LMFDB - Embedded newform 2368.2.a.a.1.1

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-3.00000 q^{3} -4.00000 q^{5} -3.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -4.00000 q^{5} -3.00000 q^{7} +6.00000 q^{9} +3.00000 q^{11} -6.00000 q^{13} +12.0000 q^{15} -4.00000 q^{17} +6.00000 q^{19} +9.00000 q^{21} +6.00000 q^{23} +11.0000 q^{25} -9.00000 q^{27} +4.00000 q^{29} -9.00000 q^{33} +12.0000 q^{35} +1.00000 q^{37} +18.0000 q^{39} -5.00000 q^{41} -6.00000 q^{43} -24.0000 q^{45} +9.00000 q^{47} +2.00000 q^{49} +12.0000 q^{51} -5.00000 q^{53} -12.0000 q^{55} -18.0000 q^{57} +6.00000 q^{59} +10.0000 q^{61} -18.0000 q^{63} +24.0000 q^{65} +12.0000 q^{67} -18.0000 q^{69} -9.00000 q^{71} +3.00000 q^{73} -33.0000 q^{75} -9.00000 q^{77} -12.0000 q^{79} +9.00000 q^{81} -3.00000 q^{83} +16.0000 q^{85} -12.0000 q^{87} -2.00000 q^{89} +18.0000 q^{91} -24.0000 q^{95} -6.00000 q^{97} +18.0000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	12.0000	3.09839
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	9.00000	1.96396
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−9.00000	−1.56670
$34$	0	0
$35$	12.0000	2.02837
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	18.0000	2.88231
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−24.0000	−3.57771
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	−18.0000	−2.38416
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−18.0000	−2.26779
$64$	0	0
$65$	24.0000	2.97683
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	0	0
$75$	−33.0000	−3.81051
$76$	0	0
$77$	−9.00000	−1.02565
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	18.0000	1.88691
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−24.0000	−2.46235
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	18.0000	1.80907
$100$	0	0
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−36.0000	−3.51324
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	0	0
$117$	−36.0000	−3.32820
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	15.0000	1.35250
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	15.0000	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$128$	0	0
$129$	18.0000	1.58481
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−18.0000	−1.56080
$134$	0	0
$135$	36.0000	3.09839
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−27.0000	−2.27381
$142$	0	0
$143$	−18.0000	−1.50524
$144$	0	0
$145$	−16.0000	−1.32873
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−24.0000	−1.94029
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	15.0000	1.18958
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	36.0000	2.80260
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	36.0000	2.75299
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	−33.0000	−2.49457
$176$	0	0
$177$	−18.0000	−1.35296
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	0	0
$183$	−30.0000	−2.21766
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	27.0000	1.96396
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	−72.0000	−5.15603
$196$	0	0
$197$	5.00000	0.356235	0.178118	−	0.984009i	$-0.442999\pi$
$197$	5.00000	0.356235	0.178118	+	0.984009i	$0.442999\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−36.0000	−2.53924
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	20.0000	1.39686
$206$	0	0
$207$	36.0000	2.50217
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	0	0
$213$	27.0000	1.85001
$214$	0	0
$215$	24.0000	1.63679
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	27.0000	1.77647
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	0	0
$237$	36.0000	2.33845
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.00000	−0.511101
$246$	0	0
$247$	−36.0000	−2.29063
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	−48.0000	−3.00588
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	24.0000	1.48556
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	0	0
$273$	−54.0000	−3.26823
$274$	0	0
$275$	33.0000	1.98997
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	72.0000	4.26491
$286$	0	0
$287$	15.0000	0.885422
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	−27.0000	−1.56670
$298$	0	0
$299$	−36.0000	−2.08193
$300$	0	0
$301$	18.0000	1.03750
$302$	0	0
$303$	−15.0000	−0.861727
$304$	0	0
$305$	−40.0000	−2.29039
$306$	0	0
$307$	27.0000	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	72.0000	4.05674
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−66.0000	−3.66102
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	−27.0000	−1.48856
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−48.0000	−2.62252
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	72.0000	3.87635
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	54.0000	2.88231
$352$	0	0
$353$	−32.0000	−1.70319	−0.851594	−	0.524202i	$-0.824364\pi$
$353$	−32.0000	−1.70319	−0.851594	+	0.524202i	$0.824364\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	6.00000	0.314918
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	15.0000	0.778761
$372$	0	0
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	0	0
$381$	−45.0000	−2.30542
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	−36.0000	−1.82998
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	48.0000	2.41514
$396$	0	0
$397$	−11.0000	−0.552074	−0.276037	−	0.961147i	$-0.589021\pi$
$397$	−11.0000	−0.552074	−0.276037	+	0.961147i	$0.589021\pi$
$398$	0	0
$399$	54.0000	2.70338
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−36.0000	−1.78885
$406$	0	0
$407$	3.00000	0.148704
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	30.0000	1.47979
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	36.0000	1.76293
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	54.0000	2.62557
$424$	0	0
$425$	−44.0000	−2.13431
$426$	0	0
$427$	−30.0000	−1.45180
$428$	0	0
$429$	54.0000	2.60714
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	0	0
$435$	48.0000	2.30142
$436$	0	0
$437$	36.0000	1.72211
$438$	0	0
$439$	18.0000	0.859093	0.429547	−	0.903045i	$-0.358673\pi$
$439$	18.0000	0.859093	0.429547	+	0.903045i	$0.358673\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	−3.00000	−0.141895
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−15.0000	−0.706322
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−72.0000	−3.37541
$456$	0	0
$457$	36.0000	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$458$	0	0
$459$	36.0000	1.68034
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−42.0000	−1.94353	−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000	−1.94353	−0.971764	+	0.235954i	$0.924178\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−15.0000	−0.691164
$472$	0	0
$473$	−18.0000	−0.827641
$474$	0	0
$475$	66.0000	3.02829
$476$	0	0
$477$	−30.0000	−1.37361
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	54.0000	2.45709
$484$	0	0
$485$	24.0000	1.08978
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	−72.0000	−3.23616
$496$	0	0
$497$	27.0000	1.21112
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	−69.0000	−3.06440
$508$	0	0
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	0	0
$513$	−54.0000	−2.38416
$514$	0	0
$515$	−24.0000	−1.05757
$516$	0	0
$517$	27.0000	1.18746
$518$	0	0
$519$	39.0000	1.71191
$520$	0	0
$521$	23.0000	1.00765	0.503824	−	0.863806i	$-0.331926\pi$
$521$	23.0000	1.00765	0.503824	+	0.863806i	$0.331926\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	99.0000	4.32071
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	36.0000	1.56227
$532$	0	0
$533$	30.0000	1.29944
$534$	0	0
$535$	48.0000	2.07522
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	63.0000	2.70359
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	60.0000	2.56074
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	36.0000	1.51992
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	0	0
$567$	−27.0000	−1.13389
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−9.00000	−0.376638	−0.188319	−	0.982108i	$-0.560304\pi$
$571$	−9.00000	−0.376638	−0.188319	+	0.982108i	$0.560304\pi$
$572$	0	0
$573$	54.0000	2.25588
$574$	0	0
$575$	66.0000	2.75239
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	−15.0000	−0.621237
$584$	0	0
$585$	144.000	5.95367
$586$	0	0
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−15.0000	−0.617018
$592$	0	0
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	0	0
$595$	−48.0000	−1.96781
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	0	0
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	72.0000	2.93207
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	−54.0000	−2.18461
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	0	0
$615$	−60.0000	−2.41943
$616$	0	0
$617$	−11.0000	−0.442843	−0.221422	−	0.975178i	$-0.571070\pi$
$617$	−11.0000	−0.442843	−0.221422	+	0.975178i	$0.571070\pi$
$618$	0	0
$619$	27.0000	1.08522	0.542611	−	0.839984i	$-0.317436\pi$
$619$	27.0000	1.08522	0.542611	+	0.839984i	$0.317436\pi$
$620$	0	0
$621$	−54.0000	−2.16695
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−54.0000	−2.15655
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	6.00000	0.238856	0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000	0.238856	0.119428	+	0.992843i	$0.461894\pi$
$632$	0	0
$633$	−45.0000	−1.78859
$634$	0	0
$635$	−60.0000	−2.38103
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−54.0000	−2.13621
$640$	0	0
$641$	−37.0000	−1.46141	−0.730706	−	0.682692i	$-0.760809\pi$
$641$	−37.0000	−1.46141	−0.730706	+	0.682692i	$0.760809\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−72.0000	−2.83500
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	−72.0000	−2.79625
$664$	0	0
$665$	72.0000	2.79204
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	−27.0000	−1.04388
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	0	0
$675$	−99.0000	−3.81051
$676$	0	0
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	18.0000	0.690777
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	0	0
$685$	40.0000	1.52832
$686$	0	0
$687$	−27.0000	−1.03011
$688$	0	0
$689$	30.0000	1.14291
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	0	0
$693$	−54.0000	−2.05129
$694$	0	0
$695$	48.0000	1.82074
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	66.0000	2.49635
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	108.000	4.06752
$706$	0	0
$707$	−15.0000	−0.564133
$708$	0	0
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	−72.0000	−2.70021
$712$	0	0
$713$	0	0
$714$	0	0
$715$	72.0000	2.69265
$716$	0	0
$717$	−72.0000	−2.68889
$718$	0	0
$719$	−39.0000	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$719$	−39.0000	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	−72.0000	−2.67771
$724$	0	0
$725$	44.0000	1.63412
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	9.00000	0.332423	0.166211	−	0.986090i	$-0.446847\pi$
$733$	9.00000	0.332423	0.166211	+	0.986090i	$0.446847\pi$
$734$	0	0
$735$	24.0000	0.885253
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	0	0
$741$	108.000	3.96748
$742$	0	0
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	0	0
$745$	−4.00000	−0.146549
$746$	0	0
$747$	−18.0000	−0.658586
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	−33.0000	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$751$	−33.0000	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	−54.0000	−1.96008
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	0	0
$765$	96.0000	3.47089
$766$	0	0
$767$	−36.0000	−1.29988
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	1.00000	0.0359675	0.0179838	−	0.999838i	$-0.494275\pi$
$773$	1.00000	0.0359675	0.0179838	+	0.999838i	$0.494275\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	−30.0000	−1.07486
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	−36.0000	−1.28654
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	−9.00000	−0.320815	−0.160408	−	0.987051i	$-0.551281\pi$
$787$	−9.00000	−0.320815	−0.160408	+	0.987051i	$0.551281\pi$
$788$	0	0
$789$	−63.0000	−2.24286
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−60.0000	−2.13066
$794$	0	0
$795$	−60.0000	−2.12798
$796$	0	0
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	9.00000	0.317603
$804$	0	0
$805$	72.0000	2.53767
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−9.00000	−0.316033	−0.158016	−	0.987436i	$-0.550510\pi$
$811$	−9.00000	−0.316033	−0.158016	+	0.987436i	$0.550510\pi$
$812$	0	0
$813$	−9.00000	−0.315644
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	108.000	3.77383
$820$	0	0
$821$	−13.0000	−0.453703	−0.226852	−	0.973929i	$-0.572843\pi$
$821$	−13.0000	−0.453703	−0.226852	+	0.973929i	$0.572843\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	−99.0000	−3.44674
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	72.0000	2.49765
$832$	0	0
$833$	−8.00000	−0.277184
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−48.0000	−1.65321
$844$	0	0
$845$	−92.0000	−3.16490
$846$	0	0
$847$	6.00000	0.206162
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−144.000	−4.92470
$856$	0	0
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	−45.0000	−1.53360
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	52.0000	1.76805
$866$	0	0
$867$	3.00000	0.101885
$868$	0	0
$869$	−36.0000	−1.22122
$870$	0	0
$871$	−72.0000	−2.43963
$872$	0	0
$873$	−36.0000	−1.21842
$874$	0	0
$875$	72.0000	2.43404
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	42.0000	1.41662
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	42.0000	1.41341	0.706706	−	0.707507i	$-0.250180\pi$
$883$	42.0000	1.41341	0.706706	+	0.707507i	$0.250180\pi$
$884$	0	0
$885$	72.0000	2.42025
$886$	0	0
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	0	0
$889$	−45.0000	−1.50925
$890$	0	0
$891$	27.0000	0.904534
$892$	0	0
$893$	54.0000	1.80704
$894$	0	0
$895$	0	0
$896$	0	0
$897$	108.000	3.60602
$898$	0	0
$899$	0	0
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	−54.0000	−1.79701
$904$	0	0
$905$	84.0000	2.79225
$906$	0	0
$907$	−42.0000	−1.39459	−0.697294	−	0.716786i	$-0.745613\pi$
$907$	−42.0000	−1.39459	−0.697294	+	0.716786i	$0.745613\pi$
$908$	0	0
$909$	30.0000	0.995037
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	0	0
$915$	120.000	3.96708
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−81.0000	−2.66904
$922$	0	0
$923$	54.0000	1.77743
$924$	0	0
$925$	11.0000	0.361678
$926$	0	0
$927$	36.0000	1.18240
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	90.0000	2.94647
$934$	0	0
$935$	48.0000	1.56977
$936$	0	0
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	−78.0000	−2.54543
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−30.0000	−0.976934
$944$	0	0
$945$	−108.000	−3.51324
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	−78.0000	−2.52932
$952$	0	0
$953$	−19.0000	−0.615470	−0.307735	−	0.951472i	$-0.599571\pi$
$953$	−19.0000	−0.615470	−0.307735	+	0.951472i	$0.599571\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	0	0
$957$	−36.0000	−1.16371
$958$	0	0
$959$	30.0000	0.968751
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−72.0000	−2.32017
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	72.0000	2.31297
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	36.0000	1.15411
$974$	0	0
$975$	198.000	6.34107
$976$	0	0
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	36.0000	1.14939
$982$	0	0
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	81.0000	2.57826
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	54.0000	1.71364
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−44.0000	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$998$	0	0
$999$	−9.00000	−0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.a.1.1		1
4.3	odd	2		2368.2.a.p.1.1		1
8.3	odd	2		1184.2.a.b.1.1	✓	1
8.5	even	2		1184.2.a.h.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.b.1.1	✓	1	8.3	odd	2
1184.2.a.h.1.1	yes	1	8.5	even	2
2368.2.a.a.1.1		1	1.1	even	1	trivial
2368.2.a.p.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more