LMFDB - Embedded newform 2312.4.a.s.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$136.412415933$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45719 q^{3} +4.99476 q^{5} -13.8475 q^{7} -20.9622 q^{9} +O(q^{10})$	$q-2.45719 q^{3} +4.99476 q^{5} -13.8475 q^{7} -20.9622 q^{9} -39.1259 q^{11} -68.8952 q^{13} -12.2731 q^{15} -73.2962 q^{19} +34.0260 q^{21} -110.765 q^{23} -100.052 q^{25} +117.852 q^{27} -70.1869 q^{29} +13.9203 q^{31} +96.1398 q^{33} -69.1650 q^{35} -201.107 q^{37} +169.289 q^{39} -358.549 q^{41} -548.617 q^{43} -104.701 q^{45} +264.683 q^{47} -151.246 q^{49} +632.190 q^{53} -195.425 q^{55} +180.103 q^{57} +186.457 q^{59} -287.643 q^{61} +290.275 q^{63} -344.115 q^{65} -4.09762 q^{67} +272.170 q^{69} +674.128 q^{71} -454.670 q^{73} +245.848 q^{75} +541.797 q^{77} -442.930 q^{79} +276.394 q^{81} +921.811 q^{83} +172.463 q^{87} -104.536 q^{89} +954.028 q^{91} -34.2048 q^{93} -366.097 q^{95} +836.291 q^{97} +820.166 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 308 q^{9}+O(q^{10}) $ 28 * q + 308 * q^9	$ 28 q + 308 q^{9} + 64 q^{13} + 120 q^{15} + 368 q^{19} - 192 q^{21} + 956 q^{25} - 504 q^{33} + 1664 q^{35} + 1760 q^{43} + 984 q^{47} + 1492 q^{49} - 1088 q^{53} + 424 q^{55} + 2288 q^{59} + 3424 q^{67} - 2920 q^{69} + 816 q^{77} + 2380 q^{81} + 3472 q^{83} + 1560 q^{87} + 2400 q^{89} - 1768 q^{93}+O(q^{100}) $ 28 * q + 308 * q^9 + 64 * q^13 + 120 * q^15 + 368 * q^19 - 192 * q^21 + 956 * q^25 - 504 * q^33 + 1664 * q^35 + 1760 * q^43 + 984 * q^47 + 1492 * q^49 - 1088 * q^53 + 424 * q^55 + 2288 * q^59 + 3424 * q^67 - 2920 * q^69 + 816 * q^77 + 2380 * q^81 + 3472 * q^83 + 1560 * q^87 + 2400 * q^89 - 1768 * q^93

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.45719	−0.472886	−0.236443	−	0.971645i	$-0.575982\pi$
$3$	−2.45719	−0.472886	−0.236443	+	0.971645i	$0.575982\pi$
$4$	0	0
$5$	4.99476	0.446745	0.223372	−	0.974733i	$-0.428293\pi$
$5$	4.99476	0.446745	0.223372	+	0.974733i	$0.428293\pi$
$6$	0	0
$7$	−13.8475	−0.747696	−0.373848	−	0.927490i	$-0.621962\pi$
$7$	−13.8475	−0.747696	−0.373848	+	0.927490i	$0.621962\pi$
$8$	0	0
$9$	−20.9622	−0.776378
$10$	0	0
$11$	−39.1259	−1.07245	−0.536223	−	0.844076i	$-0.680149\pi$
$11$	−39.1259	−1.07245	−0.536223	+	0.844076i	$0.680149\pi$
$12$	0	0
$13$	−68.8952	−1.46985	−0.734927	−	0.678146i	$-0.762784\pi$
$13$	−68.8952	−1.46985	−0.734927	+	0.678146i	$0.762784\pi$
$14$	0	0
$15$	−12.2731	−0.211260
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−73.2962	−0.885016	−0.442508	−	0.896765i	$-0.645911\pi$
$19$	−73.2962	−0.885016	−0.442508	+	0.896765i	$0.645911\pi$
$20$	0	0
$21$	34.0260	0.353575
$22$	0	0
$23$	−110.765	−1.00418	−0.502088	−	0.864816i	$-0.667435\pi$
$23$	−110.765	−1.00418	−0.502088	+	0.864816i	$0.667435\pi$
$24$	0	0
$25$	−100.052	−0.800419
$26$	0	0
$27$	117.852	0.840025
$28$	0	0
$29$	−70.1869	−0.449427	−0.224713	−	0.974425i	$-0.572145\pi$
$29$	−70.1869	−0.449427	−0.224713	+	0.974425i	$0.572145\pi$
$30$	0	0
$31$	13.9203	0.0806504	0.0403252	−	0.999187i	$-0.487161\pi$
$31$	13.9203	0.0806504	0.0403252	+	0.999187i	$0.487161\pi$
$32$	0	0
$33$	96.1398	0.507145
$34$	0	0
$35$	−69.1650	−0.334029
$36$	0	0
$37$	−201.107	−0.893562	−0.446781	−	0.894643i	$-0.647430\pi$
$37$	−201.107	−0.893562	−0.446781	+	0.894643i	$0.647430\pi$
$38$	0	0
$39$	169.289	0.695074
$40$	0	0
$41$	−358.549	−1.36576	−0.682878	−	0.730532i	$-0.739272\pi$
$41$	−358.549	−1.36576	−0.682878	+	0.730532i	$0.739272\pi$
$42$	0	0
$43$	−548.617	−1.94566	−0.972828	−	0.231527i	$-0.925628\pi$
$43$	−548.617	−1.94566	−0.972828	+	0.231527i	$0.925628\pi$
$44$	0	0
$45$	−104.701	−0.346843
$46$	0	0
$47$	264.683	0.821448	0.410724	−	0.911760i	$-0.365276\pi$
$47$	264.683	0.821448	0.410724	+	0.911760i	$0.365276\pi$
$48$	0	0
$49$	−151.246	−0.440951
$50$	0	0
$51$	0	0
$52$	0	0
$53$	632.190	1.63845	0.819226	−	0.573471i	$-0.194404\pi$
$53$	632.190	1.63845	0.819226	+	0.573471i	$0.194404\pi$
$54$	0	0
$55$	−195.425	−0.479110
$56$	0	0
$57$	180.103	0.418512
$58$	0	0
$59$	186.457	0.411435	0.205717	−	0.978611i	$-0.434047\pi$
$59$	186.457	0.411435	0.205717	+	0.978611i	$0.434047\pi$
$60$	0	0
$61$	−287.643	−0.603753	−0.301876	−	0.953347i	$-0.597613\pi$
$61$	−287.643	−0.603753	−0.301876	+	0.953347i	$0.597613\pi$
$62$	0	0
$63$	290.275	0.580495
$64$	0	0
$65$	−344.115	−0.656650
$66$	0	0
$67$	−4.09762	−0.00747171	−0.00373585	−	0.999993i	$-0.501189\pi$
$67$	−4.09762	−0.00747171	−0.00373585	+	0.999993i	$0.501189\pi$
$68$	0	0
$69$	272.170	0.474861
$70$	0	0
$71$	674.128	1.12682	0.563411	−	0.826177i	$-0.309489\pi$
$71$	674.128	1.12682	0.563411	+	0.826177i	$0.309489\pi$
$72$	0	0
$73$	−454.670	−0.728975	−0.364487	−	0.931208i	$-0.618756\pi$
$73$	−454.670	−0.728975	−0.364487	+	0.931208i	$0.618756\pi$
$74$	0	0
$75$	245.848	0.378507
$76$	0	0
$77$	541.797	0.801863
$78$	0	0
$79$	−442.930	−0.630804	−0.315402	−	0.948958i	$-0.602139\pi$
$79$	−442.930	−0.630804	−0.315402	+	0.948958i	$0.602139\pi$
$80$	0	0
$81$	276.394	0.379142
$82$	0	0
$83$	921.811	1.21906	0.609530	−	0.792763i	$-0.291358\pi$
$83$	921.811	1.21906	0.609530	+	0.792763i	$0.291358\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	172.463	0.212528
$88$	0	0
$89$	−104.536	−0.124503	−0.0622516	−	0.998060i	$-0.519828\pi$
$89$	−104.536	−0.124503	−0.0622516	+	0.998060i	$0.519828\pi$
$90$	0	0
$91$	954.028	1.09900
$92$	0	0
$93$	−34.2048	−0.0381385
$94$	0	0
$95$	−366.097	−0.395377
$96$	0	0
$97$	836.291	0.875386	0.437693	−	0.899124i	$-0.355796\pi$
$97$	836.291	0.875386	0.437693	+	0.899124i	$0.355796\pi$
$98$	0	0
$99$	820.166	0.832624
$100$	0	0
$101$	50.7667	0.0500146	0.0250073	−	0.999687i	$-0.492039\pi$
$101$	50.7667	0.0500146	0.0250073	+	0.999687i	$0.492039\pi$
$102$	0	0
$103$	−248.626	−0.237843	−0.118922	−	0.992904i	$-0.537944\pi$
$103$	−248.626	−0.237843	−0.118922	+	0.992904i	$0.537944\pi$
$104$	0	0
$105$	169.952	0.157958
$106$	0	0
$107$	−902.252	−0.815177	−0.407589	−	0.913166i	$-0.633630\pi$
$107$	−902.252	−0.815177	−0.407589	+	0.913166i	$0.633630\pi$
$108$	0	0
$109$	−475.917	−0.418207	−0.209104	−	0.977893i	$-0.567055\pi$
$109$	−475.917	−0.418207	−0.209104	+	0.977893i	$0.567055\pi$
$110$	0	0
$111$	494.158	0.422553
$112$	0	0
$113$	−1159.86	−0.965576	−0.482788	−	0.875737i	$-0.660376\pi$
$113$	−1159.86	−0.965576	−0.482788	+	0.875737i	$0.660376\pi$
$114$	0	0
$115$	−553.244	−0.448611
$116$	0	0
$117$	1444.20	1.14116
$118$	0	0
$119$	0	0
$120$	0	0
$121$	199.837	0.150140
$122$	0	0
$123$	881.024	0.645848
$124$	0	0
$125$	−1124.08	−0.804328
$126$	0	0
$127$	−1259.90	−0.880302	−0.440151	−	0.897924i	$-0.645075\pi$
$127$	−1259.90	−0.880302	−0.440151	+	0.897924i	$0.645075\pi$
$128$	0	0
$129$	1348.06	0.920075
$130$	0	0
$131$	−589.674	−0.393283	−0.196641	−	0.980475i	$-0.563003\pi$
$131$	−589.674	−0.393283	−0.196641	+	0.980475i	$0.563003\pi$
$132$	0	0
$133$	1014.97	0.661723
$134$	0	0
$135$	588.644	0.375277
$136$	0	0
$137$	−2030.76	−1.26642	−0.633209	−	0.773981i	$-0.718263\pi$
$137$	−2030.76	−1.26642	−0.633209	+	0.773981i	$0.718263\pi$
$138$	0	0
$139$	−1039.21	−0.634136	−0.317068	−	0.948403i	$-0.602698\pi$
$139$	−1039.21	−0.634136	−0.317068	+	0.948403i	$0.602698\pi$
$140$	0	0
$141$	−650.377	−0.388452
$142$	0	0
$143$	2695.59	1.57634
$144$	0	0
$145$	−350.567	−0.200779
$146$	0	0
$147$	371.641	0.208520
$148$	0	0
$149$	1611.20	0.885870	0.442935	−	0.896554i	$-0.353937\pi$
$149$	1611.20	0.885870	0.442935	+	0.896554i	$0.353937\pi$
$150$	0	0
$151$	2590.47	1.39609	0.698044	−	0.716055i	$-0.254054\pi$
$151$	2590.47	1.39609	0.698044	+	0.716055i	$0.254054\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	69.5286	0.0360301
$156$	0	0
$157$	−2882.23	−1.46514	−0.732570	−	0.680692i	$-0.761679\pi$
$157$	−2882.23	−1.46514	−0.732570	+	0.680692i	$0.761679\pi$
$158$	0	0
$159$	−1553.41	−0.774801
$160$	0	0
$161$	1533.82	0.750818
$162$	0	0
$163$	3617.97	1.73853	0.869267	−	0.494343i	$-0.164591\pi$
$163$	3617.97	1.73853	0.869267	+	0.494343i	$0.164591\pi$
$164$	0	0
$165$	480.195	0.226565
$166$	0	0
$167$	−3104.69	−1.43861	−0.719307	−	0.694693i	$-0.755540\pi$
$167$	−3104.69	−1.43861	−0.719307	+	0.694693i	$0.755540\pi$
$168$	0	0
$169$	2549.55	1.16047
$170$	0	0
$171$	1536.45	0.687107
$172$	0	0
$173$	−1555.90	−0.683775	−0.341887	−	0.939741i	$-0.611066\pi$
$173$	−1555.90	−0.683775	−0.341887	+	0.939741i	$0.611066\pi$
$174$	0	0
$175$	1385.48	0.598470
$176$	0	0
$177$	−458.161	−0.194562
$178$	0	0
$179$	3057.19	1.27656	0.638282	−	0.769803i	$-0.279645\pi$
$179$	3057.19	1.27656	0.638282	+	0.769803i	$0.279645\pi$
$180$	0	0
$181$	−1524.51	−0.626054	−0.313027	−	0.949744i	$-0.601343\pi$
$181$	−1524.51	−0.626054	−0.313027	+	0.949744i	$0.601343\pi$
$182$	0	0
$183$	706.793	0.285506
$184$	0	0
$185$	−1004.48	−0.399194
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1631.96	−0.628083
$190$	0	0
$191$	−4024.54	−1.52464	−0.762318	−	0.647203i	$-0.775939\pi$
$191$	−4024.54	−1.52464	−0.762318	+	0.647203i	$0.775939\pi$
$192$	0	0
$193$	−4622.74	−1.72410	−0.862052	−	0.506820i	$-0.830821\pi$
$193$	−4622.74	−1.72410	−0.862052	+	0.506820i	$0.830821\pi$
$194$	0	0
$195$	845.557	0.310521
$196$	0	0
$197$	−1501.31	−0.542966	−0.271483	−	0.962443i	$-0.587514\pi$
$197$	−1501.31	−0.542966	−0.271483	+	0.962443i	$0.587514\pi$
$198$	0	0
$199$	3084.94	1.09892	0.549461	−	0.835519i	$-0.314833\pi$
$199$	3084.94	1.09892	0.549461	+	0.835519i	$0.314833\pi$
$200$	0	0
$201$	10.0686	0.00353327
$202$	0	0
$203$	971.914	0.336035
$204$	0	0
$205$	−1790.87	−0.610145
$206$	0	0
$207$	2321.88	0.779621
$208$	0	0
$209$	2867.78	0.949132
$210$	0	0
$211$	−5195.11	−1.69501	−0.847503	−	0.530790i	$-0.821895\pi$
$211$	−5195.11	−1.69501	−0.847503	+	0.530790i	$0.821895\pi$
$212$	0	0
$213$	−1656.46	−0.532858
$214$	0	0
$215$	−2740.21	−0.869212
$216$	0	0
$217$	−192.762	−0.0603019
$218$	0	0
$219$	1117.21	0.344722
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4707.24	1.41354	0.706772	−	0.707442i	$-0.250151\pi$
$223$	4707.24	1.41354	0.706772	+	0.707442i	$0.250151\pi$
$224$	0	0
$225$	2097.32	0.621428
$226$	0	0
$227$	−1523.20	−0.445367	−0.222684	−	0.974891i	$-0.571482\pi$
$227$	−1523.20	−0.445367	−0.222684	+	0.974891i	$0.571482\pi$
$228$	0	0
$229$	−3458.68	−0.998060	−0.499030	−	0.866585i	$-0.666310\pi$
$229$	−3458.68	−0.998060	−0.499030	+	0.866585i	$0.666310\pi$
$230$	0	0
$231$	−1331.30	−0.379190
$232$	0	0
$233$	2932.43	0.824507	0.412253	−	0.911069i	$-0.364742\pi$
$233$	2932.43	0.824507	0.412253	+	0.911069i	$0.364742\pi$
$234$	0	0
$235$	1322.03	0.366978
$236$	0	0
$237$	1088.36	0.298299
$238$	0	0
$239$	832.038	0.225188	0.112594	−	0.993641i	$-0.464084\pi$
$239$	832.038	0.225188	0.112594	+	0.993641i	$0.464084\pi$
$240$	0	0
$241$	−3210.61	−0.858147	−0.429073	−	0.903270i	$-0.641160\pi$
$241$	−3210.61	−0.858147	−0.429073	+	0.903270i	$0.641160\pi$
$242$	0	0
$243$	−3861.17	−1.01932
$244$	0	0
$245$	−755.439	−0.196993
$246$	0	0
$247$	5049.76	1.30084
$248$	0	0
$249$	−2265.07	−0.576477
$250$	0	0
$251$	5831.41	1.46643	0.733217	−	0.679994i	$-0.238018\pi$
$251$	5831.41	1.46643	0.733217	+	0.679994i	$0.238018\pi$
$252$	0	0
$253$	4333.77	1.07692
$254$	0	0
$255$	0	0
$256$	0	0
$257$	75.7557	0.0183872	0.00919360	−	0.999958i	$-0.497074\pi$
$257$	75.7557	0.0183872	0.00919360	+	0.999958i	$0.497074\pi$
$258$	0	0
$259$	2784.83	0.668112
$260$	0	0
$261$	1471.27	0.348925
$262$	0	0
$263$	3749.67	0.879143	0.439571	−	0.898208i	$-0.355130\pi$
$263$	3749.67	0.879143	0.439571	+	0.898208i	$0.355130\pi$
$264$	0	0
$265$	3157.64	0.731970
$266$	0	0
$267$	256.864	0.0588758
$268$	0	0
$269$	3456.01	0.783334	0.391667	−	0.920107i	$-0.371898\pi$
$269$	3456.01	0.783334	0.391667	+	0.920107i	$0.371898\pi$
$270$	0	0
$271$	−3661.21	−0.820674	−0.410337	−	0.911934i	$-0.634589\pi$
$271$	−3661.21	−0.820674	−0.410337	+	0.911934i	$0.634589\pi$
$272$	0	0
$273$	−2344.23	−0.519704
$274$	0	0
$275$	3914.64	0.858406
$276$	0	0
$277$	5585.64	1.21158	0.605792	−	0.795623i	$-0.292856\pi$
$277$	5585.64	1.21158	0.605792	+	0.795623i	$0.292856\pi$
$278$	0	0
$279$	−291.801	−0.0626152
$280$	0	0
$281$	8330.80	1.76859	0.884295	−	0.466929i	$-0.154640\pi$
$281$	8330.80	1.76859	0.884295	+	0.466929i	$0.154640\pi$
$282$	0	0
$283$	−6880.46	−1.44523	−0.722616	−	0.691250i	$-0.757060\pi$
$283$	−6880.46	−1.44523	−0.722616	+	0.691250i	$0.757060\pi$
$284$	0	0
$285$	899.570	0.186968
$286$	0	0
$287$	4965.02	1.02117
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−2054.93	−0.413958
$292$	0	0
$293$	4551.61	0.907535	0.453767	−	0.891120i	$-0.350080\pi$
$293$	4551.61	0.907535	0.453767	+	0.891120i	$0.350080\pi$
$294$	0	0
$295$	931.309	0.183806
$296$	0	0
$297$	−4611.08	−0.900882
$298$	0	0
$299$	7631.17	1.47599
$300$	0	0
$301$	7596.98	1.45476
$302$	0	0
$303$	−124.743	−0.0236512
$304$	0	0
$305$	−1436.71	−0.269723
$306$	0	0
$307$	−4055.46	−0.753933	−0.376966	−	0.926227i	$-0.623033\pi$
$307$	−4055.46	−0.753933	−0.376966	+	0.926227i	$0.623033\pi$
$308$	0	0
$309$	610.921	0.112473
$310$	0	0
$311$	3479.02	0.634331	0.317166	−	0.948370i	$-0.397269\pi$
$311$	3479.02	0.634331	0.317166	+	0.948370i	$0.397269\pi$
$312$	0	0
$313$	−8434.86	−1.52322	−0.761608	−	0.648038i	$-0.775590\pi$
$313$	−8434.86	−1.52322	−0.761608	+	0.648038i	$0.775590\pi$
$314$	0	0
$315$	1449.85	0.259333
$316$	0	0
$317$	−720.967	−0.127740	−0.0638699	−	0.997958i	$-0.520344\pi$
$317$	−720.967	−0.127740	−0.0638699	+	0.997958i	$0.520344\pi$
$318$	0	0
$319$	2746.13	0.481986
$320$	0	0
$321$	2217.00	0.385486
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6893.13	1.17650
$326$	0	0
$327$	1169.42	0.197765
$328$	0	0
$329$	−3665.21	−0.614193
$330$	0	0
$331$	−2714.37	−0.450742	−0.225371	−	0.974273i	$-0.572359\pi$
$331$	−2714.37	−0.450742	−0.225371	+	0.974273i	$0.572359\pi$
$332$	0	0
$333$	4215.65	0.693742
$334$	0	0
$335$	−20.4666	−0.00333795
$336$	0	0
$337$	−10092.9	−1.63144	−0.815719	−	0.578448i	$-0.803658\pi$
$337$	−10092.9	−1.63144	−0.815719	+	0.578448i	$0.803658\pi$
$338$	0	0
$339$	2849.99	0.456608
$340$	0	0
$341$	−544.645	−0.0864932
$342$	0	0
$343$	6844.08	1.07739
$344$	0	0
$345$	1359.42	0.212142
$346$	0	0
$347$	6791.96	1.05075	0.525377	−	0.850870i	$-0.323924\pi$
$347$	6791.96	1.05075	0.525377	+	0.850870i	$0.323924\pi$
$348$	0	0
$349$	−7584.41	−1.16328	−0.581639	−	0.813447i	$-0.697588\pi$
$349$	−7584.41	−1.16328	−0.581639	+	0.813447i	$0.697588\pi$
$350$	0	0
$351$	−8119.46	−1.23471
$352$	0	0
$353$	4525.78	0.682387	0.341194	−	0.939993i	$-0.389169\pi$
$353$	4525.78	0.682387	0.341194	+	0.939993i	$0.389169\pi$
$354$	0	0
$355$	3367.11	0.503402
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9971.29	−1.46592	−0.732959	−	0.680273i	$-0.761861\pi$
$359$	−9971.29	−1.46592	−0.732959	+	0.680273i	$0.761861\pi$
$360$	0	0
$361$	−1486.66	−0.216747
$362$	0	0
$363$	−491.037	−0.0709993
$364$	0	0
$365$	−2270.97	−0.325666
$366$	0	0
$367$	8524.73	1.21250	0.606249	−	0.795275i	$-0.292673\pi$
$367$	8524.73	1.21250	0.606249	+	0.795275i	$0.292673\pi$
$368$	0	0
$369$	7515.99	1.06034
$370$	0	0
$371$	−8754.25	−1.22506
$372$	0	0
$373$	10570.4	1.46733	0.733663	−	0.679514i	$-0.237809\pi$
$373$	10570.4	1.46733	0.733663	+	0.679514i	$0.237809\pi$
$374$	0	0
$375$	2762.08	0.380356
$376$	0	0
$377$	4835.54	0.660592
$378$	0	0
$379$	−2238.89	−0.303441	−0.151721	−	0.988423i	$-0.548481\pi$
$379$	−2238.89	−0.303441	−0.151721	+	0.988423i	$0.548481\pi$
$380$	0	0
$381$	3095.82	0.416283
$382$	0	0
$383$	5696.59	0.760006	0.380003	−	0.924985i	$-0.375923\pi$
$383$	5696.59	0.760006	0.380003	+	0.924985i	$0.375923\pi$
$384$	0	0
$385$	2706.14	0.358228
$386$	0	0
$387$	11500.2	1.51057
$388$	0	0
$389$	11158.8	1.45443	0.727216	−	0.686409i	$-0.240814\pi$
$389$	11158.8	1.45443	0.727216	+	0.686409i	$0.240814\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1448.94	0.185978
$394$	0	0
$395$	−2212.33	−0.281809
$396$	0	0
$397$	−13630.3	−1.72314	−0.861571	−	0.507637i	$-0.830519\pi$
$397$	−13630.3	−1.72314	−0.861571	+	0.507637i	$0.830519\pi$
$398$	0	0
$399$	−2493.98	−0.312920
$400$	0	0
$401$	5650.43	0.703663	0.351831	−	0.936063i	$-0.385559\pi$
$401$	5650.43	0.703663	0.351831	+	0.936063i	$0.385559\pi$
$402$	0	0
$403$	−959.043	−0.118544
$404$	0	0
$405$	1380.52	0.169380
$406$	0	0
$407$	7868.49	0.958297
$408$	0	0
$409$	−2013.60	−0.243438	−0.121719	−	0.992565i	$-0.538841\pi$
$409$	−2013.60	−0.243438	−0.121719	+	0.992565i	$0.538841\pi$
$410$	0	0
$411$	4989.96	0.598872
$412$	0	0
$413$	−2581.97	−0.307628
$414$	0	0
$415$	4604.23	0.544609
$416$	0	0
$417$	2553.54	0.299874
$418$	0	0
$419$	−55.2367	−0.00644031	−0.00322015	−	0.999995i	$-0.501025\pi$
$419$	−55.2367	−0.00644031	−0.00322015	+	0.999995i	$0.501025\pi$
$420$	0	0
$421$	−3308.80	−0.383043	−0.191522	−	0.981488i	$-0.561342\pi$
$421$	−3308.80	−0.383043	−0.191522	+	0.981488i	$0.561342\pi$
$422$	0	0
$423$	−5548.35	−0.637754
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3983.14	0.451423
$428$	0	0
$429$	−6623.57	−0.745429
$430$	0	0
$431$	−13593.4	−1.51919	−0.759593	−	0.650399i	$-0.774602\pi$
$431$	−13593.4	−1.51919	−0.759593	+	0.650399i	$0.774602\pi$
$432$	0	0
$433$	3358.42	0.372737	0.186369	−	0.982480i	$-0.440328\pi$
$433$	3358.42	0.372737	0.186369	+	0.982480i	$0.440328\pi$
$434$	0	0
$435$	861.409	0.0949458
$436$	0	0
$437$	8118.64	0.888712
$438$	0	0
$439$	7282.28	0.791718	0.395859	−	0.918311i	$-0.370447\pi$
$439$	7282.28	0.791718	0.395859	+	0.918311i	$0.370447\pi$
$440$	0	0
$441$	3170.46	0.342345
$442$	0	0
$443$	5645.62	0.605489	0.302744	−	0.953072i	$-0.402097\pi$
$443$	5645.62	0.605489	0.302744	+	0.953072i	$0.402097\pi$
$444$	0	0
$445$	−522.132	−0.0556211
$446$	0	0
$447$	−3959.02	−0.418916
$448$	0	0
$449$	−12624.9	−1.32696	−0.663480	−	0.748194i	$-0.730921\pi$
$449$	−12624.9	−1.32696	−0.663480	+	0.748194i	$0.730921\pi$
$450$	0	0
$451$	14028.6	1.46470
$452$	0	0
$453$	−6365.27	−0.660191
$454$	0	0
$455$	4765.14	0.490974
$456$	0	0
$457$	−8717.64	−0.892328	−0.446164	−	0.894951i	$-0.647210\pi$
$457$	−8717.64	−0.892328	−0.446164	+	0.894951i	$0.647210\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8126.98	0.821065	0.410533	−	0.911846i	$-0.365343\pi$
$461$	8126.98	0.821065	0.410533	+	0.911846i	$0.365343\pi$
$462$	0	0
$463$	5383.50	0.540372	0.270186	−	0.962808i	$-0.412915\pi$
$463$	5383.50	0.540372	0.270186	+	0.962808i	$0.412915\pi$
$464$	0	0
$465$	−170.845	−0.0170382
$466$	0	0
$467$	−10046.7	−0.995517	−0.497759	−	0.867316i	$-0.665843\pi$
$467$	−10046.7	−0.995517	−0.497759	+	0.867316i	$0.665843\pi$
$468$	0	0
$469$	56.7419	0.00558656
$470$	0	0
$471$	7082.19	0.692845
$472$	0	0
$473$	21465.1	2.08661
$474$	0	0
$475$	7333.46	0.708384
$476$	0	0
$477$	−13252.1	−1.27206
$478$	0	0
$479$	−7384.94	−0.704439	−0.352220	−	0.935917i	$-0.614573\pi$
$479$	−7384.94	−0.704439	−0.352220	+	0.935917i	$0.614573\pi$
$480$	0	0
$481$	13855.3	1.31341
$482$	0	0
$483$	−3768.88	−0.355052
$484$	0	0
$485$	4177.07	0.391075
$486$	0	0
$487$	1765.54	0.164280	0.0821398	−	0.996621i	$-0.473825\pi$
$487$	1765.54	0.164280	0.0821398	+	0.996621i	$0.473825\pi$
$488$	0	0
$489$	−8890.03	−0.822129
$490$	0	0
$491$	9170.07	0.842850	0.421425	−	0.906863i	$-0.361530\pi$
$491$	9170.07	0.842850	0.421425	+	0.906863i	$0.361530\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4096.53	0.371971
$496$	0	0
$497$	−9335.00	−0.842519
$498$	0	0
$499$	3865.67	0.346796	0.173398	−	0.984852i	$-0.444525\pi$
$499$	3865.67	0.346796	0.173398	+	0.984852i	$0.444525\pi$
$500$	0	0
$501$	7628.82	0.680301
$502$	0	0
$503$	20182.5	1.78905	0.894526	−	0.447017i	$-0.147514\pi$
$503$	20182.5	1.78905	0.894526	+	0.447017i	$0.147514\pi$
$504$	0	0
$505$	253.568	0.0223438
$506$	0	0
$507$	−6264.74	−0.548771
$508$	0	0
$509$	−12027.1	−1.04733	−0.523665	−	0.851924i	$-0.675436\pi$
$509$	−12027.1	−1.04733	−0.523665	+	0.851924i	$0.675436\pi$
$510$	0	0
$511$	6296.06	0.545051
$512$	0	0
$513$	−8638.13	−0.743436
$514$	0	0
$515$	−1241.83	−0.106255
$516$	0	0
$517$	−10356.0	−0.880958
$518$	0	0
$519$	3823.15	0.323348
$520$	0	0
$521$	−18355.3	−1.54349	−0.771746	−	0.635931i	$-0.780616\pi$
$521$	−18355.3	−1.54349	−0.771746	+	0.635931i	$0.780616\pi$
$522$	0	0
$523$	9657.35	0.807431	0.403716	−	0.914885i	$-0.367719\pi$
$523$	9657.35	0.807431	0.403716	+	0.914885i	$0.367719\pi$
$524$	0	0
$525$	−3404.38	−0.283008
$526$	0	0
$527$	0	0
$528$	0	0
$529$	101.836	0.00836985
$530$	0	0
$531$	−3908.56	−0.319429
$532$	0	0
$533$	24702.4	2.00746
$534$	0	0
$535$	−4506.53	−0.364176
$536$	0	0
$537$	−7512.09	−0.603670
$538$	0	0
$539$	5917.65	0.472896
$540$	0	0
$541$	−3083.64	−0.245058	−0.122529	−	0.992465i	$-0.539100\pi$
$541$	−3083.64	−0.245058	−0.122529	+	0.992465i	$0.539100\pi$
$542$	0	0
$543$	3746.01	0.296053
$544$	0	0
$545$	−2377.09	−0.186832
$546$	0	0
$547$	9271.00	0.724679	0.362340	−	0.932046i	$-0.381978\pi$
$547$	9271.00	0.724679	0.362340	+	0.932046i	$0.381978\pi$
$548$	0	0
$549$	6029.64	0.468741
$550$	0	0
$551$	5144.44	0.397750
$552$	0	0
$553$	6133.48	0.471650
$554$	0	0
$555$	2468.20	0.188774
$556$	0	0
$557$	−16475.6	−1.25331	−0.626656	−	0.779296i	$-0.715577\pi$
$557$	−16475.6	−1.25331	−0.626656	+	0.779296i	$0.715577\pi$
$558$	0	0
$559$	37797.1	2.85983
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1536.51	0.115020	0.0575101	−	0.998345i	$-0.481684\pi$
$563$	1536.51	0.115020	0.0575101	+	0.998345i	$0.481684\pi$
$564$	0	0
$565$	−5793.21	−0.431366
$566$	0	0
$567$	−3827.38	−0.283483
$568$	0	0
$569$	4800.76	0.353705	0.176853	−	0.984237i	$-0.443408\pi$
$569$	4800.76	0.353705	0.176853	+	0.984237i	$0.443408\pi$
$570$	0	0
$571$	−22711.8	−1.66455	−0.832276	−	0.554361i	$-0.812963\pi$
$571$	−22711.8	−1.66455	−0.832276	+	0.554361i	$0.812963\pi$
$572$	0	0
$573$	9889.06	0.720979
$574$	0	0
$575$	11082.3	0.803762
$576$	0	0
$577$	10132.9	0.731086	0.365543	−	0.930794i	$-0.380883\pi$
$577$	10132.9	0.731086	0.365543	+	0.930794i	$0.380883\pi$
$578$	0	0
$579$	11358.9	0.815305
$580$	0	0
$581$	−12764.8	−0.911485
$582$	0	0
$583$	−24735.0	−1.75715
$584$	0	0
$585$	7213.42	0.509809
$586$	0	0
$587$	−7146.56	−0.502504	−0.251252	−	0.967922i	$-0.580842\pi$
$587$	−7146.56	−0.502504	−0.251252	+	0.967922i	$0.580842\pi$
$588$	0	0
$589$	−1020.31	−0.0713769
$590$	0	0
$591$	3689.02	0.256761
$592$	0	0
$593$	−9943.65	−0.688595	−0.344297	−	0.938861i	$-0.611883\pi$
$593$	−9943.65	−0.688595	−0.344297	+	0.938861i	$0.611883\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7580.28	−0.519665
$598$	0	0
$599$	586.728	0.0400218	0.0200109	−	0.999800i	$-0.493630\pi$
$599$	586.728	0.0400218	0.0200109	+	0.999800i	$0.493630\pi$
$600$	0	0
$601$	−8953.21	−0.607669	−0.303834	−	0.952725i	$-0.598267\pi$
$601$	−8953.21	−0.607669	−0.303834	+	0.952725i	$0.598267\pi$
$602$	0	0
$603$	85.8953	0.00580087
$604$	0	0
$605$	998.136	0.0670744
$606$	0	0
$607$	−20593.0	−1.37701	−0.688503	−	0.725233i	$-0.741732\pi$
$607$	−20593.0	−1.37701	−0.688503	+	0.725233i	$0.741732\pi$
$608$	0	0
$609$	−2388.18	−0.158906
$610$	0	0
$611$	−18235.4	−1.20741
$612$	0	0
$613$	−26155.4	−1.72334	−0.861669	−	0.507471i	$-0.830581\pi$
$613$	−26155.4	−1.72334	−0.861669	+	0.507471i	$0.830581\pi$
$614$	0	0
$615$	4400.50	0.288529
$616$	0	0
$617$	21491.6	1.40230	0.701152	−	0.713012i	$-0.252670\pi$
$617$	21491.6	1.40230	0.701152	+	0.713012i	$0.252670\pi$
$618$	0	0
$619$	11135.8	0.723081	0.361541	−	0.932356i	$-0.382251\pi$
$619$	11135.8	0.723081	0.361541	+	0.932356i	$0.382251\pi$
$620$	0	0
$621$	−13053.9	−0.843533
$622$	0	0
$623$	1447.56	0.0930904
$624$	0	0
$625$	6892.02	0.441089
$626$	0	0
$627$	−7046.68	−0.448832
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−520.105	−0.0328131	−0.0164065	−	0.999865i	$-0.505223\pi$
$631$	−520.105	−0.0328131	−0.0164065	+	0.999865i	$0.505223\pi$
$632$	0	0
$633$	12765.4	0.801545
$634$	0	0
$635$	−6292.92	−0.393270
$636$	0	0
$637$	10420.2	0.648134
$638$	0	0
$639$	−14131.2	−0.874840
$640$	0	0
$641$	14587.7	0.898875	0.449437	−	0.893312i	$-0.351624\pi$
$641$	14587.7	0.898875	0.449437	+	0.893312i	$0.351624\pi$
$642$	0	0
$643$	−31531.5	−1.93387	−0.966937	−	0.255014i	$-0.917920\pi$
$643$	−31531.5	−1.93387	−0.966937	+	0.255014i	$0.917920\pi$
$644$	0	0
$645$	6733.21	0.411039
$646$	0	0
$647$	−11711.6	−0.711637	−0.355819	−	0.934555i	$-0.615798\pi$
$647$	−11711.6	−0.711637	−0.355819	+	0.934555i	$0.615798\pi$
$648$	0	0
$649$	−7295.31	−0.441242
$650$	0	0
$651$	473.652	0.0285160
$652$	0	0
$653$	−11082.8	−0.664174	−0.332087	−	0.943249i	$-0.607753\pi$
$653$	−11082.8	−0.664174	−0.332087	+	0.943249i	$0.607753\pi$
$654$	0	0
$655$	−2945.28	−0.175697
$656$	0	0
$657$	9530.90	0.565960
$658$	0	0
$659$	3696.88	0.218528	0.109264	−	0.994013i	$-0.465151\pi$
$659$	3696.88	0.218528	0.109264	+	0.994013i	$0.465151\pi$
$660$	0	0
$661$	5803.70	0.341509	0.170755	−	0.985314i	$-0.445379\pi$
$661$	5803.70	0.341509	0.170755	+	0.985314i	$0.445379\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5069.54	0.295621
$666$	0	0
$667$	7774.24	0.451304
$668$	0	0
$669$	−11566.6	−0.668446
$670$	0	0
$671$	11254.3	0.647492
$672$	0	0
$673$	19023.1	1.08958	0.544790	−	0.838573i	$-0.316609\pi$
$673$	19023.1	1.08958	0.544790	+	0.838573i	$0.316609\pi$
$674$	0	0
$675$	−11791.4	−0.672372
$676$	0	0
$677$	−23080.4	−1.31027	−0.655133	−	0.755513i	$-0.727388\pi$
$677$	−23080.4	−1.31027	−0.655133	+	0.755513i	$0.727388\pi$
$678$	0	0
$679$	−11580.6	−0.654523
$680$	0	0
$681$	3742.79	0.210608
$682$	0	0
$683$	5783.65	0.324019	0.162010	−	0.986789i	$-0.448202\pi$
$683$	5783.65	0.324019	0.162010	+	0.986789i	$0.448202\pi$
$684$	0	0
$685$	−10143.2	−0.565766
$686$	0	0
$687$	8498.63	0.471969
$688$	0	0
$689$	−43554.9	−2.40828
$690$	0	0
$691$	2153.36	0.118550	0.0592748	−	0.998242i	$-0.481121\pi$
$691$	2153.36	0.118550	0.0592748	+	0.998242i	$0.481121\pi$
$692$	0	0
$693$	−11357.3	−0.622549
$694$	0	0
$695$	−5190.62	−0.283297
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−7205.54	−0.389898
$700$	0	0
$701$	2775.24	0.149529	0.0747643	−	0.997201i	$-0.476180\pi$
$701$	2775.24	0.149529	0.0747643	+	0.997201i	$0.476180\pi$
$702$	0	0
$703$	14740.4	0.790817
$704$	0	0
$705$	−3248.48	−0.173539
$706$	0	0
$707$	−702.993	−0.0373957
$708$	0	0
$709$	−18099.3	−0.958719	−0.479359	−	0.877619i	$-0.659131\pi$
$709$	−18099.3	−0.958719	−0.479359	+	0.877619i	$0.659131\pi$
$710$	0	0
$711$	9284.80	0.489743
$712$	0	0
$713$	−1541.88	−0.0809872
$714$	0	0
$715$	13463.8	0.704222
$716$	0	0
$717$	−2044.47	−0.106489
$718$	0	0
$719$	−24955.0	−1.29438	−0.647192	−	0.762327i	$-0.724057\pi$
$719$	−24955.0	−1.29438	−0.647192	+	0.762327i	$0.724057\pi$
$720$	0	0
$721$	3442.85	0.177834
$722$	0	0
$723$	7889.07	0.405806
$724$	0	0
$725$	7022.37	0.359730
$726$	0	0
$727$	36475.0	1.86077	0.930387	−	0.366580i	$-0.119471\pi$
$727$	36475.0	1.86077	0.930387	+	0.366580i	$0.119471\pi$
$728$	0	0
$729$	2024.96	0.102879
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8604.88	0.433600	0.216800	−	0.976216i	$-0.430438\pi$
$733$	8604.88	0.433600	0.216800	+	0.976216i	$0.430438\pi$
$734$	0	0
$735$	1856.26	0.0931552
$736$	0	0
$737$	160.323	0.00801300
$738$	0	0
$739$	−21577.7	−1.07408	−0.537042	−	0.843555i	$-0.680458\pi$
$739$	−21577.7	−1.07408	−0.537042	+	0.843555i	$0.680458\pi$
$740$	0	0
$741$	−12408.2	−0.615152
$742$	0	0
$743$	−32034.1	−1.58172	−0.790860	−	0.611997i	$-0.790366\pi$
$743$	−32034.1	−1.58172	−0.790860	+	0.611997i	$0.790366\pi$
$744$	0	0
$745$	8047.56	0.395758
$746$	0	0
$747$	−19323.2	−0.946452
$748$	0	0
$749$	12493.9	0.609504
$750$	0	0
$751$	14979.3	0.727834	0.363917	−	0.931431i	$-0.381439\pi$
$751$	14979.3	0.727834	0.363917	+	0.931431i	$0.381439\pi$
$752$	0	0
$753$	−14328.9	−0.693457
$754$	0	0
$755$	12938.8	0.623695
$756$	0	0
$757$	16598.8	0.796955	0.398477	−	0.917178i	$-0.369539\pi$
$757$	16598.8	0.796955	0.398477	+	0.917178i	$0.369539\pi$
$758$	0	0
$759$	−10648.9	−0.509263
$760$	0	0
$761$	−31711.2	−1.51055	−0.755276	−	0.655407i	$-0.772497\pi$
$761$	−31711.2	−1.51055	−0.755276	+	0.655407i	$0.772497\pi$
$762$	0	0
$763$	6590.27	0.312692
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12846.0	−0.604749
$768$	0	0
$769$	−34297.3	−1.60831	−0.804157	−	0.594418i	$-0.797383\pi$
$769$	−34297.3	−1.60831	−0.804157	+	0.594418i	$0.797383\pi$
$770$	0	0
$771$	−186.146	−0.00869506
$772$	0	0
$773$	21854.8	1.01690	0.508450	−	0.861092i	$-0.330219\pi$
$773$	21854.8	1.01690	0.508450	+	0.861092i	$0.330219\pi$
$774$	0	0
$775$	−1392.76	−0.0645541
$776$	0	0
$777$	−6842.86	−0.315941
$778$	0	0
$779$	26280.3	1.20872
$780$	0	0
$781$	−26375.9	−1.20845
$782$	0	0
$783$	−8271.69	−0.377530
$784$	0	0
$785$	−14396.1	−0.654544
$786$	0	0
$787$	2305.04	0.104404	0.0522018	−	0.998637i	$-0.483376\pi$
$787$	2305.04	0.104404	0.0522018	+	0.998637i	$0.483376\pi$
$788$	0	0
$789$	−9213.65	−0.415735
$790$	0	0
$791$	16061.1	0.721957
$792$	0	0
$793$	19817.2	0.887428
$794$	0	0
$795$	−7758.91	−0.346139
$796$	0	0
$797$	−38833.9	−1.72593	−0.862966	−	0.505262i	$-0.831396\pi$
$797$	−38833.9	−1.72593	−0.862966	+	0.505262i	$0.831396\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2191.30	0.0966615
$802$	0	0
$803$	17789.4	0.781786
$804$	0	0
$805$	7661.05	0.335424
$806$	0	0
$807$	−8492.08	−0.370428
$808$	0	0
$809$	−18173.8	−0.789811	−0.394905	−	0.918722i	$-0.629223\pi$
$809$	−18173.8	−0.789811	−0.394905	+	0.918722i	$0.629223\pi$
$810$	0	0
$811$	−5067.61	−0.219418	−0.109709	−	0.993964i	$-0.534992\pi$
$811$	−5067.61	−0.219418	−0.109709	+	0.993964i	$0.534992\pi$
$812$	0	0
$813$	8996.29	0.388086
$814$	0	0
$815$	18070.9	0.776681
$816$	0	0
$817$	40211.5	1.72194
$818$	0	0
$819$	−19998.5	−0.853243
$820$	0	0
$821$	23644.1	1.00510	0.502549	−	0.864549i	$-0.332396\pi$
$821$	23644.1	1.00510	0.502549	+	0.864549i	$0.332396\pi$
$822$	0	0
$823$	9415.97	0.398809	0.199405	−	0.979917i	$-0.436099\pi$
$823$	9415.97	0.398809	0.199405	+	0.979917i	$0.436099\pi$
$824$	0	0
$825$	−9619.01	−0.405929
$826$	0	0
$827$	−5764.85	−0.242398	−0.121199	−	0.992628i	$-0.538674\pi$
$827$	−5764.85	−0.242398	−0.121199	+	0.992628i	$0.538674\pi$
$828$	0	0
$829$	−12899.5	−0.540432	−0.270216	−	0.962800i	$-0.587095\pi$
$829$	−12899.5	−0.540432	−0.270216	+	0.962800i	$0.587095\pi$
$830$	0	0
$831$	−13725.0	−0.572941
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15507.2	−0.642693
$836$	0	0
$837$	1640.54	0.0677483
$838$	0	0
$839$	38312.2	1.57650	0.788251	−	0.615354i	$-0.210987\pi$
$839$	38312.2	1.57650	0.788251	+	0.615354i	$0.210987\pi$
$840$	0	0
$841$	−19462.8	−0.798015
$842$	0	0
$843$	−20470.3	−0.836342
$844$	0	0
$845$	12734.4	0.518435
$846$	0	0
$847$	−2767.24	−0.112259
$848$	0	0
$849$	16906.6	0.683431
$850$	0	0
$851$	22275.6	0.897294
$852$	0	0
$853$	−42669.8	−1.71276	−0.856381	−	0.516345i	$-0.827292\pi$
$853$	−42669.8	−1.71276	−0.856381	+	0.516345i	$0.827292\pi$
$854$	0	0
$855$	7674.21	0.306962
$856$	0	0
$857$	−33155.3	−1.32154	−0.660772	−	0.750587i	$-0.729771\pi$
$857$	−33155.3	−1.32154	−0.660772	+	0.750587i	$0.729771\pi$
$858$	0	0
$859$	−10795.2	−0.428785	−0.214392	−	0.976748i	$-0.568777\pi$
$859$	−10795.2	−0.428785	−0.214392	+	0.976748i	$0.568777\pi$
$860$	0	0
$861$	−12200.0	−0.482897
$862$	0	0
$863$	−1251.50	−0.0493645	−0.0246823	−	0.999695i	$-0.507857\pi$
$863$	−1251.50	−0.0493645	−0.0246823	+	0.999695i	$0.507857\pi$
$864$	0	0
$865$	−7771.36	−0.305473
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17330.0	0.676504
$870$	0	0
$871$	282.307	0.0109823
$872$	0	0
$873$	−17530.5	−0.679631
$874$	0	0
$875$	15565.8	0.601393
$876$	0	0
$877$	−10024.5	−0.385978	−0.192989	−	0.981201i	$-0.561818\pi$
$877$	−10024.5	−0.385978	−0.192989	+	0.981201i	$0.561818\pi$
$878$	0	0
$879$	−11184.2	−0.429161
$880$	0	0
$881$	1983.12	0.0758377	0.0379189	−	0.999281i	$-0.487927\pi$
$881$	1983.12	0.0758377	0.0379189	+	0.999281i	$0.487927\pi$
$882$	0	0
$883$	689.393	0.0262740	0.0131370	−	0.999914i	$-0.495818\pi$
$883$	689.393	0.0262740	0.0131370	+	0.999914i	$0.495818\pi$
$884$	0	0
$885$	−2288.40	−0.0869196
$886$	0	0
$887$	41395.4	1.56699	0.783495	−	0.621398i	$-0.213435\pi$
$887$	41395.4	1.56699	0.783495	+	0.621398i	$0.213435\pi$
$888$	0	0
$889$	17446.5	0.658198
$890$	0	0
$891$	−10814.2	−0.406609
$892$	0	0
$893$	−19400.3	−0.726995
$894$	0	0
$895$	15269.9	0.570299
$896$	0	0
$897$	−18751.2	−0.697977
$898$	0	0
$899$	−977.024	−0.0362464
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−18667.2	−0.687936
$904$	0	0
$905$	−7614.56	−0.279687
$906$	0	0
$907$	39085.1	1.43087	0.715434	−	0.698680i	$-0.246229\pi$
$907$	39085.1	1.43087	0.715434	+	0.698680i	$0.246229\pi$
$908$	0	0
$909$	−1064.18	−0.0388303
$910$	0	0
$911$	−17905.0	−0.651175	−0.325587	−	0.945512i	$-0.605562\pi$
$911$	−17905.0	−0.651175	−0.325587	+	0.945512i	$0.605562\pi$
$912$	0	0
$913$	−36066.7	−1.30738
$914$	0	0
$915$	3530.26	0.127549
$916$	0	0
$917$	8165.52	0.294056
$918$	0	0
$919$	−3048.53	−0.109425	−0.0547126	−	0.998502i	$-0.517424\pi$
$919$	−3048.53	−0.109425	−0.0547126	+	0.998502i	$0.517424\pi$
$920$	0	0
$921$	9965.04	0.356525
$922$	0	0
$923$	−46444.2	−1.65626
$924$	0	0
$925$	20121.2	0.715224
$926$	0	0
$927$	5211.75	0.184656
$928$	0	0
$929$	37785.9	1.33446	0.667230	−	0.744851i	$-0.267480\pi$
$929$	37785.9	1.33446	0.667230	+	0.744851i	$0.267480\pi$
$930$	0	0
$931$	11085.8	0.390249
$932$	0	0
$933$	−8548.61	−0.299967
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24196.5	−0.843614	−0.421807	−	0.906686i	$-0.638604\pi$
$937$	−24196.5	−0.843614	−0.421807	+	0.906686i	$0.638604\pi$
$938$	0	0
$939$	20726.0	0.720308
$940$	0	0
$941$	20692.7	0.716857	0.358429	−	0.933557i	$-0.383313\pi$
$941$	20692.7	0.716857	0.358429	+	0.933557i	$0.383313\pi$
$942$	0	0
$943$	39714.7	1.37146
$944$	0	0
$945$	−8151.26	−0.280593
$946$	0	0
$947$	28073.9	0.963337	0.481669	−	0.876353i	$-0.340031\pi$
$947$	28073.9	0.963337	0.481669	+	0.876353i	$0.340031\pi$
$948$	0	0
$949$	31324.6	1.07149
$950$	0	0
$951$	1771.55	0.0604065
$952$	0	0
$953$	−30146.7	−1.02471	−0.512354	−	0.858775i	$-0.671226\pi$
$953$	−30146.7	−1.02471	−0.512354	+	0.858775i	$0.671226\pi$
$954$	0	0
$955$	−20101.6	−0.681123
$956$	0	0
$957$	−6747.75	−0.227925
$958$	0	0
$959$	28121.0	0.946896
$960$	0	0
$961$	−29597.2	−0.993496
$962$	0	0
$963$	18913.2	0.632886
$964$	0	0
$965$	−23089.5	−0.770235
$966$	0	0
$967$	−52203.0	−1.73602	−0.868012	−	0.496542i	$-0.834603\pi$
$967$	−52203.0	−1.73602	−0.868012	+	0.496542i	$0.834603\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55402.9	1.83106	0.915532	−	0.402246i	$-0.131770\pi$
$971$	55402.9	1.83106	0.915532	+	0.402246i	$0.131770\pi$
$972$	0	0
$973$	14390.5	0.474141
$974$	0	0
$975$	−16937.7	−0.556350
$976$	0	0
$977$	−17032.8	−0.557755	−0.278877	−	0.960327i	$-0.589962\pi$
$977$	−17032.8	−0.557755	−0.278877	+	0.960327i	$0.589962\pi$
$978$	0	0
$979$	4090.06	0.133523
$980$	0	0
$981$	9976.28	0.324687
$982$	0	0
$983$	−25278.4	−0.820200	−0.410100	−	0.912041i	$-0.634506\pi$
$983$	−25278.4	−0.820200	−0.410100	+	0.912041i	$0.634506\pi$
$984$	0	0
$985$	−7498.71	−0.242567
$986$	0	0
$987$	9006.11	0.290443
$988$	0	0
$989$	60767.4	1.95378
$990$	0	0
$991$	−8235.39	−0.263981	−0.131991	−	0.991251i	$-0.542137\pi$
$991$	−8235.39	−0.263981	−0.131991	+	0.991251i	$0.542137\pi$
$992$	0	0
$993$	6669.73	0.213150
$994$	0	0
$995$	15408.5	0.490938
$996$	0	0
$997$	48625.2	1.54461	0.772304	−	0.635253i	$-0.219104\pi$
$997$	48625.2	1.54461	0.772304	+	0.635253i	$0.219104\pi$
$998$	0	0
$999$	−23700.9	−0.750614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.s.1.11		28
17.5	odd	16		136.4.n.b.25.5	✓	28
17.7	odd	16		136.4.n.b.49.5	yes	28
17.16	even	2	inner	2312.4.a.s.1.18		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.b.25.5	✓	28	17.5	odd	16
136.4.n.b.49.5	yes	28	17.7	odd	16
2312.4.a.s.1.11		28	1.1	even	1	trivial
2312.4.a.s.1.18		28	17.16	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45719 q^{3} +4.99476 q^{5} -13.8475 q^{7} -20.9622 q^{9} +O(q^{10})\)	\(q-2.45719 q^{3} +4.99476 q^{5} -13.8475 q^{7} -20.9622 q^{9} -39.1259 q^{11} -68.8952 q^{13} -12.2731 q^{15} -73.2962 q^{19} +34.0260 q^{21} -110.765 q^{23} -100.052 q^{25} +117.852 q^{27} -70.1869 q^{29} +13.9203 q^{31} +96.1398 q^{33} -69.1650 q^{35} -201.107 q^{37} +169.289 q^{39} -358.549 q^{41} -548.617 q^{43} -104.701 q^{45} +264.683 q^{47} -151.246 q^{49} +632.190 q^{53} -195.425 q^{55} +180.103 q^{57} +186.457 q^{59} -287.643 q^{61} +290.275 q^{63} -344.115 q^{65} -4.09762 q^{67} +272.170 q^{69} +674.128 q^{71} -454.670 q^{73} +245.848 q^{75} +541.797 q^{77} -442.930 q^{79} +276.394 q^{81} +921.811 q^{83} +172.463 q^{87} -104.536 q^{89} +954.028 q^{91} -34.2048 q^{93} -366.097 q^{95} +836.291 q^{97} +820.166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 308 q^{9}+O(q^{10}) \) 28 * q + 308 * q^9	\( 28 q + 308 q^{9} + 64 q^{13} + 120 q^{15} + 368 q^{19} - 192 q^{21} + 956 q^{25} - 504 q^{33} + 1664 q^{35} + 1760 q^{43} + 984 q^{47} + 1492 q^{49} - 1088 q^{53} + 424 q^{55} + 2288 q^{59} + 3424 q^{67} - 2920 q^{69} + 816 q^{77} + 2380 q^{81} + 3472 q^{83} + 1560 q^{87} + 2400 q^{89} - 1768 q^{93}+O(q^{100}) \) 28 * q + 308 * q^9 + 64 * q^13 + 120 * q^15 + 368 * q^19 - 192 * q^21 + 956 * q^25 - 504 * q^33 + 1664 * q^35 + 1760 * q^43 + 984 * q^47 + 1492 * q^49 - 1088 * q^53 + 424 * q^55 + 2288 * q^59 + 3424 * q^67 - 2920 * q^69 + 816 * q^77 + 2380 * q^81 + 3472 * q^83 + 1560 * q^87 + 2400 * q^89 - 1768 * q^93

Introduction

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