LMFDB - Embedded newform 459.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [459,2,Mod(1,459)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(459, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("459.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$459 = 3^{3} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	459.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:	$3.66513345278$
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	$\chi$	$=$	459.1

$q$ -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^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} -1.00000 q^{10} -5.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{23} -4.00000 q^{25} -5.00000 q^{26} +2.00000 q^{28} +9.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} -1.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} -1.00000 q^{38} +3.00000 q^{40} -3.00000 q^{41} +7.00000 q^{43} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -4.00000 q^{50} +5.00000 q^{52} +6.00000 q^{53} +6.00000 q^{56} +9.00000 q^{58} -10.0000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +5.00000 q^{65} +1.00000 q^{67} +1.00000 q^{68} +2.00000 q^{70} -11.0000 q^{71} +6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{76} +1.00000 q^{80} -3.00000 q^{82} +4.00000 q^{83} +1.00000 q^{85} +7.00000 q^{86} +2.00000 q^{89} +10.0000 q^{91} +1.00000 q^{92} +6.00000 q^{94} +1.00000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−5.00000	−0.980581
$27$	0	0
$28$	2.00000	0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−1.00000	−0.171499
$35$	2.00000	0.338062
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	−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
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	9.00000	1.18176
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	5.00000	0.620174
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	2.00000	0.239046
$71$	−11.0000	−1.30546	−0.652730	−	0.757591i	$-0.726376\pi$
$71$	−11.0000	−1.30546	−0.652730	+	0.757591i	$0.726376\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.00000	−0.331295
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	7.00000	0.754829
$87$	0	0
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	1.00000	0.104257
$93$	0	0
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	4.00000	0.400000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	6.00000	0.582772
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	2.00000	0.188982
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−9.00000	−0.835629
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	0	0
$124$	8.00000	0.718421
$125$	9.00000	0.804984
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	5.00000	0.438529
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	1.00000	0.0863868
$135$	0	0
$136$	3.00000	0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−11.0000	−0.923099
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	6.00000	0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	−5.00000	−0.395285
$161$	2.00000	0.157622
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	4.00000	0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	1.00000	0.0766965
$171$	0	0
$172$	−7.00000	−0.533745
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	0	0
$178$	2.00000	0.149906
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	3.00000	0.221163
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	1.00000	0.0725476
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	−14.0000	−0.985037
$203$	−18.0000	−1.26335
$204$	0	0
$205$	3.00000	0.209529
$206$	17.0000	1.18445
$207$	0	0
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−17.0000	−1.16210
$215$	−7.00000	−0.477396
$216$	0	0
$217$	16.0000	1.08615
$218$	−12.0000	−0.812743
$219$	0	0
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−27.0000	−1.77264
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	2.00000	0.129641
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	10.0000	0.640184
$245$	3.00000	0.191663
$246$	0	0
$247$	5.00000	0.318142
$248$	24.0000	1.52400
$249$	0	0
$250$	9.00000	0.569210
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	−5.00000	−0.310087
$261$	0	0
$262$	−1.00000	−0.0617802
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	2.00000	0.122628
$267$	0	0
$268$	−1.00000	−0.0610847
$269$	−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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	−6.00000	−0.358569
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	11.0000	0.652730
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	1.00000	0.0588235
$290$	−9.00000	−0.528498
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−20.0000	−1.15857
$299$	5.00000	0.289157
$300$	0	0
$301$	−14.0000	−0.806947
$302$	17.0000	0.978240
$303$	0	0
$304$	1.00000	0.0573539
$305$	10.0000	0.572598
$306$	0	0
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	0	0
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	0	0
$319$	0	0
$320$	−7.00000	−0.391312
$321$	0	0
$322$	2.00000	0.111456
$323$	1.00000	0.0556415
$324$	0	0
$325$	20.0000	1.10940
$326$	6.00000	0.332309
$327$	0	0
$328$	9.00000	0.496942
$329$	−12.0000	−0.661581
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	16.0000	0.875481
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−1.00000	−0.0542326
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	−21.0000	−1.13224
$345$	0	0
$346$	−22.0000	−1.18273
$347$	−29.0000	−1.55680	−0.778401	−	0.627768i	$-0.783969\pi$
$347$	−29.0000	−1.55680	−0.778401	+	0.627768i	$0.783969\pi$
$348$	0	0
$349$	9.00000	0.481759	0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000	0.481759	0.240879	+	0.970555i	$0.422564\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	11.0000	0.583819
$356$	−2.00000	−0.106000
$357$	0	0
$358$	24.0000	1.26844
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−20.0000	−1.05118
$363$	0	0
$364$	−10.0000	−0.524142
$365$	−6.00000	−0.314054
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	2.00000	0.103975
$371$	−12.0000	−0.623009
$372$	0	0
$373$	21.0000	1.08734	0.543669	−	0.839299i	$-0.317035\pi$
$373$	21.0000	1.08734	0.543669	+	0.839299i	$0.317035\pi$
$374$	0	0
$375$	0	0
$376$	−18.0000	−0.928279
$377$	−45.0000	−2.31762
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	−20.0000	−1.02329
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	−2.00000	−0.101535
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	9.00000	0.454569
$393$	0	0
$394$	15.0000	0.755689
$395$	0	0
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	4.00000	0.200000
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	0	0
$403$	40.0000	1.99254
$404$	14.0000	0.696526
$405$	0	0
$406$	−18.0000	−0.893325
$407$	0	0
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	3.00000	0.148159
$411$	0	0
$412$	−17.0000	−0.837530
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	−25.0000	−1.22573
$417$	0	0
$418$	0	0
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−18.0000	−0.874157
$425$	4.00000	0.194029
$426$	0	0
$427$	20.0000	0.967868
$428$	17.0000	0.821726
$429$	0	0
$430$	−7.00000	−0.337570
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	12.0000	0.574696
$437$	1.00000	0.0478365
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	5.00000	0.237826
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−21.0000	−0.994379
$447$	0	0
$448$	−14.0000	−0.661438
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	0	0
$452$	10.0000	0.470360
$453$	0	0
$454$	−21.0000	−0.985579
$455$	−10.0000	−0.468807
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−15.0000	−0.700904
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	21.0000	0.972806
$467$	42.0000	1.94353	0.971764	−	0.235954i	$-0.0758216\pi$
$467$	42.0000	1.94353	0.971764	+	0.235954i	$0.0758216\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	−6.00000	−0.276759
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	16.0000	0.731823
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	−12.0000	−0.546585
$483$	0	0
$484$	11.0000	0.500000
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	3.00000	0.135526
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	5.00000	0.224961
$495$	0	0
$496$	8.00000	0.359211
$497$	22.0000	0.986835
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	24.0000	1.07117
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	4.00000	0.177471
$509$	−40.0000	−1.77297	−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000	−1.77297	−0.886484	+	0.462758i	$0.846860\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	−11.0000	−0.486136
$513$	0	0
$514$	6.00000	0.264649
$515$	−17.0000	−0.749110
$516$	0	0
$517$	0	0
$518$	4.00000	0.175750
$519$	0	0
$520$	−15.0000	−0.657794
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	−14.0000	−0.610429
$527$	8.00000	0.348485
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	15.0000	0.649722
$534$	0	0
$535$	17.0000	0.734974
$536$	−3.00000	−0.129580
$537$	0	0
$538$	−10.0000	−0.431131
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−5.00000	−0.214373
$545$	12.0000	0.514024
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	0	0
$551$	−9.00000	−0.383413
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	2.00000	0.0848189
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	−35.0000	−1.48034
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	−4.00000	−0.168133
$567$	0	0
$568$	33.0000	1.38465
$569$	44.0000	1.84458	0.922288	−	0.386503i	$-0.126317\pi$
$569$	44.0000	1.84458	0.922288	+	0.386503i	$0.126317\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	0	0
$574$	6.00000	0.250435
$575$	4.00000	0.166812
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	9.00000	0.373705
$581$	−8.00000	−0.331896
$582$	0	0
$583$	0	0
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−12.0000	−0.495715
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	20.0000	0.819232
$597$	0	0
$598$	5.00000	0.204465
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	−14.0000	−0.570597
$603$	0	0
$604$	−17.0000	−0.691720
$605$	11.0000	0.447214
$606$	0	0
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	10.0000	0.404888
$611$	−30.0000	−1.21367
$612$	0	0
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	9.00000	0.363210
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−16.0000	−0.641542
$623$	−4.00000	−0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	14.0000	0.559553
$627$	0	0
$628$	14.0000	0.558661
$629$	2.00000	0.0797452
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	0	0
$633$	0	0
$634$	−17.0000	−0.675156
$635$	4.00000	0.158735
$636$	0	0
$637$	15.0000	0.594322
$638$	0	0
$639$	0	0
$640$	3.00000	0.118585
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	1.00000	0.0393445
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	0	0
$650$	20.0000	0.784465
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	1.00000	0.0390732
$656$	3.00000	0.117130
$657$	0	0
$658$	−12.0000	−0.467809
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	13.0000	0.505259
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−9.00000	−0.348481
$668$	−16.0000	−0.619059
$669$	0	0
$670$	−1.00000	−0.0386334
$671$	0	0
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	4.00000	0.154074
$675$	0	0
$676$	−12.0000	−0.461538
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	−3.00000	−0.115045
$681$	0	0
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	20.0000	0.763604
$687$	0	0
$688$	−7.00000	−0.266872
$689$	−30.0000	−1.14291
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	−29.0000	−1.10082
$695$	2.00000	0.0758643
$696$	0	0
$697$	3.00000	0.113633
$698$	9.00000	0.340655
$699$	0	0
$700$	−8.00000	−0.302372
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	28.0000	1.05305
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	11.0000	0.412823
$711$	0	0
$712$	−6.00000	−0.224860
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	30.0000	1.11959
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	0	0
$721$	−34.0000	−1.26623
$722$	−18.0000	−0.669891
$723$	0	0
$724$	20.0000	0.743294
$725$	−36.0000	−1.33701
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	−30.0000	−1.11187
$729$	0	0
$730$	−6.00000	−0.222070
$731$	−7.00000	−0.258904
$732$	0	0
$733$	−29.0000	−1.07114	−0.535570	−	0.844491i	$-0.679903\pi$
$733$	−29.0000	−1.07114	−0.535570	+	0.844491i	$0.679903\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−5.00000	−0.184302
$737$	0	0
$738$	0	0
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	21.0000	0.768865
$747$	0	0
$748$	0	0
$749$	34.0000	1.24233
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−45.0000	−1.63880
$755$	−17.0000	−0.618693
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	22.0000	0.799076
$759$	0	0
$760$	−3.00000	−0.108821
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	20.0000	0.723575
$765$	0	0
$766$	30.0000	1.08394
$767$	0	0
$768$	0	0
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	−6.00000	−0.215387
$777$	0	0
$778$	24.0000	0.860442
$779$	3.00000	0.107486
$780$	0	0
$781$	0	0
$782$	1.00000	0.0357599
$783$	0	0
$784$	3.00000	0.107143
$785$	14.0000	0.499681
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−15.0000	−0.534353
$789$	0	0
$790$	0	0
$791$	20.0000	0.711118
$792$	0	0
$793$	50.0000	1.77555
$794$	30.0000	1.06466
$795$	0	0
$796$	20.0000	0.708881
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	−20.0000	−0.707107
$801$	0	0
$802$	5.00000	0.176556
$803$	0	0
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	40.0000	1.40894
$807$	0	0
$808$	42.0000	1.47755
$809$	−23.0000	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$809$	−23.0000	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	18.0000	0.631676
$813$	0	0
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	−7.00000	−0.244899
$818$	−7.00000	−0.244749
$819$	0	0
$820$	−3.00000	−0.104765
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	−51.0000	−1.77667
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	−35.0000	−1.21341
$833$	3.00000	0.103944
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	0	0
$838$	7.00000	0.241811
$839$	19.0000	0.655953	0.327976	−	0.944686i	$-0.393633\pi$
$839$	19.0000	0.655953	0.327976	+	0.944686i	$0.393633\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−38.0000	−1.30957
$843$	0	0
$844$	4.00000	0.137686
$845$	−12.0000	−0.412813
$846$	0	0
$847$	22.0000	0.755929
$848$	−6.00000	−0.206041
$849$	0	0
$850$	4.00000	0.137199
$851$	2.00000	0.0685591
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	51.0000	1.74314
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	7.00000	0.238698
$861$	0	0
$862$	15.0000	0.510902
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	11.0000	0.373795
$867$	0	0
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	−5.00000	−0.169419
$872$	36.0000	1.21911
$873$	0	0
$874$	1.00000	0.0338255
$875$	−18.0000	−0.608511
$876$	0	0
$877$	54.0000	1.82345	0.911725	−	0.410801i	$-0.134751\pi$
$877$	54.0000	1.82345	0.911725	+	0.410801i	$0.134751\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	0	0
$881$	1.00000	0.0336909	0.0168454	−	0.999858i	$-0.494638\pi$
$881$	1.00000	0.0336909	0.0168454	+	0.999858i	$0.494638\pi$
$882$	0	0
$883$	−3.00000	−0.100958	−0.0504790	−	0.998725i	$-0.516075\pi$
$883$	−3.00000	−0.100958	−0.0504790	+	0.998725i	$0.516075\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	−4.00000	−0.134383
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	21.0000	0.703132
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−24.0000	−0.802232
$896$	6.00000	0.200446
$897$	0	0
$898$	26.0000	0.867631
$899$	−72.0000	−2.40133
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	0	0
$904$	30.0000	0.997785
$905$	20.0000	0.664822
$906$	0	0
$907$	−58.0000	−1.92586	−0.962929	−	0.269754i	$-0.913058\pi$
$907$	−58.0000	−1.92586	−0.962929	+	0.269754i	$0.913058\pi$
$908$	21.0000	0.696909
$909$	0	0
$910$	−10.0000	−0.331497
$911$	−11.0000	−0.364446	−0.182223	−	0.983257i	$-0.558329\pi$
$911$	−11.0000	−0.364446	−0.182223	+	0.983257i	$0.558329\pi$
$912$	0	0
$913$	0	0
$914$	18.0000	0.595387
$915$	0	0
$916$	15.0000	0.495614
$917$	2.00000	0.0660458
$918$	0	0
$919$	−45.0000	−1.48441	−0.742207	−	0.670171i	$-0.766221\pi$
$919$	−45.0000	−1.48441	−0.742207	+	0.670171i	$0.766221\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	38.0000	1.25146
$923$	55.0000	1.81035
$924$	0	0
$925$	8.00000	0.263038
$926$	−4.00000	−0.131448
$927$	0	0
$928$	45.0000	1.47720
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−21.0000	−0.687878
$933$	0	0
$934$	42.0000	1.37428
$935$	0	0
$936$	0	0
$937$	49.0000	1.60076	0.800380	−	0.599493i	$-0.204631\pi$
$937$	49.0000	1.60076	0.800380	+	0.599493i	$0.204631\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	6.00000	0.195698
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	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
$949$	−30.0000	−0.973841
$950$	4.00000	0.129777
$951$	0	0
$952$	−6.00000	−0.194461
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	−16.0000	−0.517477
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	10.0000	0.322413
$963$	0	0
$964$	12.0000	0.386494
$965$	−14.0000	−0.450676
$966$	0	0
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	33.0000	1.06066
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	32.0000	1.02535
$975$	0	0
$976$	10.0000	0.320092
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	22.0000	0.702048
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	−15.0000	−0.477940
$986$	−9.00000	−0.286618
$987$	0	0
$988$	−5.00000	−0.159071
$989$	−7.00000	−0.222587
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	22.0000	0.697798
$995$	20.0000	0.634043
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	20.0000	0.633089
$999$	0	0

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	459.2.a.f.1.1	yes	1
3.2	odd	2		459.2.a.c.1.1	✓	1
4.3	odd	2		7344.2.a.n.1.1		1
12.11	even	2		7344.2.a.y.1.1		1
17.16	even	2		7803.2.a.p.1.1		1
51.50	odd	2		7803.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.a.c.1.1	✓	1	3.2	odd	2
459.2.a.f.1.1	yes	1	1.1	even	1	trivial
7344.2.a.n.1.1		1	4.3	odd	2
7344.2.a.y.1.1		1	12.11	even	2
7803.2.a.f.1.1		1	51.50	odd	2
7803.2.a.p.1.1		1	17.16	even	2

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

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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	459.2.a.f.1.1

Level	$459$
Weight	$2$
Character	459.1
Self dual	yes
Analytic conductor	$3.665$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$