LMFDB - Embedded newform 2736.3.m.d.1711.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (of order $2$, degree $1$, 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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 61x^{10} + 1243x^{8} + 9566x^{6} + 25219x^{4} + 13245x^{2} + 841 $ x^12 + 61x^10 + 1243x^8 + 9566x^6 + 25219x^4 + 13245*x^2 + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.3
Root			$0.769253i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.d.1711.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.32253 q^{5} -1.51700i q^{7} +O(q^{10})$	$q-5.32253 q^{5} -1.51700i q^{7} +10.6649i q^{11} -20.0811 q^{13} +25.9327 q^{17} -4.35890i q^{19} -39.6351i q^{23} +3.32932 q^{25} +44.4200 q^{29} -31.8897i q^{31} +8.07425i q^{35} +1.58503 q^{37} -27.5248 q^{41} -46.4321i q^{43} +46.6506i q^{47} +46.6987 q^{49} -46.4525 q^{53} -56.7643i q^{55} +37.8483i q^{59} -90.5803 q^{61} +106.882 q^{65} +100.284i q^{67} -82.4118i q^{71} -98.7982 q^{73} +16.1786 q^{77} +103.210i q^{79} +164.542i q^{83} -138.028 q^{85} -57.8355 q^{89} +30.4629i q^{91} +23.2004i q^{95} +1.55097 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{5}+O(q^{10}) $ 12 * q + 4 * q^5	$ 12 q + 4 q^{5} - 20 q^{17} + 88 q^{25} + 24 q^{29} - 40 q^{37} + 32 q^{41} + 128 q^{49} - 184 q^{53} - 276 q^{61} + 232 q^{65} - 92 q^{73} - 308 q^{77} - 244 q^{85} - 72 q^{89} - 280 q^{97}+O(q^{100}) $ 12 * q + 4 * q^5 - 20 * q^17 + 88 * q^25 + 24 * q^29 - 40 * q^37 + 32 * q^41 + 128 * q^49 - 184 * q^53 - 276 * q^61 + 232 * q^65 - 92 * q^73 - 308 * q^77 - 244 * q^85 - 72 * q^89 - 280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	−5.32253	−1.06451	−0.532253	−	0.846585i	$-0.678654\pi$
$5$	−5.32253	−1.06451	−0.532253	+	0.846585i	$0.678654\pi$
$6$	0	0
$7$	− 1.51700i	− 0.216714i	−0.994112	−	0.108357i	$-0.965441\pi$
$7$	− 1.51700i	− 0.216714i	0.994112	−	0.108357i	$-0.0345589\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.6649i	0.969537i	0.874643	+	0.484768i	$0.161096\pi$
$11$	10.6649i	0.969537i	−0.874643	+	0.484768i	$0.838904\pi$
$12$	0	0
$13$	−20.0811	−1.54470	−0.772349	−	0.635199i	$-0.780918\pi$
$13$	−20.0811	−1.54470	−0.772349	+	0.635199i	$0.780918\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25.9327	1.52545	0.762727	−	0.646721i	$-0.223860\pi$
$17$	25.9327	1.52545	0.762727	+	0.646721i	$0.223860\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 39.6351i	− 1.72327i	−0.507532	−	0.861633i	$-0.669442\pi$
$23$	− 39.6351i	− 1.72327i	0.507532	−	0.861633i	$-0.330558\pi$
$24$	0	0
$25$	3.32932	0.133173
$26$	0	0
$27$	0	0
$28$	0	0
$29$	44.4200	1.53172	0.765861	−	0.643006i	$-0.222313\pi$
$29$	44.4200	1.53172	0.765861	+	0.643006i	$0.222313\pi$
$30$	0	0
$31$	− 31.8897i	− 1.02870i	−0.857580	−	0.514350i	$-0.828033\pi$
$31$	− 31.8897i	− 1.02870i	0.857580	−	0.514350i	$-0.171967\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.07425i	0.230693i
$36$	0	0
$37$	1.58503	0.0428387	0.0214194	−	0.999771i	$-0.493181\pi$
$37$	1.58503	0.0428387	0.0214194	+	0.999771i	$0.493181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−27.5248	−0.671336	−0.335668	−	0.941980i	$-0.608962\pi$
$41$	−27.5248	−0.671336	−0.335668	+	0.941980i	$0.608962\pi$
$42$	0	0
$43$	− 46.4321i	− 1.07982i	−0.841724	−	0.539908i	$-0.818459\pi$
$43$	− 46.4321i	− 1.07982i	0.841724	−	0.539908i	$-0.181541\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	46.6506i	0.992566i	0.868161	+	0.496283i	$0.165302\pi$
$47$	46.6506i	0.992566i	−0.868161	+	0.496283i	$0.834698\pi$
$48$	0	0
$49$	46.6987	0.953035
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−46.4525	−0.876462	−0.438231	−	0.898862i	$-0.644395\pi$
$53$	−46.4525	−0.876462	−0.438231	+	0.898862i	$0.644395\pi$
$54$	0	0
$55$	− 56.7643i	− 1.03208i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	37.8483i	0.641496i	0.947165	+	0.320748i	$0.103934\pi$
$59$	37.8483i	0.641496i	−0.947165	+	0.320748i	$0.896066\pi$
$60$	0	0
$61$	−90.5803	−1.48492	−0.742462	−	0.669888i	$-0.766342\pi$
$61$	−90.5803	−1.48492	−0.742462	+	0.669888i	$0.766342\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	106.882	1.64434
$66$	0	0
$67$	100.284i	1.49678i	0.663258	+	0.748391i	$0.269173\pi$
$67$	100.284i	1.49678i	−0.663258	+	0.748391i	$0.730827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 82.4118i	− 1.16073i	−0.814357	−	0.580365i	$-0.802910\pi$
$71$	− 82.4118i	− 1.16073i	0.814357	−	0.580365i	$-0.197090\pi$
$72$	0	0
$73$	−98.7982	−1.35340	−0.676700	−	0.736259i	$-0.736591\pi$
$73$	−98.7982	−1.35340	−0.676700	+	0.736259i	$0.736591\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.1786	0.210112
$78$	0	0
$79$	103.210i	1.30645i	0.757164	+	0.653225i	$0.226585\pi$
$79$	103.210i	1.30645i	−0.757164	+	0.653225i	$0.773415\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	164.542i	1.98243i	0.132250	+	0.991216i	$0.457780\pi$
$83$	164.542i	1.98243i	−0.132250	+	0.991216i	$0.542220\pi$
$84$	0	0
$85$	−138.028	−1.62385
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−57.8355	−0.649837	−0.324918	−	0.945742i	$-0.605337\pi$
$89$	−57.8355	−0.649837	−0.324918	+	0.945742i	$0.605337\pi$
$90$	0	0
$91$	30.4629i	0.334757i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	23.2004i	0.244214i
$96$	0	0
$97$	1.55097	0.0159893	0.00799467	−	0.999968i	$-0.497455\pi$
$97$	1.55097	0.0159893	0.00799467	+	0.999968i	$0.497455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	58.9060	0.583227	0.291614	−	0.956536i	$-0.405808\pi$
$101$	58.9060	0.583227	0.291614	+	0.956536i	$0.405808\pi$
$102$	0	0
$103$	− 154.851i	− 1.50341i	−0.659499	−	0.751705i	$-0.729232\pi$
$103$	− 154.851i	− 1.50341i	0.659499	−	0.751705i	$-0.270768\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.64113i	− 0.0620666i	−0.999518	−	0.0310333i	$-0.990120\pi$
$107$	− 6.64113i	− 0.0620666i	0.999518	−	0.0310333i	$-0.00987980\pi$
$108$	0	0
$109$	15.5942	0.143066	0.0715331	−	0.997438i	$-0.477211\pi$
$109$	15.5942	0.143066	0.0715331	+	0.997438i	$0.477211\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	101.858	0.901395	0.450698	−	0.892677i	$-0.351175\pi$
$113$	101.858	0.901395	0.450698	+	0.892677i	$0.351175\pi$
$114$	0	0
$115$	210.959i	1.83443i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 39.3398i	− 0.330587i
$120$	0	0
$121$	7.25981	0.0599985
$122$	0	0
$123$	0	0
$124$	0	0
$125$	115.343	0.922743
$126$	0	0
$127$	20.6143i	0.162317i	0.996701	+	0.0811587i	$0.0258621\pi$
$127$	20.6143i	0.162317i	−0.996701	+	0.0811587i	$0.974138\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	174.075i	1.32881i	0.747371	+	0.664407i	$0.231316\pi$
$131$	174.075i	1.32881i	−0.747371	+	0.664407i	$0.768684\pi$
$132$	0	0
$133$	−6.61243	−0.0497175
$134$	0	0
$135$	0	0
$136$	0	0
$137$	196.739	1.43605	0.718025	−	0.696017i	$-0.245046\pi$
$137$	196.739	1.43605	0.718025	+	0.696017i	$0.245046\pi$
$138$	0	0
$139$	193.372i	1.39117i	0.718446	+	0.695583i	$0.244854\pi$
$139$	193.372i	1.39117i	−0.718446	+	0.695583i	$0.755146\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 214.163i	− 1.49764i
$144$	0	0
$145$	−236.427	−1.63053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−89.5244	−0.600835	−0.300418	−	0.953808i	$-0.597126\pi$
$149$	−89.5244	−0.600835	−0.300418	+	0.953808i	$0.597126\pi$
$150$	0	0
$151$	111.282i	0.736967i	0.929634	+	0.368483i	$0.120123\pi$
$151$	111.282i	0.736967i	−0.929634	+	0.368483i	$0.879877\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	169.734i	1.09506i
$156$	0	0
$157$	206.598	1.31591	0.657955	−	0.753057i	$-0.271422\pi$
$157$	206.598	1.31591	0.657955	+	0.753057i	$0.271422\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−60.1263	−0.373455
$162$	0	0
$163$	− 270.012i	− 1.65651i	−0.560349	−	0.828256i	$-0.689333\pi$
$163$	− 270.012i	− 1.65651i	0.560349	−	0.828256i	$-0.310667\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	312.111i	1.86893i	0.356059	+	0.934463i	$0.384120\pi$
$167$	312.111i	1.86893i	−0.356059	+	0.934463i	$0.615880\pi$
$168$	0	0
$169$	234.249	1.38609
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.57648	0.0206733	0.0103367	−	0.999947i	$-0.496710\pi$
$173$	3.57648	0.0206733	0.0103367	+	0.999947i	$0.496710\pi$
$174$	0	0
$175$	− 5.05057i	− 0.0288604i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	75.6210i	0.422464i	0.977436	+	0.211232i	$0.0677475\pi$
$179$	75.6210i	0.422464i	−0.977436	+	0.211232i	$0.932252\pi$
$180$	0	0
$181$	274.805	1.51826	0.759129	−	0.650940i	$-0.225625\pi$
$181$	274.805	1.51826	0.759129	+	0.650940i	$0.225625\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.43638	−0.0456021
$186$	0	0
$187$	276.570i	1.47898i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	113.605i	0.594790i	0.954754	+	0.297395i	$0.0961179\pi$
$191$	113.605i	0.594790i	−0.954754	+	0.297395i	$0.903882\pi$
$192$	0	0
$193$	−366.062	−1.89670	−0.948348	−	0.317231i	$-0.897247\pi$
$193$	−366.062	−1.89670	−0.948348	+	0.317231i	$0.897247\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−133.490	−0.677613	−0.338807	−	0.940856i	$-0.610023\pi$
$197$	−133.490	−0.677613	−0.338807	+	0.940856i	$0.610023\pi$
$198$	0	0
$199$	63.1501i	0.317337i	0.987332	+	0.158669i	$0.0507201\pi$
$199$	63.1501i	0.317337i	−0.987332	+	0.158669i	$0.949280\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 67.3849i	− 0.331945i
$204$	0	0
$205$	146.501	0.714641
$206$	0	0
$207$	0	0
$208$	0	0
$209$	46.4872	0.222427
$210$	0	0
$211$	− 43.9449i	− 0.208270i	−0.994563	−	0.104135i	$-0.966793\pi$
$211$	− 43.9449i	− 0.208270i	0.994563	−	0.104135i	$-0.0332073\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	247.136i	1.14947i
$216$	0	0
$217$	−48.3765	−0.222933
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−520.756	−2.35636
$222$	0	0
$223$	304.568i	1.36578i	0.730523	+	0.682889i	$0.239277\pi$
$223$	304.568i	1.36578i	−0.730523	+	0.682889i	$0.760723\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	243.845i	1.07421i	0.843516	+	0.537104i	$0.180482\pi$
$227$	243.845i	1.07421i	−0.843516	+	0.537104i	$0.819518\pi$
$228$	0	0
$229$	269.966	1.17889	0.589445	−	0.807809i	$-0.299347\pi$
$229$	269.966	1.17889	0.589445	+	0.807809i	$0.299347\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	60.1384	0.258105	0.129052	−	0.991638i	$-0.458806\pi$
$233$	60.1384	0.258105	0.129052	+	0.991638i	$0.458806\pi$
$234$	0	0
$235$	− 248.299i	− 1.05659i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	364.980i	1.52711i	0.645742	+	0.763556i	$0.276548\pi$
$239$	364.980i	1.52711i	−0.645742	+	0.763556i	$0.723452\pi$
$240$	0	0
$241$	−231.191	−0.959298	−0.479649	−	0.877460i	$-0.659236\pi$
$241$	−231.191	−0.959298	−0.479649	+	0.877460i	$0.659236\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−248.555	−1.01451
$246$	0	0
$247$	87.5313i	0.354378i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.5801i	0.0780085i	0.999239	+	0.0390042i	$0.0124186\pi$
$251$	19.5801i	0.0780085i	−0.999239	+	0.0390042i	$0.987581\pi$
$252$	0	0
$253$	422.705	1.67077
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.1006	0.0665391	0.0332696	−	0.999446i	$-0.489408\pi$
$257$	17.1006	0.0665391	0.0332696	+	0.999446i	$0.489408\pi$
$258$	0	0
$259$	− 2.40449i	− 0.00928373i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	57.3106i	0.217911i	0.994047	+	0.108956i	$0.0347506\pi$
$263$	57.3106i	0.217911i	−0.994047	+	0.108956i	$0.965249\pi$
$264$	0	0
$265$	247.245	0.932999
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.7230	0.106777	0.0533885	−	0.998574i	$-0.482998\pi$
$269$	28.7230	0.106777	0.0533885	+	0.998574i	$0.482998\pi$
$270$	0	0
$271$	114.957i	0.424197i	0.977248	+	0.212099i	$0.0680298\pi$
$271$	114.957i	0.424197i	−0.977248	+	0.212099i	$0.931970\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	35.5069i	0.129116i
$276$	0	0
$277$	−464.495	−1.67688	−0.838439	−	0.544995i	$-0.816531\pi$
$277$	−464.495	−1.67688	−0.838439	+	0.544995i	$0.816531\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−131.163	−0.466772	−0.233386	−	0.972384i	$-0.574981\pi$
$281$	−131.163	−0.466772	−0.233386	+	0.972384i	$0.574981\pi$
$282$	0	0
$283$	144.862i	0.511881i	0.966693	+	0.255940i	$0.0823851\pi$
$283$	144.862i	0.511881i	−0.966693	+	0.255940i	$0.917615\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.7549i	0.145488i
$288$	0	0
$289$	383.506	1.32701
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−430.757	−1.47016	−0.735080	−	0.677981i	$-0.762855\pi$
$293$	−430.757	−1.47016	−0.735080	+	0.677981i	$0.762855\pi$
$294$	0	0
$295$	− 201.449i	− 0.682877i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	795.915i	2.66192i
$300$	0	0
$301$	−70.4373	−0.234011
$302$	0	0
$303$	0	0
$304$	0	0
$305$	482.117	1.58071
$306$	0	0
$307$	578.208i	1.88341i	0.336436	+	0.941706i	$0.390778\pi$
$307$	578.208i	1.88341i	−0.336436	+	0.941706i	$0.609222\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	99.1339i	0.318759i	0.987217	+	0.159379i	$0.0509493\pi$
$311$	99.1339i	0.318759i	−0.987217	+	0.159379i	$0.949051\pi$
$312$	0	0
$313$	−294.871	−0.942080	−0.471040	−	0.882112i	$-0.656121\pi$
$313$	−294.871	−0.942080	−0.471040	+	0.882112i	$0.656121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−440.217	−1.38870	−0.694348	−	0.719639i	$-0.744307\pi$
$317$	−440.217	−1.38870	−0.694348	+	0.719639i	$0.744307\pi$
$318$	0	0
$319$	473.735i	1.48506i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 113.038i	− 0.349963i
$324$	0	0
$325$	−66.8564	−0.205712
$326$	0	0
$327$	0	0
$328$	0	0
$329$	70.7688	0.215103
$330$	0	0
$331$	− 325.854i	− 0.984454i	−0.870467	−	0.492227i	$-0.836183\pi$
$331$	− 325.854i	− 0.984454i	0.870467	−	0.492227i	$-0.163817\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 533.767i	− 1.59333i
$336$	0	0
$337$	63.9035	0.189625	0.0948123	−	0.995495i	$-0.469775\pi$
$337$	63.9035	0.189625	0.0948123	+	0.995495i	$0.469775\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	340.101	0.997363
$342$	0	0
$343$	− 145.174i	− 0.423249i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 439.669i	− 1.26706i	−0.773720	−	0.633528i	$-0.781606\pi$
$347$	− 439.669i	− 1.26706i	0.773720	−	0.633528i	$-0.218394\pi$
$348$	0	0
$349$	459.775	1.31741	0.658703	−	0.752403i	$-0.271105\pi$
$349$	459.775	1.31741	0.658703	+	0.752403i	$0.271105\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−116.450	−0.329887	−0.164943	−	0.986303i	$-0.552744\pi$
$353$	−116.450	−0.329887	−0.164943	+	0.986303i	$0.552744\pi$
$354$	0	0
$355$	438.639i	1.23560i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 60.8754i	− 0.169569i	−0.996399	−	0.0847847i	$-0.972980\pi$
$359$	− 60.8754i	− 0.169569i	0.996399	−	0.0847847i	$-0.0270202\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	525.856	1.44070
$366$	0	0
$367$	− 653.185i	− 1.77980i	−0.456159	−	0.889898i	$-0.650775\pi$
$367$	− 653.185i	− 1.77980i	0.456159	−	0.889898i	$-0.349225\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	70.4682i	0.189941i
$372$	0	0
$373$	173.776	0.465887	0.232944	−	0.972490i	$-0.425164\pi$
$373$	173.776	0.465887	0.232944	+	0.972490i	$0.425164\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−892.000	−2.36605
$378$	0	0
$379$	− 120.525i	− 0.318007i	−0.987278	−	0.159003i	$-0.949172\pi$
$379$	− 120.525i	− 0.318007i	0.987278	−	0.159003i	$-0.0508281\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 212.557i	− 0.554980i	−0.960729	−	0.277490i	$-0.910497\pi$
$383$	− 212.557i	− 0.554980i	0.960729	−	0.277490i	$-0.0895025\pi$
$384$	0	0
$385$	−86.1111	−0.223665
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−203.069	−0.522028	−0.261014	−	0.965335i	$-0.584057\pi$
$389$	−203.069	−0.522028	−0.261014	+	0.965335i	$0.584057\pi$
$390$	0	0
$391$	− 1027.85i	− 2.62876i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 549.336i	− 1.39072i
$396$	0	0
$397$	−254.424	−0.640867	−0.320433	−	0.947271i	$-0.603828\pi$
$397$	−254.424	−0.640867	−0.320433	+	0.947271i	$0.603828\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.25127	0.00312037	0.00156019	−	0.999999i	$-0.499503\pi$
$401$	1.25127	0.00312037	0.00156019	+	0.999999i	$0.499503\pi$
$402$	0	0
$403$	640.379i	1.58903i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.9042i	0.0415337i
$408$	0	0
$409$	333.713	0.815923	0.407962	−	0.912999i	$-0.366240\pi$
$409$	333.713	0.815923	0.407962	+	0.912999i	$0.366240\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	57.4157	0.139021
$414$	0	0
$415$	− 875.779i	− 2.11031i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 87.5217i	− 0.208882i	−0.994531	−	0.104441i	$-0.966695\pi$
$419$	− 87.5217i	− 0.208882i	0.994531	−	0.104441i	$-0.0333054\pi$
$420$	0	0
$421$	390.964	0.928656	0.464328	−	0.885663i	$-0.346296\pi$
$421$	390.964	0.928656	0.464328	+	0.885663i	$0.346296\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	86.3384	0.203149
$426$	0	0
$427$	137.410i	0.321803i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	128.733i	0.298685i	0.988786	+	0.149342i	$0.0477157\pi$
$431$	128.733i	0.298685i	−0.988786	+	0.149342i	$0.952284\pi$
$432$	0	0
$433$	−718.320	−1.65894	−0.829468	−	0.558554i	$-0.811357\pi$
$433$	−718.320	−1.65894	−0.829468	+	0.558554i	$0.811357\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−172.766	−0.395344
$438$	0	0
$439$	− 31.3496i	− 0.0714113i	−0.999362	−	0.0357057i	$-0.988632\pi$
$439$	− 31.3496i	− 0.0714113i	0.999362	−	0.0357057i	$-0.0113679\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	162.714i	0.367300i	0.982992	+	0.183650i	$0.0587913\pi$
$443$	162.714i	0.367300i	−0.982992	+	0.183650i	$0.941209\pi$
$444$	0	0
$445$	307.831	0.691755
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.7009	−0.0839664	−0.0419832	−	0.999118i	$-0.513368\pi$
$449$	−37.7009	−0.0839664	−0.0419832	+	0.999118i	$0.513368\pi$
$450$	0	0
$451$	− 293.549i	− 0.650885i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 162.140i	− 0.356351i
$456$	0	0
$457$	224.412	0.491054	0.245527	−	0.969390i	$-0.421039\pi$
$457$	224.412	0.491054	0.245527	+	0.969390i	$0.421039\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	457.231	0.991825	0.495912	−	0.868373i	$-0.334834\pi$
$461$	457.231	0.991825	0.495912	+	0.868373i	$0.334834\pi$
$462$	0	0
$463$	576.614i	1.24539i	0.782466	+	0.622693i	$0.213962\pi$
$463$	576.614i	1.24539i	−0.782466	+	0.622693i	$0.786038\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	481.334i	1.03069i	0.856982	+	0.515347i	$0.172337\pi$
$467$	481.334i	1.03069i	−0.856982	+	0.515347i	$0.827663\pi$
$468$	0	0
$469$	152.131	0.324373
$470$	0	0
$471$	0	0
$472$	0	0
$473$	495.194	1.04692
$474$	0	0
$475$	− 14.5122i	− 0.0305520i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	319.707i	0.667446i	0.942671	+	0.333723i	$0.108305\pi$
$479$	319.707i	0.667446i	−0.942671	+	0.333723i	$0.891695\pi$
$480$	0	0
$481$	−31.8291	−0.0661728
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.25506	−0.0170207
$486$	0	0
$487$	209.819i	0.430840i	0.976521	+	0.215420i	$0.0691121\pi$
$487$	209.819i	0.430840i	−0.976521	+	0.215420i	$0.930888\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	494.297i	1.00671i	0.864078	+	0.503357i	$0.167902\pi$
$491$	494.297i	1.00671i	−0.864078	+	0.503357i	$0.832098\pi$
$492$	0	0
$493$	1151.93	2.33657
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−125.018	−0.251546
$498$	0	0
$499$	− 23.0217i	− 0.0461357i	−0.999734	−	0.0230678i	$-0.992657\pi$
$499$	− 23.0217i	− 0.0461357i	0.999734	−	0.0230678i	$-0.00734337\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	164.552i	0.327142i	0.986532	+	0.163571i	$0.0523012\pi$
$503$	164.552i	0.327142i	−0.986532	+	0.163571i	$0.947699\pi$
$504$	0	0
$505$	−313.529	−0.620849
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−318.792	−0.626310	−0.313155	−	0.949702i	$-0.601386\pi$
$509$	−318.792	−0.626310	−0.313155	+	0.949702i	$0.601386\pi$
$510$	0	0
$511$	149.876i	0.293300i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	824.200i	1.60039i
$516$	0	0
$517$	−497.524	−0.962330
$518$	0	0
$519$	0	0
$520$	0	0
$521$	678.757	1.30280	0.651399	−	0.758736i	$-0.274183\pi$
$521$	678.757	1.30280	0.651399	+	0.758736i	$0.274183\pi$
$522$	0	0
$523$	199.492i	0.381437i	0.981645	+	0.190719i	$0.0610818\pi$
$523$	199.492i	0.381437i	−0.981645	+	0.190719i	$0.938918\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 826.987i	− 1.56924i
$528$	0	0
$529$	−1041.94	−1.96965
$530$	0	0
$531$	0	0
$532$	0	0
$533$	552.726	1.03701
$534$	0	0
$535$	35.3476i	0.0660703i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	498.037i	0.924003i
$540$	0	0
$541$	−468.348	−0.865708	−0.432854	−	0.901464i	$-0.642493\pi$
$541$	−468.348	−0.865708	−0.432854	+	0.901464i	$0.642493\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−83.0006	−0.152295
$546$	0	0
$547$	93.4661i	0.170870i	0.996344	+	0.0854352i	$0.0272281\pi$
$547$	93.4661i	0.170870i	−0.996344	+	0.0854352i	$0.972772\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 193.622i	− 0.351401i
$552$	0	0
$553$	156.568	0.283126
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−296.185	−0.531751	−0.265876	−	0.964007i	$-0.585661\pi$
$557$	−296.185	−0.531751	−0.265876	+	0.964007i	$0.585661\pi$
$558$	0	0
$559$	932.406i	1.66799i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 292.251i	− 0.519097i	−0.965730	−	0.259548i	$-0.916426\pi$
$563$	− 292.251i	− 0.519097i	0.965730	−	0.259548i	$-0.0835737\pi$
$564$	0	0
$565$	−542.141	−0.959541
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−527.208	−0.926551	−0.463276	−	0.886214i	$-0.653326\pi$
$569$	−527.208	−0.926551	−0.463276	+	0.886214i	$0.653326\pi$
$570$	0	0
$571$	75.0971i	0.131519i	0.997836	+	0.0657593i	$0.0209470\pi$
$571$	75.0971i	0.131519i	−0.997836	+	0.0657593i	$0.979053\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 131.958i	− 0.229492i
$576$	0	0
$577$	870.821	1.50922	0.754611	−	0.656172i	$-0.227826\pi$
$577$	870.821	1.50922	0.754611	+	0.656172i	$0.227826\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	249.609	0.429620
$582$	0	0
$583$	− 495.411i	− 0.849762i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	872.310i	1.48605i	0.669265	+	0.743024i	$0.266609\pi$
$587$	872.310i	1.48605i	−0.669265	+	0.743024i	$0.733391\pi$
$588$	0	0
$589$	−139.004	−0.236000
$590$	0	0
$591$	0	0
$592$	0	0
$593$	467.746	0.788779	0.394390	−	0.918943i	$-0.370956\pi$
$593$	467.746	0.788779	0.394390	+	0.918943i	$0.370956\pi$
$594$	0	0
$595$	209.387i	0.351911i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 154.268i	− 0.257542i	−0.991674	−	0.128771i	$-0.958897\pi$
$599$	− 154.268i	− 0.257542i	0.991674	−	0.128771i	$-0.0411032\pi$
$600$	0	0
$601$	−77.3914	−0.128771	−0.0643855	−	0.997925i	$-0.520509\pi$
$601$	−77.3914	−0.128771	−0.0643855	+	0.997925i	$0.520509\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−38.6406	−0.0638687
$606$	0	0
$607$	91.0406i	0.149985i	0.997184	+	0.0749923i	$0.0238932\pi$
$607$	91.0406i	0.149985i	−0.997184	+	0.0749923i	$0.976107\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 936.794i	− 1.53321i
$612$	0	0
$613$	−327.588	−0.534402	−0.267201	−	0.963641i	$-0.586099\pi$
$613$	−327.588	−0.534402	−0.267201	+	0.963641i	$0.586099\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	902.097	1.46207	0.731034	−	0.682341i	$-0.239038\pi$
$617$	902.097	1.46207	0.731034	+	0.682341i	$0.239038\pi$
$618$	0	0
$619$	356.416i	0.575793i	0.957662	+	0.287896i	$0.0929559\pi$
$619$	356.416i	0.575793i	−0.957662	+	0.287896i	$0.907044\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	87.7361i	0.140828i
$624$	0	0
$625$	−697.149	−1.11544
$626$	0	0
$627$	0	0
$628$	0	0
$629$	41.1042	0.0653485
$630$	0	0
$631$	759.264i	1.20327i	0.798771	+	0.601635i	$0.205484\pi$
$631$	759.264i	1.20327i	−0.798771	+	0.601635i	$0.794516\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 109.720i	− 0.172788i
$636$	0	0
$637$	−937.760	−1.47215
$638$	0	0
$639$	0	0
$640$	0	0
$641$	912.838	1.42408	0.712042	−	0.702136i	$-0.247770\pi$
$641$	912.838	1.42408	0.712042	+	0.702136i	$0.247770\pi$
$642$	0	0
$643$	− 69.3051i	− 0.107784i	−0.998547	−	0.0538920i	$-0.982837\pi$
$643$	− 69.3051i	− 0.107784i	0.998547	−	0.0538920i	$-0.0171627\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 382.031i	− 0.590465i	−0.955425	−	0.295232i	$-0.904603\pi$
$647$	− 382.031i	− 0.590465i	0.955425	−	0.295232i	$-0.0953971\pi$
$648$	0	0
$649$	−403.648	−0.621954
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.546434	−0.000836806 0	−0.000418403	−	1.00000i	$-0.500133\pi$
$653$	−0.546434	−0.000836806 0	−0.000418403		1.00000i	$0.500133\pi$
$654$	0	0
$655$	− 926.517i	− 1.41453i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1196.64i	1.81584i	0.419143	+	0.907920i	$0.362330\pi$
$659$	1196.64i	1.81584i	−0.419143	+	0.907920i	$0.637670\pi$
$660$	0	0
$661$	−665.059	−1.00614	−0.503070	−	0.864245i	$-0.667796\pi$
$661$	−665.059	−1.00614	−0.503070	+	0.864245i	$0.667796\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	35.1948	0.0529246
$666$	0	0
$667$	− 1760.59i	− 2.63957i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 966.031i	− 1.43969i
$672$	0	0
$673$	178.037	0.264543	0.132271	−	0.991214i	$-0.457773\pi$
$673$	178.037	0.264543	0.132271	+	0.991214i	$0.457773\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	668.313	0.987169	0.493585	−	0.869698i	$-0.335686\pi$
$677$	668.313	0.987169	0.493585	+	0.869698i	$0.335686\pi$
$678$	0	0
$679$	− 2.35281i	− 0.00346511i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 253.264i	− 0.370811i	−0.982662	−	0.185405i	$-0.940640\pi$
$683$	− 253.264i	− 0.370811i	0.982662	−	0.185405i	$-0.0593598\pi$
$684$	0	0
$685$	−1047.15	−1.52868
$686$	0	0
$687$	0	0
$688$	0	0
$689$	932.815	1.35387
$690$	0	0
$691$	− 830.873i	− 1.20242i	−0.799091	−	0.601211i	$-0.794685\pi$
$691$	− 830.873i	− 1.20242i	0.799091	−	0.601211i	$-0.205315\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1029.23i	− 1.48090i
$696$	0	0
$697$	−713.792	−1.02409
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−426.515	−0.608438	−0.304219	−	0.952602i	$-0.598396\pi$
$701$	−426.515	−0.608438	−0.304219	+	0.952602i	$0.598396\pi$
$702$	0	0
$703$	− 6.90900i	− 0.00982787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 89.3600i	− 0.126393i
$708$	0	0
$709$	−143.702	−0.202682	−0.101341	−	0.994852i	$-0.532313\pi$
$709$	−143.702	−0.202682	−0.101341	+	0.994852i	$0.532313\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1263.95	−1.77273
$714$	0	0
$715$	1139.89i	1.59425i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	276.110i	0.384020i	0.981393	+	0.192010i	$0.0615005\pi$
$719$	276.110i	0.384020i	−0.981393	+	0.192010i	$0.938499\pi$
$720$	0	0
$721$	−234.909	−0.325809
$722$	0	0
$723$	0	0
$724$	0	0
$725$	147.888	0.203984
$726$	0	0
$727$	− 85.2768i	− 0.117300i	−0.998279	−	0.0586498i	$-0.981320\pi$
$727$	− 85.2768i	− 0.117300i	0.998279	−	0.0586498i	$-0.0186795\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1204.11i	− 1.64721i
$732$	0	0
$733$	−585.716	−0.799067	−0.399533	−	0.916719i	$-0.630828\pi$
$733$	−585.716	−0.799067	−0.399533	+	0.916719i	$0.630828\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1069.52	−1.45119
$738$	0	0
$739$	− 1297.15i	− 1.75527i	−0.479328	−	0.877636i	$-0.659120\pi$
$739$	− 1297.15i	− 1.75527i	0.479328	−	0.877636i	$-0.340880\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1384.78i	− 1.86376i	−0.362764	−	0.931881i	$-0.618167\pi$
$743$	− 1384.78i	− 1.86376i	0.362764	−	0.931881i	$-0.381833\pi$
$744$	0	0
$745$	476.496	0.639592
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.0746	−0.0134507
$750$	0	0
$751$	199.884i	0.266157i	0.991105	+	0.133079i	$0.0424863\pi$
$751$	199.884i	0.266157i	−0.991105	+	0.133079i	$0.957514\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 592.302i	− 0.784505i
$756$	0	0
$757$	599.426	0.791844	0.395922	−	0.918284i	$-0.370425\pi$
$757$	599.426	0.791844	0.395922	+	0.918284i	$0.370425\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−688.083	−0.904182	−0.452091	−	0.891972i	$-0.649322\pi$
$761$	−688.083	−0.904182	−0.452091	+	0.891972i	$0.649322\pi$
$762$	0	0
$763$	− 23.6563i	− 0.0310044i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 760.034i	− 0.990918i
$768$	0	0
$769$	1002.76	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$769$	1002.76	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−137.272	−0.177583	−0.0887915	−	0.996050i	$-0.528300\pi$
$773$	−137.272	−0.177583	−0.0887915	+	0.996050i	$0.528300\pi$
$774$	0	0
$775$	− 106.171i	− 0.136995i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	119.978i	0.154015i
$780$	0	0
$781$	878.914	1.12537
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1099.62	−1.40079
$786$	0	0
$787$	969.224i	1.23154i	0.787925	+	0.615771i	$0.211155\pi$
$787$	969.224i	1.23154i	−0.787925	+	0.615771i	$0.788845\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 154.518i	− 0.195345i
$792$	0	0
$793$	1818.95	2.29376
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1458.98	1.83058	0.915292	−	0.402790i	$-0.131960\pi$
$797$	1458.98	1.83058	0.915292	+	0.402790i	$0.131960\pi$
$798$	0	0
$799$	1209.78i	1.51411i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1053.67i	− 1.31217i
$804$	0	0
$805$	320.024	0.397545
$806$	0	0
$807$	0	0
$808$	0	0
$809$	449.947	0.556177	0.278089	−	0.960555i	$-0.410299\pi$
$809$	449.947	0.556177	0.278089	+	0.960555i	$0.410299\pi$
$810$	0	0
$811$	1086.35i	1.33952i	0.742579	+	0.669759i	$0.233602\pi$
$811$	1086.35i	1.33952i	−0.742579	+	0.669759i	$0.766398\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1437.14i	1.76337i
$816$	0	0
$817$	−202.393	−0.247727
$818$	0	0
$819$	0	0
$820$	0	0
$821$	499.306	0.608168	0.304084	−	0.952645i	$-0.401650\pi$
$821$	499.306	0.608168	0.304084	+	0.952645i	$0.401650\pi$
$822$	0	0
$823$	1139.14i	1.38413i	0.721837	+	0.692063i	$0.243298\pi$
$823$	1139.14i	1.38413i	−0.721837	+	0.692063i	$0.756702\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.60576i	− 0.0104060i	−0.999986	−	0.00520300i	$-0.998344\pi$
$827$	− 8.60576i	− 0.0104060i	0.999986	−	0.00520300i	$-0.00165617\pi$
$828$	0	0
$829$	1326.72	1.60039	0.800194	−	0.599741i	$-0.204730\pi$
$829$	1326.72	1.60039	0.800194	+	0.599741i	$0.204730\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1211.02	1.45381
$834$	0	0
$835$	− 1661.22i	− 1.98948i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	674.341i	0.803743i	0.915696	+	0.401872i	$0.131640\pi$
$839$	674.341i	0.803743i	−0.915696	+	0.401872i	$0.868360\pi$
$840$	0	0
$841$	1132.13	1.34617
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1246.80	−1.47550
$846$	0	0
$847$	− 11.0131i	− 0.0130025i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 62.8230i	− 0.0738225i
$852$	0	0
$853$	217.616	0.255118	0.127559	−	0.991831i	$-0.459286\pi$
$853$	217.616	0.255118	0.127559	+	0.991831i	$0.459286\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	881.427	1.02850	0.514251	−	0.857640i	$-0.328070\pi$
$857$	881.427	1.02850	0.514251	+	0.857640i	$0.328070\pi$
$858$	0	0
$859$	− 938.186i	− 1.09218i	−0.837725	−	0.546092i	$-0.816115\pi$
$859$	− 938.186i	− 1.09218i	0.837725	−	0.546092i	$-0.183885\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 729.700i	− 0.845539i	−0.906237	−	0.422770i	$-0.861058\pi$
$863$	− 729.700i	− 0.845539i	0.906237	−	0.422770i	$-0.138942\pi$
$864$	0	0
$865$	−19.0359	−0.0220069
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1100.72	−1.26665
$870$	0	0
$871$	− 2013.82i	− 2.31207i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 174.974i	− 0.199971i
$876$	0	0
$877$	−775.530	−0.884299	−0.442149	−	0.896941i	$-0.645784\pi$
$877$	−775.530	−0.884299	−0.442149	+	0.896941i	$0.645784\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	897.202	1.01839	0.509195	−	0.860651i	$-0.329943\pi$
$881$	897.202	1.01839	0.509195	+	0.860651i	$0.329943\pi$
$882$	0	0
$883$	− 757.581i	− 0.857962i	−0.903313	−	0.428981i	$-0.858873\pi$
$883$	− 757.581i	− 0.857962i	0.903313	−	0.428981i	$-0.141127\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 601.943i	− 0.678628i	−0.940673	−	0.339314i	$-0.889805\pi$
$887$	− 601.943i	− 0.678628i	0.940673	−	0.339314i	$-0.110195\pi$
$888$	0	0
$889$	31.2718	0.0351764
$890$	0	0
$891$	0	0
$892$	0	0
$893$	203.345	0.227710
$894$	0	0
$895$	− 402.495i	− 0.449715i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1416.54i	− 1.57568i
$900$	0	0
$901$	−1204.64	−1.33700
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1462.66	−1.61620
$906$	0	0
$907$	1426.37i	1.57263i	0.617826	+	0.786315i	$0.288013\pi$
$907$	1426.37i	1.57263i	−0.617826	+	0.786315i	$0.711987\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	502.815i	0.551937i	0.961167	+	0.275969i	$0.0889986\pi$
$911$	502.815i	0.551937i	−0.961167	+	0.275969i	$0.911001\pi$
$912$	0	0
$913$	−1754.82	−1.92204
$914$	0	0
$915$	0	0
$916$	0	0
$917$	264.070	0.287972
$918$	0	0
$919$	− 438.460i	− 0.477105i	−0.971130	−	0.238553i	$-0.923327\pi$
$919$	− 438.460i	− 0.477105i	0.971130	−	0.238553i	$-0.0766729\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1654.92i	1.79297i
$924$	0	0
$925$	5.27709	0.00570496
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−228.035	−0.245462	−0.122731	−	0.992440i	$-0.539165\pi$
$929$	−228.035	−0.245462	−0.122731	+	0.992440i	$0.539165\pi$
$930$	0	0
$931$	− 203.555i	− 0.218641i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1472.05i	− 1.57439i
$936$	0	0
$937$	−750.124	−0.800559	−0.400280	−	0.916393i	$-0.631087\pi$
$937$	−750.124	−0.800559	−0.400280	+	0.916393i	$0.631087\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1191.03	−1.26571	−0.632853	−	0.774272i	$-0.718116\pi$
$941$	−1191.03	−1.26571	−0.632853	+	0.774272i	$0.718116\pi$
$942$	0	0
$943$	1090.95i	1.15689i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	420.827i	0.444379i	0.975004	+	0.222189i	$0.0713203\pi$
$947$	420.827i	0.444379i	−0.975004	+	0.222189i	$0.928680\pi$
$948$	0	0
$949$	1983.97	2.09059
$950$	0	0
$951$	0	0
$952$	0	0
$953$	494.580	0.518971	0.259486	−	0.965747i	$-0.416447\pi$
$953$	494.580	0.518971	0.259486	+	0.965747i	$0.416447\pi$
$954$	0	0
$955$	− 604.666i	− 0.633158i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 298.452i	− 0.311212i
$960$	0	0
$961$	−55.9542	−0.0582250
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1948.38	2.01905
$966$	0	0
$967$	− 994.616i	− 1.02856i	−0.857623	−	0.514279i	$-0.828060\pi$
$967$	− 994.616i	− 1.02856i	0.857623	−	0.514279i	$-0.171940\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1512.76i	1.55794i	0.627062	+	0.778969i	$0.284257\pi$
$971$	1512.76i	1.55794i	−0.627062	+	0.778969i	$0.715743\pi$
$972$	0	0
$973$	293.345	0.301485
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−454.879	−0.465588	−0.232794	−	0.972526i	$-0.574787\pi$
$977$	−454.879	−0.465588	−0.232794	+	0.972526i	$0.574787\pi$
$978$	0	0
$979$	− 616.810i	− 0.630041i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1799.80i	− 1.83093i	−0.402401	−	0.915464i	$-0.631824\pi$
$983$	− 1799.80i	− 1.83093i	0.402401	−	0.915464i	$-0.368176\pi$
$984$	0	0
$985$	710.504	0.721324
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1840.34	−1.86081
$990$	0	0
$991$	− 1579.90i	− 1.59425i	−0.603817	−	0.797123i	$-0.706354\pi$
$991$	− 1579.90i	− 1.59425i	0.603817	−	0.797123i	$-0.293646\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 336.118i	− 0.337807i
$996$	0	0
$997$	−1577.10	−1.58184	−0.790922	−	0.611917i	$-0.790399\pi$
$997$	−1577.10	−1.58184	−0.790922	+	0.611917i	$0.790399\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.d.1711.3		12
3.2	odd	2		912.3.m.c.799.5	✓	12
4.3	odd	2	inner	2736.3.m.d.1711.4		12
12.11	even	2		912.3.m.c.799.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.c.799.5	✓	12	3.2	odd	2
912.3.m.c.799.11	yes	12	12.11	even	2
2736.3.m.d.1711.3		12	1.1	even	1	trivial
2736.3.m.d.1711.4		12	4.3	odd	2	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.32253 q^{5} -1.51700i q^{7} +O(q^{10})\)	\(q-5.32253 q^{5} -1.51700i q^{7} +10.6649i q^{11} -20.0811 q^{13} +25.9327 q^{17} -4.35890i q^{19} -39.6351i q^{23} +3.32932 q^{25} +44.4200 q^{29} -31.8897i q^{31} +8.07425i q^{35} +1.58503 q^{37} -27.5248 q^{41} -46.4321i q^{43} +46.6506i q^{47} +46.6987 q^{49} -46.4525 q^{53} -56.7643i q^{55} +37.8483i q^{59} -90.5803 q^{61} +106.882 q^{65} +100.284i q^{67} -82.4118i q^{71} -98.7982 q^{73} +16.1786 q^{77} +103.210i q^{79} +164.542i q^{83} -138.028 q^{85} -57.8355 q^{89} +30.4629i q^{91} +23.2004i q^{95} +1.55097 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{5}+O(q^{10}) \) 12 * q + 4 * q^5	\( 12 q + 4 q^{5} - 20 q^{17} + 88 q^{25} + 24 q^{29} - 40 q^{37} + 32 q^{41} + 128 q^{49} - 184 q^{53} - 276 q^{61} + 232 q^{65} - 92 q^{73} - 308 q^{77} - 244 q^{85} - 72 q^{89} - 280 q^{97}+O(q^{100}) \) 12 * q + 4 * q^5 - 20 * q^17 + 88 * q^25 + 24 * q^29 - 40 * q^37 + 32 * q^41 + 128 * q^49 - 184 * q^53 - 276 * q^61 + 232 * q^65 - 92 * q^73 - 308 * q^77 - 244 * q^85 - 72 * q^89 - 280 * q^97

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