LMFDB - Embedded newform 4356.3.f.m.1693.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4356,3,Mod(1693,4356)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4356.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4356, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4356.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,92,0,184,0,0,0, 0,0,-104,0,0,0,0,0,144,0,0,0,0,0,0,0,0,0,80,0,-172,0,0,0,32,0,0,0,0,0, -4,0,0,0,0,0,0,0,-316,0,0,0,-132,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12, 0,-656] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$118.692403155$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 45 x^{14} + 278 x^{13} + 569 x^{12} - 3118 x^{11} + 4245 x^{10} - 22868 x^{9} + \cdots + 188375625 $ x^16 - 2x^15 - 45x^14 + 278x^13 + 569x^12 - 3118x^11 + 4245x^10 - 22868x^9 + 188656x^8 - 554550x^7 + 955785x^6 - 3424950x^5 + 11074725x^4 - 11495250x^3 + 10290375x^2 - 111172500*x + 188375625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.9
Root			$-0.638487 + 2.64005i$ of defining polynomial
Character	$\chi$	$=$	4356.1693
Dual form			4356.3.f.m.1693.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.99572 q^{5} -4.27442i q^{7} +5.55938i q^{13} -12.4287i q^{17} +17.4843i q^{19} -5.57255 q^{23} -21.0171 q^{25} -21.4684i q^{29} -25.8537 q^{31} -8.53054i q^{35} +44.5086 q^{37} +34.7764i q^{41} +49.3763i q^{43} +28.6679 q^{47} +30.7294 q^{49} -34.6602 q^{53} -64.2165 q^{59} +30.3578i q^{61} +11.0950i q^{65} +89.0195 q^{67} +20.1148 q^{71} +59.4717i q^{73} +120.703i q^{79} -22.2367i q^{83} -24.8043i q^{85} -118.788 q^{89} +23.7631 q^{91} +34.8938i q^{95} +37.2961 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{5} + 92 q^{23} + 184 q^{25} - 104 q^{31} + 144 q^{37} + 80 q^{47} - 172 q^{49} + 32 q^{53} - 4 q^{59} - 316 q^{67} - 132 q^{71} - 12 q^{89} - 656 q^{91} - 296 q^{97}+O(q^{100}) $ 16 * q + 4 * q^5 + 92 * q^23 + 184 * q^25 - 104 * q^31 + 144 * q^37 + 80 * q^47 - 172 * q^49 + 32 * q^53 - 4 * q^59 - 316 * q^67 - 132 * q^71 - 12 * q^89 - 656 * q^91 - 296 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4356\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$	$2179$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	1.99572	0.399144	0.199572	−	0.979883i	$-0.436045\pi$
$5$	1.99572	0.399144	0.199572	+	0.979883i	$0.436045\pi$
$6$	0	0
$7$	− 4.27442i	− 0.610631i	−0.952251	−	0.305315i	$-0.901238\pi$
$7$	− 4.27442i	− 0.610631i	0.952251	−	0.305315i	$-0.0987619\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.55938i	0.427645i	0.976873	+	0.213822i	$0.0685913\pi$
$13$	5.55938i	0.427645i	−0.976873	+	0.213822i	$0.931409\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 12.4287i	− 0.731101i	−0.930792	−	0.365550i	$-0.880881\pi$
$17$	− 12.4287i	− 0.731101i	0.930792	−	0.365550i	$-0.119119\pi$
$18$	0	0
$19$	17.4843i	0.920225i	0.887861	+	0.460113i	$0.152191\pi$
$19$	17.4843i	0.920225i	−0.887861	+	0.460113i	$0.847809\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.57255	−0.242285	−0.121142	−	0.992635i	$-0.538656\pi$
$23$	−5.57255	−0.242285	−0.121142	+	0.992635i	$0.538656\pi$
$24$	0	0
$25$	−21.0171	−0.840684
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 21.4684i	− 0.740290i	−0.928974	−	0.370145i	$-0.879308\pi$
$29$	− 21.4684i	− 0.740290i	0.928974	−	0.370145i	$-0.120692\pi$
$30$	0	0
$31$	−25.8537	−0.833990	−0.416995	−	0.908909i	$-0.636917\pi$
$31$	−25.8537	−0.833990	−0.416995	+	0.908909i	$0.636917\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 8.53054i	− 0.243730i
$36$	0	0
$37$	44.5086	1.20294	0.601468	−	0.798897i	$-0.294583\pi$
$37$	44.5086	1.20294	0.601468	+	0.798897i	$0.294583\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	34.7764i	0.848204i	0.905614	+	0.424102i	$0.139410\pi$
$41$	34.7764i	0.848204i	−0.905614	+	0.424102i	$0.860590\pi$
$42$	0	0
$43$	49.3763i	1.14829i	0.818755	+	0.574143i	$0.194665\pi$
$43$	49.3763i	1.14829i	−0.818755	+	0.574143i	$0.805335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	28.6679	0.609955	0.304977	−	0.952360i	$-0.401351\pi$
$47$	28.6679	0.609955	0.304977	+	0.952360i	$0.401351\pi$
$48$	0	0
$49$	30.7294	0.627130
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−34.6602	−0.653966	−0.326983	−	0.945030i	$-0.606032\pi$
$53$	−34.6602	−0.653966	−0.326983	+	0.945030i	$0.606032\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−64.2165	−1.08842	−0.544208	−	0.838950i	$-0.683170\pi$
$59$	−64.2165	−1.08842	−0.544208	+	0.838950i	$0.683170\pi$
$60$	0	0
$61$	30.3578i	0.497669i	0.968546	+	0.248834i	$0.0800474\pi$
$61$	30.3578i	0.497669i	−0.968546	+	0.248834i	$0.919953\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.0950i	0.170692i
$66$	0	0
$67$	89.0195	1.32865	0.664325	−	0.747444i	$-0.268719\pi$
$67$	89.0195	1.32865	0.664325	+	0.747444i	$0.268719\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	20.1148	0.283306	0.141653	−	0.989916i	$-0.454758\pi$
$71$	20.1148	0.283306	0.141653	+	0.989916i	$0.454758\pi$
$72$	0	0
$73$	59.4717i	0.814681i	0.913276	+	0.407340i	$0.133544\pi$
$73$	59.4717i	0.814681i	−0.913276	+	0.407340i	$0.866456\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	120.703i	1.52789i	0.645283	+	0.763944i	$0.276739\pi$
$79$	120.703i	1.52789i	−0.645283	+	0.763944i	$0.723261\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 22.2367i	− 0.267912i	−0.990987	−	0.133956i	$-0.957232\pi$
$83$	− 22.2367i	− 0.267912i	0.990987	−	0.133956i	$-0.0427680\pi$
$84$	0	0
$85$	− 24.8043i	− 0.291815i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−118.788	−1.33470	−0.667350	−	0.744744i	$-0.732571\pi$
$89$	−118.788	−1.33470	−0.667350	+	0.744744i	$0.732571\pi$
$90$	0	0
$91$	23.7631	0.261133
$92$	0	0
$93$	0	0
$94$	0	0
$95$	34.8938i	0.367303i
$96$	0	0
$97$	37.2961	0.384496	0.192248	−	0.981346i	$-0.438422\pi$
$97$	37.2961	0.384496	0.192248	+	0.981346i	$0.438422\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 167.707i	− 1.66047i	−0.557415	−	0.830234i	$-0.688207\pi$
$101$	− 167.707i	− 1.66047i	0.557415	−	0.830234i	$-0.311793\pi$
$102$	0	0
$103$	−121.785	−1.18238	−0.591188	−	0.806534i	$-0.701341\pi$
$103$	−121.785	−1.18238	−0.591188	+	0.806534i	$0.701341\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	105.647i	0.987352i	0.869646	+	0.493676i	$0.164347\pi$
$107$	105.647i	0.987352i	−0.869646	+	0.493676i	$0.835653\pi$
$108$	0	0
$109$	88.3582i	0.810625i	0.914178	+	0.405313i	$0.132837\pi$
$109$	88.3582i	0.810625i	−0.914178	+	0.405313i	$0.867163\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	159.499	1.41149	0.705747	−	0.708464i	$-0.250611\pi$
$113$	159.499	1.41149	0.705747	+	0.708464i	$0.250611\pi$
$114$	0	0
$115$	−11.1213	−0.0967066
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−53.1255	−0.446433
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−91.8373	−0.734698
$126$	0	0
$127$	213.671i	1.68245i	0.540686	+	0.841225i	$0.318165\pi$
$127$	213.671i	1.68245i	−0.540686	+	0.841225i	$0.681835\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 260.251i	− 1.98665i	−0.115349	−	0.993325i	$-0.536799\pi$
$131$	− 260.251i	− 1.98665i	0.115349	−	0.993325i	$-0.463201\pi$
$132$	0	0
$133$	74.7351	0.561918
$134$	0	0
$135$	0	0
$136$	0	0
$137$	54.5129	0.397904	0.198952	−	0.980009i	$-0.436246\pi$
$137$	54.5129	0.397904	0.198952	+	0.980009i	$0.436246\pi$
$138$	0	0
$139$	16.9607i	0.122020i	0.998137	+	0.0610099i	$0.0194321\pi$
$139$	16.9607i	0.122020i	−0.998137	+	0.0610099i	$0.980568\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 42.8450i	− 0.295482i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	179.100i	1.20202i	0.799243	+	0.601008i	$0.205234\pi$
$149$	179.100i	1.20202i	−0.799243	+	0.601008i	$0.794766\pi$
$150$	0	0
$151$	− 144.706i	− 0.958315i	−0.877729	−	0.479158i	$-0.840942\pi$
$151$	− 144.706i	− 0.958315i	0.877729	−	0.479158i	$-0.159058\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−51.5967	−0.332882
$156$	0	0
$157$	−100.725	−0.641563	−0.320782	−	0.947153i	$-0.603946\pi$
$157$	−100.725	−0.641563	−0.320782	+	0.947153i	$0.603946\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.8194i	0.147947i
$162$	0	0
$163$	87.1103	0.534419	0.267209	−	0.963638i	$-0.413898\pi$
$163$	87.1103	0.534419	0.267209	+	0.963638i	$0.413898\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 30.0052i	− 0.179672i	−0.995957	−	0.0898359i	$-0.971366\pi$
$167$	− 30.0052i	− 0.179672i	0.995957	−	0.0898359i	$-0.0286343\pi$
$168$	0	0
$169$	138.093	0.817120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	220.968i	1.27727i	0.769509	+	0.638635i	$0.220501\pi$
$173$	220.968i	1.27727i	−0.769509	+	0.638635i	$0.779499\pi$
$174$	0	0
$175$	89.8358i	0.513347i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	316.793	1.76979	0.884897	−	0.465787i	$-0.154229\pi$
$179$	316.793	1.76979	0.884897	+	0.465787i	$0.154229\pi$
$180$	0	0
$181$	−70.5839	−0.389966	−0.194983	−	0.980807i	$-0.562465\pi$
$181$	−70.5839	−0.389966	−0.194983	+	0.980807i	$0.562465\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	88.8269	0.480145
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	296.917	1.55454	0.777270	−	0.629167i	$-0.216604\pi$
$191$	296.917	1.55454	0.777270	+	0.629167i	$0.216604\pi$
$192$	0	0
$193$	136.180i	0.705598i	0.935699	+	0.352799i	$0.114770\pi$
$193$	136.180i	0.705598i	−0.935699	+	0.352799i	$0.885230\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	140.944i	0.715450i	0.933827	+	0.357725i	$0.116448\pi$
$197$	140.944i	0.715450i	−0.933827	+	0.357725i	$0.883552\pi$
$198$	0	0
$199$	289.812	1.45634	0.728172	−	0.685395i	$-0.240370\pi$
$199$	289.812	1.45634	0.728172	+	0.685395i	$0.240370\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−91.7649	−0.452044
$204$	0	0
$205$	69.4040i	0.338556i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.6654i	0.0600255i	0.999550	+	0.0300128i	$0.00955480\pi$
$211$	12.6654i	0.0600255i	−0.999550	+	0.0300128i	$0.990445\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	98.5414i	0.458332i
$216$	0	0
$217$	110.509i	0.509260i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	69.0959	0.312651
$222$	0	0
$223$	238.808	1.07089	0.535445	−	0.844570i	$-0.320144\pi$
$223$	238.808	1.07089	0.535445	+	0.844570i	$0.320144\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	254.939i	1.12308i	0.827449	+	0.561541i	$0.189791\pi$
$227$	254.939i	1.12308i	−0.827449	+	0.561541i	$0.810209\pi$
$228$	0	0
$229$	−366.702	−1.60132	−0.800659	−	0.599121i	$-0.795517\pi$
$229$	−366.702	−1.60132	−0.800659	+	0.599121i	$0.795517\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 436.795i	− 1.87466i	−0.348449	−	0.937328i	$-0.613292\pi$
$233$	− 436.795i	− 1.87466i	0.348449	−	0.937328i	$-0.386708\pi$
$234$	0	0
$235$	57.2131	0.243460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	171.126i	0.716009i	0.933720	+	0.358004i	$0.116543\pi$
$239$	171.126i	0.716009i	−0.933720	+	0.358004i	$0.883457\pi$
$240$	0	0
$241$	29.5990i	0.122817i	0.998113	+	0.0614086i	$0.0195593\pi$
$241$	29.5990i	0.122817i	−0.998113	+	0.0614086i	$0.980441\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	61.3273	0.250315
$246$	0	0
$247$	−97.2017	−0.393529
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−279.128	−1.11206	−0.556032	−	0.831161i	$-0.687677\pi$
$251$	−279.128	−1.11206	−0.556032	+	0.831161i	$0.687677\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−312.004	−1.21402	−0.607012	−	0.794693i	$-0.707632\pi$
$257$	−312.004	−1.21402	−0.607012	+	0.794693i	$0.707632\pi$
$258$	0	0
$259$	− 190.248i	− 0.734550i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	345.558i	1.31391i	0.753930	+	0.656955i	$0.228156\pi$
$263$	345.558i	1.31391i	−0.753930	+	0.656955i	$0.771844\pi$
$264$	0	0
$265$	−69.1722	−0.261027
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−140.545	−0.522471	−0.261235	−	0.965275i	$-0.584130\pi$
$269$	−140.545	−0.522471	−0.261235	+	0.965275i	$0.584130\pi$
$270$	0	0
$271$	− 65.1005i	− 0.240223i	−0.992760	−	0.120112i	$-0.961675\pi$
$271$	− 65.1005i	− 0.240223i	0.992760	−	0.120112i	$-0.0383253\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 153.119i	− 0.552777i	−0.961046	−	0.276389i	$-0.910862\pi$
$277$	− 153.119i	− 0.552777i	0.961046	−	0.276389i	$-0.0891377\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 62.7887i	− 0.223447i	−0.993739	−	0.111724i	$-0.964363\pi$
$281$	− 62.7887i	− 0.223447i	0.993739	−	0.111724i	$-0.0356372\pi$
$282$	0	0
$283$	189.771i	0.670571i	0.942117	+	0.335285i	$0.108833\pi$
$283$	189.771i	0.670571i	−0.942117	+	0.335285i	$0.891167\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	148.649	0.517940
$288$	0	0
$289$	134.527	0.465491
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 171.047i	− 0.583779i	−0.956452	−	0.291890i	$-0.905716\pi$
$293$	− 171.047i	− 0.583779i	0.956452	−	0.291890i	$-0.0942840\pi$
$294$	0	0
$295$	−128.158	−0.434435
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 30.9799i	− 0.103612i
$300$	0	0
$301$	211.055	0.701179
$302$	0	0
$303$	0	0
$304$	0	0
$305$	60.5857i	0.198642i
$306$	0	0
$307$	203.453i	0.662714i	0.943505	+	0.331357i	$0.107506\pi$
$307$	203.453i	0.662714i	−0.943505	+	0.331357i	$0.892494\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	488.348	1.57025	0.785126	−	0.619336i	$-0.212598\pi$
$311$	488.348	1.57025	0.785126	+	0.619336i	$0.212598\pi$
$312$	0	0
$313$	204.475	0.653274	0.326637	−	0.945150i	$-0.394085\pi$
$313$	204.475	0.653274	0.326637	+	0.945150i	$0.394085\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−450.752	−1.42193	−0.710965	−	0.703227i	$-0.751742\pi$
$317$	−450.752	−1.42193	−0.710965	+	0.703227i	$0.751742\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	217.307	0.672778
$324$	0	0
$325$	− 116.842i	− 0.359514i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 122.538i	− 0.372457i
$330$	0	0
$331$	−59.0892	−0.178517	−0.0892587	−	0.996008i	$-0.528450\pi$
$331$	−59.0892	−0.178517	−0.0892587	+	0.996008i	$0.528450\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	177.658	0.530323
$336$	0	0
$337$	552.013i	1.63802i	0.573778	+	0.819011i	$0.305477\pi$
$337$	552.013i	1.63802i	−0.573778	+	0.819011i	$0.694523\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 340.796i	− 0.993576i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	637.931i	1.83842i	0.393772	+	0.919208i	$0.371170\pi$
$347$	637.931i	1.83842i	−0.393772	+	0.919208i	$0.628830\pi$
$348$	0	0
$349$	576.180i	1.65095i	0.564442	+	0.825473i	$0.309092\pi$
$349$	576.180i	1.65095i	−0.564442	+	0.825473i	$0.690908\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	606.017	1.71676	0.858381	−	0.513013i	$-0.171471\pi$
$353$	606.017	1.71676	0.858381	+	0.513013i	$0.171471\pi$
$354$	0	0
$355$	40.1434	0.113080
$356$	0	0
$357$	0	0
$358$	0	0
$359$	432.644i	1.20514i	0.798068	+	0.602568i	$0.205856\pi$
$359$	432.644i	1.20514i	−0.798068	+	0.602568i	$0.794144\pi$
$360$	0	0
$361$	55.3000	0.153185
$362$	0	0
$363$	0	0
$364$	0	0
$365$	118.689i	0.325175i
$366$	0	0
$367$	−78.5581	−0.214055	−0.107027	−	0.994256i	$-0.534133\pi$
$367$	−78.5581	−0.214055	−0.107027	+	0.994256i	$0.534133\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	148.152i	0.399332i
$372$	0	0
$373$	− 708.914i	− 1.90057i	−0.311375	−	0.950287i	$-0.600790\pi$
$373$	− 708.914i	− 1.90057i	0.311375	−	0.950287i	$-0.399210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	119.351	0.316581
$378$	0	0
$379$	−94.8860	−0.250359	−0.125179	−	0.992134i	$-0.539951\pi$
$379$	−94.8860	−0.250359	−0.125179	+	0.992134i	$0.539951\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.2704	0.0868679	0.0434340	−	0.999056i	$-0.486170\pi$
$383$	33.2704	0.0868679	0.0434340	+	0.999056i	$0.486170\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−243.917	−0.627036	−0.313518	−	0.949582i	$-0.601508\pi$
$389$	−243.917	−0.627036	−0.313518	+	0.949582i	$0.601508\pi$
$390$	0	0
$391$	69.2596i	0.177135i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	240.890i	0.609847i
$396$	0	0
$397$	−534.439	−1.34619	−0.673097	−	0.739555i	$-0.735036\pi$
$397$	−534.439	−1.34619	−0.673097	+	0.739555i	$0.735036\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−103.020	−0.256908	−0.128454	−	0.991715i	$-0.541002\pi$
$401$	−103.020	−0.256908	−0.128454	+	0.991715i	$0.541002\pi$
$402$	0	0
$403$	− 143.730i	− 0.356651i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8.50648i	0.0207982i	0.999946	+	0.0103991i	$0.00331020\pi$
$409$	8.50648i	0.0207982i	−0.999946	+	0.0103991i	$0.996690\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	274.488i	0.664620i
$414$	0	0
$415$	− 44.3782i	− 0.106935i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	351.102	0.837952	0.418976	−	0.907997i	$-0.362389\pi$
$419$	351.102	0.837952	0.418976	+	0.907997i	$0.362389\pi$
$420$	0	0
$421$	764.361	1.81558	0.907792	−	0.419421i	$-0.137767\pi$
$421$	764.361	1.81558	0.907792	+	0.419421i	$0.137767\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	261.216i	0.614625i
$426$	0	0
$427$	129.762	0.303892
$428$	0	0
$429$	0	0
$430$	0	0
$431$	384.778i	0.892755i	0.894845	+	0.446378i	$0.147286\pi$
$431$	384.778i	0.892755i	−0.894845	+	0.446378i	$0.852714\pi$
$432$	0	0
$433$	−520.075	−1.20110	−0.600548	−	0.799588i	$-0.705051\pi$
$433$	−520.075	−1.20110	−0.600548	+	0.799588i	$0.705051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 97.4320i	− 0.222957i
$438$	0	0
$439$	− 79.5957i	− 0.181311i	−0.995882	−	0.0906557i	$-0.971104\pi$
$439$	− 79.5957i	− 0.181311i	0.995882	−	0.0906557i	$-0.0288963\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−524.711	−1.18445	−0.592224	−	0.805773i	$-0.701750\pi$
$443$	−524.711	−1.18445	−0.592224	+	0.805773i	$0.701750\pi$
$444$	0	0
$445$	−237.068	−0.532738
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−138.191	−0.307774	−0.153887	−	0.988088i	$-0.549179\pi$
$449$	−138.191	−0.307774	−0.153887	+	0.988088i	$0.549179\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	47.4245	0.104230
$456$	0	0
$457$	647.870i	1.41766i	0.705380	+	0.708829i	$0.250776\pi$
$457$	647.870i	1.41766i	−0.705380	+	0.708829i	$0.749224\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	603.213i	1.30849i	0.756284	+	0.654244i	$0.227013\pi$
$461$	603.213i	1.30849i	−0.756284	+	0.654244i	$0.772987\pi$
$462$	0	0
$463$	−105.557	−0.227986	−0.113993	−	0.993482i	$-0.536364\pi$
$463$	−105.557	−0.227986	−0.113993	+	0.993482i	$0.536364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	280.330	0.600278	0.300139	−	0.953895i	$-0.402967\pi$
$467$	280.330	0.600278	0.300139	+	0.953895i	$0.402967\pi$
$468$	0	0
$469$	− 380.506i	− 0.811314i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 367.469i	− 0.773618i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	779.025i	1.62636i	0.582015	+	0.813178i	$0.302265\pi$
$479$	779.025i	1.62636i	−0.582015	+	0.813178i	$0.697735\pi$
$480$	0	0
$481$	247.440i	0.514429i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	74.4327	0.153469
$486$	0	0
$487$	102.065	0.209579	0.104790	−	0.994494i	$-0.466583\pi$
$487$	102.065	0.209579	0.104790	+	0.994494i	$0.466583\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	527.003i	1.07333i	0.843797	+	0.536663i	$0.180315\pi$
$491$	527.003i	1.07333i	−0.843797	+	0.536663i	$0.819685\pi$
$492$	0	0
$493$	−266.825	−0.541227
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 85.9788i	− 0.172996i
$498$	0	0
$499$	−692.086	−1.38695	−0.693473	−	0.720483i	$-0.743920\pi$
$499$	−692.086	−1.38695	−0.693473	+	0.720483i	$0.743920\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	474.578i	0.943495i	0.881734	+	0.471747i	$0.156377\pi$
$503$	474.578i	0.943495i	−0.881734	+	0.471747i	$0.843623\pi$
$504$	0	0
$505$	− 334.697i	− 0.662766i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	857.022	1.68374	0.841869	−	0.539682i	$-0.181456\pi$
$509$	857.022	1.68374	0.841869	+	0.539682i	$0.181456\pi$
$510$	0	0
$511$	254.207	0.497469
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−243.048	−0.471938
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−206.606	−0.396557	−0.198279	−	0.980146i	$-0.563535\pi$
$521$	−206.606	−0.396557	−0.198279	+	0.980146i	$0.563535\pi$
$522$	0	0
$523$	49.3219i	0.0943058i	0.998888	+	0.0471529i	$0.0150148\pi$
$523$	49.3219i	0.0943058i	−0.998888	+	0.0471529i	$0.984985\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	321.328i	0.609731i
$528$	0	0
$529$	−497.947	−0.941298
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−193.335	−0.362730
$534$	0	0
$535$	210.841i	0.394096i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.1928i	0.0761420i	0.999275	+	0.0380710i	$0.0121213\pi$
$541$	41.1928i	0.0761420i	−0.999275	+	0.0380710i	$0.987879\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	176.338i	0.323556i
$546$	0	0
$547$	− 531.357i	− 0.971402i	−0.874125	−	0.485701i	$-0.838564\pi$
$547$	− 531.357i	− 0.971402i	0.874125	−	0.485701i	$-0.161436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	375.360	0.681233
$552$	0	0
$553$	515.935	0.932975
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 148.347i	− 0.266332i	−0.991094	−	0.133166i	$-0.957486\pi$
$557$	− 148.347i	− 0.266332i	0.991094	−	0.133166i	$-0.0425143\pi$
$558$	0	0
$559$	−274.502	−0.491058
$560$	0	0
$561$	0	0
$562$	0	0
$563$	580.912i	1.03182i	0.856644	+	0.515908i	$0.172545\pi$
$563$	580.912i	1.03182i	−0.856644	+	0.515908i	$0.827455\pi$
$564$	0	0
$565$	318.315	0.563390
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5.08773i	− 0.00894154i	−0.999990	−	0.00447077i	$-0.998577\pi$
$569$	− 5.08773i	− 0.00894154i	0.999990	−	0.00447077i	$-0.00142309\pi$
$570$	0	0
$571$	351.952i	0.616378i	0.951325	+	0.308189i	$0.0997229\pi$
$571$	351.952i	0.616378i	−0.951325	+	0.308189i	$0.900277\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	117.119	0.203685
$576$	0	0
$577$	1056.85	1.83163	0.915817	−	0.401595i	$-0.131544\pi$
$577$	1056.85	1.83163	0.915817	+	0.401595i	$0.131544\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−95.0488	−0.163595
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	784.047	1.33569	0.667843	−	0.744302i	$-0.267218\pi$
$587$	784.047	1.33569	0.667843	+	0.744302i	$0.267218\pi$
$588$	0	0
$589$	− 452.033i	− 0.767458i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 378.692i	− 0.638604i	−0.947653	−	0.319302i	$-0.896551\pi$
$593$	− 378.692i	− 0.638604i	0.947653	−	0.319302i	$-0.103449\pi$
$594$	0	0
$595$	−106.024	−0.178191
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−176.013	−0.293845	−0.146922	−	0.989148i	$-0.546937\pi$
$599$	−176.013	−0.293845	−0.146922	+	0.989148i	$0.546937\pi$
$600$	0	0
$601$	− 468.533i	− 0.779590i	−0.920902	−	0.389795i	$-0.872546\pi$
$601$	− 468.533i	− 0.779590i	0.920902	−	0.389795i	$-0.127454\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1036.78i	− 1.70804i	−0.520243	−	0.854018i	$-0.674159\pi$
$607$	− 1036.78i	− 1.70804i	0.520243	−	0.854018i	$-0.325841\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	159.376i	0.260844i
$612$	0	0
$613$	1087.98i	1.77484i	0.460963	+	0.887419i	$0.347504\pi$
$613$	1087.98i	1.77484i	−0.460963	+	0.887419i	$0.652496\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−301.362	−0.488431	−0.244216	−	0.969721i	$-0.578530\pi$
$617$	−301.362	−0.488431	−0.244216	+	0.969721i	$0.578530\pi$
$618$	0	0
$619$	−67.5843	−0.109183	−0.0545915	−	0.998509i	$-0.517386\pi$
$619$	−67.5843	−0.109183	−0.0545915	+	0.998509i	$0.517386\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	507.751i	0.815009i
$624$	0	0
$625$	342.146	0.547433
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 553.185i	− 0.879468i
$630$	0	0
$631$	−252.237	−0.399742	−0.199871	−	0.979822i	$-0.564052\pi$
$631$	−252.237	−0.399742	−0.199871	+	0.979822i	$0.564052\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	426.428i	0.671540i
$636$	0	0
$637$	170.836i	0.268189i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	597.888	0.932743	0.466372	−	0.884589i	$-0.345561\pi$
$641$	597.888	0.932743	0.466372	+	0.884589i	$0.345561\pi$
$642$	0	0
$643$	−757.964	−1.17879	−0.589397	−	0.807844i	$-0.700634\pi$
$643$	−757.964	−1.17879	−0.589397	+	0.807844i	$0.700634\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	83.5746	0.129173	0.0645863	−	0.997912i	$-0.479427\pi$
$647$	83.5746	0.129173	0.0645863	+	0.997912i	$0.479427\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.23477	0.00342231	0.00171116	−	0.999999i	$-0.499455\pi$
$653$	2.23477	0.00342231	0.00171116	+	0.999999i	$0.499455\pi$
$654$	0	0
$655$	− 519.389i	− 0.792960i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	644.819i	0.978482i	0.872149	+	0.489241i	$0.162726\pi$
$659$	644.819i	0.978482i	−0.872149	+	0.489241i	$0.837274\pi$
$660$	0	0
$661$	729.227	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$661$	729.227	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	149.150	0.224286
$666$	0	0
$667$	119.634i	0.179361i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 185.906i	− 0.276235i	−0.990416	−	0.138117i	$-0.955895\pi$
$673$	− 185.906i	− 0.276235i	0.990416	−	0.138117i	$-0.0441051\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.5555i	0.0274084i	0.999906	+	0.0137042i	$0.00436232\pi$
$677$	18.5555i	0.0274084i	−0.999906	+	0.0137042i	$0.995638\pi$
$678$	0	0
$679$	− 159.419i	− 0.234785i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	410.544	0.601090	0.300545	−	0.953768i	$-0.402831\pi$
$683$	410.544	0.601090	0.300545	+	0.953768i	$0.402831\pi$
$684$	0	0
$685$	108.793	0.158821
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 192.689i	− 0.279665i
$690$	0	0
$691$	112.223	0.162406	0.0812030	−	0.996698i	$-0.474124\pi$
$691$	112.223	0.162406	0.0812030	+	0.996698i	$0.474124\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.8489i	0.0487035i
$696$	0	0
$697$	432.226	0.620123
$698$	0	0
$699$	0	0
$700$	0	0
$701$	56.2228i	0.0802038i	0.999196	+	0.0401019i	$0.0127683\pi$
$701$	56.2228i	0.0802038i	−0.999196	+	0.0401019i	$0.987232\pi$
$702$	0	0
$703$	778.202i	1.10697i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−716.850	−1.01393
$708$	0	0
$709$	−805.555	−1.13619	−0.568093	−	0.822965i	$-0.692319\pi$
$709$	−805.555	−1.13619	−0.568093	+	0.822965i	$0.692319\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	144.071	0.202063
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	294.061	0.408985	0.204493	−	0.978868i	$-0.434446\pi$
$719$	294.061	0.408985	0.204493	+	0.978868i	$0.434446\pi$
$720$	0	0
$721$	520.558i	0.721995i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	451.203i	0.622350i
$726$	0	0
$727$	675.124	0.928644	0.464322	−	0.885666i	$-0.346298\pi$
$727$	675.124	0.928644	0.464322	+	0.885666i	$0.346298\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	613.684	0.839513
$732$	0	0
$733$	360.437i	0.491728i	0.969304	+	0.245864i	$0.0790717\pi$
$733$	360.437i	0.491728i	−0.969304	+	0.245864i	$0.920928\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1112.66i	1.50563i	0.658235	+	0.752813i	$0.271303\pi$
$739$	1112.66i	1.50563i	−0.658235	+	0.752813i	$0.728697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.157513i	0 0.000211996i	1.00000		0.000105998i	$3.37401e-5\pi$
$743$	0.157513i	0 0.000211996i	−1.00000		0.000105998i	$0.999966\pi$
$744$	0	0
$745$	357.434i	0.479778i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	451.578	0.602908
$750$	0	0
$751$	−391.009	−0.520652	−0.260326	−	0.965521i	$-0.583830\pi$
$751$	−391.009	−0.520652	−0.260326	+	0.965521i	$0.583830\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 288.792i	− 0.382506i
$756$	0	0
$757$	−257.842	−0.340611	−0.170305	−	0.985391i	$-0.554475\pi$
$757$	−257.842	−0.340611	−0.170305	+	0.985391i	$0.554475\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1287.40i	− 1.69172i	−0.533402	−	0.845862i	$-0.679087\pi$
$761$	− 1287.40i	− 1.69172i	0.533402	−	0.845862i	$-0.320913\pi$
$762$	0	0
$763$	377.679	0.494993
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 357.004i	− 0.465455i
$768$	0	0
$769$	− 235.786i	− 0.306614i	−0.988179	−	0.153307i	$-0.951008\pi$
$769$	− 235.786i	− 0.306614i	0.988179	−	0.153307i	$-0.0489923\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.6732	0.0189822	0.00949110	−	0.999955i	$-0.496979\pi$
$773$	14.6732	0.0189822	0.00949110	+	0.999955i	$0.496979\pi$
$774$	0	0
$775$	543.369	0.701122
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−608.040	−0.780539
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−201.020	−0.256076
$786$	0	0
$787$	164.869i	0.209490i	0.994499	+	0.104745i	$0.0334027\pi$
$787$	164.869i	0.209490i	−0.994499	+	0.104745i	$0.966597\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 681.765i	− 0.861902i
$792$	0	0
$793$	−168.770	−0.212825
$794$	0	0
$795$	0	0
$796$	0	0
$797$	64.1926	0.0805428	0.0402714	−	0.999189i	$-0.487178\pi$
$797$	64.1926	0.0805428	0.0402714	+	0.999189i	$0.487178\pi$
$798$	0	0
$799$	− 356.305i	− 0.445938i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	47.5369i	0.0590520i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 520.853i	− 0.643824i	−0.946770	−	0.321912i	$-0.895675\pi$
$809$	− 520.853i	− 0.643824i	0.946770	−	0.321912i	$-0.104325\pi$
$810$	0	0
$811$	229.804i	0.283359i	0.989913	+	0.141680i	$0.0452503\pi$
$811$	229.804i	0.283359i	−0.989913	+	0.141680i	$0.954750\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	173.848	0.213310
$816$	0	0
$817$	−863.309	−1.05668
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 897.675i	− 1.09339i	−0.837331	−	0.546696i	$-0.815885\pi$
$821$	− 897.675i	− 1.09339i	0.837331	−	0.546696i	$-0.184115\pi$
$822$	0	0
$823$	−736.485	−0.894878	−0.447439	−	0.894314i	$-0.647664\pi$
$823$	−736.485	−0.894878	−0.447439	+	0.894314i	$0.647664\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 208.213i	− 0.251769i	−0.992045	−	0.125884i	$-0.959823\pi$
$827$	− 208.213i	− 0.251769i	0.992045	−	0.125884i	$-0.0401768\pi$
$828$	0	0
$829$	557.318	0.672277	0.336139	−	0.941813i	$-0.390879\pi$
$829$	557.318	0.672277	0.336139	+	0.941813i	$0.390879\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 381.927i	− 0.458495i
$834$	0	0
$835$	− 59.8820i	− 0.0717150i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	215.364	0.256691	0.128346	−	0.991729i	$-0.459033\pi$
$839$	215.364	0.256691	0.128346	+	0.991729i	$0.459033\pi$
$840$	0	0
$841$	380.108	0.451971
$842$	0	0
$843$	0	0
$844$	0	0
$845$	275.596	0.326149
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−248.027	−0.291453
$852$	0	0
$853$	− 207.821i	− 0.243635i	−0.992552	−	0.121818i	$-0.961128\pi$
$853$	− 207.821i	− 0.243635i	0.992552	−	0.121818i	$-0.0388723\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1643.32i	− 1.91752i	−0.284209	−	0.958762i	$-0.591731\pi$
$857$	− 1643.32i	− 1.91752i	0.284209	−	0.958762i	$-0.408269\pi$
$858$	0	0
$859$	596.456	0.694361	0.347181	−	0.937798i	$-0.387139\pi$
$859$	596.456	0.694361	0.347181	+	0.937798i	$0.387139\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−861.713	−0.998508	−0.499254	−	0.866456i	$-0.666393\pi$
$863$	−861.713	−0.998508	−0.499254	+	0.866456i	$0.666393\pi$
$864$	0	0
$865$	440.990i	0.509815i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	494.893i	0.568190i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	392.551i	0.448629i
$876$	0	0
$877$	− 1423.15i	− 1.62275i	−0.584527	−	0.811374i	$-0.698720\pi$
$877$	− 1423.15i	− 1.62275i	0.584527	−	0.811374i	$-0.301280\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1172.32	−1.33067	−0.665337	−	0.746543i	$-0.731712\pi$
$881$	−1172.32	−1.33067	−0.665337	+	0.746543i	$0.731712\pi$
$882$	0	0
$883$	24.1743	0.0273775	0.0136887	−	0.999906i	$-0.495643\pi$
$883$	24.1743	0.0273775	0.0136887	+	0.999906i	$0.495643\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	190.901i	0.215221i	0.994193	+	0.107610i	$0.0343199\pi$
$887$	190.901i	0.215221i	−0.994193	+	0.107610i	$0.965680\pi$
$888$	0	0
$889$	913.319	1.02736
$890$	0	0
$891$	0	0
$892$	0	0
$893$	501.237i	0.561296i
$894$	0	0
$895$	632.231	0.706403
$896$	0	0
$897$	0	0
$898$	0	0
$899$	555.037i	0.617394i
$900$	0	0
$901$	430.782i	0.478115i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−140.866	−0.155653
$906$	0	0
$907$	−97.9473	−0.107990	−0.0539952	−	0.998541i	$-0.517196\pi$
$907$	−97.9473	−0.107990	−0.0539952	+	0.998541i	$0.517196\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1128.92	1.23921	0.619605	−	0.784914i	$-0.287293\pi$
$911$	1128.92	1.23921	0.619605	+	0.784914i	$0.287293\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1112.42	−1.21311
$918$	0	0
$919$	1244.01i	1.35366i	0.736140	+	0.676829i	$0.236646\pi$
$919$	1244.01i	1.35366i	−0.736140	+	0.676829i	$0.763354\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	111.826i	0.121154i
$924$	0	0
$925$	−935.442	−1.01129
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1400.19	−1.50720	−0.753598	−	0.657336i	$-0.771683\pi$
$929$	−1400.19	−1.50720	−0.753598	+	0.657336i	$0.771683\pi$
$930$	0	0
$931$	537.281i	0.577101i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	520.734i	0.555746i	0.960618	+	0.277873i	$0.0896295\pi$
$937$	520.734i	0.555746i	−0.960618	+	0.277873i	$0.910371\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1618.21i	− 1.71967i	−0.510573	−	0.859835i	$-0.670567\pi$
$941$	− 1618.21i	− 1.71967i	0.510573	−	0.859835i	$-0.329433\pi$
$942$	0	0
$943$	− 193.793i	− 0.205507i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1530.96	−1.61664	−0.808320	−	0.588743i	$-0.799623\pi$
$947$	−1530.96	−1.61664	−0.808320	+	0.588743i	$0.799623\pi$
$948$	0	0
$949$	−330.626	−0.348394
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 165.927i	− 0.174110i	−0.996203	−	0.0870552i	$-0.972254\pi$
$953$	− 165.927i	− 0.174110i	0.996203	−	0.0870552i	$-0.0277456\pi$
$954$	0	0
$955$	592.564	0.620486
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 233.011i	− 0.242973i
$960$	0	0
$961$	−292.587	−0.304461
$962$	0	0
$963$	0	0
$964$	0	0
$965$	271.778i	0.281635i
$966$	0	0
$967$	− 684.767i	− 0.708135i	−0.935220	−	0.354067i	$-0.884798\pi$
$967$	− 684.767i	− 0.708135i	0.935220	−	0.354067i	$-0.115202\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	310.150	0.319413	0.159706	−	0.987165i	$-0.448945\pi$
$971$	310.150	0.319413	0.159706	+	0.987165i	$0.448945\pi$
$972$	0	0
$973$	72.4973	0.0745090
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−350.141	−0.358384	−0.179192	−	0.983814i	$-0.557348\pi$
$977$	−350.141	−0.358384	−0.179192	+	0.983814i	$0.557348\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1016.15	−1.03372	−0.516862	−	0.856069i	$-0.672900\pi$
$983$	−1016.15	−1.03372	−0.516862	+	0.856069i	$0.672900\pi$
$984$	0	0
$985$	281.284i	0.285568i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 275.152i	− 0.278212i
$990$	0	0
$991$	1230.37	1.24154	0.620770	−	0.783993i	$-0.286820\pi$
$991$	1230.37	1.24154	0.620770	+	0.783993i	$0.286820\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	578.385	0.581291
$996$	0	0
$997$	1258.19i	1.26197i	0.775794	+	0.630987i	$0.217350\pi$
$997$	1258.19i	1.26197i	−0.775794	+	0.630987i	$0.782650\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.3.f.m.1693.9		16
3.2	odd	2		1452.3.f.d.241.11		16
11.6	odd	10		396.3.t.c.217.2		16
11.9	even	5		396.3.t.c.73.2		16
11.10	odd	2	inner	4356.3.f.m.1693.10		16
33.17	even	10		132.3.l.a.85.4	yes	16
33.20	odd	10		132.3.l.a.73.4	✓	16
33.32	even	2		1452.3.f.d.241.12		16
132.83	odd	10		528.3.bf.a.481.2		16
132.119	even	10		528.3.bf.a.337.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.l.a.73.4	✓	16	33.20	odd	10
132.3.l.a.85.4	yes	16	33.17	even	10
396.3.t.c.73.2		16	11.9	even	5
396.3.t.c.217.2		16	11.6	odd	10
528.3.bf.a.337.2		16	132.119	even	10
528.3.bf.a.481.2		16	132.83	odd	10
1452.3.f.d.241.11		16	3.2	odd	2
1452.3.f.d.241.12		16	33.32	even	2
4356.3.f.m.1693.9		16	1.1	even	1	trivial
4356.3.f.m.1693.10		16	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4356.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.99572 q^{5} -4.27442i q^{7} +5.55938i q^{13} -12.4287i q^{17} +17.4843i q^{19} -5.57255 q^{23} -21.0171 q^{25} -21.4684i q^{29} -25.8537 q^{31} -8.53054i q^{35} +44.5086 q^{37} +34.7764i q^{41} +49.3763i q^{43} +28.6679 q^{47} +30.7294 q^{49} -34.6602 q^{53} -64.2165 q^{59} +30.3578i q^{61} +11.0950i q^{65} +89.0195 q^{67} +20.1148 q^{71} +59.4717i q^{73} +120.703i q^{79} -22.2367i q^{83} -24.8043i q^{85} -118.788 q^{89} +23.7631 q^{91} +34.8938i q^{95} +37.2961 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{5} + 92 q^{23} + 184 q^{25} - 104 q^{31} + 144 q^{37} + 80 q^{47} - 172 q^{49} + 32 q^{53} - 4 q^{59} - 316 q^{67} - 132 q^{71} - 12 q^{89} - 656 q^{91} - 296 q^{97}+O(q^{100}) \) 16 * q + 4 * q^5 + 92 * q^23 + 184 * q^25 - 104 * q^31 + 144 * q^37 + 80 * q^47 - 172 * q^49 + 32 * q^53 - 4 * q^59 - 316 * q^67 - 132 * q^71 - 12 * q^89 - 656 * q^91 - 296 * q^97

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