LMFDB - Embedded newform 675.4.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,30,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 21x^{4} + 5x^{3} + 101x^{2} + 29x + 2 $ x^6 - x^5 - 21x^4 + 5x^3 + 101x^2 + 29x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.113318$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25118 q^{2} -6.43456 q^{4} +5.67029 q^{7} +18.0602 q^{8} +21.2537 q^{11} -40.9479 q^{13} -7.09453 q^{14} +28.8800 q^{16} -59.1853 q^{17} +21.8800 q^{19} -26.5922 q^{22} +98.7941 q^{23} +51.2331 q^{26} -36.4858 q^{28} -159.643 q^{29} -69.4873 q^{31} -180.615 q^{32} +74.0512 q^{34} +235.618 q^{37} -27.3757 q^{38} -491.487 q^{41} +95.3080 q^{43} -136.758 q^{44} -123.609 q^{46} +548.796 q^{47} -310.848 q^{49} +263.482 q^{52} -509.829 q^{53} +102.406 q^{56} +199.742 q^{58} +741.098 q^{59} +387.157 q^{61} +86.9409 q^{62} -5.05795 q^{64} +1060.45 q^{67} +380.831 q^{68} -508.998 q^{71} +914.906 q^{73} -294.800 q^{74} -140.788 q^{76} +120.515 q^{77} +925.717 q^{79} +614.937 q^{82} -708.964 q^{83} -119.247 q^{86} +383.846 q^{88} +273.516 q^{89} -232.186 q^{91} -635.696 q^{92} -686.641 q^{94} +1450.22 q^{97} +388.926 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64}+ \cdots + 4104 q^{97}+O(q^{100}) $ 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

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 $

$2$	−1.25118	−0.442358	−0.221179	−	0.975233i	$-0.570990\pi$
$2$	−1.25118	−0.442358	−0.221179	+	0.975233i	$0.570990\pi$
$3$	0	0
$3$	0	0
$4$	−6.43456	−0.804320
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.67029	0.306167	0.153083	−	0.988213i	$-0.451080\pi$
$7$	5.67029	0.306167	0.153083	+	0.988213i	$0.451080\pi$
$8$	18.0602	0.798155
$9$	0	0
$10$	0	0
$11$	21.2537	0.582567	0.291283	−	0.956637i	$-0.405918\pi$
$11$	21.2537	0.582567	0.291283	+	0.956637i	$0.405918\pi$
$12$	0	0
$13$	−40.9479	−0.873608	−0.436804	−	0.899557i	$-0.643890\pi$
$13$	−40.9479	−0.873608	−0.436804	+	0.899557i	$0.643890\pi$
$14$	−7.09453	−0.135435
$15$	0	0
$16$	28.8800	0.451250
$17$	−59.1853	−0.844384	−0.422192	−	0.906506i	$-0.638739\pi$
$17$	−59.1853	−0.844384	−0.422192	+	0.906506i	$0.638739\pi$
$18$	0	0
$19$	21.8800	0.264190	0.132095	−	0.991237i	$-0.457830\pi$
$19$	21.8800	0.264190	0.132095	+	0.991237i	$0.457830\pi$
$20$	0	0
$21$	0	0
$22$	−26.5922	−0.257703
$23$	98.7941	0.895652	0.447826	−	0.894121i	$-0.352198\pi$
$23$	98.7941	0.895652	0.447826	+	0.894121i	$0.352198\pi$
$24$	0	0
$25$	0	0
$26$	51.2331	0.386447
$27$	0	0
$28$	−36.4858	−0.246256
$29$	−159.643	−1.02224	−0.511120	−	0.859509i	$-0.670769\pi$
$29$	−159.643	−1.02224	−0.511120	+	0.859509i	$0.670769\pi$
$30$	0	0
$31$	−69.4873	−0.402590	−0.201295	−	0.979531i	$-0.564515\pi$
$31$	−69.4873	−0.402590	−0.201295	+	0.979531i	$0.564515\pi$
$32$	−180.615	−0.997769
$33$	0	0
$34$	74.0512	0.373520
$35$	0	0
$36$	0	0
$37$	235.618	1.04690	0.523451	−	0.852056i	$-0.324644\pi$
$37$	235.618	1.04690	0.523451	+	0.852056i	$0.324644\pi$
$38$	−27.3757	−0.116866
$39$	0	0
$40$	0	0
$41$	−491.487	−1.87213	−0.936066	−	0.351825i	$-0.885561\pi$
$41$	−491.487	−1.87213	−0.936066	+	0.351825i	$0.885561\pi$
$42$	0	0
$43$	95.3080	0.338008	0.169004	−	0.985615i	$-0.445945\pi$
$43$	95.3080	0.338008	0.169004	+	0.985615i	$0.445945\pi$
$44$	−136.758	−0.468570
$45$	0	0
$46$	−123.609	−0.396199
$47$	548.796	1.70319	0.851597	−	0.524197i	$-0.175634\pi$
$47$	548.796	1.70319	0.851597	+	0.524197i	$0.175634\pi$
$48$	0	0
$49$	−310.848	−0.906262
$50$	0	0
$51$	0	0
$52$	263.482	0.702660
$53$	−509.829	−1.32133	−0.660664	−	0.750682i	$-0.729725\pi$
$53$	−509.829	−1.32133	−0.660664	+	0.750682i	$0.729725\pi$
$54$	0	0
$55$	0	0
$56$	102.406	0.244368
$57$	0	0
$58$	199.742	0.452196
$59$	741.098	1.63530	0.817650	−	0.575715i	$-0.195276\pi$
$59$	741.098	1.63530	0.817650	+	0.575715i	$0.195276\pi$
$60$	0	0
$61$	387.157	0.812629	0.406314	−	0.913733i	$-0.366814\pi$
$61$	387.157	0.812629	0.406314	+	0.913733i	$0.366814\pi$
$62$	86.9409	0.178089
$63$	0	0
$64$	−5.05795	−0.00987881
$65$	0	0
$66$	0	0
$67$	1060.45	1.93364	0.966822	−	0.255451i	$-0.0822238\pi$
$67$	1060.45	1.93364	0.966822	+	0.255451i	$0.0822238\pi$
$68$	380.831	0.679155
$69$	0	0
$70$	0	0
$71$	−508.998	−0.850802	−0.425401	−	0.905005i	$-0.639867\pi$
$71$	−508.998	−0.850802	−0.425401	+	0.905005i	$0.639867\pi$
$72$	0	0
$73$	914.906	1.46687	0.733436	−	0.679758i	$-0.237915\pi$
$73$	914.906	1.46687	0.733436	+	0.679758i	$0.237915\pi$
$74$	−294.800	−0.463105
$75$	0	0
$76$	−140.788	−0.212493
$77$	120.515	0.178363
$78$	0	0
$79$	925.717	1.31837	0.659185	−	0.751981i	$-0.270901\pi$
$79$	925.717	1.31837	0.659185	+	0.751981i	$0.270901\pi$
$80$	0	0
$81$	0	0
$82$	614.937	0.828152
$83$	−708.964	−0.937577	−0.468788	−	0.883310i	$-0.655309\pi$
$83$	−708.964	−0.937577	−0.468788	+	0.883310i	$0.655309\pi$
$84$	0	0
$85$	0	0
$86$	−119.247	−0.149520
$87$	0	0
$88$	383.846	0.464979
$89$	273.516	0.325760	0.162880	−	0.986646i	$-0.447922\pi$
$89$	273.516	0.325760	0.162880	+	0.986646i	$0.447922\pi$
$90$	0	0
$91$	−232.186	−0.267470
$92$	−635.696	−0.720390
$93$	0	0
$94$	−686.641	−0.753421
$95$	0	0
$96$	0	0
$97$	1450.22	1.51802	0.759008	−	0.651081i	$-0.225684\pi$
$97$	1450.22	1.51802	0.759008	+	0.651081i	$0.225684\pi$
$98$	388.926	0.400892
$99$	0	0
$100$	0	0
$101$	676.246	0.666228	0.333114	−	0.942887i	$-0.391901\pi$
$101$	676.246	0.666228	0.333114	+	0.942887i	$0.391901\pi$
$102$	0	0
$103$	1069.47	1.02308	0.511542	−	0.859258i	$-0.329074\pi$
$103$	1069.47	1.02308	0.511542	+	0.859258i	$0.329074\pi$
$104$	−739.527	−0.697275
$105$	0	0
$106$	637.886	0.584499
$107$	−209.566	−0.189341	−0.0946707	−	0.995509i	$-0.530180\pi$
$107$	−209.566	−0.189341	−0.0946707	+	0.995509i	$0.530180\pi$
$108$	0	0
$109$	−455.127	−0.399938	−0.199969	−	0.979802i	$-0.564084\pi$
$109$	−455.127	−0.399938	−0.199969	+	0.979802i	$0.564084\pi$
$110$	0	0
$111$	0	0
$112$	163.758	0.138158
$113$	1130.83	0.941413	0.470706	−	0.882290i	$-0.343999\pi$
$113$	1130.83	0.941413	0.470706	+	0.882290i	$0.343999\pi$
$114$	0	0
$115$	0	0
$116$	1027.23	0.822208
$117$	0	0
$118$	−927.245	−0.723388
$119$	−335.597	−0.258522
$120$	0	0
$121$	−879.280	−0.660616
$122$	−484.402	−0.359473
$123$	0	0
$124$	447.120	0.323811
$125$	0	0
$126$	0	0
$127$	1277.62	0.892682	0.446341	−	0.894863i	$-0.352727\pi$
$127$	1277.62	0.892682	0.446341	+	0.894863i	$0.352727\pi$
$128$	1451.25	1.00214
$129$	0	0
$130$	0	0
$131$	1358.29	0.905909	0.452954	−	0.891534i	$-0.350370\pi$
$131$	1358.29	0.905909	0.452954	+	0.891534i	$0.350370\pi$
$132$	0	0
$133$	124.066	0.0808862
$134$	−1326.81	−0.855363
$135$	0	0
$136$	−1068.90	−0.673949
$137$	−956.545	−0.596519	−0.298260	−	0.954485i	$-0.596406\pi$
$137$	−956.545	−0.596519	−0.298260	+	0.954485i	$0.596406\pi$
$138$	0	0
$139$	27.3274	0.0166754	0.00833769	−	0.999965i	$-0.497346\pi$
$139$	27.3274	0.0166754	0.00833769	+	0.999965i	$0.497346\pi$
$140$	0	0
$141$	0	0
$142$	636.847	0.376359
$143$	−870.295	−0.508935
$144$	0	0
$145$	0	0
$146$	−1144.71	−0.648882
$147$	0	0
$148$	−1516.10	−0.842044
$149$	−216.410	−0.118987	−0.0594933	−	0.998229i	$-0.518948\pi$
$149$	−216.410	−0.118987	−0.0594933	+	0.998229i	$0.518948\pi$
$150$	0	0
$151$	2675.41	1.44187	0.720933	−	0.693004i	$-0.243713\pi$
$151$	2675.41	1.44187	0.720933	+	0.693004i	$0.243713\pi$
$152$	395.156	0.210865
$153$	0	0
$154$	−150.785	−0.0789001
$155$	0	0
$156$	0	0
$157$	956.987	0.486471	0.243235	−	0.969967i	$-0.421791\pi$
$157$	956.987	0.486471	0.243235	+	0.969967i	$0.421791\pi$
$158$	−1158.24	−0.583192
$159$	0	0
$160$	0	0
$161$	560.191	0.274219
$162$	0	0
$163$	478.098	0.229739	0.114870	−	0.993381i	$-0.463355\pi$
$163$	478.098	0.229739	0.114870	+	0.993381i	$0.463355\pi$
$164$	3162.50	1.50579
$165$	0	0
$166$	887.039	0.414744
$167$	2666.34	1.23549	0.617746	−	0.786377i	$-0.288046\pi$
$167$	2666.34	1.23549	0.617746	+	0.786377i	$0.288046\pi$
$168$	0	0
$169$	−520.269	−0.236809
$170$	0	0
$171$	0	0
$172$	−613.265	−0.271866
$173$	492.589	0.216479	0.108239	−	0.994125i	$-0.465479\pi$
$173$	492.589	0.216479	0.108239	+	0.994125i	$0.465479\pi$
$174$	0	0
$175$	0	0
$176$	613.807	0.262883
$177$	0	0
$178$	−342.217	−0.144103
$179$	3890.48	1.62451	0.812257	−	0.583299i	$-0.198238\pi$
$179$	3890.48	1.62451	0.812257	+	0.583299i	$0.198238\pi$
$180$	0	0
$181$	−4371.58	−1.79523	−0.897616	−	0.440779i	$-0.854702\pi$
$181$	−4371.58	−1.79523	−0.897616	+	0.440779i	$0.854702\pi$
$182$	290.506	0.118317
$183$	0	0
$184$	1784.24	0.714869
$185$	0	0
$186$	0	0
$187$	−1257.91	−0.491910
$188$	−3531.26	−1.36991
$189$	0	0
$190$	0	0
$191$	−328.351	−0.124391	−0.0621954	−	0.998064i	$-0.519810\pi$
$191$	−328.351	−0.124391	−0.0621954	+	0.998064i	$0.519810\pi$
$192$	0	0
$193$	944.655	0.352320	0.176160	−	0.984362i	$-0.443632\pi$
$193$	944.655	0.352320	0.176160	+	0.984362i	$0.443632\pi$
$194$	−1814.48	−0.671506
$195$	0	0
$196$	2000.17	0.728924
$197$	−2757.66	−0.997337	−0.498668	−	0.866793i	$-0.666177\pi$
$197$	−2757.66	−0.997337	−0.498668	+	0.866793i	$0.666177\pi$
$198$	0	0
$199$	−1529.88	−0.544978	−0.272489	−	0.962159i	$-0.587847\pi$
$199$	−1529.88	−0.544978	−0.272489	+	0.962159i	$0.587847\pi$
$200$	0	0
$201$	0	0
$202$	−846.103	−0.294711
$203$	−905.222	−0.312976
$204$	0	0
$205$	0	0
$206$	−1338.09	−0.452569
$207$	0	0
$208$	−1182.57	−0.394215
$209$	465.031	0.153908
$210$	0	0
$211$	−150.761	−0.0491886	−0.0245943	−	0.999698i	$-0.507829\pi$
$211$	−150.761	−0.0491886	−0.0245943	+	0.999698i	$0.507829\pi$
$212$	3280.52	1.06277
$213$	0	0
$214$	262.204	0.0837567
$215$	0	0
$216$	0	0
$217$	−394.013	−0.123260
$218$	569.444	0.176916
$219$	0	0
$220$	0	0
$221$	2423.51	0.737661
$222$	0	0
$223$	2573.87	0.772910	0.386455	−	0.922308i	$-0.373699\pi$
$223$	2573.87	0.772910	0.386455	+	0.922308i	$0.373699\pi$
$224$	−1024.14	−0.305483
$225$	0	0
$226$	−1414.87	−0.416441
$227$	2491.62	0.728523	0.364261	−	0.931297i	$-0.381321\pi$
$227$	2491.62	0.728523	0.364261	+	0.931297i	$0.381321\pi$
$228$	0	0
$229$	1400.56	0.404154	0.202077	−	0.979370i	$-0.435231\pi$
$229$	1400.56	0.404154	0.202077	+	0.979370i	$0.435231\pi$
$230$	0	0
$231$	0	0
$232$	−2883.18	−0.815906
$233$	3258.86	0.916289	0.458144	−	0.888878i	$-0.348514\pi$
$233$	3258.86	0.916289	0.458144	+	0.888878i	$0.348514\pi$
$234$	0	0
$235$	0	0
$236$	−4768.64	−1.31530
$237$	0	0
$238$	419.892	0.114359
$239$	−1660.80	−0.449491	−0.224746	−	0.974417i	$-0.572155\pi$
$239$	−1660.80	−0.449491	−0.224746	+	0.974417i	$0.572155\pi$
$240$	0	0
$241$	5966.20	1.59467	0.797337	−	0.603534i	$-0.206241\pi$
$241$	5966.20	1.59467	0.797337	+	0.603534i	$0.206241\pi$
$242$	1100.13	0.292229
$243$	0	0
$244$	−2491.18	−0.653613
$245$	0	0
$246$	0	0
$247$	−895.939	−0.230799
$248$	−1254.95	−0.321329
$249$	0	0
$250$	0	0
$251$	−3051.45	−0.767354	−0.383677	−	0.923467i	$-0.625342\pi$
$251$	−3051.45	−0.767354	−0.383677	+	0.923467i	$0.625342\pi$
$252$	0	0
$253$	2099.74	0.521777
$254$	−1598.53	−0.394885
$255$	0	0
$256$	−1775.31	−0.433425
$257$	−5559.16	−1.34930	−0.674652	−	0.738136i	$-0.735706\pi$
$257$	−5559.16	−1.34930	−0.674652	+	0.738136i	$0.735706\pi$
$258$	0	0
$259$	1336.02	0.320527
$260$	0	0
$261$	0	0
$262$	−1699.46	−0.400736
$263$	956.467	0.224252	0.112126	−	0.993694i	$-0.464234\pi$
$263$	956.467	0.224252	0.112126	+	0.993694i	$0.464234\pi$
$264$	0	0
$265$	0	0
$266$	−155.228	−0.0357806
$267$	0	0
$268$	−6823.50	−1.55527
$269$	−4740.41	−1.07445	−0.537226	−	0.843438i	$-0.680528\pi$
$269$	−4740.41	−1.07445	−0.537226	+	0.843438i	$0.680528\pi$
$270$	0	0
$271$	−3814.78	−0.855097	−0.427548	−	0.903992i	$-0.640623\pi$
$271$	−3814.78	−0.855097	−0.427548	+	0.903992i	$0.640623\pi$
$272$	−1709.27	−0.381028
$273$	0	0
$274$	1196.81	0.263875
$275$	0	0
$276$	0	0
$277$	−1579.65	−0.342642	−0.171321	−	0.985215i	$-0.554804\pi$
$277$	−1579.65	−0.342642	−0.171321	+	0.985215i	$0.554804\pi$
$278$	−34.1914	−0.00737649
$279$	0	0
$280$	0	0
$281$	7876.16	1.67207	0.836036	−	0.548674i	$-0.184867\pi$
$281$	7876.16	1.67207	0.836036	+	0.548674i	$0.184867\pi$
$282$	0	0
$283$	−905.509	−0.190201	−0.0951006	−	0.995468i	$-0.530317\pi$
$283$	−905.509	−0.190201	−0.0951006	+	0.995468i	$0.530317\pi$
$284$	3275.18	0.684317
$285$	0	0
$286$	1088.89	0.225131
$287$	−2786.87	−0.573184
$288$	0	0
$289$	−1410.11	−0.287015
$290$	0	0
$291$	0	0
$292$	−5887.02	−1.17983
$293$	7691.27	1.53355	0.766773	−	0.641919i	$-0.221861\pi$
$293$	7691.27	1.53355	0.766773	+	0.641919i	$0.221861\pi$
$294$	0	0
$295$	0	0
$296$	4255.31	0.835590
$297$	0	0
$298$	270.767	0.0526346
$299$	−4045.41	−0.782449
$300$	0	0
$301$	540.424	0.103487
$302$	−3347.41	−0.637821
$303$	0	0
$304$	631.893	0.119216
$305$	0	0
$306$	0	0
$307$	−8464.71	−1.57364	−0.786818	−	0.617185i	$-0.788273\pi$
$307$	−8464.71	−1.57364	−0.786818	+	0.617185i	$0.788273\pi$
$308$	−775.458	−0.143460
$309$	0	0
$310$	0	0
$311$	−8388.01	−1.52939	−0.764695	−	0.644392i	$-0.777111\pi$
$311$	−8388.01	−1.52939	−0.764695	+	0.644392i	$0.777111\pi$
$312$	0	0
$313$	2384.69	0.430640	0.215320	−	0.976544i	$-0.430920\pi$
$313$	2384.69	0.430640	0.215320	+	0.976544i	$0.430920\pi$
$314$	−1197.36	−0.215194
$315$	0	0
$316$	−5956.58	−1.06039
$317$	−3211.58	−0.569023	−0.284511	−	0.958673i	$-0.591831\pi$
$317$	−3211.58	−0.569023	−0.284511	+	0.958673i	$0.591831\pi$
$318$	0	0
$319$	−3393.01	−0.595523
$320$	0	0
$321$	0	0
$322$	−700.898	−0.121303
$323$	−1294.97	−0.223078
$324$	0	0
$325$	0	0
$326$	−598.185	−0.101627
$327$	0	0
$328$	−8876.35	−1.49425
$329$	3111.83	0.521461
$330$	0	0
$331$	7193.90	1.19460	0.597300	−	0.802018i	$-0.296240\pi$
$331$	7193.90	1.19460	0.597300	+	0.802018i	$0.296240\pi$
$332$	4561.87	0.754111
$333$	0	0
$334$	−3336.06	−0.546530
$335$	0	0
$336$	0	0
$337$	−6913.84	−1.11757	−0.558785	−	0.829313i	$-0.688732\pi$
$337$	−6913.84	−1.11757	−0.558785	+	0.829313i	$0.688732\pi$
$338$	650.948	0.104754
$339$	0	0
$340$	0	0
$341$	−1476.86	−0.234536
$342$	0	0
$343$	−3707.50	−0.583634
$344$	1721.28	0.269783
$345$	0	0
$346$	−616.316	−0.0957611
$347$	4009.84	0.620344	0.310172	−	0.950681i	$-0.399613\pi$
$347$	4009.84	0.620344	0.310172	+	0.950681i	$0.399613\pi$
$348$	0	0
$349$	6884.76	1.05597	0.527984	−	0.849254i	$-0.322948\pi$
$349$	6884.76	1.05597	0.527984	+	0.849254i	$0.322948\pi$
$350$	0	0
$351$	0	0
$352$	−3838.75	−0.581267
$353$	−7783.83	−1.17363	−0.586815	−	0.809721i	$-0.699618\pi$
$353$	−7783.83	−1.17363	−0.586815	+	0.809721i	$0.699618\pi$
$354$	0	0
$355$	0	0
$356$	−1759.96	−0.262016
$357$	0	0
$358$	−4867.68	−0.718617
$359$	8188.19	1.20378	0.601889	−	0.798580i	$-0.294415\pi$
$359$	8188.19	1.20378	0.601889	+	0.798580i	$0.294415\pi$
$360$	0	0
$361$	−6380.27	−0.930204
$362$	5469.62	0.794135
$363$	0	0
$364$	1494.02	0.215131
$365$	0	0
$366$	0	0
$367$	−3688.79	−0.524668	−0.262334	−	0.964977i	$-0.584492\pi$
$367$	−3688.79	−0.524668	−0.262334	+	0.964977i	$0.584492\pi$
$368$	2853.17	0.404162
$369$	0	0
$370$	0	0
$371$	−2890.87	−0.404546
$372$	0	0
$373$	4127.16	0.572912	0.286456	−	0.958093i	$-0.407523\pi$
$373$	4127.16	0.572912	0.286456	+	0.958093i	$0.407523\pi$
$374$	1573.86	0.217600
$375$	0	0
$376$	9911.36	1.35941
$377$	6537.05	0.893038
$378$	0	0
$379$	4145.52	0.561850	0.280925	−	0.959730i	$-0.409359\pi$
$379$	4145.52	0.561850	0.280925	+	0.959730i	$0.409359\pi$
$380$	0	0
$381$	0	0
$382$	410.825	0.0550253
$383$	−9983.95	−1.33200	−0.666000	−	0.745952i	$-0.731995\pi$
$383$	−9983.95	−1.33200	−0.666000	+	0.745952i	$0.731995\pi$
$384$	0	0
$385$	0	0
$386$	−1181.93	−0.155852
$387$	0	0
$388$	−9331.53	−1.22097
$389$	−7868.51	−1.02558	−0.512788	−	0.858515i	$-0.671387\pi$
$389$	−7868.51	−1.02558	−0.512788	+	0.858515i	$0.671387\pi$
$390$	0	0
$391$	−5847.15	−0.756274
$392$	−5613.97	−0.723337
$393$	0	0
$394$	3450.32	0.441180
$395$	0	0
$396$	0	0
$397$	8137.21	1.02870	0.514351	−	0.857580i	$-0.328033\pi$
$397$	8137.21	1.02870	0.514351	+	0.857580i	$0.328033\pi$
$398$	1914.16	0.241075
$399$	0	0
$400$	0	0
$401$	−4204.86	−0.523643	−0.261822	−	0.965116i	$-0.584323\pi$
$401$	−4204.86	−0.523643	−0.261822	+	0.965116i	$0.584323\pi$
$402$	0	0
$403$	2845.36	0.351706
$404$	−4351.34	−0.535860
$405$	0	0
$406$	1132.59	0.138447
$407$	5007.76	0.609891
$408$	0	0
$409$	−8318.77	−1.00571	−0.502857	−	0.864370i	$-0.667718\pi$
$409$	−8318.77	−1.00571	−0.502857	+	0.864370i	$0.667718\pi$
$410$	0	0
$411$	0	0
$412$	−6881.54	−0.822886
$413$	4202.24	0.500675
$414$	0	0
$415$	0	0
$416$	7395.82	0.871659
$417$	0	0
$418$	−581.836	−0.0680826
$419$	6938.90	0.809040	0.404520	−	0.914529i	$-0.367439\pi$
$419$	6938.90	0.809040	0.404520	+	0.914529i	$0.367439\pi$
$420$	0	0
$421$	9856.88	1.14108	0.570541	−	0.821269i	$-0.306734\pi$
$421$	9856.88	1.14108	0.570541	+	0.821269i	$0.306734\pi$
$422$	188.628	0.0217590
$423$	0	0
$424$	−9207.60	−1.05462
$425$	0	0
$426$	0	0
$427$	2195.29	0.248800
$428$	1348.47	0.152291
$429$	0	0
$430$	0	0
$431$	3013.40	0.336775	0.168388	−	0.985721i	$-0.446144\pi$
$431$	3013.40	0.336775	0.168388	+	0.985721i	$0.446144\pi$
$432$	0	0
$433$	3902.19	0.433088	0.216544	−	0.976273i	$-0.430521\pi$
$433$	3902.19	0.433088	0.216544	+	0.976273i	$0.430521\pi$
$434$	492.980	0.0545249
$435$	0	0
$436$	2928.54	0.321678
$437$	2161.61	0.236622
$438$	0	0
$439$	3169.12	0.344542	0.172271	−	0.985050i	$-0.444890\pi$
$439$	3169.12	0.344542	0.172271	+	0.985050i	$0.444890\pi$
$440$	0	0
$441$	0	0
$442$	−3032.24	−0.326310
$443$	−10732.6	−1.15107	−0.575534	−	0.817778i	$-0.695206\pi$
$443$	−10732.6	−1.15107	−0.575534	+	0.817778i	$0.695206\pi$
$444$	0	0
$445$	0	0
$446$	−3220.36	−0.341903
$447$	0	0
$448$	−28.6800	−0.00302456
$449$	−294.508	−0.0309547	−0.0154774	−	0.999880i	$-0.504927\pi$
$449$	−294.508	−0.0309547	−0.0154774	+	0.999880i	$0.504927\pi$
$450$	0	0
$451$	−10445.9	−1.09064
$452$	−7276.40	−0.757197
$453$	0	0
$454$	−3117.46	−0.322268
$455$	0	0
$456$	0	0
$457$	5419.66	0.554751	0.277376	−	0.960762i	$-0.410535\pi$
$457$	5419.66	0.554751	0.277376	+	0.960762i	$0.410535\pi$
$458$	−1752.34	−0.178781
$459$	0	0
$460$	0	0
$461$	1002.66	0.101299	0.0506493	−	0.998717i	$-0.483871\pi$
$461$	1002.66	0.101299	0.0506493	+	0.998717i	$0.483871\pi$
$462$	0	0
$463$	13615.3	1.36665	0.683323	−	0.730116i	$-0.260534\pi$
$463$	13615.3	1.36665	0.683323	+	0.730116i	$0.260534\pi$
$464$	−4610.49	−0.461285
$465$	0	0
$466$	−4077.42	−0.405328
$467$	1335.52	0.132336	0.0661678	−	0.997809i	$-0.478923\pi$
$467$	1335.52	0.132336	0.0661678	+	0.997809i	$0.478923\pi$
$468$	0	0
$469$	6013.04	0.592017
$470$	0	0
$471$	0	0
$472$	13384.4	1.30522
$473$	2025.65	0.196912
$474$	0	0
$475$	0	0
$476$	2159.42	0.207935
$477$	0	0
$478$	2077.96	0.198836
$479$	13852.1	1.32134	0.660668	−	0.750678i	$-0.270273\pi$
$479$	13852.1	1.32134	0.660668	+	0.750678i	$0.270273\pi$
$480$	0	0
$481$	−9648.07	−0.914583
$482$	−7464.77	−0.705417
$483$	0	0
$484$	5657.77	0.531346
$485$	0	0
$486$	0	0
$487$	6113.26	0.568826	0.284413	−	0.958702i	$-0.408201\pi$
$487$	6113.26	0.568826	0.284413	+	0.958702i	$0.408201\pi$
$488$	6992.12	0.648603
$489$	0	0
$490$	0	0
$491$	16356.5	1.50338	0.751691	−	0.659515i	$-0.229238\pi$
$491$	16356.5	1.50338	0.751691	+	0.659515i	$0.229238\pi$
$492$	0	0
$493$	9448.51	0.863164
$494$	1120.98	0.102096
$495$	0	0
$496$	−2006.79	−0.181669
$497$	−2886.17	−0.260487
$498$	0	0
$499$	−9262.82	−0.830983	−0.415492	−	0.909597i	$-0.636390\pi$
$499$	−9262.82	−0.830983	−0.415492	+	0.909597i	$0.636390\pi$
$500$	0	0
$501$	0	0
$502$	3817.90	0.339445
$503$	12498.5	1.10792	0.553959	−	0.832544i	$-0.313116\pi$
$503$	12498.5	1.10792	0.553959	+	0.832544i	$0.313116\pi$
$504$	0	0
$505$	0	0
$506$	−2627.15	−0.230812
$507$	0	0
$508$	−8220.93	−0.718002
$509$	−7370.25	−0.641809	−0.320904	−	0.947112i	$-0.603987\pi$
$509$	−7370.25	−0.641809	−0.320904	+	0.947112i	$0.603987\pi$
$510$	0	0
$511$	5187.78	0.449107
$512$	−9388.79	−0.810410
$513$	0	0
$514$	6955.49	0.596875
$515$	0	0
$516$	0	0
$517$	11664.0	0.992224
$518$	−1671.60	−0.141787
$519$	0	0
$520$	0	0
$521$	5426.50	0.456313	0.228157	−	0.973624i	$-0.426730\pi$
$521$	5426.50	0.456313	0.228157	+	0.973624i	$0.426730\pi$
$522$	0	0
$523$	19681.7	1.64555	0.822773	−	0.568371i	$-0.192426\pi$
$523$	19681.7	1.64555	0.822773	+	0.568371i	$0.192426\pi$
$524$	−8739.97	−0.728640
$525$	0	0
$526$	−1196.71	−0.0991996
$527$	4112.62	0.339941
$528$	0	0
$529$	−2406.73	−0.197808
$530$	0	0
$531$	0	0
$532$	−798.308	−0.0650583
$533$	20125.4	1.63551
$534$	0	0
$535$	0	0
$536$	19151.9	1.54335
$537$	0	0
$538$	5931.09	0.475293
$539$	−6606.67	−0.527958
$540$	0	0
$541$	16726.1	1.32923	0.664613	−	0.747188i	$-0.268596\pi$
$541$	16726.1	1.32923	0.664613	+	0.747188i	$0.268596\pi$
$542$	4772.96	0.378259
$543$	0	0
$544$	10689.8	0.842500
$545$	0	0
$546$	0	0
$547$	19292.6	1.50803	0.754014	−	0.656859i	$-0.228115\pi$
$547$	19292.6	1.50803	0.754014	+	0.656859i	$0.228115\pi$
$548$	6154.94	0.479792
$549$	0	0
$550$	0	0
$551$	−3492.99	−0.270066
$552$	0	0
$553$	5249.08	0.403641
$554$	1976.42	0.151570
$555$	0	0
$556$	−175.840	−0.0134123
$557$	−7161.62	−0.544789	−0.272395	−	0.962186i	$-0.587816\pi$
$557$	−7161.62	−0.544789	−0.272395	+	0.962186i	$0.587816\pi$
$558$	0	0
$559$	−3902.66	−0.295286
$560$	0	0
$561$	0	0
$562$	−9854.47	−0.739655
$563$	−17096.8	−1.27983	−0.639914	−	0.768446i	$-0.721030\pi$
$563$	−17096.8	−1.27983	−0.639914	+	0.768446i	$0.721030\pi$
$564$	0	0
$565$	0	0
$566$	1132.95	0.0841369
$567$	0	0
$568$	−9192.60	−0.679072
$569$	11508.9	0.847942	0.423971	−	0.905676i	$-0.360636\pi$
$569$	11508.9	0.847942	0.423971	+	0.905676i	$0.360636\pi$
$570$	0	0
$571$	10695.3	0.783863	0.391931	−	0.919994i	$-0.371807\pi$
$571$	10695.3	0.783863	0.391931	+	0.919994i	$0.371807\pi$
$572$	5599.96	0.409347
$573$	0	0
$574$	3486.87	0.253553
$575$	0	0
$576$	0	0
$577$	23047.9	1.66291	0.831453	−	0.555595i	$-0.187509\pi$
$577$	23047.9	1.66291	0.831453	+	0.555595i	$0.187509\pi$
$578$	1764.29	0.126963
$579$	0	0
$580$	0	0
$581$	−4020.03	−0.287055
$582$	0	0
$583$	−10835.8	−0.769762
$584$	16523.4	1.17079
$585$	0	0
$586$	−9623.14	−0.678376
$587$	−12449.5	−0.875378	−0.437689	−	0.899126i	$-0.644203\pi$
$587$	−12449.5	−0.875378	−0.437689	+	0.899126i	$0.644203\pi$
$588$	0	0
$589$	−1520.38	−0.106360
$590$	0	0
$591$	0	0
$592$	6804.65	0.472414
$593$	−18331.7	−1.26946	−0.634732	−	0.772732i	$-0.718890\pi$
$593$	−18331.7	−1.26946	−0.634732	+	0.772732i	$0.718890\pi$
$594$	0	0
$595$	0	0
$596$	1392.50	0.0957032
$597$	0	0
$598$	5061.53	0.346122
$599$	−23905.9	−1.63067	−0.815333	−	0.578992i	$-0.803446\pi$
$599$	−23905.9	−1.63067	−0.815333	+	0.578992i	$0.803446\pi$
$600$	0	0
$601$	−16524.6	−1.12155	−0.560777	−	0.827967i	$-0.689498\pi$
$601$	−16524.6	−1.12155	−0.560777	+	0.827967i	$0.689498\pi$
$602$	−676.165	−0.0457782
$603$	0	0
$604$	−17215.1	−1.15972
$605$	0	0
$606$	0	0
$607$	2610.92	0.174587	0.0872933	−	0.996183i	$-0.472178\pi$
$607$	2610.92	0.174587	0.0872933	+	0.996183i	$0.472178\pi$
$608$	−3951.86	−0.263600
$609$	0	0
$610$	0	0
$611$	−22472.0	−1.48792
$612$	0	0
$613$	5036.27	0.331832	0.165916	−	0.986140i	$-0.446942\pi$
$613$	5036.27	0.331832	0.165916	+	0.986140i	$0.446942\pi$
$614$	10590.8	0.696110
$615$	0	0
$616$	2176.52	0.142361
$617$	15805.7	1.03130	0.515650	−	0.856800i	$-0.327551\pi$
$617$	15805.7	1.03130	0.515650	+	0.856800i	$0.327551\pi$
$618$	0	0
$619$	−5929.52	−0.385020	−0.192510	−	0.981295i	$-0.561663\pi$
$619$	−5929.52	−0.385020	−0.192510	+	0.981295i	$0.561663\pi$
$620$	0	0
$621$	0	0
$622$	10494.9	0.676538
$623$	1550.92	0.0997370
$624$	0	0
$625$	0	0
$626$	−2983.66	−0.190497
$627$	0	0
$628$	−6157.79	−0.391278
$629$	−13945.1	−0.883988
$630$	0	0
$631$	12427.8	0.784060	0.392030	−	0.919952i	$-0.371773\pi$
$631$	12427.8	0.784060	0.392030	+	0.919952i	$0.371773\pi$
$632$	16718.6	1.05226
$633$	0	0
$634$	4018.25	0.251712
$635$	0	0
$636$	0	0
$637$	12728.6	0.791718
$638$	4245.25	0.263434
$639$	0	0
$640$	0	0
$641$	−28418.8	−1.75113	−0.875566	−	0.483098i	$-0.839511\pi$
$641$	−28418.8	−1.75113	−0.875566	+	0.483098i	$0.839511\pi$
$642$	0	0
$643$	−13882.4	−0.851431	−0.425715	−	0.904857i	$-0.639978\pi$
$643$	−13882.4	−0.851431	−0.425715	+	0.904857i	$0.639978\pi$
$644$	−3604.58	−0.220559
$645$	0	0
$646$	1620.24	0.0986802
$647$	13037.5	0.792207	0.396104	−	0.918206i	$-0.370362\pi$
$647$	13037.5	0.792207	0.396104	+	0.918206i	$0.370362\pi$
$648$	0	0
$649$	15751.1	0.952672
$650$	0	0
$651$	0	0
$652$	−3076.35	−0.184784
$653$	−4574.74	−0.274155	−0.137078	−	0.990560i	$-0.543771\pi$
$653$	−4574.74	−0.274155	−0.137078	+	0.990560i	$0.543771\pi$
$654$	0	0
$655$	0	0
$656$	−14194.1	−0.844798
$657$	0	0
$658$	−3893.45	−0.230672
$659$	−7182.71	−0.424581	−0.212290	−	0.977207i	$-0.568092\pi$
$659$	−7182.71	−0.424581	−0.212290	+	0.977207i	$0.568092\pi$
$660$	0	0
$661$	1611.10	0.0948026	0.0474013	−	0.998876i	$-0.484906\pi$
$661$	1611.10	0.0948026	0.0474013	+	0.998876i	$0.484906\pi$
$662$	−9000.84	−0.528441
$663$	0	0
$664$	−12804.0	−0.748332
$665$	0	0
$666$	0	0
$667$	−15771.8	−0.915571
$668$	−17156.7	−0.993731
$669$	0	0
$670$	0	0
$671$	8228.52	0.473410
$672$	0	0
$673$	17721.1	1.01501	0.507503	−	0.861650i	$-0.330569\pi$
$673$	17721.1	1.01501	0.507503	+	0.861650i	$0.330569\pi$
$674$	8650.44	0.494365
$675$	0	0
$676$	3347.70	0.190470
$677$	−12774.3	−0.725196	−0.362598	−	0.931946i	$-0.618110\pi$
$677$	−12774.3	−0.725196	−0.362598	+	0.931946i	$0.618110\pi$
$678$	0	0
$679$	8223.17	0.464766
$680$	0	0
$681$	0	0
$682$	1847.82	0.103749
$683$	7065.27	0.395820	0.197910	−	0.980220i	$-0.436585\pi$
$683$	7065.27	0.395820	0.197910	+	0.980220i	$0.436585\pi$
$684$	0	0
$685$	0	0
$686$	4638.74	0.258175
$687$	0	0
$688$	2752.49	0.152526
$689$	20876.4	1.15432
$690$	0	0
$691$	−26270.4	−1.44627	−0.723136	−	0.690706i	$-0.757300\pi$
$691$	−26270.4	−1.44627	−0.723136	+	0.690706i	$0.757300\pi$
$692$	−3169.59	−0.174118
$693$	0	0
$694$	−5017.01	−0.274414
$695$	0	0
$696$	0	0
$697$	29088.8	1.58080
$698$	−8614.05	−0.467116
$699$	0	0
$700$	0	0
$701$	32274.9	1.73895	0.869476	−	0.493975i	$-0.164456\pi$
$701$	32274.9	1.73895	0.869476	+	0.493975i	$0.164456\pi$
$702$	0	0
$703$	5155.32	0.276581
$704$	−107.500	−0.00575507
$705$	0	0
$706$	9738.94	0.519164
$707$	3834.51	0.203977
$708$	0	0
$709$	−13697.1	−0.725539	−0.362770	−	0.931879i	$-0.618169\pi$
$709$	−13697.1	−0.725539	−0.362770	+	0.931879i	$0.618169\pi$
$710$	0	0
$711$	0	0
$712$	4939.76	0.260007
$713$	−6864.94	−0.360580
$714$	0	0
$715$	0	0
$716$	−25033.5	−1.30663
$717$	0	0
$718$	−10244.9	−0.532501
$719$	6826.36	0.354076	0.177038	−	0.984204i	$-0.443349\pi$
$719$	6826.36	0.354076	0.177038	+	0.984204i	$0.443349\pi$
$720$	0	0
$721$	6064.18	0.313234
$722$	7982.84	0.411483
$723$	0	0
$724$	28129.2	1.44394
$725$	0	0
$726$	0	0
$727$	−22413.2	−1.14341	−0.571706	−	0.820458i	$-0.693718\pi$
$727$	−22413.2	−1.14341	−0.571706	+	0.820458i	$0.693718\pi$
$728$	−4193.33	−0.213482
$729$	0	0
$730$	0	0
$731$	−5640.83	−0.285408
$732$	0	0
$733$	167.208	0.00842562	0.00421281	−	0.999991i	$-0.498659\pi$
$733$	167.208	0.00842562	0.00421281	+	0.999991i	$0.498659\pi$
$734$	4615.33	0.232091
$735$	0	0
$736$	−17843.7	−0.893653
$737$	22538.4	1.12648
$738$	0	0
$739$	27695.8	1.37863	0.689313	−	0.724463i	$-0.257912\pi$
$739$	27695.8	1.37863	0.689313	+	0.724463i	$0.257912\pi$
$740$	0	0
$741$	0	0
$742$	3616.99	0.178954
$743$	35379.4	1.74689	0.873447	−	0.486919i	$-0.161879\pi$
$743$	35379.4	1.74689	0.873447	+	0.486919i	$0.161879\pi$
$744$	0	0
$745$	0	0
$746$	−5163.81	−0.253432
$747$	0	0
$748$	8094.07	0.395653
$749$	−1188.30	−0.0579700
$750$	0	0
$751$	−5408.34	−0.262787	−0.131394	−	0.991330i	$-0.541945\pi$
$751$	−5408.34	−0.262787	−0.131394	+	0.991330i	$0.541945\pi$
$752$	15849.2	0.768566
$753$	0	0
$754$	−8179.00	−0.395042
$755$	0	0
$756$	0	0
$757$	−701.285	−0.0336706	−0.0168353	−	0.999858i	$-0.505359\pi$
$757$	−701.285	−0.0336706	−0.0168353	+	0.999858i	$0.505359\pi$
$758$	−5186.78	−0.248539
$759$	0	0
$760$	0	0
$761$	−23271.0	−1.10850	−0.554252	−	0.832349i	$-0.686996\pi$
$761$	−23271.0	−1.10850	−0.554252	+	0.832349i	$0.686996\pi$
$762$	0	0
$763$	−2580.70	−0.122448
$764$	2112.79	0.100050
$765$	0	0
$766$	12491.7	0.589221
$767$	−30346.4	−1.42861
$768$	0	0
$769$	−10456.3	−0.490332	−0.245166	−	0.969481i	$-0.578842\pi$
$769$	−10456.3	−0.490332	−0.245166	+	0.969481i	$0.578842\pi$
$770$	0	0
$771$	0	0
$772$	−6078.44	−0.283378
$773$	−18324.8	−0.852650	−0.426325	−	0.904570i	$-0.640192\pi$
$773$	−18324.8	−0.852650	−0.426325	+	0.904570i	$0.640192\pi$
$774$	0	0
$775$	0	0
$776$	26191.3	1.21161
$777$	0	0
$778$	9844.89	0.453672
$779$	−10753.7	−0.494598
$780$	0	0
$781$	−10818.1	−0.495649
$782$	7315.82	0.334544
$783$	0	0
$784$	−8977.28	−0.408950
$785$	0	0
$786$	0	0
$787$	−14839.7	−0.672147	−0.336073	−	0.941836i	$-0.609099\pi$
$787$	−14839.7	−0.672147	−0.336073	+	0.941836i	$0.609099\pi$
$788$	17744.3	0.802178
$789$	0	0
$790$	0	0
$791$	6412.14	0.288229
$792$	0	0
$793$	−15853.3	−0.709919
$794$	−10181.1	−0.455055
$795$	0	0
$796$	9844.13	0.438337
$797$	−38086.1	−1.69269	−0.846347	−	0.532631i	$-0.821203\pi$
$797$	−38086.1	−1.69269	−0.846347	+	0.532631i	$0.821203\pi$
$798$	0	0
$799$	−32480.6	−1.43815
$800$	0	0
$801$	0	0
$802$	5261.03	0.231638
$803$	19445.2	0.854551
$804$	0	0
$805$	0	0
$806$	−3560.05	−0.155580
$807$	0	0
$808$	12213.1	0.531753
$809$	−37170.3	−1.61537	−0.807686	−	0.589612i	$-0.799281\pi$
$809$	−37170.3	−1.61537	−0.807686	+	0.589612i	$0.799281\pi$
$810$	0	0
$811$	37402.9	1.61947	0.809736	−	0.586794i	$-0.199610\pi$
$811$	37402.9	1.61947	0.809736	+	0.586794i	$0.199610\pi$
$812$	5824.70	0.251733
$813$	0	0
$814$	−6265.60	−0.269790
$815$	0	0
$816$	0	0
$817$	2085.34	0.0892983
$818$	10408.3	0.444885
$819$	0	0
$820$	0	0
$821$	15002.4	0.637743	0.318872	−	0.947798i	$-0.396696\pi$
$821$	15002.4	0.637743	0.318872	+	0.947798i	$0.396696\pi$
$822$	0	0
$823$	11648.5	0.493366	0.246683	−	0.969096i	$-0.420659\pi$
$823$	11648.5	0.493366	0.246683	+	0.969096i	$0.420659\pi$
$824$	19314.8	0.816579
$825$	0	0
$826$	−5257.74	−0.221477
$827$	−46483.9	−1.95454	−0.977269	−	0.212003i	$-0.932001\pi$
$827$	−46483.9	−1.95454	−0.977269	+	0.212003i	$0.932001\pi$
$828$	0	0
$829$	1565.64	0.0655933	0.0327967	−	0.999462i	$-0.489559\pi$
$829$	1565.64	0.0655933	0.0327967	+	0.999462i	$0.489559\pi$
$830$	0	0
$831$	0	0
$832$	207.113	0.00863021
$833$	18397.6	0.765233
$834$	0	0
$835$	0	0
$836$	−2992.27	−0.123791
$837$	0	0
$838$	−8681.79	−0.357885
$839$	−32926.2	−1.35487	−0.677436	−	0.735582i	$-0.736909\pi$
$839$	−32926.2	−1.35487	−0.677436	+	0.735582i	$0.736909\pi$
$840$	0	0
$841$	1096.90	0.0449754
$842$	−12332.7	−0.504766
$843$	0	0
$844$	970.078	0.0395633
$845$	0	0
$846$	0	0
$847$	−4985.77	−0.202259
$848$	−14723.8	−0.596248
$849$	0	0
$850$	0	0
$851$	23277.7	0.937660
$852$	0	0
$853$	14963.0	0.600613	0.300306	−	0.953843i	$-0.402911\pi$
$853$	14963.0	0.600613	0.300306	+	0.953843i	$0.402911\pi$
$854$	−2746.70	−0.110059
$855$	0	0
$856$	−3784.81	−0.151124
$857$	21868.7	0.871668	0.435834	−	0.900027i	$-0.356454\pi$
$857$	21868.7	0.871668	0.435834	+	0.900027i	$0.356454\pi$
$858$	0	0
$859$	13542.5	0.537909	0.268954	−	0.963153i	$-0.413322\pi$
$859$	13542.5	0.537909	0.268954	+	0.963153i	$0.413322\pi$
$860$	0	0
$861$	0	0
$862$	−3770.29	−0.148975
$863$	−2080.25	−0.0820539	−0.0410269	−	0.999158i	$-0.513063\pi$
$863$	−2080.25	−0.0820539	−0.0410269	+	0.999158i	$0.513063\pi$
$864$	0	0
$865$	0	0
$866$	−4882.33	−0.191580
$867$	0	0
$868$	2535.30	0.0991401
$869$	19674.9	0.768039
$870$	0	0
$871$	−43423.1	−1.68925
$872$	−8219.67	−0.319212
$873$	0	0
$874$	−2704.56	−0.104672
$875$	0	0
$876$	0	0
$877$	−28076.9	−1.08106	−0.540530	−	0.841325i	$-0.681776\pi$
$877$	−28076.9	−1.08106	−0.540530	+	0.841325i	$0.681776\pi$
$878$	−3965.13	−0.152411
$879$	0	0
$880$	0	0
$881$	−2105.32	−0.0805110	−0.0402555	−	0.999189i	$-0.512817\pi$
$881$	−2105.32	−0.0805110	−0.0402555	+	0.999189i	$0.512817\pi$
$882$	0	0
$883$	−23829.7	−0.908190	−0.454095	−	0.890953i	$-0.650037\pi$
$883$	−23829.7	−0.908190	−0.454095	+	0.890953i	$0.650037\pi$
$884$	−15594.2	−0.593315
$885$	0	0
$886$	13428.4	0.509184
$887$	12634.7	0.478277	0.239139	−	0.970985i	$-0.423135\pi$
$887$	12634.7	0.478277	0.239139	+	0.970985i	$0.423135\pi$
$888$	0	0
$889$	7244.48	0.273310
$890$	0	0
$891$	0	0
$892$	−16561.7	−0.621667
$893$	12007.6	0.449967
$894$	0	0
$895$	0	0
$896$	8229.01	0.306821
$897$	0	0
$898$	368.481	0.0136931
$899$	11093.2	0.411544
$900$	0	0
$901$	30174.3	1.11571
$902$	13069.7	0.482454
$903$	0	0
$904$	20423.0	0.751393
$905$	0	0
$906$	0	0
$907$	−31942.7	−1.16939	−0.584697	−	0.811252i	$-0.698787\pi$
$907$	−31942.7	−1.16939	−0.584697	+	0.811252i	$0.698787\pi$
$908$	−16032.5	−0.585965
$909$	0	0
$910$	0	0
$911$	22605.6	0.822128	0.411064	−	0.911607i	$-0.365157\pi$
$911$	22605.6	0.822128	0.411064	+	0.911607i	$0.365157\pi$
$912$	0	0
$913$	−15068.1	−0.546201
$914$	−6780.96	−0.245398
$915$	0	0
$916$	−9011.95	−0.325069
$917$	7701.87	0.277359
$918$	0	0
$919$	45168.0	1.62128	0.810638	−	0.585547i	$-0.199120\pi$
$919$	45168.0	1.62128	0.810638	+	0.585547i	$0.199120\pi$
$920$	0	0
$921$	0	0
$922$	−1254.51	−0.0448102
$923$	20842.4	0.743268
$924$	0	0
$925$	0	0
$926$	−17035.2	−0.604547
$927$	0	0
$928$	28834.0	1.01996
$929$	32891.6	1.16161	0.580806	−	0.814042i	$-0.302738\pi$
$929$	32891.6	1.16161	0.580806	+	0.814042i	$0.302738\pi$
$930$	0	0
$931$	−6801.34	−0.239425
$932$	−20969.3	−0.736989
$933$	0	0
$934$	−1670.98	−0.0585397
$935$	0	0
$936$	0	0
$937$	−34702.8	−1.20992	−0.604958	−	0.796257i	$-0.706810\pi$
$937$	−34702.8	−1.20992	−0.604958	+	0.796257i	$0.706810\pi$
$938$	−7523.37	−0.261884
$939$	0	0
$940$	0	0
$941$	−27172.9	−0.941351	−0.470675	−	0.882306i	$-0.655990\pi$
$941$	−27172.9	−0.941351	−0.470675	+	0.882306i	$0.655990\pi$
$942$	0	0
$943$	−48556.0	−1.67678
$944$	21402.9	0.737929
$945$	0	0
$946$	−2534.44	−0.0871056
$947$	39846.1	1.36729	0.683645	−	0.729815i	$-0.260394\pi$
$947$	39846.1	1.36729	0.683645	+	0.729815i	$0.260394\pi$
$948$	0	0
$949$	−37463.5	−1.28147
$950$	0	0
$951$	0	0
$952$	−6060.95	−0.206341
$953$	−45983.5	−1.56301	−0.781507	−	0.623896i	$-0.785549\pi$
$953$	−45983.5	−1.56301	−0.781507	+	0.623896i	$0.785549\pi$
$954$	0	0
$955$	0	0
$956$	10686.5	0.361534
$957$	0	0
$958$	−17331.5	−0.584503
$959$	−5423.88	−0.182634
$960$	0	0
$961$	−24962.5	−0.837921
$962$	12071.4	0.404573
$963$	0	0
$964$	−38389.8	−1.28263
$965$	0	0
$966$	0	0
$967$	−52188.1	−1.73553	−0.867765	−	0.496976i	$-0.834444\pi$
$967$	−52188.1	−1.73553	−0.867765	+	0.496976i	$0.834444\pi$
$968$	−15880.0	−0.527274
$969$	0	0
$970$	0	0
$971$	2777.37	0.0917921	0.0458961	−	0.998946i	$-0.485386\pi$
$971$	2777.37	0.0917921	0.0458961	+	0.998946i	$0.485386\pi$
$972$	0	0
$973$	154.954	0.00510545
$974$	−7648.77	−0.251625
$975$	0	0
$976$	11181.1	0.366698
$977$	33242.3	1.08855	0.544275	−	0.838907i	$-0.316805\pi$
$977$	33242.3	1.08855	0.544275	+	0.838907i	$0.316805\pi$
$978$	0	0
$979$	5813.24	0.189777
$980$	0	0
$981$	0	0
$982$	−20464.9	−0.665033
$983$	−18130.3	−0.588268	−0.294134	−	0.955764i	$-0.595031\pi$
$983$	−18130.3	−0.588268	−0.294134	+	0.955764i	$0.595031\pi$
$984$	0	0
$985$	0	0
$986$	−11821.8	−0.381827
$987$	0	0
$988$	5764.97	0.185636
$989$	9415.87	0.302737
$990$	0	0
$991$	30597.2	0.980778	0.490389	−	0.871504i	$-0.336855\pi$
$991$	30597.2	0.980778	0.490389	+	0.871504i	$0.336855\pi$
$992$	12550.5	0.401692
$993$	0	0
$994$	3611.10	0.115229
$995$	0	0
$996$	0	0
$997$	23839.4	0.757273	0.378637	−	0.925545i	$-0.376393\pi$
$997$	23839.4	0.757273	0.378637	+	0.925545i	$0.376393\pi$
$998$	11589.4	0.367592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.bc.1.3		6
3.2	odd	2	inner	675.4.a.bc.1.4		6
5.2	odd	4		135.4.b.c.109.6	yes	12
5.3	odd	4		135.4.b.c.109.8	yes	12
5.4	even	2		675.4.a.bb.1.4		6
15.2	even	4		135.4.b.c.109.7	yes	12
15.8	even	4		135.4.b.c.109.5	✓	12
15.14	odd	2		675.4.a.bb.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.c.109.5	✓	12	15.8	even	4
135.4.b.c.109.6	yes	12	5.2	odd	4
135.4.b.c.109.7	yes	12	15.2	even	4
135.4.b.c.109.8	yes	12	5.3	odd	4
675.4.a.bb.1.3		6	15.14	odd	2
675.4.a.bb.1.4		6	5.4	even	2
675.4.a.bc.1.3		6	1.1	even	1	trivial
675.4.a.bc.1.4		6	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25118 q^{2} -6.43456 q^{4} +5.67029 q^{7} +18.0602 q^{8} +21.2537 q^{11} -40.9479 q^{13} -7.09453 q^{14} +28.8800 q^{16} -59.1853 q^{17} +21.8800 q^{19} -26.5922 q^{22} +98.7941 q^{23} +51.2331 q^{26} -36.4858 q^{28} -159.643 q^{29} -69.4873 q^{31} -180.615 q^{32} +74.0512 q^{34} +235.618 q^{37} -27.3757 q^{38} -491.487 q^{41} +95.3080 q^{43} -136.758 q^{44} -123.609 q^{46} +548.796 q^{47} -310.848 q^{49} +263.482 q^{52} -509.829 q^{53} +102.406 q^{56} +199.742 q^{58} +741.098 q^{59} +387.157 q^{61} +86.9409 q^{62} -5.05795 q^{64} +1060.45 q^{67} +380.831 q^{68} -508.998 q^{71} +914.906 q^{73} -294.800 q^{74} -140.788 q^{76} +120.515 q^{77} +925.717 q^{79} +614.937 q^{82} -708.964 q^{83} -119.247 q^{86} +383.846 q^{88} +273.516 q^{89} -232.186 q^{91} -635.696 q^{92} -686.641 q^{94} +1450.22 q^{97} +388.926 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64}+ \cdots + 4104 q^{97}+O(q^{100}) \) 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

Introduction

L-functions

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