\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16.0173	−	27.7429i	0.599854	−	1.03898i
$714$		0			0
$715$	0.996111	+	1.72531i	0.0372524	+	0.0645231i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.59102	+	6.21983i	0.133923	+	0.231961i	0.925185	−	0.379516i	$-0.123909\pi$
$719$	3.59102	+	6.21983i	0.133923	+	0.231961i	−0.791263	+	0.611476i	$0.790576\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$			0.675052i			0.0250708i
$726$		0			0
$727$	9.90439	+	5.71830i	0.367333	+	0.212080i	0.672293	−	0.740285i	$-0.265310\pi$
$727$	9.90439	+	5.71830i	0.367333	+	0.212080i	−0.304959	+	0.952365i	$0.598643\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.55596	+	6.15910i	0.131522	+	0.227803i
$732$		0			0
$733$	11.5803	+	6.68589i	0.427728	+	0.246949i	0.698378	−	0.715729i	$-0.253905\pi$
$733$	11.5803	+	6.68589i	0.427728	+	0.246949i	−0.270650	+	0.962678i	$0.587239\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−51.0826	+	29.4926i	−1.88165	+	1.08637i
$738$		0			0
$739$	−8.75603	+	15.1659i	−0.322096	+	0.557886i	−0.980920	−	0.194410i	$-0.937721\pi$
$739$	−8.75603	+	15.1659i	−0.322096	+	0.557886i	0.658824	+	0.752297i	$0.271054\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.36210	+	4.25051i	−0.270089	+	0.155936i	−0.628928	−	0.777463i	$-0.716506\pi$
$743$	−7.36210	+	4.25051i	−0.270089	+	0.155936i	0.358839	+	0.933399i	$0.383173\pi$
$744$		0			0
$745$	−2.31188	+	1.33476i	−0.0847008	+	0.0489020i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11.3608	+	19.6774i	−0.414560	+	0.718038i	−0.995382	−	0.0959918i	$-0.969398\pi$
$751$	−11.3608	+	19.6774i	−0.414560	+	0.718038i	0.580822	+	0.814030i	$0.302731\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.495827			0.0180450
$756$		0			0
$757$	2.31506			0.0841425			0.0420712	−	0.999115i	$-0.486604\pi$
$757$	2.31506			0.0841425			0.0420712	+	0.999115i	$0.486604\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.5656	+	23.4963i	−0.491752	+	0.851739i	−0.999955	−	0.00949790i	$-0.996977\pi$
$761$	−13.5656	+	23.4963i	−0.491752	+	0.851739i	0.508203	+	0.861237i	$0.330310\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31.0509	−	17.9272i	1.12118	−	0.647315i
$768$		0			0
$769$	19.1716	−	11.0687i	0.691345	−	0.399148i	−0.112771	−	0.993621i	$-0.535973\pi$
$769$	19.1716	−	11.0687i	0.691345	−	0.399148i	0.804116	+	0.594473i	$0.202639\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.2570	+	42.0143i	−0.872462	+	1.51115i	−0.0130206	+	0.999915i	$0.504145\pi$
$773$	−24.2570	+	42.0143i	−0.872462	+	1.51115i	−0.859442	+	0.511234i	$0.829189\pi$
$774$		0			0
$775$	29.4901	−	17.0261i	1.05931	−	0.611595i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.62310	+	4.97855i	0.308954	+	0.178375i
$780$		0			0
$781$	35.0492	+	60.7070i	1.25416	+	2.17227i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.38160	+	0.797668i	0.0493115	+	0.0284700i
$786$		0			0
$787$			4.33003i			0.154349i	0.997018	+	0.0771744i	$0.0245898\pi$
$787$			4.33003i			0.154349i	−0.997018	+	0.0771744i	$0.975410\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0604905	+	0.104773i	0.00214808	+	0.00372059i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.401963	+	0.696221i	0.0142383	+	0.0246614i	0.873057	−	0.487619i	$-0.162134\pi$
$797$	0.401963	+	0.696221i	0.0142383	+	0.0246614i	−0.858818	+	0.512280i	$0.828801\pi$
$798$		0			0
$799$	−3.27914	+	5.67963i	−0.116008	+	0.200931i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−73.8688			−2.60677
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.89142	−	1.66936i	−0.101657	−	0.0586916i	0.448310	−	0.893878i	$-0.352026\pi$
$809$	−2.89142	−	1.66936i	−0.101657	−	0.0586916i	−0.549966	+	0.835187i	$0.685360\pi$
$810$		0			0
$811$			23.9391i			0.840616i	0.907382	+	0.420308i	$0.138078\pi$
$811$			23.9391i			0.840616i	−0.907382	+	0.420308i	$0.861922\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.53896			0.0539074
$816$		0			0
$817$			22.6089i			0.790985i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.2527i		−	0.706823i	−0.935468	−	0.353411i	$-0.885022\pi$
$821$		−	20.2527i		−	0.706823i	0.935468	−	0.353411i	$-0.114978\pi$
$822$		0			0
$823$	−27.3727			−0.954153			−0.477076	−	0.878862i	$-0.658303\pi$
$823$	−27.3727			−0.954153			−0.477076	+	0.878862i	$0.658303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			0.732736i			0.0254797i	0.999919	+	0.0127399i	$0.00405533\pi$
$827$			0.732736i			0.0254797i	−0.999919	+	0.0127399i	$0.995945\pi$
$828$		0			0
$829$	24.2088	+	13.9770i	0.840806	+	0.485440i	0.857538	−	0.514420i	$-0.171993\pi$
$829$	24.2088	+	13.9770i	0.840806	+	0.485440i	−0.0167319	+	0.999860i	$0.505326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.92128			−0.0664885
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.23496	+	9.06722i	−0.180731	+	0.313035i	−0.942130	−	0.335249i	$-0.891180\pi$
$839$	−5.23496	+	9.06722i	−0.180731	+	0.313035i	0.761399	+	0.648284i	$0.224513\pi$
$840$		0			0
$841$	−14.4908	−	25.0989i	−0.499684	−	0.865478i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.180239	−	0.312183i	−0.00620040	−	0.0107394i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	31.8270i		−	1.09101i
$852$		0			0
$853$	5.60649	+	3.23691i	0.191963	+	0.110830i	0.592901	−	0.805275i	$-0.297983\pi$
$853$	5.60649	+	3.23691i	0.191963	+	0.110830i	−0.400938	+	0.916105i	$0.631316\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.3518	+	26.5901i	0.524407	+	0.908299i	0.999596	+	0.0284155i	$0.00904614\pi$
$857$	15.3518	+	26.5901i	0.524407	+	0.908299i	−0.475190	+	0.879883i	$0.657621\pi$
$858$		0			0
$859$	20.2979	+	11.7190i	0.692556	+	0.399848i	0.804569	−	0.593859i	$-0.202396\pi$
$859$	20.2979	+	11.7190i	0.692556	+	0.399848i	−0.112013	+	0.993707i	$0.535730\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.6023	+	18.2456i	−1.07575	+	0.621086i	−0.929748	−	0.368197i	$-0.879975\pi$
$863$	−31.6023	+	18.2456i	−1.07575	+	0.621086i	−0.146006	+	0.989284i	$0.546642\pi$
$864$		0			0
$865$	−0.100172	+	0.173504i	−0.00340597	+	0.00589931i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	54.6554	−	31.5553i	1.85406	−	1.07044i
$870$		0			0
$871$	−28.4577	+	16.4301i	−0.964252	+	0.556711i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.8083	+	37.7731i	−0.736415	+	1.27551i	0.217685	+	0.976019i	$0.430150\pi$
$877$	−21.8083	+	37.7731i	−0.736415	+	1.27551i	−0.954100	+	0.299489i	$0.903184\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.3722			−0.618974			−0.309487	−	0.950904i	$-0.600157\pi$
$881$	−18.3722			−0.618974			−0.309487	+	0.950904i	$0.600157\pi$
$882$		0			0
$883$	−38.9500			−1.31077			−0.655385	−	0.755295i	$-0.727494\pi$
$883$	−38.9500			−1.31077			−0.655385	+	0.755295i	$0.727494\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.7267	+	34.1676i	−0.662357	+	1.14724i	0.317637	+	0.948212i	$0.397111\pi$
$887$	−19.7267	+	34.1676i	−0.662357	+	1.14724i	−0.979995	+	0.199024i	$0.936223\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.0556	+	10.4244i	−0.604209	+	0.348840i
$894$		0			0
$895$	−0.807106	+	0.465983i	−0.0269786	+	0.0155761i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.462101	−	0.800383i	0.0154119	−	0.0266943i
$900$		0			0
$901$	−3.06505	+	1.76961i	−0.102112	+	0.0589542i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.0915812	−	0.0528744i	−0.00304426	−	0.00175761i
$906$		0			0
$907$	−10.3062	−	17.8509i	−0.342213	−	0.592731i	0.642630	−	0.766177i	$-0.277843\pi$
$907$	−10.3062	−	17.8509i	−0.342213	−	0.592731i	−0.984843	+	0.173446i	$0.944510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.40092	−	2.54087i	−0.145809	−	0.0841829i	0.425321	−	0.905043i	$-0.360161\pi$
$911$	−4.40092	−	2.54087i	−0.145809	−	0.0841829i	−0.571130	+	0.820860i	$0.693495\pi$
$912$		0			0
$913$		−	3.87653i		−	0.128294i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	11.7798	+	20.4032i	0.388579	+	0.673038i	0.992259	−	0.124189i	$-0.0396328\pi$
$919$	11.7798	+	20.4032i	0.388579	+	0.673038i	−0.603680	+	0.797227i	$0.706300\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.5256	+	33.8193i	0.642693	+	1.11318i
$924$		0			0
$925$	16.9157	−	29.2988i	0.556185	−	0.963340i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.0420			0.952836			0.476418	−	0.879219i	$-0.341935\pi$
$929$	29.0420			0.952836			0.476418	+	0.879219i	$0.341935\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.678145	+	0.391527i	0.0221777	+	0.0128043i
$936$		0			0
$937$		−	48.5954i		−	1.58754i	−0.608217	−	0.793771i	$-0.708115\pi$
$937$		−	48.5954i		−	1.58754i	0.608217	−	0.793771i	$-0.291885\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.6218			−0.346261			−0.173130	−	0.984899i	$-0.555388\pi$
$941$	−10.6218			−0.346261			−0.173130	+	0.984899i	$0.555388\pi$
$942$		0			0
$943$			11.9350i			0.388657i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.9282i		−	0.550091i	−0.961431	−	0.275046i	$-0.911307\pi$
$947$		−	16.9282i		−	0.550091i	0.961431	−	0.275046i	$-0.0886930\pi$
$948$		0			0
$949$	−41.1517			−1.33584
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20.2937i		−	0.657378i	−0.944438	−	0.328689i	$-0.893393\pi$
$953$		−	20.2937i		−	0.657378i	0.944438	−	0.328689i	$-0.106607\pi$
$954$		0			0
$955$	1.10512	+	0.638040i	0.0357608	+	0.0206465i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.6203			−0.503880
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.18693	+	2.05582i	−0.0382085	+	0.0661791i
$966$		0			0
$967$	−9.30930	−	16.1242i	−0.299367	−	0.518519i	0.676624	−	0.736328i	$-0.263442\pi$
$967$	−9.30930	−	16.1242i	−0.299367	−	0.518519i	−0.975991	+	0.217810i	$0.930109\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.819339	+	1.41914i	0.0262938	+	0.0455423i	0.878873	−	0.477056i	$-0.158296\pi$
$971$	0.819339	+	1.41914i	0.0262938	+	0.0455423i	−0.852579	+	0.522598i	$0.824963\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	13.0308i		−	0.416893i	−0.978034	−	0.208447i	$-0.933159\pi$
$977$		−	13.0308i		−	0.416893i	0.978034	−	0.208447i	$-0.0668408\pi$
$978$		0			0
$979$	−52.3077	−	30.1999i	−1.67176	−	0.965192i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.6364	+	32.2792i	0.594410	+	1.02955i	0.993630	+	0.112693i	$0.0359476\pi$
$983$	18.6364	+	32.2792i	0.594410	+	1.02955i	−0.399220	+	0.916855i	$0.630719\pi$
$984$		0			0
$985$	−1.69320	−	0.977569i	−0.0539498	−	0.0311479i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−23.4693	+	13.5500i	−0.746279	+	0.430864i
$990$		0			0
$991$	−12.2935	+	21.2929i	−0.390515	+	0.676392i	−0.992517	−	0.122103i	$-0.961036\pi$
$991$	−12.2935	+	21.2929i	−0.390515	+	0.676392i	0.602003	+	0.798494i	$0.294370\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.509171	−	0.293970i	0.0161418	−	0.00931947i
$996$		0			0
$997$	−47.2434	+	27.2760i	−1.49621	+	0.863840i	−0.999991	−	0.00435443i	$-0.998614\pi$
$997$	−47.2434	+	27.2760i	−1.49621	+	0.863840i	−0.496224	+	0.868194i	$0.665281\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.w.c.521.12		48
3.2	odd	2		1764.2.w.c.1109.3		48
7.2	even	3		5292.2.bm.c.4625.13		48
7.3	odd	6		5292.2.x.c.4409.13		48
7.4	even	3		5292.2.x.c.4409.12		48
7.5	odd	6		5292.2.bm.c.4625.12		48
7.6	odd	2	inner	5292.2.w.c.521.13		48
9.4	even	3		1764.2.bm.c.1697.5		48
9.5	odd	6		5292.2.bm.c.2285.12		48
21.2	odd	6		1764.2.bm.c.1685.20		48
21.5	even	6		1764.2.bm.c.1685.5		48
21.11	odd	6		1764.2.x.c.1469.15	yes	48
21.17	even	6		1764.2.x.c.1469.10	yes	48
21.20	even	2		1764.2.w.c.1109.22		48
63.4	even	3		1764.2.x.c.293.10	✓	48
63.5	even	6	inner	5292.2.w.c.1097.12		48
63.13	odd	6		1764.2.bm.c.1697.20		48
63.23	odd	6	inner	5292.2.w.c.1097.13		48
63.31	odd	6		1764.2.x.c.293.15	yes	48
63.32	odd	6		5292.2.x.c.881.13		48
63.40	odd	6		1764.2.w.c.509.3		48
63.41	even	6		5292.2.bm.c.2285.13		48
63.58	even	3		1764.2.w.c.509.22		48
63.59	even	6		5292.2.x.c.881.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.w.c.509.3		48	63.40	odd	6
1764.2.w.c.509.22		48	63.58	even	3
1764.2.w.c.1109.3		48	3.2	odd	2
1764.2.w.c.1109.22		48	21.20	even	2
1764.2.x.c.293.10	✓	48	63.4	even	3
1764.2.x.c.293.15	yes	48	63.31	odd	6
1764.2.x.c.1469.10	yes	48	21.17	even	6
1764.2.x.c.1469.15	yes	48	21.11	odd	6
1764.2.bm.c.1685.5		48	21.5	even	6
1764.2.bm.c.1685.20		48	21.2	odd	6
1764.2.bm.c.1697.5		48	9.4	even	3
1764.2.bm.c.1697.20		48	63.13	odd	6
5292.2.w.c.521.12		48	1.1	even	1	trivial
5292.2.w.c.521.13		48	7.6	odd	2	inner
5292.2.w.c.1097.12		48	63.5	even	6	inner
5292.2.w.c.1097.13		48	63.23	odd	6	inner
5292.2.x.c.881.12		48	63.59	even	6
5292.2.x.c.881.13		48	63.32	odd	6
5292.2.x.c.4409.12		48	7.4	even	3
5292.2.x.c.4409.13		48	7.3	odd	6
5292.2.bm.c.2285.12		48	9.5	odd	6
5292.2.bm.c.2285.13		48	63.41	even	6
5292.2.bm.c.4625.12		48	7.5	odd	6
5292.2.bm.c.4625.13		48	7.2	even	3

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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.0565510 + 0.0979493i) q^{5} +O(q^{10})\)	\(q+(-0.0565510 + 0.0979493i) q^{5} +(4.86969 - 2.81151i) q^{11} +(2.71286 - 1.56627i) q^{13} +(0.615632 - 1.06631i) q^{17} +(3.38980 - 1.95710i) q^{19} +(4.06316 + 2.34587i) q^{23} +(2.49360 + 4.31905i) q^{25} +(0.117222 + 0.0676783i) q^{29} -6.82790i q^{31} +(-3.39182 - 5.87480i) q^{37} +(1.27192 + 2.20303i) q^{41} +(-2.88806 + 5.00226i) q^{43} -5.32646 q^{47} +(-2.48935 - 1.43723i) q^{53} +0.635976i q^{55} +11.4458 q^{59} +0.0386207i q^{61} +0.354297i q^{65} -10.4899 q^{67} +12.4663i q^{71} +(-11.3768 - 6.56842i) q^{73} +11.2236 q^{79} +(0.344701 - 0.597040i) q^{83} +(0.0696293 + 0.120601i) q^{85} +(-5.37075 - 9.30241i) q^{89} +0.442705i q^{95} +(-5.89327 - 3.40248i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 24 q^{11} + 48 q^{23} - 24 q^{25} + 48 q^{53} + 48 q^{79} - 24 q^{85}+O(q^{100}) \) 48 * q - 24 * q^11 + 48 * q^23 - 24 * q^25 + 48 * q^53 + 48 * q^79 - 24 * q^85

Introduction

L-functions

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