LMFDB - Embedded newform 2535.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2535 = 3 \cdot 5 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2535.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:	$20.2420769124$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2535.1

$q$ -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} -2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{20} -3.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} +3.00000 q^{33} -3.00000 q^{35} -2.00000 q^{36} -9.00000 q^{37} -3.00000 q^{41} +10.0000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -12.0000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +3.00000 q^{51} -3.00000 q^{53} +3.00000 q^{55} +12.0000 q^{59} -2.00000 q^{60} +1.00000 q^{61} +3.00000 q^{63} -8.00000 q^{64} +6.00000 q^{68} -3.00000 q^{69} +9.00000 q^{71} +6.00000 q^{73} -1.00000 q^{75} -9.00000 q^{77} +1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} +6.00000 q^{84} +3.00000 q^{85} +6.00000 q^{87} +15.0000 q^{89} -6.00000 q^{92} -6.00000 q^{93} +9.00000 q^{97} -3.00000 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	−3.00000	−0.654654
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−6.00000	−1.13389
$29$	−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
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−3.00000	−0.507093
$36$	−2.00000	−0.333333
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	6.00000	0.727607
$69$	−3.00000	−0.361158
$70$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−9.00000	−1.02565
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	6.00000	0.654654
$85$	3.00000	0.325396
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−6.00000	−0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−2.00000	−0.200000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	2.00000	0.192450
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	12.0000	1.13389
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	12.0000	1.11417
$117$	0	0
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	3.00000	0.270501
$124$	−12.0000	−1.07763
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	6.00000	0.507093
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	6.00000	0.498273
$146$	0	0
$147$	−2.00000	−0.164957
$148$	18.0000	1.47959
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	6.00000	0.468521
$165$	−3.00000	−0.233550
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−20.0000	−1.52499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	−12.0000	−0.904534
$177$	−12.0000	−0.901975
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	2.00000	0.149071
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	9.00000	0.658145
$188$	24.0000	1.75038
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	8.00000	0.577350
$193$	3.00000	0.215945	0.107972	−	0.994154i	$-0.465564\pi$
$193$	3.00000	0.215945	0.107972	+	0.994154i	$0.465564\pi$
$194$	0	0
$195$	0	0
$196$	−4.00000	−0.285714
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.0000	−1.26335
$204$	−6.00000	−0.420084
$205$	3.00000	0.209529
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	−9.00000	−0.616670
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	−6.00000	−0.405442
$220$	−6.00000	−0.404520
$221$	0	0
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−24.0000	−1.56227
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	4.00000	0.258199
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−2.00000	−0.127775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−6.00000	−0.377964
$253$	−9.00000	−0.565825
$254$	0	0
$255$	−3.00000	−0.187867
$256$	16.0000	1.00000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−27.0000	−1.67770
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	−15.0000	−0.917985
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	6.00000	0.361158
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−18.0000	−1.06810
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−9.00000	−0.527589
$292$	−12.0000	−0.702247
$293$	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
$295$	−12.0000	−0.698667
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	0	0
$300$	2.00000	0.115470
$301$	30.0000	1.72917
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	15.0000	0.856095	0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000	0.856095	0.428048	+	0.903756i	$0.359202\pi$
$308$	18.0000	1.02565
$309$	−4.00000	−0.227552
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	−2.00000	−0.112509
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	8.00000	0.447214
$321$	9.00000	0.502331
$322$	0	0
$323$	0	0
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	−36.0000	−1.98474
$330$	0	0
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	−12.0000	−0.658586
$333$	−9.00000	−0.493197
$334$	0	0
$335$	0	0
$336$	−12.0000	−0.654654
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	−6.00000	−0.325396
$341$	−18.0000	−0.974755
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	−12.0000	−0.643268
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	−30.0000	−1.59000
$357$	9.00000	0.476331
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	12.0000	0.625543
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−9.00000	−0.467257
$372$	12.0000	0.622171
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	10.0000	0.508329
$388$	−18.0000	−0.913812
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−1.00000	−0.0503155
$396$	6.00000	0.301511
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	36.0000	1.79107
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	27.0000	1.33834
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−8.00000	−0.394132
$413$	36.0000	1.77144
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	13.0000	0.636613
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−6.00000	−0.292770
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	3.00000	0.145180
$428$	18.0000	0.870063
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−4.00000	−0.192450
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	−36.0000	−1.72409
$437$	0	0
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	−18.0000	−0.854242
$445$	−15.0000	−0.711068
$446$	0	0
$447$	−3.00000	−0.141895
$448$	−24.0000	−1.13389
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	−36.0000	−1.69330
$453$	−6.00000	−0.281905
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	6.00000	0.279751
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	15.0000	0.697109	0.348555	−	0.937288i	$-0.386673\pi$
$463$	15.0000	0.697109	0.348555	+	0.937288i	$0.386673\pi$
$464$	−24.0000	−1.11417
$465$	6.00000	0.278243
$466$	0	0
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	−30.0000	−1.37940
$474$	0	0
$475$	0	0
$476$	18.0000	0.825029
$477$	−3.00000	−0.137361
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−9.00000	−0.409514
$484$	4.00000	0.181818
$485$	−9.00000	−0.408669
$486$	0	0
$487$	−15.0000	−0.679715	−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000	−0.679715	−0.339857	+	0.940477i	$0.610379\pi$
$488$	0	0
$489$	−3.00000	−0.135665
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	18.0000	0.810679
$494$	0	0
$495$	3.00000	0.134840
$496$	24.0000	1.07763
$497$	27.0000	1.21112
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	2.00000	0.0894427
$501$	−12.0000	−0.536120
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	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
$511$	18.0000	0.796273
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	20.0000	0.880451
$517$	36.0000	1.58328
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−24.0000	−1.04844
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−18.0000	−0.784092
$528$	12.0000	0.522233
$529$	−14.0000	−0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	−6.00000	−0.258438
$540$	−2.00000	−0.0860663
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	0	0
$543$	−25.0000	−1.07285
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−36.0000	−1.53784
$549$	1.00000	0.0426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.00000	0.127573
$554$	0	0
$555$	−9.00000	−0.382029
$556$	26.0000	1.10265
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	0	0
$560$	−12.0000	−0.507093
$561$	−9.00000	−0.379980
$562$	0	0
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	−24.0000	−1.01058
$565$	−18.0000	−0.757266
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	3.00000	0.125109
$576$	−8.00000	−0.333333
$577$	21.0000	0.874241	0.437121	−	0.899403i	$-0.355998\pi$
$577$	21.0000	0.874241	0.437121	+	0.899403i	$0.355998\pi$
$578$	0	0
$579$	−3.00000	−0.124676
$580$	−12.0000	−0.498273
$581$	18.0000	0.746766
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	4.00000	0.164957
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−36.0000	−1.47959
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	−6.00000	−0.245770
$597$	16.0000	0.654836
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	0	0
$604$	−12.0000	−0.488273
$605$	2.00000	0.0813116
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	0	0
$612$	6.00000	0.242536
$613$	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
$615$	−3.00000	−0.120972
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	30.0000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$620$	12.0000	0.481932
$621$	−3.00000	−0.120386
$622$	0	0
$623$	45.0000	1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	8.00000	0.319235
$629$	27.0000	1.07656
$630$	0	0
$631$	6.00000	0.238856	0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000	0.238856	0.119428	+	0.992843i	$0.461894\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	9.00000	0.356034
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	27.0000	1.06478	0.532388	−	0.846500i	$-0.321295\pi$
$643$	27.0000	1.06478	0.532388	+	0.846500i	$0.321295\pi$
$644$	−18.0000	−0.709299
$645$	10.0000	0.393750
$646$	0	0
$647$	−39.0000	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$647$	−39.0000	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−18.0000	−0.705476
$652$	−6.00000	−0.234978
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	−12.0000	−0.468521
$657$	6.00000	0.234082
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	6.00000	0.233550
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.0000	−0.696963
$668$	−24.0000	−0.928588
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−3.00000	−0.115814
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−39.0000	−1.49889	−0.749446	−	0.662066i	$-0.769680\pi$
$677$	−39.0000	−1.49889	−0.749446	+	0.662066i	$0.769680\pi$
$678$	0	0
$679$	27.0000	1.03616
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−6.00000	−0.228914
$688$	40.0000	1.52499
$689$	0	0
$690$	0	0
$691$	−30.0000	−1.14125	−0.570627	−	0.821209i	$-0.693300\pi$
$691$	−30.0000	−1.14125	−0.570627	+	0.821209i	$0.693300\pi$
$692$	−12.0000	−0.456172
$693$	−9.00000	−0.341882
$694$	0	0
$695$	13.0000	0.493118
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	−15.0000	−0.567352
$700$	−6.00000	−0.226779
$701$	−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
$703$	0	0
$704$	24.0000	0.904534
$705$	−12.0000	−0.451946
$706$	0	0
$707$	−54.0000	−2.03088
$708$	24.0000	0.901975
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	1.00000	0.0375029
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	0	0
$716$	−48.0000	−1.79384
$717$	27.0000	1.00833
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	−4.00000	−0.149071
$721$	12.0000	0.446903
$722$	0	0
$723$	−30.0000	−1.11571
$724$	−50.0000	−1.85824
$725$	−6.00000	−0.222834
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.0000	−1.10959
$732$	2.00000	0.0739221
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	0	0
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	−18.0000	−0.661693
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	6.00000	0.219529
$748$	−18.0000	−0.658145
$749$	−27.0000	−0.986559
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	−48.0000	−1.75038
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−6.00000	−0.218362
$756$	6.00000	0.218218
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	54.0000	1.95493
$764$	24.0000	0.868290
$765$	3.00000	0.108465
$766$	0	0
$767$	0	0
$768$	−16.0000	−0.577350
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−6.00000	−0.215945
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	27.0000	0.968620
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	6.00000	0.214423
$784$	8.00000	0.285714
$785$	4.00000	0.142766
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−12.0000	−0.427482
$789$	−24.0000	−0.854423
$790$	0	0
$791$	54.0000	1.92002
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.00000	−0.106399
$796$	32.0000	1.13421
$797$	51.0000	1.80651	0.903256	−	0.429101i	$-0.141170\pi$
$797$	51.0000	1.80651	0.903256	+	0.429101i	$0.141170\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	15.0000	0.529999
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	36.0000	1.26335
$813$	−6.00000	−0.210429
$814$	0	0
$815$	−3.00000	−0.105085
$816$	12.0000	0.420084
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−6.00000	−0.209529
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−6.00000	−0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	6.00000	0.206651
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	−6.00000	−0.206162
$848$	−12.0000	−0.412082
$849$	22.0000	0.755038
$850$	0	0
$851$	−27.0000	−0.925548
$852$	18.0000	0.616670
$853$	9.00000	0.308154	0.154077	−	0.988059i	$-0.450760\pi$
$853$	9.00000	0.308154	0.154077	+	0.988059i	$0.450760\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	20.0000	0.681994
$861$	9.00000	0.306719
$862$	0	0
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	8.00000	0.271694
$868$	−36.0000	−1.22192
$869$	−3.00000	−0.101768
$870$	0	0
$871$	0	0
$872$	0	0
$873$	9.00000	0.304604
$874$	0	0
$875$	−3.00000	−0.101419
$876$	12.0000	0.405442
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	12.0000	0.404520
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−48.0000	−1.60716
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−36.0000	−1.20067
$900$	−2.00000	−0.0666667
$901$	9.00000	0.299833
$902$	0	0
$903$	−30.0000	−0.998337
$904$	0	0
$905$	−25.0000	−0.831028
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	1.00000	0.0330590
$916$	−12.0000	−0.396491
$917$	36.0000	1.18882
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	−15.0000	−0.494267
$922$	0	0
$923$	0	0
$924$	−18.0000	−0.592157
$925$	−9.00000	−0.295918
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	0	0
$932$	−30.0000	−0.982683
$933$	−6.00000	−0.196431
$934$	0	0
$935$	−9.00000	−0.294331
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	−24.0000	−0.782794
$941$	−9.00000	−0.293392	−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000	−0.293392	−0.146696	+	0.989182i	$0.546864\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	48.0000	1.56227
$945$	3.00000	0.0975900
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	2.00000	0.0649570
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−27.0000	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	54.0000	1.74648
$957$	−18.0000	−0.581857
$958$	0	0
$959$	54.0000	1.74375
$960$	−8.00000	−0.258199
$961$	5.00000	0.161290
$962$	0	0
$963$	−9.00000	−0.290021
$964$	−60.0000	−1.93247
$965$	−3.00000	−0.0965734
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	2.00000	0.0641500
$973$	−39.0000	−1.25028
$974$	0	0
$975$	0	0
$976$	4.00000	0.128037
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	0	0
$979$	−45.0000	−1.43821
$980$	4.00000	0.127775
$981$	18.0000	0.574696
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	36.0000	1.14589
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	16.0000	0.507234
$996$	12.0000	0.380235
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	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	2535.2.a.f.1.1		1
3.2	odd	2		7605.2.a.n.1.1		1
13.5	odd	4		195.2.b.a.181.2	yes	2
13.8	odd	4		195.2.b.a.181.1	✓	2
13.12	even	2		2535.2.a.g.1.1		1
39.5	even	4		585.2.b.d.181.1		2
39.8	even	4		585.2.b.d.181.2		2
39.38	odd	2		7605.2.a.i.1.1		1
52.31	even	4		3120.2.g.j.961.2		2
52.47	even	4		3120.2.g.j.961.1		2
65.8	even	4		975.2.h.b.649.1		2
65.18	even	4		975.2.h.c.649.1		2
65.34	odd	4		975.2.b.e.376.2		2
65.44	odd	4		975.2.b.e.376.1		2
65.47	even	4		975.2.h.c.649.2		2
65.57	even	4		975.2.h.b.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.b.a.181.1	✓	2	13.8	odd	4
195.2.b.a.181.2	yes	2	13.5	odd	4
585.2.b.d.181.1		2	39.5	even	4
585.2.b.d.181.2		2	39.8	even	4
975.2.b.e.376.1		2	65.44	odd	4
975.2.b.e.376.2		2	65.34	odd	4
975.2.h.b.649.1		2	65.8	even	4
975.2.h.b.649.2		2	65.57	even	4
975.2.h.c.649.1		2	65.18	even	4
975.2.h.c.649.2		2	65.47	even	4
2535.2.a.f.1.1		1	1.1	even	1	trivial
2535.2.a.g.1.1		1	13.12	even	2
3120.2.g.j.961.1		2	52.47	even	4
3120.2.g.j.961.2		2	52.31	even	4
7605.2.a.i.1.1		1	39.38	odd	2
7605.2.a.n.1.1		1	3.2	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

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Fields

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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	2535.2.a.f.1.1

Level	$2535$
Weight	$2$
Character	2535.1
Self dual	yes
Analytic conductor	$20.242$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$