LMFDB - Embedded newform 2400.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2400 = 2^{5} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2400.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:	$19.1640964851$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2400.1

$q$ -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} +1.00000 q^{9} -4.00000 q^{11} -3.00000 q^{13} -4.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} +1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{31} -4.00000 q^{33} +2.00000 q^{37} -3.00000 q^{39} +2.00000 q^{41} -11.0000 q^{43} -2.00000 q^{47} -6.00000 q^{49} -4.00000 q^{51} -10.0000 q^{53} -1.00000 q^{57} -6.00000 q^{59} +11.0000 q^{61} +1.00000 q^{63} +9.00000 q^{67} +6.00000 q^{71} -14.0000 q^{73} -4.00000 q^{77} +16.0000 q^{79} +1.00000 q^{81} +2.00000 q^{83} -8.00000 q^{87} -3.00000 q^{91} +1.00000 q^{93} -11.0000 q^{97} -4.00000 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
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	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
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.0000	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	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
$141$	−2.00000	−0.168430
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	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
$181$	−15.0000	−1.11494	−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000	−1.11494	−0.557471	+	0.830197i	$0.688228\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000	1.17004
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.00000	0.0678844
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	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
$231$	−4.00000	−0.263181
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.00000	0.190885
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−8.00000	−0.495188
$262$	0	0
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−11.0000	−0.644831
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.0000	−0.634029
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	0	0
$313$	15.0000	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$313$	15.0000	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	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
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−10.0000	−0.519174
$372$	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
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	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
$381$	0	0
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.0000	−0.559161
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−3.00000	−0.149441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.0000	0.532327
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	23.0000	1.08063
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	−5.00000	−0.230388
$472$	0	0
$473$	44.0000	2.02312
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	32.0000	1.44121
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	−22.0000	−0.982888
$502$	0	0
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	24.0000	1.03375
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	−15.0000	−0.643712
$544$	0	0
$545$	0	0
$546$	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
$549$	11.0000	0.469469
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	16.0000	0.680389
$554$	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
$559$	33.0000	1.39575
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	0	0
$573$	20.0000	0.835512
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	−13.0000	−0.540262
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	40.0000	1.65663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.0000	0.450200
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	1.00000	0.0391931
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−19.0000	−0.734582
$670$	0	0
$671$	−44.0000	−1.69860
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.0000	1.69106	0.845529	−	0.533930i	$-0.179285\pi$
$677$	44.0000	1.69106	0.845529	+	0.533930i	$0.179285\pi$
$678$	0	0
$679$	−11.0000	−0.422141
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.00000	0.190762
$688$	0	0
$689$	30.0000	1.14291
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000	0.526524
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.0000	0.373457
$718$	0	0
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	21.0000	0.780998
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.0000	1.62740
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000	0.0731762
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.0000	−0.399802	−0.199901	−	0.979816i	$-0.564062\pi$
$757$	−11.0000	−0.399802	−0.199901	+	0.979816i	$0.564062\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0000	0.649942
$768$	0	0
$769$	−9.00000	−0.324548	−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000	−0.324548	−0.162274	+	0.986746i	$0.551883\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	0	0
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.0000	−1.88925	−0.944623	−	0.328158i	$-0.893572\pi$
$787$	−53.0000	−1.88925	−0.944623	+	0.328158i	$0.893572\pi$
$788$	0	0
$789$	26.0000	0.925625
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−33.0000	−1.17186
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	56.0000	1.97620
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	43.0000	1.50993	0.754967	−	0.655763i	$-0.227653\pi$
$811$	43.0000	1.50993	0.754967	+	0.655763i	$0.227653\pi$
$812$	0	0
$813$	28.0000	0.982003
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.0000	0.384841
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	0	0
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−16.0000	−0.551069
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−17.0000	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−27.0000	−0.914860
$872$	0	0
$873$	−11.0000	−0.372294
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−52.0000	−1.75192	−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000	−1.75192	−0.875962	+	0.482380i	$0.839773\pi$
$882$	0	0
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.0000	1.14161	0.570804	−	0.821086i	$-0.306632\pi$
$887$	34.0000	1.14161	0.570804	+	0.821086i	$0.306632\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	2.00000	0.0669274
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	−11.0000	−0.366057
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.0000	−0.462321
$918$	0	0
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	−18.0000	−0.592477
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	−26.0000	−0.851202
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−53.0000	−1.73143	−0.865717	−	0.500533i	$-0.833137\pi$
$937$	−53.0000	−1.73143	−0.865717	+	0.500533i	$0.833137\pi$
$938$	0	0
$939$	15.0000	0.489506
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.0000	1.03441
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−11.0000	−0.351203
$982$	0	0
$983$	44.0000	1.40338	0.701691	−	0.712481i	$-0.252429\pi$
$983$	44.0000	1.40338	0.701691	+	0.712481i	$0.252429\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	2.00000	0.0632772

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	2400.2.a.ba.1.1	yes	1
3.2	odd	2		7200.2.a.bk.1.1		1
4.3	odd	2		2400.2.a.h.1.1	yes	1
5.2	odd	4		2400.2.f.c.1249.1		2
5.3	odd	4		2400.2.f.c.1249.2		2
5.4	even	2		2400.2.a.e.1.1	✓	1
8.3	odd	2		4800.2.a.bv.1.1		1
8.5	even	2		4800.2.a.y.1.1		1
12.11	even	2		7200.2.a.q.1.1		1
15.2	even	4		7200.2.f.z.6049.2		2
15.8	even	4		7200.2.f.z.6049.1		2
15.14	odd	2		7200.2.a.t.1.1		1
20.3	even	4		2400.2.f.p.1249.1		2
20.7	even	4		2400.2.f.p.1249.2		2
20.19	odd	2		2400.2.a.bd.1.1	yes	1
40.3	even	4		4800.2.f.g.3649.2		2
40.13	odd	4		4800.2.f.bd.3649.1		2
40.19	odd	2		4800.2.a.v.1.1		1
40.27	even	4		4800.2.f.g.3649.1		2
40.29	even	2		4800.2.a.by.1.1		1
40.37	odd	4		4800.2.f.bd.3649.2		2
60.23	odd	4		7200.2.f.d.6049.2		2
60.47	odd	4		7200.2.f.d.6049.1		2
60.59	even	2		7200.2.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.a.e.1.1	✓	1	5.4	even	2
2400.2.a.h.1.1	yes	1	4.3	odd	2
2400.2.a.ba.1.1	yes	1	1.1	even	1	trivial
2400.2.a.bd.1.1	yes	1	20.19	odd	2
2400.2.f.c.1249.1		2	5.2	odd	4
2400.2.f.c.1249.2		2	5.3	odd	4
2400.2.f.p.1249.1		2	20.3	even	4
2400.2.f.p.1249.2		2	20.7	even	4
4800.2.a.v.1.1		1	40.19	odd	2
4800.2.a.y.1.1		1	8.5	even	2
4800.2.a.bv.1.1		1	8.3	odd	2
4800.2.a.by.1.1		1	40.29	even	2
4800.2.f.g.3649.1		2	40.27	even	4
4800.2.f.g.3649.2		2	40.3	even	4
4800.2.f.bd.3649.1		2	40.13	odd	4
4800.2.f.bd.3649.2		2	40.37	odd	4
7200.2.a.q.1.1		1	12.11	even	2
7200.2.a.t.1.1		1	15.14	odd	2
7200.2.a.bh.1.1		1	60.59	even	2
7200.2.a.bk.1.1		1	3.2	odd	2
7200.2.f.d.6049.1		2	60.47	odd	4
7200.2.f.d.6049.2		2	60.23	odd	4
7200.2.f.z.6049.1		2	15.8	even	4
7200.2.f.z.6049.2		2	15.2	even	4

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

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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	2400.2.a.ba.1.1

Level	$2400$
Weight	$2$
Character	2400.1
Self dual	yes
Analytic conductor	$19.164$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$