LMFDB - Embedded newform 2368.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46260 q^{3} -0.462598 q^{5} -0.323404 q^{7} -0.860806 q^{9} +O(q^{10})$	$q-1.46260 q^{3} -0.462598 q^{5} -0.323404 q^{7} -0.860806 q^{9} -5.58242 q^{11} -6.58242 q^{13} +0.676596 q^{15} -6.64681 q^{17} +2.64681 q^{19} +0.473011 q^{21} +4.86081 q^{23} -4.78600 q^{25} +5.64681 q^{27} +4.86081 q^{29} +6.46260 q^{31} +8.16484 q^{33} +0.149606 q^{35} +1.00000 q^{37} +9.62743 q^{39} -0.815790 q^{41} +1.87122 q^{43} +0.398207 q^{45} -1.11982 q^{47} -6.89541 q^{49} +9.72161 q^{51} +12.6918 q^{53} +2.58242 q^{55} -3.87122 q^{57} +0.128782 q^{59} +1.10941 q^{61} +0.278388 q^{63} +3.04502 q^{65} -13.4778 q^{67} -7.10941 q^{69} -8.04502 q^{71} +3.58242 q^{73} +7.00000 q^{75} +1.80538 q^{77} -3.78600 q^{79} -5.67660 q^{81} +14.6918 q^{83} +3.07480 q^{85} -7.10941 q^{87} +2.14961 q^{89} +2.12878 q^{91} -9.45219 q^{93} -1.22441 q^{95} +1.05398 q^{97} +4.80538 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41} + 4 q^{43} - 2 q^{45} + 11 q^{47} + 8 q^{49} + 18 q^{51} + 3 q^{53} - 9 q^{55} - 10 q^{57} + 2 q^{59} - 15 q^{61} + 12 q^{63} - 10 q^{65} + 5 q^{67} - 3 q^{69} - 5 q^{71} - 6 q^{73} + 21 q^{75} + 15 q^{77} - q^{79} - 25 q^{81} + 9 q^{83} + 14 q^{85} - 3 q^{87} + 16 q^{89} + 8 q^{91} - 22 q^{93} - 18 q^{95} + 24 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * q^99

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
$3$	−1.46260	−0.844432	−0.422216	−	0.906495i	$-0.638748\pi$
$3$	−1.46260	−0.844432	−0.422216	+	0.906495i	$0.638748\pi$
$4$	0	0
$5$	−0.462598	−0.206880	−0.103440	−	0.994636i	$-0.532985\pi$
$5$	−0.462598	−0.206880	−0.103440	+	0.994636i	$0.532985\pi$
$6$	0	0
$7$	−0.323404	−0.122235	−0.0611177	−	0.998131i	$-0.519466\pi$
$7$	−0.323404	−0.122235	−0.0611177	+	0.998131i	$0.519466\pi$
$8$	0	0
$9$	−0.860806	−0.286935
$10$	0	0
$11$	−5.58242	−1.68316	−0.841581	−	0.540131i	$-0.818375\pi$
$11$	−5.58242	−1.68316	−0.841581	+	0.540131i	$0.818375\pi$
$12$	0	0
$13$	−6.58242	−1.82563	−0.912817	−	0.408369i	$-0.866098\pi$
$13$	−6.58242	−1.82563	−0.912817	+	0.408369i	$0.866098\pi$
$14$	0	0
$15$	0.676596	0.174696
$16$	0	0
$17$	−6.64681	−1.61209	−0.806044	−	0.591856i	$-0.798396\pi$
$17$	−6.64681	−1.61209	−0.806044	+	0.591856i	$0.798396\pi$
$18$	0	0
$19$	2.64681	0.607220	0.303610	−	0.952796i	$-0.401808\pi$
$19$	2.64681	0.607220	0.303610	+	0.952796i	$0.401808\pi$
$20$	0	0
$21$	0.473011	0.103219
$22$	0	0
$23$	4.86081	1.01355	0.506774	−	0.862079i	$-0.330838\pi$
$23$	4.86081	1.01355	0.506774	+	0.862079i	$0.330838\pi$
$24$	0	0
$25$	−4.78600	−0.957201
$26$	0	0
$27$	5.64681	1.08673
$28$	0	0
$29$	4.86081	0.902629	0.451314	−	0.892365i	$-0.350955\pi$
$29$	4.86081	0.902629	0.451314	+	0.892365i	$0.350955\pi$
$30$	0	0
$31$	6.46260	1.16072	0.580358	−	0.814361i	$-0.302912\pi$
$31$	6.46260	1.16072	0.580358	+	0.814361i	$0.302912\pi$
$32$	0	0
$33$	8.16484	1.42132
$34$	0	0
$35$	0.149606	0.0252881
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	9.62743	1.54162
$40$	0	0
$41$	−0.815790	−0.127405	−0.0637025	−	0.997969i	$-0.520291\pi$
$41$	−0.815790	−0.127405	−0.0637025	+	0.997969i	$0.520291\pi$
$42$	0	0
$43$	1.87122	0.285358	0.142679	−	0.989769i	$-0.454428\pi$
$43$	1.87122	0.285358	0.142679	+	0.989769i	$0.454428\pi$
$44$	0	0
$45$	0.398207	0.0593613
$46$	0	0
$47$	−1.11982	−0.163342	−0.0816712	−	0.996659i	$-0.526026\pi$
$47$	−1.11982	−0.163342	−0.0816712	+	0.996659i	$0.526026\pi$
$48$	0	0
$49$	−6.89541	−0.985059
$50$	0	0
$51$	9.72161	1.36130
$52$	0	0
$53$	12.6918	1.74336	0.871678	−	0.490079i	$-0.163032\pi$
$53$	12.6918	1.74336	0.871678	+	0.490079i	$0.163032\pi$
$54$	0	0
$55$	2.58242	0.348213
$56$	0	0
$57$	−3.87122	−0.512755
$58$	0	0
$59$	0.128782	0.0167660	0.00838299	−	0.999965i	$-0.497332\pi$
$59$	0.128782	0.0167660	0.00838299	+	0.999965i	$0.497332\pi$
$60$	0	0
$61$	1.10941	0.142045	0.0710225	−	0.997475i	$-0.477374\pi$
$61$	1.10941	0.142045	0.0710225	+	0.997475i	$0.477374\pi$
$62$	0	0
$63$	0.278388	0.0350736
$64$	0	0
$65$	3.04502	0.377688
$66$	0	0
$67$	−13.4778	−1.64658	−0.823289	−	0.567622i	$-0.807864\pi$
$67$	−13.4778	−1.64658	−0.823289	+	0.567622i	$0.807864\pi$
$68$	0	0
$69$	−7.10941	−0.855872
$70$	0	0
$71$	−8.04502	−0.954768	−0.477384	−	0.878695i	$-0.658415\pi$
$71$	−8.04502	−0.954768	−0.477384	+	0.878695i	$0.658415\pi$
$72$	0	0
$73$	3.58242	0.419290	0.209645	−	0.977778i	$-0.432769\pi$
$73$	3.58242	0.419290	0.209645	+	0.977778i	$0.432769\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	1.80538	0.205742
$78$	0	0
$79$	−3.78600	−0.425959	−0.212979	−	0.977057i	$-0.568317\pi$
$79$	−3.78600	−0.425959	−0.212979	+	0.977057i	$0.568317\pi$
$80$	0	0
$81$	−5.67660	−0.630733
$82$	0	0
$83$	14.6918	1.61264	0.806319	−	0.591481i	$-0.201457\pi$
$83$	14.6918	1.61264	0.806319	+	0.591481i	$0.201457\pi$
$84$	0	0
$85$	3.07480	0.333509
$86$	0	0
$87$	−7.10941	−0.762208
$88$	0	0
$89$	2.14961	0.227858	0.113929	−	0.993489i	$-0.463656\pi$
$89$	2.14961	0.227858	0.113929	+	0.993489i	$0.463656\pi$
$90$	0	0
$91$	2.12878	0.223157
$92$	0	0
$93$	−9.45219	−0.980146
$94$	0	0
$95$	−1.22441	−0.125622
$96$	0	0
$97$	1.05398	0.107015	0.0535077	−	0.998567i	$-0.482960\pi$
$97$	1.05398	0.107015	0.0535077	+	0.998567i	$0.482960\pi$
$98$	0	0
$99$	4.80538	0.482959
$100$	0	0
$101$	−8.69182	−0.864869	−0.432434	−	0.901665i	$-0.642345\pi$
$101$	−8.69182	−0.864869	−0.432434	+	0.901665i	$0.642345\pi$
$102$	0	0
$103$	3.66618	0.361240	0.180620	−	0.983553i	$-0.442190\pi$
$103$	3.66618	0.361240	0.180620	+	0.983553i	$0.442190\pi$
$104$	0	0
$105$	−0.218814	−0.0213541
$106$	0	0
$107$	18.0048	1.74059	0.870296	−	0.492530i	$-0.163928\pi$
$107$	18.0048	1.74059	0.870296	+	0.492530i	$0.163928\pi$
$108$	0	0
$109$	−16.7964	−1.60880	−0.804402	−	0.594085i	$-0.797514\pi$
$109$	−16.7964	−1.60880	−0.804402	+	0.594085i	$0.797514\pi$
$110$	0	0
$111$	−1.46260	−0.138824
$112$	0	0
$113$	−7.59283	−0.714273	−0.357137	−	0.934052i	$-0.616247\pi$
$113$	−7.59283	−0.714273	−0.357137	+	0.934052i	$0.616247\pi$
$114$	0	0
$115$	−2.24860	−0.209683
$116$	0	0
$117$	5.66618	0.523839
$118$	0	0
$119$	2.14961	0.197054
$120$	0	0
$121$	20.1634	1.83304
$122$	0	0
$123$	1.19317	0.107585
$124$	0	0
$125$	4.52699	0.404906
$126$	0	0
$127$	18.2847	1.62250	0.811250	−	0.584699i	$-0.198787\pi$
$127$	18.2847	1.62250	0.811250	+	0.584699i	$0.198787\pi$
$128$	0	0
$129$	−2.73684	−0.240965
$130$	0	0
$131$	−9.16484	−0.800735	−0.400368	−	0.916355i	$-0.631118\pi$
$131$	−9.16484	−0.800735	−0.400368	+	0.916355i	$0.631118\pi$
$132$	0	0
$133$	−0.855989	−0.0742237
$134$	0	0
$135$	−2.61220	−0.224823
$136$	0	0
$137$	15.4778	1.32236	0.661180	−	0.750227i	$-0.270056\pi$
$137$	15.4778	1.32236	0.661180	+	0.750227i	$0.270056\pi$
$138$	0	0
$139$	6.73202	0.571003	0.285501	−	0.958378i	$-0.407840\pi$
$139$	6.73202	0.571003	0.285501	+	0.958378i	$0.407840\pi$
$140$	0	0
$141$	1.63785	0.137931
$142$	0	0
$143$	36.7458	3.07284
$144$	0	0
$145$	−2.24860	−0.186736
$146$	0	0
$147$	10.0852	0.831815
$148$	0	0
$149$	−0.323404	−0.0264943	−0.0132472	−	0.999912i	$-0.504217\pi$
$149$	−0.323404	−0.0264943	−0.0132472	+	0.999912i	$0.504217\pi$
$150$	0	0
$151$	−11.8116	−0.961218	−0.480609	−	0.876935i	$-0.659584\pi$
$151$	−11.8116	−0.961218	−0.480609	+	0.876935i	$0.659584\pi$
$152$	0	0
$153$	5.72161	0.462565
$154$	0	0
$155$	−2.98959	−0.240129
$156$	0	0
$157$	−12.6918	−1.01292	−0.506459	−	0.862264i	$-0.669046\pi$
$157$	−12.6918	−1.01292	−0.506459	+	0.862264i	$0.669046\pi$
$158$	0	0
$159$	−18.5630	−1.47215
$160$	0	0
$161$	−1.57201	−0.123891
$162$	0	0
$163$	−11.8504	−0.928194	−0.464097	−	0.885784i	$-0.653621\pi$
$163$	−11.8504	−0.928194	−0.464097	+	0.885784i	$0.653621\pi$
$164$	0	0
$165$	−3.77704	−0.294042
$166$	0	0
$167$	6.20503	0.480160	0.240080	−	0.970753i	$-0.422826\pi$
$167$	6.20503	0.480160	0.240080	+	0.970753i	$0.422826\pi$
$168$	0	0
$169$	30.3282	2.33294
$170$	0	0
$171$	−2.27839	−0.174233
$172$	0	0
$173$	7.39821	0.562475	0.281238	−	0.959638i	$-0.409255\pi$
$173$	7.39821	0.562475	0.281238	+	0.959638i	$0.409255\pi$
$174$	0	0
$175$	1.54781	0.117004
$176$	0	0
$177$	−0.188356	−0.0141577
$178$	0	0
$179$	−7.97918	−0.596392	−0.298196	−	0.954505i	$-0.596385\pi$
$179$	−7.97918	−0.596392	−0.298196	+	0.954505i	$0.596385\pi$
$180$	0	0
$181$	9.48824	0.705255	0.352628	−	0.935764i	$-0.385288\pi$
$181$	9.48824	0.705255	0.352628	+	0.935764i	$0.385288\pi$
$182$	0	0
$183$	−1.62262	−0.119947
$184$	0	0
$185$	−0.462598	−0.0340109
$186$	0	0
$187$	37.1053	2.71341
$188$	0	0
$189$	−1.82620	−0.132837
$190$	0	0
$191$	19.1994	1.38922	0.694611	−	0.719385i	$-0.255576\pi$
$191$	19.1994	1.38922	0.694611	+	0.719385i	$0.255576\pi$
$192$	0	0
$193$	−16.0900	−1.15819	−0.579093	−	0.815262i	$-0.696593\pi$
$193$	−16.0900	−1.15819	−0.579093	+	0.815262i	$0.696593\pi$
$194$	0	0
$195$	−4.45364	−0.318931
$196$	0	0
$197$	−0.691825	−0.0492905	−0.0246452	−	0.999696i	$-0.507846\pi$
$197$	−0.691825	−0.0492905	−0.0246452	+	0.999696i	$0.507846\pi$
$198$	0	0
$199$	2.79641	0.198233	0.0991163	−	0.995076i	$-0.468398\pi$
$199$	2.79641	0.198233	0.0991163	+	0.995076i	$0.468398\pi$
$200$	0	0
$201$	19.7126	1.39042
$202$	0	0
$203$	−1.57201	−0.110333
$204$	0	0
$205$	0.377383	0.0263576
$206$	0	0
$207$	−4.18421	−0.290823
$208$	0	0
$209$	−14.7756	−1.02205
$210$	0	0
$211$	−13.2140	−0.909689	−0.454845	−	0.890571i	$-0.650305\pi$
$211$	−13.2140	−0.909689	−0.454845	+	0.890571i	$0.650305\pi$
$212$	0	0
$213$	11.7666	0.806236
$214$	0	0
$215$	−0.865623	−0.0590350
$216$	0	0
$217$	−2.09003	−0.141881
$218$	0	0
$219$	−5.23964	−0.354062
$220$	0	0
$221$	43.7521	2.94308
$222$	0	0
$223$	15.6170	1.04579	0.522897	−	0.852396i	$-0.324851\pi$
$223$	15.6170	1.04579	0.522897	+	0.852396i	$0.324851\pi$
$224$	0	0
$225$	4.11982	0.274655
$226$	0	0
$227$	3.07480	0.204082	0.102041	−	0.994780i	$-0.467463\pi$
$227$	3.07480	0.204082	0.102041	+	0.994780i	$0.467463\pi$
$228$	0	0
$229$	12.7306	0.841260	0.420630	−	0.907232i	$-0.361809\pi$
$229$	12.7306	0.841260	0.420630	+	0.907232i	$0.361809\pi$
$230$	0	0
$231$	−2.64054	−0.173735
$232$	0	0
$233$	−18.8221	−1.23307	−0.616537	−	0.787326i	$-0.711465\pi$
$233$	−18.8221	−1.23307	−0.616537	+	0.787326i	$0.711465\pi$
$234$	0	0
$235$	0.518027	0.0337923
$236$	0	0
$237$	5.53740	0.359693
$238$	0	0
$239$	−6.18421	−0.400023	−0.200012	−	0.979794i	$-0.564098\pi$
$239$	−6.18421	−0.400023	−0.200012	+	0.979794i	$0.564098\pi$
$240$	0	0
$241$	17.9404	1.15564	0.577822	−	0.816163i	$-0.303903\pi$
$241$	17.9404	1.15564	0.577822	+	0.816163i	$0.303903\pi$
$242$	0	0
$243$	−8.63785	−0.554118
$244$	0	0
$245$	3.18981	0.203789
$246$	0	0
$247$	−17.4224	−1.10856
$248$	0	0
$249$	−21.4882	−1.36176
$250$	0	0
$251$	5.20359	0.328447	0.164224	−	0.986423i	$-0.447488\pi$
$251$	5.20359	0.328447	0.164224	+	0.986423i	$0.447488\pi$
$252$	0	0
$253$	−27.1350	−1.70597
$254$	0	0
$255$	−4.49720	−0.281626
$256$	0	0
$257$	−6.38924	−0.398550	−0.199275	−	0.979944i	$-0.563859\pi$
$257$	−6.38924	−0.398550	−0.199275	+	0.979944i	$0.563859\pi$
$258$	0	0
$259$	−0.323404	−0.0200954
$260$	0	0
$261$	−4.18421	−0.258996
$262$	0	0
$263$	−20.0063	−1.23364	−0.616820	−	0.787105i	$-0.711579\pi$
$263$	−20.0063	−1.23364	−0.616820	+	0.787105i	$0.711579\pi$
$264$	0	0
$265$	−5.87122	−0.360666
$266$	0	0
$267$	−3.14401	−0.192410
$268$	0	0
$269$	−27.7908	−1.69444	−0.847218	−	0.531245i	$-0.821724\pi$
$269$	−27.7908	−1.69444	−0.847218	+	0.531245i	$0.821724\pi$
$270$	0	0
$271$	−16.8414	−1.02304	−0.511522	−	0.859270i	$-0.670918\pi$
$271$	−16.8414	−1.02304	−0.511522	+	0.859270i	$0.670918\pi$
$272$	0	0
$273$	−3.11355	−0.188441
$274$	0	0
$275$	26.7175	1.61112
$276$	0	0
$277$	−4.55263	−0.273541	−0.136771	−	0.990603i	$-0.543672\pi$
$277$	−4.55263	−0.273541	−0.136771	+	0.990603i	$0.543672\pi$
$278$	0	0
$279$	−5.56304	−0.333051
$280$	0	0
$281$	18.8864	1.12667	0.563335	−	0.826228i	$-0.309518\pi$
$281$	18.8864	1.12667	0.563335	+	0.826228i	$0.309518\pi$
$282$	0	0
$283$	−27.8116	−1.65323	−0.826615	−	0.562767i	$-0.809737\pi$
$283$	−27.8116	−1.65323	−0.826615	+	0.562767i	$0.809737\pi$
$284$	0	0
$285$	1.79082	0.106079
$286$	0	0
$287$	0.263830	0.0155734
$288$	0	0
$289$	27.1801	1.59883
$290$	0	0
$291$	−1.54155	−0.0903671
$292$	0	0
$293$	22.1288	1.29278	0.646389	−	0.763008i	$-0.276279\pi$
$293$	22.1288	1.29278	0.646389	+	0.763008i	$0.276279\pi$
$294$	0	0
$295$	−0.0595743	−0.00346855
$296$	0	0
$297$	−31.5228	−1.82914
$298$	0	0
$299$	−31.9959	−1.85037
$300$	0	0
$301$	−0.605160	−0.0348808
$302$	0	0
$303$	12.7126	0.730323
$304$	0	0
$305$	−0.513210	−0.0293863
$306$	0	0
$307$	15.3130	0.873959	0.436979	−	0.899472i	$-0.356048\pi$
$307$	15.3130	0.873959	0.436979	+	0.899472i	$0.356048\pi$
$308$	0	0
$309$	−5.36215	−0.305042
$310$	0	0
$311$	22.0644	1.25116	0.625578	−	0.780161i	$-0.284863\pi$
$311$	22.0644	1.25116	0.625578	+	0.780161i	$0.284863\pi$
$312$	0	0
$313$	−24.2188	−1.36893	−0.684464	−	0.729047i	$-0.739964\pi$
$313$	−24.2188	−1.36893	−0.684464	+	0.729047i	$0.739964\pi$
$314$	0	0
$315$	−0.128782	−0.00725604
$316$	0	0
$317$	13.3836	0.751701	0.375850	−	0.926680i	$-0.377351\pi$
$317$	13.3836	0.751701	0.375850	+	0.926680i	$0.377351\pi$
$318$	0	0
$319$	−27.1350	−1.51927
$320$	0	0
$321$	−26.3338	−1.46981
$322$	0	0
$323$	−17.5928	−0.978891
$324$	0	0
$325$	31.5035	1.74750
$326$	0	0
$327$	24.5664	1.35853
$328$	0	0
$329$	0.362154	0.0199662
$330$	0	0
$331$	−8.09003	−0.444668	−0.222334	−	0.974971i	$-0.571368\pi$
$331$	−8.09003	−0.444668	−0.222334	+	0.974971i	$0.571368\pi$
$332$	0	0
$333$	−0.860806	−0.0471719
$334$	0	0
$335$	6.23482	0.340645
$336$	0	0
$337$	−12.4176	−0.676429	−0.338214	−	0.941069i	$-0.609823\pi$
$337$	−12.4176	−0.676429	−0.338214	+	0.941069i	$0.609823\pi$
$338$	0	0
$339$	11.1053	0.603155
$340$	0	0
$341$	−36.0769	−1.95367
$342$	0	0
$343$	4.49383	0.242644
$344$	0	0
$345$	3.28880	0.177063
$346$	0	0
$347$	31.7216	1.70291	0.851453	−	0.524431i	$-0.175722\pi$
$347$	31.7216	1.70291	0.851453	+	0.524431i	$0.175722\pi$
$348$	0	0
$349$	−19.7216	−1.05567	−0.527837	−	0.849346i	$-0.676997\pi$
$349$	−19.7216	−1.05567	−0.527837	+	0.849346i	$0.676997\pi$
$350$	0	0
$351$	−37.1697	−1.98397
$352$	0	0
$353$	7.07480	0.376554	0.188277	−	0.982116i	$-0.439710\pi$
$353$	7.07480	0.376554	0.188277	+	0.982116i	$0.439710\pi$
$354$	0	0
$355$	3.72161	0.197523
$356$	0	0
$357$	−3.14401	−0.166399
$358$	0	0
$359$	20.9910	1.10786	0.553932	−	0.832562i	$-0.313127\pi$
$359$	20.9910	1.10786	0.553932	+	0.832562i	$0.313127\pi$
$360$	0	0
$361$	−11.9944	−0.631284
$362$	0	0
$363$	−29.4909	−1.54787
$364$	0	0
$365$	−1.65722	−0.0867429
$366$	0	0
$367$	12.8656	0.671580	0.335790	−	0.941937i	$-0.390997\pi$
$367$	12.8656	0.671580	0.335790	+	0.941937i	$0.390997\pi$
$368$	0	0
$369$	0.702237	0.0365570
$370$	0	0
$371$	−4.10459	−0.213100
$372$	0	0
$373$	27.7071	1.43462	0.717308	−	0.696756i	$-0.245374\pi$
$373$	27.7071	1.43462	0.717308	+	0.696756i	$0.245374\pi$
$374$	0	0
$375$	−6.62117	−0.341916
$376$	0	0
$377$	−31.9959	−1.64787
$378$	0	0
$379$	18.0194	0.925593	0.462797	−	0.886465i	$-0.346846\pi$
$379$	18.0194	0.925593	0.462797	+	0.886465i	$0.346846\pi$
$380$	0	0
$381$	−26.7431	−1.37009
$382$	0	0
$383$	24.1801	1.23554	0.617772	−	0.786357i	$-0.288036\pi$
$383$	24.1801	1.23554	0.617772	+	0.786357i	$0.288036\pi$
$384$	0	0
$385$	−0.835165	−0.0425639
$386$	0	0
$387$	−1.61076	−0.0818793
$388$	0	0
$389$	−23.7860	−1.20600	−0.602999	−	0.797742i	$-0.706028\pi$
$389$	−23.7860	−1.20600	−0.602999	+	0.797742i	$0.706028\pi$
$390$	0	0
$391$	−32.3088	−1.63393
$392$	0	0
$393$	13.4045	0.676166
$394$	0	0
$395$	1.75140	0.0881224
$396$	0	0
$397$	17.2099	0.863738	0.431869	−	0.901936i	$-0.357854\pi$
$397$	17.2099	0.863738	0.431869	+	0.901936i	$0.357854\pi$
$398$	0	0
$399$	1.25197	0.0626768
$400$	0	0
$401$	−4.21881	−0.210678	−0.105339	−	0.994436i	$-0.533593\pi$
$401$	−4.21881	−0.210678	−0.105339	+	0.994436i	$0.533593\pi$
$402$	0	0
$403$	−42.5395	−2.11904
$404$	0	0
$405$	2.62598	0.130486
$406$	0	0
$407$	−5.58242	−0.276710
$408$	0	0
$409$	37.4045	1.84953	0.924766	−	0.380536i	$-0.124261\pi$
$409$	37.4045	1.84953	0.924766	+	0.380536i	$0.124261\pi$
$410$	0	0
$411$	−22.6378	−1.11664
$412$	0	0
$413$	−0.0416486	−0.00204940
$414$	0	0
$415$	−6.79641	−0.333623
$416$	0	0
$417$	−9.84625	−0.482173
$418$	0	0
$419$	6.62743	0.323771	0.161886	−	0.986810i	$-0.448242\pi$
$419$	6.62743	0.323771	0.161886	+	0.986810i	$0.448242\pi$
$420$	0	0
$421$	33.5589	1.63556	0.817780	−	0.575531i	$-0.195204\pi$
$421$	33.5589	1.63556	0.817780	+	0.575531i	$0.195204\pi$
$422$	0	0
$423$	0.963947	0.0468687
$424$	0	0
$425$	31.8116	1.54309
$426$	0	0
$427$	−0.358787	−0.0173629
$428$	0	0
$429$	−53.7444	−2.59480
$430$	0	0
$431$	−6.53885	−0.314965	−0.157483	−	0.987522i	$-0.550338\pi$
$431$	−6.53885	−0.314965	−0.157483	+	0.987522i	$0.550338\pi$
$432$	0	0
$433$	−18.3699	−0.882800	−0.441400	−	0.897311i	$-0.645518\pi$
$433$	−18.3699	−0.882800	−0.441400	+	0.897311i	$0.645518\pi$
$434$	0	0
$435$	3.28880	0.157686
$436$	0	0
$437$	12.8656	0.615446
$438$	0	0
$439$	19.3580	0.923907	0.461954	−	0.886904i	$-0.347149\pi$
$439$	19.3580	0.923907	0.461954	+	0.886904i	$0.347149\pi$
$440$	0	0
$441$	5.93561	0.282648
$442$	0	0
$443$	5.28320	0.251013	0.125506	−	0.992093i	$-0.459944\pi$
$443$	5.28320	0.251013	0.125506	+	0.992093i	$0.459944\pi$
$444$	0	0
$445$	−0.994404	−0.0471393
$446$	0	0
$447$	0.473011	0.0223726
$448$	0	0
$449$	−10.3476	−0.488333	−0.244167	−	0.969733i	$-0.578514\pi$
$449$	−10.3476	−0.488333	−0.244167	+	0.969733i	$0.578514\pi$
$450$	0	0
$451$	4.55408	0.214443
$452$	0	0
$453$	17.2757	0.811683
$454$	0	0
$455$	−0.984771	−0.0461668
$456$	0	0
$457$	26.3684	1.23346	0.616731	−	0.787174i	$-0.288456\pi$
$457$	26.3684	1.23346	0.616731	+	0.787174i	$0.288456\pi$
$458$	0	0
$459$	−37.5333	−1.75190
$460$	0	0
$461$	3.89204	0.181270	0.0906352	−	0.995884i	$-0.471110\pi$
$461$	3.89204	0.181270	0.0906352	+	0.995884i	$0.471110\pi$
$462$	0	0
$463$	−33.4689	−1.55543	−0.777715	−	0.628617i	$-0.783621\pi$
$463$	−33.4689	−1.55543	−0.777715	+	0.628617i	$0.783621\pi$
$464$	0	0
$465$	4.37257	0.202773
$466$	0	0
$467$	−29.1261	−1.34779	−0.673897	−	0.738825i	$-0.735381\pi$
$467$	−29.1261	−1.34779	−0.673897	+	0.738825i	$0.735381\pi$
$468$	0	0
$469$	4.35879	0.201270
$470$	0	0
$471$	18.5630	0.855340
$472$	0	0
$473$	−10.4459	−0.480304
$474$	0	0
$475$	−12.6676	−0.581231
$476$	0	0
$477$	−10.9252	−0.500230
$478$	0	0
$479$	−8.00482	−0.365749	−0.182875	−	0.983136i	$-0.558540\pi$
$479$	−8.00482	−0.365749	−0.182875	+	0.983136i	$0.558540\pi$
$480$	0	0
$481$	−6.58242	−0.300132
$482$	0	0
$483$	2.29921	0.104618
$484$	0	0
$485$	−0.487569	−0.0221394
$486$	0	0
$487$	−10.1496	−0.459923	−0.229961	−	0.973200i	$-0.573860\pi$
$487$	−10.1496	−0.459923	−0.229961	+	0.973200i	$0.573860\pi$
$488$	0	0
$489$	17.3324	0.783797
$490$	0	0
$491$	4.88163	0.220305	0.110152	−	0.993915i	$-0.464866\pi$
$491$	4.88163	0.220305	0.110152	+	0.993915i	$0.464866\pi$
$492$	0	0
$493$	−32.3088	−1.45512
$494$	0	0
$495$	−2.22296	−0.0999146
$496$	0	0
$497$	2.60179	0.116706
$498$	0	0
$499$	−13.1648	−0.589339	−0.294669	−	0.955599i	$-0.595210\pi$
$499$	−13.1648	−0.589339	−0.294669	+	0.955599i	$0.595210\pi$
$500$	0	0
$501$	−9.07547	−0.405462
$502$	0	0
$503$	16.9806	0.757129	0.378564	−	0.925575i	$-0.376418\pi$
$503$	16.9806	0.757129	0.378564	+	0.925575i	$0.376418\pi$
$504$	0	0
$505$	4.02082	0.178924
$506$	0	0
$507$	−44.3580	−1.97001
$508$	0	0
$509$	−24.1142	−1.06884	−0.534422	−	0.845218i	$-0.679471\pi$
$509$	−24.1142	−1.06884	−0.534422	+	0.845218i	$0.679471\pi$
$510$	0	0
$511$	−1.15857	−0.0512521
$512$	0	0
$513$	14.9460	0.659883
$514$	0	0
$515$	−1.69597	−0.0747334
$516$	0	0
$517$	6.25130	0.274932
$518$	0	0
$519$	−10.8206	−0.474972
$520$	0	0
$521$	14.1046	0.617933	0.308967	−	0.951073i	$-0.400017\pi$
$521$	14.1046	0.617933	0.308967	+	0.951073i	$0.400017\pi$
$522$	0	0
$523$	41.2549	1.80395	0.901975	−	0.431789i	$-0.142117\pi$
$523$	41.2549	1.80395	0.901975	+	0.431789i	$0.142117\pi$
$524$	0	0
$525$	−2.26383	−0.0988016
$526$	0	0
$527$	−42.9557	−1.87118
$528$	0	0
$529$	0.627434	0.0272797
$530$	0	0
$531$	−0.110856	−0.00481075
$532$	0	0
$533$	5.36987	0.232595
$534$	0	0
$535$	−8.32900	−0.360094
$536$	0	0
$537$	11.6703	0.503612
$538$	0	0
$539$	38.4931	1.65801
$540$	0	0
$541$	1.10941	0.0476971	0.0238486	−	0.999716i	$-0.492408\pi$
$541$	1.10941	0.0476971	0.0238486	+	0.999716i	$0.492408\pi$
$542$	0	0
$543$	−13.8775	−0.595540
$544$	0	0
$545$	7.77000	0.332830
$546$	0	0
$547$	22.7756	0.973814	0.486907	−	0.873454i	$-0.338125\pi$
$547$	22.7756	0.973814	0.486907	+	0.873454i	$0.338125\pi$
$548$	0	0
$549$	−0.954984	−0.0407577
$550$	0	0
$551$	12.8656	0.548094
$552$	0	0
$553$	1.22441	0.0520672
$554$	0	0
$555$	0.676596	0.0287199
$556$	0	0
$557$	4.72016	0.200000	0.0999998	−	0.994987i	$-0.468116\pi$
$557$	4.72016	0.200000	0.0999998	+	0.994987i	$0.468116\pi$
$558$	0	0
$559$	−12.3171	−0.520959
$560$	0	0
$561$	−54.2701	−2.29129
$562$	0	0
$563$	44.7673	1.88672	0.943358	−	0.331776i	$-0.107648\pi$
$563$	44.7673	1.88672	0.943358	+	0.331776i	$0.107648\pi$
$564$	0	0
$565$	3.51243	0.147769
$566$	0	0
$567$	1.83584	0.0770978
$568$	0	0
$569$	23.7008	0.993589	0.496794	−	0.867868i	$-0.334510\pi$
$569$	23.7008	0.993589	0.496794	+	0.867868i	$0.334510\pi$
$570$	0	0
$571$	−32.2590	−1.35000	−0.674999	−	0.737819i	$-0.735856\pi$
$571$	−32.2590	−1.35000	−0.674999	+	0.737819i	$0.735856\pi$
$572$	0	0
$573$	−28.0811	−1.17310
$574$	0	0
$575$	−23.2638	−0.970169
$576$	0	0
$577$	0.427995	0.0178176	0.00890882	−	0.999960i	$-0.497164\pi$
$577$	0.427995	0.0178176	0.00890882	+	0.999960i	$0.497164\pi$
$578$	0	0
$579$	23.5333	0.978009
$580$	0	0
$581$	−4.75140	−0.197121
$582$	0	0
$583$	−70.8511	−2.93435
$584$	0	0
$585$	−2.62117	−0.108372
$586$	0	0
$587$	−35.3836	−1.46044	−0.730220	−	0.683212i	$-0.760582\pi$
$587$	−35.3836	−1.46044	−0.730220	+	0.683212i	$0.760582\pi$
$588$	0	0
$589$	17.1053	0.704810
$590$	0	0
$591$	1.01186	0.0416224
$592$	0	0
$593$	−1.27357	−0.0522993	−0.0261497	−	0.999658i	$-0.508325\pi$
$593$	−1.27357	−0.0522993	−0.0261497	+	0.999658i	$0.508325\pi$
$594$	0	0
$595$	−0.994404	−0.0407666
$596$	0	0
$597$	−4.09003	−0.167394
$598$	0	0
$599$	27.2791	1.11459	0.557296	−	0.830314i	$-0.311839\pi$
$599$	27.2791	1.11459	0.557296	+	0.830314i	$0.311839\pi$
$600$	0	0
$601$	−25.2590	−1.03034	−0.515168	−	0.857089i	$-0.672271\pi$
$601$	−25.2590	−1.03034	−0.515168	+	0.857089i	$0.672271\pi$
$602$	0	0
$603$	11.6018	0.472462
$604$	0	0
$605$	−9.32755	−0.379219
$606$	0	0
$607$	17.9356	0.727984	0.363992	−	0.931402i	$-0.381414\pi$
$607$	17.9356	0.727984	0.363992	+	0.931402i	$0.381414\pi$
$608$	0	0
$609$	2.29921	0.0931688
$610$	0	0
$611$	7.37112	0.298203
$612$	0	0
$613$	−8.30548	−0.335455	−0.167728	−	0.985833i	$-0.553643\pi$
$613$	−8.30548	−0.335455	−0.167728	+	0.985833i	$0.553643\pi$
$614$	0	0
$615$	−0.551960	−0.0222572
$616$	0	0
$617$	−10.5762	−0.425780	−0.212890	−	0.977076i	$-0.568288\pi$
$617$	−10.5762	−0.425780	−0.212890	+	0.977076i	$0.568288\pi$
$618$	0	0
$619$	−6.39676	−0.257107	−0.128554	−	0.991703i	$-0.541033\pi$
$619$	−6.39676	−0.257107	−0.128554	+	0.991703i	$0.541033\pi$
$620$	0	0
$621$	27.4480	1.10145
$622$	0	0
$623$	−0.695192	−0.0278523
$624$	0	0
$625$	21.8358	0.873433
$626$	0	0
$627$	21.6108	0.863050
$628$	0	0
$629$	−6.64681	−0.265026
$630$	0	0
$631$	−18.4924	−0.736170	−0.368085	−	0.929792i	$-0.619987\pi$
$631$	−18.4924	−0.736170	−0.368085	+	0.929792i	$0.619987\pi$
$632$	0	0
$633$	19.3268	0.768170
$634$	0	0
$635$	−8.45845	−0.335663
$636$	0	0
$637$	45.3885	1.79836
$638$	0	0
$639$	6.92520	0.273957
$640$	0	0
$641$	5.59138	0.220846	0.110423	−	0.993885i	$-0.464779\pi$
$641$	5.59138	0.220846	0.110423	+	0.993885i	$0.464779\pi$
$642$	0	0
$643$	−6.36842	−0.251146	−0.125573	−	0.992084i	$-0.540077\pi$
$643$	−6.36842	−0.251146	−0.125573	+	0.992084i	$0.540077\pi$
$644$	0	0
$645$	1.26606	0.0498510
$646$	0	0
$647$	4.34278	0.170732	0.0853661	−	0.996350i	$-0.472794\pi$
$647$	4.34278	0.170732	0.0853661	+	0.996350i	$0.472794\pi$
$648$	0	0
$649$	−0.718915	−0.0282199
$650$	0	0
$651$	3.05688	0.119808
$652$	0	0
$653$	15.6572	0.612714	0.306357	−	0.951917i	$-0.400890\pi$
$653$	15.6572	0.612714	0.306357	+	0.951917i	$0.400890\pi$
$654$	0	0
$655$	4.23964	0.165656
$656$	0	0
$657$	−3.08377	−0.120309
$658$	0	0
$659$	2.36697	0.0922041	0.0461020	−	0.998937i	$-0.485320\pi$
$659$	2.36697	0.0922041	0.0461020	+	0.998937i	$0.485320\pi$
$660$	0	0
$661$	−20.1365	−0.783219	−0.391609	−	0.920132i	$-0.628082\pi$
$661$	−20.1365	−0.783219	−0.391609	+	0.920132i	$0.628082\pi$
$662$	0	0
$663$	−63.9917	−2.48523
$664$	0	0
$665$	0.395979	0.0153554
$666$	0	0
$667$	23.6274	0.914858
$668$	0	0
$669$	−22.8414	−0.883101
$670$	0	0
$671$	−6.19317	−0.239085
$672$	0	0
$673$	−26.3670	−1.01637	−0.508186	−	0.861247i	$-0.669684\pi$
$673$	−26.3670	−1.01637	−0.508186	+	0.861247i	$0.669684\pi$
$674$	0	0
$675$	−27.0256	−1.04022
$676$	0	0
$677$	43.9050	1.68741	0.843704	−	0.536809i	$-0.180370\pi$
$677$	43.9050	1.68741	0.843704	+	0.536809i	$0.180370\pi$
$678$	0	0
$679$	−0.340861	−0.0130811
$680$	0	0
$681$	−4.49720	−0.172333
$682$	0	0
$683$	−16.3476	−0.625523	−0.312762	−	0.949832i	$-0.601254\pi$
$683$	−16.3476	−0.625523	−0.312762	+	0.949832i	$0.601254\pi$
$684$	0	0
$685$	−7.16002	−0.273570
$686$	0	0
$687$	−18.6197	−0.710387
$688$	0	0
$689$	−83.5429	−3.18273
$690$	0	0
$691$	−32.1801	−1.22419	−0.612094	−	0.790785i	$-0.709672\pi$
$691$	−32.1801	−1.22419	−0.612094	+	0.790785i	$0.709672\pi$
$692$	0	0
$693$	−1.55408	−0.0590346
$694$	0	0
$695$	−3.11422	−0.118129
$696$	0	0
$697$	5.42240	0.205388
$698$	0	0
$699$	27.5291	1.04125
$700$	0	0
$701$	−22.9598	−0.867180	−0.433590	−	0.901110i	$-0.642753\pi$
$701$	−22.9598	−0.867180	−0.433590	+	0.901110i	$0.642753\pi$
$702$	0	0
$703$	2.64681	0.0998263
$704$	0	0
$705$	−0.757665	−0.0285353
$706$	0	0
$707$	2.81097	0.105718
$708$	0	0
$709$	11.8158	0.443751	0.221876	−	0.975075i	$-0.428782\pi$
$709$	11.8158	0.443751	0.221876	+	0.975075i	$0.428782\pi$
$710$	0	0
$711$	3.25901	0.122223
$712$	0	0
$713$	31.4134	1.17644
$714$	0	0
$715$	−16.9986	−0.635710
$716$	0	0
$717$	9.04502	0.337792
$718$	0	0
$719$	20.7998	0.775701	0.387850	−	0.921722i	$-0.373218\pi$
$719$	20.7998	0.775701	0.387850	+	0.921722i	$0.373218\pi$
$720$	0	0
$721$	−1.18566	−0.0441563
$722$	0	0
$723$	−26.2396	−0.975863
$724$	0	0
$725$	−23.2638	−0.863997
$726$	0	0
$727$	−8.79227	−0.326087	−0.163044	−	0.986619i	$-0.552131\pi$
$727$	−8.79227	−0.326087	−0.163044	+	0.986619i	$0.552131\pi$
$728$	0	0
$729$	29.6635	1.09865
$730$	0	0
$731$	−12.4376	−0.460022
$732$	0	0
$733$	20.9010	0.771996	0.385998	−	0.922500i	$-0.373857\pi$
$733$	20.9010	0.771996	0.385998	+	0.922500i	$0.373857\pi$
$734$	0	0
$735$	−4.66540	−0.172086
$736$	0	0
$737$	75.2389	2.77146
$738$	0	0
$739$	26.3401	0.968936	0.484468	−	0.874809i	$-0.339013\pi$
$739$	26.3401	0.968936	0.484468	+	0.874809i	$0.339013\pi$
$740$	0	0
$741$	25.4820	0.936104
$742$	0	0
$743$	32.5630	1.19462	0.597311	−	0.802010i	$-0.296236\pi$
$743$	32.5630	1.19462	0.597311	+	0.802010i	$0.296236\pi$
$744$	0	0
$745$	0.149606	0.00548115
$746$	0	0
$747$	−12.6468	−0.462723
$748$	0	0
$749$	−5.82283	−0.212762
$750$	0	0
$751$	1.54781	0.0564805	0.0282403	−	0.999601i	$-0.491010\pi$
$751$	1.54781	0.0564805	0.0282403	+	0.999601i	$0.491010\pi$
$752$	0	0
$753$	−7.61076	−0.277351
$754$	0	0
$755$	5.46405	0.198857
$756$	0	0
$757$	−51.2299	−1.86198	−0.930991	−	0.365042i	$-0.881055\pi$
$757$	−51.2299	−1.86198	−0.930991	+	0.365042i	$0.881055\pi$
$758$	0	0
$759$	39.6877	1.44057
$760$	0	0
$761$	20.4391	0.740916	0.370458	−	0.928849i	$-0.379201\pi$
$761$	20.4391	0.740916	0.370458	+	0.928849i	$0.379201\pi$
$762$	0	0
$763$	5.43203	0.196653
$764$	0	0
$765$	−2.64681	−0.0956956
$766$	0	0
$767$	−0.847697	−0.0306086
$768$	0	0
$769$	−24.8864	−0.897428	−0.448714	−	0.893675i	$-0.648118\pi$
$769$	−24.8864	−0.897428	−0.448714	+	0.893675i	$0.648118\pi$
$770$	0	0
$771$	9.34490	0.336548
$772$	0	0
$773$	−28.1530	−1.01259	−0.506296	−	0.862360i	$-0.668986\pi$
$773$	−28.1530	−1.01259	−0.506296	+	0.862360i	$0.668986\pi$
$774$	0	0
$775$	−30.9300	−1.11104
$776$	0	0
$777$	0.473011	0.0169692
$778$	0	0
$779$	−2.15924	−0.0773628
$780$	0	0
$781$	44.9106	1.60703
$782$	0	0
$783$	27.4480	0.980913
$784$	0	0
$785$	5.87122	0.209553
$786$	0	0
$787$	24.8898	0.887226	0.443613	−	0.896218i	$-0.353696\pi$
$787$	24.8898	0.887226	0.443613	+	0.896218i	$0.353696\pi$
$788$	0	0
$789$	29.2611	1.04172
$790$	0	0
$791$	2.45555	0.0873094
$792$	0	0
$793$	−7.30258	−0.259322
$794$	0	0
$795$	8.58723	0.304558
$796$	0	0
$797$	28.6420	1.01455	0.507276	−	0.861784i	$-0.330653\pi$
$797$	28.6420	1.01455	0.507276	+	0.861784i	$0.330653\pi$
$798$	0	0
$799$	7.44322	0.263322
$800$	0	0
$801$	−1.85039	−0.0653804
$802$	0	0
$803$	−19.9986	−0.705734
$804$	0	0
$805$	0.727207	0.0256307
$806$	0	0
$807$	40.6468	1.43084
$808$	0	0
$809$	−32.9557	−1.15866	−0.579330	−	0.815093i	$-0.696686\pi$
$809$	−32.9557	−1.15866	−0.579330	+	0.815093i	$0.696686\pi$
$810$	0	0
$811$	−42.5768	−1.49507	−0.747537	−	0.664220i	$-0.768764\pi$
$811$	−42.5768	−1.49507	−0.747537	+	0.664220i	$0.768764\pi$
$812$	0	0
$813$	24.6323	0.863891
$814$	0	0
$815$	5.48197	0.192025
$816$	0	0
$817$	4.95276	0.173275
$818$	0	0
$819$	−1.83247	−0.0640316
$820$	0	0
$821$	44.5727	1.55560	0.777799	−	0.628514i	$-0.216336\pi$
$821$	44.5727	1.55560	0.777799	+	0.628514i	$0.216336\pi$
$822$	0	0
$823$	−1.11355	−0.0388160	−0.0194080	−	0.999812i	$-0.506178\pi$
$823$	−1.11355	−0.0388160	−0.0194080	+	0.999812i	$0.506178\pi$
$824$	0	0
$825$	−39.0769	−1.36048
$826$	0	0
$827$	26.0692	0.906515	0.453258	−	0.891380i	$-0.350262\pi$
$827$	26.0692	0.906515	0.453258	+	0.891380i	$0.350262\pi$
$828$	0	0
$829$	3.78600	0.131493	0.0657467	−	0.997836i	$-0.479057\pi$
$829$	3.78600	0.131493	0.0657467	+	0.997836i	$0.479057\pi$
$830$	0	0
$831$	6.65867	0.230987
$832$	0	0
$833$	45.8325	1.58800
$834$	0	0
$835$	−2.87044	−0.0993356
$836$	0	0
$837$	36.4931	1.26138
$838$	0	0
$839$	34.2605	1.18280	0.591401	−	0.806377i	$-0.298575\pi$
$839$	34.2605	1.18280	0.591401	+	0.806377i	$0.298575\pi$
$840$	0	0
$841$	−5.37257	−0.185261
$842$	0	0
$843$	−27.6233	−0.951397
$844$	0	0
$845$	−14.0298	−0.482639
$846$	0	0
$847$	−6.52093	−0.224062
$848$	0	0
$849$	40.6773	1.39604
$850$	0	0
$851$	4.86081	0.166626
$852$	0	0
$853$	36.0167	1.23319	0.616594	−	0.787281i	$-0.288512\pi$
$853$	36.0167	1.23319	0.616594	+	0.787281i	$0.288512\pi$
$854$	0	0
$855$	1.05398	0.0360453
$856$	0	0
$857$	41.9709	1.43370	0.716849	−	0.697228i	$-0.245584\pi$
$857$	41.9709	1.43370	0.716849	+	0.697228i	$0.245584\pi$
$858$	0	0
$859$	−46.7077	−1.59365	−0.796823	−	0.604212i	$-0.793488\pi$
$859$	−46.7077	−1.59365	−0.796823	+	0.604212i	$0.793488\pi$
$860$	0	0
$861$	−0.385877	−0.0131507
$862$	0	0
$863$	22.0305	0.749925	0.374963	−	0.927040i	$-0.377655\pi$
$863$	22.0305	0.749925	0.374963	+	0.927040i	$0.377655\pi$
$864$	0	0
$865$	−3.42240	−0.116365
$866$	0	0
$867$	−39.7535	−1.35010
$868$	0	0
$869$	21.1350	0.716957
$870$	0	0
$871$	88.7167	3.00605
$872$	0	0
$873$	−0.907271	−0.0307065
$874$	0	0
$875$	−1.46405	−0.0494938
$876$	0	0
$877$	−7.89204	−0.266495	−0.133248	−	0.991083i	$-0.542541\pi$
$877$	−7.89204	−0.266495	−0.133248	+	0.991083i	$0.542541\pi$
$878$	0	0
$879$	−32.3655	−1.09166
$880$	0	0
$881$	−3.73539	−0.125849	−0.0629243	−	0.998018i	$-0.520043\pi$
$881$	−3.73539	−0.125849	−0.0629243	+	0.998018i	$0.520043\pi$
$882$	0	0
$883$	−34.0305	−1.14522	−0.572608	−	0.819829i	$-0.694068\pi$
$883$	−34.0305	−1.14522	−0.572608	+	0.819829i	$0.694068\pi$
$884$	0	0
$885$	0.0871333	0.00292896
$886$	0	0
$887$	−12.2251	−0.410478	−0.205239	−	0.978712i	$-0.565797\pi$
$887$	−12.2251	−0.410478	−0.205239	+	0.978712i	$0.565797\pi$
$888$	0	0
$889$	−5.91334	−0.198327
$890$	0	0
$891$	31.6891	1.06163
$892$	0	0
$893$	−2.96395	−0.0991847
$894$	0	0
$895$	3.69115	0.123382
$896$	0	0
$897$	46.7971	1.56251
$898$	0	0
$899$	31.4134	1.04770
$900$	0	0
$901$	−84.3601	−2.81044
$902$	0	0
$903$	0.885106	0.0294545
$904$	0	0
$905$	−4.38924	−0.145903
$906$	0	0
$907$	11.3836	0.377988	0.188994	−	0.981978i	$-0.439477\pi$
$907$	11.3836	0.377988	0.188994	+	0.981978i	$0.439477\pi$
$908$	0	0
$909$	7.48197	0.248161
$910$	0	0
$911$	−21.6829	−0.718385	−0.359193	−	0.933263i	$-0.616948\pi$
$911$	−21.6829	−0.718385	−0.359193	+	0.933263i	$0.616948\pi$
$912$	0	0
$913$	−82.0159	−2.71433
$914$	0	0
$915$	0.750620	0.0248147
$916$	0	0
$917$	2.96395	0.0978781
$918$	0	0
$919$	−29.0844	−0.959407	−0.479704	−	0.877431i	$-0.659256\pi$
$919$	−29.0844	−0.959407	−0.479704	+	0.877431i	$0.659256\pi$
$920$	0	0
$921$	−22.3968	−0.737998
$922$	0	0
$923$	52.9557	1.74306
$924$	0	0
$925$	−4.78600	−0.157363
$926$	0	0
$927$	−3.15587	−0.103652
$928$	0	0
$929$	11.4086	0.374305	0.187152	−	0.982331i	$-0.440074\pi$
$929$	11.4086	0.374305	0.187152	+	0.982331i	$0.440074\pi$
$930$	0	0
$931$	−18.2508	−0.598147
$932$	0	0
$933$	−32.2713	−1.05652
$934$	0	0
$935$	−17.1648	−0.561350
$936$	0	0
$937$	29.9806	0.979424	0.489712	−	0.871884i	$-0.337102\pi$
$937$	29.9806	0.979424	0.489712	+	0.871884i	$0.337102\pi$
$938$	0	0
$939$	35.4224	1.15597
$940$	0	0
$941$	−45.1953	−1.47332	−0.736662	−	0.676261i	$-0.763599\pi$
$941$	−45.1953	−1.47332	−0.736662	+	0.676261i	$0.763599\pi$
$942$	0	0
$943$	−3.96540	−0.129131
$944$	0	0
$945$	0.844798	0.0274813
$946$	0	0
$947$	−30.5872	−0.993952	−0.496976	−	0.867764i	$-0.665556\pi$
$947$	−30.5872	−0.993952	−0.496976	+	0.867764i	$0.665556\pi$
$948$	0	0
$949$	−23.5810	−0.765471
$950$	0	0
$951$	−19.5749	−0.634760
$952$	0	0
$953$	−53.1253	−1.72090	−0.860449	−	0.509537i	$-0.829817\pi$
$953$	−53.1253	−1.72090	−0.860449	+	0.509537i	$0.829817\pi$
$954$	0	0
$955$	−8.88163	−0.287403
$956$	0	0
$957$	39.6877	1.28292
$958$	0	0
$959$	−5.00560	−0.161639
$960$	0	0
$961$	10.7652	0.347264
$962$	0	0
$963$	−15.4987	−0.499437
$964$	0	0
$965$	7.44322	0.239606
$966$	0	0
$967$	4.14546	0.133309	0.0666545	−	0.997776i	$-0.478767\pi$
$967$	4.14546	0.133309	0.0666545	+	0.997776i	$0.478767\pi$
$968$	0	0
$969$	25.7312	0.826607
$970$	0	0
$971$	1.51948	0.0487623	0.0243812	−	0.999703i	$-0.492238\pi$
$971$	1.51948	0.0487623	0.0243812	+	0.999703i	$0.492238\pi$
$972$	0	0
$973$	−2.17717	−0.0697967
$974$	0	0
$975$	−46.0769	−1.47564
$976$	0	0
$977$	−7.76036	−0.248276	−0.124138	−	0.992265i	$-0.539617\pi$
$977$	−7.76036	−0.248276	−0.124138	+	0.992265i	$0.539617\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	14.4585	0.461623
$982$	0	0
$983$	50.2251	1.60193	0.800966	−	0.598710i	$-0.204320\pi$
$983$	50.2251	1.60193	0.800966	+	0.598710i	$0.204320\pi$
$984$	0	0
$985$	0.320037	0.0101972
$986$	0	0
$987$	−0.529686	−0.0168601
$988$	0	0
$989$	9.09563	0.289224
$990$	0	0
$991$	38.3761	1.21906	0.609529	−	0.792764i	$-0.291359\pi$
$991$	38.3761	1.21906	0.609529	+	0.792764i	$0.291359\pi$
$992$	0	0
$993$	11.8325	0.375492
$994$	0	0
$995$	−1.29362	−0.0410104
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	5.64681	0.178657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bb.1.2		3
4.3	odd	2		2368.2.a.be.1.2		3
8.3	odd	2		592.2.a.i.1.2		3
8.5	even	2		296.2.a.c.1.2	✓	3
24.5	odd	2		2664.2.a.p.1.2		3
24.11	even	2		5328.2.a.bn.1.2		3
40.29	even	2		7400.2.a.k.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.c.1.2	✓	3	8.5	even	2
592.2.a.i.1.2		3	8.3	odd	2
2368.2.a.bb.1.2		3	1.1	even	1	trivial
2368.2.a.be.1.2		3	4.3	odd	2
2664.2.a.p.1.2		3	24.5	odd	2
5328.2.a.bn.1.2		3	24.11	even	2
7400.2.a.k.1.2		3	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46260 q^{3} -0.462598 q^{5} -0.323404 q^{7} -0.860806 q^{9} +O(q^{10})\)	\(q-1.46260 q^{3} -0.462598 q^{5} -0.323404 q^{7} -0.860806 q^{9} -5.58242 q^{11} -6.58242 q^{13} +0.676596 q^{15} -6.64681 q^{17} +2.64681 q^{19} +0.473011 q^{21} +4.86081 q^{23} -4.78600 q^{25} +5.64681 q^{27} +4.86081 q^{29} +6.46260 q^{31} +8.16484 q^{33} +0.149606 q^{35} +1.00000 q^{37} +9.62743 q^{39} -0.815790 q^{41} +1.87122 q^{43} +0.398207 q^{45} -1.11982 q^{47} -6.89541 q^{49} +9.72161 q^{51} +12.6918 q^{53} +2.58242 q^{55} -3.87122 q^{57} +0.128782 q^{59} +1.10941 q^{61} +0.278388 q^{63} +3.04502 q^{65} -13.4778 q^{67} -7.10941 q^{69} -8.04502 q^{71} +3.58242 q^{73} +7.00000 q^{75} +1.80538 q^{77} -3.78600 q^{79} -5.67660 q^{81} +14.6918 q^{83} +3.07480 q^{85} -7.10941 q^{87} +2.14961 q^{89} +2.12878 q^{91} -9.45219 q^{93} -1.22441 q^{95} +1.05398 q^{97} +4.80538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41} + 4 q^{43} - 2 q^{45} + 11 q^{47} + 8 q^{49} + 18 q^{51} + 3 q^{53} - 9 q^{55} - 10 q^{57} + 2 q^{59} - 15 q^{61} + 12 q^{63} - 10 q^{65} + 5 q^{67} - 3 q^{69} - 5 q^{71} - 6 q^{73} + 21 q^{75} + 15 q^{77} - q^{79} - 25 q^{81} + 9 q^{83} + 14 q^{85} - 3 q^{87} + 16 q^{89} + 8 q^{91} - 22 q^{93} - 18 q^{95} + 24 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * q^99

Introduction

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