LMFDB - Embedded newform 2368.2.a.r.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:	$0$
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} -1.00000 q^{7} +6.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} +12.0000 q^{15} +8.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} +11.0000 q^{25} +9.00000 q^{27} -8.00000 q^{31} -9.00000 q^{33} -4.00000 q^{35} +1.00000 q^{37} -6.00000 q^{39} -5.00000 q^{41} +2.00000 q^{43} +24.0000 q^{45} +11.0000 q^{47} -6.00000 q^{49} +24.0000 q^{51} -9.00000 q^{53} -12.0000 q^{55} +6.00000 q^{57} +6.00000 q^{59} +6.00000 q^{61} -6.00000 q^{63} -8.00000 q^{65} -4.00000 q^{67} -18.0000 q^{69} +5.00000 q^{71} +11.0000 q^{73} +33.0000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +9.00000 q^{81} -5.00000 q^{83} +32.0000 q^{85} -18.0000 q^{89} +2.00000 q^{91} -24.0000 q^{93} +8.00000 q^{95} -2.00000 q^{97} -18.0000 q^{99} +O(q^{100})

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	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	12.0000	3.09839
$16$	0	0
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−9.00000	−1.56670
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−6.00000	−0.960769
$40$	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
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	24.0000	3.57771
$46$	0	0
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	24.0000	3.36067
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	6.00000	0.794719
$58$	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
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	33.0000	3.81051
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	32.0000	3.47089
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−24.0000	−2.48868
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−18.0000	−1.80907
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−18.0000	−1.77359	−0.886796	−	0.462160i	$-0.847074\pi$
$103$	−18.0000	−1.77359	−0.886796	+	0.462160i	$0.847074\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	−8.00000	−0.733359
$120$	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
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	36.0000	3.09839
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	33.0000	2.77910
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	48.0000	3.88057
$154$	0	0
$155$	−32.0000	−2.57030
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	0	0
$159$	−27.0000	−2.14124
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	−36.0000	−2.80260
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	12.0000	0.917663
$172$	0	0
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	0	0
$183$	18.0000	1.33060
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	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
$195$	−24.0000	−1.71868
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−20.0000	−1.39686
$206$	0	0
$207$	−36.0000	−2.50217
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	0	0
$213$	15.0000	1.02778
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	33.0000	2.22993
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	0	0
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	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
$235$	44.0000	2.87024
$236$	0	0
$237$	24.0000	1.55897
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−24.0000	−1.53330
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	96.0000	6.01175
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.0000	−1.54157	−0.770783	−	0.637098i	$-0.780135\pi$
$263$	−25.0000	−1.54157	−0.770783	+	0.637098i	$0.780135\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	0	0
$267$	−54.0000	−3.30475
$268$	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
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−33.0000	−1.98997
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−48.0000	−2.87368
$280$	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
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	5.00000	0.295141
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	−27.0000	−1.56670
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	27.0000	1.55111
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	0	0
$309$	−54.0000	−3.07195
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−24.0000	−1.35225
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−36.0000	−2.00932
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−22.0000	−1.22034
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	−11.0000	−0.606450
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	−24.0000	−1.30350
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−72.0000	−3.87635
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	−18.0000	−0.960769
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	−24.0000	−1.27021
$358$	0	0
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−6.00000	−0.314918
$364$	0	0
$365$	44.0000	2.30307
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−27.0000	−1.38690	−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000	−1.38690	−0.693448	+	0.720506i	$0.743909\pi$
$380$	0	0
$381$	−33.0000	−1.69064
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	−36.0000	−1.81596
$394$	0	0
$395$	32.0000	1.61009
$396$	0	0
$397$	17.0000	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$397$	17.0000	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	36.0000	1.78885
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	60.0000	2.93821
$418$	0	0
$419$	17.0000	0.830504	0.415252	−	0.909706i	$-0.363693\pi$
$419$	17.0000	0.830504	0.415252	+	0.909706i	$0.363693\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	66.0000	3.20903
$424$	0	0
$425$	88.0000	4.26863
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	18.0000	0.869048
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	−15.0000	−0.720854	−0.360427	−	0.932787i	$-0.617369\pi$
$433$	−15.0000	−0.720854	−0.360427	+	0.932787i	$0.617369\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	−5.00000	−0.237557	−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000	−0.237557	−0.118779	+	0.992921i	$0.537898\pi$
$444$	0	0
$445$	−72.0000	−3.41313
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	15.0000	0.706322
$452$	0	0
$453$	−24.0000	−1.12762
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	72.0000	3.36067
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−96.0000	−4.45189
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	3.00000	0.138233
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	22.0000	1.00943
$476$	0	0
$477$	−54.0000	−2.47249
$478$	0	0
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	18.0000	0.819028
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	48.0000	2.17064
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−72.0000	−3.23616
$496$	0	0
$497$	−5.00000	−0.224281
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	0	0
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	−27.0000	−1.19911
$508$	0	0
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	18.0000	0.794719
$514$	0	0
$515$	−72.0000	−3.17270
$516$	0	0
$517$	−33.0000	−1.45134
$518$	0	0
$519$	21.0000	0.921798
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−33.0000	−1.44024
$526$	0	0
$527$	−64.0000	−2.78788
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	36.0000	1.56227
$532$	0	0
$533$	10.0000	0.433148
$534$	0	0
$535$	−48.0000	−2.07522
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	69.0000	2.96107
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	36.0000	1.53644
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	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
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−72.0000	−3.03984
$562$	0	0
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	−32.0000	−1.34625
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	−66.0000	−2.75239
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	5.00000	0.207435
$582$	0	0
$583$	27.0000	1.11823
$584$	0	0
$585$	−48.0000	−1.98456
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−45.0000	−1.85105
$592$	0	0
$593$	43.0000	1.76580	0.882899	−	0.469563i	$-0.155588\pi$
$593$	43.0000	1.76580	0.882899	+	0.469563i	$0.155588\pi$
$594$	0	0
$595$	−32.0000	−1.31187
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	−8.00000	−0.325246
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.0000	−0.890025
$612$	0	0
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	0	0
$615$	−60.0000	−2.41943
$616$	0	0
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	0	0
$621$	−54.0000	−2.16695
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	51.0000	2.02707
$634$	0	0
$635$	−44.0000	−1.74609
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	30.0000	1.18678
$640$	0	0
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	0	0
$651$	24.0000	0.940634
$652$	0	0
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	66.0000	2.57491
$658$	0	0
$659$	43.0000	1.67504	0.837521	−	0.546405i	$-0.184004\pi$
$659$	43.0000	1.67504	0.837521	+	0.546405i	$0.184004\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	0	0
$663$	−48.0000	−1.86417
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−15.0000	−0.579934
$670$	0	0
$671$	−18.0000	−0.694882
$672$	0	0
$673$	−21.0000	−0.809491	−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000	−0.809491	−0.404745	+	0.914429i	$0.632640\pi$
$674$	0	0
$675$	99.0000	3.81051
$676$	0	0
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	84.0000	3.21889
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	15.0000	0.572286
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	0	0
$693$	18.0000	0.683763
$694$	0	0
$695$	80.0000	3.03457
$696$	0	0
$697$	−40.0000	−1.51511
$698$	0	0
$699$	−66.0000	−2.49635
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	132.000	4.97141
$706$	0	0
$707$	−9.00000	−0.338480
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	48.0000	1.80014
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	60.0000	2.24074
$718$	0	0
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	−60.0000	−2.23142
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	0	0
$735$	−72.0000	−2.65576
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	−30.0000	−1.09764
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−19.0000	−0.693320	−0.346660	−	0.937991i	$-0.612684\pi$
$751$	−19.0000	−0.693320	−0.346660	+	0.937991i	$0.612684\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−32.0000	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$758$	0	0
$759$	54.0000	1.96008
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	192.000	6.94177
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	54.0000	1.94476
$772$	0	0
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	0	0
$775$	−88.0000	−3.16105
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−15.0000	−0.536742
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	0	0
$789$	−75.0000	−2.67007
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	−108.000	−3.83037
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	88.0000	3.11322
$800$	0	0
$801$	−108.000	−3.81599
$802$	0	0
$803$	−33.0000	−1.16454
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	9.00000	0.316033	0.158016	−	0.987436i	$-0.449490\pi$
$811$	9.00000	0.316033	0.158016	+	0.987436i	$0.449490\pi$
$812$	0	0
$813$	75.0000	2.63036
$814$	0	0
$815$	64.0000	2.24182
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	23.0000	0.802706	0.401353	−	0.915924i	$-0.368540\pi$
$821$	23.0000	0.802706	0.401353	+	0.915924i	$0.368540\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	−99.0000	−3.44674
$826$	0	0
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−24.0000	−0.832551
$832$	0	0
$833$	−48.0000	−1.66310
$834$	0	0
$835$	−56.0000	−1.93796
$836$	0	0
$837$	−72.0000	−2.48868
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	48.0000	1.65321
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	48.0000	1.64157
$856$	0	0
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	15.0000	0.511199
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	0	0
$867$	141.000	4.78861
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	0	0
$885$	72.0000	2.42025
$886$	0	0
$887$	−43.0000	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$887$	−43.0000	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$888$	0	0
$889$	11.0000	0.368928
$890$	0	0
$891$	−27.0000	−0.904534
$892$	0	0
$893$	22.0000	0.736202
$894$	0	0
$895$	−48.0000	−1.60446
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−72.0000	−2.39867
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	92.0000	3.05818
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	0	0
$909$	54.0000	1.79107
$910$	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
$913$	15.0000	0.496428
$914$	0	0
$915$	72.0000	2.38025
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−33.0000	−1.08739
$922$	0	0
$923$	−10.0000	−0.329154
$924$	0	0
$925$	11.0000	0.361678
$926$	0	0
$927$	−108.000	−3.54719
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	−54.0000	−1.76788
$934$	0	0
$935$	−96.0000	−3.13954
$936$	0	0
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	0	0
$939$	−30.0000	−0.979013
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	−36.0000	−1.17108
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	−22.0000	−0.714150
$950$	0	0
$951$	54.0000	1.75107
$952$	0	0
$953$	13.0000	0.421111	0.210556	−	0.977582i	$-0.432473\pi$
$953$	13.0000	0.421111	0.210556	+	0.977582i	$0.432473\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−72.0000	−2.32017
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	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
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−66.0000	−2.11369
$976$	0	0
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	−47.0000	−1.49907	−0.749534	−	0.661966i	$-0.769722\pi$
$983$	−47.0000	−1.49907	−0.749534	+	0.661966i	$0.769722\pi$
$984$	0	0
$985$	−60.0000	−1.91176
$986$	0	0
$987$	−33.0000	−1.05040
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	−42.0000	−1.33283
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−24.0000	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$998$	0	0
$999$	9.00000	0.284747

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

)

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2368.2.a.r.1.1

Level	$2368$
Weight	$2$
Character	2368.1
Self dual	yes
Analytic conductor	$18.909$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$