LMFDB - Embedded newform 2368.2.a.e.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-1.00000 q^{3} -1.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +2.00000 q^{13} -4.00000 q^{17} +6.00000 q^{19} +1.00000 q^{21} +2.00000 q^{23} -5.00000 q^{25} +5.00000 q^{27} +8.00000 q^{31} -1.00000 q^{33} +1.00000 q^{37} -2.00000 q^{39} -5.00000 q^{41} -10.0000 q^{43} -13.0000 q^{47} -6.00000 q^{49} +4.00000 q^{51} -5.00000 q^{53} -6.00000 q^{57} -2.00000 q^{59} +6.00000 q^{61} +2.00000 q^{63} -4.00000 q^{67} -2.00000 q^{69} +13.0000 q^{71} -5.00000 q^{73} +5.00000 q^{75} -1.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -1.00000 q^{83} -14.0000 q^{89} -2.00000 q^{91} -8.00000 q^{93} -2.00000 q^{97} -2.00000 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	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
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	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
$15$	0	0
$16$	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
$21$	1.00000	0.218218
$22$	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
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.00000	0.962250
$28$	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	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.00000	−0.320256
$40$	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
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	4.00000	0.560112
$52$	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
$55$	0	0
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	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
$63$	2.00000	0.251976
$64$	0	0
$65$	0	0
$66$	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
$69$	−2.00000	−0.240772
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	−1.00000	−0.113961
$78$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	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
$99$	−2.00000	−0.201008
$100$	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
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	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	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	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
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	5.00000	0.450835
$124$	0	0
$125$	0	0
$126$	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
$129$	10.0000	0.880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	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
$139$	−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
$141$	13.0000	1.09480
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−23.0000	−1.88423	−0.942117	−	0.335285i	$-0.891167\pi$
$149$	−23.0000	−1.88423	−0.942117	+	0.335285i	$0.891167\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	5.00000	0.396526
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−12.0000	−0.917663
$172$	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
$175$	5.00000	0.377964
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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
$183$	−6.00000	−0.443533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−5.00000	−0.363696
$190$	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
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	0	0
$196$	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
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	−13.0000	−0.890745
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	5.00000	0.337869
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	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
$229$	−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
$231$	1.00000	0.0657952
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	1.00000	0.0633724
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.0000	0.856786
$268$	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
$273$	2.00000	0.121046
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	−16.0000	−0.957895
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.00000	0.295141
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	10.0000	0.576390
$302$	0	0
$303$	3.00000	0.172345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	13.0000	0.716713
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.00000	−0.211702
$358$	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
$361$	17.0000	0.894737
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	5.00000	0.259587
$372$	0	0
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	33.0000	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$379$	33.0000	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000	1.01666
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	37.0000	1.85698	0.928488	−	0.371361i	$-0.121109\pi$
$397$	37.0000	1.85698	0.928488	+	0.371361i	$0.121109\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	2.00000	0.0984136
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	26.0000	1.26416
$424$	0	0
$425$	20.0000	0.970143
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	0	0
$433$	33.0000	1.58588	0.792939	−	0.609301i	$-0.208550\pi$
$433$	33.0000	1.58588	0.792939	+	0.609301i	$0.208550\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−33.0000	−1.56788	−0.783939	−	0.620838i	$-0.786792\pi$
$443$	−33.0000	−1.56788	−0.783939	+	0.620838i	$0.786792\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.0000	1.08786
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	0	0
$459$	−20.0000	−0.933520
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	−30.0000	−1.37649
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.0000	−0.583130
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	−14.0000	−0.625474
$502$	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
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	0	0
$513$	30.0000	1.32453
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.0000	−0.571739
$518$	0	0
$519$	13.0000	0.570637
$520$	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
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	−32.0000	−1.39394
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	0	0
$543$	5.00000	0.214571
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−12.0000	−0.512148
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	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
$571$	37.0000	1.54840	0.774201	−	0.632940i	$-0.218152\pi$
$571$	37.0000	1.54840	0.774201	+	0.632940i	$0.218152\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−10.0000	−0.417029
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	1.00000	0.0414870
$582$	0	0
$583$	−5.00000	−0.207079
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	3.00000	0.123404
$592$	0	0
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−11.0000	−0.449448	−0.224724	−	0.974422i	$-0.572148\pi$
$599$	−11.0000	−0.449448	−0.224724	+	0.974422i	$0.572148\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.0000	−1.05185
$612$	0	0
$613$	21.0000	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$613$	21.0000	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	0	0
$633$	19.0000	0.755182
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−26.0000	−1.02854
$640$	0	0
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−1.00000	−0.0389545	−0.0194772	−	0.999810i	$-0.506200\pi$
$659$	−1.00000	−0.0389545	−0.0194772	+	0.999810i	$0.506200\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	13.0000	0.502609
$670$	0	0
$671$	6.00000	0.231627
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−25.0000	−0.962250
$676$	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
$679$	2.00000	0.0767530
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.00000	0.267067
$688$	0	0
$689$	−10.0000	−0.380970
$690$	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
$693$	2.00000	0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.00000	0.149383
$718$	0	0
$719$	11.0000	0.410231	0.205115	−	0.978738i	$-0.434243\pi$
$719$	11.0000	0.410231	0.205115	+	0.978738i	$0.434243\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	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
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000	0.0731762
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	37.0000	1.35015	0.675075	−	0.737749i	$-0.264111\pi$
$751$	37.0000	1.35015	0.675075	+	0.737749i	$0.264111\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	−30.0000	−1.07486
$780$	0	0
$781$	13.0000	0.465177
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	52.0000	1.83963
$800$	0	0
$801$	28.0000	0.989331
$802$	0	0
$803$	−5.00000	−0.176446
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−51.0000	−1.79085	−0.895426	−	0.445210i	$-0.853129\pi$
$811$	−51.0000	−1.79085	−0.895426	+	0.445210i	$0.853129\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−60.0000	−2.09913
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−21.0000	−0.732905	−0.366453	−	0.930437i	$-0.619428\pi$
$821$	−21.0000	−0.732905	−0.366453	+	0.930437i	$0.619428\pi$
$822$	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
$825$	5.00000	0.174078
$826$	0	0
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	0	0
$836$	0	0
$837$	40.0000	1.38260
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−8.00000	−0.275535
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−22.0000	−0.755038
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	−5.00000	−0.170400
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	18.0000	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.0000	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$887$	−51.0000	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$888$	0	0
$889$	11.0000	0.368928
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−78.0000	−2.61017
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	0	0
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	−10.0000	−0.332779
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	46.0000	1.52405	0.762024	−	0.647549i	$-0.224206\pi$
$911$	46.0000	1.52405	0.762024	+	0.647549i	$0.224206\pi$
$912$	0	0
$913$	−1.00000	−0.0330952
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	26.0000	0.855800
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	12.0000	0.394132
$928$	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
$931$	−36.0000	−1.17985
$932$	0	0
$933$	−18.0000	−0.589294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−10.0000	−0.325645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	10.0000	0.320256
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	−14.0000	−0.447442
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.0000	−0.413795
$988$	0	0
$989$	−20.0000	−0.635963
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	22.0000	0.698149
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	0	0
$999$	5.00000	0.158193

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a_n

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Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.c.1.1	✓	1	8.3	odd	2
1184.2.a.f.1.1	yes	1	8.5	even	2
2368.2.a.e.1.1		1	1.1	even	1	trivial
2368.2.a.l.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 $\F_{q}$

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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.e.1.1

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