LMFDB - Embedded newform 4732.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4732 = 2^{2} \cdot 7 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 19x^{6} - 2x^{5} + 113x^{4} + 40x^{3} - 232x^{2} - 136x + 52$ x^8 - 19x^6 - 2x^5 + 113x^4 + 40x^3 - 232x^2 - 136x + 52
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.42977$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42977 q^{3} +3.63781 q^{5} -1.00000 q^{7} +2.90380 q^{9} -3.86676 q^{11} -8.83905 q^{15} +1.63539 q^{17} -2.01179 q^{19} +2.42977 q^{21} -2.41411 q^{23} +8.23365 q^{25} +0.233744 q^{27} +4.02983 q^{29} +9.12236 q^{31} +9.39535 q^{33} -3.63781 q^{35} -12.0257 q^{37} +9.56193 q^{41} -9.86092 q^{43} +10.5635 q^{45} -8.94009 q^{47} +1.00000 q^{49} -3.97364 q^{51} -6.91625 q^{53} -14.0665 q^{55} +4.88819 q^{57} +0.115252 q^{59} -4.94075 q^{61} -2.90380 q^{63} +9.11849 q^{67} +5.86575 q^{69} +0.292994 q^{71} -1.89250 q^{73} -20.0059 q^{75} +3.86676 q^{77} +5.12890 q^{79} -9.27934 q^{81} -11.8010 q^{83} +5.94925 q^{85} -9.79157 q^{87} -2.46394 q^{89} -22.1653 q^{93} -7.31850 q^{95} +17.4756 q^{97} -11.2283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100})$ 8 * q - 6 * q^5 - 8 * q^7 + 14 * q^9 - 12 * q^11 - 12 * q^15 + 2 * q^17 - 6 * q^19 + 22 * q^25 - 6 * q^27 + 22 * q^29 - 14 * q^31 + 28 * q^33 + 6 * q^35 - 12 * q^37 - 4 * q^41 + 6 * q^43 - 20 * q^45 - 42 * q^47 + 8 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 - 2 * q^59 - 4 * q^61 - 14 * q^63 - 24 * q^67 - 52 * q^69 - 28 * q^71 - 28 * q^73 - 10 * q^75 + 12 * q^77 + 4 * q^79 - 38 * q^83 + 36 * q^85 - 26 * q^87 + 22 * q^89 - 56 * q^93 - 30 * q^95 + 4 * q^97 - 80 * q^99

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
$3$	−2.42977	−1.40283	−0.701415	−	0.712753i	$-0.747448\pi$
$3$	−2.42977	−1.40283	−0.701415	+	0.712753i	$0.747448\pi$
$4$	0	0
$5$	3.63781	1.62688	0.813439	−	0.581651i	$-0.197593\pi$
$5$	3.63781	1.62688	0.813439	+	0.581651i	$0.197593\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.90380	0.967933
$10$	0	0
$11$	−3.86676	−1.16587	−0.582936	−	0.812518i	$-0.698096\pi$
$11$	−3.86676	−1.16587	−0.582936	+	0.812518i	$0.698096\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−8.83905	−2.28223
$16$	0	0
$17$	1.63539	0.396641	0.198321	−	0.980137i	$-0.436451\pi$
$17$	1.63539	0.396641	0.198321	+	0.980137i	$0.436451\pi$
$18$	0	0
$19$	−2.01179	−0.461536	−0.230768	−	0.973009i	$-0.574124\pi$
$19$	−2.01179	−0.461536	−0.230768	+	0.973009i	$0.574124\pi$
$20$	0	0
$21$	2.42977	0.530220
$22$	0	0
$23$	−2.41411	−0.503377	−0.251689	−	0.967808i	$-0.580986\pi$
$23$	−2.41411	−0.503377	−0.251689	+	0.967808i	$0.580986\pi$
$24$	0	0
$25$	8.23365	1.64673
$26$	0	0
$27$	0.233744	0.0449840
$28$	0	0
$29$	4.02983	0.748320	0.374160	−	0.927364i	$-0.377931\pi$
$29$	4.02983	0.748320	0.374160	+	0.927364i	$0.377931\pi$
$30$	0	0
$31$	9.12236	1.63842	0.819212	−	0.573490i	$-0.194411\pi$
$31$	9.12236	1.63842	0.819212	+	0.573490i	$0.194411\pi$
$32$	0	0
$33$	9.39535	1.63552
$34$	0	0
$35$	−3.63781	−0.614902
$36$	0	0
$37$	−12.0257	−1.97701	−0.988507	−	0.151175i	$-0.951694\pi$
$37$	−12.0257	−1.97701	−0.988507	+	0.151175i	$0.951694\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.56193	1.49332	0.746661	−	0.665204i	$-0.231656\pi$
$41$	9.56193	1.49332	0.746661	+	0.665204i	$0.231656\pi$
$42$	0	0
$43$	−9.86092	−1.50378	−0.751888	−	0.659291i	$-0.770857\pi$
$43$	−9.86092	−1.50378	−0.751888	+	0.659291i	$0.770857\pi$
$44$	0	0
$45$	10.5635	1.57471
$46$	0	0
$47$	−8.94009	−1.30405	−0.652023	−	0.758199i	$-0.726079\pi$
$47$	−8.94009	−1.30405	−0.652023	+	0.758199i	$0.726079\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.97364	−0.556421
$52$	0	0
$53$	−6.91625	−0.950020	−0.475010	−	0.879980i	$-0.657556\pi$
$53$	−6.91625	−0.950020	−0.475010	+	0.879980i	$0.657556\pi$
$54$	0	0
$55$	−14.0665	−1.89673
$56$	0	0
$57$	4.88819	0.647457
$58$	0	0
$59$	0.115252	0.0150045	0.00750224	−	0.999972i	$-0.497612\pi$
$59$	0.115252	0.0150045	0.00750224	+	0.999972i	$0.497612\pi$
$60$	0	0
$61$	−4.94075	−0.632598	−0.316299	−	0.948660i	$-0.602440\pi$
$61$	−4.94075	−0.632598	−0.316299	+	0.948660i	$0.602440\pi$
$62$	0	0
$63$	−2.90380	−0.365844
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.11849	1.11400	0.557000	−	0.830512i	$-0.311952\pi$
$67$	9.11849	1.11400	0.557000	+	0.830512i	$0.311952\pi$
$68$	0	0
$69$	5.86575	0.706153
$70$	0	0
$71$	0.292994	0.0347720	0.0173860	−	0.999849i	$-0.494466\pi$
$71$	0.292994	0.0347720	0.0173860	+	0.999849i	$0.494466\pi$
$72$	0	0
$73$	−1.89250	−0.221500	−0.110750	−	0.993848i	$-0.535325\pi$
$73$	−1.89250	−0.221500	−0.110750	+	0.993848i	$0.535325\pi$
$74$	0	0
$75$	−20.0059	−2.31008
$76$	0	0
$77$	3.86676	0.440658
$78$	0	0
$79$	5.12890	0.577046	0.288523	−	0.957473i	$-0.406836\pi$
$79$	5.12890	0.577046	0.288523	+	0.957473i	$0.406836\pi$
$80$	0	0
$81$	−9.27934	−1.03104
$82$	0	0
$83$	−11.8010	−1.29533	−0.647664	−	0.761926i	$-0.724254\pi$
$83$	−11.8010	−1.29533	−0.647664	+	0.761926i	$0.724254\pi$
$84$	0	0
$85$	5.94925	0.645287
$86$	0	0
$87$	−9.79157	−1.04977
$88$	0	0
$89$	−2.46394	−0.261177	−0.130588	−	0.991437i	$-0.541687\pi$
$89$	−2.46394	−0.261177	−0.130588	+	0.991437i	$0.541687\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−22.1653	−2.29843
$94$	0	0
$95$	−7.31850	−0.750862
$96$	0	0
$97$	17.4756	1.77438	0.887190	−	0.461404i	$-0.152654\pi$
$97$	17.4756	1.77438	0.887190	+	0.461404i	$0.152654\pi$
$98$	0	0
$99$	−11.2283	−1.12849
$100$	0	0
$101$	−10.2825	−1.02315	−0.511575	−	0.859239i	$-0.670938\pi$
$101$	−10.2825	−1.02315	−0.511575	+	0.859239i	$0.670938\pi$
$102$	0	0
$103$	−7.17129	−0.706608	−0.353304	−	0.935509i	$-0.614942\pi$
$103$	−7.17129	−0.706608	−0.353304	+	0.935509i	$0.614942\pi$
$104$	0	0
$105$	8.83905	0.862603
$106$	0	0
$107$	14.9827	1.44843	0.724216	−	0.689573i	$-0.242202\pi$
$107$	14.9827	1.44843	0.724216	+	0.689573i	$0.242202\pi$
$108$	0	0
$109$	−8.82413	−0.845198	−0.422599	−	0.906317i	$-0.638882\pi$
$109$	−8.82413	−0.845198	−0.422599	+	0.906317i	$0.638882\pi$
$110$	0	0
$111$	29.2197	2.77342
$112$	0	0
$113$	5.48631	0.516108	0.258054	−	0.966130i	$-0.416919\pi$
$113$	5.48631	0.516108	0.258054	+	0.966130i	$0.416919\pi$
$114$	0	0
$115$	−8.78208	−0.818933
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.63539	−0.149916
$120$	0	0
$121$	3.95183	0.359258
$122$	0	0
$123$	−23.2333	−2.09488
$124$	0	0
$125$	11.7634	1.05215
$126$	0	0
$127$	−12.0811	−1.07202	−0.536012	−	0.844210i	$-0.680070\pi$
$127$	−12.0811	−1.07202	−0.536012	+	0.844210i	$0.680070\pi$
$128$	0	0
$129$	23.9598	2.10954
$130$	0	0
$131$	3.82950	0.334585	0.167293	−	0.985907i	$-0.446498\pi$
$131$	3.82950	0.334585	0.167293	+	0.985907i	$0.446498\pi$
$132$	0	0
$133$	2.01179	0.174444
$134$	0	0
$135$	0.850315	0.0731834
$136$	0	0
$137$	−6.31957	−0.539917	−0.269959	−	0.962872i	$-0.587010\pi$
$137$	−6.31957	−0.539917	−0.269959	+	0.962872i	$0.587010\pi$
$138$	0	0
$139$	−2.68186	−0.227472	−0.113736	−	0.993511i	$-0.536282\pi$
$139$	−2.68186	−0.227472	−0.113736	+	0.993511i	$0.536282\pi$
$140$	0	0
$141$	21.7224	1.82935
$142$	0	0
$143$	0	0
$144$	0	0
$145$	14.6597	1.21743
$146$	0	0
$147$	−2.42977	−0.200404
$148$	0	0
$149$	−15.5711	−1.27563	−0.637817	−	0.770188i	$-0.720163\pi$
$149$	−15.5711	−1.27563	−0.637817	+	0.770188i	$0.720163\pi$
$150$	0	0
$151$	−15.7761	−1.28384	−0.641920	−	0.766771i	$-0.721862\pi$
$151$	−15.7761	−1.28384	−0.641920	+	0.766771i	$0.721862\pi$
$152$	0	0
$153$	4.74886	0.383922
$154$	0	0
$155$	33.1854	2.66552
$156$	0	0
$157$	−9.67555	−0.772193	−0.386096	−	0.922458i	$-0.626177\pi$
$157$	−9.67555	−0.772193	−0.386096	+	0.922458i	$0.626177\pi$
$158$	0	0
$159$	16.8049	1.33272
$160$	0	0
$161$	2.41411	0.190259
$162$	0	0
$163$	12.8297	1.00490	0.502448	−	0.864607i	$-0.332433\pi$
$163$	12.8297	1.00490	0.502448	+	0.864607i	$0.332433\pi$
$164$	0	0
$165$	34.1785	2.66079
$166$	0	0
$167$	10.4124	0.805736	0.402868	−	0.915258i	$-0.368013\pi$
$167$	10.4124	0.805736	0.402868	+	0.915258i	$0.368013\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−5.84183	−0.446736
$172$	0	0
$173$	12.1778	0.925862	0.462931	−	0.886394i	$-0.346798\pi$
$173$	12.1778	0.925862	0.462931	+	0.886394i	$0.346798\pi$
$174$	0	0
$175$	−8.23365	−0.622405
$176$	0	0
$177$	−0.280035	−0.0210488
$178$	0	0
$179$	0.468018	0.0349813	0.0174907	−	0.999847i	$-0.494432\pi$
$179$	0.468018	0.0349813	0.0174907	+	0.999847i	$0.494432\pi$
$180$	0	0
$181$	20.4110	1.51714	0.758570	−	0.651592i	$-0.225898\pi$
$181$	20.4110	1.51714	0.758570	+	0.651592i	$0.225898\pi$
$182$	0	0
$183$	12.0049	0.887427
$184$	0	0
$185$	−43.7472	−3.21636
$186$	0	0
$187$	−6.32368	−0.462433
$188$	0	0
$189$	−0.233744	−0.0170024
$190$	0	0
$191$	−3.48905	−0.252459	−0.126230	−	0.992001i	$-0.540288\pi$
$191$	−3.48905	−0.252459	−0.126230	+	0.992001i	$0.540288\pi$
$192$	0	0
$193$	−3.44525	−0.247994	−0.123997	−	0.992283i	$-0.539571\pi$
$193$	−3.44525	−0.247994	−0.123997	+	0.992283i	$0.539571\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.6229	−0.899347	−0.449674	−	0.893193i	$-0.648460\pi$
$197$	−12.6229	−0.899347	−0.449674	+	0.893193i	$0.648460\pi$
$198$	0	0
$199$	−11.0467	−0.783080	−0.391540	−	0.920161i	$-0.628058\pi$
$199$	−11.0467	−0.783080	−0.391540	+	0.920161i	$0.628058\pi$
$200$	0	0
$201$	−22.1559	−1.56275
$202$	0	0
$203$	−4.02983	−0.282838
$204$	0	0
$205$	34.7845	2.42945
$206$	0	0
$207$	−7.01010	−0.487236
$208$	0	0
$209$	7.77910	0.538092
$210$	0	0
$211$	−1.00569	−0.0692342	−0.0346171	−	0.999401i	$-0.511021\pi$
$211$	−1.00569	−0.0692342	−0.0346171	+	0.999401i	$0.511021\pi$
$212$	0	0
$213$	−0.711910	−0.0487793
$214$	0	0
$215$	−35.8721	−2.44646
$216$	0	0
$217$	−9.12236	−0.619266
$218$	0	0
$219$	4.59835	0.310728
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−22.3034	−1.49355	−0.746773	−	0.665079i	$-0.768398\pi$
$223$	−22.3034	−1.49355	−0.746773	+	0.665079i	$0.768398\pi$
$224$	0	0
$225$	23.9089	1.59392
$226$	0	0
$227$	−16.0475	−1.06511	−0.532555	−	0.846395i	$-0.678768\pi$
$227$	−16.0475	−1.06511	−0.532555	+	0.846395i	$0.678768\pi$
$228$	0	0
$229$	−12.2285	−0.808082	−0.404041	−	0.914741i	$-0.632395\pi$
$229$	−12.2285	−0.808082	−0.404041	+	0.914741i	$0.632395\pi$
$230$	0	0
$231$	−9.39535	−0.618169
$232$	0	0
$233$	7.66467	0.502129	0.251065	−	0.967970i	$-0.419219\pi$
$233$	7.66467	0.502129	0.251065	+	0.967970i	$0.419219\pi$
$234$	0	0
$235$	−32.5223	−2.12152
$236$	0	0
$237$	−12.4621	−0.809498
$238$	0	0
$239$	−9.63081	−0.622965	−0.311483	−	0.950252i	$-0.600826\pi$
$239$	−9.63081	−0.622965	−0.311483	+	0.950252i	$0.600826\pi$
$240$	0	0
$241$	−0.143077	−0.00921638	−0.00460819	−	0.999989i	$-0.501467\pi$
$241$	−0.143077	−0.00921638	−0.00460819	+	0.999989i	$0.501467\pi$
$242$	0	0
$243$	21.8455	1.40139
$244$	0	0
$245$	3.63781	0.232411
$246$	0	0
$247$	0	0
$248$	0	0
$249$	28.6738	1.81713
$250$	0	0
$251$	10.2047	0.644117	0.322059	−	0.946720i	$-0.395625\pi$
$251$	10.2047	0.644117	0.322059	+	0.946720i	$0.395625\pi$
$252$	0	0
$253$	9.33479	0.586873
$254$	0	0
$255$	−14.4553	−0.905228
$256$	0	0
$257$	−20.1595	−1.25751	−0.628756	−	0.777602i	$-0.716436\pi$
$257$	−20.1595	−1.25751	−0.628756	+	0.777602i	$0.716436\pi$
$258$	0	0
$259$	12.0257	0.747241
$260$	0	0
$261$	11.7018	0.724324
$262$	0	0
$263$	28.5916	1.76303	0.881516	−	0.472154i	$-0.156523\pi$
$263$	28.5916	1.76303	0.881516	+	0.472154i	$0.156523\pi$
$264$	0	0
$265$	−25.1600	−1.54557
$266$	0	0
$267$	5.98681	0.366387
$268$	0	0
$269$	−32.2457	−1.96606	−0.983028	−	0.183453i	$-0.941272\pi$
$269$	−32.2457	−1.96606	−0.983028	+	0.183453i	$0.941272\pi$
$270$	0	0
$271$	−24.1675	−1.46807	−0.734035	−	0.679112i	$-0.762365\pi$
$271$	−24.1675	−1.46807	−0.734035	+	0.679112i	$0.762365\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−31.8375	−1.91988
$276$	0	0
$277$	−20.1726	−1.21205	−0.606027	−	0.795444i	$-0.707238\pi$
$277$	−20.1726	−1.21205	−0.606027	+	0.795444i	$0.707238\pi$
$278$	0	0
$279$	26.4895	1.58589
$280$	0	0
$281$	24.0074	1.43216	0.716080	−	0.698018i	$-0.245934\pi$
$281$	24.0074	1.43216	0.716080	+	0.698018i	$0.245934\pi$
$282$	0	0
$283$	13.5536	0.805680	0.402840	−	0.915270i	$-0.368023\pi$
$283$	13.5536	0.805680	0.402840	+	0.915270i	$0.368023\pi$
$284$	0	0
$285$	17.7823	1.05333
$286$	0	0
$287$	−9.56193	−0.564423
$288$	0	0
$289$	−14.3255	−0.842676
$290$	0	0
$291$	−42.4618	−2.48915
$292$	0	0
$293$	−14.2739	−0.833891	−0.416945	−	0.908931i	$-0.636899\pi$
$293$	−14.2739	−0.833891	−0.416945	+	0.908931i	$0.636899\pi$
$294$	0	0
$295$	0.419263	0.0244105
$296$	0	0
$297$	−0.903831	−0.0524456
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.86092	0.568374
$302$	0	0
$303$	24.9842	1.43531
$304$	0	0
$305$	−17.9735	−1.02916
$306$	0	0
$307$	−28.2784	−1.61393	−0.806966	−	0.590598i	$-0.798892\pi$
$307$	−28.2784	−1.61393	−0.806966	+	0.590598i	$0.798892\pi$
$308$	0	0
$309$	17.4246	0.991251
$310$	0	0
$311$	−6.86783	−0.389439	−0.194719	−	0.980859i	$-0.562380\pi$
$311$	−6.86783	−0.389439	−0.194719	+	0.980859i	$0.562380\pi$
$312$	0	0
$313$	−19.8476	−1.12185	−0.560927	−	0.827865i	$-0.689555\pi$
$313$	−19.8476	−1.12185	−0.560927	+	0.827865i	$0.689555\pi$
$314$	0	0
$315$	−10.5635	−0.595184
$316$	0	0
$317$	−33.1727	−1.86317	−0.931584	−	0.363527i	$-0.881572\pi$
$317$	−33.1727	−1.86317	−0.931584	+	0.363527i	$0.881572\pi$
$318$	0	0
$319$	−15.5824	−0.872446
$320$	0	0
$321$	−36.4046	−2.03191
$322$	0	0
$323$	−3.29007	−0.183064
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.4406	1.18567
$328$	0	0
$329$	8.94009	0.492883
$330$	0	0
$331$	16.6643	0.915954	0.457977	−	0.888964i	$-0.348574\pi$
$331$	16.6643	0.915954	0.457977	+	0.888964i	$0.348574\pi$
$332$	0	0
$333$	−34.9203	−1.91362
$334$	0	0
$335$	33.1713	1.81234
$336$	0	0
$337$	−2.84905	−0.155198	−0.0775989	−	0.996985i	$-0.524725\pi$
$337$	−2.84905	−0.155198	−0.0775989	+	0.996985i	$0.524725\pi$
$338$	0	0
$339$	−13.3305	−0.724013
$340$	0	0
$341$	−35.2740	−1.91019
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.3385	1.14882
$346$	0	0
$347$	−13.6349	−0.731958	−0.365979	−	0.930623i	$-0.619266\pi$
$347$	−13.6349	−0.731958	−0.365979	+	0.930623i	$0.619266\pi$
$348$	0	0
$349$	−2.37578	−0.127173	−0.0635864	−	0.997976i	$-0.520254\pi$
$349$	−2.37578	−0.127173	−0.0635864	+	0.997976i	$0.520254\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.272700	−0.0145143	−0.00725717	−	0.999974i	$-0.502310\pi$
$353$	−0.272700	−0.0145143	−0.00725717	+	0.999974i	$0.502310\pi$
$354$	0	0
$355$	1.06586	0.0565698
$356$	0	0
$357$	3.97364	0.210307
$358$	0	0
$359$	−2.41260	−0.127332	−0.0636661	−	0.997971i	$-0.520279\pi$
$359$	−2.41260	−0.127332	−0.0636661	+	0.997971i	$0.520279\pi$
$360$	0	0
$361$	−14.9527	−0.786985
$362$	0	0
$363$	−9.60206	−0.503977
$364$	0	0
$365$	−6.88455	−0.360354
$366$	0	0
$367$	27.1383	1.41661	0.708304	−	0.705908i	$-0.249461\pi$
$367$	27.1383	1.41661	0.708304	+	0.705908i	$0.249461\pi$
$368$	0	0
$369$	27.7659	1.44544
$370$	0	0
$371$	6.91625	0.359074
$372$	0	0
$373$	28.3127	1.46598	0.732988	−	0.680242i	$-0.238125\pi$
$373$	28.3127	1.46598	0.732988	+	0.680242i	$0.238125\pi$
$374$	0	0
$375$	−28.5824	−1.47599
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−18.8560	−0.968570	−0.484285	−	0.874910i	$-0.660920\pi$
$379$	−18.8560	−0.968570	−0.484285	+	0.874910i	$0.660920\pi$
$380$	0	0
$381$	29.3544	1.50387
$382$	0	0
$383$	−38.4766	−1.96606	−0.983031	−	0.183437i	$-0.941278\pi$
$383$	−38.4766	−1.96606	−0.983031	+	0.183437i	$0.941278\pi$
$384$	0	0
$385$	14.0665	0.716897
$386$	0	0
$387$	−28.6341	−1.45556
$388$	0	0
$389$	−25.6279	−1.29938	−0.649692	−	0.760198i	$-0.725102\pi$
$389$	−25.6279	−1.29938	−0.649692	+	0.760198i	$0.725102\pi$
$390$	0	0
$391$	−3.94802	−0.199660
$392$	0	0
$393$	−9.30483	−0.469366
$394$	0	0
$395$	18.6579	0.938783
$396$	0	0
$397$	6.69787	0.336156	0.168078	−	0.985774i	$-0.446244\pi$
$397$	6.69787	0.336156	0.168078	+	0.985774i	$0.446244\pi$
$398$	0	0
$399$	−4.88819	−0.244716
$400$	0	0
$401$	20.5402	1.02573	0.512865	−	0.858469i	$-0.328584\pi$
$401$	20.5402	1.02573	0.512865	+	0.858469i	$0.328584\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−33.7565	−1.67737
$406$	0	0
$407$	46.5005	2.30495
$408$	0	0
$409$	14.6682	0.725296	0.362648	−	0.931926i	$-0.381873\pi$
$409$	14.6682	0.725296	0.362648	+	0.931926i	$0.381873\pi$
$410$	0	0
$411$	15.3551	0.757413
$412$	0	0
$413$	−0.115252	−0.00567116
$414$	0	0
$415$	−42.9298	−2.10734
$416$	0	0
$417$	6.51631	0.319105
$418$	0	0
$419$	26.8818	1.31326	0.656631	−	0.754212i	$-0.271981\pi$
$419$	26.8818	1.31326	0.656631	+	0.754212i	$0.271981\pi$
$420$	0	0
$421$	−4.54437	−0.221479	−0.110739	−	0.993849i	$-0.535322\pi$
$421$	−4.54437	−0.221479	−0.110739	+	0.993849i	$0.535322\pi$
$422$	0	0
$423$	−25.9602	−1.26223
$424$	0	0
$425$	13.4653	0.653161
$426$	0	0
$427$	4.94075	0.239099
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.3205	−0.593458	−0.296729	−	0.954962i	$-0.595896\pi$
$431$	−12.3205	−0.593458	−0.296729	+	0.954962i	$0.595896\pi$
$432$	0	0
$433$	−24.4463	−1.17482	−0.587408	−	0.809291i	$-0.699851\pi$
$433$	−24.4463	−1.17482	−0.587408	+	0.809291i	$0.699851\pi$
$434$	0	0
$435$	−35.6199	−1.70784
$436$	0	0
$437$	4.85668	0.232327
$438$	0	0
$439$	5.30566	0.253226	0.126613	−	0.991952i	$-0.459589\pi$
$439$	5.30566	0.253226	0.126613	+	0.991952i	$0.459589\pi$
$440$	0	0
$441$	2.90380	0.138276
$442$	0	0
$443$	−1.26200	−0.0599596	−0.0299798	−	0.999551i	$-0.509544\pi$
$443$	−1.26200	−0.0599596	−0.0299798	+	0.999551i	$0.509544\pi$
$444$	0	0
$445$	−8.96333	−0.424903
$446$	0	0
$447$	37.8343	1.78950
$448$	0	0
$449$	23.5399	1.11092	0.555458	−	0.831544i	$-0.312543\pi$
$449$	23.5399	1.11092	0.555458	+	0.831544i	$0.312543\pi$
$450$	0	0
$451$	−36.9737	−1.74102
$452$	0	0
$453$	38.3323	1.80101
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.1800	−1.08431	−0.542157	−	0.840277i	$-0.682392\pi$
$457$	−23.1800	−1.08431	−0.542157	+	0.840277i	$0.682392\pi$
$458$	0	0
$459$	0.382263	0.0178425
$460$	0	0
$461$	35.9400	1.67389	0.836945	−	0.547286i	$-0.184339\pi$
$461$	35.9400	1.67389	0.836945	+	0.547286i	$0.184339\pi$
$462$	0	0
$463$	−12.9147	−0.600197	−0.300098	−	0.953908i	$-0.597019\pi$
$463$	−12.9147	−0.600197	−0.300098	+	0.953908i	$0.597019\pi$
$464$	0	0
$465$	−80.6330	−3.73927
$466$	0	0
$467$	13.6014	0.629398	0.314699	−	0.949192i	$-0.398096\pi$
$467$	13.6014	0.629398	0.314699	+	0.949192i	$0.398096\pi$
$468$	0	0
$469$	−9.11849	−0.421053
$470$	0	0
$471$	23.5094	1.08326
$472$	0	0
$473$	38.1298	1.75321
$474$	0	0
$475$	−16.5644	−0.760025
$476$	0	0
$477$	−20.0834	−0.919556
$478$	0	0
$479$	19.9491	0.911499	0.455750	−	0.890108i	$-0.349371\pi$
$479$	19.9491	0.911499	0.455750	+	0.890108i	$0.349371\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−5.86575	−0.266901
$484$	0	0
$485$	63.5729	2.88670
$486$	0	0
$487$	−19.4990	−0.883583	−0.441791	−	0.897118i	$-0.645657\pi$
$487$	−19.4990	−0.883583	−0.441791	+	0.897118i	$0.645657\pi$
$488$	0	0
$489$	−31.1732	−1.40970
$490$	0	0
$491$	19.0547	0.859925	0.429962	−	0.902847i	$-0.358527\pi$
$491$	19.0547	0.859925	0.429962	+	0.902847i	$0.358527\pi$
$492$	0	0
$493$	6.59036	0.296815
$494$	0	0
$495$	−40.8464	−1.83591
$496$	0	0
$497$	−0.292994	−0.0131426
$498$	0	0
$499$	−13.6662	−0.611785	−0.305892	−	0.952066i	$-0.598955\pi$
$499$	−13.6662	−0.611785	−0.305892	+	0.952066i	$0.598955\pi$
$500$	0	0
$501$	−25.2998	−1.13031
$502$	0	0
$503$	−2.30153	−0.102620	−0.0513100	−	0.998683i	$-0.516340\pi$
$503$	−2.30153	−0.102620	−0.0513100	+	0.998683i	$0.516340\pi$
$504$	0	0
$505$	−37.4059	−1.66454
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.2554	−0.986452	−0.493226	−	0.869901i	$-0.664182\pi$
$509$	−22.2554	−0.986452	−0.493226	+	0.869901i	$0.664182\pi$
$510$	0	0
$511$	1.89250	0.0837193
$512$	0	0
$513$	−0.470243	−0.0207617
$514$	0	0
$515$	−26.0878	−1.14956
$516$	0	0
$517$	34.5692	1.52035
$518$	0	0
$519$	−29.5893	−1.29883
$520$	0	0
$521$	13.1581	0.576465	0.288233	−	0.957560i	$-0.406932\pi$
$521$	13.1581	0.576465	0.288233	+	0.957560i	$0.406932\pi$
$522$	0	0
$523$	−39.8057	−1.74058	−0.870291	−	0.492538i	$-0.836069\pi$
$523$	−39.8057	−1.74058	−0.870291	+	0.492538i	$0.836069\pi$
$524$	0	0
$525$	20.0059	0.873129
$526$	0	0
$527$	14.9187	0.649867
$528$	0	0
$529$	−17.1721	−0.746611
$530$	0	0
$531$	0.334668	0.0145233
$532$	0	0
$533$	0	0
$534$	0	0
$535$	54.5042	2.35642
$536$	0	0
$537$	−1.13718	−0.0490729
$538$	0	0
$539$	−3.86676	−0.166553
$540$	0	0
$541$	18.7716	0.807055	0.403527	−	0.914968i	$-0.367784\pi$
$541$	18.7716	0.807055	0.403527	+	0.914968i	$0.367784\pi$
$542$	0	0
$543$	−49.5942	−2.12829
$544$	0	0
$545$	−32.1005	−1.37503
$546$	0	0
$547$	43.1197	1.84366	0.921832	−	0.387589i	$-0.126692\pi$
$547$	43.1197	1.84366	0.921832	+	0.387589i	$0.126692\pi$
$548$	0	0
$549$	−14.3469	−0.612312
$550$	0	0
$551$	−8.10716	−0.345377
$552$	0	0
$553$	−5.12890	−0.218103
$554$	0	0
$555$	106.296	4.51201
$556$	0	0
$557$	−23.5716	−0.998763	−0.499381	−	0.866382i	$-0.666439\pi$
$557$	−23.5716	−0.998763	−0.499381	+	0.866382i	$0.666439\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	15.3651	0.648715
$562$	0	0
$563$	−24.2726	−1.02297	−0.511484	−	0.859293i	$-0.670904\pi$
$563$	−24.2726	−1.02297	−0.511484	+	0.859293i	$0.670904\pi$
$564$	0	0
$565$	19.9581	0.839645
$566$	0	0
$567$	9.27934	0.389696
$568$	0	0
$569$	43.0307	1.80394	0.901969	−	0.431800i	$-0.142121\pi$
$569$	43.0307	1.80394	0.901969	+	0.431800i	$0.142121\pi$
$570$	0	0
$571$	10.1365	0.424201	0.212101	−	0.977248i	$-0.431970\pi$
$571$	10.1365	0.424201	0.212101	+	0.977248i	$0.431970\pi$
$572$	0	0
$573$	8.47761	0.354157
$574$	0	0
$575$	−19.8770	−0.828926
$576$	0	0
$577$	11.8664	0.494005	0.247003	−	0.969015i	$-0.420554\pi$
$577$	11.8664	0.494005	0.247003	+	0.969015i	$0.420554\pi$
$578$	0	0
$579$	8.37117	0.347894
$580$	0	0
$581$	11.8010	0.489588
$582$	0	0
$583$	26.7435	1.10760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.7030	−0.937054	−0.468527	−	0.883449i	$-0.655215\pi$
$587$	−22.7030	−0.937054	−0.468527	+	0.883449i	$0.655215\pi$
$588$	0	0
$589$	−18.3523	−0.756192
$590$	0	0
$591$	30.6709	1.26163
$592$	0	0
$593$	−41.0665	−1.68640	−0.843200	−	0.537600i	$-0.819331\pi$
$593$	−41.0665	−1.68640	−0.843200	+	0.537600i	$0.819331\pi$
$594$	0	0
$595$	−5.94925	−0.243895
$596$	0	0
$597$	26.8410	1.09853
$598$	0	0
$599$	−20.8122	−0.850362	−0.425181	−	0.905108i	$-0.639789\pi$
$599$	−20.8122	−0.850362	−0.425181	+	0.905108i	$0.639789\pi$
$600$	0	0
$601$	−5.03451	−0.205362	−0.102681	−	0.994714i	$-0.532742\pi$
$601$	−5.03451	−0.205362	−0.102681	+	0.994714i	$0.532742\pi$
$602$	0	0
$603$	26.4783	1.07828
$604$	0	0
$605$	14.3760	0.584468
$606$	0	0
$607$	−9.73875	−0.395284	−0.197642	−	0.980274i	$-0.563328\pi$
$607$	−9.73875	−0.395284	−0.197642	+	0.980274i	$0.563328\pi$
$608$	0	0
$609$	9.79157	0.396774
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17.1955	0.694521	0.347260	−	0.937769i	$-0.387112\pi$
$613$	17.1955	0.694521	0.347260	+	0.937769i	$0.387112\pi$
$614$	0	0
$615$	−84.5184	−3.40811
$616$	0	0
$617$	17.8752	0.719628	0.359814	−	0.933024i	$-0.382840\pi$
$617$	17.8752	0.719628	0.359814	+	0.933024i	$0.382840\pi$
$618$	0	0
$619$	21.3112	0.856568	0.428284	−	0.903644i	$-0.359118\pi$
$619$	21.3112	0.856568	0.428284	+	0.903644i	$0.359118\pi$
$620$	0	0
$621$	−0.564283	−0.0226439
$622$	0	0
$623$	2.46394	0.0987155
$624$	0	0
$625$	1.62474	0.0649895
$626$	0	0
$627$	−18.9015	−0.754851
$628$	0	0
$629$	−19.6668	−0.784166
$630$	0	0
$631$	−12.6066	−0.501861	−0.250930	−	0.968005i	$-0.580736\pi$
$631$	−12.6066	−0.501861	−0.250930	+	0.968005i	$0.580736\pi$
$632$	0	0
$633$	2.44359	0.0971239
$634$	0	0
$635$	−43.9487	−1.74405
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.850797	0.0336570
$640$	0	0
$641$	1.90705	0.0753238	0.0376619	−	0.999291i	$-0.488009\pi$
$641$	1.90705	0.0753238	0.0376619	+	0.999291i	$0.488009\pi$
$642$	0	0
$643$	11.0489	0.435726	0.217863	−	0.975979i	$-0.430091\pi$
$643$	11.0489	0.435726	0.217863	+	0.975979i	$0.430091\pi$
$644$	0	0
$645$	87.1612	3.43197
$646$	0	0
$647$	44.7891	1.76084	0.880420	−	0.474195i	$-0.157261\pi$
$647$	44.7891	1.76084	0.880420	+	0.474195i	$0.157261\pi$
$648$	0	0
$649$	−0.445651	−0.0174933
$650$	0	0
$651$	22.1653	0.868726
$652$	0	0
$653$	20.8774	0.816997	0.408499	−	0.912759i	$-0.366052\pi$
$653$	20.8774	0.816997	0.408499	+	0.912759i	$0.366052\pi$
$654$	0	0
$655$	13.9310	0.544329
$656$	0	0
$657$	−5.49544	−0.214398
$658$	0	0
$659$	1.80102	0.0701577	0.0350788	−	0.999385i	$-0.488832\pi$
$659$	1.80102	0.0701577	0.0350788	+	0.999385i	$0.488832\pi$
$660$	0	0
$661$	−9.77342	−0.380142	−0.190071	−	0.981770i	$-0.560872\pi$
$661$	−9.77342	−0.380142	−0.190071	+	0.981770i	$0.560872\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.31850	0.283799
$666$	0	0
$667$	−9.72846	−0.376687
$668$	0	0
$669$	54.1922	2.09519
$670$	0	0
$671$	19.1047	0.737528
$672$	0	0
$673$	−16.6334	−0.641169	−0.320584	−	0.947220i	$-0.603879\pi$
$673$	−16.6334	−0.641169	−0.320584	+	0.947220i	$0.603879\pi$
$674$	0	0
$675$	1.92456	0.0740765
$676$	0	0
$677$	33.5232	1.28840	0.644201	−	0.764856i	$-0.277190\pi$
$677$	33.5232	1.28840	0.644201	+	0.764856i	$0.277190\pi$
$678$	0	0
$679$	−17.4756	−0.670653
$680$	0	0
$681$	38.9918	1.49417
$682$	0	0
$683$	−23.7649	−0.909339	−0.454670	−	0.890660i	$-0.650243\pi$
$683$	−23.7649	−0.909339	−0.454670	+	0.890660i	$0.650243\pi$
$684$	0	0
$685$	−22.9894	−0.878379
$686$	0	0
$687$	29.7125	1.13360
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−5.25581	−0.199940	−0.0999702	−	0.994990i	$-0.531875\pi$
$691$	−5.25581	−0.199940	−0.0999702	+	0.994990i	$0.531875\pi$
$692$	0	0
$693$	11.2283	0.426528
$694$	0	0
$695$	−9.75608	−0.370069
$696$	0	0
$697$	15.6375	0.592314
$698$	0	0
$699$	−18.6234	−0.704402
$700$	0	0
$701$	−10.3164	−0.389646	−0.194823	−	0.980838i	$-0.562413\pi$
$701$	−10.3164	−0.389646	−0.194823	+	0.980838i	$0.562413\pi$
$702$	0	0
$703$	24.1932	0.912463
$704$	0	0
$705$	79.0219	2.97614
$706$	0	0
$707$	10.2825	0.386714
$708$	0	0
$709$	19.3457	0.726544	0.363272	−	0.931683i	$-0.381660\pi$
$709$	19.3457	0.726544	0.363272	+	0.931683i	$0.381660\pi$
$710$	0	0
$711$	14.8933	0.558542
$712$	0	0
$713$	−22.0224	−0.824746
$714$	0	0
$715$	0	0
$716$	0	0
$717$	23.4007	0.873915
$718$	0	0
$719$	−48.0744	−1.79287	−0.896437	−	0.443171i	$-0.853853\pi$
$719$	−48.0744	−1.79287	−0.896437	+	0.443171i	$0.853853\pi$
$720$	0	0
$721$	7.17129	0.267073
$722$	0	0
$723$	0.347644	0.0129290
$724$	0	0
$725$	33.1802	1.23228
$726$	0	0
$727$	−11.6338	−0.431475	−0.215738	−	0.976451i	$-0.569216\pi$
$727$	−11.6338	−0.431475	−0.215738	+	0.976451i	$0.569216\pi$
$728$	0	0
$729$	−25.2415	−0.934872
$730$	0	0
$731$	−16.1265	−0.596460
$732$	0	0
$733$	−30.3258	−1.12011	−0.560055	−	0.828456i	$-0.689220\pi$
$733$	−30.3258	−1.12011	−0.560055	+	0.828456i	$0.689220\pi$
$734$	0	0
$735$	−8.83905	−0.326033
$736$	0	0
$737$	−35.2590	−1.29878
$738$	0	0
$739$	9.70934	0.357164	0.178582	−	0.983925i	$-0.442849\pi$
$739$	9.70934	0.357164	0.178582	+	0.983925i	$0.442849\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.0979	1.54442	0.772211	−	0.635366i	$-0.219151\pi$
$743$	42.0979	1.54442	0.772211	+	0.635366i	$0.219151\pi$
$744$	0	0
$745$	−56.6447	−2.07530
$746$	0	0
$747$	−34.2677	−1.25379
$748$	0	0
$749$	−14.9827	−0.547456
$750$	0	0
$751$	24.5710	0.896609	0.448305	−	0.893881i	$-0.352028\pi$
$751$	24.5710	0.896609	0.448305	+	0.893881i	$0.352028\pi$
$752$	0	0
$753$	−24.7952	−0.903587
$754$	0	0
$755$	−57.3904	−2.08865
$756$	0	0
$757$	36.1120	1.31251	0.656257	−	0.754538i	$-0.272139\pi$
$757$	36.1120	1.31251	0.656257	+	0.754538i	$0.272139\pi$
$758$	0	0
$759$	−22.6814	−0.823284
$760$	0	0
$761$	−15.4059	−0.558462	−0.279231	−	0.960224i	$-0.590080\pi$
$761$	−15.4059	−0.558462	−0.279231	+	0.960224i	$0.590080\pi$
$762$	0	0
$763$	8.82413	0.319455
$764$	0	0
$765$	17.2754	0.624595
$766$	0	0
$767$	0	0
$768$	0	0
$769$	20.4142	0.736155	0.368078	−	0.929795i	$-0.380016\pi$
$769$	20.4142	0.736155	0.368078	+	0.929795i	$0.380016\pi$
$770$	0	0
$771$	48.9829	1.76408
$772$	0	0
$773$	−49.1342	−1.76723	−0.883616	−	0.468211i	$-0.844899\pi$
$773$	−49.1342	−1.76723	−0.883616	+	0.468211i	$0.844899\pi$
$774$	0	0
$775$	75.1103	2.69804
$776$	0	0
$777$	−29.2197	−1.04825
$778$	0	0
$779$	−19.2366	−0.689222
$780$	0	0
$781$	−1.13294	−0.0405398
$782$	0	0
$783$	0.941947	0.0336624
$784$	0	0
$785$	−35.1978	−1.25626
$786$	0	0
$787$	−12.3426	−0.439968	−0.219984	−	0.975503i	$-0.570601\pi$
$787$	−12.3426	−0.439968	−0.219984	+	0.975503i	$0.570601\pi$
$788$	0	0
$789$	−69.4711	−2.47324
$790$	0	0
$791$	−5.48631	−0.195071
$792$	0	0
$793$	0	0
$794$	0	0
$795$	61.1331	2.16817
$796$	0	0
$797$	−20.5033	−0.726265	−0.363133	−	0.931737i	$-0.618293\pi$
$797$	−20.5033	−0.726265	−0.363133	+	0.931737i	$0.618293\pi$
$798$	0	0
$799$	−14.6206	−0.517238
$800$	0	0
$801$	−7.15478	−0.252802
$802$	0	0
$803$	7.31784	0.258241
$804$	0	0
$805$	8.78208	0.309528
$806$	0	0
$807$	78.3498	2.75804
$808$	0	0
$809$	−7.48733	−0.263241	−0.131620	−	0.991300i	$-0.542018\pi$
$809$	−7.48733	−0.263241	−0.131620	+	0.991300i	$0.542018\pi$
$810$	0	0
$811$	−22.8692	−0.803048	−0.401524	−	0.915849i	$-0.631519\pi$
$811$	−22.8692	−0.803048	−0.401524	+	0.915849i	$0.631519\pi$
$812$	0	0
$813$	58.7215	2.05945
$814$	0	0
$815$	46.6719	1.63484
$816$	0	0
$817$	19.8381	0.694047
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.4801	−1.41276	−0.706382	−	0.707830i	$-0.749674\pi$
$821$	−40.4801	−1.41276	−0.706382	+	0.707830i	$0.749674\pi$
$822$	0	0
$823$	−22.6962	−0.791139	−0.395570	−	0.918436i	$-0.629453\pi$
$823$	−22.6962	−0.791139	−0.395570	+	0.918436i	$0.629453\pi$
$824$	0	0
$825$	77.3580	2.69326
$826$	0	0
$827$	−15.7414	−0.547383	−0.273691	−	0.961818i	$-0.588245\pi$
$827$	−15.7414	−0.547383	−0.273691	+	0.961818i	$0.588245\pi$
$828$	0	0
$829$	19.6878	0.683785	0.341893	−	0.939739i	$-0.388932\pi$
$829$	19.6878	0.683785	0.341893	+	0.939739i	$0.388932\pi$
$830$	0	0
$831$	49.0149	1.70031
$832$	0	0
$833$	1.63539	0.0566630
$834$	0	0
$835$	37.8783	1.31083
$836$	0	0
$837$	2.13229	0.0737029
$838$	0	0
$839$	−2.20420	−0.0760974	−0.0380487	−	0.999276i	$-0.512114\pi$
$839$	−2.20420	−0.0760974	−0.0380487	+	0.999276i	$0.512114\pi$
$840$	0	0
$841$	−12.7605	−0.440017
$842$	0	0
$843$	−58.3325	−2.00908
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.95183	−0.135787
$848$	0	0
$849$	−32.9323	−1.13023
$850$	0	0
$851$	29.0314	0.995184
$852$	0	0
$853$	−2.60526	−0.0892026	−0.0446013	−	0.999005i	$-0.514202\pi$
$853$	−2.60526	−0.0892026	−0.0446013	+	0.999005i	$0.514202\pi$
$854$	0	0
$855$	−21.2515	−0.726785
$856$	0	0
$857$	33.0023	1.12734	0.563668	−	0.826001i	$-0.309390\pi$
$857$	33.0023	1.12734	0.563668	+	0.826001i	$0.309390\pi$
$858$	0	0
$859$	17.0404	0.581411	0.290706	−	0.956813i	$-0.406110\pi$
$859$	17.0404	0.581411	0.290706	+	0.956813i	$0.406110\pi$
$860$	0	0
$861$	23.2333	0.791790
$862$	0	0
$863$	1.85114	0.0630134	0.0315067	−	0.999504i	$-0.489969\pi$
$863$	1.85114	0.0630134	0.0315067	+	0.999504i	$0.489969\pi$
$864$	0	0
$865$	44.3006	1.50626
$866$	0	0
$867$	34.8077	1.18213
$868$	0	0
$869$	−19.8322	−0.672762
$870$	0	0
$871$	0	0
$872$	0	0
$873$	50.7457	1.71748
$874$	0	0
$875$	−11.7634	−0.397675
$876$	0	0
$877$	−11.1915	−0.377910	−0.188955	−	0.981986i	$-0.560510\pi$
$877$	−11.1915	−0.377910	−0.188955	+	0.981986i	$0.560510\pi$
$878$	0	0
$879$	34.6824	1.16981
$880$	0	0
$881$	18.7951	0.633222	0.316611	−	0.948555i	$-0.397455\pi$
$881$	18.7951	0.633222	0.316611	+	0.948555i	$0.397455\pi$
$882$	0	0
$883$	7.53316	0.253511	0.126756	−	0.991934i	$-0.459544\pi$
$883$	7.53316	0.253511	0.126756	+	0.991934i	$0.459544\pi$
$884$	0	0
$885$	−1.01872	−0.0342437
$886$	0	0
$887$	27.7461	0.931622	0.465811	−	0.884884i	$-0.345763\pi$
$887$	27.7461	0.931622	0.465811	+	0.884884i	$0.345763\pi$
$888$	0	0
$889$	12.0811	0.405187
$890$	0	0
$891$	35.8810	1.20206
$892$	0	0
$893$	17.9856	0.601864
$894$	0	0
$895$	1.70256	0.0569103
$896$	0	0
$897$	0	0
$898$	0	0
$899$	36.7616	1.22607
$900$	0	0
$901$	−11.3108	−0.376817
$902$	0	0
$903$	−23.9598	−0.797333
$904$	0	0
$905$	74.2514	2.46820
$906$	0	0
$907$	12.6722	0.420775	0.210388	−	0.977618i	$-0.432527\pi$
$907$	12.6722	0.420775	0.210388	+	0.977618i	$0.432527\pi$
$908$	0	0
$909$	−29.8584	−0.990341
$910$	0	0
$911$	53.8605	1.78448	0.892239	−	0.451564i	$-0.149134\pi$
$911$	53.8605	1.78448	0.892239	+	0.451564i	$0.149134\pi$
$912$	0	0
$913$	45.6316	1.51019
$914$	0	0
$915$	43.6715	1.44374
$916$	0	0
$917$	−3.82950	−0.126461
$918$	0	0
$919$	29.6096	0.976730	0.488365	−	0.872639i	$-0.337593\pi$
$919$	29.6096	0.976730	0.488365	+	0.872639i	$0.337593\pi$
$920$	0	0
$921$	68.7100	2.26407
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−99.0155	−3.25561
$926$	0	0
$927$	−20.8240	−0.683949
$928$	0	0
$929$	−46.5045	−1.52576	−0.762881	−	0.646538i	$-0.776216\pi$
$929$	−46.5045	−1.52576	−0.762881	+	0.646538i	$0.776216\pi$
$930$	0	0
$931$	−2.01179	−0.0659337
$932$	0	0
$933$	16.6873	0.546317
$934$	0	0
$935$	−23.0043	−0.752322
$936$	0	0
$937$	25.7685	0.841821	0.420911	−	0.907102i	$-0.361711\pi$
$937$	25.7685	0.841821	0.420911	+	0.907102i	$0.361711\pi$
$938$	0	0
$939$	48.2252	1.57377
$940$	0	0
$941$	−51.0205	−1.66322	−0.831610	−	0.555360i	$-0.812580\pi$
$941$	−51.0205	−1.66322	−0.831610	+	0.555360i	$0.812580\pi$
$942$	0	0
$943$	−23.0836	−0.751705
$944$	0	0
$945$	−0.850315	−0.0276607
$946$	0	0
$947$	−9.50717	−0.308941	−0.154471	−	0.987997i	$-0.549367\pi$
$947$	−9.50717	−0.308941	−0.154471	+	0.987997i	$0.549367\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	80.6023	2.61371
$952$	0	0
$953$	−13.8901	−0.449946	−0.224973	−	0.974365i	$-0.572229\pi$
$953$	−13.8901	−0.449946	−0.224973	+	0.974365i	$0.572229\pi$
$954$	0	0
$955$	−12.6925	−0.410720
$956$	0	0
$957$	37.8616	1.22389
$958$	0	0
$959$	6.31957	0.204070
$960$	0	0
$961$	52.2175	1.68444
$962$	0	0
$963$	43.5068	1.40199
$964$	0	0
$965$	−12.5331	−0.403456
$966$	0	0
$967$	30.4473	0.979120	0.489560	−	0.871970i	$-0.337157\pi$
$967$	30.4473	0.979120	0.489560	+	0.871970i	$0.337157\pi$
$968$	0	0
$969$	7.99412	0.256808
$970$	0	0
$971$	13.7799	0.442218	0.221109	−	0.975249i	$-0.429032\pi$
$971$	13.7799	0.442218	0.221109	+	0.975249i	$0.429032\pi$
$972$	0	0
$973$	2.68186	0.0859764
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.43321	−0.0458524	−0.0229262	−	0.999737i	$-0.507298\pi$
$977$	−1.43321	−0.0458524	−0.0229262	+	0.999737i	$0.507298\pi$
$978$	0	0
$979$	9.52745	0.304499
$980$	0	0
$981$	−25.6235	−0.818096
$982$	0	0
$983$	−31.6454	−1.00933	−0.504665	−	0.863315i	$-0.668384\pi$
$983$	−31.6454	−1.00933	−0.504665	+	0.863315i	$0.668384\pi$
$984$	0	0
$985$	−45.9198	−1.46313
$986$	0	0
$987$	−21.7224	−0.691431
$988$	0	0
$989$	23.8054	0.756967
$990$	0	0
$991$	−5.49055	−0.174413	−0.0872066	−	0.996190i	$-0.527794\pi$
$991$	−5.49055	−0.174413	−0.0872066	+	0.996190i	$0.527794\pi$
$992$	0	0
$993$	−40.4905	−1.28493
$994$	0	0
$995$	−40.1858	−1.27397
$996$	0	0
$997$	24.4403	0.774033	0.387016	−	0.922073i	$-0.373506\pi$
$997$	24.4403	0.774033	0.387016	+	0.922073i	$0.373506\pi$
$998$	0	0
$999$	−2.81093	−0.0889340

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	4732.2.a.s.1.2		8
13.2	odd	12		364.2.u.a.225.7	✓	16
13.5	odd	4		4732.2.g.k.337.3		16
13.7	odd	12		364.2.u.a.309.7	yes	16
13.8	odd	4		4732.2.g.k.337.4		16
13.12	even	2		4732.2.a.t.1.2		8
39.2	even	12		3276.2.cf.c.2773.8		16
39.20	even	12		3276.2.cf.c.1765.1		16
52.7	even	12		1456.2.cc.f.673.2		16
52.15	even	12		1456.2.cc.f.225.2		16
91.2	odd	12		2548.2.bb.d.1733.7		16
91.20	even	12		2548.2.u.c.1765.2		16
91.33	even	12		2548.2.bq.c.361.7		16
91.41	even	12		2548.2.u.c.589.2		16
91.46	odd	12		2548.2.bb.d.569.7		16
91.54	even	12		2548.2.bb.c.1733.2		16
91.59	even	12		2548.2.bb.c.569.2		16
91.67	odd	12		2548.2.bq.e.1941.2		16
91.72	odd	12		2548.2.bq.e.361.2		16
91.80	even	12		2548.2.bq.c.1941.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.7	✓	16	13.2	odd	12
364.2.u.a.309.7	yes	16	13.7	odd	12
1456.2.cc.f.225.2		16	52.15	even	12
1456.2.cc.f.673.2		16	52.7	even	12
2548.2.u.c.589.2		16	91.41	even	12
2548.2.u.c.1765.2		16	91.20	even	12
2548.2.bb.c.569.2		16	91.59	even	12
2548.2.bb.c.1733.2		16	91.54	even	12
2548.2.bb.d.569.7		16	91.46	odd	12
2548.2.bb.d.1733.7		16	91.2	odd	12
2548.2.bq.c.361.7		16	91.33	even	12
2548.2.bq.c.1941.7		16	91.80	even	12
2548.2.bq.e.361.2		16	91.72	odd	12
2548.2.bq.e.1941.2		16	91.67	odd	12
3276.2.cf.c.1765.1		16	39.20	even	12
3276.2.cf.c.2773.8		16	39.2	even	12
4732.2.a.s.1.2		8	1.1	even	1	trivial
4732.2.a.t.1.2		8	13.12	even	2
4732.2.g.k.337.3		16	13.5	odd	4
4732.2.g.k.337.4		16	13.8	odd	4

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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

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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	4732.2.a.s.1.2

Level	$4732$
Weight	$2$
Character	4732.1
Self dual	yes
Analytic conductor	$37.785$
Analytic rank	$1$
Dimension	$8$
CM	no
Inner twists	$1$