LMFDB - Embedded newform 5292.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5292.1");

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.a (trivial)

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:	yes
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 756)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{5} +O(q^{10})$	$q+3.16228 q^{5} -6.32456 q^{11} -3.16228 q^{17} +7.00000 q^{19} +3.16228 q^{23} +5.00000 q^{25} -3.16228 q^{29} -3.00000 q^{31} -4.00000 q^{37} +9.48683 q^{41} +5.00000 q^{43} +9.48683 q^{47} -9.48683 q^{53} -20.0000 q^{55} +12.6491 q^{59} +3.00000 q^{61} +10.0000 q^{67} +12.6491 q^{71} +5.00000 q^{73} +12.0000 q^{79} +6.32456 q^{83} -10.0000 q^{85} -9.48683 q^{89} +22.1359 q^{95} +5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 14 q^{19} + 10 q^{25} - 6 q^{31} - 8 q^{37} + 10 q^{43} - 40 q^{55} + 6 q^{61} + 20 q^{67} + 10 q^{73} + 24 q^{79} - 20 q^{85} + 10 q^{97}+O(q^{100}) $ 2 * q + 14 * q^19 + 10 * q^25 - 6 * q^31 - 8 * q^37 + 10 * q^43 - 40 * q^55 + 6 * q^61 + 20 * q^67 + 10 * q^73 + 24 * q^79 - 20 * q^85 + 10 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.16228	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	3.16228	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.32456	−1.90693	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	−6.32456	−1.90693	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.16228	−0.766965	−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228	−0.766965	−0.383482	+	0.923548i	$0.625275\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.16228	0.659380	0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228	0.659380	0.329690	+	0.944089i	$0.393056\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.16228	−0.587220	−0.293610	−	0.955925i	$-0.594857\pi$
$29$	−3.16228	−0.587220	−0.293610	+	0.955925i	$0.594857\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.48683	1.48159	0.740797	−	0.671729i	$-0.234448\pi$
$41$	9.48683	1.48159	0.740797	+	0.671729i	$0.234448\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.48683	1.38380	0.691898	−	0.721995i	$-0.256775\pi$
$47$	9.48683	1.38380	0.691898	+	0.721995i	$0.256775\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.48683	−1.30312	−0.651558	−	0.758599i	$-0.725884\pi$
$53$	−9.48683	−1.30312	−0.651558	+	0.758599i	$0.725884\pi$
$54$	0	0
$55$	−20.0000	−2.69680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.6491	1.64677	0.823387	−	0.567480i	$-0.192082\pi$
$59$	12.6491	1.64677	0.823387	+	0.567480i	$0.192082\pi$
$60$	0	0
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.6491	1.50117	0.750587	−	0.660772i	$-0.229771\pi$
$71$	12.6491	1.50117	0.750587	+	0.660772i	$0.229771\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.32456	0.694210	0.347105	−	0.937826i	$-0.387165\pi$
$83$	6.32456	0.694210	0.347105	+	0.937826i	$0.387165\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.48683	−1.00560	−0.502801	−	0.864402i	$-0.667697\pi$
$89$	−9.48683	−1.00560	−0.502801	+	0.864402i	$0.667697\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	22.1359	2.27110
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.32456	−0.629317	−0.314658	−	0.949205i	$-0.601890\pi$
$101$	−6.32456	−0.629317	−0.314658	+	0.949205i	$0.601890\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.32456	0.594964	0.297482	−	0.954727i	$-0.403853\pi$
$113$	6.32456	0.594964	0.297482	+	0.954727i	$0.403853\pi$
$114$	0	0
$115$	10.0000	0.932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	29.0000	2.63636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.0000	1.86345	0.931724	−	0.363166i	$-0.118304\pi$
$127$	21.0000	1.86345	0.931724	+	0.363166i	$0.118304\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.8114	−1.38145	−0.690724	−	0.723119i	$-0.742708\pi$
$131$	−15.8114	−1.38145	−0.690724	+	0.723119i	$0.742708\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.16228	−0.270172	−0.135086	−	0.990834i	$-0.543131\pi$
$137$	−3.16228	−0.270172	−0.135086	+	0.990834i	$0.543131\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.48683	0.777192	0.388596	−	0.921408i	$-0.372960\pi$
$149$	9.48683	0.777192	0.388596	+	0.921408i	$0.372960\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.48683	−0.762001
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.32456	−0.489409	−0.244704	−	0.969598i	$-0.578691\pi$
$167$	−6.32456	−0.489409	−0.244704	+	0.969598i	$0.578691\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.16228	0.240424	0.120212	−	0.992748i	$-0.461643\pi$
$173$	3.16228	0.240424	0.120212	+	0.992748i	$0.461643\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.48683	0.709079	0.354540	−	0.935041i	$-0.384638\pi$
$179$	9.48683	0.709079	0.354540	+	0.935041i	$0.384638\pi$
$180$	0	0
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.6491	−0.929981
$186$	0	0
$187$	20.0000	1.46254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.48683	−0.686443	−0.343222	−	0.939254i	$-0.611518\pi$
$191$	−9.48683	−0.686443	−0.343222	+	0.939254i	$0.611518\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.2982	−1.80242	−0.901212	−	0.433379i	$-0.857321\pi$
$197$	−25.2982	−1.80242	−0.901212	+	0.433379i	$0.857321\pi$
$198$	0	0
$199$	13.0000	0.921546	0.460773	−	0.887518i	$-0.347572\pi$
$199$	13.0000	0.921546	0.460773	+	0.887518i	$0.347572\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	30.0000	2.09529
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−44.2719	−3.06235
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.8114	1.07833
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.48683	0.629663	0.314832	−	0.949148i	$-0.398052\pi$
$227$	9.48683	0.629663	0.314832	+	0.949148i	$0.398052\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.8114	−1.03584	−0.517919	−	0.855430i	$-0.673293\pi$
$233$	−15.8114	−1.03584	−0.517919	+	0.855430i	$0.673293\pi$
$234$	0	0
$235$	30.0000	1.95698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.48683	0.613652	0.306826	−	0.951766i	$-0.400733\pi$
$239$	9.48683	0.613652	0.306826	+	0.951766i	$0.400733\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.48683	−0.598804	−0.299402	−	0.954127i	$-0.596787\pi$
$251$	−9.48683	−0.598804	−0.299402	+	0.954127i	$0.596787\pi$
$252$	0	0
$253$	−20.0000	−1.25739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.9737	−1.18354	−0.591772	−	0.806105i	$-0.701572\pi$
$257$	−18.9737	−1.18354	−0.591772	+	0.806105i	$0.701572\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.9737	−1.16997	−0.584983	−	0.811045i	$-0.698899\pi$
$263$	−18.9737	−1.16997	−0.584983	+	0.811045i	$0.698899\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.32456	−0.385615	−0.192807	−	0.981237i	$-0.561759\pi$
$269$	−6.32456	−0.385615	−0.192807	+	0.981237i	$0.561759\pi$
$270$	0	0
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−31.6228	−1.90693
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.32456	−0.377291	−0.188646	−	0.982045i	$-0.560410\pi$
$281$	−6.32456	−0.377291	−0.188646	+	0.982045i	$0.560410\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	40.0000	2.32889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.48683	0.543214
$306$	0	0
$307$	−25.0000	−1.42683	−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000	−1.42683	−0.713413	+	0.700744i	$0.752851\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.32456	−0.358633	−0.179316	−	0.983791i	$-0.557389\pi$
$311$	−6.32456	−0.358633	−0.179316	+	0.983791i	$0.557389\pi$
$312$	0	0
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.9737	1.06567	0.532834	−	0.846220i	$-0.321127\pi$
$317$	18.9737	1.06567	0.532834	+	0.846220i	$0.321127\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−22.1359	−1.23168
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	31.6228	1.72774
$336$	0	0
$337$	21.0000	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.9737	1.02748
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.9737	−1.00987	−0.504933	−	0.863158i	$-0.668483\pi$
$353$	−18.9737	−1.00987	−0.504933	+	0.863158i	$0.668483\pi$
$354$	0	0
$355$	40.0000	2.12298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.2982	1.33519	0.667595	−	0.744525i	$-0.267324\pi$
$359$	25.2982	1.33519	0.667595	+	0.744525i	$0.267324\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.8114	0.827606
$366$	0	0
$367$	19.0000	0.991792	0.495896	−	0.868382i	$-0.334840\pi$
$367$	19.0000	0.991792	0.495896	+	0.868382i	$0.334840\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.0000	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.6491	−0.646339	−0.323170	−	0.946341i	$-0.604748\pi$
$383$	−12.6491	−0.646339	−0.323170	+	0.946341i	$0.604748\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.7851	1.76367	0.881836	−	0.471556i	$-0.156307\pi$
$389$	34.7851	1.76367	0.881836	+	0.471556i	$0.156307\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	0	0
$393$	0	0
$394$	0	0
$395$	37.9473	1.90934
$396$	0	0
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.9737	0.947500	0.473750	−	0.880659i	$-0.342900\pi$
$401$	18.9737	0.947500	0.473750	+	0.880659i	$0.342900\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.2982	1.25399
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.0000	0.981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.48683	−0.463462	−0.231731	−	0.972780i	$-0.574439\pi$
$419$	−9.48683	−0.463462	−0.231731	+	0.972780i	$0.574439\pi$
$420$	0	0
$421$	3.00000	0.146211	0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000	0.146211	0.0731055	+	0.997324i	$0.476709\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.8114	−0.766965
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.1359	1.05891
$438$	0	0
$439$	18.0000	0.859093	0.429547	−	0.903045i	$-0.358673\pi$
$439$	18.0000	0.859093	0.429547	+	0.903045i	$0.358673\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.7851	−1.65269	−0.826344	−	0.563166i	$-0.809583\pi$
$443$	−34.7851	−1.65269	−0.826344	+	0.563166i	$0.809583\pi$
$444$	0	0
$445$	−30.0000	−1.42214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.7851	1.64161	0.820804	−	0.571210i	$-0.193526\pi$
$449$	34.7851	1.64161	0.820804	+	0.571210i	$0.193526\pi$
$450$	0	0
$451$	−60.0000	−2.82529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.6491	0.589128	0.294564	−	0.955632i	$-0.404826\pi$
$461$	12.6491	0.589128	0.294564	+	0.955632i	$0.404826\pi$
$462$	0	0
$463$	25.0000	1.16185	0.580924	−	0.813958i	$-0.302691\pi$
$463$	25.0000	1.16185	0.580924	+	0.813958i	$0.302691\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.32456	−0.292666	−0.146333	−	0.989235i	$-0.546747\pi$
$467$	−6.32456	−0.292666	−0.146333	+	0.989235i	$0.546747\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−31.6228	−1.45402
$474$	0	0
$475$	35.0000	1.60591
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.1359	−1.01142	−0.505709	−	0.862704i	$-0.668769\pi$
$479$	−22.1359	−1.01142	−0.505709	+	0.862704i	$0.668769\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.8114	0.717958
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.9737	−0.856270	−0.428135	−	0.903715i	$-0.640829\pi$
$491$	−18.9737	−0.856270	−0.428135	+	0.903715i	$0.640829\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.4605	1.26899	0.634495	−	0.772927i	$-0.281208\pi$
$503$	28.4605	1.26899	0.634495	+	0.772927i	$0.281208\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.9473	1.68199	0.840993	−	0.541046i	$-0.181972\pi$
$509$	37.9473	1.68199	0.840993	+	0.541046i	$0.181972\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.32456	−0.278693
$516$	0	0
$517$	−60.0000	−2.63880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.48683	0.413253
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.0000	−0.902861	−0.451430	−	0.892306i	$-0.649086\pi$
$541$	−21.0000	−0.902861	−0.451430	+	0.892306i	$0.649086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.1359	0.948200
$546$	0	0
$547$	21.0000	0.897895	0.448948	−	0.893558i	$-0.351799\pi$
$547$	21.0000	0.897895	0.448948	+	0.893558i	$0.351799\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−22.1359	−0.943023
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.16228	−0.133990	−0.0669950	−	0.997753i	$-0.521341\pi$
$557$	−3.16228	−0.133990	−0.0669950	+	0.997753i	$0.521341\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.1096	−1.73256	−0.866282	−	0.499556i	$-0.833497\pi$
$563$	−41.1096	−1.73256	−0.866282	+	0.499556i	$0.833497\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.9473	1.59083	0.795417	−	0.606062i	$-0.207252\pi$
$569$	37.9473	1.59083	0.795417	+	0.606062i	$0.207252\pi$
$570$	0	0
$571$	39.0000	1.63210	0.816050	−	0.577982i	$-0.196160\pi$
$571$	39.0000	1.63210	0.816050	+	0.577982i	$0.196160\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.8114	0.659380
$576$	0	0
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	60.0000	2.48495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.7851	1.43573	0.717866	−	0.696181i	$-0.245119\pi$
$587$	34.7851	1.43573	0.717866	+	0.696181i	$0.245119\pi$
$588$	0	0
$589$	−21.0000	−0.865290
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18.9737	−0.779155	−0.389578	−	0.920994i	$-0.627379\pi$
$593$	−18.9737	−0.779155	−0.389578	+	0.920994i	$0.627379\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.4605	−1.16286	−0.581432	−	0.813595i	$-0.697507\pi$
$599$	−28.4605	−1.16286	−0.581432	+	0.813595i	$0.697507\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	91.7061	3.72838
$606$	0	0
$607$	−15.0000	−0.608831	−0.304416	−	0.952539i	$-0.598461\pi$
$607$	−15.0000	−0.608831	−0.304416	+	0.952539i	$0.598461\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	33.0000	1.33286	0.666429	−	0.745569i	$-0.267822\pi$
$613$	33.0000	1.33286	0.666429	+	0.745569i	$0.267822\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.2982	−1.01847	−0.509234	−	0.860628i	$-0.670071\pi$
$617$	−25.2982	−1.01847	−0.509234	+	0.860628i	$0.670071\pi$
$618$	0	0
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.6491	0.504353
$630$	0	0
$631$	−31.0000	−1.23409	−0.617045	−	0.786928i	$-0.711670\pi$
$631$	−31.0000	−1.23409	−0.617045	+	0.786928i	$0.711670\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	66.4078	2.63531
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.16228	0.124902	0.0624512	−	0.998048i	$-0.480108\pi$
$641$	3.16228	0.124902	0.0624512	+	0.998048i	$0.480108\pi$
$642$	0	0
$643$	37.0000	1.45914	0.729569	−	0.683907i	$-0.239721\pi$
$643$	37.0000	1.45914	0.729569	+	0.683907i	$0.239721\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.9473	−1.49186	−0.745932	−	0.666022i	$-0.767996\pi$
$647$	−37.9473	−1.49186	−0.745932	+	0.666022i	$0.767996\pi$
$648$	0	0
$649$	−80.0000	−3.14027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.4605	−1.11375	−0.556873	−	0.830598i	$-0.687999\pi$
$653$	−28.4605	−1.11375	−0.556873	+	0.830598i	$0.687999\pi$
$654$	0	0
$655$	−50.0000	−1.95366
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.16228	0.123185	0.0615924	−	0.998101i	$-0.480382\pi$
$659$	3.16228	0.123185	0.0615924	+	0.998101i	$0.480382\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.0000	−0.387202
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−18.9737	−0.732470
$672$	0	0
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.9473	1.45843	0.729217	−	0.684282i	$-0.239884\pi$
$677$	37.9473	1.45843	0.729217	+	0.684282i	$0.239884\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.9737	−0.726007	−0.363004	−	0.931788i	$-0.618249\pi$
$683$	−18.9737	−0.726007	−0.363004	+	0.931788i	$0.618249\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.32456	0.239904
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.16228	−0.119438	−0.0597188	−	0.998215i	$-0.519020\pi$
$701$	−3.16228	−0.119438	−0.0597188	+	0.998215i	$0.519020\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.00000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.48683	−0.355285
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.8114	−0.587220
$726$	0	0
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.8114	−0.584805
$732$	0	0
$733$	−3.00000	−0.110808	−0.0554038	−	0.998464i	$-0.517645\pi$
$733$	−3.00000	−0.110808	−0.0554038	+	0.998464i	$0.517645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−63.2456	−2.32968
$738$	0	0
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.4605	−1.04411	−0.522057	−	0.852910i	$-0.674835\pi$
$743$	−28.4605	−1.04411	−0.522057	+	0.852910i	$0.674835\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3.00000	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−53.7587	−1.95648
$756$	0	0
$757$	15.0000	0.545184	0.272592	−	0.962130i	$-0.412119\pi$
$757$	15.0000	0.545184	0.272592	+	0.962130i	$0.412119\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.2982	0.917060	0.458530	−	0.888679i	$-0.348376\pi$
$761$	25.2982	0.917060	0.458530	+	0.888679i	$0.348376\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.7851	−1.25113	−0.625566	−	0.780171i	$-0.715132\pi$
$773$	−34.7851	−1.25113	−0.625566	+	0.780171i	$0.715132\pi$
$774$	0	0
$775$	−15.0000	−0.538816
$776$	0	0
$777$	0	0
$778$	0	0
$779$	66.4078	2.37931
$780$	0	0
$781$	−80.0000	−2.86263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	50.5964	1.80586
$786$	0	0
$787$	−15.0000	−0.534692	−0.267346	−	0.963601i	$-0.586147\pi$
$787$	−15.0000	−0.534692	−0.267346	+	0.963601i	$0.586147\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.9737	−0.672082	−0.336041	−	0.941847i	$-0.609088\pi$
$797$	−18.9737	−0.672082	−0.336041	+	0.941847i	$0.609088\pi$
$798$	0	0
$799$	−30.0000	−1.06132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31.6228	−1.11594
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31.6228	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$809$	−31.6228	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	53.7587	1.88309
$816$	0	0
$817$	35.0000	1.22449
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.2982	−0.882914	−0.441457	−	0.897282i	$-0.645538\pi$
$821$	−25.2982	−0.882914	−0.441457	+	0.897282i	$0.645538\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.9737	0.659779	0.329890	−	0.944020i	$-0.392989\pi$
$827$	18.9737	0.659779	0.329890	+	0.944020i	$0.392989\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.0000	−0.692129
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.6228	−1.09174	−0.545870	−	0.837870i	$-0.683801\pi$
$839$	−31.6228	−1.09174	−0.545870	+	0.837870i	$0.683801\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41.1096	−1.41421
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.6491	−0.433606
$852$	0	0
$853$	43.0000	1.47229	0.736146	−	0.676823i	$-0.236644\pi$
$853$	43.0000	1.47229	0.736146	+	0.676823i	$0.236644\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.9473	−1.29626	−0.648128	−	0.761531i	$-0.724448\pi$
$857$	−37.9473	−1.29626	−0.648128	+	0.761531i	$0.724448\pi$
$858$	0	0
$859$	−17.0000	−0.580033	−0.290016	−	0.957022i	$-0.593661\pi$
$859$	−17.0000	−0.580033	−0.290016	+	0.957022i	$0.593661\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.8114	0.538226	0.269113	−	0.963109i	$-0.413270\pi$
$863$	15.8114	0.538226	0.269113	+	0.963109i	$0.413270\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−75.8947	−2.57455
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.00000	−0.168838	−0.0844190	−	0.996430i	$-0.526903\pi$
$877$	−5.00000	−0.168838	−0.0844190	+	0.996430i	$0.526903\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.4342	1.59810	0.799049	−	0.601266i	$-0.205337\pi$
$881$	47.4342	1.59810	0.799049	+	0.601266i	$0.205337\pi$
$882$	0	0
$883$	23.0000	0.774012	0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000	0.774012	0.387006	+	0.922077i	$0.373509\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	66.4078	2.22225
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.48683	0.316404
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−66.4078	−2.20747
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.4342	−1.57156	−0.785782	−	0.618504i	$-0.787739\pi$
$911$	−47.4342	−1.57156	−0.785782	+	0.618504i	$0.787739\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.4605	−0.933759	−0.466879	−	0.884321i	$-0.654622\pi$
$929$	−28.4605	−0.933759	−0.466879	+	0.884321i	$0.654622\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	63.2456	2.06835
$936$	0	0
$937$	−44.0000	−1.43742	−0.718709	−	0.695311i	$-0.755266\pi$
$937$	−44.0000	−1.43742	−0.718709	+	0.695311i	$0.755266\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.2982	−0.824698	−0.412349	−	0.911026i	$-0.635292\pi$
$941$	−25.2982	−0.824698	−0.412349	+	0.911026i	$0.635292\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.7851	1.13036	0.565181	−	0.824967i	$-0.308806\pi$
$947$	34.7851	1.13036	0.565181	+	0.824967i	$0.308806\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.16228	0.102436	0.0512181	−	0.998687i	$-0.483690\pi$
$953$	3.16228	0.102436	0.0512181	+	0.998687i	$0.483690\pi$
$954$	0	0
$955$	−30.0000	−0.970777
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.2982	0.814379
$966$	0	0
$967$	26.0000	0.836104	0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000	0.836104	0.418052	+	0.908423i	$0.362713\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.4342	−1.52223	−0.761117	−	0.648614i	$-0.775349\pi$
$971$	−47.4342	−1.52223	−0.761117	+	0.648614i	$0.775349\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.9473	1.21404	0.607021	−	0.794686i	$-0.292364\pi$
$977$	37.9473	1.21404	0.607021	+	0.794686i	$0.292364\pi$
$978$	0	0
$979$	60.0000	1.91761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−80.0000	−2.54901
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.8114	0.502773
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	41.1096	1.30326
$996$	0	0
$997$	−41.0000	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$997$	−41.0000	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.r.1.2		2
3.2	odd	2	inner	5292.2.a.r.1.1		2
7.3	odd	6		756.2.k.d.541.2	yes	4
7.5	odd	6		756.2.k.d.109.2	yes	4
7.6	odd	2		5292.2.a.q.1.1		2
21.5	even	6		756.2.k.d.109.1	✓	4
21.17	even	6		756.2.k.d.541.1	yes	4
21.20	even	2		5292.2.a.q.1.2		2
63.5	even	6		2268.2.i.i.865.1		4
63.31	odd	6		2268.2.l.i.541.1		4
63.38	even	6		2268.2.i.i.2053.1		4
63.40	odd	6		2268.2.i.i.865.2		4
63.47	even	6		2268.2.l.i.109.2		4
63.52	odd	6		2268.2.i.i.2053.2		4
63.59	even	6		2268.2.l.i.541.2		4
63.61	odd	6		2268.2.l.i.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.d.109.1	✓	4	21.5	even	6
756.2.k.d.109.2	yes	4	7.5	odd	6
756.2.k.d.541.1	yes	4	21.17	even	6
756.2.k.d.541.2	yes	4	7.3	odd	6
2268.2.i.i.865.1		4	63.5	even	6
2268.2.i.i.865.2		4	63.40	odd	6
2268.2.i.i.2053.1		4	63.38	even	6
2268.2.i.i.2053.2		4	63.52	odd	6
2268.2.l.i.109.1		4	63.61	odd	6
2268.2.l.i.109.2		4	63.47	even	6
2268.2.l.i.541.1		4	63.31	odd	6
2268.2.l.i.541.2		4	63.59	even	6
5292.2.a.q.1.1		2	7.6	odd	2
5292.2.a.q.1.2		2	21.20	even	2
5292.2.a.r.1.1		2	3.2	odd	2	inner
5292.2.a.r.1.2		2	1.1	even	1	trivial

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.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16228 q^{5} +O(q^{10})\)	\(q+3.16228 q^{5} -6.32456 q^{11} -3.16228 q^{17} +7.00000 q^{19} +3.16228 q^{23} +5.00000 q^{25} -3.16228 q^{29} -3.00000 q^{31} -4.00000 q^{37} +9.48683 q^{41} +5.00000 q^{43} +9.48683 q^{47} -9.48683 q^{53} -20.0000 q^{55} +12.6491 q^{59} +3.00000 q^{61} +10.0000 q^{67} +12.6491 q^{71} +5.00000 q^{73} +12.0000 q^{79} +6.32456 q^{83} -10.0000 q^{85} -9.48683 q^{89} +22.1359 q^{95} +5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 14 q^{19} + 10 q^{25} - 6 q^{31} - 8 q^{37} + 10 q^{43} - 40 q^{55} + 6 q^{61} + 20 q^{67} + 10 q^{73} + 24 q^{79} - 20 q^{85} + 10 q^{97}+O(q^{100}) \) 2 * q + 14 * q^19 + 10 * q^25 - 6 * q^31 - 8 * q^37 + 10 * q^43 - 40 * q^55 + 6 * q^61 + 20 * q^67 + 10 * q^73 + 24 * q^79 - 20 * q^85 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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