LMFDB - Embedded newform 2736.3.h.c.305.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(305,2736)]

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([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2736.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$74.5506003290$
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} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 $ x^12 + 156x^10 + 8721x^8 + 208784x^6 + 2024760x^4 + 7117056*x^2 + 6533136
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.7
Root			$1.18282i$ of defining polynomial
Character	$\chi$	$=$	2736.305
Dual form			2736.3.h.c.305.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18282i q^{5} -11.5550 q^{7} +O(q^{10})$	$q+1.18282i q^{5} -11.5550 q^{7} +16.8047i q^{11} -14.6782 q^{13} -23.3981i q^{17} -4.35890 q^{19} +11.6340i q^{23} +23.6009 q^{25} +43.4775i q^{29} -7.00373 q^{31} -13.6675i q^{35} -71.2290 q^{37} -5.74273i q^{41} -78.3909 q^{43} +59.9521i q^{47} +84.5187 q^{49} -96.4896i q^{53} -19.8769 q^{55} +20.9608i q^{59} +29.1844 q^{61} -17.3616i q^{65} +5.31239 q^{67} +84.5651i q^{71} +61.3966 q^{73} -194.179i q^{77} -16.8583 q^{79} -123.534i q^{83} +27.6756 q^{85} -129.705i q^{89} +169.607 q^{91} -5.15578i q^{95} +116.532 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{7}+O(q^{10}) $ 12 * q - 16 * q^7	$ 12 q - 16 q^{7} - 16 q^{13} - 12 q^{25} + 40 q^{31} - 32 q^{37} - 92 q^{43} + 84 q^{55} - 48 q^{61} + 88 q^{67} + 148 q^{73} + 56 q^{79} + 228 q^{85} + 8 q^{91} + 72 q^{97}+O(q^{100}) $ 12 * q - 16 * q^7 - 16 * q^13 - 12 * q^25 + 40 * q^31 - 32 * q^37 - 92 * q^43 + 84 * q^55 - 48 * q^61 + 88 * q^67 + 148 * q^73 + 56 * q^79 + 228 * q^85 + 8 * q^91 + 72 * 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
$5$	1.18282i	0.236563i	0.992980	+	0.118282i	$0.0377386\pi$
$5$	1.18282i	0.236563i	−0.992980	+	0.118282i	$0.962261\pi$
$6$	0	0
$7$	−11.5550	−1.65072	−0.825359	−	0.564608i	$-0.809027\pi$
$7$	−11.5550	−1.65072	−0.825359	+	0.564608i	$0.809027\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.8047i	1.52770i	0.645394	+	0.763850i	$0.276693\pi$
$11$	16.8047i	1.52770i	−0.645394	+	0.763850i	$0.723307\pi$
$12$	0	0
$13$	−14.6782	−1.12909	−0.564546	−	0.825402i	$-0.690949\pi$
$13$	−14.6782	−1.12909	−0.564546	+	0.825402i	$0.690949\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 23.3981i	− 1.37636i	−0.725542	−	0.688178i	$-0.758411\pi$
$17$	− 23.3981i	− 1.37636i	0.725542	−	0.688178i	$-0.241589\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.6340i	0.505827i	0.967489	+	0.252913i	$0.0813888\pi$
$23$	11.6340i	0.505827i	−0.967489	+	0.252913i	$0.918611\pi$
$24$	0	0
$25$	23.6009	0.944038
$26$	0	0
$27$	0	0
$28$	0	0
$29$	43.4775i	1.49922i	0.661878	+	0.749611i	$0.269760\pi$
$29$	43.4775i	1.49922i	−0.661878	+	0.749611i	$0.730240\pi$
$30$	0	0
$31$	−7.00373	−0.225927	−0.112963	−	0.993599i	$-0.536034\pi$
$31$	−7.00373	−0.225927	−0.112963	+	0.993599i	$0.536034\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 13.6675i	− 0.390499i
$36$	0	0
$37$	−71.2290	−1.92511	−0.962554	−	0.271090i	$-0.912616\pi$
$37$	−71.2290	−1.92511	−0.962554	+	0.271090i	$0.912616\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.74273i	− 0.140067i	−0.997545	−	0.0700334i	$-0.977689\pi$
$41$	− 5.74273i	− 0.140067i	0.997545	−	0.0700334i	$-0.0223106\pi$
$42$	0	0
$43$	−78.3909	−1.82305	−0.911523	−	0.411250i	$-0.865092\pi$
$43$	−78.3909	−1.82305	−0.911523	+	0.411250i	$0.865092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	59.9521i	1.27558i	0.770211	+	0.637789i	$0.220151\pi$
$47$	59.9521i	1.27558i	−0.770211	+	0.637789i	$0.779849\pi$
$48$	0	0
$49$	84.5187	1.72487
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 96.4896i	− 1.82056i	−0.413995	−	0.910279i	$-0.635867\pi$
$53$	− 96.4896i	− 1.82056i	0.413995	−	0.910279i	$-0.364133\pi$
$54$	0	0
$55$	−19.8769	−0.361397
$56$	0	0
$57$	0	0
$58$	0	0
$59$	20.9608i	0.355268i	0.984097	+	0.177634i	$0.0568444\pi$
$59$	20.9608i	0.355268i	−0.984097	+	0.177634i	$0.943156\pi$
$60$	0	0
$61$	29.1844	0.478433	0.239216	−	0.970966i	$-0.423109\pi$
$61$	29.1844	0.478433	0.239216	+	0.970966i	$0.423109\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 17.3616i	− 0.267102i
$66$	0	0
$67$	5.31239	0.0792894	0.0396447	−	0.999214i	$-0.487377\pi$
$67$	5.31239	0.0792894	0.0396447	+	0.999214i	$0.487377\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84.5651i	1.19106i	0.803334	+	0.595529i	$0.203057\pi$
$71$	84.5651i	1.19106i	−0.803334	+	0.595529i	$0.796943\pi$
$72$	0	0
$73$	61.3966	0.841050	0.420525	−	0.907281i	$-0.361846\pi$
$73$	61.3966	0.841050	0.420525	+	0.907281i	$0.361846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 194.179i	− 2.52180i
$78$	0	0
$79$	−16.8583	−0.213396	−0.106698	−	0.994291i	$-0.534028\pi$
$79$	−16.8583	−0.213396	−0.106698	+	0.994291i	$0.534028\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 123.534i	− 1.48836i	−0.667980	−	0.744179i	$-0.732841\pi$
$83$	− 123.534i	− 1.48836i	0.667980	−	0.744179i	$-0.267159\pi$
$84$	0	0
$85$	27.6756	0.325595
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 129.705i	− 1.45736i	−0.684854	−	0.728680i	$-0.740134\pi$
$89$	− 129.705i	− 1.45736i	0.684854	−	0.728680i	$-0.259866\pi$
$90$	0	0
$91$	169.607	1.86381
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5.15578i	− 0.0542713i
$96$	0	0
$97$	116.532	1.20136	0.600681	−	0.799489i	$-0.294896\pi$
$97$	116.532	1.20136	0.600681	+	0.799489i	$0.294896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	185.316i	1.83481i	0.397957	+	0.917404i	$0.369719\pi$
$101$	185.316i	1.83481i	−0.397957	+	0.917404i	$0.630281\pi$
$102$	0	0
$103$	150.646	1.46258	0.731291	−	0.682066i	$-0.238918\pi$
$103$	150.646	1.46258	0.731291	+	0.682066i	$0.238918\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 195.599i	− 1.82803i	−0.405683	−	0.914014i	$-0.632966\pi$
$107$	− 195.599i	− 1.82803i	0.405683	−	0.914014i	$-0.367034\pi$
$108$	0	0
$109$	166.873	1.53095	0.765474	−	0.643467i	$-0.222504\pi$
$109$	166.873	1.53095	0.765474	+	0.643467i	$0.222504\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	64.2081i	0.568213i	0.958793	+	0.284107i	$0.0916970\pi$
$113$	64.2081i	0.568213i	−0.958793	+	0.284107i	$0.908303\pi$
$114$	0	0
$115$	−13.7609	−0.119660
$116$	0	0
$117$	0	0
$118$	0	0
$119$	270.365i	2.27198i
$120$	0	0
$121$	−161.398	−1.33386
$122$	0	0
$123$	0	0
$124$	0	0
$125$	57.4860i	0.459888i
$126$	0	0
$127$	−49.2467	−0.387770	−0.193885	−	0.981024i	$-0.562109\pi$
$127$	−49.2467	−0.387770	−0.193885	+	0.981024i	$0.562109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 93.9712i	− 0.717338i	−0.933465	−	0.358669i	$-0.883231\pi$
$131$	− 93.9712i	− 0.717338i	0.933465	−	0.358669i	$-0.116769\pi$
$132$	0	0
$133$	50.3672	0.378701
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 133.054i	− 0.971195i	−0.874183	−	0.485597i	$-0.838602\pi$
$137$	− 133.054i	− 0.971195i	0.874183	−	0.485597i	$-0.161398\pi$
$138$	0	0
$139$	−122.453	−0.880956	−0.440478	−	0.897763i	$-0.645191\pi$
$139$	−122.453	−0.880956	−0.440478	+	0.897763i	$0.645191\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 246.662i	− 1.72491i
$144$	0	0
$145$	−51.4258	−0.354661
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 72.6965i	− 0.487896i	−0.969788	−	0.243948i	$-0.921557\pi$
$149$	− 72.6965i	− 0.487896i	0.969788	−	0.243948i	$-0.0784426\pi$
$150$	0	0
$151$	241.294	1.59797	0.798986	−	0.601350i	$-0.205370\pi$
$151$	241.294	1.59797	0.798986	+	0.601350i	$0.205370\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.28412i	− 0.0534460i
$156$	0	0
$157$	48.6451	0.309841	0.154921	−	0.987927i	$-0.450488\pi$
$157$	48.6451	0.309841	0.154921	+	0.987927i	$0.450488\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 134.431i	− 0.834978i
$162$	0	0
$163$	94.2841	0.578430	0.289215	−	0.957264i	$-0.406606\pi$
$163$	94.2841	0.578430	0.289215	+	0.957264i	$0.406606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 141.605i	− 0.847937i	−0.905677	−	0.423968i	$-0.860637\pi$
$167$	− 141.605i	− 0.847937i	0.905677	−	0.423968i	$-0.139363\pi$
$168$	0	0
$169$	46.4492	0.274848
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 138.037i	− 0.797900i	−0.916973	−	0.398950i	$-0.869375\pi$
$173$	− 138.037i	− 0.797900i	0.916973	−	0.398950i	$-0.130625\pi$
$174$	0	0
$175$	−272.710	−1.55834
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 68.9735i	− 0.385327i	−0.981265	−	0.192664i	$-0.938287\pi$
$179$	− 68.9735i	− 0.385327i	0.981265	−	0.192664i	$-0.0617126\pi$
$180$	0	0
$181$	−274.416	−1.51611	−0.758054	−	0.652192i	$-0.773850\pi$
$181$	−274.416	−1.51611	−0.758054	+	0.652192i	$0.773850\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 84.2508i	− 0.455410i
$186$	0	0
$187$	393.197	2.10266
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 171.950i	− 0.900261i	−0.892963	−	0.450130i	$-0.851378\pi$
$191$	− 171.950i	− 0.900261i	0.892963	−	0.450130i	$-0.148622\pi$
$192$	0	0
$193$	84.5350	0.438005	0.219003	−	0.975724i	$-0.429720\pi$
$193$	84.5350	0.438005	0.219003	+	0.975724i	$0.429720\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.7423i	− 0.135748i	−0.997694	−	0.0678738i	$-0.978378\pi$
$197$	− 26.7423i	− 0.135748i	0.997694	−	0.0678738i	$-0.0216215\pi$
$198$	0	0
$199$	−39.1603	−0.196786	−0.0983928	−	0.995148i	$-0.531370\pi$
$199$	−39.1603	−0.196786	−0.0983928	+	0.995148i	$0.531370\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 502.383i	− 2.47479i
$204$	0	0
$205$	6.79260	0.0331346
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 73.2499i	− 0.350478i
$210$	0	0
$211$	−167.542	−0.794037	−0.397018	−	0.917811i	$-0.629955\pi$
$211$	−167.542	−0.794037	−0.397018	+	0.917811i	$0.629955\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 92.7221i	− 0.431265i
$216$	0	0
$217$	80.9283	0.372941
$218$	0	0
$219$	0	0
$220$	0	0
$221$	343.441i	1.55403i
$222$	0	0
$223$	41.1052	0.184328	0.0921641	−	0.995744i	$-0.470622\pi$
$223$	41.1052	0.184328	0.0921641	+	0.995744i	$0.470622\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 120.298i	− 0.529947i	−0.964256	−	0.264974i	$-0.914637\pi$
$227$	− 120.298i	− 0.529947i	0.964256	−	0.264974i	$-0.0853633\pi$
$228$	0	0
$229$	−54.0963	−0.236228	−0.118114	−	0.993000i	$-0.537685\pi$
$229$	−54.0963	−0.236228	−0.118114	+	0.993000i	$0.537685\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	57.0923i	0.245031i	0.992467	+	0.122516i	$0.0390962\pi$
$233$	57.0923i	0.245031i	−0.992467	+	0.122516i	$0.960904\pi$
$234$	0	0
$235$	−70.9123	−0.301755
$236$	0	0
$237$	0	0
$238$	0	0
$239$	173.405i	0.725543i	0.931878	+	0.362772i	$0.118170\pi$
$239$	173.405i	0.725543i	−0.931878	+	0.362772i	$0.881830\pi$
$240$	0	0
$241$	−90.7079	−0.376381	−0.188191	−	0.982133i	$-0.560262\pi$
$241$	−90.7079	−0.376381	−0.188191	+	0.982133i	$0.560262\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	99.9700i	0.408041i
$246$	0	0
$247$	63.9807	0.259031
$248$	0	0
$249$	0	0
$250$	0	0
$251$	59.5477i	0.237242i	0.992940	+	0.118621i	$0.0378474\pi$
$251$	59.5477i	0.237242i	−0.992940	+	0.118621i	$0.962153\pi$
$252$	0	0
$253$	−195.506	−0.772751
$254$	0	0
$255$	0	0
$256$	0	0
$257$	425.688i	1.65637i	0.560453	+	0.828186i	$0.310627\pi$
$257$	425.688i	1.65637i	−0.560453	+	0.828186i	$0.689373\pi$
$258$	0	0
$259$	823.053	3.17781
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 162.639i	− 0.618400i	−0.950997	−	0.309200i	$-0.899939\pi$
$263$	− 162.639i	− 0.618400i	0.950997	−	0.309200i	$-0.100061\pi$
$264$	0	0
$265$	114.129	0.430677
$266$	0	0
$267$	0	0
$268$	0	0
$269$	184.141i	0.684539i	0.939602	+	0.342269i	$0.111196\pi$
$269$	184.141i	0.684539i	−0.939602	+	0.342269i	$0.888804\pi$
$270$	0	0
$271$	467.251	1.72417	0.862087	−	0.506761i	$-0.169157\pi$
$271$	467.251	1.72417	0.862087	+	0.506761i	$0.169157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	396.607i	1.44221i
$276$	0	0
$277$	−108.683	−0.392359	−0.196179	−	0.980568i	$-0.562853\pi$
$277$	−108.683	−0.392359	−0.196179	+	0.980568i	$0.562853\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	135.042i	0.480577i	0.970701	+	0.240289i	$0.0772421\pi$
$281$	135.042i	0.480577i	−0.970701	+	0.240289i	$0.922758\pi$
$282$	0	0
$283$	311.236	1.09977	0.549886	−	0.835240i	$-0.314671\pi$
$283$	311.236	1.09977	0.549886	+	0.835240i	$0.314671\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	66.3575i	0.231211i
$288$	0	0
$289$	−258.469	−0.894358
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 252.775i	− 0.862714i	−0.902181	−	0.431357i	$-0.858035\pi$
$293$	− 252.775i	− 0.862714i	0.902181	−	0.431357i	$-0.141965\pi$
$294$	0	0
$295$	−24.7928	−0.0840434
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 170.766i	− 0.571125i
$300$	0	0
$301$	905.810	3.00933
$302$	0	0
$303$	0	0
$304$	0	0
$305$	34.5198i	0.113180i
$306$	0	0
$307$	−189.189	−0.616252	−0.308126	−	0.951346i	$-0.599702\pi$
$307$	−189.189	−0.616252	−0.308126	+	0.951346i	$0.599702\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 270.953i	− 0.871232i	−0.900133	−	0.435616i	$-0.856531\pi$
$311$	− 270.953i	− 0.871232i	0.900133	−	0.435616i	$-0.143469\pi$
$312$	0	0
$313$	326.309	1.04252	0.521260	−	0.853398i	$-0.325462\pi$
$313$	326.309	1.04252	0.521260	+	0.853398i	$0.325462\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 205.699i	− 0.648892i	−0.945904	−	0.324446i	$-0.894822\pi$
$317$	− 205.699i	− 0.648892i	0.945904	−	0.324446i	$-0.105178\pi$
$318$	0	0
$319$	−730.625	−2.29036
$320$	0	0
$321$	0	0
$322$	0	0
$323$	101.990i	0.315758i
$324$	0	0
$325$	−346.419	−1.06591
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 692.749i	− 2.10562i
$330$	0	0
$331$	−64.7809	−0.195713	−0.0978564	−	0.995201i	$-0.531199\pi$
$331$	−64.7809	−0.195713	−0.0978564	+	0.995201i	$0.531199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.28358i	0.0187570i
$336$	0	0
$337$	189.465	0.562212	0.281106	−	0.959677i	$-0.409299\pi$
$337$	189.465	0.562212	0.281106	+	0.959677i	$0.409299\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 117.696i	− 0.345148i
$342$	0	0
$343$	−410.419	−1.19656
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 484.445i	− 1.39609i	−0.716052	−	0.698047i	$-0.754053\pi$
$347$	− 484.445i	− 1.39609i	0.716052	−	0.698047i	$-0.245947\pi$
$348$	0	0
$349$	−121.051	−0.346852	−0.173426	−	0.984847i	$-0.555484\pi$
$349$	−121.051	−0.346852	−0.173426	+	0.984847i	$0.555484\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	461.204i	1.30653i	0.757131	+	0.653263i	$0.226600\pi$
$353$	461.204i	1.30653i	−0.757131	+	0.653263i	$0.773400\pi$
$354$	0	0
$355$	−100.025	−0.281760
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 272.559i	− 0.759216i	−0.925147	−	0.379608i	$-0.876059\pi$
$359$	− 272.559i	− 0.759216i	0.925147	−	0.379608i	$-0.123941\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	72.6209i	0.198961i
$366$	0	0
$367$	428.161	1.16665	0.583325	−	0.812239i	$-0.301751\pi$
$367$	428.161	1.16665	0.583325	+	0.812239i	$0.301751\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1114.94i	3.00523i
$372$	0	0
$373$	301.903	0.809392	0.404696	−	0.914451i	$-0.367377\pi$
$373$	301.903	0.809392	0.404696	+	0.914451i	$0.367377\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 638.170i	− 1.69276i
$378$	0	0
$379$	−350.479	−0.924746	−0.462373	−	0.886686i	$-0.653002\pi$
$379$	−350.479	−0.924746	−0.462373	+	0.886686i	$0.653002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	680.296i	1.77623i	0.459621	+	0.888115i	$0.347985\pi$
$383$	680.296i	1.77623i	−0.459621	+	0.888115i	$0.652015\pi$
$384$	0	0
$385$	229.678	0.596565
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 41.4655i	− 0.106595i	−0.998579	−	0.0532976i	$-0.983027\pi$
$389$	− 41.4655i	− 0.106595i	0.998579	−	0.0532976i	$-0.0169732\pi$
$390$	0	0
$391$	272.213	0.696198
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 19.9402i	− 0.0504816i
$396$	0	0
$397$	−636.189	−1.60249	−0.801246	−	0.598335i	$-0.795829\pi$
$397$	−636.189	−1.60249	−0.801246	+	0.598335i	$0.795829\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 39.3996i	− 0.0982535i	−0.998793	−	0.0491267i	$-0.984356\pi$
$401$	− 39.3996i	− 0.0982535i	0.998793	−	0.0491267i	$-0.0156438\pi$
$402$	0	0
$403$	102.802	0.255092
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1196.98i	− 2.94099i
$408$	0	0
$409$	421.712	1.03108	0.515540	−	0.856865i	$-0.327591\pi$
$409$	421.712	1.03108	0.515540	+	0.856865i	$0.327591\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 242.203i	− 0.586448i
$414$	0	0
$415$	146.118	0.352091
$416$	0	0
$417$	0	0
$418$	0	0
$419$	25.5688i	0.0610235i	0.999534	+	0.0305117i	$0.00971369\pi$
$419$	25.5688i	0.0610235i	−0.999534	+	0.0305117i	$0.990286\pi$
$420$	0	0
$421$	−384.575	−0.913480	−0.456740	−	0.889600i	$-0.650983\pi$
$421$	−384.575	−0.913480	−0.456740	+	0.889600i	$0.650983\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 552.216i	− 1.29933i
$426$	0	0
$427$	−337.227	−0.789758
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 672.168i	− 1.55955i	−0.626058	−	0.779777i	$-0.715333\pi$
$431$	− 672.168i	− 1.55955i	0.626058	−	0.779777i	$-0.284667\pi$
$432$	0	0
$433$	−537.952	−1.24238	−0.621192	−	0.783658i	$-0.713351\pi$
$433$	−537.952	−1.24238	−0.621192	+	0.783658i	$0.713351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 50.7115i	− 0.116045i
$438$	0	0
$439$	156.236	0.355890	0.177945	−	0.984040i	$-0.443055\pi$
$439$	156.236	0.355890	0.177945	+	0.984040i	$0.443055\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	89.3191i	0.201623i	0.994906	+	0.100812i	$0.0321439\pi$
$443$	89.3191i	0.201623i	−0.994906	+	0.100812i	$0.967856\pi$
$444$	0	0
$445$	153.417	0.344758
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.3171i	− 0.0229780i	−0.999934	−	0.0114890i	$-0.996343\pi$
$449$	− 10.3171i	− 0.0229780i	0.999934	−	0.0114890i	$-0.00365715\pi$
$450$	0	0
$451$	96.5049	0.213980
$452$	0	0
$453$	0	0
$454$	0	0
$455$	200.614i	0.440909i
$456$	0	0
$457$	116.289	0.254462	0.127231	−	0.991873i	$-0.459391\pi$
$457$	116.289	0.254462	0.127231	+	0.991873i	$0.459391\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	303.341i	0.658007i	0.944329	+	0.329003i	$0.106713\pi$
$461$	303.341i	0.658007i	−0.944329	+	0.329003i	$0.893287\pi$
$462$	0	0
$463$	−140.485	−0.303424	−0.151712	−	0.988425i	$-0.548479\pi$
$463$	−140.485	−0.303424	−0.151712	+	0.988425i	$0.548479\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.8034i	− 0.0638189i	−0.999491	−	0.0319095i	$-0.989841\pi$
$467$	− 29.8034i	− 0.0638189i	0.999491	−	0.0319095i	$-0.0101588\pi$
$468$	0	0
$469$	−61.3848	−0.130884
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1317.34i	− 2.78506i
$474$	0	0
$475$	−102.874	−0.216577
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 285.569i	− 0.596177i	−0.954538	−	0.298088i	$-0.903651\pi$
$479$	− 285.569i	− 0.596177i	0.954538	−	0.298088i	$-0.0963490\pi$
$480$	0	0
$481$	1045.51	2.17362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	137.836i	0.284198i
$486$	0	0
$487$	−290.413	−0.596330	−0.298165	−	0.954514i	$-0.596375\pi$
$487$	−290.413	−0.596330	−0.298165	+	0.954514i	$0.596375\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	146.360i	0.298086i	0.988831	+	0.149043i	$0.0476194\pi$
$491$	146.360i	0.298086i	−0.988831	+	0.149043i	$0.952381\pi$
$492$	0	0
$493$	1017.29	2.06346
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 977.152i	− 1.96610i
$498$	0	0
$499$	−76.5115	−0.153330	−0.0766648	−	0.997057i	$-0.524427\pi$
$499$	−76.5115	−0.153330	−0.0766648	+	0.997057i	$0.524427\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	261.880i	0.520636i	0.965523	+	0.260318i	$0.0838273\pi$
$503$	261.880i	0.520636i	−0.965523	+	0.260318i	$0.916173\pi$
$504$	0	0
$505$	−219.194	−0.434048
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 183.329i	− 0.360174i	−0.983651	−	0.180087i	$-0.942362\pi$
$509$	− 183.329i	− 0.360174i	0.983651	−	0.180087i	$-0.0576380\pi$
$510$	0	0
$511$	−709.440	−1.38834
$512$	0	0
$513$	0	0
$514$	0	0
$515$	178.186i	0.345993i
$516$	0	0
$517$	−1007.48	−1.94870
$518$	0	0
$519$	0	0
$520$	0	0
$521$	384.597i	0.738190i	0.929392	+	0.369095i	$0.120332\pi$
$521$	384.597i	0.738190i	−0.929392	+	0.369095i	$0.879668\pi$
$522$	0	0
$523$	−359.471	−0.687325	−0.343663	−	0.939093i	$-0.611668\pi$
$523$	−359.471	−0.687325	−0.343663	+	0.939093i	$0.611668\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	163.874i	0.310956i
$528$	0	0
$529$	393.650	0.744139
$530$	0	0
$531$	0	0
$532$	0	0
$533$	84.2930i	0.158148i
$534$	0	0
$535$	231.358	0.432444
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1420.31i	2.63508i
$540$	0	0
$541$	562.194	1.03918	0.519588	−	0.854417i	$-0.326085\pi$
$541$	562.194	1.03918	0.519588	+	0.854417i	$0.326085\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	197.381i	0.362166i
$546$	0	0
$547$	−701.537	−1.28252	−0.641259	−	0.767325i	$-0.721588\pi$
$547$	−701.537	−1.28252	−0.641259	+	0.767325i	$0.721588\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 189.514i	− 0.343945i
$552$	0	0
$553$	194.798	0.352256
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 912.042i	− 1.63742i	−0.574208	−	0.818709i	$-0.694690\pi$
$557$	− 912.042i	− 1.63742i	0.574208	−	0.818709i	$-0.305310\pi$
$558$	0	0
$559$	1150.64	2.05838
$560$	0	0
$561$	0	0
$562$	0	0
$563$	563.674i	1.00120i	0.865680	+	0.500599i	$0.166887\pi$
$563$	563.674i	1.00120i	−0.865680	+	0.500599i	$0.833113\pi$
$564$	0	0
$565$	−75.9464	−0.134418
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 499.895i	− 0.878551i	−0.898352	−	0.439275i	$-0.855235\pi$
$569$	− 499.895i	− 0.878551i	0.898352	−	0.439275i	$-0.144765\pi$
$570$	0	0
$571$	−16.8910	−0.0295814	−0.0147907	−	0.999891i	$-0.504708\pi$
$571$	−16.8910	−0.0295814	−0.0147907	+	0.999891i	$0.504708\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	274.574i	0.477520i
$576$	0	0
$577$	642.505	1.11353	0.556763	−	0.830671i	$-0.312043\pi$
$577$	642.505	1.11353	0.556763	+	0.830671i	$0.312043\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1427.44i	2.45686i
$582$	0	0
$583$	1621.48	2.78127
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 547.322i	− 0.932406i	−0.884678	−	0.466203i	$-0.845622\pi$
$587$	− 547.322i	− 0.932406i	0.884678	−	0.466203i	$-0.154378\pi$
$588$	0	0
$589$	30.5286	0.0518312
$590$	0	0
$591$	0	0
$592$	0	0
$593$	739.380i	1.24685i	0.781884	+	0.623424i	$0.214259\pi$
$593$	739.380i	1.24685i	−0.781884	+	0.623424i	$0.785741\pi$
$594$	0	0
$595$	−319.792	−0.537466
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 435.313i	− 0.726733i	−0.931646	−	0.363366i	$-0.881627\pi$
$599$	− 435.313i	− 0.726733i	0.931646	−	0.363366i	$-0.118373\pi$
$600$	0	0
$601$	511.457	0.851009	0.425505	−	0.904956i	$-0.360097\pi$
$601$	511.457	0.851009	0.425505	+	0.904956i	$0.360097\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 190.904i	− 0.315543i
$606$	0	0
$607$	−739.350	−1.21804	−0.609020	−	0.793155i	$-0.708437\pi$
$607$	−739.350	−1.21804	−0.609020	+	0.793155i	$0.708437\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 879.989i	− 1.44024i
$612$	0	0
$613$	−528.345	−0.861900	−0.430950	−	0.902376i	$-0.641821\pi$
$613$	−528.345	−0.861900	−0.430950	+	0.902376i	$0.641821\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	653.241i	1.05874i	0.848392	+	0.529369i	$0.177571\pi$
$617$	653.241i	1.05874i	−0.848392	+	0.529369i	$0.822429\pi$
$618$	0	0
$619$	115.083	0.185918	0.0929591	−	0.995670i	$-0.470367\pi$
$619$	115.083	0.185918	0.0929591	+	0.995670i	$0.470367\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1498.75i	2.40569i
$624$	0	0
$625$	522.028	0.835245
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1666.62i	2.64964i
$630$	0	0
$631$	894.388	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$631$	894.388	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 58.2498i	− 0.0917320i
$636$	0	0
$637$	−1240.58	−1.94754
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1164.18i	− 1.81619i	−0.418764	−	0.908095i	$-0.637537\pi$
$641$	− 1164.18i	− 1.81619i	0.418764	−	0.908095i	$-0.362463\pi$
$642$	0	0
$643$	−9.54128	−0.0148387	−0.00741934	−	0.999972i	$-0.502362\pi$
$643$	−9.54128	−0.0148387	−0.00741934	+	0.999972i	$0.502362\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	774.112i	1.19646i	0.801323	+	0.598232i	$0.204130\pi$
$647$	774.112i	1.19646i	−0.801323	+	0.598232i	$0.795870\pi$
$648$	0	0
$649$	−352.240	−0.542743
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.3044i	0.0295626i	0.999891	+	0.0147813i	$0.00470520\pi$
$653$	19.3044i	0.0295626i	−0.999891	+	0.0147813i	$0.995295\pi$
$654$	0	0
$655$	111.151	0.169696
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1089.39i	− 1.65309i	−0.562867	−	0.826547i	$-0.690302\pi$
$659$	− 1089.39i	− 1.65309i	0.562867	−	0.826547i	$-0.309698\pi$
$660$	0	0
$661$	−757.463	−1.14593	−0.572967	−	0.819578i	$-0.694208\pi$
$661$	−757.463	−1.14593	−0.572967	+	0.819578i	$0.694208\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	59.5751i	0.0895867i
$666$	0	0
$667$	−505.817	−0.758347
$668$	0	0
$669$	0	0
$670$	0	0
$671$	490.435i	0.730902i
$672$	0	0
$673$	−517.955	−0.769621	−0.384810	−	0.922996i	$-0.625733\pi$
$673$	−517.955	−0.769621	−0.384810	+	0.922996i	$0.625733\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	705.089i	1.04149i	0.853712	+	0.520745i	$0.174346\pi$
$677$	705.089i	1.04149i	−0.853712	+	0.520745i	$0.825654\pi$
$678$	0	0
$679$	−1346.53	−1.98311
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1165.61i	1.70661i	0.521412	+	0.853305i	$0.325406\pi$
$683$	1165.61i	1.70661i	−0.521412	+	0.853305i	$0.674594\pi$
$684$	0	0
$685$	157.378	0.229749
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1416.29i	2.05558i
$690$	0	0
$691$	−327.194	−0.473508	−0.236754	−	0.971570i	$-0.576084\pi$
$691$	−327.194	−0.473508	−0.236754	+	0.971570i	$0.576084\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 144.839i	− 0.208402i
$696$	0	0
$697$	−134.369	−0.192782
$698$	0	0
$699$	0	0
$700$	0	0
$701$	486.285i	0.693702i	0.937920	+	0.346851i	$0.112749\pi$
$701$	486.285i	0.693702i	−0.937920	+	0.346851i	$0.887251\pi$
$702$	0	0
$703$	310.480	0.441650
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2141.33i	− 3.02875i
$708$	0	0
$709$	218.065	0.307567	0.153783	−	0.988105i	$-0.450854\pi$
$709$	218.065	0.307567	0.153783	+	0.988105i	$0.450854\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 81.4815i	− 0.114280i
$714$	0	0
$715$	291.756	0.408051
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.2847i	0.0699369i	0.999388	+	0.0349685i	$0.0111331\pi$
$719$	50.2847i	0.0699369i	−0.999388	+	0.0349685i	$0.988867\pi$
$720$	0	0
$721$	−1740.72	−2.41431
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1026.11i	1.41532i
$726$	0	0
$727$	−443.742	−0.610374	−0.305187	−	0.952292i	$-0.598719\pi$
$727$	−443.742	−0.610374	−0.305187	+	0.952292i	$0.598719\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1834.20i	2.50916i
$732$	0	0
$733$	−149.384	−0.203797	−0.101899	−	0.994795i	$-0.532492\pi$
$733$	−149.384	−0.203797	−0.101899	+	0.994795i	$0.532492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	89.2731i	0.121130i
$738$	0	0
$739$	570.189	0.771568	0.385784	−	0.922589i	$-0.373931\pi$
$739$	570.189	0.771568	0.385784	+	0.922589i	$0.373931\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	576.839i	0.776364i	0.921583	+	0.388182i	$0.126897\pi$
$743$	576.839i	0.776364i	−0.921583	+	0.388182i	$0.873103\pi$
$744$	0	0
$745$	85.9866	0.115418
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2260.15i	3.01756i
$750$	0	0
$751$	−1245.80	−1.65885	−0.829426	−	0.558616i	$-0.811333\pi$
$751$	−1245.80	−1.65885	−0.829426	+	0.558616i	$0.811333\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	285.406i	0.378021i
$756$	0	0
$757$	213.255	0.281711	0.140855	−	0.990030i	$-0.455015\pi$
$757$	213.255	0.281711	0.140855	+	0.990030i	$0.455015\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1088.31i	− 1.43011i	−0.699068	−	0.715055i	$-0.746402\pi$
$761$	− 1088.31i	− 1.43011i	0.699068	−	0.715055i	$-0.253598\pi$
$762$	0	0
$763$	−1928.23	−2.52716
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 307.667i	− 0.401130i
$768$	0	0
$769$	−947.668	−1.23234	−0.616169	−	0.787614i	$-0.711316\pi$
$769$	−947.668	−1.23234	−0.616169	+	0.787614i	$0.711316\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 126.680i	− 0.163881i	−0.996637	−	0.0819404i	$-0.973888\pi$
$773$	− 126.680i	− 0.163881i	0.996637	−	0.0819404i	$-0.0261117\pi$
$774$	0	0
$775$	−165.295	−0.213283
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.0320i	0.0321335i
$780$	0	0
$781$	−1421.09	−1.81958
$782$	0	0
$783$	0	0
$784$	0	0
$785$	57.5381i	0.0732970i
$786$	0	0
$787$	1023.40	1.30039	0.650194	−	0.759769i	$-0.274688\pi$
$787$	1023.40	1.30039	0.650194	+	0.759769i	$0.274688\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 741.926i	− 0.937960i
$792$	0	0
$793$	−428.374	−0.540195
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 203.733i	− 0.255625i	−0.991798	−	0.127813i	$-0.959204\pi$
$797$	− 203.733i	− 0.255625i	0.991798	−	0.127813i	$-0.0407956\pi$
$798$	0	0
$799$	1402.76	1.75565
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1031.75i	1.28487i
$804$	0	0
$805$	159.008	0.197525
$806$	0	0
$807$	0	0
$808$	0	0
$809$	405.365i	0.501069i	0.968108	+	0.250534i	$0.0806063\pi$
$809$	405.365i	0.501069i	−0.968108	+	0.250534i	$0.919394\pi$
$810$	0	0
$811$	248.205	0.306049	0.153024	−	0.988222i	$-0.451099\pi$
$811$	248.205	0.306049	0.153024	+	0.988222i	$0.451099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	111.521i	0.136835i
$816$	0	0
$817$	341.698	0.418235
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1275.86i	1.55403i	0.629482	+	0.777015i	$0.283267\pi$
$821$	1275.86i	1.55403i	−0.629482	+	0.777015i	$0.716733\pi$
$822$	0	0
$823$	−1207.10	−1.46671	−0.733355	−	0.679845i	$-0.762047\pi$
$823$	−1207.10	−1.46671	−0.733355	+	0.679845i	$0.762047\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	689.696i	0.833974i	0.908912	+	0.416987i	$0.136914\pi$
$827$	689.696i	0.833974i	−0.908912	+	0.416987i	$0.863086\pi$
$828$	0	0
$829$	−812.726	−0.980369	−0.490184	−	0.871619i	$-0.663071\pi$
$829$	−812.726	−0.980369	−0.490184	+	0.871619i	$0.663071\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1977.57i	− 2.37404i
$834$	0	0
$835$	167.493	0.200591
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 78.7405i	− 0.0938504i	−0.998898	−	0.0469252i	$-0.985058\pi$
$839$	− 78.7405i	− 0.0938504i	0.998898	−	0.0469252i	$-0.0149422\pi$
$840$	0	0
$841$	−1049.29	−1.24767
$842$	0	0
$843$	0	0
$844$	0	0
$845$	54.9409i	0.0650188i
$846$	0	0
$847$	1864.95	2.20183
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 828.679i	− 0.973771i
$852$	0	0
$853$	1460.81	1.71256	0.856280	−	0.516511i	$-0.172770\pi$
$853$	1460.81	1.71256	0.856280	+	0.516511i	$0.172770\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 262.458i	− 0.306253i	−0.988207	−	0.153126i	$-0.951066\pi$
$857$	− 262.458i	− 0.306253i	0.988207	−	0.153126i	$-0.0489341\pi$
$858$	0	0
$859$	143.790	0.167392	0.0836960	−	0.996491i	$-0.473328\pi$
$859$	143.790	0.167392	0.0836960	+	0.996491i	$0.473328\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1162.82i	− 1.34742i	−0.738996	−	0.673710i	$-0.764700\pi$
$863$	− 1162.82i	− 1.34742i	0.738996	−	0.673710i	$-0.235300\pi$
$864$	0	0
$865$	163.272	0.188754
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 283.298i	− 0.326005i
$870$	0	0
$871$	−77.9763	−0.0895250
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 664.252i	− 0.759145i
$876$	0	0
$877$	839.844	0.957633	0.478816	−	0.877915i	$-0.341066\pi$
$877$	839.844	0.957633	0.478816	+	0.877915i	$0.341066\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1520.31i	1.72566i	0.505491	+	0.862832i	$0.331312\pi$
$881$	1520.31i	1.72566i	−0.505491	+	0.862832i	$0.668688\pi$
$882$	0	0
$883$	−966.317	−1.09436	−0.547178	−	0.837016i	$-0.684298\pi$
$883$	−966.317	−1.09436	−0.547178	+	0.837016i	$0.684298\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 363.989i	− 0.410359i	−0.978724	−	0.205180i	$-0.934222\pi$
$887$	− 363.989i	− 0.410359i	0.978724	−	0.205180i	$-0.0657779\pi$
$888$	0	0
$889$	569.047	0.640098
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 261.325i	− 0.292638i
$894$	0	0
$895$	81.5830	0.0911542
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 304.504i	− 0.338715i
$900$	0	0
$901$	−2257.67	−2.50574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 324.583i	− 0.358655i
$906$	0	0
$907$	346.160	0.381654	0.190827	−	0.981624i	$-0.438883\pi$
$907$	346.160	0.381654	0.190827	+	0.981624i	$0.438883\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 108.937i	− 0.119580i	−0.998211	−	0.0597900i	$-0.980957\pi$
$911$	− 108.937i	− 0.119580i	0.998211	−	0.0597900i	$-0.0190431\pi$
$912$	0	0
$913$	2075.95	2.27376
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1085.84i	1.18412i
$918$	0	0
$919$	−972.113	−1.05779	−0.528897	−	0.848686i	$-0.677394\pi$
$919$	−972.113	−1.05779	−0.528897	+	0.848686i	$0.677394\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1241.26i	− 1.34481i
$924$	0	0
$925$	−1681.07	−1.81737
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1247.54i	1.34289i	0.741057	+	0.671443i	$0.234325\pi$
$929$	1247.54i	1.34289i	−0.741057	+	0.671443i	$0.765675\pi$
$930$	0	0
$931$	−368.408	−0.395712
$932$	0	0
$933$	0	0
$934$	0	0
$935$	465.080i	0.497412i
$936$	0	0
$937$	826.848	0.882442	0.441221	−	0.897398i	$-0.354545\pi$
$937$	826.848	0.882442	0.441221	+	0.897398i	$0.354545\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	995.652i	1.05808i	0.848597	+	0.529039i	$0.177448\pi$
$941$	995.652i	1.05808i	−0.848597	+	0.529039i	$0.822552\pi$
$942$	0	0
$943$	66.8111	0.0708495
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 676.969i	− 0.714856i	−0.933941	−	0.357428i	$-0.883654\pi$
$947$	− 676.969i	− 0.714856i	0.933941	−	0.357428i	$-0.116346\pi$
$948$	0	0
$949$	−901.191	−0.949622
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1047.94i	− 1.09962i	−0.835289	−	0.549810i	$-0.814700\pi$
$953$	− 1047.94i	− 1.09962i	0.835289	−	0.549810i	$-0.185300\pi$
$954$	0	0
$955$	203.385	0.212969
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1537.44i	1.60317i
$960$	0	0
$961$	−911.948	−0.948957
$962$	0	0
$963$	0	0
$964$	0	0
$965$	99.9893i	0.103616i
$966$	0	0
$967$	−173.455	−0.179375	−0.0896874	−	0.995970i	$-0.528587\pi$
$967$	−173.455	−0.179375	−0.0896874	+	0.995970i	$0.528587\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 249.875i	− 0.257338i	−0.991688	−	0.128669i	$-0.958929\pi$
$971$	− 249.875i	− 0.257338i	0.991688	−	0.128669i	$-0.0410705\pi$
$972$	0	0
$973$	1414.95	1.45421
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 697.966i	− 0.714397i	−0.934029	−	0.357198i	$-0.883732\pi$
$977$	− 697.966i	− 0.714397i	0.934029	−	0.357198i	$-0.116268\pi$
$978$	0	0
$979$	2179.65	2.22641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 739.100i	− 0.751881i	−0.926644	−	0.375941i	$-0.877320\pi$
$983$	− 739.100i	− 0.751881i	0.926644	−	0.375941i	$-0.122680\pi$
$984$	0	0
$985$	31.6312	0.0321129
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 912.002i	− 0.922145i
$990$	0	0
$991$	−264.660	−0.267064	−0.133532	−	0.991045i	$-0.542632\pi$
$991$	−264.660	−0.267064	−0.133532	+	0.991045i	$0.542632\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 46.3195i	− 0.0465522i
$996$	0	0
$997$	−35.5526	−0.0356596	−0.0178298	−	0.999841i	$-0.505676\pi$
$997$	−35.5526	−0.0356596	−0.0178298	+	0.999841i	$0.505676\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.h.c.305.7		12
3.2	odd	2	inner	2736.3.h.c.305.6		12
4.3	odd	2		684.3.e.a.305.7	yes	12
12.11	even	2		684.3.e.a.305.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.6	✓	12	12.11	even	2
684.3.e.a.305.7	yes	12	4.3	odd	2
2736.3.h.c.305.6		12	3.2	odd	2	inner
2736.3.h.c.305.7		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.18282i q^{5} -11.5550 q^{7} +O(q^{10})\)	\(q+1.18282i q^{5} -11.5550 q^{7} +16.8047i q^{11} -14.6782 q^{13} -23.3981i q^{17} -4.35890 q^{19} +11.6340i q^{23} +23.6009 q^{25} +43.4775i q^{29} -7.00373 q^{31} -13.6675i q^{35} -71.2290 q^{37} -5.74273i q^{41} -78.3909 q^{43} +59.9521i q^{47} +84.5187 q^{49} -96.4896i q^{53} -19.8769 q^{55} +20.9608i q^{59} +29.1844 q^{61} -17.3616i q^{65} +5.31239 q^{67} +84.5651i q^{71} +61.3966 q^{73} -194.179i q^{77} -16.8583 q^{79} -123.534i q^{83} +27.6756 q^{85} -129.705i q^{89} +169.607 q^{91} -5.15578i q^{95} +116.532 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{7}+O(q^{10}) \) 12 * q - 16 * q^7	\( 12 q - 16 q^{7} - 16 q^{13} - 12 q^{25} + 40 q^{31} - 32 q^{37} - 92 q^{43} + 84 q^{55} - 48 q^{61} + 88 q^{67} + 148 q^{73} + 56 q^{79} + 228 q^{85} + 8 q^{91} + 72 q^{97}+O(q^{100}) \) 12 * q - 16 * q^7 - 16 * q^13 - 12 * q^25 + 40 * q^31 - 32 * q^37 - 92 * q^43 + 84 * q^55 - 48 * q^61 + 88 * q^67 + 148 * q^73 + 56 * q^79 + 228 * q^85 + 8 * q^91 + 72 * q^97

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