LMFDB - Embedded newform 3249.2.a.w.1.2

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 = [3,0,0,0,3,0,-3,-3,0,6,3,0,3,6,0,-6,9,0,0,-3,0,3,-6,0,-6,12,0, -3,-9,0,24,9,0,-15,3,0,6,0,0,-9,6,0,-21,12,0,9,3,0,-12,9,0,-6,18,0,6,-3, 0,-12,15,0,-9,3,0,-3,15,0,-6,3,0,-3,9,0,-6,3,0,0,0,0,9,-3,0,21,15,0,-6, 18,0,-6,0,0,9] 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:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+0.347296 q^{2} -1.87939 q^{4} +1.34730 q^{5} -0.652704 q^{7} -1.34730 q^{8} +0.467911 q^{10} -3.94356 q^{11} +1.69459 q^{13} -0.226682 q^{14} +3.29086 q^{16} +3.83750 q^{17} -2.53209 q^{20} -1.36959 q^{22} +3.63816 q^{23} -3.18479 q^{25} +0.588526 q^{26} +1.22668 q^{28} -10.5175 q^{29} +6.46791 q^{31} +3.83750 q^{32} +1.33275 q^{34} -0.879385 q^{35} -2.94356 q^{37} -1.81521 q^{40} +1.50980 q^{41} -9.36959 q^{43} +7.41147 q^{44} +1.26352 q^{46} +12.7665 q^{47} -6.57398 q^{49} -1.10607 q^{50} -3.18479 q^{52} +10.8007 q^{53} -5.31315 q^{55} +0.879385 q^{56} -3.65270 q^{58} +2.77332 q^{59} +4.31315 q^{61} +2.24628 q^{62} -5.24897 q^{64} +2.28312 q^{65} +7.88713 q^{67} -7.21213 q^{68} -0.305407 q^{70} +5.36959 q^{71} +5.86484 q^{73} -1.02229 q^{74} +2.57398 q^{77} -3.12836 q^{79} +4.43376 q^{80} +0.524348 q^{82} +4.50980 q^{83} +5.17024 q^{85} -3.25402 q^{86} +5.31315 q^{88} +11.1334 q^{89} -1.10607 q^{91} -6.83750 q^{92} +4.43376 q^{94} +6.70233 q^{97} -2.28312 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} + 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 9 q^{29} + 24 q^{31} + 9 q^{32} - 15 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 + 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 + 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 9 * q^29 + 24 * q^31 + 9 * q^32 - 15 * q^34 + 3 * q^35 + 6 * q^37 - 9 * q^40 + 6 * q^41 - 21 * q^43 + 12 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 + 9 * q^50 - 6 * q^52 + 18 * q^53 + 6 * q^55 - 3 * q^56 - 12 * q^58 + 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 + 15 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 9 * q^71 - 6 * q^73 + 3 * q^74 + 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 + 18 * q^86 - 6 * q^88 + 9 * q^91 - 18 * q^92 - 3 * q^94 - 6 * q^97 - 15 * q^98

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 $

$2$	0.347296	0.245576	0.122788	−	0.992433i	$-0.460817\pi$
$2$	0.347296	0.245576	0.122788	+	0.992433i	$0.460817\pi$
$3$	0	0
$3$	0	0
$4$	−1.87939	−0.939693
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	−0.652704	−0.246699	−0.123349	−	0.992363i	$-0.539364\pi$
$7$	−0.652704	−0.246699	−0.123349	+	0.992363i	$0.539364\pi$
$8$	−1.34730	−0.476341
$9$	0	0
$10$	0.467911	0.147966
$11$	−3.94356	−1.18903	−0.594514	−	0.804085i	$-0.702656\pi$
$11$	−3.94356	−1.18903	−0.594514	+	0.804085i	$0.702656\pi$
$12$	0	0
$13$	1.69459	0.469995	0.234998	−	0.971996i	$-0.424492\pi$
$13$	1.69459	0.469995	0.234998	+	0.971996i	$0.424492\pi$
$14$	−0.226682	−0.0605832
$15$	0	0
$16$	3.29086	0.822715
$17$	3.83750	0.930730	0.465365	−	0.885119i	$-0.345923\pi$
$17$	3.83750	0.930730	0.465365	+	0.885119i	$0.345923\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−2.53209	−0.566192
$21$	0	0
$22$	−1.36959	−0.291997
$23$	3.63816	0.758608	0.379304	−	0.925272i	$-0.376163\pi$
$23$	3.63816	0.758608	0.379304	+	0.925272i	$0.376163\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	0.588526	0.115419
$27$	0	0
$28$	1.22668	0.231821
$29$	−10.5175	−1.95306	−0.976529	−	0.215385i	$-0.930899\pi$
$29$	−10.5175	−1.95306	−0.976529	+	0.215385i	$0.930899\pi$
$30$	0	0
$31$	6.46791	1.16167	0.580836	−	0.814021i	$-0.302726\pi$
$31$	6.46791	1.16167	0.580836	+	0.814021i	$0.302726\pi$
$32$	3.83750	0.678380
$33$	0	0
$34$	1.33275	0.228564
$35$	−0.879385	−0.148643
$36$	0	0
$37$	−2.94356	−0.483919	−0.241959	−	0.970286i	$-0.577790\pi$
$37$	−2.94356	−0.483919	−0.241959	+	0.970286i	$0.577790\pi$
$38$	0	0
$39$	0	0
$40$	−1.81521	−0.287010
$41$	1.50980	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$41$	1.50980	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$42$	0	0
$43$	−9.36959	−1.42885	−0.714424	−	0.699713i	$-0.753311\pi$
$43$	−9.36959	−1.42885	−0.714424	+	0.699713i	$0.753311\pi$
$44$	7.41147	1.11732
$45$	0	0
$46$	1.26352	0.186296
$47$	12.7665	1.86219	0.931094	−	0.364781i	$-0.118856\pi$
$47$	12.7665	1.86219	0.931094	+	0.364781i	$0.118856\pi$
$48$	0	0
$49$	−6.57398	−0.939140
$50$	−1.10607	−0.156421
$51$	0	0
$52$	−3.18479	−0.441651
$53$	10.8007	1.48358	0.741792	−	0.670630i	$-0.233976\pi$
$53$	10.8007	1.48358	0.741792	+	0.670630i	$0.233976\pi$
$54$	0	0
$55$	−5.31315	−0.716425
$56$	0.879385	0.117513
$57$	0	0
$58$	−3.65270	−0.479623
$59$	2.77332	0.361055	0.180528	−	0.983570i	$-0.442219\pi$
$59$	2.77332	0.361055	0.180528	+	0.983570i	$0.442219\pi$
$60$	0	0
$61$	4.31315	0.552242	0.276121	−	0.961123i	$-0.410951\pi$
$61$	4.31315	0.552242	0.276121	+	0.961123i	$0.410951\pi$
$62$	2.24628	0.285278
$63$	0	0
$64$	−5.24897	−0.656121
$65$	2.28312	0.283186
$66$	0	0
$67$	7.88713	0.963566	0.481783	−	0.876291i	$-0.339989\pi$
$67$	7.88713	0.963566	0.481783	+	0.876291i	$0.339989\pi$
$68$	−7.21213	−0.874600
$69$	0	0
$70$	−0.305407	−0.0365032
$71$	5.36959	0.637253	0.318626	−	0.947880i	$-0.396779\pi$
$71$	5.36959	0.637253	0.318626	+	0.947880i	$0.396779\pi$
$72$	0	0
$73$	5.86484	0.686427	0.343214	−	0.939257i	$-0.388485\pi$
$73$	5.86484	0.686427	0.343214	+	0.939257i	$0.388485\pi$
$74$	−1.02229	−0.118839
$75$	0	0
$76$	0	0
$77$	2.57398	0.293332
$78$	0	0
$79$	−3.12836	−0.351967	−0.175984	−	0.984393i	$-0.556311\pi$
$79$	−3.12836	−0.351967	−0.175984	+	0.984393i	$0.556311\pi$
$80$	4.43376	0.495710
$81$	0	0
$82$	0.524348	0.0579046
$83$	4.50980	0.495015	0.247507	−	0.968886i	$-0.420389\pi$
$83$	4.50980	0.495015	0.247507	+	0.968886i	$0.420389\pi$
$84$	0	0
$85$	5.17024	0.560792
$86$	−3.25402	−0.350890
$87$	0	0
$88$	5.31315	0.566383
$89$	11.1334	1.18014	0.590069	−	0.807352i	$-0.299100\pi$
$89$	11.1334	1.18014	0.590069	+	0.807352i	$0.299100\pi$
$90$	0	0
$91$	−1.10607	−0.115947
$92$	−6.83750	−0.712858
$93$	0	0
$94$	4.43376	0.457308
$95$	0	0
$96$	0	0
$97$	6.70233	0.680519	0.340259	−	0.940332i	$-0.389485\pi$
$97$	6.70233	0.680519	0.340259	+	0.940332i	$0.389485\pi$
$98$	−2.28312	−0.230630
$99$	0	0
$100$	5.98545	0.598545
$101$	1.93582	0.192622	0.0963108	−	0.995351i	$-0.469296\pi$
$101$	1.93582	0.192622	0.0963108	+	0.995351i	$0.469296\pi$
$102$	0	0
$103$	1.03684	0.102163	0.0510813	−	0.998694i	$-0.483733\pi$
$103$	1.03684	0.102163	0.0510813	+	0.998694i	$0.483733\pi$
$104$	−2.28312	−0.223878
$105$	0	0
$106$	3.75103	0.364332
$107$	12.2121	1.18059	0.590296	−	0.807187i	$-0.299011\pi$
$107$	12.2121	1.18059	0.590296	+	0.807187i	$0.299011\pi$
$108$	0	0
$109$	−4.02734	−0.385749	−0.192875	−	0.981223i	$-0.561781\pi$
$109$	−4.02734	−0.385749	−0.192875	+	0.981223i	$0.561781\pi$
$110$	−1.84524	−0.175936
$111$	0	0
$112$	−2.14796	−0.202963
$113$	13.1925	1.24105	0.620525	−	0.784187i	$-0.286920\pi$
$113$	13.1925	1.24105	0.620525	+	0.784187i	$0.286920\pi$
$114$	0	0
$115$	4.90167	0.457083
$116$	19.7665	1.83527
$117$	0	0
$118$	0.963163	0.0886664
$119$	−2.50475	−0.229610
$120$	0	0
$121$	4.55169	0.413790
$122$	1.49794	0.135617
$123$	0	0
$124$	−12.1557	−1.09161
$125$	−11.0273	−0.986315
$126$	0	0
$127$	13.2490	1.17566	0.587828	−	0.808986i	$-0.299983\pi$
$127$	13.2490	1.17566	0.587828	+	0.808986i	$0.299983\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	0.792919	0.0695436
$131$	−17.1780	−1.50085	−0.750424	−	0.660957i	$-0.770151\pi$
$131$	−17.1780	−1.50085	−0.750424	+	0.660957i	$0.770151\pi$
$132$	0	0
$133$	0	0
$134$	2.73917	0.236628
$135$	0	0
$136$	−5.17024	−0.443345
$137$	10.5466	0.901060	0.450530	−	0.892761i	$-0.351235\pi$
$137$	10.5466	0.901060	0.450530	+	0.892761i	$0.351235\pi$
$138$	0	0
$139$	10.7811	0.914438	0.457219	−	0.889354i	$-0.348846\pi$
$139$	10.7811	0.914438	0.457219	+	0.889354i	$0.348846\pi$
$140$	1.65270	0.139679
$141$	0	0
$142$	1.86484	0.156494
$143$	−6.68273	−0.558838
$144$	0	0
$145$	−14.1702	−1.17677
$146$	2.03684	0.168570
$147$	0	0
$148$	5.53209	0.454735
$149$	19.7520	1.61814	0.809072	−	0.587710i	$-0.199970\pi$
$149$	19.7520	1.61814	0.809072	+	0.587710i	$0.199970\pi$
$150$	0	0
$151$	3.10607	0.252768	0.126384	−	0.991981i	$-0.459663\pi$
$151$	3.10607	0.252768	0.126384	+	0.991981i	$0.459663\pi$
$152$	0	0
$153$	0	0
$154$	0.893933	0.0720352
$155$	8.71419	0.699941
$156$	0	0
$157$	0.736482	0.0587776	0.0293888	−	0.999568i	$-0.490644\pi$
$157$	0.736482	0.0587776	0.0293888	+	0.999568i	$0.490644\pi$
$158$	−1.08647	−0.0864346
$159$	0	0
$160$	5.17024	0.408744
$161$	−2.37464	−0.187148
$162$	0	0
$163$	−12.6236	−0.988757	−0.494379	−	0.869247i	$-0.664604\pi$
$163$	−12.6236	−0.988757	−0.494379	+	0.869247i	$0.664604\pi$
$164$	−2.83750	−0.221571
$165$	0	0
$166$	1.56624	0.121564
$167$	5.51754	0.426960	0.213480	−	0.976947i	$-0.431520\pi$
$167$	5.51754	0.426960	0.213480	+	0.976947i	$0.431520\pi$
$168$	0	0
$169$	−10.1284	−0.779104
$170$	1.79561	0.137717
$171$	0	0
$172$	17.6091	1.34268
$173$	−12.9290	−0.982975	−0.491487	−	0.870885i	$-0.663547\pi$
$173$	−12.9290	−0.982975	−0.491487	+	0.870885i	$0.663547\pi$
$174$	0	0
$175$	2.07873	0.157137
$176$	−12.9777	−0.978232
$177$	0	0
$178$	3.86659	0.289813
$179$	15.0865	1.12762	0.563808	−	0.825906i	$-0.309336\pi$
$179$	15.0865	1.12762	0.563808	+	0.825906i	$0.309336\pi$
$180$	0	0
$181$	−18.6827	−1.38868	−0.694338	−	0.719649i	$-0.744303\pi$
$181$	−18.6827	−1.38868	−0.694338	+	0.719649i	$0.744303\pi$
$182$	−0.384133	−0.0284738
$183$	0	0
$184$	−4.90167	−0.361356
$185$	−3.96585	−0.291575
$186$	0	0
$187$	−15.1334	−1.10666
$188$	−23.9932	−1.74988
$189$	0	0
$190$	0	0
$191$	8.55169	0.618779	0.309389	−	0.950935i	$-0.399875\pi$
$191$	8.55169	0.618779	0.309389	+	0.950935i	$0.399875\pi$
$192$	0	0
$193$	16.9590	1.22074	0.610369	−	0.792117i	$-0.291021\pi$
$193$	16.9590	1.22074	0.610369	+	0.792117i	$0.291021\pi$
$194$	2.32770	0.167119
$195$	0	0
$196$	12.3550	0.882503
$197$	−4.51754	−0.321861	−0.160931	−	0.986966i	$-0.551450\pi$
$197$	−4.51754	−0.321861	−0.160931	+	0.986966i	$0.551450\pi$
$198$	0	0
$199$	−8.04694	−0.570433	−0.285216	−	0.958463i	$-0.592065\pi$
$199$	−8.04694	−0.570433	−0.285216	+	0.958463i	$0.592065\pi$
$200$	4.29086	0.303410
$201$	0	0
$202$	0.672304	0.0473031
$203$	6.86484	0.481817
$204$	0	0
$205$	2.03415	0.142071
$206$	0.360090	0.0250886
$207$	0	0
$208$	5.57667	0.386672
$209$	0	0
$210$	0	0
$211$	20.0838	1.38262	0.691312	−	0.722556i	$-0.257033\pi$
$211$	20.0838	1.38262	0.691312	+	0.722556i	$0.257033\pi$
$212$	−20.2986	−1.39411
$213$	0	0
$214$	4.24123	0.289924
$215$	−12.6236	−0.860923
$216$	0	0
$217$	−4.22163	−0.286583
$218$	−1.39868	−0.0947306
$219$	0	0
$220$	9.98545	0.673219
$221$	6.50299	0.437439
$222$	0	0
$223$	19.5895	1.31181	0.655904	−	0.754845i	$-0.272288\pi$
$223$	19.5895	1.31181	0.655904	+	0.754845i	$0.272288\pi$
$224$	−2.50475	−0.167355
$225$	0	0
$226$	4.58172	0.304771
$227$	18.4979	1.22775	0.613876	−	0.789403i	$-0.289610\pi$
$227$	18.4979	1.22775	0.613876	+	0.789403i	$0.289610\pi$
$228$	0	0
$229$	−28.1634	−1.86109	−0.930546	−	0.366175i	$-0.880667\pi$
$229$	−28.1634	−1.86109	−0.930546	+	0.366175i	$0.880667\pi$
$230$	1.70233	0.112249
$231$	0	0
$232$	14.1702	0.930322
$233$	−0.155697	−0.0102000	−0.00510001	−	0.999987i	$-0.501623\pi$
$233$	−0.155697	−0.0102000	−0.00510001	+	0.999987i	$0.501623\pi$
$234$	0	0
$235$	17.2003	1.12202
$236$	−5.21213	−0.339281
$237$	0	0
$238$	−0.869890	−0.0563866
$239$	−6.63041	−0.428886	−0.214443	−	0.976737i	$-0.568794\pi$
$239$	−6.63041	−0.428886	−0.214443	+	0.976737i	$0.568794\pi$
$240$	0	0
$241$	−5.26083	−0.338880	−0.169440	−	0.985541i	$-0.554196\pi$
$241$	−5.26083	−0.338880	−0.169440	+	0.985541i	$0.554196\pi$
$242$	1.58079	0.101617
$243$	0	0
$244$	−8.10607	−0.518938
$245$	−8.85710	−0.565859
$246$	0	0
$247$	0	0
$248$	−8.71419	−0.553352
$249$	0	0
$250$	−3.82976	−0.242215
$251$	−14.8152	−0.935128	−0.467564	−	0.883959i	$-0.654868\pi$
$251$	−14.8152	−0.935128	−0.467564	+	0.883959i	$0.654868\pi$
$252$	0	0
$253$	−14.3473	−0.902007
$254$	4.60132	0.288712
$255$	0	0
$256$	7.19934	0.449959
$257$	19.1634	1.19538	0.597691	−	0.801726i	$-0.296085\pi$
$257$	19.1634	1.19538	0.597691	+	0.801726i	$0.296085\pi$
$258$	0	0
$259$	1.92127	0.119382
$260$	−4.29086	−0.266108
$261$	0	0
$262$	−5.96585	−0.368572
$263$	16.4979	1.01731	0.508653	−	0.860971i	$-0.330144\pi$
$263$	16.4979	1.01731	0.508653	+	0.860971i	$0.330144\pi$
$264$	0	0
$265$	14.5517	0.893903
$266$	0	0
$267$	0	0
$268$	−14.8229	−0.905456
$269$	−4.21213	−0.256818	−0.128409	−	0.991721i	$-0.540987\pi$
$269$	−4.21213	−0.256818	−0.128409	+	0.991721i	$0.540987\pi$
$270$	0	0
$271$	9.16250	0.556582	0.278291	−	0.960497i	$-0.410232\pi$
$271$	9.16250	0.556582	0.278291	+	0.960497i	$0.410232\pi$
$272$	12.6287	0.765725
$273$	0	0
$274$	3.66281	0.221278
$275$	12.5594	0.757362
$276$	0	0
$277$	−22.4953	−1.35161	−0.675804	−	0.737081i	$-0.736204\pi$
$277$	−22.4953	−1.35161	−0.675804	+	0.737081i	$0.736204\pi$
$278$	3.74422	0.224564
$279$	0	0
$280$	1.18479	0.0708049
$281$	23.5280	1.40356	0.701781	−	0.712393i	$-0.252389\pi$
$281$	23.5280	1.40356	0.701781	+	0.712393i	$0.252389\pi$
$282$	0	0
$283$	26.7469	1.58994	0.794969	−	0.606650i	$-0.207487\pi$
$283$	26.7469	1.58994	0.794969	+	0.606650i	$0.207487\pi$
$284$	−10.0915	−0.598821
$285$	0	0
$286$	−2.32089	−0.137237
$287$	−0.985452	−0.0581694
$288$	0	0
$289$	−2.27362	−0.133743
$290$	−4.92127	−0.288987
$291$	0	0
$292$	−11.0223	−0.645031
$293$	0.207081	0.0120978	0.00604891	−	0.999982i	$-0.498075\pi$
$293$	0.207081	0.0120978	0.00604891	+	0.999982i	$0.498075\pi$
$294$	0	0
$295$	3.73648	0.217546
$296$	3.96585	0.230510
$297$	0	0
$298$	6.85978	0.397377
$299$	6.16519	0.356542
$300$	0	0
$301$	6.11556	0.352495
$302$	1.07873	0.0620737
$303$	0	0
$304$	0	0
$305$	5.81109	0.332742
$306$	0	0
$307$	15.3182	0.874256	0.437128	−	0.899399i	$-0.355996\pi$
$307$	15.3182	0.874256	0.437128	+	0.899399i	$0.355996\pi$
$308$	−4.83750	−0.275642
$309$	0	0
$310$	3.02641	0.171888
$311$	−15.6432	−0.887045	−0.443522	−	0.896263i	$-0.646271\pi$
$311$	−15.6432	−0.887045	−0.443522	+	0.896263i	$0.646271\pi$
$312$	0	0
$313$	10.1625	0.574419	0.287209	−	0.957868i	$-0.407272\pi$
$313$	10.1625	0.574419	0.287209	+	0.957868i	$0.407272\pi$
$314$	0.255777	0.0144344
$315$	0	0
$316$	5.87939	0.330741
$317$	7.64765	0.429535	0.214767	−	0.976665i	$-0.431101\pi$
$317$	7.64765	0.429535	0.214767	+	0.976665i	$0.431101\pi$
$318$	0	0
$319$	41.4766	2.32224
$320$	−7.07192	−0.395332
$321$	0	0
$322$	−0.824703	−0.0459589
$323$	0	0
$324$	0	0
$325$	−5.39693	−0.299368
$326$	−4.38413	−0.242815
$327$	0	0
$328$	−2.03415	−0.112317
$329$	−8.33275	−0.459399
$330$	0	0
$331$	−4.10607	−0.225690	−0.112845	−	0.993613i	$-0.535996\pi$
$331$	−4.10607	−0.225690	−0.112845	+	0.993613i	$0.535996\pi$
$332$	−8.47565	−0.465162
$333$	0	0
$334$	1.91622	0.104851
$335$	10.6263	0.580577
$336$	0	0
$337$	−12.0915	−0.658667	−0.329334	−	0.944214i	$-0.606824\pi$
$337$	−12.0915	−0.658667	−0.329334	+	0.944214i	$0.606824\pi$
$338$	−3.51754	−0.191329
$339$	0	0
$340$	−9.71688	−0.526972
$341$	−25.5066	−1.38126
$342$	0	0
$343$	8.85978	0.478383
$344$	12.6236	0.680619
$345$	0	0
$346$	−4.49020	−0.241395
$347$	4.14290	0.222403	0.111201	−	0.993798i	$-0.464530\pi$
$347$	4.14290	0.222403	0.111201	+	0.993798i	$0.464530\pi$
$348$	0	0
$349$	−0.0392007	−0.00209837	−0.00104918	−	0.999999i	$-0.500334\pi$
$349$	−0.0392007	−0.00209837	−0.00104918	+	0.999999i	$0.500334\pi$
$350$	0.721934	0.0385890
$351$	0	0
$352$	−15.1334	−0.806613
$353$	−7.16250	−0.381222	−0.190611	−	0.981666i	$-0.561047\pi$
$353$	−7.16250	−0.381222	−0.190611	+	0.981666i	$0.561047\pi$
$354$	0	0
$355$	7.23442	0.383963
$356$	−20.9240	−1.10897
$357$	0	0
$358$	5.23947	0.276915
$359$	20.4415	1.07886	0.539431	−	0.842030i	$-0.318640\pi$
$359$	20.4415	1.07886	0.539431	+	0.842030i	$0.318640\pi$
$360$	0	0
$361$	0	0
$362$	−6.48845	−0.341025
$363$	0	0
$364$	2.07873	0.108955
$365$	7.90167	0.413593
$366$	0	0
$367$	−1.59533	−0.0832757	−0.0416379	−	0.999133i	$-0.513258\pi$
$367$	−1.59533	−0.0832757	−0.0416379	+	0.999133i	$0.513258\pi$
$368$	11.9727	0.624118
$369$	0	0
$370$	−1.37733	−0.0716038
$371$	−7.04963	−0.365999
$372$	0	0
$373$	−12.8520	−0.665454	−0.332727	−	0.943023i	$-0.607969\pi$
$373$	−12.8520	−0.665454	−0.332727	+	0.943023i	$0.607969\pi$
$374$	−5.25578	−0.271770
$375$	0	0
$376$	−17.2003	−0.887036
$377$	−17.8229	−0.917929
$378$	0	0
$379$	−15.7023	−0.806575	−0.403287	−	0.915073i	$-0.632132\pi$
$379$	−15.7023	−0.806575	−0.403287	+	0.915073i	$0.632132\pi$
$380$	0	0
$381$	0	0
$382$	2.96997	0.151957
$383$	16.4483	0.840469	0.420235	−	0.907415i	$-0.361948\pi$
$383$	16.4483	0.840469	0.420235	+	0.907415i	$0.361948\pi$
$384$	0	0
$385$	3.46791	0.176741
$386$	5.88981	0.299784
$387$	0	0
$388$	−12.5963	−0.639479
$389$	−7.53302	−0.381939	−0.190970	−	0.981596i	$-0.561163\pi$
$389$	−7.53302	−0.381939	−0.190970	+	0.981596i	$0.561163\pi$
$390$	0	0
$391$	13.9614	0.706059
$392$	8.85710	0.447351
$393$	0	0
$394$	−1.56893	−0.0790413
$395$	−4.21482	−0.212071
$396$	0	0
$397$	−11.2189	−0.563062	−0.281531	−	0.959552i	$-0.590842\pi$
$397$	−11.2189	−0.563062	−0.281531	+	0.959552i	$0.590842\pi$
$398$	−2.79467	−0.140084
$399$	0	0
$400$	−10.4807	−0.524035
$401$	−24.2550	−1.21123	−0.605617	−	0.795756i	$-0.707074\pi$
$401$	−24.2550	−1.21123	−0.605617	+	0.795756i	$0.707074\pi$
$402$	0	0
$403$	10.9605	0.545980
$404$	−3.63816	−0.181005
$405$	0	0
$406$	2.38413	0.118323
$407$	11.6081	0.575393
$408$	0	0
$409$	26.2841	1.29966	0.649831	−	0.760078i	$-0.274840\pi$
$409$	26.2841	1.29966	0.649831	+	0.760078i	$0.274840\pi$
$410$	0.706452	0.0348892
$411$	0	0
$412$	−1.94862	−0.0960014
$413$	−1.81016	−0.0890719
$414$	0	0
$415$	6.07604	0.298261
$416$	6.50299	0.318835
$417$	0	0
$418$	0	0
$419$	−13.2226	−0.645964	−0.322982	−	0.946405i	$-0.604685\pi$
$419$	−13.2226	−0.645964	−0.322982	+	0.946405i	$0.604685\pi$
$420$	0	0
$421$	−37.0360	−1.80502	−0.902512	−	0.430664i	$-0.858280\pi$
$421$	−37.0360	−1.80502	−0.902512	+	0.430664i	$0.858280\pi$
$422$	6.97502	0.339539
$423$	0	0
$424$	−14.5517	−0.706693
$425$	−12.2216	−0.592836
$426$	0	0
$427$	−2.81521	−0.136237
$428$	−22.9513	−1.10939
$429$	0	0
$430$	−4.38413	−0.211422
$431$	−9.47296	−0.456297	−0.228148	−	0.973626i	$-0.573267\pi$
$431$	−9.47296	−0.456297	−0.228148	+	0.973626i	$0.573267\pi$
$432$	0	0
$433$	−22.5226	−1.08237	−0.541183	−	0.840905i	$-0.682024\pi$
$433$	−22.5226	−1.08237	−0.541183	+	0.840905i	$0.682024\pi$
$434$	−1.46616	−0.0703778
$435$	0	0
$436$	7.56893	0.362486
$437$	0	0
$438$	0	0
$439$	−33.2327	−1.58611	−0.793054	−	0.609151i	$-0.791510\pi$
$439$	−33.2327	−1.58611	−0.793054	+	0.609151i	$0.791510\pi$
$440$	7.15839	0.341263
$441$	0	0
$442$	2.25847	0.107424
$443$	−30.6290	−1.45523	−0.727613	−	0.685987i	$-0.759371\pi$
$443$	−30.6290	−1.45523	−0.727613	+	0.685987i	$0.759371\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	6.80335	0.322148
$447$	0	0
$448$	3.42602	0.161864
$449$	−15.0925	−0.712257	−0.356128	−	0.934437i	$-0.615903\pi$
$449$	−15.0925	−0.712257	−0.356128	+	0.934437i	$0.615903\pi$
$450$	0	0
$451$	−5.95399	−0.280363
$452$	−24.7939	−1.16620
$453$	0	0
$454$	6.42427	0.301506
$455$	−1.49020	−0.0698616
$456$	0	0
$457$	19.8462	0.928365	0.464182	−	0.885740i	$-0.346348\pi$
$457$	19.8462	0.928365	0.464182	+	0.885740i	$0.346348\pi$
$458$	−9.78106	−0.457039
$459$	0	0
$460$	−9.21213	−0.429518
$461$	34.7648	1.61916	0.809578	−	0.587012i	$-0.199696\pi$
$461$	34.7648	1.61916	0.809578	+	0.587012i	$0.199696\pi$
$462$	0	0
$463$	−3.42602	−0.159221	−0.0796104	−	0.996826i	$-0.525368\pi$
$463$	−3.42602	−0.159221	−0.0796104	+	0.996826i	$0.525368\pi$
$464$	−34.6117	−1.60681
$465$	0	0
$466$	−0.0540729	−0.00250488
$467$	11.3250	0.524059	0.262029	−	0.965060i	$-0.415608\pi$
$467$	11.3250	0.524059	0.262029	+	0.965060i	$0.415608\pi$
$468$	0	0
$469$	−5.14796	−0.237711
$470$	5.97359	0.275541
$471$	0	0
$472$	−3.73648	−0.171986
$473$	36.9495	1.69894
$474$	0	0
$475$	0	0
$476$	4.70739	0.215763
$477$	0	0
$478$	−2.30272	−0.105324
$479$	−11.5672	−0.528518	−0.264259	−	0.964452i	$-0.585127\pi$
$479$	−11.5672	−0.528518	−0.264259	+	0.964452i	$0.585127\pi$
$480$	0	0
$481$	−4.98814	−0.227440
$482$	−1.82707	−0.0832206
$483$	0	0
$484$	−8.55438	−0.388835
$485$	9.03003	0.410033
$486$	0	0
$487$	11.7469	0.532303	0.266152	−	0.963931i	$-0.414248\pi$
$487$	11.7469	0.532303	0.266152	+	0.963931i	$0.414248\pi$
$488$	−5.81109	−0.263056
$489$	0	0
$490$	−3.07604	−0.138961
$491$	−10.2540	−0.462758	−0.231379	−	0.972864i	$-0.574324\pi$
$491$	−10.2540	−0.462758	−0.231379	+	0.972864i	$0.574324\pi$
$492$	0	0
$493$	−40.3610	−1.81777
$494$	0	0
$495$	0	0
$496$	21.2850	0.955724
$497$	−3.50475	−0.157209
$498$	0	0
$499$	−6.45842	−0.289118	−0.144559	−	0.989496i	$-0.546176\pi$
$499$	−6.45842	−0.289118	−0.144559	+	0.989496i	$0.546176\pi$
$500$	20.7246	0.926833
$501$	0	0
$502$	−5.14527	−0.229645
$503$	18.3081	0.816318	0.408159	−	0.912911i	$-0.366171\pi$
$503$	18.3081	0.816318	0.408159	+	0.912911i	$0.366171\pi$
$504$	0	0
$505$	2.60813	0.116060
$506$	−4.98276	−0.221511
$507$	0	0
$508$	−24.8999	−1.10476
$509$	−12.0692	−0.534959	−0.267480	−	0.963564i	$-0.586191\pi$
$509$	−12.0692	−0.534959	−0.267480	+	0.963564i	$0.586191\pi$
$510$	0	0
$511$	−3.82800	−0.169341
$512$	21.4962	0.950006
$513$	0	0
$514$	6.65539	0.293557
$515$	1.39693	0.0615559
$516$	0	0
$517$	−50.3455	−2.21419
$518$	0.667252	0.0293174
$519$	0	0
$520$	−3.07604	−0.134893
$521$	−28.5699	−1.25167	−0.625834	−	0.779956i	$-0.715241\pi$
$521$	−28.5699	−1.25167	−0.625834	+	0.779956i	$0.715241\pi$
$522$	0	0
$523$	18.6459	0.815328	0.407664	−	0.913132i	$-0.366343\pi$
$523$	18.6459	0.815328	0.407664	+	0.913132i	$0.366343\pi$
$524$	32.2841	1.41034
$525$	0	0
$526$	5.72967	0.249826
$527$	24.8206	1.08120
$528$	0	0
$529$	−9.76382	−0.424514
$530$	5.05375	0.219521
$531$	0	0
$532$	0	0
$533$	2.55850	0.110821
$534$	0	0
$535$	16.4534	0.711341
$536$	−10.6263	−0.458986
$537$	0	0
$538$	−1.46286	−0.0630683
$539$	25.9249	1.11666
$540$	0	0
$541$	−17.3628	−0.746484	−0.373242	−	0.927734i	$-0.621754\pi$
$541$	−17.3628	−0.746484	−0.373242	+	0.927734i	$0.621754\pi$
$542$	3.18210	0.136683
$543$	0	0
$544$	14.7264	0.631388
$545$	−5.42602	−0.232425
$546$	0	0
$547$	5.05232	0.216022	0.108011	−	0.994150i	$-0.465552\pi$
$547$	5.05232	0.216022	0.108011	+	0.994150i	$0.465552\pi$
$548$	−19.8212	−0.846719
$549$	0	0
$550$	4.36184	0.185990
$551$	0	0
$552$	0	0
$553$	2.04189	0.0868300
$554$	−7.81252	−0.331922
$555$	0	0
$556$	−20.2618	−0.859290
$557$	23.9763	1.01591	0.507954	−	0.861384i	$-0.330402\pi$
$557$	23.9763	1.01591	0.507954	+	0.861384i	$0.330402\pi$
$558$	0	0
$559$	−15.8776	−0.671552
$560$	−2.89393	−0.122291
$561$	0	0
$562$	8.17118	0.344680
$563$	−42.6837	−1.79890	−0.899451	−	0.437022i	$-0.856033\pi$
$563$	−42.6837	−1.79890	−0.899451	+	0.437022i	$0.856033\pi$
$564$	0	0
$565$	17.7743	0.747768
$566$	9.28910	0.390450
$567$	0	0
$568$	−7.23442	−0.303550
$569$	−21.9358	−0.919598	−0.459799	−	0.888023i	$-0.652078\pi$
$569$	−21.9358	−0.919598	−0.459799	+	0.888023i	$0.652078\pi$
$570$	0	0
$571$	−32.3063	−1.35198	−0.675989	−	0.736912i	$-0.736283\pi$
$571$	−32.3063	−1.35198	−0.675989	+	0.736912i	$0.736283\pi$
$572$	12.5594	0.525136
$573$	0	0
$574$	−0.342244	−0.0142850
$575$	−11.5868	−0.483202
$576$	0	0
$577$	−46.8120	−1.94881	−0.974405	−	0.224800i	$-0.927827\pi$
$577$	−46.8120	−1.94881	−0.974405	+	0.224800i	$0.927827\pi$
$578$	−0.789621	−0.0328439
$579$	0	0
$580$	26.6313	1.10581
$581$	−2.94356	−0.122120
$582$	0	0
$583$	−42.5931	−1.76403
$584$	−7.90167	−0.326974
$585$	0	0
$586$	0.0719186	0.00297093
$587$	24.7820	1.02286	0.511431	−	0.859324i	$-0.329116\pi$
$587$	24.7820	1.02286	0.511431	+	0.859324i	$0.329116\pi$
$588$	0	0
$589$	0	0
$590$	1.29767	0.0534241
$591$	0	0
$592$	−9.68685	−0.398127
$593$	3.65364	0.150037	0.0750185	−	0.997182i	$-0.476098\pi$
$593$	3.65364	0.150037	0.0750185	+	0.997182i	$0.476098\pi$
$594$	0	0
$595$	−3.37464	−0.138347
$596$	−37.1215	−1.52056
$597$	0	0
$598$	2.14115	0.0875581
$599$	−1.49701	−0.0611660	−0.0305830	−	0.999532i	$-0.509736\pi$
$599$	−1.49701	−0.0611660	−0.0305830	+	0.999532i	$0.509736\pi$
$600$	0	0
$601$	23.8239	0.971796	0.485898	−	0.874015i	$-0.338493\pi$
$601$	23.8239	0.971796	0.485898	+	0.874015i	$0.338493\pi$
$602$	2.12391	0.0865642
$603$	0	0
$604$	−5.83750	−0.237524
$605$	6.13247	0.249321
$606$	0	0
$607$	16.4757	0.668726	0.334363	−	0.942444i	$-0.391479\pi$
$607$	16.4757	0.668726	0.334363	+	0.942444i	$0.391479\pi$
$608$	0	0
$609$	0	0
$610$	2.01817	0.0817133
$611$	21.6340	0.875219
$612$	0	0
$613$	1.33368	0.0538669	0.0269335	−	0.999637i	$-0.491426\pi$
$613$	1.33368	0.0538669	0.0269335	+	0.999637i	$0.491426\pi$
$614$	5.31996	0.214696
$615$	0	0
$616$	−3.46791	−0.139726
$617$	−36.1421	−1.45503	−0.727513	−	0.686094i	$-0.759324\pi$
$617$	−36.1421	−1.45503	−0.727513	+	0.686094i	$0.759324\pi$
$618$	0	0
$619$	19.3158	0.776369	0.388185	−	0.921582i	$-0.373102\pi$
$619$	19.3158	0.776369	0.388185	+	0.921582i	$0.373102\pi$
$620$	−16.3773	−0.657729
$621$	0	0
$622$	−5.43283	−0.217837
$623$	−7.26682	−0.291139
$624$	0	0
$625$	1.06687	0.0426746
$626$	3.52940	0.141063
$627$	0	0
$628$	−1.38413	−0.0552329
$629$	−11.2959	−0.450397
$630$	0	0
$631$	28.5226	1.13547	0.567733	−	0.823213i	$-0.307821\pi$
$631$	28.5226	1.13547	0.567733	+	0.823213i	$0.307821\pi$
$632$	4.21482	0.167657
$633$	0	0
$634$	2.65600	0.105483
$635$	17.8503	0.708367
$636$	0	0
$637$	−11.1402	−0.441391
$638$	14.4047	0.570286
$639$	0	0
$640$	−12.7965	−0.505828
$641$	31.0019	1.22450	0.612250	−	0.790664i	$-0.290265\pi$
$641$	31.0019	1.22450	0.612250	+	0.790664i	$0.290265\pi$
$642$	0	0
$643$	23.6628	0.933170	0.466585	−	0.884476i	$-0.345484\pi$
$643$	23.6628	0.933170	0.466585	+	0.884476i	$0.345484\pi$
$644$	4.46286	0.175861
$645$	0	0
$646$	0	0
$647$	−36.0865	−1.41871	−0.709353	−	0.704854i	$-0.751013\pi$
$647$	−36.0865	−1.41871	−0.709353	+	0.704854i	$0.751013\pi$
$648$	0	0
$649$	−10.9368	−0.429305
$650$	−1.87433	−0.0735174
$651$	0	0
$652$	23.7246	0.929128
$653$	37.4489	1.46549	0.732745	−	0.680504i	$-0.238239\pi$
$653$	37.4489	1.46549	0.732745	+	0.680504i	$0.238239\pi$
$654$	0	0
$655$	−23.1438	−0.904305
$656$	4.96854	0.193989
$657$	0	0
$658$	−2.89393	−0.112817
$659$	−19.3310	−0.753029	−0.376514	−	0.926411i	$-0.622877\pi$
$659$	−19.3310	−0.753029	−0.376514	+	0.926411i	$0.622877\pi$
$660$	0	0
$661$	−24.6236	−0.957747	−0.478874	−	0.877884i	$-0.658955\pi$
$661$	−24.6236	−0.957747	−0.478874	+	0.877884i	$0.658955\pi$
$662$	−1.42602	−0.0554239
$663$	0	0
$664$	−6.07604	−0.235796
$665$	0	0
$666$	0	0
$667$	−38.2645	−1.48161
$668$	−10.3696	−0.401211
$669$	0	0
$670$	3.69047	0.142575
$671$	−17.0092	−0.656632
$672$	0	0
$673$	42.4347	1.63574	0.817869	−	0.575405i	$-0.195156\pi$
$673$	42.4347	1.63574	0.817869	+	0.575405i	$0.195156\pi$
$674$	−4.19934	−0.161753
$675$	0	0
$676$	19.0351	0.732119
$677$	−14.5963	−0.560980	−0.280490	−	0.959857i	$-0.590497\pi$
$677$	−14.5963	−0.560980	−0.280490	+	0.959857i	$0.590497\pi$
$678$	0	0
$679$	−4.37464	−0.167883
$680$	−6.96585	−0.267128
$681$	0	0
$682$	−8.85835	−0.339204
$683$	−17.8043	−0.681262	−0.340631	−	0.940197i	$-0.610641\pi$
$683$	−17.8043	−0.681262	−0.340631	+	0.940197i	$0.610641\pi$
$684$	0	0
$685$	14.2094	0.542915
$686$	3.07697	0.117479
$687$	0	0
$688$	−30.8340	−1.17553
$689$	18.3027	0.697278
$690$	0	0
$691$	12.4730	0.474494	0.237247	−	0.971449i	$-0.423755\pi$
$691$	12.4730	0.474494	0.237247	+	0.971449i	$0.423755\pi$
$692$	24.2986	0.923694
$693$	0	0
$694$	1.43882	0.0546167
$695$	14.5253	0.550975
$696$	0	0
$697$	5.79385	0.219458
$698$	−0.0136143	−0.000515308 0
$699$	0	0
$700$	−3.90673	−0.147660
$701$	3.23349	0.122127	0.0610636	−	0.998134i	$-0.480551\pi$
$701$	3.23349	0.122127	0.0610636	+	0.998134i	$0.480551\pi$
$702$	0	0
$703$	0	0
$704$	20.6996	0.780147
$705$	0	0
$706$	−2.48751	−0.0936187
$707$	−1.26352	−0.0475195
$708$	0	0
$709$	−21.0256	−0.789632	−0.394816	−	0.918760i	$-0.629192\pi$
$709$	−21.0256	−0.789632	−0.394816	+	0.918760i	$0.629192\pi$
$710$	2.51249	0.0942920
$711$	0	0
$712$	−15.0000	−0.562149
$713$	23.5313	0.881253
$714$	0	0
$715$	−9.00362	−0.336716
$716$	−28.3533	−1.05961
$717$	0	0
$718$	7.09926	0.264942
$719$	4.05232	0.151126	0.0755630	−	0.997141i	$-0.475925\pi$
$719$	4.05232	0.151126	0.0755630	+	0.997141i	$0.475925\pi$
$720$	0	0
$721$	−0.676747	−0.0252034
$722$	0	0
$723$	0	0
$724$	35.1121	1.30493
$725$	33.4962	1.24402
$726$	0	0
$727$	−40.9231	−1.51776	−0.758878	−	0.651233i	$-0.774252\pi$
$727$	−40.9231	−1.51776	−0.758878	+	0.651233i	$0.774252\pi$
$728$	1.49020	0.0552305
$729$	0	0
$730$	2.74422	0.101568
$731$	−35.9557	−1.32987
$732$	0	0
$733$	7.26857	0.268471	0.134235	−	0.990949i	$-0.457142\pi$
$733$	7.26857	0.268471	0.134235	+	0.990949i	$0.457142\pi$
$734$	−0.554053	−0.0204505
$735$	0	0
$736$	13.9614	0.514624
$737$	−31.1034	−1.14571
$738$	0	0
$739$	4.65002	0.171054	0.0855268	−	0.996336i	$-0.472743\pi$
$739$	4.65002	0.171054	0.0855268	+	0.996336i	$0.472743\pi$
$740$	7.45336	0.273991
$741$	0	0
$742$	−2.44831	−0.0898803
$743$	2.68241	0.0984080	0.0492040	−	0.998789i	$-0.484332\pi$
$743$	2.68241	0.0984080	0.0492040	+	0.998789i	$0.484332\pi$
$744$	0	0
$745$	26.6117	0.974979
$746$	−4.46347	−0.163419
$747$	0	0
$748$	28.4415	1.03992
$749$	−7.97090	−0.291250
$750$	0	0
$751$	32.4320	1.18346	0.591730	−	0.806136i	$-0.298445\pi$
$751$	32.4320	1.18346	0.591730	+	0.806136i	$0.298445\pi$
$752$	42.0128	1.53205
$753$	0	0
$754$	−6.18984	−0.225421
$755$	4.18479	0.152300
$756$	0	0
$757$	4.75970	0.172994	0.0864972	−	0.996252i	$-0.472433\pi$
$757$	4.75970	0.172994	0.0864972	+	0.996252i	$0.472433\pi$
$758$	−5.45336	−0.198075
$759$	0	0
$760$	0	0
$761$	30.2481	1.09649	0.548247	−	0.836316i	$-0.315295\pi$
$761$	30.2481	1.09649	0.548247	+	0.836316i	$0.315295\pi$
$762$	0	0
$763$	2.62866	0.0951639
$764$	−16.0719	−0.581462
$765$	0	0
$766$	5.71244	0.206399
$767$	4.69965	0.169694
$768$	0	0
$769$	31.6049	1.13970	0.569852	−	0.821748i	$-0.307001\pi$
$769$	31.6049	1.13970	0.569852	+	0.821748i	$0.307001\pi$
$770$	1.20439	0.0434033
$771$	0	0
$772$	−31.8726	−1.14712
$773$	−46.3996	−1.66888	−0.834439	−	0.551100i	$-0.814208\pi$
$773$	−46.3996	−1.66888	−0.834439	+	0.551100i	$0.814208\pi$
$774$	0	0
$775$	−20.5990	−0.739936
$776$	−9.03003	−0.324159
$777$	0	0
$778$	−2.61619	−0.0937950
$779$	0	0
$780$	0	0
$781$	−21.1753	−0.757712
$782$	4.84875	0.173391
$783$	0	0
$784$	−21.6340	−0.772644
$785$	0.992259	0.0354152
$786$	0	0
$787$	−36.8489	−1.31352	−0.656760	−	0.754100i	$-0.728074\pi$
$787$	−36.8489	−1.31352	−0.656760	+	0.754100i	$0.728074\pi$
$788$	8.49020	0.302451
$789$	0	0
$790$	−1.46379	−0.0520794
$791$	−8.61081	−0.306165
$792$	0	0
$793$	7.30903	0.259551
$794$	−3.89630	−0.138274
$795$	0	0
$796$	15.1233	0.536031
$797$	−18.7939	−0.665712	−0.332856	−	0.942978i	$-0.608012\pi$
$797$	−18.7939	−0.665712	−0.332856	+	0.942978i	$0.608012\pi$
$798$	0	0
$799$	48.9914	1.73319
$800$	−12.2216	−0.432100
$801$	0	0
$802$	−8.42366	−0.297450
$803$	−23.1284	−0.816182
$804$	0	0
$805$	−3.19934	−0.112762
$806$	3.80653	0.134079
$807$	0	0
$808$	−2.60813	−0.0917536
$809$	−6.04551	−0.212549	−0.106274	−	0.994337i	$-0.533892\pi$
$809$	−6.04551	−0.212549	−0.106274	+	0.994337i	$0.533892\pi$
$810$	0	0
$811$	8.95367	0.314406	0.157203	−	0.987566i	$-0.449752\pi$
$811$	8.95367	0.314406	0.157203	+	0.987566i	$0.449752\pi$
$812$	−12.9017	−0.452760
$813$	0	0
$814$	4.03146	0.141303
$815$	−17.0077	−0.595755
$816$	0	0
$817$	0	0
$818$	9.12836	0.319165
$819$	0	0
$820$	−3.82295	−0.133503
$821$	−8.14384	−0.284222	−0.142111	−	0.989851i	$-0.545389\pi$
$821$	−8.14384	−0.284222	−0.142111	+	0.989851i	$0.545389\pi$
$822$	0	0
$823$	−2.64321	−0.0921364	−0.0460682	−	0.998938i	$-0.514669\pi$
$823$	−2.64321	−0.0921364	−0.0460682	+	0.998938i	$0.514669\pi$
$824$	−1.39693	−0.0486642
$825$	0	0
$826$	−0.628660	−0.0218739
$827$	36.7425	1.27766	0.638830	−	0.769348i	$-0.279419\pi$
$827$	36.7425	1.27766	0.638830	+	0.769348i	$0.279419\pi$
$828$	0	0
$829$	−43.7434	−1.51927	−0.759636	−	0.650349i	$-0.774623\pi$
$829$	−43.7434	−1.51927	−0.759636	+	0.650349i	$0.774623\pi$
$830$	2.11019	0.0732456
$831$	0	0
$832$	−8.89487	−0.308374
$833$	−25.2276	−0.874085
$834$	0	0
$835$	7.43376	0.257256
$836$	0	0
$837$	0	0
$838$	−4.59215	−0.158633
$839$	−13.7820	−0.475807	−0.237904	−	0.971289i	$-0.576460\pi$
$839$	−13.7820	−0.475807	−0.237904	+	0.971289i	$0.576460\pi$
$840$	0	0
$841$	81.6187	2.81444
$842$	−12.8625	−0.443270
$843$	0	0
$844$	−37.7452	−1.29924
$845$	−13.6459	−0.469433
$846$	0	0
$847$	−2.97090	−0.102081
$848$	35.5435	1.22057
$849$	0	0
$850$	−4.24453	−0.145586
$851$	−10.7091	−0.367105
$852$	0	0
$853$	10.5695	0.361894	0.180947	−	0.983493i	$-0.442084\pi$
$853$	10.5695	0.361894	0.180947	+	0.983493i	$0.442084\pi$
$854$	−0.977711	−0.0334566
$855$	0	0
$856$	−16.4534	−0.562364
$857$	−36.8571	−1.25901	−0.629507	−	0.776995i	$-0.716743\pi$
$857$	−36.8571	−1.25901	−0.629507	+	0.776995i	$0.716743\pi$
$858$	0	0
$859$	−35.1138	−1.19807	−0.599034	−	0.800724i	$-0.704449\pi$
$859$	−35.1138	−1.19807	−0.599034	+	0.800724i	$0.704449\pi$
$860$	23.7246	0.809003
$861$	0	0
$862$	−3.28993	−0.112055
$863$	−3.01691	−0.102697	−0.0513484	−	0.998681i	$-0.516352\pi$
$863$	−3.01691	−0.102697	−0.0513484	+	0.998681i	$0.516352\pi$
$864$	0	0
$865$	−17.4192	−0.592271
$866$	−7.82201	−0.265803
$867$	0	0
$868$	7.93407	0.269300
$869$	12.3369	0.418500
$870$	0	0
$871$	13.3655	0.452872
$872$	5.42602	0.183748
$873$	0	0
$874$	0	0
$875$	7.19759	0.243323
$876$	0	0
$877$	58.1976	1.96519	0.982596	−	0.185753i	$-0.0594725\pi$
$877$	58.1976	1.96519	0.982596	+	0.185753i	$0.0594725\pi$
$878$	−11.5416	−0.389510
$879$	0	0
$880$	−17.4848	−0.589413
$881$	−25.6382	−0.863771	−0.431886	−	0.901928i	$-0.642152\pi$
$881$	−25.6382	−0.863771	−0.431886	+	0.901928i	$0.642152\pi$
$882$	0	0
$883$	20.2189	0.680422	0.340211	−	0.940349i	$-0.389502\pi$
$883$	20.2189	0.680422	0.340211	+	0.940349i	$0.389502\pi$
$884$	−12.2216	−0.411058
$885$	0	0
$886$	−10.6373	−0.357368
$887$	21.1061	0.708672	0.354336	−	0.935118i	$-0.384707\pi$
$887$	21.1061	0.708672	0.354336	+	0.935118i	$0.384707\pi$
$888$	0	0
$889$	−8.64765	−0.290033
$890$	5.20945	0.174621
$891$	0	0
$892$	−36.8161	−1.23270
$893$	0	0
$894$	0	0
$895$	20.3259	0.679421
$896$	6.19934	0.207105
$897$	0	0
$898$	−5.24155	−0.174913
$899$	−68.0265	−2.26881
$900$	0	0
$901$	41.4475	1.38082
$902$	−2.06780	−0.0688502
$903$	0	0
$904$	−17.7743	−0.591163
$905$	−25.1712	−0.836718
$906$	0	0
$907$	25.8289	0.857636	0.428818	−	0.903391i	$-0.358930\pi$
$907$	25.8289	0.857636	0.428818	+	0.903391i	$0.358930\pi$
$908$	−34.7648	−1.15371
$909$	0	0
$910$	−0.517541	−0.0171563
$911$	−15.4415	−0.511600	−0.255800	−	0.966730i	$-0.582339\pi$
$911$	−15.4415	−0.511600	−0.255800	+	0.966730i	$0.582339\pi$
$912$	0	0
$913$	−17.7847	−0.588587
$914$	6.89250	0.227984
$915$	0	0
$916$	52.9299	1.74885
$917$	11.2121	0.370257
$918$	0	0
$919$	9.26445	0.305606	0.152803	−	0.988257i	$-0.451170\pi$
$919$	9.26445	0.305606	0.152803	+	0.988257i	$0.451170\pi$
$920$	−6.60401	−0.217728
$921$	0	0
$922$	12.0737	0.397625
$923$	9.09926	0.299506
$924$	0	0
$925$	9.37464	0.308236
$926$	−1.18984	−0.0391007
$927$	0	0
$928$	−40.3610	−1.32492
$929$	18.4029	0.603780	0.301890	−	0.953343i	$-0.402382\pi$
$929$	18.4029	0.603780	0.301890	+	0.953343i	$0.402382\pi$
$930$	0	0
$931$	0	0
$932$	0.292614	0.00958489
$933$	0	0
$934$	3.93313	0.128696
$935$	−20.3892	−0.666798
$936$	0	0
$937$	7.85204	0.256515	0.128258	−	0.991741i	$-0.459062\pi$
$937$	7.85204	0.256515	0.128258	+	0.991741i	$0.459062\pi$
$938$	−1.78787	−0.0583759
$939$	0	0
$940$	−32.3259	−1.05436
$941$	58.4374	1.90500	0.952502	−	0.304532i	$-0.0985000\pi$
$941$	58.4374	1.90500	0.952502	+	0.304532i	$0.0985000\pi$
$942$	0	0
$943$	5.49289	0.178873
$944$	9.12660	0.297046
$945$	0	0
$946$	12.8324	0.417219
$947$	36.0286	1.17077	0.585386	−	0.810755i	$-0.300943\pi$
$947$	36.0286	1.17077	0.585386	+	0.810755i	$0.300943\pi$
$948$	0	0
$949$	9.93851	0.322618
$950$	0	0
$951$	0	0
$952$	3.37464	0.109373
$953$	−12.9976	−0.421035	−0.210517	−	0.977590i	$-0.567515\pi$
$953$	−12.9976	−0.421035	−0.210517	+	0.977590i	$0.567515\pi$
$954$	0	0
$955$	11.5217	0.372832
$956$	12.4611	0.403021
$957$	0	0
$958$	−4.01724	−0.129791
$959$	−6.88383	−0.222290
$960$	0	0
$961$	10.8339	0.349480
$962$	−1.73236	−0.0558536
$963$	0	0
$964$	9.88713	0.318443
$965$	22.8489	0.735531
$966$	0	0
$967$	50.3141	1.61799	0.808996	−	0.587814i	$-0.200011\pi$
$967$	50.3141	1.61799	0.808996	+	0.587814i	$0.200011\pi$
$968$	−6.13247	−0.197105
$969$	0	0
$970$	3.13610	0.100694
$971$	30.6973	0.985123	0.492561	−	0.870278i	$-0.336061\pi$
$971$	30.6973	0.985123	0.492561	+	0.870278i	$0.336061\pi$
$972$	0	0
$973$	−7.03684	−0.225591
$974$	4.07966	0.130721
$975$	0	0
$976$	14.1940	0.454338
$977$	35.0806	1.12233	0.561164	−	0.827705i	$-0.310354\pi$
$977$	35.0806	1.12233	0.561164	+	0.827705i	$0.310354\pi$
$978$	0	0
$979$	−43.9053	−1.40322
$980$	16.6459	0.531734
$981$	0	0
$982$	−3.56118	−0.113642
$983$	−18.7811	−0.599023	−0.299511	−	0.954093i	$-0.596824\pi$
$983$	−18.7811	−0.599023	−0.299511	+	0.954093i	$0.596824\pi$
$984$	0	0
$985$	−6.08647	−0.193931
$986$	−14.0172	−0.446400
$987$	0	0
$988$	0	0
$989$	−34.0880	−1.08394
$990$	0	0
$991$	−13.5371	−0.430021	−0.215011	−	0.976612i	$-0.568979\pi$
$991$	−13.5371	−0.430021	−0.215011	+	0.976612i	$0.568979\pi$
$992$	24.8206	0.788054
$993$	0	0
$994$	−1.21719	−0.0386068
$995$	−10.8416	−0.343702
$996$	0	0
$997$	6.90673	0.218738	0.109369	−	0.994001i	$-0.465117\pi$
$997$	6.90673	0.218738	0.109369	+	0.994001i	$0.465117\pi$
$998$	−2.24298	−0.0710004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.w.1.2		3
3.2	odd	2		1083.2.a.m.1.2		3
19.4	even	9		171.2.u.a.73.1		6
19.5	even	9		171.2.u.a.82.1		6
19.18	odd	2		3249.2.a.x.1.2		3
57.5	odd	18		57.2.i.a.25.1	yes	6
57.23	odd	18		57.2.i.a.16.1	✓	6
57.56	even	2		1083.2.a.n.1.2		3
228.23	even	18		912.2.bo.b.529.1		6
228.119	even	18		912.2.bo.b.481.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.i.a.16.1	✓	6	57.23	odd	18
57.2.i.a.25.1	yes	6	57.5	odd	18
171.2.u.a.73.1		6	19.4	even	9
171.2.u.a.82.1		6	19.5	even	9
912.2.bo.b.481.1		6	228.119	even	18
912.2.bo.b.529.1		6	228.23	even	18
1083.2.a.m.1.2		3	3.2	odd	2
1083.2.a.n.1.2		3	57.56	even	2
3249.2.a.w.1.2		3	1.1	even	1	trivial
3249.2.a.x.1.2		3	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)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{4} +1.34730 q^{5} -0.652704 q^{7} -1.34730 q^{8} +0.467911 q^{10} -3.94356 q^{11} +1.69459 q^{13} -0.226682 q^{14} +3.29086 q^{16} +3.83750 q^{17} -2.53209 q^{20} -1.36959 q^{22} +3.63816 q^{23} -3.18479 q^{25} +0.588526 q^{26} +1.22668 q^{28} -10.5175 q^{29} +6.46791 q^{31} +3.83750 q^{32} +1.33275 q^{34} -0.879385 q^{35} -2.94356 q^{37} -1.81521 q^{40} +1.50980 q^{41} -9.36959 q^{43} +7.41147 q^{44} +1.26352 q^{46} +12.7665 q^{47} -6.57398 q^{49} -1.10607 q^{50} -3.18479 q^{52} +10.8007 q^{53} -5.31315 q^{55} +0.879385 q^{56} -3.65270 q^{58} +2.77332 q^{59} +4.31315 q^{61} +2.24628 q^{62} -5.24897 q^{64} +2.28312 q^{65} +7.88713 q^{67} -7.21213 q^{68} -0.305407 q^{70} +5.36959 q^{71} +5.86484 q^{73} -1.02229 q^{74} +2.57398 q^{77} -3.12836 q^{79} +4.43376 q^{80} +0.524348 q^{82} +4.50980 q^{83} +5.17024 q^{85} -3.25402 q^{86} +5.31315 q^{88} +11.1334 q^{89} -1.10607 q^{91} -6.83750 q^{92} +4.43376 q^{94} +6.70233 q^{97} -2.28312 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} + 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 9 q^{29} + 24 q^{31} + 9 q^{32} - 15 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 3 * q^11 + 3 * q^13 + 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 + 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 9 * q^29 + 24 * q^31 + 9 * q^32 - 15 * q^34 + 3 * q^35 + 6 * q^37 - 9 * q^40 + 6 * q^41 - 21 * q^43 + 12 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 + 9 * q^50 - 6 * q^52 + 18 * q^53 + 6 * q^55 - 3 * q^56 - 12 * q^58 + 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 + 15 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 9 * q^71 - 6 * q^73 + 3 * q^74 + 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 + 18 * q^86 - 6 * q^88 + 9 * q^91 - 18 * q^92 - 3 * q^94 - 6 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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