LMFDB - Embedded newform 1881.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.15351$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.484093 q^{2} -1.76565 q^{4} -0.637602 q^{5} +2.66942 q^{7} +1.82293 q^{8} +0.308658 q^{10} -1.00000 q^{11} -1.20725 q^{13} -1.29225 q^{14} +2.64884 q^{16} -0.222022 q^{17} -1.00000 q^{19} +1.12578 q^{20} +0.484093 q^{22} +9.48717 q^{23} -4.59346 q^{25} +0.584420 q^{26} -4.71327 q^{28} -4.94518 q^{29} -3.05563 q^{31} -4.92814 q^{32} +0.107479 q^{34} -1.70203 q^{35} +7.18015 q^{37} +0.484093 q^{38} -1.16230 q^{40} +6.92650 q^{41} +1.53538 q^{43} +1.76565 q^{44} -4.59267 q^{46} -5.94581 q^{47} +0.125785 q^{49} +2.22366 q^{50} +2.13158 q^{52} +9.63879 q^{53} +0.637602 q^{55} +4.86615 q^{56} +2.39392 q^{58} -2.65817 q^{59} -0.809792 q^{61} +1.47921 q^{62} -2.91201 q^{64} +0.769744 q^{65} -2.22447 q^{67} +0.392015 q^{68} +0.823938 q^{70} -2.58912 q^{71} +16.6108 q^{73} -3.47586 q^{74} +1.76565 q^{76} -2.66942 q^{77} +5.69298 q^{79} -1.68891 q^{80} -3.35307 q^{82} +2.93784 q^{83} +0.141562 q^{85} -0.743267 q^{86} -1.82293 q^{88} +5.54136 q^{89} -3.22265 q^{91} -16.7511 q^{92} +2.87832 q^{94} +0.637602 q^{95} +13.7728 q^{97} -0.0608914 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

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 $

$2$	−0.484093	−0.342305	−0.171153	−	0.985245i	$-0.554749\pi$
$2$	−0.484093	−0.342305	−0.171153	+	0.985245i	$0.554749\pi$
$3$	0	0
$3$	0	0
$4$	−1.76565	−0.882827
$5$	−0.637602	−0.285144	−0.142572	−	0.989784i	$-0.545537\pi$
$5$	−0.637602	−0.285144	−0.142572	+	0.989784i	$0.545537\pi$
$6$	0	0
$7$	2.66942	1.00894	0.504472	−	0.863428i	$-0.331687\pi$
$7$	2.66942	1.00894	0.504472	+	0.863428i	$0.331687\pi$
$8$	1.82293	0.644502
$9$	0	0
$10$	0.308658	0.0976064
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.20725	−0.334831	−0.167415	−	0.985886i	$-0.553542\pi$
$13$	−1.20725	−0.334831	−0.167415	+	0.985886i	$0.553542\pi$
$14$	−1.29225	−0.345367
$15$	0	0
$16$	2.64884	0.662211
$17$	−0.222022	−0.0538483	−0.0269242	−	0.999637i	$-0.508571\pi$
$17$	−0.222022	−0.0538483	−0.0269242	+	0.999637i	$0.508571\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.12578	0.251733
$21$	0	0
$22$	0.484093	0.103209
$23$	9.48717	1.97821	0.989106	−	0.147206i	$-0.0470279\pi$
$23$	9.48717	1.97821	0.989106	+	0.147206i	$0.0470279\pi$
$24$	0	0
$25$	−4.59346	−0.918693
$26$	0.584420	0.114614
$27$	0	0
$28$	−4.71327	−0.890724
$29$	−4.94518	−0.918297	−0.459148	−	0.888360i	$-0.651845\pi$
$29$	−4.94518	−0.918297	−0.459148	+	0.888360i	$0.651845\pi$
$30$	0	0
$31$	−3.05563	−0.548808	−0.274404	−	0.961615i	$-0.588480\pi$
$31$	−3.05563	−0.548808	−0.274404	+	0.961615i	$0.588480\pi$
$32$	−4.92814	−0.871180
$33$	0	0
$34$	0.107479	0.0184326
$35$	−1.70203	−0.287695
$36$	0	0
$37$	7.18015	1.18041	0.590205	−	0.807254i	$-0.299047\pi$
$37$	7.18015	1.18041	0.590205	+	0.807254i	$0.299047\pi$
$38$	0.484093	0.0785302
$39$	0	0
$40$	−1.16230	−0.183776
$41$	6.92650	1.08174	0.540869	−	0.841107i	$-0.318096\pi$
$41$	6.92650	1.08174	0.540869	+	0.841107i	$0.318096\pi$
$42$	0	0
$43$	1.53538	0.234144	0.117072	−	0.993123i	$-0.462649\pi$
$43$	1.53538	0.234144	0.117072	+	0.993123i	$0.462649\pi$
$44$	1.76565	0.266182
$45$	0	0
$46$	−4.59267	−0.677152
$47$	−5.94581	−0.867285	−0.433642	−	0.901085i	$-0.642772\pi$
$47$	−5.94581	−0.867285	−0.433642	+	0.901085i	$0.642772\pi$
$48$	0	0
$49$	0.125785	0.0179692
$50$	2.22366	0.314473
$51$	0	0
$52$	2.13158	0.295598
$53$	9.63879	1.32399	0.661995	−	0.749509i	$-0.269710\pi$
$53$	9.63879	1.32399	0.661995	+	0.749509i	$0.269710\pi$
$54$	0	0
$55$	0.637602	0.0859742
$56$	4.86615	0.650266
$57$	0	0
$58$	2.39392	0.314338
$59$	−2.65817	−0.346065	−0.173032	−	0.984916i	$-0.555357\pi$
$59$	−2.65817	−0.346065	−0.173032	+	0.984916i	$0.555357\pi$
$60$	0	0
$61$	−0.809792	−0.103683	−0.0518416	−	0.998655i	$-0.516509\pi$
$61$	−0.809792	−0.103683	−0.0518416	+	0.998655i	$0.516509\pi$
$62$	1.47921	0.187860
$63$	0	0
$64$	−2.91201	−0.364001
$65$	0.769744	0.0954750
$66$	0	0
$67$	−2.22447	−0.271763	−0.135881	−	0.990725i	$-0.543387\pi$
$67$	−2.22447	−0.271763	−0.135881	+	0.990725i	$0.543387\pi$
$68$	0.392015	0.0475387
$69$	0	0
$70$	0.823938	0.0984794
$71$	−2.58912	−0.307272	−0.153636	−	0.988127i	$-0.549098\pi$
$71$	−2.58912	−0.307272	−0.153636	+	0.988127i	$0.549098\pi$
$72$	0	0
$73$	16.6108	1.94414	0.972071	−	0.234687i	$-0.0754065\pi$
$73$	16.6108	1.94414	0.972071	+	0.234687i	$0.0754065\pi$
$74$	−3.47586	−0.404060
$75$	0	0
$76$	1.76565	0.202534
$77$	−2.66942	−0.304208
$78$	0	0
$79$	5.69298	0.640510	0.320255	−	0.947331i	$-0.396231\pi$
$79$	5.69298	0.640510	0.320255	+	0.947331i	$0.396231\pi$
$80$	−1.68891	−0.188826
$81$	0	0
$82$	−3.35307	−0.370284
$83$	2.93784	0.322470	0.161235	−	0.986916i	$-0.448452\pi$
$83$	2.93784	0.322470	0.161235	+	0.986916i	$0.448452\pi$
$84$	0	0
$85$	0.141562	0.0153545
$86$	−0.743267	−0.0801486
$87$	0	0
$88$	−1.82293	−0.194325
$89$	5.54136	0.587383	0.293692	−	0.955900i	$-0.405116\pi$
$89$	5.54136	0.587383	0.293692	+	0.955900i	$0.405116\pi$
$90$	0	0
$91$	−3.22265	−0.337826
$92$	−16.7511	−1.74642
$93$	0	0
$94$	2.87832	0.296876
$95$	0.637602	0.0654166
$96$	0	0
$97$	13.7728	1.39842	0.699209	−	0.714917i	$-0.253536\pi$
$97$	13.7728	1.39842	0.699209	+	0.714917i	$0.253536\pi$
$98$	−0.0608914	−0.00615096
$99$	0	0
$100$	8.11047	0.811047
$101$	−12.0308	−1.19711	−0.598555	−	0.801082i	$-0.704258\pi$
$101$	−12.0308	−1.19711	−0.598555	+	0.801082i	$0.704258\pi$
$102$	0	0
$103$	8.53068	0.840553	0.420276	−	0.907396i	$-0.361933\pi$
$103$	8.53068	0.840553	0.420276	+	0.907396i	$0.361933\pi$
$104$	−2.20073	−0.215799
$105$	0	0
$106$	−4.66607	−0.453208
$107$	−1.64585	−0.159110	−0.0795552	−	0.996830i	$-0.525350\pi$
$107$	−1.64585	−0.159110	−0.0795552	+	0.996830i	$0.525350\pi$
$108$	0	0
$109$	16.8065	1.60977	0.804886	−	0.593430i	$-0.202226\pi$
$109$	16.8065	1.60977	0.804886	+	0.593430i	$0.202226\pi$
$110$	−0.308658	−0.0294294
$111$	0	0
$112$	7.07087	0.668134
$113$	11.2947	1.06252	0.531258	−	0.847210i	$-0.321720\pi$
$113$	11.2947	1.06252	0.531258	+	0.847210i	$0.321720\pi$
$114$	0	0
$115$	−6.04904	−0.564076
$116$	8.73148	0.810697
$117$	0	0
$118$	1.28680	0.118460
$119$	−0.592670	−0.0543300
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.392015	0.0354913
$123$	0	0
$124$	5.39519	0.484502
$125$	6.11681	0.547104
$126$	0	0
$127$	18.5894	1.64954	0.824771	−	0.565467i	$-0.191304\pi$
$127$	18.5894	1.64954	0.824771	+	0.565467i	$0.191304\pi$
$128$	11.2660	0.995779
$129$	0	0
$130$	−0.372628	−0.0326816
$131$	3.09098	0.270060	0.135030	−	0.990842i	$-0.456887\pi$
$131$	3.09098	0.270060	0.135030	+	0.990842i	$0.456887\pi$
$132$	0	0
$133$	−2.66942	−0.231468
$134$	1.07685	0.0930257
$135$	0	0
$136$	−0.404730	−0.0347053
$137$	8.67619	0.741257	0.370629	−	0.928781i	$-0.379142\pi$
$137$	8.67619	0.741257	0.370629	+	0.928781i	$0.379142\pi$
$138$	0	0
$139$	14.5426	1.23348	0.616742	−	0.787166i	$-0.288452\pi$
$139$	14.5426	1.23348	0.616742	+	0.787166i	$0.288452\pi$
$140$	3.00519	0.253985
$141$	0	0
$142$	1.25338	0.105181
$143$	1.20725	0.100955
$144$	0	0
$145$	3.15306	0.261847
$146$	−8.04115	−0.665490
$147$	0	0
$148$	−12.6777	−1.04210
$149$	18.4716	1.51325	0.756625	−	0.653849i	$-0.226847\pi$
$149$	18.4716	1.51325	0.756625	+	0.653849i	$0.226847\pi$
$150$	0	0
$151$	−0.913992	−0.0743796	−0.0371898	−	0.999308i	$-0.511841\pi$
$151$	−0.913992	−0.0743796	−0.0371898	+	0.999308i	$0.511841\pi$
$152$	−1.82293	−0.147859
$153$	0	0
$154$	1.29225	0.104132
$155$	1.94828	0.156489
$156$	0	0
$157$	−2.39421	−0.191079	−0.0955395	−	0.995426i	$-0.530458\pi$
$157$	−2.39421	−0.191079	−0.0955395	+	0.995426i	$0.530458\pi$
$158$	−2.75593	−0.219250
$159$	0	0
$160$	3.14219	0.248412
$161$	25.3252	1.99591
$162$	0	0
$163$	14.1986	1.11212	0.556061	−	0.831141i	$-0.312312\pi$
$163$	14.1986	1.11212	0.556061	+	0.831141i	$0.312312\pi$
$164$	−12.2298	−0.954987
$165$	0	0
$166$	−1.42219	−0.110383
$167$	−9.25056	−0.715830	−0.357915	−	0.933754i	$-0.616512\pi$
$167$	−9.25056	−0.715830	−0.357915	+	0.933754i	$0.616512\pi$
$168$	0	0
$169$	−11.5426	−0.887888
$170$	−0.0685290	−0.00525594
$171$	0	0
$172$	−2.71095	−0.206708
$173$	−5.39376	−0.410080	−0.205040	−	0.978754i	$-0.565732\pi$
$173$	−5.39376	−0.410080	−0.205040	+	0.978754i	$0.565732\pi$
$174$	0	0
$175$	−12.2619	−0.926910
$176$	−2.64884	−0.199664
$177$	0	0
$178$	−2.68253	−0.201064
$179$	−1.30720	−0.0977048	−0.0488524	−	0.998806i	$-0.515556\pi$
$179$	−1.30720	−0.0977048	−0.0488524	+	0.998806i	$0.515556\pi$
$180$	0	0
$181$	−10.7624	−0.799961	−0.399980	−	0.916524i	$-0.630983\pi$
$181$	−10.7624	−0.799961	−0.399980	+	0.916524i	$0.630983\pi$
$182$	1.56006	0.115639
$183$	0	0
$184$	17.2944	1.27496
$185$	−4.57808	−0.336587
$186$	0	0
$187$	0.222022	0.0162359
$188$	10.4982	0.765663
$189$	0	0
$190$	−0.308658	−0.0223924
$191$	−15.3041	−1.10737	−0.553684	−	0.832727i	$-0.686778\pi$
$191$	−15.3041	−1.10737	−0.553684	+	0.832727i	$0.686778\pi$
$192$	0	0
$193$	−18.3032	−1.31749	−0.658747	−	0.752364i	$-0.728913\pi$
$193$	−18.3032	−1.31749	−0.658747	+	0.752364i	$0.728913\pi$
$194$	−6.66732	−0.478686
$195$	0	0
$196$	−0.222092	−0.0158637
$197$	0.576171	0.0410505	0.0205252	−	0.999789i	$-0.493466\pi$
$197$	0.576171	0.0410505	0.0205252	+	0.999789i	$0.493466\pi$
$198$	0	0
$199$	−10.1678	−0.720778	−0.360389	−	0.932802i	$-0.617356\pi$
$199$	−10.1678	−0.720778	−0.360389	+	0.932802i	$0.617356\pi$
$200$	−8.37354	−0.592099
$201$	0	0
$202$	5.82402	0.409777
$203$	−13.2007	−0.926510
$204$	0	0
$205$	−4.41635	−0.308451
$206$	−4.12964	−0.287726
$207$	0	0
$208$	−3.19781	−0.221728
$209$	1.00000	0.0691714
$210$	0	0
$211$	−7.25837	−0.499687	−0.249843	−	0.968286i	$-0.580379\pi$
$211$	−7.25837	−0.499687	−0.249843	+	0.968286i	$0.580379\pi$
$212$	−17.0188	−1.16885
$213$	0	0
$214$	0.796745	0.0544643
$215$	−0.978963	−0.0667647
$216$	0	0
$217$	−8.15675	−0.553716
$218$	−8.13591	−0.551033
$219$	0	0
$220$	−1.12578	−0.0759004
$221$	0.268036	0.0180301
$222$	0	0
$223$	16.2772	1.09000	0.545000	−	0.838436i	$-0.316529\pi$
$223$	16.2772	1.09000	0.545000	+	0.838436i	$0.316529\pi$
$224$	−13.1553	−0.878972
$225$	0	0
$226$	−5.46768	−0.363705
$227$	20.9260	1.38890	0.694452	−	0.719539i	$-0.255647\pi$
$227$	20.9260	1.38890	0.694452	+	0.719539i	$0.255647\pi$
$228$	0	0
$229$	−10.8083	−0.714234	−0.357117	−	0.934060i	$-0.616240\pi$
$229$	−10.8083	−0.714234	−0.357117	+	0.934060i	$0.616240\pi$
$230$	2.92830	0.193086
$231$	0	0
$232$	−9.01469	−0.591844
$233$	25.4173	1.66514	0.832570	−	0.553919i	$-0.186868\pi$
$233$	25.4173	1.66514	0.832570	+	0.553919i	$0.186868\pi$
$234$	0	0
$235$	3.79106	0.247301
$236$	4.69342	0.305515
$237$	0	0
$238$	0.286907	0.0185974
$239$	−0.352238	−0.0227844	−0.0113922	−	0.999935i	$-0.503626\pi$
$239$	−0.352238	−0.0227844	−0.0113922	+	0.999935i	$0.503626\pi$
$240$	0	0
$241$	−15.6832	−1.01024	−0.505122	−	0.863048i	$-0.668552\pi$
$241$	−15.6832	−1.01024	−0.505122	+	0.863048i	$0.668552\pi$
$242$	−0.484093	−0.0311187
$243$	0	0
$244$	1.42981	0.0915344
$245$	−0.0802004	−0.00512382
$246$	0	0
$247$	1.20725	0.0768154
$248$	−5.57019	−0.353707
$249$	0	0
$250$	−2.96110	−0.187277
$251$	−1.16332	−0.0734282	−0.0367141	−	0.999326i	$-0.511689\pi$
$251$	−1.16332	−0.0734282	−0.0367141	+	0.999326i	$0.511689\pi$
$252$	0	0
$253$	−9.48717	−0.596453
$254$	−8.99899	−0.564647
$255$	0	0
$256$	0.370256	0.0231410
$257$	15.0510	0.938857	0.469428	−	0.882971i	$-0.344460\pi$
$257$	15.0510	0.938857	0.469428	+	0.882971i	$0.344460\pi$
$258$	0	0
$259$	19.1668	1.19097
$260$	−1.35910	−0.0842879
$261$	0	0
$262$	−1.49632	−0.0924429
$263$	20.6609	1.27400	0.637002	−	0.770862i	$-0.280174\pi$
$263$	20.6609	1.27400	0.637002	+	0.770862i	$0.280174\pi$
$264$	0	0
$265$	−6.14571	−0.377528
$266$	1.29225	0.0792326
$267$	0	0
$268$	3.92765	0.239919
$269$	2.82662	0.172342	0.0861711	−	0.996280i	$-0.472537\pi$
$269$	2.82662	0.172342	0.0861711	+	0.996280i	$0.472537\pi$
$270$	0	0
$271$	23.5970	1.43341	0.716707	−	0.697375i	$-0.245649\pi$
$271$	23.5970	1.43341	0.716707	+	0.697375i	$0.245649\pi$
$272$	−0.588102	−0.0356589
$273$	0	0
$274$	−4.20008	−0.253736
$275$	4.59346	0.276996
$276$	0	0
$277$	−18.7773	−1.12822	−0.564110	−	0.825700i	$-0.690781\pi$
$277$	−18.7773	−1.12822	−0.564110	+	0.825700i	$0.690781\pi$
$278$	−7.03994	−0.422228
$279$	0	0
$280$	−3.10267	−0.185420
$281$	−27.2632	−1.62639	−0.813194	−	0.581993i	$-0.802273\pi$
$281$	−27.2632	−1.62639	−0.813194	+	0.581993i	$0.802273\pi$
$282$	0	0
$283$	−27.2601	−1.62044	−0.810221	−	0.586124i	$-0.800653\pi$
$283$	−27.2601	−1.62044	−0.810221	+	0.586124i	$0.800653\pi$
$284$	4.57149	0.271268
$285$	0	0
$286$	−0.584420	−0.0345575
$287$	18.4897	1.09141
$288$	0	0
$289$	−16.9507	−0.997100
$290$	−1.52637	−0.0896316
$291$	0	0
$292$	−29.3289	−1.71634
$293$	−30.9996	−1.81101	−0.905507	−	0.424332i	$-0.860509\pi$
$293$	−30.9996	−1.81101	−0.905507	+	0.424332i	$0.860509\pi$
$294$	0	0
$295$	1.69486	0.0986784
$296$	13.0889	0.760776
$297$	0	0
$298$	−8.94195	−0.517993
$299$	−11.4534	−0.662366
$300$	0	0
$301$	4.09858	0.236238
$302$	0.442457	0.0254605
$303$	0	0
$304$	−2.64884	−0.151922
$305$	0.516325	0.0295647
$306$	0	0
$307$	−18.2495	−1.04156	−0.520778	−	0.853692i	$-0.674358\pi$
$307$	−18.2495	−1.04156	−0.520778	+	0.853692i	$0.674358\pi$
$308$	4.71327	0.268563
$309$	0	0
$310$	−0.943146	−0.0535671
$311$	−30.8931	−1.75179	−0.875894	−	0.482503i	$-0.839728\pi$
$311$	−30.8931	−1.75179	−0.875894	+	0.482503i	$0.839728\pi$
$312$	0	0
$313$	28.7973	1.62772	0.813859	−	0.581062i	$-0.197363\pi$
$313$	28.7973	1.62772	0.813859	+	0.581062i	$0.197363\pi$
$314$	1.15902	0.0654073
$315$	0	0
$316$	−10.0518	−0.565460
$317$	−5.96040	−0.334769	−0.167385	−	0.985892i	$-0.553532\pi$
$317$	−5.96040	−0.334769	−0.167385	+	0.985892i	$0.553532\pi$
$318$	0	0
$319$	4.94518	0.276877
$320$	1.85670	0.103793
$321$	0	0
$322$	−12.2597	−0.683209
$323$	0.222022	0.0123536
$324$	0	0
$325$	5.54545	0.307606
$326$	−6.87345	−0.380685
$327$	0	0
$328$	12.6265	0.697181
$329$	−15.8718	−0.875043
$330$	0	0
$331$	33.4010	1.83589	0.917944	−	0.396711i	$-0.129849\pi$
$331$	33.4010	1.83589	0.917944	+	0.396711i	$0.129849\pi$
$332$	−5.18722	−0.284686
$333$	0	0
$334$	4.47813	0.245032
$335$	1.41833	0.0774915
$336$	0	0
$337$	24.9988	1.36177	0.680885	−	0.732390i	$-0.261595\pi$
$337$	24.9988	1.36177	0.680885	+	0.732390i	$0.261595\pi$
$338$	5.58766	0.303929
$339$	0	0
$340$	−0.249949	−0.0135554
$341$	3.05563	0.165472
$342$	0	0
$343$	−18.3501	−0.990815
$344$	2.79889	0.150906
$345$	0	0
$346$	2.61108	0.140372
$347$	8.35577	0.448561	0.224281	−	0.974525i	$-0.427997\pi$
$347$	8.35577	0.448561	0.224281	+	0.974525i	$0.427997\pi$
$348$	0	0
$349$	−28.3911	−1.51974	−0.759871	−	0.650074i	$-0.774738\pi$
$349$	−28.3911	−1.51974	−0.759871	+	0.650074i	$0.774738\pi$
$350$	5.93588	0.317286
$351$	0	0
$352$	4.92814	0.262671
$353$	12.4082	0.660423	0.330212	−	0.943907i	$-0.392880\pi$
$353$	12.4082	0.660423	0.330212	+	0.943907i	$0.392880\pi$
$354$	0	0
$355$	1.65083	0.0876169
$356$	−9.78413	−0.518558
$357$	0	0
$358$	0.632806	0.0334448
$359$	−12.1134	−0.639319	−0.319659	−	0.947533i	$-0.603568\pi$
$359$	−12.1134	−0.639319	−0.319659	+	0.947533i	$0.603568\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	5.20999	0.273831
$363$	0	0
$364$	5.69009	0.298242
$365$	−10.5911	−0.554361
$366$	0	0
$367$	−4.06064	−0.211964	−0.105982	−	0.994368i	$-0.533799\pi$
$367$	−4.06064	−0.211964	−0.105982	+	0.994368i	$0.533799\pi$
$368$	25.1300	1.30999
$369$	0	0
$370$	2.21621	0.115216
$371$	25.7299	1.33583
$372$	0	0
$373$	35.4898	1.83759	0.918796	−	0.394732i	$-0.129163\pi$
$373$	35.4898	1.83759	0.918796	+	0.394732i	$0.129163\pi$
$374$	−0.107479	−0.00555763
$375$	0	0
$376$	−10.8388	−0.558967
$377$	5.97006	0.307474
$378$	0	0
$379$	13.5584	0.696449	0.348224	−	0.937411i	$-0.386785\pi$
$379$	13.5584	0.696449	0.348224	+	0.937411i	$0.386785\pi$
$380$	−1.12578	−0.0577515
$381$	0	0
$382$	7.40862	0.379058
$383$	23.8326	1.21779	0.608894	−	0.793251i	$-0.291613\pi$
$383$	23.8326	1.21779	0.608894	+	0.793251i	$0.291613\pi$
$384$	0	0
$385$	1.70203	0.0867432
$386$	8.86045	0.450985
$387$	0	0
$388$	−24.3180	−1.23456
$389$	12.7567	0.646789	0.323395	−	0.946264i	$-0.395176\pi$
$389$	12.7567	0.646789	0.323395	+	0.946264i	$0.395176\pi$
$390$	0	0
$391$	−2.10636	−0.106523
$392$	0.229296	0.0115812
$393$	0	0
$394$	−0.278920	−0.0140518
$395$	−3.62986	−0.182638
$396$	0	0
$397$	−30.3169	−1.52156	−0.760780	−	0.649009i	$-0.775184\pi$
$397$	−30.3169	−1.52156	−0.760780	+	0.649009i	$0.775184\pi$
$398$	4.92217	0.246726
$399$	0	0
$400$	−12.1674	−0.608368
$401$	−32.6370	−1.62982	−0.814908	−	0.579591i	$-0.803212\pi$
$401$	−32.6370	−1.62982	−0.814908	+	0.579591i	$0.803212\pi$
$402$	0	0
$403$	3.68891	0.183758
$404$	21.2422	1.05684
$405$	0	0
$406$	6.39038	0.317149
$407$	−7.18015	−0.355907
$408$	0	0
$409$	−23.1587	−1.14512	−0.572562	−	0.819862i	$-0.694050\pi$
$409$	−23.1587	−1.14512	−0.572562	+	0.819862i	$0.694050\pi$
$410$	2.13792	0.105584
$411$	0	0
$412$	−15.0622	−0.742063
$413$	−7.09578	−0.349160
$414$	0	0
$415$	−1.87318	−0.0919506
$416$	5.94949	0.291698
$417$	0	0
$418$	−0.484093	−0.0236777
$419$	14.2213	0.694755	0.347377	−	0.937725i	$-0.387072\pi$
$419$	14.2213	0.694755	0.347377	+	0.937725i	$0.387072\pi$
$420$	0	0
$421$	15.9605	0.777866	0.388933	−	0.921266i	$-0.372844\pi$
$421$	15.9605	0.777866	0.388933	+	0.921266i	$0.372844\pi$
$422$	3.51373	0.171045
$423$	0	0
$424$	17.5708	0.853313
$425$	1.01985	0.0494701
$426$	0	0
$427$	−2.16167	−0.104611
$428$	2.90600	0.140467
$429$	0	0
$430$	0.473909	0.0228539
$431$	−24.5100	−1.18061	−0.590304	−	0.807181i	$-0.700992\pi$
$431$	−24.5100	−1.18061	−0.590304	+	0.807181i	$0.700992\pi$
$432$	0	0
$433$	−19.1839	−0.921921	−0.460961	−	0.887421i	$-0.652495\pi$
$433$	−19.1839	−0.921921	−0.460961	+	0.887421i	$0.652495\pi$
$434$	3.94862	0.189540
$435$	0	0
$436$	−29.6745	−1.42115
$437$	−9.48717	−0.453833
$438$	0	0
$439$	13.7961	0.658450	0.329225	−	0.944251i	$-0.393212\pi$
$439$	13.7961	0.658450	0.329225	+	0.944251i	$0.393212\pi$
$440$	1.16230	0.0554105
$441$	0	0
$442$	−0.129754	−0.00617178
$443$	−14.1259	−0.671140	−0.335570	−	0.942015i	$-0.608929\pi$
$443$	−14.1259	−0.671140	−0.335570	+	0.942015i	$0.608929\pi$
$444$	0	0
$445$	−3.53318	−0.167489
$446$	−7.87967	−0.373113
$447$	0	0
$448$	−7.77337	−0.367257
$449$	21.0016	0.991129	0.495565	−	0.868571i	$-0.334961\pi$
$449$	21.0016	0.991129	0.495565	+	0.868571i	$0.334961\pi$
$450$	0	0
$451$	−6.92650	−0.326156
$452$	−19.9425	−0.938018
$453$	0	0
$454$	−10.1301	−0.475429
$455$	2.05477	0.0963290
$456$	0	0
$457$	−22.4621	−1.05073	−0.525367	−	0.850876i	$-0.676072\pi$
$457$	−22.4621	−1.05073	−0.525367	+	0.850876i	$0.676072\pi$
$458$	5.23223	0.244486
$459$	0	0
$460$	10.6805	0.497981
$461$	31.1213	1.44946	0.724732	−	0.689031i	$-0.241964\pi$
$461$	31.1213	1.44946	0.724732	+	0.689031i	$0.241964\pi$
$462$	0	0
$463$	31.6932	1.47291	0.736455	−	0.676487i	$-0.236498\pi$
$463$	31.6932	1.47291	0.736455	+	0.676487i	$0.236498\pi$
$464$	−13.0990	−0.608106
$465$	0	0
$466$	−12.3043	−0.569986
$467$	1.75474	0.0811995	0.0405997	−	0.999175i	$-0.487073\pi$
$467$	1.75474	0.0811995	0.0405997	+	0.999175i	$0.487073\pi$
$468$	0	0
$469$	−5.93804	−0.274193
$470$	−1.83522	−0.0846525
$471$	0	0
$472$	−4.84566	−0.223039
$473$	−1.53538	−0.0705970
$474$	0	0
$475$	4.59346	0.210763
$476$	1.04645	0.0479640
$477$	0	0
$478$	0.170516	0.00779922
$479$	11.1294	0.508515	0.254257	−	0.967137i	$-0.418169\pi$
$479$	11.1294	0.508515	0.254257	+	0.967137i	$0.418169\pi$
$480$	0	0
$481$	−8.66823	−0.395237
$482$	7.59212	0.345812
$483$	0	0
$484$	−1.76565	−0.0802570
$485$	−8.78158	−0.398751
$486$	0	0
$487$	3.27007	0.148181	0.0740905	−	0.997252i	$-0.476395\pi$
$487$	3.27007	0.148181	0.0740905	+	0.997252i	$0.476395\pi$
$488$	−1.47619	−0.0668240
$489$	0	0
$490$	0.0388244	0.00175391
$491$	−29.2686	−1.32087	−0.660437	−	0.750882i	$-0.729629\pi$
$491$	−29.2686	−1.32087	−0.660437	+	0.750882i	$0.729629\pi$
$492$	0	0
$493$	1.09794	0.0494487
$494$	−0.584420	−0.0262943
$495$	0	0
$496$	−8.09389	−0.363426
$497$	−6.91145	−0.310021
$498$	0	0
$499$	−1.04828	−0.0469275	−0.0234638	−	0.999725i	$-0.507469\pi$
$499$	−1.04828	−0.0469275	−0.0234638	+	0.999725i	$0.507469\pi$
$500$	−10.8002	−0.482998
$501$	0	0
$502$	0.563155	0.0251348
$503$	−28.4724	−1.26952	−0.634761	−	0.772709i	$-0.718901\pi$
$503$	−28.4724	−1.26952	−0.634761	+	0.772709i	$0.718901\pi$
$504$	0	0
$505$	7.67086	0.341349
$506$	4.59267	0.204169
$507$	0	0
$508$	−32.8224	−1.45626
$509$	−39.6801	−1.75879	−0.879396	−	0.476091i	$-0.842053\pi$
$509$	−39.6801	−1.75879	−0.879396	+	0.476091i	$0.842053\pi$
$510$	0	0
$511$	44.3410	1.96153
$512$	−22.7112	−1.00370
$513$	0	0
$514$	−7.28609	−0.321376
$515$	−5.43918	−0.239679
$516$	0	0
$517$	5.94581	0.261496
$518$	−9.27852	−0.407675
$519$	0	0
$520$	1.40319	0.0615338
$521$	15.3168	0.671043	0.335522	−	0.942033i	$-0.391087\pi$
$521$	15.3168	0.671043	0.335522	+	0.942033i	$0.391087\pi$
$522$	0	0
$523$	−4.17850	−0.182713	−0.0913566	−	0.995818i	$-0.529120\pi$
$523$	−4.17850	−0.182713	−0.0913566	+	0.995818i	$0.529120\pi$
$524$	−5.45760	−0.238416
$525$	0	0
$526$	−10.0018	−0.436098
$527$	0.678418	0.0295524
$528$	0	0
$529$	67.0064	2.91332
$530$	2.97509	0.129230
$531$	0	0
$532$	4.71327	0.204346
$533$	−8.36201	−0.362199
$534$	0	0
$535$	1.04940	0.0453694
$536$	−4.05505	−0.175151
$537$	0	0
$538$	−1.36835	−0.0589937
$539$	−0.125785	−0.00541792
$540$	0	0
$541$	−14.6201	−0.628565	−0.314283	−	0.949329i	$-0.601764\pi$
$541$	−14.6201	−0.628565	−0.314283	+	0.949329i	$0.601764\pi$
$542$	−11.4231	−0.490665
$543$	0	0
$544$	1.09416	0.0469116
$545$	−10.7159	−0.459017
$546$	0	0
$547$	−34.7220	−1.48461	−0.742303	−	0.670064i	$-0.766267\pi$
$547$	−34.7220	−1.48461	−0.742303	+	0.670064i	$0.766267\pi$
$548$	−15.3192	−0.654402
$549$	0	0
$550$	−2.22366	−0.0948173
$551$	4.94518	0.210672
$552$	0	0
$553$	15.1969	0.646240
$554$	9.08997	0.386196
$555$	0	0
$556$	−25.6771	−1.08895
$557$	7.07603	0.299821	0.149910	−	0.988700i	$-0.452101\pi$
$557$	7.07603	0.299821	0.149910	+	0.988700i	$0.452101\pi$
$558$	0	0
$559$	−1.85359	−0.0783985
$560$	−4.50840	−0.190515
$561$	0	0
$562$	13.1979	0.556721
$563$	32.5712	1.37271	0.686357	−	0.727265i	$-0.259209\pi$
$563$	32.5712	1.37271	0.686357	+	0.727265i	$0.259209\pi$
$564$	0	0
$565$	−7.20152	−0.302970
$566$	13.1964	0.554686
$567$	0	0
$568$	−4.71978	−0.198037
$569$	−32.6457	−1.36858	−0.684290	−	0.729210i	$-0.739888\pi$
$569$	−32.6457	−1.36858	−0.684290	+	0.729210i	$0.739888\pi$
$570$	0	0
$571$	−35.8059	−1.49843	−0.749216	−	0.662326i	$-0.769569\pi$
$571$	−35.8059	−1.49843	−0.749216	+	0.662326i	$0.769569\pi$
$572$	−2.13158	−0.0891260
$573$	0	0
$574$	−8.95073	−0.373596
$575$	−43.5790	−1.81737
$576$	0	0
$577$	15.7218	0.654509	0.327254	−	0.944936i	$-0.393877\pi$
$577$	15.7218	0.654509	0.327254	+	0.944936i	$0.393877\pi$
$578$	8.20571	0.341313
$579$	0	0
$580$	−5.56721	−0.231166
$581$	7.84233	0.325355
$582$	0	0
$583$	−9.63879	−0.399198
$584$	30.2802	1.25300
$585$	0	0
$586$	15.0067	0.619919
$587$	−24.4810	−1.01044	−0.505220	−	0.862991i	$-0.668588\pi$
$587$	−24.4810	−1.01044	−0.505220	+	0.862991i	$0.668588\pi$
$588$	0	0
$589$	3.05563	0.125905
$590$	−0.820468	−0.0337781
$591$	0	0
$592$	19.0191	0.781680
$593$	21.2995	0.874666	0.437333	−	0.899300i	$-0.355923\pi$
$593$	21.2995	0.874666	0.437333	+	0.899300i	$0.355923\pi$
$594$	0	0
$595$	0.377887	0.0154919
$596$	−32.6144	−1.33594
$597$	0	0
$598$	5.54450	0.226731
$599$	8.90839	0.363987	0.181994	−	0.983300i	$-0.441745\pi$
$599$	8.90839	0.363987	0.181994	+	0.983300i	$0.441745\pi$
$600$	0	0
$601$	−8.36794	−0.341335	−0.170668	−	0.985329i	$-0.554592\pi$
$601$	−8.36794	−0.341335	−0.170668	+	0.985329i	$0.554592\pi$
$602$	−1.98409	−0.0808655
$603$	0	0
$604$	1.61379	0.0656643
$605$	−0.637602	−0.0259222
$606$	0	0
$607$	43.2321	1.75474	0.877368	−	0.479819i	$-0.159298\pi$
$607$	43.2321	1.75474	0.877368	+	0.479819i	$0.159298\pi$
$608$	4.92814	0.199862
$609$	0	0
$610$	−0.249949	−0.0101201
$611$	7.17807	0.290394
$612$	0	0
$613$	−12.4893	−0.504438	−0.252219	−	0.967670i	$-0.581160\pi$
$613$	−12.4893	−0.504438	−0.252219	+	0.967670i	$0.581160\pi$
$614$	8.83447	0.356530
$615$	0	0
$616$	−4.86615	−0.196063
$617$	18.4079	0.741073	0.370536	−	0.928818i	$-0.379174\pi$
$617$	18.4079	0.741073	0.370536	+	0.928818i	$0.379174\pi$
$618$	0	0
$619$	−7.30900	−0.293773	−0.146887	−	0.989153i	$-0.546925\pi$
$619$	−7.30900	−0.293773	−0.146887	+	0.989153i	$0.546925\pi$
$620$	−3.43998	−0.138153
$621$	0	0
$622$	14.9551	0.599646
$623$	14.7922	0.592637
$624$	0	0
$625$	19.0672	0.762689
$626$	−13.9406	−0.557177
$627$	0	0
$628$	4.22735	0.168690
$629$	−1.59415	−0.0635631
$630$	0	0
$631$	−11.8397	−0.471329	−0.235665	−	0.971834i	$-0.575727\pi$
$631$	−11.8397	−0.471329	−0.235665	+	0.971834i	$0.575727\pi$
$632$	10.3779	0.412810
$633$	0	0
$634$	2.88538	0.114593
$635$	−11.8526	−0.470357
$636$	0	0
$637$	−0.151853	−0.00601664
$638$	−2.39392	−0.0947764
$639$	0	0
$640$	−7.18320	−0.283941
$641$	36.5974	1.44551	0.722756	−	0.691104i	$-0.242875\pi$
$641$	36.5974	1.44551	0.722756	+	0.691104i	$0.242875\pi$
$642$	0	0
$643$	−14.0452	−0.553888	−0.276944	−	0.960886i	$-0.589322\pi$
$643$	−14.0452	−0.553888	−0.276944	+	0.960886i	$0.589322\pi$
$644$	−44.7156	−1.76204
$645$	0	0
$646$	−0.107479	−0.00422872
$647$	43.5266	1.71121	0.855604	−	0.517631i	$-0.173186\pi$
$647$	43.5266	1.71121	0.855604	+	0.517631i	$0.173186\pi$
$648$	0	0
$649$	2.65817	0.104342
$650$	−2.68451	−0.105295
$651$	0	0
$652$	−25.0699	−0.981812
$653$	−44.0328	−1.72314	−0.861568	−	0.507642i	$-0.830517\pi$
$653$	−44.0328	−1.72314	−0.861568	+	0.507642i	$0.830517\pi$
$654$	0	0
$655$	−1.97081	−0.0770060
$656$	18.3472	0.716338
$657$	0	0
$658$	7.68344	0.299532
$659$	3.86784	0.150670	0.0753349	−	0.997158i	$-0.475997\pi$
$659$	3.86784	0.150670	0.0753349	+	0.997158i	$0.475997\pi$
$660$	0	0
$661$	7.96999	0.309997	0.154998	−	0.987915i	$-0.450463\pi$
$661$	7.96999	0.309997	0.154998	+	0.987915i	$0.450463\pi$
$662$	−16.1692	−0.628434
$663$	0	0
$664$	5.35547	0.207833
$665$	1.70203	0.0660017
$666$	0	0
$667$	−46.9158	−1.81659
$668$	16.3333	0.631954
$669$	0	0
$670$	−0.686602	−0.0265258
$671$	0.809792	0.0312617
$672$	0	0
$673$	21.7872	0.839835	0.419917	−	0.907562i	$-0.362059\pi$
$673$	21.7872	0.839835	0.419917	+	0.907562i	$0.362059\pi$
$674$	−12.1017	−0.466141
$675$	0	0
$676$	20.3802	0.783852
$677$	−33.2004	−1.27599	−0.637997	−	0.770039i	$-0.720237\pi$
$677$	−33.2004	−1.27599	−0.637997	+	0.770039i	$0.720237\pi$
$678$	0	0
$679$	36.7654	1.41093
$680$	0.258057	0.00989602
$681$	0	0
$682$	−1.47921	−0.0566418
$683$	34.9595	1.33769	0.668843	−	0.743403i	$-0.266790\pi$
$683$	34.9595	1.33769	0.668843	+	0.743403i	$0.266790\pi$
$684$	0	0
$685$	−5.53196	−0.211365
$686$	8.88317	0.339161
$687$	0	0
$688$	4.06699	0.155052
$689$	−11.6364	−0.443312
$690$	0	0
$691$	−30.4626	−1.15885	−0.579426	−	0.815025i	$-0.696723\pi$
$691$	−30.4626	−1.15885	−0.579426	+	0.815025i	$0.696723\pi$
$692$	9.52351	0.362030
$693$	0	0
$694$	−4.04497	−0.153545
$695$	−9.27236	−0.351721
$696$	0	0
$697$	−1.53784	−0.0582497
$698$	13.7439	0.520215
$699$	0	0
$700$	21.6502	0.818301
$701$	−10.3329	−0.390269	−0.195135	−	0.980776i	$-0.562514\pi$
$701$	−10.3329	−0.390269	−0.195135	+	0.980776i	$0.562514\pi$
$702$	0	0
$703$	−7.18015	−0.270805
$704$	2.91201	0.109751
$705$	0	0
$706$	−6.00673	−0.226066
$707$	−32.1152	−1.20782
$708$	0	0
$709$	−45.4499	−1.70691	−0.853454	−	0.521169i	$-0.825496\pi$
$709$	−45.4499	−1.70691	−0.853454	+	0.521169i	$0.825496\pi$
$710$	−0.799154	−0.0299917
$711$	0	0
$712$	10.1015	0.378570
$713$	−28.9893	−1.08566
$714$	0	0
$715$	−0.769744	−0.0287868
$716$	2.30806	0.0862564
$717$	0	0
$718$	5.86399	0.218842
$719$	7.51239	0.280165	0.140082	−	0.990140i	$-0.455263\pi$
$719$	7.51239	0.280165	0.140082	+	0.990140i	$0.455263\pi$
$720$	0	0
$721$	22.7719	0.848071
$722$	−0.484093	−0.0180161
$723$	0	0
$724$	19.0026	0.706227
$725$	22.7155	0.843632
$726$	0	0
$727$	4.79898	0.177984	0.0889922	−	0.996032i	$-0.471635\pi$
$727$	4.79898	0.177984	0.0889922	+	0.996032i	$0.471635\pi$
$728$	−5.87465	−0.217729
$729$	0	0
$730$	5.12705	0.189761
$731$	−0.340889	−0.0126082
$732$	0	0
$733$	−8.55077	−0.315830	−0.157915	−	0.987453i	$-0.550477\pi$
$733$	−8.55077	−0.315830	−0.157915	+	0.987453i	$0.550477\pi$
$734$	1.96572	0.0725562
$735$	0	0
$736$	−46.7541	−1.72338
$737$	2.22447	0.0819395
$738$	0	0
$739$	38.8820	1.43030	0.715149	−	0.698972i	$-0.246359\pi$
$739$	38.8820	1.43030	0.715149	+	0.698972i	$0.246359\pi$
$740$	8.08330	0.297148
$741$	0	0
$742$	−12.4557	−0.457262
$743$	−28.4350	−1.04318	−0.521589	−	0.853197i	$-0.674660\pi$
$743$	−28.4350	−1.04318	−0.521589	+	0.853197i	$0.674660\pi$
$744$	0	0
$745$	−11.7775	−0.431494
$746$	−17.1804	−0.629018
$747$	0	0
$748$	−0.392015	−0.0143335
$749$	−4.39346	−0.160534
$750$	0	0
$751$	18.2732	0.666799	0.333400	−	0.942786i	$-0.391804\pi$
$751$	18.2732	0.666799	0.333400	+	0.942786i	$0.391804\pi$
$752$	−15.7495	−0.574326
$753$	0	0
$754$	−2.89006	−0.105250
$755$	0.582763	0.0212089
$756$	0	0
$757$	1.10637	0.0402117	0.0201058	−	0.999798i	$-0.493600\pi$
$757$	1.10637	0.0402117	0.0201058	+	0.999798i	$0.493600\pi$
$758$	−6.56352	−0.238398
$759$	0	0
$760$	1.16230	0.0421611
$761$	−1.10110	−0.0399147	−0.0199573	−	0.999801i	$-0.506353\pi$
$761$	−1.10110	−0.0399147	−0.0199573	+	0.999801i	$0.506353\pi$
$762$	0	0
$763$	44.8636	1.62417
$764$	27.0218	0.977615
$765$	0	0
$766$	−11.5372	−0.416855
$767$	3.20908	0.115873
$768$	0	0
$769$	30.2914	1.09234	0.546169	−	0.837675i	$-0.316086\pi$
$769$	30.2914	1.09234	0.546169	+	0.837675i	$0.316086\pi$
$770$	−0.823938	−0.0296927
$771$	0	0
$772$	32.3171	1.16312
$773$	43.9677	1.58141	0.790704	−	0.612199i	$-0.209715\pi$
$773$	43.9677	1.58141	0.790704	+	0.612199i	$0.209715\pi$
$774$	0	0
$775$	14.0359	0.504186
$776$	25.1068	0.901283
$777$	0	0
$778$	−6.17542	−0.221399
$779$	−6.92650	−0.248168
$780$	0	0
$781$	2.58912	0.0926461
$782$	1.01967	0.0364635
$783$	0	0
$784$	0.333183	0.0118994
$785$	1.52655	0.0544851
$786$	0	0
$787$	−37.8350	−1.34867	−0.674337	−	0.738424i	$-0.735570\pi$
$787$	−37.8350	−1.34867	−0.674337	+	0.738424i	$0.735570\pi$
$788$	−1.01732	−0.0362405
$789$	0	0
$790$	1.75719	0.0625179
$791$	30.1502	1.07202
$792$	0	0
$793$	0.977621	0.0347163
$794$	14.6762	0.520838
$795$	0	0
$796$	17.9529	0.636323
$797$	−11.2902	−0.399918	−0.199959	−	0.979804i	$-0.564081\pi$
$797$	−11.2902	−0.399918	−0.199959	+	0.979804i	$0.564081\pi$
$798$	0	0
$799$	1.32010	0.0467018
$800$	22.6372	0.800347
$801$	0	0
$802$	15.7993	0.557894
$803$	−16.6108	−0.586181
$804$	0	0
$805$	−16.1474	−0.569121
$806$	−1.78577	−0.0629012
$807$	0	0
$808$	−21.9313	−0.771539
$809$	−17.8104	−0.626180	−0.313090	−	0.949723i	$-0.601364\pi$
$809$	−17.8104	−0.626180	−0.313090	+	0.949723i	$0.601364\pi$
$810$	0	0
$811$	28.1638	0.988965	0.494482	−	0.869188i	$-0.335358\pi$
$811$	28.1638	0.988965	0.494482	+	0.869188i	$0.335358\pi$
$812$	23.3079	0.817949
$813$	0	0
$814$	3.47586	0.121829
$815$	−9.05307	−0.317115
$816$	0	0
$817$	−1.53538	−0.0537162
$818$	11.2110	0.391982
$819$	0	0
$820$	7.79774	0.272309
$821$	44.2644	1.54484	0.772418	−	0.635114i	$-0.219047\pi$
$821$	44.2644	1.54484	0.772418	+	0.635114i	$0.219047\pi$
$822$	0	0
$823$	−22.4258	−0.781716	−0.390858	−	0.920451i	$-0.627822\pi$
$823$	−22.4258	−0.781716	−0.390858	+	0.920451i	$0.627822\pi$
$824$	15.5508	0.541738
$825$	0	0
$826$	3.43501	0.119519
$827$	−19.7303	−0.686090	−0.343045	−	0.939319i	$-0.611458\pi$
$827$	−19.7303	−0.686090	−0.343045	+	0.939319i	$0.611458\pi$
$828$	0	0
$829$	5.70836	0.198259	0.0991297	−	0.995075i	$-0.468394\pi$
$829$	5.70836	0.198259	0.0991297	+	0.995075i	$0.468394\pi$
$830$	0.906791	0.0314752
$831$	0	0
$832$	3.51552	0.121879
$833$	−0.0279270	−0.000967612 0
$834$	0	0
$835$	5.89817	0.204115
$836$	−1.76565	−0.0610664
$837$	0	0
$838$	−6.88441	−0.237818
$839$	0.0666011	0.00229932	0.00114966	−	0.999999i	$-0.499634\pi$
$839$	0.0666011	0.00229932	0.00114966	+	0.999999i	$0.499634\pi$
$840$	0	0
$841$	−4.54521	−0.156731
$842$	−7.72635	−0.266268
$843$	0	0
$844$	12.8158	0.441137
$845$	7.35955	0.253176
$846$	0	0
$847$	2.66942	0.0917222
$848$	25.5316	0.876760
$849$	0	0
$850$	−0.493703	−0.0169339
$851$	68.1193	2.33510
$852$	0	0
$853$	−14.8615	−0.508848	−0.254424	−	0.967093i	$-0.581886\pi$
$853$	−14.8615	−0.508848	−0.254424	+	0.967093i	$0.581886\pi$
$854$	1.04645	0.0358088
$855$	0	0
$856$	−3.00026	−0.102547
$857$	−12.5169	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$857$	−12.5169	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$858$	0	0
$859$	−21.2738	−0.725854	−0.362927	−	0.931818i	$-0.618223\pi$
$859$	−21.2738	−0.725854	−0.362927	+	0.931818i	$0.618223\pi$
$860$	1.72851	0.0589417
$861$	0	0
$862$	11.8651	0.404128
$863$	34.4402	1.17236	0.586179	−	0.810181i	$-0.300631\pi$
$863$	34.4402	1.17236	0.586179	+	0.810181i	$0.300631\pi$
$864$	0	0
$865$	3.43907	0.116932
$866$	9.28680	0.315578
$867$	0	0
$868$	14.4020	0.488836
$869$	−5.69298	−0.193121
$870$	0	0
$871$	2.68549	0.0909944
$872$	30.6370	1.03750
$873$	0	0
$874$	4.59267	0.155349
$875$	16.3283	0.551998
$876$	0	0
$877$	−0.910260	−0.0307373	−0.0153687	−	0.999882i	$-0.504892\pi$
$877$	−0.910260	−0.0307373	−0.0153687	+	0.999882i	$0.504892\pi$
$878$	−6.67857	−0.225391
$879$	0	0
$880$	1.68891	0.0569331
$881$	−33.7502	−1.13707	−0.568537	−	0.822658i	$-0.692490\pi$
$881$	−33.7502	−1.13707	−0.568537	+	0.822658i	$0.692490\pi$
$882$	0	0
$883$	39.8800	1.34207	0.671035	−	0.741426i	$-0.265850\pi$
$883$	39.8800	1.34207	0.671035	+	0.741426i	$0.265850\pi$
$884$	−0.473259	−0.0159174
$885$	0	0
$886$	6.83822	0.229735
$887$	16.1868	0.543500	0.271750	−	0.962368i	$-0.412398\pi$
$887$	16.1868	0.543500	0.271750	+	0.962368i	$0.412398\pi$
$888$	0	0
$889$	49.6228	1.66430
$890$	1.71039	0.0573324
$891$	0	0
$892$	−28.7399	−0.962282
$893$	5.94581	0.198969
$894$	0	0
$895$	0.833474	0.0278599
$896$	30.0735	1.00469
$897$	0	0
$898$	−10.1667	−0.339269
$899$	15.1106	0.503968
$900$	0	0
$901$	−2.14003	−0.0712946
$902$	3.35307	0.111645
$903$	0	0
$904$	20.5894	0.684793
$905$	6.86211	0.228104
$906$	0	0
$907$	−14.3005	−0.474839	−0.237419	−	0.971407i	$-0.576302\pi$
$907$	−14.3005	−0.474839	−0.237419	+	0.971407i	$0.576302\pi$
$908$	−36.9480	−1.22616
$909$	0	0
$910$	−0.994698	−0.0329739
$911$	8.50108	0.281653	0.140827	−	0.990034i	$-0.455024\pi$
$911$	8.50108	0.281653	0.140827	+	0.990034i	$0.455024\pi$
$912$	0	0
$913$	−2.93784	−0.0972285
$914$	10.8738	0.359672
$915$	0	0
$916$	19.0838	0.630545
$917$	8.25111	0.272476
$918$	0	0
$919$	−26.0156	−0.858177	−0.429088	−	0.903263i	$-0.641165\pi$
$919$	−26.0156	−0.858177	−0.429088	+	0.903263i	$0.641165\pi$
$920$	−11.0269	−0.363548
$921$	0	0
$922$	−15.0656	−0.496159
$923$	3.12571	0.102884
$924$	0	0
$925$	−32.9818	−1.08443
$926$	−15.3425	−0.504185
$927$	0	0
$928$	24.3705	0.800001
$929$	−5.99773	−0.196779	−0.0983896	−	0.995148i	$-0.531369\pi$
$929$	−5.99773	−0.196779	−0.0983896	+	0.995148i	$0.531369\pi$
$930$	0	0
$931$	−0.125785	−0.00412242
$932$	−44.8781	−1.47003
$933$	0	0
$934$	−0.849454	−0.0277950
$935$	−0.141562	−0.00462957
$936$	0	0
$937$	−50.5847	−1.65253	−0.826265	−	0.563282i	$-0.809539\pi$
$937$	−50.5847	−1.65253	−0.826265	+	0.563282i	$0.809539\pi$
$938$	2.87456	0.0938578
$939$	0	0
$940$	−6.69370	−0.218324
$941$	21.8640	0.712747	0.356373	−	0.934344i	$-0.384013\pi$
$941$	21.8640	0.712747	0.356373	+	0.934344i	$0.384013\pi$
$942$	0	0
$943$	65.7129	2.13991
$944$	−7.04109	−0.229168
$945$	0	0
$946$	0.743267	0.0241657
$947$	33.1722	1.07795	0.538976	−	0.842321i	$-0.318812\pi$
$947$	33.1722	1.07795	0.538976	+	0.842321i	$0.318812\pi$
$948$	0	0
$949$	−20.0533	−0.650958
$950$	−2.22366	−0.0721451
$951$	0	0
$952$	−1.08039	−0.0350157
$953$	−22.4494	−0.727209	−0.363604	−	0.931553i	$-0.618454\pi$
$953$	−22.4494	−0.727209	−0.363604	+	0.931553i	$0.618454\pi$
$954$	0	0
$955$	9.75794	0.315760
$956$	0.621931	0.0201147
$957$	0	0
$958$	−5.38765	−0.174067
$959$	23.1604	0.747887
$960$	0	0
$961$	−21.6631	−0.698810
$962$	4.19623	0.135292
$963$	0	0
$964$	27.6911	0.891870
$965$	11.6702	0.375676
$966$	0	0
$967$	−0.790013	−0.0254051	−0.0127026	−	0.999919i	$-0.504043\pi$
$967$	−0.790013	−0.0254051	−0.0127026	+	0.999919i	$0.504043\pi$
$968$	1.82293	0.0585911
$969$	0	0
$970$	4.25110	0.136495
$971$	−51.6970	−1.65904	−0.829518	−	0.558479i	$-0.811385\pi$
$971$	−51.6970	−1.65904	−0.829518	+	0.558479i	$0.811385\pi$
$972$	0	0
$973$	38.8201	1.24452
$974$	−1.58302	−0.0507231
$975$	0	0
$976$	−2.14501	−0.0686602
$977$	−21.4536	−0.686360	−0.343180	−	0.939270i	$-0.611504\pi$
$977$	−21.4536	−0.686360	−0.343180	+	0.939270i	$0.611504\pi$
$978$	0	0
$979$	−5.54136	−0.177103
$980$	0.141606	0.00452345
$981$	0	0
$982$	14.1687	0.452142
$983$	42.9916	1.37122	0.685610	−	0.727969i	$-0.259536\pi$
$983$	42.9916	1.37122	0.685610	+	0.727969i	$0.259536\pi$
$984$	0	0
$985$	−0.367368	−0.0117053
$986$	−0.531505	−0.0169266
$987$	0	0
$988$	−2.13158	−0.0678147
$989$	14.5664	0.463186
$990$	0	0
$991$	57.7319	1.83392	0.916958	−	0.398985i	$-0.130637\pi$
$991$	57.7319	1.83392	0.916958	+	0.398985i	$0.130637\pi$
$992$	15.0586	0.478110
$993$	0	0
$994$	3.34578	0.106122
$995$	6.48303	0.205526
$996$	0	0
$997$	−30.3487	−0.961152	−0.480576	−	0.876953i	$-0.659572\pi$
$997$	−30.3487	−0.961152	−0.480576	+	0.876953i	$0.659572\pi$
$998$	0.507466	0.0160635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.k.1.3		5
3.2	odd	2		209.2.a.c.1.3	✓	5
12.11	even	2		3344.2.a.t.1.1		5
15.14	odd	2		5225.2.a.h.1.3		5
33.32	even	2		2299.2.a.n.1.3		5
57.56	even	2		3971.2.a.h.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.3	✓	5	3.2	odd	2
1881.2.a.k.1.3		5	1.1	even	1	trivial
2299.2.a.n.1.3		5	33.32	even	2
3344.2.a.t.1.1		5	12.11	even	2
3971.2.a.h.1.3		5	57.56	even	2
5225.2.a.h.1.3		5	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-0.484093 q^{2} -1.76565 q^{4} -0.637602 q^{5} +2.66942 q^{7} +1.82293 q^{8} +0.308658 q^{10} -1.00000 q^{11} -1.20725 q^{13} -1.29225 q^{14} +2.64884 q^{16} -0.222022 q^{17} -1.00000 q^{19} +1.12578 q^{20} +0.484093 q^{22} +9.48717 q^{23} -4.59346 q^{25} +0.584420 q^{26} -4.71327 q^{28} -4.94518 q^{29} -3.05563 q^{31} -4.92814 q^{32} +0.107479 q^{34} -1.70203 q^{35} +7.18015 q^{37} +0.484093 q^{38} -1.16230 q^{40} +6.92650 q^{41} +1.53538 q^{43} +1.76565 q^{44} -4.59267 q^{46} -5.94581 q^{47} +0.125785 q^{49} +2.22366 q^{50} +2.13158 q^{52} +9.63879 q^{53} +0.637602 q^{55} +4.86615 q^{56} +2.39392 q^{58} -2.65817 q^{59} -0.809792 q^{61} +1.47921 q^{62} -2.91201 q^{64} +0.769744 q^{65} -2.22447 q^{67} +0.392015 q^{68} +0.823938 q^{70} -2.58912 q^{71} +16.6108 q^{73} -3.47586 q^{74} +1.76565 q^{76} -2.66942 q^{77} +5.69298 q^{79} -1.68891 q^{80} -3.35307 q^{82} +2.93784 q^{83} +0.141562 q^{85} -0.743267 q^{86} -1.82293 q^{88} +5.54136 q^{89} -3.22265 q^{91} -16.7511 q^{92} +2.87832 q^{94} +0.637602 q^{95} +13.7728 q^{97} -0.0608914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

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Database

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