LMFDB - Embedded newform 2736.2.d.a.2015.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2736,2,Mod(2015,2736)] 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("2736.2015"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$21.8470699930$
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} - 12x^{10} + 47x^{8} - 44x^{6} + 81x^{4} + 848x^{2} + 1600 $ x^12 - 12x^10 + 47x^8 - 44x^6 + 81x^4 + 848*x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.4
Root			$0.366740 - 1.25423i$ of defining polynomial
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.a.2015.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09425i q^{5} +1.25511i q^{7} -2.92198 q^{11} -4.58480 q^{13} -0.680734i q^{17} -1.00000i q^{19} +6.07037 q^{23} +3.80261 q^{25} -1.82773i q^{29} +8.58480i q^{31} +1.37340 q^{35} -0.510210 q^{37} -3.18920i q^{41} +5.38741i q^{43} +10.9937 q^{47} +5.42471 q^{49} +9.03317i q^{53} +3.19739i q^{55} +3.54997 q^{59} +4.29240 q^{61} +5.01693i q^{65} +11.0950i q^{67} +12.1407 q^{71} -5.25511 q^{73} -3.66740i q^{77} -8.26462i q^{79} +6.07037 q^{83} -0.744895 q^{85} +13.9685i q^{89} -5.75441i q^{91} -1.09425 q^{95} -7.67982 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) $ 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

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
$5$	− 1.09425i	− 0.489365i	−0.969603	−	0.244682i	$-0.921316\pi$
$5$	− 1.09425i	− 0.489365i	0.969603	−	0.244682i	$-0.0786837\pi$
$6$	0	0
$7$	1.25511i	0.474385i	0.971463	+	0.237193i	$0.0762272\pi$
$7$	1.25511i	0.474385i	−0.971463	+	0.237193i	$0.923773\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.92198	−0.881011	−0.440506	−	0.897750i	$-0.645201\pi$
$11$	−2.92198	−0.881011	−0.440506	+	0.897750i	$0.645201\pi$
$12$	0	0
$13$	−4.58480	−1.27160	−0.635798	−	0.771856i	$-0.719329\pi$
$13$	−4.58480	−1.27160	−0.635798	+	0.771856i	$0.719329\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.680734i	− 0.165102i	−0.996587	−	0.0825511i	$-0.973693\pi$
$17$	− 0.680734i	− 0.165102i	0.996587	−	0.0825511i	$-0.0263068\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.07037	1.26576	0.632880	−	0.774250i	$-0.281873\pi$
$23$	6.07037	1.26576	0.632880	+	0.774250i	$0.281873\pi$
$24$	0	0
$25$	3.80261	0.760522
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.82773i	− 0.339401i	−0.985496	−	0.169701i	$-0.945720\pi$
$29$	− 1.82773i	− 0.339401i	0.985496	−	0.169701i	$-0.0542801\pi$
$30$	0	0
$31$	8.58480i	1.54188i	0.636910	+	0.770938i	$0.280212\pi$
$31$	8.58480i	1.54188i	−0.636910	+	0.770938i	$0.719788\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.37340	0.232147
$36$	0	0
$37$	−0.510210	−0.0838780	−0.0419390	−	0.999120i	$-0.513354\pi$
$37$	−0.510210	−0.0838780	−0.0419390	+	0.999120i	$0.513354\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.18920i	− 0.498069i	−0.968495	−	0.249035i	$-0.919887\pi$
$41$	− 3.18920i	− 0.498069i	0.968495	−	0.249035i	$-0.0801133\pi$
$42$	0	0
$43$	5.38741i	0.821573i	0.911732	+	0.410787i	$0.134746\pi$
$43$	5.38741i	0.821573i	−0.911732	+	0.410787i	$0.865254\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.9937	1.60360	0.801801	−	0.597591i	$-0.203875\pi$
$47$	10.9937	1.60360	0.801801	+	0.597591i	$0.203875\pi$
$48$	0	0
$49$	5.42471	0.774959
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.03317i	1.24080i	0.784285	+	0.620401i	$0.213030\pi$
$53$	9.03317i	1.24080i	−0.784285	+	0.620401i	$0.786970\pi$
$54$	0	0
$55$	3.19739i	0.431136i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.54997	0.462167	0.231084	−	0.972934i	$-0.425773\pi$
$59$	3.54997	0.462167	0.231084	+	0.972934i	$0.425773\pi$
$60$	0	0
$61$	4.29240	0.549586	0.274793	−	0.961503i	$-0.411391\pi$
$61$	4.29240	0.549586	0.274793	+	0.961503i	$0.411391\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.01693i	0.622274i
$66$	0	0
$67$	11.0950i	1.35547i	0.735306	+	0.677736i	$0.237039\pi$
$67$	11.0950i	1.35547i	−0.735306	+	0.677736i	$0.762961\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.1407	1.44084	0.720421	−	0.693537i	$-0.243949\pi$
$71$	12.1407	1.44084	0.720421	+	0.693537i	$0.243949\pi$
$72$	0	0
$73$	−5.25511	−0.615064	−0.307532	−	0.951538i	$-0.599503\pi$
$73$	−5.25511	−0.615064	−0.307532	+	0.951538i	$0.599503\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.66740i	− 0.417939i
$78$	0	0
$79$	− 8.26462i	− 0.929842i	−0.885352	−	0.464921i	$-0.846083\pi$
$79$	− 8.26462i	− 0.929842i	0.885352	−	0.464921i	$-0.153917\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.07037	0.666310	0.333155	−	0.942872i	$-0.391887\pi$
$83$	6.07037	0.666310	0.333155	+	0.942872i	$0.391887\pi$
$84$	0	0
$85$	−0.744895	−0.0807952
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.9685i	1.48066i	0.672246	+	0.740328i	$0.265330\pi$
$89$	13.9685i	1.48066i	−0.672246	+	0.740328i	$0.734670\pi$
$90$	0	0
$91$	− 5.75441i	− 0.603226i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.09425	−0.112268
$96$	0	0
$97$	−7.67982	−0.779767	−0.389884	−	0.920864i	$-0.627485\pi$
$97$	−7.67982	−0.779767	−0.389884	+	0.920864i	$0.627485\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.96280i	− 0.294809i	−0.989076	−	0.147405i	$-0.952908\pi$
$101$	− 2.96280i	− 0.294809i	0.989076	−	0.147405i	$-0.0470919\pi$
$102$	0	0
$103$	14.6594i	1.44443i	0.691667	+	0.722217i	$0.256877\pi$
$103$	14.6594i	1.44443i	−0.691667	+	0.722217i	$0.743123\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.84397	−0.564958	−0.282479	−	0.959274i	$-0.591157\pi$
$107$	−5.84397	−0.564958	−0.282479	+	0.959274i	$0.591157\pi$
$108$	0	0
$109$	−5.02042	−0.480869	−0.240435	−	0.970665i	$-0.577290\pi$
$109$	−5.02042	−0.480869	−0.240435	+	0.970665i	$0.577290\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 11.1400i	− 1.04797i	−0.851728	−	0.523984i	$-0.824445\pi$
$113$	− 11.1400i	− 1.04797i	0.851728	−	0.523984i	$-0.175555\pi$
$114$	0	0
$115$	− 6.64252i	− 0.619418i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.854393	0.0783221
$120$	0	0
$121$	−2.46201	−0.223819
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.63228i	− 0.861537i
$126$	0	0
$127$	0.320184i	0.0284117i	0.999899	+	0.0142059i	$0.00452201\pi$
$127$	0.320184i	0.0284117i	−0.999899	+	0.0142059i	$0.995478\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.2877	1.16096	0.580478	−	0.814276i	$-0.302866\pi$
$131$	13.2877	1.16096	0.580478	+	0.814276i	$0.302866\pi$
$132$	0	0
$133$	1.25511	0.108831
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.9074i	− 1.87167i	−0.352432	−	0.935837i	$-0.614645\pi$
$137$	− 21.9074i	− 1.87167i	0.352432	−	0.935837i	$-0.385355\pi$
$138$	0	0
$139$	− 1.25511i	− 0.106457i	−0.998582	−	0.0532283i	$-0.983049\pi$
$139$	− 1.25511i	− 0.106457i	0.998582	−	0.0532283i	$-0.0169511\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.3967	1.12029
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.3040i	1.41760i	0.705411	+	0.708799i	$0.250763\pi$
$149$	17.3040i	1.41760i	−0.705411	+	0.708799i	$0.749237\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.39394	0.754540
$156$	0	0
$157$	−1.67982	−0.134064	−0.0670320	−	0.997751i	$-0.521353\pi$
$157$	−1.67982	−0.134064	−0.0670320	+	0.997751i	$0.521353\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.61896i	0.600458i
$162$	0	0
$163$	19.3596i	1.51636i	0.652043	+	0.758182i	$0.273912\pi$
$163$	19.3596i	1.51636i	−0.652043	+	0.758182i	$0.726088\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.0847	1.94111	0.970555	−	0.240881i	$-0.0774364\pi$
$167$	25.0847	1.94111	0.970555	+	0.240881i	$0.0774364\pi$
$168$	0	0
$169$	8.02042	0.616955
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.59147i	0.729226i	0.931159	+	0.364613i	$0.118799\pi$
$173$	9.59147i	0.729226i	−0.931159	+	0.364613i	$0.881201\pi$
$174$	0	0
$175$	4.77268i	0.360780i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.39394	−0.702136	−0.351068	−	0.936350i	$-0.614181\pi$
$179$	−9.39394	−0.702136	−0.351068	+	0.936350i	$0.614181\pi$
$180$	0	0
$181$	10.7748	0.800887	0.400443	−	0.916322i	$-0.368856\pi$
$181$	10.7748	0.800887	0.400443	+	0.916322i	$0.368856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.558299i	0.0410469i
$186$	0	0
$187$	1.98909i	0.145457i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.89380	0.281746	0.140873	−	0.990028i	$-0.455009\pi$
$191$	3.89380	0.281746	0.140873	+	0.990028i	$0.455009\pi$
$192$	0	0
$193$	10.8494	0.780959	0.390479	−	0.920612i	$-0.372309\pi$
$193$	10.8494	0.780959	0.390479	+	0.920612i	$0.372309\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.61965i	− 0.614125i	−0.951689	−	0.307062i	$-0.900654\pi$
$197$	− 8.61965i	− 0.614125i	0.951689	−	0.307062i	$-0.0993460\pi$
$198$	0	0
$199$	10.4078i	0.737792i	0.929471	+	0.368896i	$0.120264\pi$
$199$	10.4078i	0.737792i	−0.929471	+	0.368896i	$0.879736\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.29400	0.161007
$204$	0	0
$205$	−3.48979	−0.243738
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.92198i	0.202118i
$210$	0	0
$211$	16.1154i	1.10943i	0.832040	+	0.554716i	$0.187173\pi$
$211$	16.1154i	1.10943i	−0.832040	+	0.554716i	$0.812827\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.89519	0.402049
$216$	0	0
$217$	−10.7748	−0.731443
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.12103i	0.209943i
$222$	0	0
$223$	− 8.26462i	− 0.553440i	−0.960951	−	0.276720i	$-0.910753\pi$
$223$	− 8.26462i	− 0.553440i	0.960951	−	0.276720i	$-0.0892474\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.74680	0.182312	0.0911559	−	0.995837i	$-0.470944\pi$
$227$	2.74680	0.182312	0.0911559	+	0.995837i	$0.470944\pi$
$228$	0	0
$229$	20.1622	1.33236	0.666179	−	0.745792i	$-0.267929\pi$
$229$	20.1622	1.33236	0.666179	+	0.745792i	$0.267929\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.33759i	− 0.415189i	−0.978215	−	0.207595i	$-0.933437\pi$
$233$	− 6.33759i	− 0.415189i	0.978215	−	0.207595i	$-0.0665635\pi$
$234$	0	0
$235$	− 12.0299i	− 0.784746i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.40298	0.155436	0.0777178	−	0.996975i	$-0.475237\pi$
$239$	2.40298	0.155436	0.0777178	+	0.996975i	$0.475237\pi$
$240$	0	0
$241$	−14.0746	−0.906624	−0.453312	−	0.891352i	$-0.649758\pi$
$241$	−14.0746	−0.906624	−0.453312	+	0.891352i	$0.649758\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.93600i	− 0.379237i
$246$	0	0
$247$	4.58480i	0.291724i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.92477	−0.437087	−0.218544	−	0.975827i	$-0.570131\pi$
$251$	−6.92477	−0.437087	−0.218544	+	0.975827i	$0.570131\pi$
$252$	0	0
$253$	−17.7375	−1.11515
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.3960i	0.773243i	0.922238	+	0.386621i	$0.126358\pi$
$257$	12.3960i	0.773243i	−0.922238	+	0.386621i	$0.873642\pi$
$258$	0	0
$259$	− 0.640367i	− 0.0397905i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.0219	−0.617979	−0.308989	−	0.951065i	$-0.599991\pi$
$263$	−10.0219	−0.617979	−0.308989	+	0.951065i	$0.599991\pi$
$264$	0	0
$265$	9.88457	0.607204
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.76304i	− 0.412350i	−0.978515	−	0.206175i	$-0.933898\pi$
$269$	− 6.76304i	− 0.412350i	0.978515	−	0.206175i	$-0.0661016\pi$
$270$	0	0
$271$	27.9444i	1.69750i	0.528791	+	0.848752i	$0.322645\pi$
$271$	27.9444i	1.69750i	−0.528791	+	0.848752i	$0.677355\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.1112	−0.670029
$276$	0	0
$277$	15.5774	0.935958	0.467979	−	0.883740i	$-0.344982\pi$
$277$	15.5774	0.935958	0.467979	+	0.883740i	$0.344982\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.20613i	− 0.489537i	−0.969582	−	0.244768i	$-0.921288\pi$
$281$	− 8.20613i	− 0.489537i	0.969582	−	0.244768i	$-0.0787119\pi$
$282$	0	0
$283$	− 22.5570i	− 1.34088i	−0.741966	−	0.670438i	$-0.766106\pi$
$283$	− 22.5570i	− 1.34088i	0.741966	−	0.670438i	$-0.233894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00278	0.236277
$288$	0	0
$289$	16.5366	0.972741
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 10.3946i	− 0.607261i	−0.952790	−	0.303631i	$-0.901801\pi$
$293$	− 10.3946i	− 0.607261i	0.952790	−	0.303631i	$-0.0981989\pi$
$294$	0	0
$295$	− 3.88457i	− 0.226168i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.8315	−1.60954
$300$	0	0
$301$	−6.76177	−0.389742
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.69697i	− 0.268948i
$306$	0	0
$307$	5.02042i	0.286531i	0.989684	+	0.143265i	$0.0457602\pi$
$307$	5.02042i	0.286531i	−0.989684	+	0.143265i	$0.954240\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.40298	0.136260	0.0681301	−	0.997676i	$-0.478297\pi$
$311$	2.40298	0.136260	0.0681301	+	0.997676i	$0.478297\pi$
$312$	0	0
$313$	5.67982	0.321042	0.160521	−	0.987032i	$-0.448683\pi$
$313$	5.67982	0.321042	0.160521	+	0.987032i	$0.448683\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.3652i	1.53698i	0.639860	+	0.768491i	$0.278992\pi$
$317$	27.3652i	1.53698i	−0.639860	+	0.768491i	$0.721008\pi$
$318$	0	0
$319$	5.34060i	0.299016i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.680734	−0.0378771
$324$	0	0
$325$	−17.4342	−0.967077
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.7983i	0.760725i
$330$	0	0
$331$	− 26.0746i	− 1.43319i	−0.697490	−	0.716595i	$-0.745700\pi$
$331$	− 26.0746i	− 1.43319i	0.697490	−	0.716595i	$-0.254300\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.1407	0.663320
$336$	0	0
$337$	−31.0394	−1.69083	−0.845413	−	0.534113i	$-0.820646\pi$
$337$	−31.0394	−1.69083	−0.845413	+	0.534113i	$0.820646\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 25.0847i	− 1.35841i
$342$	0	0
$343$	15.5943i	0.842014i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2877	−0.713323	−0.356662	−	0.934234i	$-0.616085\pi$
$347$	−13.2877	−0.713323	−0.356662	+	0.934234i	$0.616085\pi$
$348$	0	0
$349$	1.21426	0.0649981	0.0324991	−	0.999472i	$-0.489653\pi$
$349$	1.21426	0.0649981	0.0324991	+	0.999472i	$0.489653\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.3751i	1.03123i	0.856820	+	0.515615i	$0.172437\pi$
$353$	19.3751i	1.03123i	−0.856820	+	0.515615i	$0.827563\pi$
$354$	0	0
$355$	− 13.2850i	− 0.705097i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.0784	−1.90415	−0.952073	−	0.305872i	$-0.901052\pi$
$359$	−36.0784	−1.90415	−0.952073	+	0.305872i	$0.901052\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.75041i	0.300990i
$366$	0	0
$367$	5.75441i	0.300378i	0.988657	+	0.150189i	$0.0479882\pi$
$367$	5.75441i	0.300378i	−0.988657	+	0.150189i	$0.952012\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.3376	−0.588618
$372$	0	0
$373$	2.58480	0.133836	0.0669180	−	0.997758i	$-0.478683\pi$
$373$	2.58480	0.133836	0.0669180	+	0.997758i	$0.478683\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.37979i	0.431581i
$378$	0	0
$379$	− 7.53063i	− 0.386822i	−0.981118	−	0.193411i	$-0.938045\pi$
$379$	− 7.53063i	− 0.386822i	0.981118	−	0.193411i	$-0.0619552\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.5755	1.35794	0.678972	−	0.734164i	$-0.262426\pi$
$383$	26.5755	1.35794	0.678972	+	0.734164i	$0.262426\pi$
$384$	0	0
$385$	−4.01306	−0.204524
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7.61743i	− 0.386219i	−0.981177	−	0.193110i	$-0.938143\pi$
$389$	− 7.61743i	− 0.386219i	0.981177	−	0.193110i	$-0.0618573\pi$
$390$	0	0
$391$	− 4.13231i	− 0.208980i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.04358	−0.455032
$396$	0	0
$397$	11.7245	0.588435	0.294217	−	0.955739i	$-0.404941\pi$
$397$	11.7245	0.588435	0.294217	+	0.955739i	$0.404941\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.73778i	− 0.236594i	−0.992978	−	0.118297i	$-0.962257\pi$
$401$	− 4.73778i	− 0.236594i	0.992978	−	0.118297i	$-0.0377434\pi$
$402$	0	0
$403$	− 39.3596i	− 1.96064i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.49083	0.0738975
$408$	0	0
$409$	−25.7544	−1.27347	−0.636737	−	0.771081i	$-0.719716\pi$
$409$	−25.7544	−1.27347	−0.636737	+	0.771081i	$0.719716\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.45559i	0.219245i
$414$	0	0
$415$	− 6.64252i	− 0.326068i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.0732	−0.492106	−0.246053	−	0.969256i	$-0.579134\pi$
$419$	−10.0732	−0.492106	−0.246053	+	0.969256i	$0.579134\pi$
$420$	0	0
$421$	−2.77483	−0.135237	−0.0676185	−	0.997711i	$-0.521540\pi$
$421$	−2.77483	−0.135237	−0.0676185	+	0.997711i	$0.521540\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2.58857i	− 0.125564i
$426$	0	0
$427$	5.38741i	0.260715i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.5347	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$431$	21.5347	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$432$	0	0
$433$	24.3055	1.16805	0.584023	−	0.811737i	$-0.301478\pi$
$433$	24.3055	1.16805	0.584023	+	0.811737i	$0.301478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.07037i	− 0.290385i
$438$	0	0
$439$	− 6.39478i	− 0.305206i	−0.988288	−	0.152603i	$-0.951234\pi$
$439$	− 6.39478i	− 0.305206i	0.988288	−	0.152603i	$-0.0487656\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.5873	−1.12067	−0.560333	−	0.828267i	$-0.689327\pi$
$443$	−23.5873	−1.12067	−0.560333	+	0.828267i	$0.689327\pi$
$444$	0	0
$445$	15.2850	0.724580
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.57314i	− 0.121434i	−0.998155	−	0.0607171i	$-0.980661\pi$
$449$	− 2.57314i	− 0.121434i	0.998155	−	0.0607171i	$-0.0193387\pi$
$450$	0	0
$451$	9.31879i	0.438805i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.29678	−0.295197
$456$	0	0
$457$	−7.06508	−0.330490	−0.165245	−	0.986253i	$-0.552842\pi$
$457$	−7.06508	−0.330490	−0.165245	+	0.986253i	$0.552842\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 23.0425i	− 1.07319i	−0.843839	−	0.536597i	$-0.819710\pi$
$461$	− 23.0425i	− 1.07319i	0.843839	−	0.536597i	$-0.180290\pi$
$462$	0	0
$463$	27.5774i	1.28163i	0.767694	+	0.640816i	$0.221404\pi$
$463$	27.5774i	1.28163i	−0.767694	+	0.640816i	$0.778596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.5128	0.532747	0.266373	−	0.963870i	$-0.414175\pi$
$467$	11.5128	0.532747	0.266373	+	0.963870i	$0.414175\pi$
$468$	0	0
$469$	−13.9254	−0.643016
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 15.7419i	− 0.723815i
$474$	0	0
$475$	− 3.80261i	− 0.174476i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.52040	−0.115160	−0.0575800	−	0.998341i	$-0.518338\pi$
$479$	−2.52040	−0.115160	−0.0575800	+	0.998341i	$0.518338\pi$
$480$	0	0
$481$	2.33921	0.106659
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.40366i	0.381590i
$486$	0	0
$487$	− 10.4546i	− 0.473745i	−0.971541	−	0.236873i	$-0.923878\pi$
$487$	− 10.4546i	− 0.473745i	0.971541	−	0.236873i	$-0.0761224\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−41.0018	−1.85038	−0.925192	−	0.379499i	$-0.876096\pi$
$491$	−41.0018	−1.85038	−0.925192	+	0.379499i	$0.876096\pi$
$492$	0	0
$493$	−1.24420	−0.0560359
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.2379i	0.683514i
$498$	0	0
$499$	− 33.0841i	− 1.48105i	−0.672030	−	0.740524i	$-0.734577\pi$
$499$	− 33.0841i	− 1.48105i	0.672030	−	0.740524i	$-0.265423\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.6611	−0.653708	−0.326854	−	0.945075i	$-0.605989\pi$
$503$	−14.6611	−0.653708	−0.326854	+	0.945075i	$0.605989\pi$
$504$	0	0
$505$	−3.24205	−0.144269
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.92351i	0.129582i	0.997899	+	0.0647911i	$0.0206381\pi$
$509$	2.92351i	0.129582i	−0.997899	+	0.0647911i	$0.979362\pi$
$510$	0	0
$511$	− 6.59571i	− 0.291777i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.0411	0.706854
$516$	0	0
$517$	−32.1236	−1.41279
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3.91075i	− 0.171333i	−0.996324	−	0.0856664i	$-0.972698\pi$
$521$	− 3.91075i	− 0.171333i	0.996324	−	0.0856664i	$-0.0273019\pi$
$522$	0	0
$523$	− 11.8290i	− 0.517246i	−0.965978	−	0.258623i	$-0.916731\pi$
$523$	− 11.8290i	− 0.517246i	0.965978	−	0.258623i	$-0.0832688\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.84397	0.254567
$528$	0	0
$529$	13.8494	0.602149
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.6219i	0.633343i
$534$	0	0
$535$	6.39478i	0.276470i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.8509	−0.682748
$540$	0	0
$541$	44.9948	1.93448	0.967239	−	0.253869i	$-0.0817033\pi$
$541$	44.9948	1.93448	0.967239	+	0.253869i	$0.0817033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.49361i	0.235320i
$546$	0	0
$547$	− 11.4152i	− 0.488079i	−0.969765	−	0.244039i	$-0.921527\pi$
$547$	− 11.4152i	− 0.488079i	0.969765	−	0.244039i	$-0.0784726\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.82773	−0.0778640
$552$	0	0
$553$	10.3730	0.441103
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.6234i	1.67890i	0.543440	+	0.839448i	$0.317122\pi$
$557$	39.6234i	1.67890i	−0.543440	+	0.839448i	$0.682878\pi$
$558$	0	0
$559$	− 24.7002i	− 1.04471i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.8315	−1.17296	−0.586478	−	0.809965i	$-0.699486\pi$
$563$	−27.8315	−1.17296	−0.586478	+	0.809965i	$0.699486\pi$
$564$	0	0
$565$	−12.1900	−0.512838
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 34.6523i	− 1.45270i	−0.687326	−	0.726349i	$-0.741216\pi$
$569$	− 34.6523i	− 1.45270i	0.687326	−	0.726349i	$-0.258784\pi$
$570$	0	0
$571$	− 27.9444i	− 1.16944i	−0.811236	−	0.584719i	$-0.801205\pi$
$571$	− 27.9444i	− 1.16944i	0.811236	−	0.584719i	$-0.198795\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	23.0833	0.962639
$576$	0	0
$577$	16.2924	0.678262	0.339131	−	0.940739i	$-0.389867\pi$
$577$	16.2924	0.678262	0.339131	+	0.940739i	$0.389867\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.61896i	0.316088i
$582$	0	0
$583$	− 26.3948i	− 1.09316i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.01915	0.248437	0.124218	−	0.992255i	$-0.460358\pi$
$587$	6.01915	0.248437	0.124218	+	0.992255i	$0.460358\pi$
$588$	0	0
$589$	8.58480	0.353731
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7.52388i	− 0.308969i	−0.987995	−	0.154484i	$-0.950628\pi$
$593$	− 7.52388i	− 0.308969i	0.987995	−	0.154484i	$-0.0493716\pi$
$594$	0	0
$595$	− 0.934921i	− 0.0383280i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.1407	−0.496057	−0.248029	−	0.968753i	$-0.579783\pi$
$599$	−12.1407	−0.496057	−0.248029	+	0.968753i	$0.579783\pi$
$600$	0	0
$601$	28.6256	1.16766	0.583832	−	0.811874i	$-0.301553\pi$
$601$	28.6256	1.16766	0.583832	+	0.811874i	$0.301553\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.69406i	0.109529i
$606$	0	0
$607$	30.7748i	1.24911i	0.780980	+	0.624556i	$0.214720\pi$
$607$	30.7748i	1.24911i	−0.780980	+	0.624556i	$0.785280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−50.4042	−2.03913
$612$	0	0
$613$	8.14322	0.328901	0.164451	−	0.986385i	$-0.447415\pi$
$613$	8.14322	0.328901	0.164451	+	0.986385i	$0.447415\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.93822i	− 0.279322i	−0.990199	−	0.139661i	$-0.955399\pi$
$617$	− 6.93822i	− 0.279322i	0.990199	−	0.139661i	$-0.0446013\pi$
$618$	0	0
$619$	− 42.1900i	− 1.69576i	−0.530188	−	0.847880i	$-0.677879\pi$
$619$	− 42.1900i	− 1.69576i	0.530188	−	0.847880i	$-0.322121\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.5319	−0.702401
$624$	0	0
$625$	8.47291	0.338917
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.347317i	0.0138485i
$630$	0	0
$631$	0.614738i	0.0244723i	0.999925	+	0.0122362i	$0.00389499\pi$
$631$	0.614738i	0.0244723i	−0.999925	+	0.0122362i	$0.996105\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.350362	0.0139037
$636$	0	0
$637$	−24.8712	−0.985434
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.0784i	1.62250i	0.584700	+	0.811250i	$0.301212\pi$
$641$	41.0784i	1.62250i	−0.584700	+	0.811250i	$0.698788\pi$
$642$	0	0
$643$	17.5366i	0.691576i	0.938313	+	0.345788i	$0.112388\pi$
$643$	17.5366i	0.691576i	−0.938313	+	0.345788i	$0.887612\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.3006	−1.19124	−0.595621	−	0.803266i	$-0.703094\pi$
$647$	−30.3006	−1.19124	−0.595621	+	0.803266i	$0.703094\pi$
$648$	0	0
$649$	−10.3730	−0.407175
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.6895i	− 1.43577i	−0.696162	−	0.717885i	$-0.745111\pi$
$653$	− 36.6895i	− 1.43577i	0.696162	−	0.717885i	$-0.254889\pi$
$654$	0	0
$655$	− 14.5401i	− 0.568130i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.7754	−1.58838	−0.794192	−	0.607667i	$-0.792105\pi$
$659$	−40.7754	−1.58838	−0.794192	+	0.607667i	$0.792105\pi$
$660$	0	0
$661$	20.7748	0.808047	0.404024	−	0.914749i	$-0.367611\pi$
$661$	20.7748	0.808047	0.404024	+	0.914749i	$0.367611\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.37340i	− 0.0532582i
$666$	0	0
$667$	− 11.0950i	− 0.429601i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.5423	−0.484191
$672$	0	0
$673$	0.394777	0.0152175	0.00760876	−	0.999971i	$-0.497578\pi$
$673$	0.394777	0.0152175	0.00760876	+	0.999971i	$0.497578\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 20.5579i	− 0.790103i	−0.918659	−	0.395051i	$-0.870727\pi$
$677$	− 20.5579i	− 0.790103i	0.918659	−	0.395051i	$-0.129273\pi$
$678$	0	0
$679$	− 9.63898i	− 0.369910i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.4403	−0.858654	−0.429327	−	0.903149i	$-0.641249\pi$
$683$	−22.4403	−0.858654	−0.429327	+	0.903149i	$0.641249\pi$
$684$	0	0
$685$	−23.9722	−0.915931
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 41.4153i	− 1.57780i
$690$	0	0
$691$	− 23.1974i	− 0.882470i	−0.897392	−	0.441235i	$-0.854540\pi$
$691$	− 23.1974i	− 0.882470i	0.897392	−	0.441235i	$-0.145460\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.37340	−0.0520961
$696$	0	0
$697$	−2.17100	−0.0822324
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 45.3892i	− 1.71433i	−0.515045	−	0.857163i	$-0.672225\pi$
$701$	− 45.3892i	− 1.71433i	0.515045	−	0.857163i	$-0.327775\pi$
$702$	0	0
$703$	0.510210i	0.0192429i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.71862	0.139853
$708$	0	0
$709$	−40.8494	−1.53413	−0.767066	−	0.641568i	$-0.778284\pi$
$709$	−40.8494	−1.53413	−0.767066	+	0.641568i	$0.778284\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	52.1130i	1.95165i
$714$	0	0
$715$	− 14.6594i	− 0.548230i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.3006	1.13002	0.565012	−	0.825083i	$-0.308872\pi$
$719$	30.3006	1.13002	0.565012	+	0.825083i	$0.308872\pi$
$720$	0	0
$721$	−18.3991	−0.685218
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6.95015i	− 0.258122i
$726$	0	0
$727$	− 17.5366i	− 0.650397i	−0.945646	−	0.325198i	$-0.894569\pi$
$727$	− 17.5366i	− 0.650397i	0.945646	−	0.325198i	$-0.105431\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.66740	0.135644
$732$	0	0
$733$	6.51021	0.240460	0.120230	−	0.992746i	$-0.461637\pi$
$733$	6.51021	0.240460	0.120230	+	0.992746i	$0.461637\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 32.4195i	− 1.19419i
$738$	0	0
$739$	20.9349i	0.770104i	0.922895	+	0.385052i	$0.125816\pi$
$739$	20.9349i	0.770104i	−0.922895	+	0.385052i	$0.874184\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.6935	−0.722484	−0.361242	−	0.932472i	$-0.617647\pi$
$743$	−19.6935	−0.722484	−0.361242	+	0.932472i	$0.617647\pi$
$744$	0	0
$745$	18.9349	0.693722
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 7.33479i	− 0.268008i
$750$	0	0
$751$	34.3392i	1.25306i	0.779399	+	0.626528i	$0.215525\pi$
$751$	34.3392i	1.25306i	−0.779399	+	0.626528i	$0.784475\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.2178	−0.589446	−0.294723	−	0.955583i	$-0.595227\pi$
$757$	−16.2178	−0.589446	−0.294723	+	0.955583i	$0.595227\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.6038i	1.76189i	0.473222	+	0.880943i	$0.343091\pi$
$761$	48.6038i	1.76189i	−0.473222	+	0.880943i	$0.656909\pi$
$762$	0	0
$763$	− 6.30115i	− 0.228117i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.2759	−0.587690
$768$	0	0
$769$	−37.9912	−1.37000	−0.685000	−	0.728543i	$-0.740198\pi$
$769$	−37.9912	−1.37000	−0.685000	+	0.728543i	$0.740198\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.36494i	0.228931i	0.993427	+	0.114466i	$0.0365156\pi$
$773$	6.36494i	0.228931i	−0.993427	+	0.114466i	$0.963484\pi$
$774$	0	0
$775$	32.6447i	1.17263i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.18920	−0.114265
$780$	0	0
$781$	−35.4751	−1.26940
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.83814i	0.0656061i
$786$	0	0
$787$	28.9050i	1.03035i	0.857085	+	0.515176i	$0.172273\pi$
$787$	28.9050i	1.03035i	−0.857085	+	0.515176i	$0.827727\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.9819	0.497140
$792$	0	0
$793$	−19.6798	−0.698851
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.7705i	1.19621i	0.801417	+	0.598106i	$0.204080\pi$
$797$	33.7705i	1.19621i	−0.801417	+	0.598106i	$0.795920\pi$
$798$	0	0
$799$	− 7.48382i	− 0.264758i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.3553	0.541878
$804$	0	0
$805$	8.33706	0.293843
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.5841i	1.39170i	0.718186	+	0.695851i	$0.244973\pi$
$809$	39.5841i	1.39170i	−0.718186	+	0.695851i	$0.755027\pi$
$810$	0	0
$811$	19.0394i	0.668565i	0.942473	+	0.334283i	$0.108494\pi$
$811$	19.0394i	0.668565i	−0.942473	+	0.334283i	$0.891506\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.1843	0.742055
$816$	0	0
$817$	5.38741	0.188482
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.8413i	0.866968i	0.901161	+	0.433484i	$0.142716\pi$
$821$	24.8413i	0.866968i	−0.901161	+	0.433484i	$0.857284\pi$
$822$	0	0
$823$	1.25511i	0.0437502i	0.999761	+	0.0218751i	$0.00696362\pi$
$823$	1.25511i	0.0437502i	−0.999761	+	0.0218751i	$0.993036\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.4195	1.12733	0.563667	−	0.826002i	$-0.309390\pi$
$827$	32.4195	1.12733	0.563667	+	0.826002i	$0.309390\pi$
$828$	0	0
$829$	8.90499	0.309283	0.154641	−	0.987971i	$-0.450578\pi$
$829$	8.90499	0.309283	0.154641	+	0.987971i	$0.450578\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.69279i	− 0.127947i
$834$	0	0
$835$	− 27.4489i	− 0.949910i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.0311	−1.07131	−0.535656	−	0.844436i	$-0.679936\pi$
$839$	−31.0311	−1.07131	−0.535656	+	0.844436i	$0.679936\pi$
$840$	0	0
$841$	25.6594	0.884807
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 8.77636i	− 0.301916i
$846$	0	0
$847$	− 3.09008i	− 0.106176i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.09717	−0.106169
$852$	0	0
$853$	45.5088	1.55819	0.779096	−	0.626904i	$-0.215678\pi$
$853$	45.5088	1.55819	0.779096	+	0.626904i	$0.215678\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30.4046i	− 1.03860i	−0.854592	−	0.519301i	$-0.826192\pi$
$857$	− 30.4046i	− 1.03860i	0.854592	−	0.519301i	$-0.173808\pi$
$858$	0	0
$859$	52.1236i	1.77843i	0.457487	+	0.889216i	$0.348750\pi$
$859$	52.1236i	1.77843i	−0.457487	+	0.889216i	$0.651250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.6879	−0.397862	−0.198931	−	0.980013i	$-0.563747\pi$
$863$	−11.6879	−0.397862	−0.198931	+	0.980013i	$0.563747\pi$
$864$	0	0
$865$	10.4955	0.356857
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.1491i	0.819202i
$870$	0	0
$871$	− 50.8685i	− 1.72361i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0895	0.408700
$876$	0	0
$877$	46.7939	1.58012	0.790058	−	0.613032i	$-0.210050\pi$
$877$	46.7939	1.58012	0.790058	+	0.613032i	$0.210050\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43.2550i	− 1.45730i	−0.684888	−	0.728648i	$-0.740149\pi$
$881$	− 43.2550i	− 1.45730i	0.684888	−	0.728648i	$-0.259851\pi$
$882$	0	0
$883$	8.78574i	0.295664i	0.989013	+	0.147832i	$0.0472294\pi$
$883$	8.78574i	0.295664i	−0.989013	+	0.147832i	$0.952771\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.68516	−0.258042	−0.129021	−	0.991642i	$-0.541184\pi$
$887$	−7.68516	−0.258042	−0.129021	+	0.991642i	$0.541184\pi$
$888$	0	0
$889$	−0.401864	−0.0134781
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.9937i	− 0.367892i
$894$	0	0
$895$	10.2793i	0.343601i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.6907	0.523315
$900$	0	0
$901$	6.14919	0.204859
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 11.7904i	− 0.391926i
$906$	0	0
$907$	21.2295i	0.704913i	0.935828	+	0.352457i	$0.114654\pi$
$907$	21.2295i	0.704913i	−0.935828	+	0.352457i	$0.885346\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.74958	0.223624	0.111812	−	0.993729i	$-0.464335\pi$
$911$	6.74958	0.223624	0.111812	+	0.993729i	$0.464335\pi$
$912$	0	0
$913$	−17.7375	−0.587027
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.6775i	0.550740i
$918$	0	0
$919$	− 2.83039i	− 0.0933661i	−0.998910	−	0.0466830i	$-0.985135\pi$
$919$	− 2.83039i	− 0.0933661i	0.998910	−	0.0466830i	$-0.0148651\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−55.6629	−1.83217
$924$	0	0
$925$	−1.94013	−0.0637911
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 39.7323i	− 1.30358i	−0.758401	−	0.651788i	$-0.774019\pi$
$929$	− 39.7323i	− 1.30358i	0.758401	−	0.651788i	$-0.225981\pi$
$930$	0	0
$931$	− 5.42471i	− 0.177788i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.17657	0.0711815
$936$	0	0
$937$	−15.7245	−0.513696	−0.256848	−	0.966452i	$-0.582684\pi$
$937$	−15.7245	−0.513696	−0.256848	+	0.966452i	$0.582684\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44.1756i	1.44008i	0.693931	+	0.720041i	$0.255877\pi$
$941$	44.1756i	1.44008i	−0.693931	+	0.720041i	$0.744123\pi$
$942$	0	0
$943$	− 19.3596i	− 0.630436i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.1703	0.427978	0.213989	−	0.976836i	$-0.431354\pi$
$947$	13.1703	0.427978	0.213989	+	0.976836i	$0.431354\pi$
$948$	0	0
$949$	24.0936	0.782112
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.88407i	0.320177i	0.987103	+	0.160088i	$0.0511779\pi$
$953$	9.88407i	0.320177i	−0.987103	+	0.160088i	$0.948822\pi$
$954$	0	0
$955$	− 4.26080i	− 0.137876i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	27.4961	0.887895
$960$	0	0
$961$	−42.6988	−1.37738
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 11.8720i	− 0.382173i
$966$	0	0
$967$	31.5088i	1.01326i	0.862165	+	0.506628i	$0.169108\pi$
$967$	31.5088i	1.01326i	−0.862165	+	0.506628i	$0.830892\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.5166	−1.13978	−0.569891	−	0.821720i	$-0.693015\pi$
$971$	−35.5166	−1.13978	−0.569891	+	0.821720i	$0.693015\pi$
$972$	0	0
$973$	1.57529	0.0505014
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 14.4790i	− 0.463226i	−0.972808	−	0.231613i	$-0.925600\pi$
$977$	− 14.4790i	− 0.463226i	0.972808	−	0.231613i	$-0.0744002\pi$
$978$	0	0
$979$	− 40.8157i	− 1.30447i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.6907	0.500456	0.250228	−	0.968187i	$-0.419494\pi$
$983$	15.6907	0.500456	0.250228	+	0.968187i	$0.419494\pi$
$984$	0	0
$985$	−9.43207	−0.300531
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.7036i	1.03991i
$990$	0	0
$991$	− 11.0950i	− 0.352445i	−0.984350	−	0.176222i	$-0.943612\pi$
$991$	− 11.0950i	− 0.352445i	0.984350	−	0.176222i	$-0.0563878\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.3888	0.361049
$996$	0	0
$997$	24.8772	0.787869	0.393934	−	0.919139i	$-0.371114\pi$
$997$	24.8772	0.787869	0.393934	+	0.919139i	$0.371114\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.a.2015.4	yes	12
3.2	odd	2	inner	2736.2.d.a.2015.10	yes	12
4.3	odd	2	inner	2736.2.d.a.2015.3	✓	12
12.11	even	2	inner	2736.2.d.a.2015.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.a.2015.3	✓	12	4.3	odd	2	inner
2736.2.d.a.2015.4	yes	12	1.1	even	1	trivial
2736.2.d.a.2015.9	yes	12	12.11	even	2	inner
2736.2.d.a.2015.10	yes	12	3.2	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.09425i q^{5} +1.25511i q^{7} -2.92198 q^{11} -4.58480 q^{13} -0.680734i q^{17} -1.00000i q^{19} +6.07037 q^{23} +3.80261 q^{25} -1.82773i q^{29} +8.58480i q^{31} +1.37340 q^{35} -0.510210 q^{37} -3.18920i q^{41} +5.38741i q^{43} +10.9937 q^{47} +5.42471 q^{49} +9.03317i q^{53} +3.19739i q^{55} +3.54997 q^{59} +4.29240 q^{61} +5.01693i q^{65} +11.0950i q^{67} +12.1407 q^{71} -5.25511 q^{73} -3.66740i q^{77} -8.26462i q^{79} +6.07037 q^{83} -0.744895 q^{85} +13.9685i q^{89} -5.75441i q^{91} -1.09425 q^{95} -7.67982 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) \) 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

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