LMFDB - Embedded newform 3645.2.a.l.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3645,2,Mod(1,3645)] 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("3645.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3645, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21,3,0,27,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1054715368$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	3645.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.10205 q^{2} -0.785475 q^{4} +1.00000 q^{5} -2.98606 q^{7} +3.06975 q^{8} -1.10205 q^{10} +6.32459 q^{11} -2.14633 q^{13} +3.29080 q^{14} -1.81208 q^{16} +2.38077 q^{17} -2.11446 q^{19} -0.785475 q^{20} -6.97004 q^{22} -1.32036 q^{23} +1.00000 q^{25} +2.36538 q^{26} +2.34547 q^{28} -1.47300 q^{29} +1.25775 q^{31} -4.14248 q^{32} -2.62374 q^{34} -2.98606 q^{35} +10.6656 q^{37} +2.33025 q^{38} +3.06975 q^{40} -4.41373 q^{41} +11.9163 q^{43} -4.96780 q^{44} +1.45511 q^{46} +4.46003 q^{47} +1.91653 q^{49} -1.10205 q^{50} +1.68589 q^{52} +4.32679 q^{53} +6.32459 q^{55} -9.16644 q^{56} +1.62333 q^{58} -9.57413 q^{59} -13.1281 q^{61} -1.38611 q^{62} +8.18940 q^{64} -2.14633 q^{65} -0.565126 q^{67} -1.87004 q^{68} +3.29080 q^{70} -9.69275 q^{71} +1.46433 q^{73} -11.7541 q^{74} +1.66086 q^{76} -18.8856 q^{77} -12.0754 q^{79} -1.81208 q^{80} +4.86417 q^{82} +11.5810 q^{83} +2.38077 q^{85} -13.1324 q^{86} +19.4149 q^{88} +9.49624 q^{89} +6.40907 q^{91} +1.03711 q^{92} -4.91520 q^{94} -2.11446 q^{95} -14.6234 q^{97} -2.11213 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8} + 3 q^{10} + 12 q^{13} + 39 q^{16} + 12 q^{17} + 24 q^{19} + 27 q^{20} + 18 q^{22} + 18 q^{23} + 21 q^{25} - 9 q^{26} + 30 q^{28} - 3 q^{29} + 24 q^{31}+ \cdots + 15 q^{98}+O(q^{100}) $ 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8 + 3 * q^10 + 12 * q^13 + 39 * q^16 + 12 * q^17 + 24 * q^19 + 27 * q^20 + 18 * q^22 + 18 * q^23 + 21 * q^25 - 9 * q^26 + 30 * q^28 - 3 * q^29 + 24 * q^31 + 6 * q^32 + 18 * q^34 + 12 * q^35 + 24 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 42 * q^43 - 12 * q^44 + 30 * q^46 + 21 * q^47 + 39 * q^49 + 3 * q^50 + 36 * q^52 + 18 * q^53 - 30 * q^56 + 30 * q^58 - 6 * q^59 + 33 * q^61 + 18 * q^62 + 63 * q^64 + 12 * q^65 + 63 * q^67 + 36 * q^68 - 12 * q^71 + 39 * q^73 - 21 * q^74 + 48 * q^76 + 9 * q^77 + 48 * q^79 + 39 * q^80 + 24 * q^82 + 33 * q^83 + 12 * q^85 - 69 * q^86 + 54 * q^88 - 9 * q^89 + 69 * q^91 + 21 * q^92 + 30 * q^94 + 24 * q^95 + 30 * 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 $

$2$	−1.10205	−0.779270	−0.389635	−	0.920969i	$-0.627399\pi$
$2$	−1.10205	−0.779270	−0.389635	+	0.920969i	$0.627399\pi$
$3$	0	0
$3$	0	0
$4$	−0.785475	−0.392738
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.98606	−1.12862	−0.564312	−	0.825562i	$-0.690858\pi$
$7$	−2.98606	−1.12862	−0.564312	+	0.825562i	$0.690858\pi$
$8$	3.06975	1.08532
$9$	0	0
$10$	−1.10205	−0.348500
$11$	6.32459	1.90693	0.953467	−	0.301497i	$-0.0974863\pi$
$11$	6.32459	1.90693	0.953467	+	0.301497i	$0.0974863\pi$
$12$	0	0
$13$	−2.14633	−0.595286	−0.297643	−	0.954677i	$-0.596200\pi$
$13$	−2.14633	−0.595286	−0.297643	+	0.954677i	$0.596200\pi$
$14$	3.29080	0.879503
$15$	0	0
$16$	−1.81208	−0.453020
$17$	2.38077	0.577422	0.288711	−	0.957416i	$-0.406773\pi$
$17$	2.38077	0.577422	0.288711	+	0.957416i	$0.406773\pi$
$18$	0	0
$19$	−2.11446	−0.485091	−0.242545	−	0.970140i	$-0.577982\pi$
$19$	−2.11446	−0.485091	−0.242545	+	0.970140i	$0.577982\pi$
$20$	−0.785475	−0.175638
$21$	0	0
$22$	−6.97004	−1.48602
$23$	−1.32036	−0.275314	−0.137657	−	0.990480i	$-0.543957\pi$
$23$	−1.32036	−0.275314	−0.137657	+	0.990480i	$0.543957\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.36538	0.463888
$27$	0	0
$28$	2.34547	0.443253
$29$	−1.47300	−0.273530	−0.136765	−	0.990604i	$-0.543670\pi$
$29$	−1.47300	−0.273530	−0.136765	+	0.990604i	$0.543670\pi$
$30$	0	0
$31$	1.25775	0.225899	0.112949	−	0.993601i	$-0.463970\pi$
$31$	1.25775	0.225899	0.112949	+	0.993601i	$0.463970\pi$
$32$	−4.14248	−0.732294
$33$	0	0
$34$	−2.62374	−0.449968
$35$	−2.98606	−0.504736
$36$	0	0
$37$	10.6656	1.75341	0.876707	−	0.481024i	$-0.159735\pi$
$37$	10.6656	1.75341	0.876707	+	0.481024i	$0.159735\pi$
$38$	2.33025	0.378017
$39$	0	0
$40$	3.06975	0.485370
$41$	−4.41373	−0.689309	−0.344655	−	0.938730i	$-0.612004\pi$
$41$	−4.41373	−0.689309	−0.344655	+	0.938730i	$0.612004\pi$
$42$	0	0
$43$	11.9163	1.81721	0.908606	−	0.417654i	$-0.137148\pi$
$43$	11.9163	1.81721	0.908606	+	0.417654i	$0.137148\pi$
$44$	−4.96780	−0.748925
$45$	0	0
$46$	1.45511	0.214544
$47$	4.46003	0.650562	0.325281	−	0.945617i	$-0.394541\pi$
$47$	4.46003	0.650562	0.325281	+	0.945617i	$0.394541\pi$
$48$	0	0
$49$	1.91653	0.273791
$50$	−1.10205	−0.155854
$51$	0	0
$52$	1.68589	0.233791
$53$	4.32679	0.594331	0.297165	−	0.954826i	$-0.403959\pi$
$53$	4.32679	0.594331	0.297165	+	0.954826i	$0.403959\pi$
$54$	0	0
$55$	6.32459	0.852807
$56$	−9.16644	−1.22492
$57$	0	0
$58$	1.62333	0.213154
$59$	−9.57413	−1.24645	−0.623223	−	0.782044i	$-0.714177\pi$
$59$	−9.57413	−1.24645	−0.623223	+	0.782044i	$0.714177\pi$
$60$	0	0
$61$	−13.1281	−1.68088	−0.840441	−	0.541902i	$-0.817704\pi$
$61$	−13.1281	−1.68088	−0.840441	+	0.541902i	$0.817704\pi$
$62$	−1.38611	−0.176036
$63$	0	0
$64$	8.18940	1.02368
$65$	−2.14633	−0.266220
$66$	0	0
$67$	−0.565126	−0.0690411	−0.0345205	−	0.999404i	$-0.510990\pi$
$67$	−0.565126	−0.0690411	−0.0345205	+	0.999404i	$0.510990\pi$
$68$	−1.87004	−0.226775
$69$	0	0
$70$	3.29080	0.393326
$71$	−9.69275	−1.15032	−0.575159	−	0.818042i	$-0.695060\pi$
$71$	−9.69275	−1.15032	−0.575159	+	0.818042i	$0.695060\pi$
$72$	0	0
$73$	1.46433	0.171387	0.0856937	−	0.996322i	$-0.472689\pi$
$73$	1.46433	0.171387	0.0856937	+	0.996322i	$0.472689\pi$
$74$	−11.7541	−1.36638
$75$	0	0
$76$	1.66086	0.190513
$77$	−18.8856	−2.15221
$78$	0	0
$79$	−12.0754	−1.35858	−0.679292	−	0.733868i	$-0.737713\pi$
$79$	−12.0754	−1.35858	−0.679292	+	0.733868i	$0.737713\pi$
$80$	−1.81208	−0.202597
$81$	0	0
$82$	4.86417	0.537158
$83$	11.5810	1.27118	0.635591	−	0.772026i	$-0.280756\pi$
$83$	11.5810	1.27118	0.635591	+	0.772026i	$0.280756\pi$
$84$	0	0
$85$	2.38077	0.258231
$86$	−13.1324	−1.41610
$87$	0	0
$88$	19.4149	2.06963
$89$	9.49624	1.00660	0.503300	−	0.864112i	$-0.332119\pi$
$89$	9.49624	1.00660	0.503300	+	0.864112i	$0.332119\pi$
$90$	0	0
$91$	6.40907	0.671853
$92$	1.03711	0.108126
$93$	0	0
$94$	−4.91520	−0.506964
$95$	−2.11446	−0.216939
$96$	0	0
$97$	−14.6234	−1.48479	−0.742393	−	0.669965i	$-0.766309\pi$
$97$	−14.6234	−1.48479	−0.742393	+	0.669965i	$0.766309\pi$
$98$	−2.11213	−0.213357
$99$	0	0
$100$	−0.785475	−0.0785475
$101$	13.2719	1.32060	0.660300	−	0.751002i	$-0.270429\pi$
$101$	13.2719	1.32060	0.660300	+	0.751002i	$0.270429\pi$
$102$	0	0
$103$	−1.76885	−0.174290	−0.0871450	−	0.996196i	$-0.527774\pi$
$103$	−1.76885	−0.174290	−0.0871450	+	0.996196i	$0.527774\pi$
$104$	−6.58870	−0.646075
$105$	0	0
$106$	−4.76837	−0.463145
$107$	−7.20592	−0.696622	−0.348311	−	0.937379i	$-0.613245\pi$
$107$	−7.20592	−0.696622	−0.348311	+	0.937379i	$0.613245\pi$
$108$	0	0
$109$	−5.49119	−0.525960	−0.262980	−	0.964801i	$-0.584705\pi$
$109$	−5.49119	−0.525960	−0.262980	+	0.964801i	$0.584705\pi$
$110$	−6.97004	−0.664567
$111$	0	0
$112$	5.41097	0.511289
$113$	11.3260	1.06546	0.532730	−	0.846285i	$-0.321166\pi$
$113$	11.3260	1.06546	0.532730	+	0.846285i	$0.321166\pi$
$114$	0	0
$115$	−1.32036	−0.123124
$116$	1.15701	0.107425
$117$	0	0
$118$	10.5512	0.971318
$119$	−7.10912	−0.651692
$120$	0	0
$121$	29.0004	2.63640
$122$	14.4679	1.30986
$123$	0	0
$124$	−0.987933	−0.0887190
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0223	1.06680	0.533402	−	0.845862i	$-0.320913\pi$
$127$	12.0223	1.06680	0.533402	+	0.845862i	$0.320913\pi$
$128$	−0.740203	−0.0654253
$129$	0	0
$130$	2.36538	0.207457
$131$	1.86898	0.163294	0.0816468	−	0.996661i	$-0.473982\pi$
$131$	1.86898	0.163294	0.0816468	+	0.996661i	$0.473982\pi$
$132$	0	0
$133$	6.31390	0.547485
$134$	0.622800	0.0538017
$135$	0	0
$136$	7.30836	0.626687
$137$	5.52754	0.472250	0.236125	−	0.971723i	$-0.424123\pi$
$137$	5.52754	0.472250	0.236125	+	0.971723i	$0.424123\pi$
$138$	0	0
$139$	1.85048	0.156955	0.0784776	−	0.996916i	$-0.474994\pi$
$139$	1.85048	0.156955	0.0784776	+	0.996916i	$0.474994\pi$
$140$	2.34547	0.198229
$141$	0	0
$142$	10.6819	0.896409
$143$	−13.5747	−1.13517
$144$	0	0
$145$	−1.47300	−0.122326
$146$	−1.61378	−0.133557
$147$	0	0
$148$	−8.37757	−0.688632
$149$	−10.9328	−0.895646	−0.447823	−	0.894122i	$-0.647801\pi$
$149$	−10.9328	−0.895646	−0.447823	+	0.894122i	$0.647801\pi$
$150$	0	0
$151$	19.1695	1.55999	0.779995	−	0.625786i	$-0.215222\pi$
$151$	19.1695	1.55999	0.779995	+	0.625786i	$0.215222\pi$
$152$	−6.49086	−0.526479
$153$	0	0
$154$	20.8129	1.67715
$155$	1.25775	0.101025
$156$	0	0
$157$	−8.40393	−0.670706	−0.335353	−	0.942093i	$-0.608856\pi$
$157$	−8.40393	−0.670706	−0.335353	+	0.942093i	$0.608856\pi$
$158$	13.3077	1.05870
$159$	0	0
$160$	−4.14248	−0.327492
$161$	3.94267	0.310726
$162$	0	0
$163$	8.05469	0.630892	0.315446	−	0.948944i	$-0.397846\pi$
$163$	8.05469	0.630892	0.315446	+	0.948944i	$0.397846\pi$
$164$	3.46688	0.270718
$165$	0	0
$166$	−12.7629	−0.990595
$167$	8.18879	0.633667	0.316834	−	0.948481i	$-0.397380\pi$
$167$	8.18879	0.633667	0.316834	+	0.948481i	$0.397380\pi$
$168$	0	0
$169$	−8.39326	−0.645635
$170$	−2.62374	−0.201232
$171$	0	0
$172$	−9.35992	−0.713687
$173$	−3.31363	−0.251931	−0.125965	−	0.992035i	$-0.540203\pi$
$173$	−3.31363	−0.251931	−0.125965	+	0.992035i	$0.540203\pi$
$174$	0	0
$175$	−2.98606	−0.225725
$176$	−11.4606	−0.863879
$177$	0	0
$178$	−10.4654	−0.784413
$179$	14.7586	1.10311	0.551554	−	0.834139i	$-0.314035\pi$
$179$	14.7586	1.10311	0.551554	+	0.834139i	$0.314035\pi$
$180$	0	0
$181$	12.4562	0.925859	0.462929	−	0.886395i	$-0.346798\pi$
$181$	12.4562	0.925859	0.462929	+	0.886395i	$0.346798\pi$
$182$	−7.06315	−0.523555
$183$	0	0
$184$	−4.05317	−0.298803
$185$	10.6656	0.784151
$186$	0	0
$187$	15.0574	1.10111
$188$	−3.50324	−0.255500
$189$	0	0
$190$	2.33025	0.169054
$191$	−5.22247	−0.377884	−0.188942	−	0.981988i	$-0.560506\pi$
$191$	−5.22247	−0.377884	−0.188942	+	0.981988i	$0.560506\pi$
$192$	0	0
$193$	12.4843	0.898636	0.449318	−	0.893372i	$-0.351667\pi$
$193$	12.4843	0.898636	0.449318	+	0.893372i	$0.351667\pi$
$194$	16.1158	1.15705
$195$	0	0
$196$	−1.50539	−0.107528
$197$	14.8163	1.05562	0.527809	−	0.849363i	$-0.323014\pi$
$197$	14.8163	1.05562	0.527809	+	0.849363i	$0.323014\pi$
$198$	0	0
$199$	13.1014	0.928732	0.464366	−	0.885643i	$-0.346282\pi$
$199$	13.1014	0.928732	0.464366	+	0.885643i	$0.346282\pi$
$200$	3.06975	0.217064
$201$	0	0
$202$	−14.6263	−1.02910
$203$	4.39847	0.308712
$204$	0	0
$205$	−4.41373	−0.308268
$206$	1.94937	0.135819
$207$	0	0
$208$	3.88932	0.269676
$209$	−13.3731	−0.925037
$210$	0	0
$211$	18.9059	1.30154	0.650768	−	0.759277i	$-0.274447\pi$
$211$	18.9059	1.30154	0.650768	+	0.759277i	$0.274447\pi$
$212$	−3.39859	−0.233416
$213$	0	0
$214$	7.94131	0.542857
$215$	11.9163	0.812682
$216$	0	0
$217$	−3.75572	−0.254955
$218$	6.05159	0.409865
$219$	0	0
$220$	−4.96780	−0.334929
$221$	−5.10993	−0.343731
$222$	0	0
$223$	3.52199	0.235850	0.117925	−	0.993023i	$-0.462376\pi$
$223$	3.52199	0.235850	0.117925	+	0.993023i	$0.462376\pi$
$224$	12.3697	0.826485
$225$	0	0
$226$	−12.4819	−0.830282
$227$	−7.65761	−0.508253	−0.254127	−	0.967171i	$-0.581788\pi$
$227$	−7.65761	−0.508253	−0.254127	+	0.967171i	$0.581788\pi$
$228$	0	0
$229$	−3.01587	−0.199295	−0.0996473	−	0.995023i	$-0.531771\pi$
$229$	−3.01587	−0.199295	−0.0996473	+	0.995023i	$0.531771\pi$
$230$	1.45511	0.0959469
$231$	0	0
$232$	−4.52174	−0.296867
$233$	18.8533	1.23512	0.617561	−	0.786523i	$-0.288121\pi$
$233$	18.8533	1.23512	0.617561	+	0.786523i	$0.288121\pi$
$234$	0	0
$235$	4.46003	0.290940
$236$	7.52024	0.489526
$237$	0	0
$238$	7.83464	0.507844
$239$	9.09243	0.588140	0.294070	−	0.955784i	$-0.404990\pi$
$239$	9.09243	0.588140	0.294070	+	0.955784i	$0.404990\pi$
$240$	0	0
$241$	11.9662	0.770810	0.385405	−	0.922747i	$-0.374062\pi$
$241$	11.9662	0.770810	0.385405	+	0.922747i	$0.374062\pi$
$242$	−31.9600	−2.05447
$243$	0	0
$244$	10.3118	0.660146
$245$	1.91653	0.122443
$246$	0	0
$247$	4.53834	0.288768
$248$	3.86098	0.245172
$249$	0	0
$250$	−1.10205	−0.0697001
$251$	−11.8579	−0.748466	−0.374233	−	0.927335i	$-0.622094\pi$
$251$	−11.8579	−0.748466	−0.374233	+	0.927335i	$0.622094\pi$
$252$	0	0
$253$	−8.35072	−0.525005
$254$	−13.2492	−0.831329
$255$	0	0
$256$	−15.5631	−0.972691
$257$	−15.1159	−0.942907	−0.471453	−	0.881891i	$-0.656270\pi$
$257$	−15.1159	−0.942907	−0.471453	+	0.881891i	$0.656270\pi$
$258$	0	0
$259$	−31.8481	−1.97894
$260$	1.68589	0.104555
$261$	0	0
$262$	−2.05972	−0.127250
$263$	3.85290	0.237580	0.118790	−	0.992919i	$-0.462099\pi$
$263$	3.85290	0.237580	0.118790	+	0.992919i	$0.462099\pi$
$264$	0	0
$265$	4.32679	0.265793
$266$	−6.95827	−0.426639
$267$	0	0
$268$	0.443892	0.0271150
$269$	−4.09700	−0.249798	−0.124899	−	0.992169i	$-0.539861\pi$
$269$	−4.09700	−0.249798	−0.124899	+	0.992169i	$0.539861\pi$
$270$	0	0
$271$	27.2630	1.65611	0.828053	−	0.560650i	$-0.189448\pi$
$271$	27.2630	1.65611	0.828053	+	0.560650i	$0.189448\pi$
$272$	−4.31414	−0.261583
$273$	0	0
$274$	−6.09166	−0.368010
$275$	6.32459	0.381387
$276$	0	0
$277$	7.56826	0.454733	0.227366	−	0.973809i	$-0.426988\pi$
$277$	7.56826	0.454733	0.227366	+	0.973809i	$0.426988\pi$
$278$	−2.03933	−0.122311
$279$	0	0
$280$	−9.16644	−0.547799
$281$	2.15365	0.128476	0.0642381	−	0.997935i	$-0.479538\pi$
$281$	2.15365	0.128476	0.0642381	+	0.997935i	$0.479538\pi$
$282$	0	0
$283$	−6.24260	−0.371084	−0.185542	−	0.982636i	$-0.559404\pi$
$283$	−6.24260	−0.371084	−0.185542	+	0.982636i	$0.559404\pi$
$284$	7.61342	0.451773
$285$	0	0
$286$	14.9600	0.884605
$287$	13.1797	0.777970
$288$	0	0
$289$	−11.3319	−0.666584
$290$	1.62333	0.0953252
$291$	0	0
$292$	−1.15020	−0.0673103
$293$	16.0504	0.937676	0.468838	−	0.883284i	$-0.344673\pi$
$293$	16.0504	0.937676	0.468838	+	0.883284i	$0.344673\pi$
$294$	0	0
$295$	−9.57413	−0.557427
$296$	32.7407	1.90301
$297$	0	0
$298$	12.0485	0.697951
$299$	2.83393	0.163890
$300$	0	0
$301$	−35.5826	−2.05095
$302$	−21.1258	−1.21565
$303$	0	0
$304$	3.83157	0.219756
$305$	−13.1281	−0.751714
$306$	0	0
$307$	31.6554	1.80667	0.903336	−	0.428934i	$-0.141111\pi$
$307$	31.6554	1.80667	0.903336	+	0.428934i	$0.141111\pi$
$308$	14.8341	0.845254
$309$	0	0
$310$	−1.38611	−0.0787258
$311$	−26.2069	−1.48606	−0.743029	−	0.669259i	$-0.766612\pi$
$311$	−26.2069	−1.48606	−0.743029	+	0.669259i	$0.766612\pi$
$312$	0	0
$313$	11.6850	0.660473	0.330237	−	0.943898i	$-0.392871\pi$
$313$	11.6850	0.660473	0.330237	+	0.943898i	$0.392871\pi$
$314$	9.26159	0.522662
$315$	0	0
$316$	9.48489	0.533567
$317$	−10.9375	−0.614313	−0.307156	−	0.951659i	$-0.599377\pi$
$317$	−10.9375	−0.614313	−0.307156	+	0.951659i	$0.599377\pi$
$318$	0	0
$319$	−9.31613	−0.521603
$320$	8.18940	0.457801
$321$	0	0
$322$	−4.34503	−0.242139
$323$	−5.03405	−0.280102
$324$	0	0
$325$	−2.14633	−0.119057
$326$	−8.87671	−0.491635
$327$	0	0
$328$	−13.5490	−0.748120
$329$	−13.3179	−0.734240
$330$	0	0
$331$	25.7747	1.41670	0.708352	−	0.705859i	$-0.249439\pi$
$331$	25.7747	1.41670	0.708352	+	0.705859i	$0.249439\pi$
$332$	−9.09660	−0.499241
$333$	0	0
$334$	−9.02449	−0.493798
$335$	−0.565126	−0.0308761
$336$	0	0
$337$	16.5714	0.902699	0.451350	−	0.892347i	$-0.350943\pi$
$337$	16.5714	0.902699	0.451350	+	0.892347i	$0.350943\pi$
$338$	9.24983	0.503124
$339$	0	0
$340$	−1.87004	−0.101417
$341$	7.95476	0.430774
$342$	0	0
$343$	15.1795	0.819617
$344$	36.5799	1.97226
$345$	0	0
$346$	3.65180	0.196322
$347$	13.3005	0.714010	0.357005	−	0.934103i	$-0.383798\pi$
$347$	13.3005	0.714010	0.357005	+	0.934103i	$0.383798\pi$
$348$	0	0
$349$	−6.02053	−0.322272	−0.161136	−	0.986932i	$-0.551516\pi$
$349$	−6.02053	−0.322272	−0.161136	+	0.986932i	$0.551516\pi$
$350$	3.29080	0.175901
$351$	0	0
$352$	−26.1995	−1.39644
$353$	9.01971	0.480071	0.240035	−	0.970764i	$-0.422841\pi$
$353$	9.01971	0.480071	0.240035	+	0.970764i	$0.422841\pi$
$354$	0	0
$355$	−9.69275	−0.514438
$356$	−7.45906	−0.395329
$357$	0	0
$358$	−16.2648	−0.859619
$359$	−27.8517	−1.46996	−0.734979	−	0.678090i	$-0.762808\pi$
$359$	−27.8517	−1.46996	−0.734979	+	0.678090i	$0.762808\pi$
$360$	0	0
$361$	−14.5290	−0.764687
$362$	−13.7274	−0.721494
$363$	0	0
$364$	−5.03417	−0.263862
$365$	1.46433	0.0766468
$366$	0	0
$367$	7.13633	0.372514	0.186257	−	0.982501i	$-0.440364\pi$
$367$	7.13633	0.372514	0.186257	+	0.982501i	$0.440364\pi$
$368$	2.39259	0.124723
$369$	0	0
$370$	−11.7541	−0.611066
$371$	−12.9201	−0.670776
$372$	0	0
$373$	6.79183	0.351668	0.175834	−	0.984420i	$-0.443738\pi$
$373$	6.79183	0.351668	0.175834	+	0.984420i	$0.443738\pi$
$374$	−16.5941	−0.858059
$375$	0	0
$376$	13.6912	0.706068
$377$	3.16155	0.162828
$378$	0	0
$379$	0.676912	0.0347706	0.0173853	−	0.999849i	$-0.494466\pi$
$379$	0.676912	0.0347706	0.0173853	+	0.999849i	$0.494466\pi$
$380$	1.66086	0.0852002
$381$	0	0
$382$	5.75544	0.294474
$383$	30.6023	1.56370	0.781852	−	0.623464i	$-0.214275\pi$
$383$	30.6023	1.56370	0.781852	+	0.623464i	$0.214275\pi$
$384$	0	0
$385$	−18.8856	−0.962498
$386$	−13.7583	−0.700281
$387$	0	0
$388$	11.4864	0.583131
$389$	21.9112	1.11094	0.555471	−	0.831536i	$-0.312538\pi$
$389$	21.9112	1.11094	0.555471	+	0.831536i	$0.312538\pi$
$390$	0	0
$391$	−3.14347	−0.158972
$392$	5.88327	0.297150
$393$	0	0
$394$	−16.3284	−0.822611
$395$	−12.0754	−0.607577
$396$	0	0
$397$	−1.83744	−0.0922184	−0.0461092	−	0.998936i	$-0.514682\pi$
$397$	−1.83744	−0.0922184	−0.0461092	+	0.998936i	$0.514682\pi$
$398$	−14.4384	−0.723733
$399$	0	0
$400$	−1.81208	−0.0906039
$401$	23.3160	1.16434	0.582172	−	0.813065i	$-0.302203\pi$
$401$	23.3160	1.16434	0.582172	+	0.813065i	$0.302203\pi$
$402$	0	0
$403$	−2.69955	−0.134474
$404$	−10.4247	−0.518649
$405$	0	0
$406$	−4.84735	−0.240570
$407$	67.4555	3.34365
$408$	0	0
$409$	33.7068	1.66670	0.833348	−	0.552748i	$-0.186421\pi$
$409$	33.7068	1.66670	0.833348	+	0.552748i	$0.186421\pi$
$410$	4.86417	0.240224
$411$	0	0
$412$	1.38939	0.0684502
$413$	28.5889	1.40677
$414$	0	0
$415$	11.5810	0.568490
$416$	8.89115	0.435924
$417$	0	0
$418$	14.7379	0.720854
$419$	23.8504	1.16517	0.582585	−	0.812770i	$-0.302041\pi$
$419$	23.8504	1.16517	0.582585	+	0.812770i	$0.302041\pi$
$420$	0	0
$421$	19.8913	0.969444	0.484722	−	0.874668i	$-0.338921\pi$
$421$	19.8913	0.969444	0.484722	+	0.874668i	$0.338921\pi$
$422$	−20.8353	−1.01425
$423$	0	0
$424$	13.2822	0.645039
$425$	2.38077	0.115484
$426$	0	0
$427$	39.2013	1.89708
$428$	5.66007	0.273590
$429$	0	0
$430$	−13.1324	−0.633299
$431$	−12.6250	−0.608124	−0.304062	−	0.952652i	$-0.598343\pi$
$431$	−12.6250	−0.608124	−0.304062	+	0.952652i	$0.598343\pi$
$432$	0	0
$433$	−3.03476	−0.145841	−0.0729207	−	0.997338i	$-0.523232\pi$
$433$	−3.03476	−0.145841	−0.0729207	+	0.997338i	$0.523232\pi$
$434$	4.13901	0.198679
$435$	0	0
$436$	4.31319	0.206564
$437$	2.79185	0.133552
$438$	0	0
$439$	3.13184	0.149474	0.0747372	−	0.997203i	$-0.476188\pi$
$439$	3.13184	0.149474	0.0747372	+	0.997203i	$0.476188\pi$
$440$	19.4149	0.925568
$441$	0	0
$442$	5.63142	0.267859
$443$	−37.0348	−1.75958	−0.879788	−	0.475366i	$-0.842316\pi$
$443$	−37.0348	−1.75958	−0.879788	+	0.475366i	$0.842316\pi$
$444$	0	0
$445$	9.49624	0.450165
$446$	−3.88143	−0.183791
$447$	0	0
$448$	−24.4540	−1.15534
$449$	−26.4419	−1.24787	−0.623934	−	0.781477i	$-0.714467\pi$
$449$	−26.4419	−1.24787	−0.623934	+	0.781477i	$0.714467\pi$
$450$	0	0
$451$	−27.9150	−1.31447
$452$	−8.89629	−0.418446
$453$	0	0
$454$	8.43911	0.396067
$455$	6.40907	0.300462
$456$	0	0
$457$	0.180552	0.00844585	0.00422293	−	0.999991i	$-0.498656\pi$
$457$	0.180552	0.00844585	0.00422293	+	0.999991i	$0.498656\pi$
$458$	3.32366	0.155304
$459$	0	0
$460$	1.03711	0.0483554
$461$	−24.4642	−1.13941	−0.569706	−	0.821849i	$-0.692943\pi$
$461$	−24.4642	−1.13941	−0.569706	+	0.821849i	$0.692943\pi$
$462$	0	0
$463$	−2.61337	−0.121454	−0.0607268	−	0.998154i	$-0.519342\pi$
$463$	−2.61337	−0.121454	−0.0607268	+	0.998154i	$0.519342\pi$
$464$	2.66920	0.123914
$465$	0	0
$466$	−20.7774	−0.962493
$467$	−7.82295	−0.362003	−0.181001	−	0.983483i	$-0.557934\pi$
$467$	−7.82295	−0.362003	−0.181001	+	0.983483i	$0.557934\pi$
$468$	0	0
$469$	1.68750	0.0779214
$470$	−4.91520	−0.226721
$471$	0	0
$472$	−29.3902	−1.35279
$473$	75.3654	3.46530
$474$	0	0
$475$	−2.11446	−0.0970182
$476$	5.58403	0.255944
$477$	0	0
$478$	−10.0204	−0.458320
$479$	18.0144	0.823097	0.411548	−	0.911388i	$-0.364988\pi$
$479$	18.0144	0.823097	0.411548	+	0.911388i	$0.364988\pi$
$480$	0	0
$481$	−22.8919	−1.04378
$482$	−13.1874	−0.600670
$483$	0	0
$484$	−22.7791	−1.03541
$485$	−14.6234	−0.664017
$486$	0	0
$487$	−27.2884	−1.23656	−0.618278	−	0.785960i	$-0.712169\pi$
$487$	−27.2884	−1.23656	−0.618278	+	0.785960i	$0.712169\pi$
$488$	−40.3000	−1.82429
$489$	0	0
$490$	−2.11213	−0.0954161
$491$	−4.85541	−0.219122	−0.109561	−	0.993980i	$-0.534944\pi$
$491$	−4.85541	−0.219122	−0.109561	+	0.993980i	$0.534944\pi$
$492$	0	0
$493$	−3.50688	−0.157942
$494$	−5.00150	−0.225028
$495$	0	0
$496$	−2.27914	−0.102337
$497$	28.9431	1.29828
$498$	0	0
$499$	21.3770	0.956967	0.478484	−	0.878097i	$-0.341187\pi$
$499$	21.3770	0.956967	0.478484	+	0.878097i	$0.341187\pi$
$500$	−0.785475	−0.0351275
$501$	0	0
$502$	13.0681	0.583257
$503$	−22.4759	−1.00215	−0.501076	−	0.865403i	$-0.667062\pi$
$503$	−22.4759	−1.00215	−0.501076	+	0.865403i	$0.667062\pi$
$504$	0	0
$505$	13.2719	0.590590
$506$	9.20295	0.409121
$507$	0	0
$508$	−9.44319	−0.418974
$509$	−12.5542	−0.556453	−0.278227	−	0.960515i	$-0.589747\pi$
$509$	−12.5542	−0.556453	−0.278227	+	0.960515i	$0.589747\pi$
$510$	0	0
$511$	−4.37259	−0.193432
$512$	18.6317	0.823415
$513$	0	0
$514$	16.6586	0.734779
$515$	−1.76885	−0.0779448
$516$	0	0
$517$	28.2078	1.24058
$518$	35.0984	1.54213
$519$	0	0
$520$	−6.58870	−0.288933
$521$	7.05754	0.309196	0.154598	−	0.987977i	$-0.450592\pi$
$521$	7.05754	0.309196	0.154598	+	0.987977i	$0.450592\pi$
$522$	0	0
$523$	−32.0118	−1.39978	−0.699890	−	0.714251i	$-0.746768\pi$
$523$	−32.0118	−1.39978	−0.699890	+	0.714251i	$0.746768\pi$
$524$	−1.46804	−0.0641315
$525$	0	0
$526$	−4.24610	−0.185139
$527$	2.99442	0.130439
$528$	0	0
$529$	−21.2567	−0.924202
$530$	−4.76837	−0.207125
$531$	0	0
$532$	−4.95941	−0.215018
$533$	9.47334	0.410336
$534$	0	0
$535$	−7.20592	−0.311539
$536$	−1.73479	−0.0749316
$537$	0	0
$538$	4.51511	0.194660
$539$	12.1213	0.522101
$540$	0	0
$541$	−5.23538	−0.225087	−0.112543	−	0.993647i	$-0.535900\pi$
$541$	−5.23538	−0.225087	−0.112543	+	0.993647i	$0.535900\pi$
$542$	−30.0453	−1.29055
$543$	0	0
$544$	−9.86230	−0.422843
$545$	−5.49119	−0.235217
$546$	0	0
$547$	26.0365	1.11324	0.556620	−	0.830767i	$-0.312098\pi$
$547$	26.0365	1.11324	0.556620	+	0.830767i	$0.312098\pi$
$548$	−4.34175	−0.185470
$549$	0	0
$550$	−6.97004	−0.297204
$551$	3.11461	0.132687
$552$	0	0
$553$	36.0577	1.53333
$554$	−8.34064	−0.354360
$555$	0	0
$556$	−1.45350	−0.0616422
$557$	−3.08656	−0.130782	−0.0653909	−	0.997860i	$-0.520829\pi$
$557$	−3.08656	−0.130782	−0.0653909	+	0.997860i	$0.520829\pi$
$558$	0	0
$559$	−25.5762	−1.08176
$560$	5.41097	0.228655
$561$	0	0
$562$	−2.37344	−0.100118
$563$	17.4421	0.735098	0.367549	−	0.930004i	$-0.380197\pi$
$563$	17.4421	0.735098	0.367549	+	0.930004i	$0.380197\pi$
$564$	0	0
$565$	11.3260	0.476488
$566$	6.87968	0.289175
$567$	0	0
$568$	−29.7543	−1.24846
$569$	−10.4586	−0.438446	−0.219223	−	0.975675i	$-0.570352\pi$
$569$	−10.4586	−0.438446	−0.219223	+	0.975675i	$0.570352\pi$
$570$	0	0
$571$	35.9938	1.50629	0.753146	−	0.657853i	$-0.228535\pi$
$571$	35.9938	1.50629	0.753146	+	0.657853i	$0.228535\pi$
$572$	10.6626	0.445824
$573$	0	0
$574$	−14.5247	−0.606249
$575$	−1.32036	−0.0550628
$576$	0	0
$577$	−16.1141	−0.670839	−0.335419	−	0.942069i	$-0.608878\pi$
$577$	−16.1141	−0.670839	−0.335419	+	0.942069i	$0.608878\pi$
$578$	12.4884	0.519449
$579$	0	0
$580$	1.15701	0.0480421
$581$	−34.5816	−1.43469
$582$	0	0
$583$	27.3652	1.13335
$584$	4.49514	0.186010
$585$	0	0
$586$	−17.6885	−0.730703
$587$	−23.0827	−0.952724	−0.476362	−	0.879249i	$-0.658045\pi$
$587$	−23.0827	−0.952724	−0.476362	+	0.879249i	$0.658045\pi$
$588$	0	0
$589$	−2.65947	−0.109581
$590$	10.5512	0.434387
$591$	0	0
$592$	−19.3269	−0.794331
$593$	−14.7166	−0.604338	−0.302169	−	0.953254i	$-0.597711\pi$
$593$	−14.7166	−0.604338	−0.302169	+	0.953254i	$0.597711\pi$
$594$	0	0
$595$	−7.10912	−0.291445
$596$	8.58741	0.351754
$597$	0	0
$598$	−3.12314	−0.127715
$599$	34.2273	1.39849	0.699245	−	0.714882i	$-0.253520\pi$
$599$	34.2273	1.39849	0.699245	+	0.714882i	$0.253520\pi$
$600$	0	0
$601$	8.76721	0.357622	0.178811	−	0.983883i	$-0.442775\pi$
$601$	8.76721	0.357622	0.178811	+	0.983883i	$0.442775\pi$
$602$	39.2140	1.59824
$603$	0	0
$604$	−15.0571	−0.612666
$605$	29.0004	1.17903
$606$	0	0
$607$	12.7090	0.515843	0.257922	−	0.966166i	$-0.416962\pi$
$607$	12.7090	0.515843	0.257922	+	0.966166i	$0.416962\pi$
$608$	8.75912	0.355229
$609$	0	0
$610$	14.4679	0.585788
$611$	−9.57271	−0.387270
$612$	0	0
$613$	−26.8798	−1.08566	−0.542832	−	0.839842i	$-0.682648\pi$
$613$	−26.8798	−1.08566	−0.542832	+	0.839842i	$0.682648\pi$
$614$	−34.8860	−1.40789
$615$	0	0
$616$	−57.9739	−2.33584
$617$	−21.9813	−0.884935	−0.442467	−	0.896785i	$-0.645897\pi$
$617$	−21.9813	−0.884935	−0.442467	+	0.896785i	$0.645897\pi$
$618$	0	0
$619$	10.5269	0.423111	0.211556	−	0.977366i	$-0.432147\pi$
$619$	10.5269	0.423111	0.211556	+	0.977366i	$0.432147\pi$
$620$	−0.987933	−0.0396763
$621$	0	0
$622$	28.8815	1.15804
$623$	−28.3563	−1.13607
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.8775	−0.514687
$627$	0	0
$628$	6.60107	0.263412
$629$	25.3924	1.01246
$630$	0	0
$631$	1.62235	0.0645845	0.0322923	−	0.999478i	$-0.489719\pi$
$631$	1.62235	0.0645845	0.0322923	+	0.999478i	$0.489719\pi$
$632$	−37.0683	−1.47450
$633$	0	0
$634$	12.0538	0.478716
$635$	12.0223	0.477089
$636$	0	0
$637$	−4.11352	−0.162984
$638$	10.2669	0.406470
$639$	0	0
$640$	−0.740203	−0.0292591
$641$	11.0381	0.435977	0.217989	−	0.975951i	$-0.430050\pi$
$641$	11.0381	0.435977	0.217989	+	0.975951i	$0.430050\pi$
$642$	0	0
$643$	−37.3045	−1.47114	−0.735572	−	0.677446i	$-0.763087\pi$
$643$	−37.3045	−1.47114	−0.735572	+	0.677446i	$0.763087\pi$
$644$	−3.09687	−0.122034
$645$	0	0
$646$	5.54780	0.218275
$647$	16.0085	0.629358	0.314679	−	0.949198i	$-0.398103\pi$
$647$	16.0085	0.629358	0.314679	+	0.949198i	$0.398103\pi$
$648$	0	0
$649$	−60.5524	−2.37689
$650$	2.36538	0.0927777
$651$	0	0
$652$	−6.32676	−0.247775
$653$	−23.9070	−0.935551	−0.467776	−	0.883847i	$-0.654944\pi$
$653$	−23.9070	−0.935551	−0.467776	+	0.883847i	$0.654944\pi$
$654$	0	0
$655$	1.86898	0.0730271
$656$	7.99803	0.312271
$657$	0	0
$658$	14.6771	0.572171
$659$	26.1865	1.02008	0.510040	−	0.860150i	$-0.329630\pi$
$659$	26.1865	1.02008	0.510040	+	0.860150i	$0.329630\pi$
$660$	0	0
$661$	−10.2638	−0.399214	−0.199607	−	0.979876i	$-0.563967\pi$
$661$	−10.2638	−0.399214	−0.199607	+	0.979876i	$0.563967\pi$
$662$	−28.4051	−1.10400
$663$	0	0
$664$	35.5508	1.37964
$665$	6.31390	0.244843
$666$	0	0
$667$	1.94489	0.0753065
$668$	−6.43209	−0.248865
$669$	0	0
$670$	0.622800	0.0240608
$671$	−83.0299	−3.20533
$672$	0	0
$673$	37.6444	1.45109	0.725543	−	0.688176i	$-0.241589\pi$
$673$	37.6444	1.45109	0.725543	+	0.688176i	$0.241589\pi$
$674$	−18.2625	−0.703447
$675$	0	0
$676$	6.59269	0.253565
$677$	−27.5834	−1.06012	−0.530058	−	0.847961i	$-0.677830\pi$
$677$	−27.5834	−1.06012	−0.530058	+	0.847961i	$0.677830\pi$
$678$	0	0
$679$	43.6664	1.67576
$680$	7.30836	0.280263
$681$	0	0
$682$	−8.76658	−0.335690
$683$	24.4114	0.934077	0.467038	−	0.884237i	$-0.345321\pi$
$683$	24.4114	0.934077	0.467038	+	0.884237i	$0.345321\pi$
$684$	0	0
$685$	5.52754	0.211197
$686$	−16.7287	−0.638703
$687$	0	0
$688$	−21.5932	−0.823233
$689$	−9.28674	−0.353797
$690$	0	0
$691$	−1.25298	−0.0476658	−0.0238329	−	0.999716i	$-0.507587\pi$
$691$	−1.25298	−0.0476658	−0.0238329	+	0.999716i	$0.507587\pi$
$692$	2.60277	0.0989427
$693$	0	0
$694$	−14.6579	−0.556407
$695$	1.85048	0.0701925
$696$	0	0
$697$	−10.5081	−0.398022
$698$	6.63495	0.251137
$699$	0	0
$700$	2.34547	0.0886506
$701$	−20.1909	−0.762598	−0.381299	−	0.924452i	$-0.624523\pi$
$701$	−20.1909	−0.762598	−0.381299	+	0.924452i	$0.624523\pi$
$702$	0	0
$703$	−22.5520	−0.850566
$704$	51.7946	1.95208
$705$	0	0
$706$	−9.94021	−0.374105
$707$	−39.6305	−1.49046
$708$	0	0
$709$	−45.2979	−1.70120	−0.850600	−	0.525814i	$-0.823761\pi$
$709$	−45.2979	−1.70120	−0.850600	+	0.525814i	$0.823761\pi$
$710$	10.6819	0.400886
$711$	0	0
$712$	29.1510	1.09248
$713$	−1.66068	−0.0621931
$714$	0	0
$715$	−13.5747	−0.507664
$716$	−11.5925	−0.433232
$717$	0	0
$718$	30.6941	1.14549
$719$	50.2214	1.87294	0.936472	−	0.350744i	$-0.114071\pi$
$719$	50.2214	1.87294	0.936472	+	0.350744i	$0.114071\pi$
$720$	0	0
$721$	5.28189	0.196708
$722$	16.0118	0.595898
$723$	0	0
$724$	−9.78400	−0.363619
$725$	−1.47300	−0.0547059
$726$	0	0
$727$	−26.1403	−0.969491	−0.484745	−	0.874655i	$-0.661088\pi$
$727$	−26.1403	−0.969491	−0.484745	+	0.874655i	$0.661088\pi$
$728$	19.6742	0.729175
$729$	0	0
$730$	−1.61378	−0.0597286
$731$	28.3699	1.04930
$732$	0	0
$733$	7.44658	0.275046	0.137523	−	0.990499i	$-0.456086\pi$
$733$	7.44658	0.275046	0.137523	+	0.990499i	$0.456086\pi$
$734$	−7.86463	−0.290289
$735$	0	0
$736$	5.46956	0.201611
$737$	−3.57419	−0.131657
$738$	0	0
$739$	−13.4071	−0.493190	−0.246595	−	0.969119i	$-0.579312\pi$
$739$	−13.4071	−0.493190	−0.246595	+	0.969119i	$0.579312\pi$
$740$	−8.37757	−0.307966
$741$	0	0
$742$	14.2386	0.522716
$743$	4.61377	0.169263	0.0846314	−	0.996412i	$-0.473029\pi$
$743$	4.61377	0.169263	0.0846314	+	0.996412i	$0.473029\pi$
$744$	0	0
$745$	−10.9328	−0.400545
$746$	−7.48497	−0.274044
$747$	0	0
$748$	−11.8272	−0.432445
$749$	21.5173	0.786224
$750$	0	0
$751$	47.1671	1.72115	0.860576	−	0.509323i	$-0.170104\pi$
$751$	47.1671	1.72115	0.860576	+	0.509323i	$0.170104\pi$
$752$	−8.08193	−0.294717
$753$	0	0
$754$	−3.48420	−0.126887
$755$	19.1695	0.697648
$756$	0	0
$757$	10.4087	0.378309	0.189155	−	0.981947i	$-0.439425\pi$
$757$	10.4087	0.378309	0.189155	+	0.981947i	$0.439425\pi$
$758$	−0.745994	−0.0270957
$759$	0	0
$760$	−6.49086	−0.235448
$761$	−17.6905	−0.641278	−0.320639	−	0.947201i	$-0.603898\pi$
$761$	−17.6905	−0.641278	−0.320639	+	0.947201i	$0.603898\pi$
$762$	0	0
$763$	16.3970	0.593611
$764$	4.10212	0.148409
$765$	0	0
$766$	−33.7254	−1.21855
$767$	20.5493	0.741991
$768$	0	0
$769$	−32.0669	−1.15636	−0.578182	−	0.815908i	$-0.696238\pi$
$769$	−32.0669	−1.15636	−0.578182	+	0.815908i	$0.696238\pi$
$770$	20.8129	0.750046
$771$	0	0
$772$	−9.80607	−0.352928
$773$	−50.0261	−1.79931	−0.899656	−	0.436599i	$-0.856183\pi$
$773$	−50.0261	−1.79931	−0.899656	+	0.436599i	$0.856183\pi$
$774$	0	0
$775$	1.25775	0.0451798
$776$	−44.8903	−1.61147
$777$	0	0
$778$	−24.1473	−0.865724
$779$	9.33267	0.334378
$780$	0	0
$781$	−61.3026	−2.19358
$782$	3.46428	0.123882
$783$	0	0
$784$	−3.47291	−0.124033
$785$	−8.40393	−0.299949
$786$	0	0
$787$	−23.1591	−0.825533	−0.412767	−	0.910837i	$-0.635437\pi$
$787$	−23.1591	−0.825533	−0.412767	+	0.910837i	$0.635437\pi$
$788$	−11.6378	−0.414580
$789$	0	0
$790$	13.3077	0.473467
$791$	−33.8201	−1.20250
$792$	0	0
$793$	28.1773	1.00061
$794$	2.02496	0.0718631
$795$	0	0
$796$	−10.2908	−0.364748
$797$	17.8879	0.633622	0.316811	−	0.948489i	$-0.397388\pi$
$797$	17.8879	0.633622	0.316811	+	0.948489i	$0.397388\pi$
$798$	0	0
$799$	10.6183	0.375649
$800$	−4.14248	−0.146459
$801$	0	0
$802$	−25.6955	−0.907339
$803$	9.26131	0.326825
$804$	0	0
$805$	3.94267	0.138961
$806$	2.97506	0.104792
$807$	0	0
$808$	40.7412	1.43327
$809$	18.4115	0.647315	0.323658	−	0.946174i	$-0.395087\pi$
$809$	18.4115	0.647315	0.323658	+	0.946174i	$0.395087\pi$
$810$	0	0
$811$	16.5890	0.582519	0.291259	−	0.956644i	$-0.405926\pi$
$811$	16.5890	0.582519	0.291259	+	0.956644i	$0.405926\pi$
$812$	−3.45489	−0.121243
$813$	0	0
$814$	−74.3397	−2.60561
$815$	8.05469	0.282143
$816$	0	0
$817$	−25.1965	−0.881513
$818$	−37.1468	−1.29881
$819$	0	0
$820$	3.46688	0.121069
$821$	−1.98361	−0.0692286	−0.0346143	−	0.999401i	$-0.511020\pi$
$821$	−1.98361	−0.0692286	−0.0346143	+	0.999401i	$0.511020\pi$
$822$	0	0
$823$	1.37974	0.0480949	0.0240474	−	0.999711i	$-0.492345\pi$
$823$	1.37974	0.0480949	0.0240474	+	0.999711i	$0.492345\pi$
$824$	−5.42992	−0.189160
$825$	0	0
$826$	−31.5065	−1.09625
$827$	17.5290	0.609544	0.304772	−	0.952425i	$-0.401420\pi$
$827$	17.5290	0.609544	0.304772	+	0.952425i	$0.401420\pi$
$828$	0	0
$829$	−24.7693	−0.860275	−0.430138	−	0.902763i	$-0.641535\pi$
$829$	−24.7693	−0.860275	−0.430138	+	0.902763i	$0.641535\pi$
$830$	−12.7629	−0.443007
$831$	0	0
$832$	−17.5772	−0.609379
$833$	4.56283	0.158093
$834$	0	0
$835$	8.18879	0.283385
$836$	10.5042	0.363297
$837$	0	0
$838$	−26.2845	−0.907982
$839$	−38.6617	−1.33475	−0.667375	−	0.744722i	$-0.732582\pi$
$839$	−38.6617	−1.33475	−0.667375	+	0.744722i	$0.732582\pi$
$840$	0	0
$841$	−26.8303	−0.925182
$842$	−21.9213	−0.755459
$843$	0	0
$844$	−14.8501	−0.511162
$845$	−8.39326	−0.288737
$846$	0	0
$847$	−86.5968	−2.97550
$848$	−7.84049	−0.269244
$849$	0	0
$850$	−2.62374	−0.0899935
$851$	−14.0824	−0.482739
$852$	0	0
$853$	1.04257	0.0356969	0.0178485	−	0.999841i	$-0.494318\pi$
$853$	1.04257	0.0356969	0.0178485	+	0.999841i	$0.494318\pi$
$854$	−43.2020	−1.47834
$855$	0	0
$856$	−22.1203	−0.756057
$857$	46.0501	1.57304	0.786521	−	0.617564i	$-0.211880\pi$
$857$	46.0501	1.57304	0.786521	+	0.617564i	$0.211880\pi$
$858$	0	0
$859$	22.3369	0.762126	0.381063	−	0.924549i	$-0.375558\pi$
$859$	22.3369	0.762126	0.381063	+	0.924549i	$0.375558\pi$
$860$	−9.35992	−0.319171
$861$	0	0
$862$	13.9134	0.473893
$863$	7.44818	0.253539	0.126770	−	0.991932i	$-0.459539\pi$
$863$	7.44818	0.253539	0.126770	+	0.991932i	$0.459539\pi$
$864$	0	0
$865$	−3.31363	−0.112667
$866$	3.34448	0.113650
$867$	0	0
$868$	2.95002	0.100130
$869$	−76.3716	−2.59073
$870$	0	0
$871$	1.21295	0.0410992
$872$	−16.8566	−0.570835
$873$	0	0
$874$	−3.07677	−0.104073
$875$	−2.98606	−0.100947
$876$	0	0
$877$	−3.74467	−0.126449	−0.0632243	−	0.997999i	$-0.520138\pi$
$877$	−3.74467	−0.126449	−0.0632243	+	0.997999i	$0.520138\pi$
$878$	−3.45146	−0.116481
$879$	0	0
$880$	−11.4606	−0.386338
$881$	−19.2395	−0.648196	−0.324098	−	0.946024i	$-0.605061\pi$
$881$	−19.2395	−0.648196	−0.324098	+	0.946024i	$0.605061\pi$
$882$	0	0
$883$	5.31811	0.178969	0.0894844	−	0.995988i	$-0.471478\pi$
$883$	5.31811	0.178969	0.0894844	+	0.995988i	$0.471478\pi$
$884$	4.01372	0.134996
$885$	0	0
$886$	40.8144	1.37119
$887$	6.29425	0.211340	0.105670	−	0.994401i	$-0.466301\pi$
$887$	6.29425	0.211340	0.105670	+	0.994401i	$0.466301\pi$
$888$	0	0
$889$	−35.8992	−1.20402
$890$	−10.4654	−0.350800
$891$	0	0
$892$	−2.76644	−0.0926272
$893$	−9.43057	−0.315582
$894$	0	0
$895$	14.7586	0.493325
$896$	2.21029	0.0738405
$897$	0	0
$898$	29.1404	0.972427
$899$	−1.85267	−0.0617900
$900$	0	0
$901$	10.3011	0.343180
$902$	30.7639	1.02433
$903$	0	0
$904$	34.7680	1.15636
$905$	12.4562	0.414057
$906$	0	0
$907$	24.5506	0.815190	0.407595	−	0.913163i	$-0.366368\pi$
$907$	24.5506	0.815190	0.407595	+	0.913163i	$0.366368\pi$
$908$	6.01486	0.199610
$909$	0	0
$910$	−7.06315	−0.234141
$911$	−25.7040	−0.851611	−0.425806	−	0.904815i	$-0.640009\pi$
$911$	−25.7040	−0.851611	−0.425806	+	0.904815i	$0.640009\pi$
$912$	0	0
$913$	73.2452	2.42406
$914$	−0.198978	−0.00658160
$915$	0	0
$916$	2.36889	0.0782705
$917$	−5.58088	−0.184297
$918$	0	0
$919$	32.3013	1.06552	0.532761	−	0.846266i	$-0.321154\pi$
$919$	32.3013	1.06552	0.532761	+	0.846266i	$0.321154\pi$
$920$	−4.05317	−0.133629
$921$	0	0
$922$	26.9609	0.887910
$923$	20.8039	0.684768
$924$	0	0
$925$	10.6656	0.350683
$926$	2.88008	0.0946452
$927$	0	0
$928$	6.10189	0.200304
$929$	−25.8875	−0.849341	−0.424670	−	0.905348i	$-0.639610\pi$
$929$	−25.8875	−0.849341	−0.424670	+	0.905348i	$0.639610\pi$
$930$	0	0
$931$	−4.05244	−0.132813
$932$	−14.8088	−0.485078
$933$	0	0
$934$	8.62132	0.282098
$935$	15.0574	0.492429
$936$	0	0
$937$	−14.9948	−0.489860	−0.244930	−	0.969541i	$-0.578765\pi$
$937$	−14.9948	−0.489860	−0.244930	+	0.969541i	$0.578765\pi$
$938$	−1.85971	−0.0607218
$939$	0	0
$940$	−3.50324	−0.114263
$941$	−52.8090	−1.72153	−0.860763	−	0.509006i	$-0.830013\pi$
$941$	−52.8090	−1.72153	−0.860763	+	0.509006i	$0.830013\pi$
$942$	0	0
$943$	5.82771	0.189776
$944$	17.3491	0.564664
$945$	0	0
$946$	−83.0568	−2.70041
$947$	−15.6157	−0.507443	−0.253722	−	0.967277i	$-0.581655\pi$
$947$	−15.6157	−0.507443	−0.253722	+	0.967277i	$0.581655\pi$
$948$	0	0
$949$	−3.14295	−0.102024
$950$	2.33025	0.0756034
$951$	0	0
$952$	−21.8232	−0.707294
$953$	−20.3578	−0.659455	−0.329727	−	0.944076i	$-0.606957\pi$
$953$	−20.3578	−0.659455	−0.329727	+	0.944076i	$0.606957\pi$
$954$	0	0
$955$	−5.22247	−0.168995
$956$	−7.14188	−0.230985
$957$	0	0
$958$	−19.8528	−0.641415
$959$	−16.5056	−0.532992
$960$	0	0
$961$	−29.4181	−0.948970
$962$	25.2282	0.813389
$963$	0	0
$964$	−9.39915	−0.302726
$965$	12.4843	0.401882
$966$	0	0
$967$	−3.76239	−0.120990	−0.0604952	−	0.998168i	$-0.519268\pi$
$967$	−3.76239	−0.120990	−0.0604952	+	0.998168i	$0.519268\pi$
$968$	89.0238	2.86133
$969$	0	0
$970$	16.1158	0.517449
$971$	−13.8225	−0.443586	−0.221793	−	0.975094i	$-0.571191\pi$
$971$	−13.8225	−0.443586	−0.221793	+	0.975094i	$0.571191\pi$
$972$	0	0
$973$	−5.52562	−0.177143
$974$	30.0733	0.963612
$975$	0	0
$976$	23.7892	0.761473
$977$	46.3648	1.48334	0.741671	−	0.670764i	$-0.234034\pi$
$977$	46.3648	1.48334	0.741671	+	0.670764i	$0.234034\pi$
$978$	0	0
$979$	60.0598	1.91952
$980$	−1.50539	−0.0480879
$981$	0	0
$982$	5.35093	0.170755
$983$	43.2210	1.37854	0.689268	−	0.724507i	$-0.257932\pi$
$983$	43.2210	1.37854	0.689268	+	0.724507i	$0.257932\pi$
$984$	0	0
$985$	14.8163	0.472086
$986$	3.86477	0.123079
$987$	0	0
$988$	−3.56475	−0.113410
$989$	−15.7337	−0.500304
$990$	0	0
$991$	22.1879	0.704821	0.352410	−	0.935846i	$-0.385362\pi$
$991$	22.1879	0.704821	0.352410	+	0.935846i	$0.385362\pi$
$992$	−5.21021	−0.165424
$993$	0	0
$994$	−31.8969	−1.01171
$995$	13.1014	0.415342
$996$	0	0
$997$	48.5158	1.53651	0.768256	−	0.640143i	$-0.221125\pi$
$997$	48.5158	1.53651	0.768256	+	0.640143i	$0.221125\pi$
$998$	−23.5587	−0.745736
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.l.1.7		21
3.2	odd	2		3645.2.a.k.1.15		21
27.4	even	9		135.2.k.b.16.3	✓	42
27.7	even	9		135.2.k.b.76.3	yes	42
27.20	odd	18		405.2.k.b.361.5		42
27.23	odd	18		405.2.k.b.46.5		42
135.4	even	18		675.2.l.e.151.5		42
135.7	odd	36		675.2.u.d.49.5		84
135.34	even	18		675.2.l.e.76.5		42
135.58	odd	36		675.2.u.d.124.5		84
135.88	odd	36		675.2.u.d.49.10		84
135.112	odd	36		675.2.u.d.124.10		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.k.b.16.3	✓	42	27.4	even	9
135.2.k.b.76.3	yes	42	27.7	even	9
405.2.k.b.46.5		42	27.23	odd	18
405.2.k.b.361.5		42	27.20	odd	18
675.2.l.e.76.5		42	135.34	even	18
675.2.l.e.151.5		42	135.4	even	18
675.2.u.d.49.5		84	135.7	odd	36
675.2.u.d.49.10		84	135.88	odd	36
675.2.u.d.124.5		84	135.58	odd	36
675.2.u.d.124.10		84	135.112	odd	36
3645.2.a.k.1.15		21	3.2	odd	2
3645.2.a.l.1.7		21	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10205 q^{2} -0.785475 q^{4} +1.00000 q^{5} -2.98606 q^{7} +3.06975 q^{8} -1.10205 q^{10} +6.32459 q^{11} -2.14633 q^{13} +3.29080 q^{14} -1.81208 q^{16} +2.38077 q^{17} -2.11446 q^{19} -0.785475 q^{20} -6.97004 q^{22} -1.32036 q^{23} +1.00000 q^{25} +2.36538 q^{26} +2.34547 q^{28} -1.47300 q^{29} +1.25775 q^{31} -4.14248 q^{32} -2.62374 q^{34} -2.98606 q^{35} +10.6656 q^{37} +2.33025 q^{38} +3.06975 q^{40} -4.41373 q^{41} +11.9163 q^{43} -4.96780 q^{44} +1.45511 q^{46} +4.46003 q^{47} +1.91653 q^{49} -1.10205 q^{50} +1.68589 q^{52} +4.32679 q^{53} +6.32459 q^{55} -9.16644 q^{56} +1.62333 q^{58} -9.57413 q^{59} -13.1281 q^{61} -1.38611 q^{62} +8.18940 q^{64} -2.14633 q^{65} -0.565126 q^{67} -1.87004 q^{68} +3.29080 q^{70} -9.69275 q^{71} +1.46433 q^{73} -11.7541 q^{74} +1.66086 q^{76} -18.8856 q^{77} -12.0754 q^{79} -1.81208 q^{80} +4.86417 q^{82} +11.5810 q^{83} +2.38077 q^{85} -13.1324 q^{86} +19.4149 q^{88} +9.49624 q^{89} +6.40907 q^{91} +1.03711 q^{92} -4.91520 q^{94} -2.11446 q^{95} -14.6234 q^{97} -2.11213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8} + 3 q^{10} + 12 q^{13} + 39 q^{16} + 12 q^{17} + 24 q^{19} + 27 q^{20} + 18 q^{22} + 18 q^{23} + 21 q^{25} - 9 q^{26} + 30 q^{28} - 3 q^{29} + 24 q^{31}+ \cdots + 15 q^{98}+O(q^{100}) \) 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8 + 3 * q^10 + 12 * q^13 + 39 * q^16 + 12 * q^17 + 24 * q^19 + 27 * q^20 + 18 * q^22 + 18 * q^23 + 21 * q^25 - 9 * q^26 + 30 * q^28 - 3 * q^29 + 24 * q^31 + 6 * q^32 + 18 * q^34 + 12 * q^35 + 24 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 42 * q^43 - 12 * q^44 + 30 * q^46 + 21 * q^47 + 39 * q^49 + 3 * q^50 + 36 * q^52 + 18 * q^53 - 30 * q^56 + 30 * q^58 - 6 * q^59 + 33 * q^61 + 18 * q^62 + 63 * q^64 + 12 * q^65 + 63 * q^67 + 36 * q^68 - 12 * q^71 + 39 * q^73 - 21 * q^74 + 48 * q^76 + 9 * q^77 + 48 * q^79 + 39 * q^80 + 24 * q^82 + 33 * q^83 + 12 * q^85 - 69 * q^86 + 54 * q^88 - 9 * q^89 + 69 * q^91 + 21 * q^92 + 30 * q^94 + 24 * q^95 + 30 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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