LMFDB - Embedded newform 3249.2.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,12,0,0,-6,0,0,-12,0,0,-18,0,0,-12,0,0,0,0,0,18,0,0,12, 0,0,6,0,0,-24,0,0,-30,0,0,-24,0,0,-42,0,0,18,0,0,-6,0,0,0,0,0,-60,0,0, 0,0,0,-24,0,0,-18,0,0,-30,0,0,0,0,0,-42,0,0,0,0,0,0,0,0,-54,0,0,-66,0, 0,-48,0,0,0,0,0,-18] 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:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.21415104.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 45x^{2} - 51 $ x^6 - 12x^4 + 45x^2 - 51
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 171)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.45623$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.45623 q^{2} +0.120615 q^{4} -2.73682 q^{5} -4.41147 q^{7} +2.73682 q^{8} +3.98545 q^{10} -3.68731 q^{11} +0.758770 q^{13} +6.42413 q^{14} -4.22668 q^{16} +6.15503 q^{17} -0.330101 q^{20} +5.36959 q^{22} -0.505744 q^{23} +2.49020 q^{25} -1.10495 q^{26} -0.532089 q^{28} +3.18157 q^{29} -1.77332 q^{31} +0.681388 q^{32} -8.96316 q^{34} +12.0734 q^{35} -2.81521 q^{37} -7.49020 q^{40} +11.8043 q^{41} +2.30541 q^{43} -0.444744 q^{44} +0.736482 q^{46} +8.14947 q^{47} +12.4611 q^{49} -3.62631 q^{50} +0.0915189 q^{52} -9.84235 q^{53} +10.0915 q^{55} -12.0734 q^{56} -4.63310 q^{58} +9.60570 q^{59} -7.24897 q^{61} +2.58236 q^{62} +7.46110 q^{64} -2.07662 q^{65} +6.12836 q^{67} +0.742388 q^{68} -17.5817 q^{70} +2.83028 q^{71} +4.80066 q^{73} +4.09960 q^{74} +16.2665 q^{77} -16.5175 q^{79} +11.5677 q^{80} -17.1898 q^{82} +0.236643 q^{83} -16.8452 q^{85} -3.35721 q^{86} -10.0915 q^{88} +7.78691 q^{89} -3.34730 q^{91} -0.0610002 q^{92} -11.8675 q^{94} +4.94356 q^{97} -18.1463 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 6 q^{7} - 12 q^{10} - 18 q^{13} - 12 q^{16} + 18 q^{22} + 12 q^{25} + 6 q^{28} - 24 q^{31} - 30 q^{34} - 24 q^{37} - 42 q^{40} + 18 q^{43} - 6 q^{46} - 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots - 6 q^{94}+O(q^{100}) $ 6 * q + 12 * q^4 - 6 * q^7 - 12 * q^10 - 18 * q^13 - 12 * q^16 + 18 * q^22 + 12 * q^25 + 6 * q^28 - 24 * q^31 - 30 * q^34 - 24 * q^37 - 42 * q^40 + 18 * q^43 - 6 * q^46 - 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 - 42 * q^70 - 54 * q^79 - 66 * q^82 - 48 * q^85 - 18 * q^91 - 6 * q^94

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 $

$2$	−1.45623	−1.02971	−0.514856	−	0.857277i	$-0.672155\pi$
$2$	−1.45623	−1.02971	−0.514856	+	0.857277i	$0.672155\pi$
$3$	0	0
$3$	0	0
$4$	0.120615	0.0603074
$5$	−2.73682	−1.22394	−0.611972	−	0.790879i	$-0.709624\pi$
$5$	−2.73682	−1.22394	−0.611972	+	0.790879i	$0.709624\pi$
$6$	0	0
$7$	−4.41147	−1.66738	−0.833690	−	0.552232i	$-0.813776\pi$
$7$	−4.41147	−1.66738	−0.833690	+	0.552232i	$0.813776\pi$
$8$	2.73682	0.967613
$9$	0	0
$10$	3.98545	1.26031
$11$	−3.68731	−1.11177	−0.555883	−	0.831260i	$-0.687620\pi$
$11$	−3.68731	−1.11177	−0.555883	+	0.831260i	$0.687620\pi$
$12$	0	0
$13$	0.758770	0.210445	0.105223	−	0.994449i	$-0.466445\pi$
$13$	0.758770	0.210445	0.105223	+	0.994449i	$0.466445\pi$
$14$	6.42413	1.71692
$15$	0	0
$16$	−4.22668	−1.05667
$17$	6.15503	1.49281	0.746407	−	0.665489i	$-0.231777\pi$
$17$	6.15503	1.49281	0.746407	+	0.665489i	$0.231777\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−0.330101	−0.0738129
$21$	0	0
$22$	5.36959	1.14480
$23$	−0.505744	−0.105455	−0.0527275	−	0.998609i	$-0.516791\pi$
$23$	−0.505744	−0.105455	−0.0527275	+	0.998609i	$0.516791\pi$
$24$	0	0
$25$	2.49020	0.498040
$26$	−1.10495	−0.216698
$27$	0	0
$28$	−0.532089	−0.100555
$29$	3.18157	0.590802	0.295401	−	0.955373i	$-0.404547\pi$
$29$	3.18157	0.590802	0.295401	+	0.955373i	$0.404547\pi$
$30$	0	0
$31$	−1.77332	−0.318497	−0.159249	−	0.987238i	$-0.550907\pi$
$31$	−1.77332	−0.318497	−0.159249	+	0.987238i	$0.550907\pi$
$32$	0.681388	0.120453
$33$	0	0
$34$	−8.96316	−1.53717
$35$	12.0734	2.04078
$36$	0	0
$37$	−2.81521	−0.462817	−0.231409	−	0.972857i	$-0.574333\pi$
$37$	−2.81521	−0.462817	−0.231409	+	0.972857i	$0.574333\pi$
$38$	0	0
$39$	0	0
$40$	−7.49020	−1.18430
$41$	11.8043	1.84353	0.921763	−	0.387754i	$-0.126749\pi$
$41$	11.8043	1.84353	0.921763	+	0.387754i	$0.126749\pi$
$42$	0	0
$43$	2.30541	0.351571	0.175786	−	0.984428i	$-0.443753\pi$
$43$	2.30541	0.351571	0.175786	+	0.984428i	$0.443753\pi$
$44$	−0.444744	−0.0670477
$45$	0	0
$46$	0.736482	0.108588
$47$	8.14947	1.18872	0.594361	−	0.804198i	$-0.297405\pi$
$47$	8.14947	1.18872	0.594361	+	0.804198i	$0.297405\pi$
$48$	0	0
$49$	12.4611	1.78016
$50$	−3.62631	−0.512838
$51$	0	0
$52$	0.0915189	0.0126914
$53$	−9.84235	−1.35195	−0.675975	−	0.736925i	$-0.736277\pi$
$53$	−9.84235	−1.35195	−0.675975	+	0.736925i	$0.736277\pi$
$54$	0	0
$55$	10.0915	1.36074
$56$	−12.0734	−1.61338
$57$	0	0
$58$	−4.63310	−0.608356
$59$	9.60570	1.25056	0.625278	−	0.780402i	$-0.284986\pi$
$59$	9.60570	1.25056	0.625278	+	0.780402i	$0.284986\pi$
$60$	0	0
$61$	−7.24897	−0.928136	−0.464068	−	0.885800i	$-0.653611\pi$
$61$	−7.24897	−0.928136	−0.464068	+	0.885800i	$0.653611\pi$
$62$	2.58236	0.327961
$63$	0	0
$64$	7.46110	0.932638
$65$	−2.07662	−0.257573
$66$	0	0
$67$	6.12836	0.748698	0.374349	−	0.927288i	$-0.377866\pi$
$67$	6.12836	0.748698	0.374349	+	0.927288i	$0.377866\pi$
$68$	0.742388	0.0900278
$69$	0	0
$70$	−17.5817	−2.10142
$71$	2.83028	0.335893	0.167946	−	0.985796i	$-0.446286\pi$
$71$	2.83028	0.335893	0.167946	+	0.985796i	$0.446286\pi$
$72$	0	0
$73$	4.80066	0.561875	0.280937	−	0.959726i	$-0.409355\pi$
$73$	4.80066	0.561875	0.280937	+	0.959726i	$0.409355\pi$
$74$	4.09960	0.476569
$75$	0	0
$76$	0	0
$77$	16.2665	1.85374
$78$	0	0
$79$	−16.5175	−1.85837	−0.929184	−	0.369617i	$-0.879489\pi$
$79$	−16.5175	−1.85837	−0.929184	+	0.369617i	$0.879489\pi$
$80$	11.5677	1.29331
$81$	0	0
$82$	−17.1898	−1.89830
$83$	0.236643	0.0259750	0.0129875	−	0.999916i	$-0.495866\pi$
$83$	0.236643	0.0259750	0.0129875	+	0.999916i	$0.495866\pi$
$84$	0	0
$85$	−16.8452	−1.82712
$86$	−3.35721	−0.362017
$87$	0	0
$88$	−10.0915	−1.07576
$89$	7.78691	0.825411	0.412705	−	0.910865i	$-0.364584\pi$
$89$	7.78691	0.825411	0.412705	+	0.910865i	$0.364584\pi$
$90$	0	0
$91$	−3.34730	−0.350892
$92$	−0.0610002	−0.00635971
$93$	0	0
$94$	−11.8675	−1.22404
$95$	0	0
$96$	0	0
$97$	4.94356	0.501943	0.250971	−	0.967995i	$-0.419250\pi$
$97$	4.94356	0.501943	0.250971	+	0.967995i	$0.419250\pi$
$98$	−18.1463	−1.83305
$99$	0	0
$100$	0.300355	0.0300355
$101$	5.74275	0.571425	0.285712	−	0.958315i	$-0.407770\pi$
$101$	5.74275	0.571425	0.285712	+	0.958315i	$0.407770\pi$
$102$	0	0
$103$	−15.0077	−1.47876	−0.739378	−	0.673290i	$-0.764880\pi$
$103$	−15.0077	−1.47876	−0.739378	+	0.673290i	$0.764880\pi$
$104$	2.07662	0.203629
$105$	0	0
$106$	14.3327	1.39212
$107$	2.56118	0.247599	0.123799	−	0.992307i	$-0.460492\pi$
$107$	2.56118	0.247599	0.123799	+	0.992307i	$0.460492\pi$
$108$	0	0
$109$	10.3131	0.987820	0.493910	−	0.869513i	$-0.335567\pi$
$109$	10.3131	0.987820	0.493910	+	0.869513i	$0.335567\pi$
$110$	−14.6956	−1.40117
$111$	0	0
$112$	18.6459	1.76187
$113$	−9.93580	−0.934682	−0.467341	−	0.884077i	$-0.654788\pi$
$113$	−9.93580	−0.934682	−0.467341	+	0.884077i	$0.654788\pi$
$114$	0	0
$115$	1.38413	0.129071
$116$	0.383744	0.0356297
$117$	0	0
$118$	−13.9881	−1.28771
$119$	−27.1528	−2.48909
$120$	0	0
$121$	2.59627	0.236024
$122$	10.5562	0.955713
$123$	0	0
$124$	−0.213888	−0.0192077
$125$	6.86888	0.614371
$126$	0	0
$127$	−14.8949	−1.32171	−0.660853	−	0.750515i	$-0.729805\pi$
$127$	−14.8949	−1.32171	−0.660853	+	0.750515i	$0.729805\pi$
$128$	−12.2279	−1.08080
$129$	0	0
$130$	3.02404	0.265226
$131$	0.0610002	0.00532962	0.00266481	−	0.999996i	$-0.499152\pi$
$131$	0.0610002	0.00532962	0.00266481	+	0.999996i	$0.499152\pi$
$132$	0	0
$133$	0	0
$134$	−8.92431	−0.770943
$135$	0	0
$136$	16.8452	1.44447
$137$	−8.11701	−0.693483	−0.346742	−	0.937961i	$-0.612712\pi$
$137$	−8.11701	−0.693483	−0.346742	+	0.937961i	$0.612712\pi$
$138$	0	0
$139$	3.88444	0.329474	0.164737	−	0.986338i	$-0.447323\pi$
$139$	3.88444	0.329474	0.164737	+	0.986338i	$0.447323\pi$
$140$	1.45623	0.123074
$141$	0	0
$142$	−4.12155	−0.345873
$143$	−2.79782	−0.233966
$144$	0	0
$145$	−8.70739	−0.723109
$146$	−6.99088	−0.578569
$147$	0	0
$148$	−0.339556	−0.0279113
$149$	−16.2055	−1.32760	−0.663802	−	0.747908i	$-0.731058\pi$
$149$	−16.2055	−1.32760	−0.663802	+	0.747908i	$0.731058\pi$
$150$	0	0
$151$	−15.1506	−1.23294	−0.616471	−	0.787378i	$-0.711438\pi$
$151$	−15.1506	−1.23294	−0.616471	+	0.787378i	$0.711438\pi$
$152$	0	0
$153$	0	0
$154$	−23.6878	−1.90882
$155$	4.85326	0.389823
$156$	0	0
$157$	0.539830	0.0430831	0.0215415	−	0.999768i	$-0.493143\pi$
$157$	0.539830	0.0430831	0.0215415	+	0.999768i	$0.493143\pi$
$158$	24.0534	1.91358
$159$	0	0
$160$	−1.86484	−0.147428
$161$	2.23108	0.175834
$162$	0	0
$163$	9.67230	0.757593	0.378797	−	0.925480i	$-0.376338\pi$
$163$	9.67230	0.757593	0.378797	+	0.925480i	$0.376338\pi$
$164$	1.42378	0.111178
$165$	0	0
$166$	−0.344608	−0.0267468
$167$	10.9473	0.847127	0.423563	−	0.905866i	$-0.360779\pi$
$167$	10.9473	0.847127	0.423563	+	0.905866i	$0.360779\pi$
$168$	0	0
$169$	−12.4243	−0.955713
$170$	24.5306	1.88141
$171$	0	0
$172$	0.278066	0.0212023
$173$	13.7663	1.04663	0.523316	−	0.852139i	$-0.324695\pi$
$173$	13.7663	1.04663	0.523316	+	0.852139i	$0.324695\pi$
$174$	0	0
$175$	−10.9855	−0.830422
$176$	15.5851	1.17477
$177$	0	0
$178$	−11.3396	−0.849936
$179$	−20.3163	−1.51852	−0.759258	−	0.650790i	$-0.774438\pi$
$179$	−20.3163	−1.51852	−0.759258	+	0.650790i	$0.774438\pi$
$180$	0	0
$181$	16.0273	1.19130	0.595651	−	0.803243i	$-0.296894\pi$
$181$	16.0273	1.19130	0.595651	+	0.803243i	$0.296894\pi$
$182$	4.87444	0.361318
$183$	0	0
$184$	−1.38413	−0.102040
$185$	7.70472	0.566463
$186$	0	0
$187$	−22.6955	−1.65966
$188$	0.982946	0.0716887
$189$	0	0
$190$	0	0
$191$	−8.13820	−0.588859	−0.294430	−	0.955673i	$-0.595130\pi$
$191$	−8.13820	−0.588859	−0.294430	+	0.955673i	$0.595130\pi$
$192$	0	0
$193$	16.5057	1.18810	0.594052	−	0.804426i	$-0.297527\pi$
$193$	16.5057	1.18810	0.594052	+	0.804426i	$0.297527\pi$
$194$	−7.19898	−0.516857
$195$	0	0
$196$	1.50299	0.107357
$197$	−19.9538	−1.42165	−0.710824	−	0.703370i	$-0.751678\pi$
$197$	−19.9538	−1.42165	−0.710824	+	0.703370i	$0.751678\pi$
$198$	0	0
$199$	−1.59358	−0.112966	−0.0564829	−	0.998404i	$-0.517989\pi$
$199$	−1.59358	−0.112966	−0.0564829	+	0.998404i	$0.517989\pi$
$200$	6.81524	0.481910
$201$	0	0
$202$	−8.36278	−0.588403
$203$	−14.0354	−0.985092
$204$	0	0
$205$	−32.3063	−2.25637
$206$	21.8548	1.52269
$207$	0	0
$208$	−3.20708	−0.222371
$209$	0	0
$210$	0	0
$211$	0.128356	0.00883636	0.00441818	−	0.999990i	$-0.498594\pi$
$211$	0.128356	0.00883636	0.00441818	+	0.999990i	$0.498594\pi$
$212$	−1.18713	−0.0815326
$213$	0	0
$214$	−3.72967	−0.254955
$215$	−6.30949	−0.430304
$216$	0	0
$217$	7.82295	0.531056
$218$	−15.0183	−1.01717
$219$	0	0
$220$	1.21719	0.0820627
$221$	4.67026	0.314156
$222$	0	0
$223$	−19.2490	−1.28901	−0.644503	−	0.764602i	$-0.722936\pi$
$223$	−19.2490	−1.28901	−0.644503	+	0.764602i	$0.722936\pi$
$224$	−3.00592	−0.200842
$225$	0	0
$226$	14.4688	0.962453
$227$	3.03447	0.201405	0.100702	−	0.994917i	$-0.467891\pi$
$227$	3.03447	0.201405	0.100702	+	0.994917i	$0.467891\pi$
$228$	0	0
$229$	5.82295	0.384791	0.192396	−	0.981317i	$-0.438374\pi$
$229$	5.82295	0.384791	0.192396	+	0.981317i	$0.438374\pi$
$230$	−2.01562	−0.132906
$231$	0	0
$232$	8.70739	0.571668
$233$	10.0504	0.658427	0.329213	−	0.944256i	$-0.393216\pi$
$233$	10.0504	0.658427	0.329213	+	0.944256i	$0.393216\pi$
$234$	0	0
$235$	−22.3037	−1.45493
$236$	1.15859	0.0754177
$237$	0	0
$238$	39.5408	2.56305
$239$	−4.00614	−0.259136	−0.129568	−	0.991571i	$-0.541359\pi$
$239$	−4.00614	−0.259136	−0.129568	+	0.991571i	$0.541359\pi$
$240$	0	0
$241$	−7.51754	−0.484247	−0.242124	−	0.970245i	$-0.577844\pi$
$241$	−7.51754	−0.484247	−0.242124	+	0.970245i	$0.577844\pi$
$242$	−3.78077	−0.243037
$243$	0	0
$244$	−0.874333	−0.0559734
$245$	−34.1038	−2.17881
$246$	0	0
$247$	0	0
$248$	−4.85326	−0.308182
$249$	0	0
$250$	−10.0027	−0.632626
$251$	2.52872	0.159612	0.0798058	−	0.996810i	$-0.474570\pi$
$251$	2.52872	0.159612	0.0798058	+	0.996810i	$0.474570\pi$
$252$	0	0
$253$	1.86484	0.117241
$254$	21.6904	1.36098
$255$	0	0
$256$	2.88444	0.180277
$257$	−19.3334	−1.20598	−0.602992	−	0.797747i	$-0.706025\pi$
$257$	−19.3334	−1.20598	−0.602992	+	0.797747i	$0.706025\pi$
$258$	0	0
$259$	12.4192	0.771692
$260$	−0.250471	−0.0155336
$261$	0	0
$262$	−0.0888306	−0.00548797
$263$	−18.3219	−1.12978	−0.564889	−	0.825167i	$-0.691081\pi$
$263$	−18.3219	−1.12978	−0.564889	+	0.825167i	$0.691081\pi$
$264$	0	0
$265$	26.9368	1.65471
$266$	0	0
$267$	0	0
$268$	0.739170	0.0451520
$269$	10.2871	0.627215	0.313607	−	0.949553i	$-0.398462\pi$
$269$	10.2871	0.627215	0.313607	+	0.949553i	$0.398462\pi$
$270$	0	0
$271$	8.35504	0.507532	0.253766	−	0.967266i	$-0.418331\pi$
$271$	8.35504	0.507532	0.253766	+	0.967266i	$0.418331\pi$
$272$	−26.0154	−1.57741
$273$	0	0
$274$	11.8203	0.714088
$275$	−9.18214	−0.553704
$276$	0	0
$277$	28.3432	1.70298	0.851488	−	0.524374i	$-0.175700\pi$
$277$	28.3432	1.70298	0.851488	+	0.524374i	$0.175700\pi$
$278$	−5.65665	−0.339263
$279$	0	0
$280$	33.0428	1.97469
$281$	22.9273	1.36773	0.683863	−	0.729611i	$-0.260299\pi$
$281$	22.9273	1.36773	0.683863	+	0.729611i	$0.260299\pi$
$282$	0	0
$283$	16.9436	1.00719	0.503595	−	0.863940i	$-0.332010\pi$
$283$	16.9436	1.00719	0.503595	+	0.863940i	$0.332010\pi$
$284$	0.341374	0.0202568
$285$	0	0
$286$	4.07428	0.240917
$287$	−52.0745	−3.07386
$288$	0	0
$289$	20.8844	1.22850
$290$	12.6800	0.744594
$291$	0	0
$292$	0.579030	0.0338852
$293$	0.681388	0.0398071	0.0199035	−	0.999802i	$-0.493664\pi$
$293$	0.681388	0.0398071	0.0199035	+	0.999802i	$0.493664\pi$
$294$	0	0
$295$	−26.2891	−1.53061
$296$	−7.70472	−0.447828
$297$	0	0
$298$	23.5990	1.36705
$299$	−0.383744	−0.0221925
$300$	0	0
$301$	−10.1702	−0.586203
$302$	22.0629	1.26958
$303$	0	0
$304$	0	0
$305$	19.8391	1.13599
$306$	0	0
$307$	−12.3824	−0.706700	−0.353350	−	0.935491i	$-0.614957\pi$
$307$	−12.3824	−0.706700	−0.353350	+	0.935491i	$0.614957\pi$
$308$	1.96198	0.111794
$309$	0	0
$310$	−7.06748	−0.401406
$311$	4.28651	0.243066	0.121533	−	0.992587i	$-0.461219\pi$
$311$	4.28651	0.243066	0.121533	+	0.992587i	$0.461219\pi$
$312$	0	0
$313$	−9.01455	−0.509532	−0.254766	−	0.967003i	$-0.581999\pi$
$313$	−9.01455	−0.509532	−0.254766	+	0.967003i	$0.581999\pi$
$314$	−0.786118	−0.0443632
$315$	0	0
$316$	−1.99226	−0.112073
$317$	16.4136	0.921879	0.460939	−	0.887432i	$-0.347513\pi$
$317$	16.4136	0.921879	0.460939	+	0.887432i	$0.347513\pi$
$318$	0	0
$319$	−11.7314	−0.656834
$320$	−20.4197	−1.14150
$321$	0	0
$322$	−3.24897	−0.181058
$323$	0	0
$324$	0	0
$325$	1.88949	0.104810
$326$	−14.0851	−0.780103
$327$	0	0
$328$	32.3063	1.78382
$329$	−35.9512	−1.98205
$330$	0	0
$331$	−8.13516	−0.447149	−0.223574	−	0.974687i	$-0.571773\pi$
$331$	−8.13516	−0.447149	−0.223574	+	0.974687i	$0.571773\pi$
$332$	0.0285427	0.00156648
$333$	0	0
$334$	−15.9418	−0.872297
$335$	−16.7722	−0.916364
$336$	0	0
$337$	−23.3577	−1.27238	−0.636188	−	0.771534i	$-0.719490\pi$
$337$	−23.3577	−1.27238	−0.636188	+	0.771534i	$0.719490\pi$
$338$	18.0926	0.984109
$339$	0	0
$340$	−2.03178	−0.110189
$341$	6.53878	0.354095
$342$	0	0
$343$	−24.0915	−1.30082
$344$	6.30949	0.340185
$345$	0	0
$346$	−20.0469	−1.07773
$347$	16.0087	0.859389	0.429695	−	0.902974i	$-0.358621\pi$
$347$	16.0087	0.859389	0.429695	+	0.902974i	$0.358621\pi$
$348$	0	0
$349$	26.8229	1.43580	0.717900	−	0.696146i	$-0.245104\pi$
$349$	26.8229	1.43580	0.717900	+	0.696146i	$0.245104\pi$
$350$	15.9974	0.855096
$351$	0	0
$352$	−2.51249	−0.133916
$353$	−33.6054	−1.78864	−0.894319	−	0.447430i	$-0.852339\pi$
$353$	−33.6054	−1.78864	−0.894319	+	0.447430i	$0.852339\pi$
$354$	0	0
$355$	−7.74598	−0.411114
$356$	0.939216	0.0497784
$357$	0	0
$358$	29.5853	1.56363
$359$	−7.73327	−0.408146	−0.204073	−	0.978956i	$-0.565418\pi$
$359$	−7.73327	−0.408146	−0.204073	+	0.978956i	$0.565418\pi$
$360$	0	0
$361$	0	0
$362$	−23.3395	−1.22670
$363$	0	0
$364$	−0.403733	−0.0211614
$365$	−13.1386	−0.687703
$366$	0	0
$367$	19.8675	1.03708	0.518538	−	0.855054i	$-0.326476\pi$
$367$	19.8675	1.03708	0.518538	+	0.855054i	$0.326476\pi$
$368$	2.13762	0.111431
$369$	0	0
$370$	−11.2199	−0.583293
$371$	43.4193	2.25422
$372$	0	0
$373$	−1.91622	−0.0992182	−0.0496091	−	0.998769i	$-0.515798\pi$
$373$	−1.91622	−0.0992182	−0.0496091	+	0.998769i	$0.515798\pi$
$374$	33.0500	1.70897
$375$	0	0
$376$	22.3037	1.15022
$377$	2.41408	0.124331
$378$	0	0
$379$	−27.2591	−1.40020	−0.700102	−	0.714043i	$-0.746862\pi$
$379$	−27.2591	−1.40020	−0.700102	+	0.714043i	$0.746862\pi$
$380$	0	0
$381$	0	0
$382$	11.8511	0.606356
$383$	−1.69288	−0.0865019	−0.0432510	−	0.999064i	$-0.513772\pi$
$383$	−1.69288	−0.0865019	−0.0432510	+	0.999064i	$0.513772\pi$
$384$	0	0
$385$	−44.5185	−2.26887
$386$	−24.0361	−1.22341
$387$	0	0
$388$	0.596267	0.0302709
$389$	−20.3873	−1.03368	−0.516838	−	0.856083i	$-0.672891\pi$
$389$	−20.3873	−1.03368	−0.516838	+	0.856083i	$0.672891\pi$
$390$	0	0
$391$	−3.11287	−0.157425
$392$	34.1038	1.72250
$393$	0	0
$394$	29.0574	1.46389
$395$	45.2056	2.27454
$396$	0	0
$397$	−27.3236	−1.37133	−0.685666	−	0.727917i	$-0.740489\pi$
$397$	−27.3236	−1.37133	−0.685666	+	0.727917i	$0.740489\pi$
$398$	2.32062	0.116322
$399$	0	0
$400$	−10.5253	−0.526264
$401$	−11.6001	−0.579283	−0.289642	−	0.957135i	$-0.593536\pi$
$401$	−11.6001	−0.579283	−0.289642	+	0.957135i	$0.593536\pi$
$402$	0	0
$403$	−1.34554	−0.0670262
$404$	0.692660	0.0344611
$405$	0	0
$406$	20.4388	1.01436
$407$	10.3805	0.514545
$408$	0	0
$409$	−27.3327	−1.35152	−0.675759	−	0.737123i	$-0.736184\pi$
$409$	−27.3327	−1.35152	−0.675759	+	0.737123i	$0.736184\pi$
$410$	47.0456	2.32341
$411$	0	0
$412$	−1.81016	−0.0891799
$413$	−42.3753	−2.08515
$414$	0	0
$415$	−0.647651	−0.0317919
$416$	0.517017	0.0253488
$417$	0	0
$418$	0	0
$419$	−31.4605	−1.53694	−0.768472	−	0.639883i	$-0.778983\pi$
$419$	−31.4605	−1.53694	−0.768472	+	0.639883i	$0.778983\pi$
$420$	0	0
$421$	−2.63547	−0.128445	−0.0642224	−	0.997936i	$-0.520457\pi$
$421$	−2.63547	−0.128445	−0.0642224	+	0.997936i	$0.520457\pi$
$422$	−0.186916	−0.00909891
$423$	0	0
$424$	−26.9368	−1.30816
$425$	15.3273	0.743481
$426$	0	0
$427$	31.9786	1.54756
$428$	0.308916	0.0149320
$429$	0	0
$430$	9.18809	0.443089
$431$	−32.4110	−1.56118	−0.780590	−	0.625043i	$-0.785082\pi$
$431$	−32.4110	−1.56118	−0.780590	+	0.625043i	$0.785082\pi$
$432$	0	0
$433$	17.9855	0.864326	0.432163	−	0.901796i	$-0.357751\pi$
$433$	17.9855	0.864326	0.432163	+	0.901796i	$0.357751\pi$
$434$	−11.3920	−0.546835
$435$	0	0
$436$	1.24392	0.0595729
$437$	0	0
$438$	0	0
$439$	−16.4679	−0.785971	−0.392985	−	0.919545i	$-0.628558\pi$
$439$	−16.4679	−0.785971	−0.392985	+	0.919545i	$0.628558\pi$
$440$	27.6187	1.31667
$441$	0	0
$442$	−6.80098	−0.323490
$443$	29.9831	1.42454	0.712269	−	0.701907i	$-0.247668\pi$
$443$	29.9831	1.42454	0.712269	+	0.701907i	$0.247668\pi$
$444$	0	0
$445$	−21.3114	−1.01026
$446$	28.0310	1.32731
$447$	0	0
$448$	−32.9145	−1.55506
$449$	−12.1344	−0.572659	−0.286329	−	0.958131i	$-0.592435\pi$
$449$	−12.1344	−0.572659	−0.286329	+	0.958131i	$0.592435\pi$
$450$	0	0
$451$	−43.5262	−2.04957
$452$	−1.19840	−0.0563682
$453$	0	0
$454$	−4.41889	−0.207389
$455$	9.16096	0.429472
$456$	0	0
$457$	30.1848	1.41199	0.705993	−	0.708219i	$-0.250501\pi$
$457$	30.1848	1.41199	0.705993	+	0.708219i	$0.250501\pi$
$458$	−8.47957	−0.396224
$459$	0	0
$460$	0.166947	0.00778394
$461$	24.1468	1.12463	0.562315	−	0.826923i	$-0.309911\pi$
$461$	24.1468	1.12463	0.562315	+	0.826923i	$0.309911\pi$
$462$	0	0
$463$	−7.11650	−0.330732	−0.165366	−	0.986232i	$-0.552880\pi$
$463$	−7.11650	−0.330732	−0.165366	+	0.986232i	$0.552880\pi$
$464$	−13.4475	−0.624283
$465$	0	0
$466$	−14.6358	−0.677990
$467$	−1.75388	−0.0811597	−0.0405799	−	0.999176i	$-0.512921\pi$
$467$	−1.75388	−0.0811597	−0.0405799	+	0.999176i	$0.512921\pi$
$468$	0	0
$469$	−27.0351	−1.24836
$470$	32.4793	1.49816
$471$	0	0
$472$	26.2891	1.21005
$473$	−8.50075	−0.390865
$474$	0	0
$475$	0	0
$476$	−3.27502	−0.150111
$477$	0	0
$478$	5.83387	0.266835
$479$	12.3538	0.564459	0.282230	−	0.959347i	$-0.408926\pi$
$479$	12.3538	0.564459	0.282230	+	0.959347i	$0.408926\pi$
$480$	0	0
$481$	−2.13610	−0.0973976
$482$	10.9473	0.498635
$483$	0	0
$484$	0.313148	0.0142340
$485$	−13.5297	−0.614350
$486$	0	0
$487$	−7.61318	−0.344986	−0.172493	−	0.985011i	$-0.555182\pi$
$487$	−7.61318	−0.344986	−0.172493	+	0.985011i	$0.555182\pi$
$488$	−19.8391	−0.898076
$489$	0	0
$490$	49.6631	2.24355
$491$	5.52729	0.249443	0.124722	−	0.992192i	$-0.460196\pi$
$491$	5.52729	0.249443	0.124722	+	0.992192i	$0.460196\pi$
$492$	0	0
$493$	19.5827	0.881958
$494$	0	0
$495$	0	0
$496$	7.49525	0.336547
$497$	−12.4857	−0.560061
$498$	0	0
$499$	−5.09833	−0.228232	−0.114116	−	0.993467i	$-0.536404\pi$
$499$	−5.09833	−0.228232	−0.114116	+	0.993467i	$0.536404\pi$
$500$	0.828488	0.0370511
$501$	0	0
$502$	−3.68241	−0.164354
$503$	23.1639	1.03283	0.516414	−	0.856339i	$-0.327267\pi$
$503$	23.1639	1.03283	0.516414	+	0.856339i	$0.327267\pi$
$504$	0	0
$505$	−15.7169	−0.699392
$506$	−2.71564	−0.120725
$507$	0	0
$508$	−1.79654	−0.0797086
$509$	−9.10731	−0.403675	−0.201837	−	0.979419i	$-0.564691\pi$
$509$	−9.10731	−0.403675	−0.201837	+	0.979419i	$0.564691\pi$
$510$	0	0
$511$	−21.1780	−0.936859
$512$	20.2553	0.895168
$513$	0	0
$514$	28.1539	1.24182
$515$	41.0735	1.80992
$516$	0	0
$517$	−30.0496	−1.32158
$518$	−18.0853	−0.794621
$519$	0	0
$520$	−5.68334	−0.249231
$521$	−23.5589	−1.03213	−0.516067	−	0.856548i	$-0.672605\pi$
$521$	−23.5589	−1.03213	−0.516067	+	0.856548i	$0.672605\pi$
$522$	0	0
$523$	−21.2567	−0.929491	−0.464746	−	0.885444i	$-0.653854\pi$
$523$	−21.2567	−0.929491	−0.464746	+	0.885444i	$0.653854\pi$
$524$	0.00735753	0.000321415 0
$525$	0	0
$526$	26.6810	1.16335
$527$	−10.9148	−0.475458
$528$	0	0
$529$	−22.7442	−0.988879
$530$	−39.2262	−1.70388
$531$	0	0
$532$	0	0
$533$	8.95677	0.387961
$534$	0	0
$535$	−7.00950	−0.303047
$536$	16.7722	0.724450
$537$	0	0
$538$	−14.9804	−0.645851
$539$	−45.9480	−1.97912
$540$	0	0
$541$	22.5553	0.969729	0.484864	−	0.874589i	$-0.338869\pi$
$541$	22.5553	0.969729	0.484864	+	0.874589i	$0.338869\pi$
$542$	−12.1669	−0.522612
$543$	0	0
$544$	4.19396	0.179815
$545$	−28.2253	−1.20904
$546$	0	0
$547$	10.7510	0.459681	0.229840	−	0.973228i	$-0.426180\pi$
$547$	10.7510	0.459681	0.229840	+	0.973228i	$0.426180\pi$
$548$	−0.979031	−0.0418221
$549$	0	0
$550$	13.3713	0.570156
$551$	0	0
$552$	0	0
$553$	72.8667	3.09861
$554$	−41.2743	−1.75358
$555$	0	0
$556$	0.468521	0.0198697
$557$	29.3116	1.24197	0.620986	−	0.783822i	$-0.286732\pi$
$557$	29.3116	1.24197	0.620986	+	0.783822i	$0.286732\pi$
$558$	0	0
$559$	1.74928	0.0739864
$560$	−51.0305	−2.15643
$561$	0	0
$562$	−33.3874	−1.40836
$563$	20.7075	0.872715	0.436358	−	0.899773i	$-0.356268\pi$
$563$	20.7075	0.872715	0.436358	+	0.899773i	$0.356268\pi$
$564$	0	0
$565$	27.1925	1.14400
$566$	−24.6738	−1.03712
$567$	0	0
$568$	7.74598	0.325014
$569$	−29.0001	−1.21575	−0.607874	−	0.794034i	$-0.707977\pi$
$569$	−29.0001	−1.21575	−0.607874	+	0.794034i	$0.707977\pi$
$570$	0	0
$571$	−33.5773	−1.40517	−0.702583	−	0.711602i	$-0.747970\pi$
$571$	−33.5773	−1.40517	−0.702583	+	0.711602i	$0.747970\pi$
$572$	−0.337459	−0.0141099
$573$	0	0
$574$	75.8326	3.16519
$575$	−1.25940	−0.0525208
$576$	0	0
$577$	8.98309	0.373971	0.186985	−	0.982363i	$-0.440128\pi$
$577$	8.98309	0.373971	0.186985	+	0.982363i	$0.440128\pi$
$578$	−30.4126	−1.26500
$579$	0	0
$580$	−1.05024	−0.0436088
$581$	−1.04395	−0.0433102
$582$	0	0
$583$	36.2918	1.50305
$584$	13.1386	0.543677
$585$	0	0
$586$	−0.992259	−0.0409899
$587$	9.32533	0.384897	0.192449	−	0.981307i	$-0.438357\pi$
$587$	9.32533	0.384897	0.192449	+	0.981307i	$0.438357\pi$
$588$	0	0
$589$	0	0
$590$	38.2831	1.57609
$591$	0	0
$592$	11.8990	0.489045
$593$	−27.2787	−1.12020	−0.560101	−	0.828425i	$-0.689238\pi$
$593$	−27.2787	−1.12020	−0.560101	+	0.828425i	$0.689238\pi$
$594$	0	0
$595$	74.3123	3.04651
$596$	−1.95462	−0.0800644
$597$	0	0
$598$	0.558821	0.0228519
$599$	−13.8386	−0.565429	−0.282714	−	0.959204i	$-0.591235\pi$
$599$	−13.8386	−0.565429	−0.282714	+	0.959204i	$0.591235\pi$
$600$	0	0
$601$	17.7793	0.725233	0.362616	−	0.931938i	$-0.381883\pi$
$601$	17.7793	0.725233	0.362616	+	0.931938i	$0.381883\pi$
$602$	14.8102	0.603621
$603$	0	0
$604$	−1.82739	−0.0743555
$605$	−7.10552	−0.288881
$606$	0	0
$607$	17.2249	0.699138	0.349569	−	0.936911i	$-0.386328\pi$
$607$	17.2249	0.699138	0.349569	+	0.936911i	$0.386328\pi$
$608$	0	0
$609$	0	0
$610$	−28.8904	−1.16974
$611$	6.18358	0.250161
$612$	0	0
$613$	7.54933	0.304914	0.152457	−	0.988310i	$-0.451281\pi$
$613$	7.54933	0.304914	0.152457	+	0.988310i	$0.451281\pi$
$614$	18.0316	0.727697
$615$	0	0
$616$	44.5185	1.79370
$617$	−26.9447	−1.08475	−0.542376	−	0.840136i	$-0.682475\pi$
$617$	−26.9447	−1.08475	−0.542376	+	0.840136i	$0.682475\pi$
$618$	0	0
$619$	−26.3509	−1.05913	−0.529566	−	0.848268i	$-0.677645\pi$
$619$	−26.3509	−1.05913	−0.529566	+	0.848268i	$0.677645\pi$
$620$	0.585375	0.0235092
$621$	0	0
$622$	−6.24216	−0.250288
$623$	−34.3518	−1.37627
$624$	0	0
$625$	−31.2499	−1.25000
$626$	13.1273	0.524672
$627$	0	0
$628$	0.0651114	0.00259823
$629$	−17.3277	−0.690900
$630$	0	0
$631$	22.5381	0.897227	0.448613	−	0.893726i	$-0.351918\pi$
$631$	22.5381	0.897227	0.448613	+	0.893726i	$0.351918\pi$
$632$	−45.2056	−1.79818
$633$	0	0
$634$	−23.9020	−0.949270
$635$	40.7646	1.61769
$636$	0	0
$637$	9.45512	0.374625
$638$	17.0837	0.676350
$639$	0	0
$640$	33.4655	1.32284
$641$	−11.3384	−0.447840	−0.223920	−	0.974608i	$-0.571885\pi$
$641$	−11.3384	−0.447840	−0.223920	+	0.974608i	$0.571885\pi$
$642$	0	0
$643$	14.9682	0.590289	0.295144	−	0.955453i	$-0.404632\pi$
$643$	14.9682	0.590289	0.295144	+	0.955453i	$0.404632\pi$
$644$	0.269101	0.0106041
$645$	0	0
$646$	0	0
$647$	22.2346	0.874132	0.437066	−	0.899429i	$-0.356018\pi$
$647$	22.2346	0.874132	0.437066	+	0.899429i	$0.356018\pi$
$648$	0	0
$649$	−35.4192	−1.39033
$650$	−2.75154	−0.107924
$651$	0	0
$652$	1.16662	0.0456885
$653$	−3.07820	−0.120459	−0.0602296	−	0.998185i	$-0.519183\pi$
$653$	−3.07820	−0.120459	−0.0602296	+	0.998185i	$0.519183\pi$
$654$	0	0
$655$	−0.166947	−0.00652315
$656$	−49.8931	−1.94800
$657$	0	0
$658$	52.3533	2.04094
$659$	39.2474	1.52886	0.764430	−	0.644706i	$-0.223020\pi$
$659$	39.2474	1.52886	0.764430	+	0.644706i	$0.223020\pi$
$660$	0	0
$661$	−27.8607	−1.08366	−0.541828	−	0.840489i	$-0.682268\pi$
$661$	−27.8607	−1.08366	−0.541828	+	0.840489i	$0.682268\pi$
$662$	11.8467	0.460435
$663$	0	0
$664$	0.647651	0.0251337
$665$	0	0
$666$	0	0
$667$	−1.60906	−0.0623030
$668$	1.32040	0.0510880
$669$	0	0
$670$	24.4243	0.943592
$671$	26.7292	1.03187
$672$	0	0
$673$	15.3105	0.590175	0.295087	−	0.955470i	$-0.404651\pi$
$673$	15.3105	0.590175	0.295087	+	0.955470i	$0.404651\pi$
$674$	34.0143	1.31018
$675$	0	0
$676$	−1.49855	−0.0576365
$677$	−40.0335	−1.53861	−0.769306	−	0.638880i	$-0.779398\pi$
$677$	−40.0335	−1.53861	−0.769306	+	0.638880i	$0.779398\pi$
$678$	0	0
$679$	−21.8084	−0.836930
$680$	−46.1024	−1.76795
$681$	0	0
$682$	−9.52198	−0.364616
$683$	−0.599202	−0.0229278	−0.0114639	−	0.999934i	$-0.503649\pi$
$683$	−0.599202	−0.0229278	−0.0114639	+	0.999934i	$0.503649\pi$
$684$	0	0
$685$	22.2148	0.848785
$686$	35.0829	1.33947
$687$	0	0
$688$	−9.74422	−0.371495
$689$	−7.46808	−0.284511
$690$	0	0
$691$	1.08378	0.0412289	0.0206144	−	0.999788i	$-0.493438\pi$
$691$	1.08378	0.0412289	0.0206144	+	0.999788i	$0.493438\pi$
$692$	1.66042	0.0631197
$693$	0	0
$694$	−23.3123	−0.884924
$695$	−10.6310	−0.403258
$696$	0	0
$697$	72.6560	2.75204
$698$	−39.0605	−1.47846
$699$	0	0
$700$	−1.32501	−0.0500806
$701$	45.6179	1.72296	0.861482	−	0.507789i	$-0.169537\pi$
$701$	45.6179	1.72296	0.861482	+	0.507789i	$0.169537\pi$
$702$	0	0
$703$	0	0
$704$	−27.5114	−1.03688
$705$	0	0
$706$	48.9374	1.84178
$707$	−25.3340	−0.952782
$708$	0	0
$709$	−4.26621	−0.160221	−0.0801104	−	0.996786i	$-0.525527\pi$
$709$	−4.26621	−0.160221	−0.0801104	+	0.996786i	$0.525527\pi$
$710$	11.2799	0.423329
$711$	0	0
$712$	21.3114	0.798678
$713$	0.896846	0.0335871
$714$	0	0
$715$	7.65715	0.286361
$716$	−2.45045	−0.0915777
$717$	0	0
$718$	11.2614	0.420273
$719$	−16.8047	−0.626709	−0.313354	−	0.949636i	$-0.601453\pi$
$719$	−16.8047	−0.626709	−0.313354	+	0.949636i	$0.601453\pi$
$720$	0	0
$721$	66.2063	2.46565
$722$	0	0
$723$	0	0
$724$	1.93313	0.0718444
$725$	7.92274	0.294243
$726$	0	0
$727$	17.2618	0.640203	0.320102	−	0.947383i	$-0.396283\pi$
$727$	17.2618	0.640203	0.320102	+	0.947383i	$0.396283\pi$
$728$	−9.16096	−0.339528
$729$	0	0
$730$	19.1328	0.708137
$731$	14.1899	0.524831
$732$	0	0
$733$	38.2746	1.41370	0.706852	−	0.707362i	$-0.250115\pi$
$733$	38.2746	1.41370	0.706852	+	0.707362i	$0.250115\pi$
$734$	−28.9317	−1.06789
$735$	0	0
$736$	−0.344608	−0.0127024
$737$	−22.5972	−0.832377
$738$	0	0
$739$	−46.4243	−1.70774	−0.853872	−	0.520483i	$-0.825752\pi$
$739$	−46.4243	−1.70774	−0.853872	+	0.520483i	$0.825752\pi$
$740$	0.929303	0.0341619
$741$	0	0
$742$	−63.2285	−2.32119
$743$	19.4692	0.714257	0.357128	−	0.934055i	$-0.383756\pi$
$743$	19.4692	0.714257	0.357128	+	0.934055i	$0.383756\pi$
$744$	0	0
$745$	44.3515	1.62491
$746$	2.79047	0.102166
$747$	0	0
$748$	−2.73742	−0.100090
$749$	−11.2986	−0.412841
$750$	0	0
$751$	2.32264	0.0847545	0.0423772	−	0.999102i	$-0.486507\pi$
$751$	2.32264	0.0847545	0.0423772	+	0.999102i	$0.486507\pi$
$752$	−34.4452	−1.25609
$753$	0	0
$754$	−3.51546	−0.128026
$755$	41.4646	1.50905
$756$	0	0
$757$	−9.11381	−0.331247	−0.165623	−	0.986189i	$-0.552964\pi$
$757$	−9.11381	−0.331247	−0.165623	+	0.986189i	$0.552964\pi$
$758$	39.6956	1.44181
$759$	0	0
$760$	0	0
$761$	5.49483	0.199187	0.0995937	−	0.995028i	$-0.468246\pi$
$761$	5.49483	0.199187	0.0995937	+	0.995028i	$0.468246\pi$
$762$	0	0
$763$	−45.4962	−1.64707
$764$	−0.981587	−0.0355126
$765$	0	0
$766$	2.46522	0.0890721
$767$	7.28852	0.263173
$768$	0	0
$769$	21.1516	0.762745	0.381373	−	0.924421i	$-0.375451\pi$
$769$	21.1516	0.762745	0.381373	+	0.924421i	$0.375451\pi$
$770$	64.8293	2.33628
$771$	0	0
$772$	1.99083	0.0716515
$773$	13.0736	0.470226	0.235113	−	0.971968i	$-0.424454\pi$
$773$	13.0736	0.470226	0.235113	+	0.971968i	$0.424454\pi$
$774$	0	0
$775$	−4.41592	−0.158624
$776$	13.5297	0.485686
$777$	0	0
$778$	29.6886	1.06439
$779$	0	0
$780$	0	0
$781$	−10.4361	−0.373434
$782$	4.53307	0.162102
$783$	0	0
$784$	−52.6691	−1.88104
$785$	−1.47742	−0.0527313
$786$	0	0
$787$	20.7205	0.738606	0.369303	−	0.929309i	$-0.379596\pi$
$787$	20.7205	0.738606	0.369303	+	0.929309i	$0.379596\pi$
$788$	−2.40672	−0.0857359
$789$	0	0
$790$	−65.8299	−2.34212
$791$	43.8315	1.55847
$792$	0	0
$793$	−5.50030	−0.195322
$794$	39.7895	1.41208
$795$	0	0
$796$	−0.192209	−0.00681267
$797$	−34.3119	−1.21539	−0.607696	−	0.794170i	$-0.707906\pi$
$797$	−34.3119	−1.21539	−0.607696	+	0.794170i	$0.707906\pi$
$798$	0	0
$799$	50.1603	1.77454
$800$	1.69679	0.0599906
$801$	0	0
$802$	16.8925	0.596495
$803$	−17.7015	−0.624673
$804$	0	0
$805$	−6.10607	−0.215211
$806$	1.95942	0.0690177
$807$	0	0
$808$	15.7169	0.552918
$809$	48.3547	1.70006	0.850030	−	0.526734i	$-0.176584\pi$
$809$	48.3547	1.70006	0.850030	+	0.526734i	$0.176584\pi$
$810$	0	0
$811$	−16.5348	−0.580615	−0.290307	−	0.956933i	$-0.593758\pi$
$811$	−16.5348	−0.580615	−0.290307	+	0.956933i	$0.593758\pi$
$812$	−1.69288	−0.0594083
$813$	0	0
$814$	−15.1165	−0.529833
$815$	−26.4714	−0.927252
$816$	0	0
$817$	0	0
$818$	39.8029	1.39167
$819$	0	0
$820$	−3.89662	−0.136076
$821$	41.8086	1.45913	0.729564	−	0.683912i	$-0.239723\pi$
$821$	41.8086	1.45913	0.729564	+	0.683912i	$0.239723\pi$
$822$	0	0
$823$	26.5303	0.924789	0.462395	−	0.886674i	$-0.346990\pi$
$823$	26.5303	0.924789	0.462395	+	0.886674i	$0.346990\pi$
$824$	−41.0735	−1.43086
$825$	0	0
$826$	61.7083	2.14711
$827$	19.0756	0.663323	0.331661	−	0.943399i	$-0.392391\pi$
$827$	19.0756	0.663323	0.331661	+	0.943399i	$0.392391\pi$
$828$	0	0
$829$	9.24359	0.321043	0.160522	−	0.987032i	$-0.448682\pi$
$829$	9.24359	0.321043	0.160522	+	0.987032i	$0.448682\pi$
$830$	0.943131	0.0327366
$831$	0	0
$832$	5.66127	0.196269
$833$	76.6985	2.65745
$834$	0	0
$835$	−29.9608	−1.03684
$836$	0	0
$837$	0	0
$838$	45.8138	1.58261
$839$	8.76986	0.302769	0.151385	−	0.988475i	$-0.451627\pi$
$839$	8.76986	0.302769	0.151385	+	0.988475i	$0.451627\pi$
$840$	0	0
$841$	−18.8776	−0.650953
$842$	3.83785	0.132261
$843$	0	0
$844$	0.0154816	0.000532898 0
$845$	34.0030	1.16974
$846$	0	0
$847$	−11.4534	−0.393542
$848$	41.6005	1.42857
$849$	0	0
$850$	−22.3201	−0.765572
$851$	1.42378	0.0488064
$852$	0	0
$853$	20.6176	0.705934	0.352967	−	0.935636i	$-0.385173\pi$
$853$	20.6176	0.705934	0.352967	+	0.935636i	$0.385173\pi$
$854$	−46.5684	−1.59354
$855$	0	0
$856$	7.00950	0.239580
$857$	35.9624	1.22845	0.614227	−	0.789130i	$-0.289468\pi$
$857$	35.9624	1.22845	0.614227	+	0.789130i	$0.289468\pi$
$858$	0	0
$859$	−13.5672	−0.462906	−0.231453	−	0.972846i	$-0.574348\pi$
$859$	−13.5672	−0.462906	−0.231453	+	0.972846i	$0.574348\pi$
$860$	−0.761018	−0.0259505
$861$	0	0
$862$	47.1979	1.60757
$863$	45.3925	1.54518	0.772589	−	0.634906i	$-0.218961\pi$
$863$	45.3925	1.54518	0.772589	+	0.634906i	$0.218961\pi$
$864$	0	0
$865$	−37.6759	−1.28102
$866$	−26.1910	−0.890007
$867$	0	0
$868$	0.943563	0.0320266
$869$	60.9053	2.06607
$870$	0	0
$871$	4.65002	0.157560
$872$	28.2253	0.955828
$873$	0	0
$874$	0	0
$875$	−30.3019	−1.02439
$876$	0	0
$877$	34.1875	1.15443	0.577215	−	0.816592i	$-0.304140\pi$
$877$	34.1875	1.15443	0.577215	+	0.816592i	$0.304140\pi$
$878$	23.9811	0.809323
$879$	0	0
$880$	−42.6536	−1.43785
$881$	20.3548	0.685771	0.342885	−	0.939377i	$-0.388596\pi$
$881$	20.3548	0.685771	0.342885	+	0.939377i	$0.388596\pi$
$882$	0	0
$883$	−46.0188	−1.54866	−0.774328	−	0.632785i	$-0.781912\pi$
$883$	−46.0188	−1.54866	−0.774328	+	0.632785i	$0.781912\pi$
$884$	0.563302	0.0189459
$885$	0	0
$886$	−43.6623	−1.46686
$887$	−15.4631	−0.519200	−0.259600	−	0.965716i	$-0.583591\pi$
$887$	−15.4631	−0.519200	−0.259600	+	0.965716i	$0.583591\pi$
$888$	0	0
$889$	65.7083	2.20379
$890$	31.0344	1.04027
$891$	0	0
$892$	−2.32171	−0.0777366
$893$	0	0
$894$	0	0
$895$	55.6023	1.85858
$896$	53.9430	1.80211
$897$	0	0
$898$	17.6705	0.589674
$899$	−5.64193	−0.188169
$900$	0	0
$901$	−60.5800	−2.01821
$902$	63.3843	2.11047
$903$	0	0
$904$	−27.1925	−0.904410
$905$	−43.8640	−1.45809
$906$	0	0
$907$	37.2026	1.23529	0.617647	−	0.786456i	$-0.288086\pi$
$907$	37.2026	1.23529	0.617647	+	0.786456i	$0.288086\pi$
$908$	0.366001	0.0121462
$909$	0	0
$910$	−13.3405	−0.442233
$911$	−46.5472	−1.54218	−0.771088	−	0.636728i	$-0.780287\pi$
$911$	−46.5472	−1.54218	−0.771088	+	0.636728i	$0.780287\pi$
$912$	0	0
$913$	−0.872578	−0.0288781
$914$	−43.9561	−1.45394
$915$	0	0
$916$	0.702333	0.0232058
$917$	−0.269101	−0.00888650
$918$	0	0
$919$	−30.2003	−0.996215	−0.498108	−	0.867115i	$-0.665972\pi$
$919$	−30.2003	−0.996215	−0.498108	+	0.867115i	$0.665972\pi$
$920$	3.78813	0.124891
$921$	0	0
$922$	−35.1634	−1.15805
$923$	2.14753	0.0706869
$924$	0	0
$925$	−7.01043	−0.230501
$926$	10.3633	0.340558
$927$	0	0
$928$	2.16788	0.0711642
$929$	10.1326	0.332441	0.166220	−	0.986089i	$-0.446844\pi$
$929$	10.1326	0.332441	0.166220	+	0.986089i	$0.446844\pi$
$930$	0	0
$931$	0	0
$932$	1.21223	0.0397080
$933$	0	0
$934$	2.55405	0.0835712
$935$	62.1136	2.03133
$936$	0	0
$937$	6.03508	0.197158	0.0985788	−	0.995129i	$-0.468570\pi$
$937$	6.03508	0.197158	0.0985788	+	0.995129i	$0.468570\pi$
$938$	39.3694	1.28546
$939$	0	0
$940$	−2.69015	−0.0877430
$941$	7.34217	0.239348	0.119674	−	0.992813i	$-0.461815\pi$
$941$	7.34217	0.239348	0.119674	+	0.992813i	$0.461815\pi$
$942$	0	0
$943$	−5.96997	−0.194409
$944$	−40.6002	−1.32143
$945$	0	0
$946$	12.3791	0.402479
$947$	−15.1901	−0.493611	−0.246806	−	0.969065i	$-0.579381\pi$
$947$	−15.1901	−0.493611	−0.246806	+	0.969065i	$0.579381\pi$
$948$	0	0
$949$	3.64260	0.118244
$950$	0	0
$951$	0	0
$952$	−74.3123	−2.40848
$953$	−12.9629	−0.419910	−0.209955	−	0.977711i	$-0.567332\pi$
$953$	−12.9629	−0.419910	−0.209955	+	0.977711i	$0.567332\pi$
$954$	0	0
$955$	22.2728	0.720731
$956$	−0.483200	−0.0156278
$957$	0	0
$958$	−17.9900	−0.581231
$959$	35.8080	1.15630
$960$	0	0
$961$	−27.8553	−0.898559
$962$	3.11065	0.100292
$963$	0	0
$964$	−0.906726	−0.0292037
$965$	−45.1731	−1.45417
$966$	0	0
$967$	−4.63722	−0.149123	−0.0745615	−	0.997216i	$-0.523756\pi$
$967$	−4.63722	−0.149123	−0.0745615	+	0.997216i	$0.523756\pi$
$968$	7.10552	0.228380
$969$	0	0
$970$	19.7023	0.632604
$971$	−11.7831	−0.378139	−0.189069	−	0.981964i	$-0.560547\pi$
$971$	−11.7831	−0.378139	−0.189069	+	0.981964i	$0.560547\pi$
$972$	0	0
$973$	−17.1361	−0.549358
$974$	11.0866	0.355236
$975$	0	0
$976$	30.6391	0.980733
$977$	−22.8874	−0.732234	−0.366117	−	0.930569i	$-0.619313\pi$
$977$	−22.8874	−0.732234	−0.366117	+	0.930569i	$0.619313\pi$
$978$	0	0
$979$	−28.7128	−0.917664
$980$	−4.11343	−0.131399
$981$	0	0
$982$	−8.04902	−0.256855
$983$	28.1244	0.897030	0.448515	−	0.893775i	$-0.351953\pi$
$983$	28.1244	0.897030	0.448515	+	0.893775i	$0.351953\pi$
$984$	0	0
$985$	54.6100	1.74002
$986$	−28.5169	−0.908163
$987$	0	0
$988$	0	0
$989$	−1.16595	−0.0370750
$990$	0	0
$991$	−28.5175	−0.905890	−0.452945	−	0.891539i	$-0.649627\pi$
$991$	−28.5175	−0.905890	−0.452945	+	0.891539i	$0.649627\pi$
$992$	−1.20832	−0.0383641
$993$	0	0
$994$	18.1821	0.576701
$995$	4.36134	0.138264
$996$	0	0
$997$	−25.6168	−0.811292	−0.405646	−	0.914030i	$-0.632953\pi$
$997$	−25.6168	−0.811292	−0.405646	+	0.914030i	$0.632953\pi$
$998$	7.42435	0.235014
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bi.1.3		6
3.2	odd	2	inner	3249.2.a.bi.1.4		6
19.4	even	9		171.2.u.d.73.1	✓	12
19.5	even	9		171.2.u.d.82.1	yes	12
19.18	odd	2		3249.2.a.bj.1.4		6
57.5	odd	18		171.2.u.d.82.2	yes	12
57.23	odd	18		171.2.u.d.73.2	yes	12
57.56	even	2		3249.2.a.bj.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.u.d.73.1	✓	12	19.4	even	9
171.2.u.d.73.2	yes	12	57.23	odd	18
171.2.u.d.82.1	yes	12	19.5	even	9
171.2.u.d.82.2	yes	12	57.5	odd	18
3249.2.a.bi.1.3		6	1.1	even	1	trivial
3249.2.a.bi.1.4		6	3.2	odd	2	inner
3249.2.a.bj.1.3		6	57.56	even	2
3249.2.a.bj.1.4		6	19.18	odd	2

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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45623 q^{2} +0.120615 q^{4} -2.73682 q^{5} -4.41147 q^{7} +2.73682 q^{8} +3.98545 q^{10} -3.68731 q^{11} +0.758770 q^{13} +6.42413 q^{14} -4.22668 q^{16} +6.15503 q^{17} -0.330101 q^{20} +5.36959 q^{22} -0.505744 q^{23} +2.49020 q^{25} -1.10495 q^{26} -0.532089 q^{28} +3.18157 q^{29} -1.77332 q^{31} +0.681388 q^{32} -8.96316 q^{34} +12.0734 q^{35} -2.81521 q^{37} -7.49020 q^{40} +11.8043 q^{41} +2.30541 q^{43} -0.444744 q^{44} +0.736482 q^{46} +8.14947 q^{47} +12.4611 q^{49} -3.62631 q^{50} +0.0915189 q^{52} -9.84235 q^{53} +10.0915 q^{55} -12.0734 q^{56} -4.63310 q^{58} +9.60570 q^{59} -7.24897 q^{61} +2.58236 q^{62} +7.46110 q^{64} -2.07662 q^{65} +6.12836 q^{67} +0.742388 q^{68} -17.5817 q^{70} +2.83028 q^{71} +4.80066 q^{73} +4.09960 q^{74} +16.2665 q^{77} -16.5175 q^{79} +11.5677 q^{80} -17.1898 q^{82} +0.236643 q^{83} -16.8452 q^{85} -3.35721 q^{86} -10.0915 q^{88} +7.78691 q^{89} -3.34730 q^{91} -0.0610002 q^{92} -11.8675 q^{94} +4.94356 q^{97} -18.1463 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 6 q^{7} - 12 q^{10} - 18 q^{13} - 12 q^{16} + 18 q^{22} + 12 q^{25} + 6 q^{28} - 24 q^{31} - 30 q^{34} - 24 q^{37} - 42 q^{40} + 18 q^{43} - 6 q^{46} - 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots - 6 q^{94}+O(q^{100}) \) 6 * q + 12 * q^4 - 6 * q^7 - 12 * q^10 - 18 * q^13 - 12 * q^16 + 18 * q^22 + 12 * q^25 + 6 * q^28 - 24 * q^31 - 30 * q^34 - 24 * q^37 - 42 * q^40 + 18 * q^43 - 6 * q^46 - 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 - 42 * q^70 - 54 * q^79 - 66 * q^82 - 48 * q^85 - 18 * q^91 - 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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