LMFDB - Embedded newform 2368.2.a.q.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 37)
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} +2.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} +2.00000 q^{13} +6.00000 q^{15} -3.00000 q^{21} +2.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +15.0000 q^{33} -2.00000 q^{35} +1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -2.00000 q^{43} +12.0000 q^{45} -9.00000 q^{47} -6.00000 q^{49} -1.00000 q^{53} +10.0000 q^{55} -8.00000 q^{59} +8.00000 q^{61} -6.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} +6.00000 q^{69} +9.00000 q^{71} -1.00000 q^{73} -3.00000 q^{75} -5.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +15.0000 q^{83} -18.0000 q^{87} +4.00000 q^{89} -2.00000 q^{91} -12.0000 q^{93} +4.00000 q^{97} +30.0000 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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	6.00000	1.54919
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	15.0000	2.61116
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	12.0000	1.78885
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	10.0000	1.34840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.0000	−1.92980
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	30.0000	3.01511
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	12.0000	1.10940
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−27.0000	−2.43451
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	18.0000	1.54919
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−27.0000	−2.27381
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	0	0
$165$	30.0000	2.33550
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−24.0000	−1.80395
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	−18.0000	−1.25717
$206$	0	0
$207$	12.0000	0.834058
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	27.0000	1.85001
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	0	0
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	45.0000	2.85176
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−36.0000	−2.22834
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−31.0000	−1.88312	−0.941558	−	0.336851i	$-0.890638\pi$
$271$	−31.0000	−1.88312	−0.941558	+	0.336851i	$0.890638\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	45.0000	2.61116
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	−9.00000	−0.517036
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	0	0
$309$	54.0000	3.07195
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	48.0000	2.65441
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	−54.0000	−2.93288
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	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
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	42.0000	2.20443
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−54.0000	−2.81113
$370$	0	0
$371$	1.00000	0.0519174
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	−36.0000	−1.85903
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	3.00000	0.153695
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	36.0000	1.81596
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	5.00000	0.247841
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	30.0000	1.47264
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−7.00000	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$419$	−7.00000	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$420$	0	0
$421$	24.0000	1.16969	0.584844	−	0.811146i	$-0.301156\pi$
$421$	24.0000	1.16969	0.584844	+	0.811146i	$0.301156\pi$
$422$	0	0
$423$	−54.0000	−2.62557
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	30.0000	1.44841
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	9.00000	0.432512	0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000	0.432512	0.216256	+	0.976337i	$0.430615\pi$
$434$	0	0
$435$	−36.0000	−1.72607
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	15.0000	0.709476
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	0	0
$453$	48.0000	2.25524
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−69.0000	−3.17935
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	8.00000	0.363261
$486$	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
$489$	54.0000	2.44196
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	60.0000	2.69680
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−36.0000	−1.60836
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−27.0000	−1.19911
$508$	0	0
$509$	31.0000	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$509$	31.0000	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.0000	1.58635
$516$	0	0
$517$	−45.0000	−1.97910
$518$	0	0
$519$	−27.0000	−1.18517
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−48.0000	−2.08302
$532$	0	0
$533$	−18.0000	−0.779667
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	−54.0000	−2.33027
$538$	0	0
$539$	−30.0000	−1.29219
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	−15.0000	−0.643712
$544$	0	0
$545$	32.0000	1.37073
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	48.0000	2.04859
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	0	0
$565$	−36.0000	−1.51453
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	−78.0000	−3.24157
$580$	0	0
$581$	−15.0000	−0.622305
$582$	0	0
$583$	−5.00000	−0.207079
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	−5.00000	−0.205325	−0.102663	−	0.994716i	$-0.532736\pi$
$593$	−5.00000	−0.205325	−0.102663	+	0.994716i	$0.532736\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.00000	0.245564
$598$	0	0
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−48.0000	−1.95471
$604$	0	0
$605$	28.0000	1.13836
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	0	0
$615$	−54.0000	−2.17749
$616$	0	0
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	39.0000	1.55011
$634$	0	0
$635$	2.00000	0.0793676
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	54.0000	2.13621
$640$	0	0
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	24.0000	0.937758
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−51.0000	−1.97177
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	27.0000	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$673$	27.0000	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	48.0000	1.83936
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−21.0000	−0.801200
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−30.0000	−1.13961
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−54.0000	−2.03376
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	20.0000	0.747958
$716$	0	0
$717$	−18.0000	−0.672222
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	42.0000	1.56200
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	90.0000	3.29293
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	0	0
$759$	30.0000	1.08893
$760$	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
$763$	−16.0000	−0.579239
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	0	0
$780$	0	0
$781$	45.0000	1.61023
$782$	0	0
$783$	−54.0000	−1.92980
$784$	0	0
$785$	−46.0000	−1.64181
$786$	0	0
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	0	0
$789$	57.0000	2.02925
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	−5.00000	−0.176446
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	18.0000	0.633630
$808$	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
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	0	0
$813$	−93.0000	−3.26165
$814$	0	0
$815$	36.0000	1.26102
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	47.0000	1.64031	0.820156	−	0.572140i	$-0.193887\pi$
$821$	47.0000	1.64031	0.820156	+	0.572140i	$0.193887\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	−15.0000	−0.522233
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	−36.0000	−1.24883
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	−36.0000	−1.24434
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	36.0000	1.23991
$844$	0	0
$845$	−18.0000	−0.619219
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	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
$861$	27.0000	0.920158
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	−51.0000	−1.73205
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	24.0000	0.812277
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	−48.0000	−1.61350
$886$	0	0
$887$	25.0000	0.839418	0.419709	−	0.907659i	$-0.362132\pi$
$887$	25.0000	0.839418	0.419709	+	0.907659i	$0.362132\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	45.0000	1.50756
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	6.00000	0.199667
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	0	0
$913$	75.0000	2.48214
$914$	0	0
$915$	48.0000	1.58683
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−58.0000	−1.91324	−0.956622	−	0.291333i	$-0.905901\pi$
$919$	−58.0000	−1.91324	−0.956622	+	0.291333i	$0.905901\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	0	0
$923$	18.0000	0.592477
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	108.000	3.54719
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	0	0
$939$	66.0000	2.15383
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	−18.0000	−0.585540
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−66.0000	−2.14020
$952$	0	0
$953$	61.0000	1.97598	0.987992	−	0.154506i	$-0.0493785\pi$
$953$	61.0000	1.97598	0.987992	+	0.154506i	$0.0493785\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	−90.0000	−2.90929
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	72.0000	2.32017
$964$	0	0
$965$	−52.0000	−1.67394
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	96.0000	3.06504
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	27.0000	0.859419
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	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.q.1.1		1
4.3	odd	2		2368.2.a.b.1.1		1
8.3	odd	2		592.2.a.e.1.1		1
8.5	even	2		37.2.a.a.1.1	✓	1
24.5	odd	2		333.2.a.d.1.1		1
24.11	even	2		5328.2.a.r.1.1		1
40.13	odd	4		925.2.b.b.149.2		2
40.29	even	2		925.2.a.e.1.1		1
40.37	odd	4		925.2.b.b.149.1		2
56.13	odd	2		1813.2.a.a.1.1		1
88.21	odd	2		4477.2.a.b.1.1		1
104.77	even	2		6253.2.a.c.1.1		1
120.29	odd	2		8325.2.a.e.1.1		1
296.117	odd	4		1369.2.b.c.1368.2		2
296.221	even	2		1369.2.a.e.1.1		1
296.253	odd	4		1369.2.b.c.1368.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.a.a.1.1	✓	1	8.5	even	2
333.2.a.d.1.1		1	24.5	odd	2
592.2.a.e.1.1		1	8.3	odd	2
925.2.a.e.1.1		1	40.29	even	2
925.2.b.b.149.1		2	40.37	odd	4
925.2.b.b.149.2		2	40.13	odd	4
1369.2.a.e.1.1		1	296.221	even	2
1369.2.b.c.1368.1		2	296.253	odd	4
1369.2.b.c.1368.2		2	296.117	odd	4
1813.2.a.a.1.1		1	56.13	odd	2
2368.2.a.b.1.1		1	4.3	odd	2
2368.2.a.q.1.1		1	1.1	even	1	trivial
4477.2.a.b.1.1		1	88.21	odd	2
5328.2.a.r.1.1		1	24.11	even	2
6253.2.a.c.1.1		1	104.77	even	2
8325.2.a.e.1.1		1	120.29	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

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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.q.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	$1$