LMFDB - Embedded newform 8281.2.a.ci.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.8446345216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 $ x^8 - 8x^6 + 19x^4 - 14*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.282452$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52077 q^{2} +2.12621 q^{3} +0.312752 q^{4} -0.589391 q^{5} -3.23349 q^{6} +2.56592 q^{8} +1.52077 q^{9} +O(q^{10})$	\(q-1.52077 q^{2} +2.12621 q^{3} +0.312752 q^{4} -0.589391 q^{5} -3.23349 q^{6} +2.56592 q^{8} +1.52077 q^{9} +0.896331 q^{10} +1.52077 q^{11} +0.664976 q^{12} -1.25317 q^{15} -4.52769 q^{16} -4.79479 q^{17} -2.31275 q^{18} -1.68391 q^{19} -0.184333 q^{20} -2.31275 q^{22} +1.77394 q^{23} +5.45569 q^{24} -4.65262 q^{25} -3.14515 q^{27} +6.89251 q^{29} +1.90579 q^{30} +6.08640 q^{31} +1.75375 q^{32} +3.23349 q^{33} +7.29179 q^{34} +0.475625 q^{36} +1.40913 q^{37} +2.56085 q^{38} -1.51233 q^{40} -1.35546 q^{41} -11.5596 q^{43} +0.475625 q^{44} -0.896331 q^{45} -2.69777 q^{46} +0.464832 q^{47} -9.62682 q^{48} +7.07558 q^{50} -10.1947 q^{51} +8.24681 q^{53} +4.78306 q^{54} -0.896331 q^{55} -3.58035 q^{57} -10.4819 q^{58} +11.8756 q^{59} -0.391931 q^{60} -2.48017 q^{61} -9.25603 q^{62} +6.38833 q^{64} -4.91740 q^{66} -7.57284 q^{67} -1.49958 q^{68} +3.77178 q^{69} +6.60471 q^{71} +3.90219 q^{72} +16.3712 q^{73} -2.14296 q^{74} -9.89245 q^{75} -0.526647 q^{76} -14.9623 q^{79} +2.66858 q^{80} -11.2496 q^{81} +2.06134 q^{82} -10.1222 q^{83} +2.82601 q^{85} +17.5795 q^{86} +14.6549 q^{87} +3.90219 q^{88} -16.4850 q^{89} +1.36312 q^{90} +0.554804 q^{92} +12.9410 q^{93} -0.706904 q^{94} +0.992484 q^{95} +3.72883 q^{96} +0.973869 q^{97} +2.31275 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9	$ 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{15} + 4 q^{16} - 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} + 8 q^{37} - 32 q^{43} - 4 q^{44} + 4 q^{46} - 36 q^{50} - 44 q^{51} - 4 q^{53} - 48 q^{57} + 48 q^{58} + 64 q^{60} - 32 q^{64} - 20 q^{67} - 8 q^{71} - 28 q^{72} - 76 q^{74} - 4 q^{79} - 56 q^{81} - 36 q^{85} + 4 q^{86} - 28 q^{88} - 80 q^{92} - 8 q^{93} - 52 q^{95} + 28 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^15 + 4 * q^16 - 28 * q^18 - 28 * q^22 - 12 * q^23 - 12 * q^25 - 8 * q^29 - 28 * q^30 - 4 * q^36 + 8 * q^37 - 32 * q^43 - 4 * q^44 + 4 * q^46 - 36 * q^50 - 44 * q^51 - 4 * q^53 - 48 * q^57 + 48 * q^58 + 64 * q^60 - 32 * q^64 - 20 * q^67 - 8 * q^71 - 28 * q^72 - 76 * q^74 - 4 * q^79 - 56 * q^81 - 36 * q^85 + 4 * q^86 - 28 * q^88 - 80 * q^92 - 8 * q^93 - 52 * q^95 + 28 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.52077	−1.07535	−0.537675	−	0.843152i	$-0.680697\pi$
$2$	−1.52077	−1.07535	−0.537675	+	0.843152i	$0.680697\pi$
$3$	2.12621	1.22757	0.613784	−	0.789474i	$-0.289646\pi$
$3$	2.12621	1.22757	0.613784	+	0.789474i	$0.289646\pi$
$4$	0.312752	0.156376
$5$	−0.589391	−0.263584	−0.131792	−	0.991277i	$-0.542073\pi$
$5$	−0.589391	−0.263584	−0.131792	+	0.991277i	$0.542073\pi$
$6$	−3.23349	−1.32006
$7$	0	0
$7$	0	0
$8$	2.56592	0.907190
$9$	1.52077	0.506924
$10$	0.896331	0.283445
$11$	1.52077	0.458530	0.229265	−	0.973364i	$-0.426368\pi$
$11$	1.52077	0.458530	0.229265	+	0.973364i	$0.426368\pi$
$12$	0.664976	0.191962
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.25317	−0.323567
$16$	−4.52769	−1.13192
$17$	−4.79479	−1.16291	−0.581454	−	0.813579i	$-0.697516\pi$
$17$	−4.79479	−1.16291	−0.581454	+	0.813579i	$0.697516\pi$
$18$	−2.31275	−0.545121
$19$	−1.68391	−0.386316	−0.193158	−	0.981168i	$-0.561873\pi$
$19$	−1.68391	−0.386316	−0.193158	+	0.981168i	$0.561873\pi$
$20$	−0.184333	−0.0412182
$21$	0	0
$22$	−2.31275	−0.493080
$23$	1.77394	0.369893	0.184946	−	0.982749i	$-0.440789\pi$
$23$	1.77394	0.369893	0.184946	+	0.982749i	$0.440789\pi$
$24$	5.45569	1.11364
$25$	−4.65262	−0.930524
$26$	0	0
$27$	−3.14515	−0.605284
$28$	0	0
$29$	6.89251	1.27991	0.639953	−	0.768414i	$-0.278954\pi$
$29$	6.89251	1.27991	0.639953	+	0.768414i	$0.278954\pi$
$30$	1.90579	0.347948
$31$	6.08640	1.09315	0.546575	−	0.837410i	$-0.315931\pi$
$31$	6.08640	1.09315	0.546575	+	0.837410i	$0.315931\pi$
$32$	1.75375	0.310021
$33$	3.23349	0.562878
$34$	7.29179	1.25053
$35$	0	0
$36$	0.475625	0.0792708
$37$	1.40913	0.231659	0.115830	−	0.993269i	$-0.463047\pi$
$37$	1.40913	0.231659	0.115830	+	0.993269i	$0.463047\pi$
$38$	2.56085	0.415425
$39$	0	0
$40$	−1.51233	−0.239121
$41$	−1.35546	−0.211687	−0.105843	−	0.994383i	$-0.533754\pi$
$41$	−1.35546	−0.211687	−0.105843	+	0.994383i	$0.533754\pi$
$42$	0	0
$43$	−11.5596	−1.76282	−0.881408	−	0.472356i	$-0.843404\pi$
$43$	−11.5596	−1.76282	−0.881408	+	0.472356i	$0.843404\pi$
$44$	0.475625	0.0717031
$45$	−0.896331	−0.133617
$46$	−2.69777	−0.397764
$47$	0.464832	0.0678027	0.0339013	−	0.999425i	$-0.489207\pi$
$47$	0.464832	0.0678027	0.0339013	+	0.999425i	$0.489207\pi$
$48$	−9.62682	−1.38951
$49$	0	0
$50$	7.07558	1.00064
$51$	−10.1947	−1.42755
$52$	0	0
$53$	8.24681	1.13279	0.566393	−	0.824135i	$-0.308338\pi$
$53$	8.24681	1.13279	0.566393	+	0.824135i	$0.308338\pi$
$54$	4.78306	0.650892
$55$	−0.896331	−0.120861
$56$	0	0
$57$	−3.58035	−0.474230
$58$	−10.4819	−1.37635
$59$	11.8756	1.54608	0.773038	−	0.634359i	$-0.218736\pi$
$59$	11.8756	1.54608	0.773038	+	0.634359i	$0.218736\pi$
$60$	−0.391931	−0.0505981
$61$	−2.48017	−0.317553	−0.158777	−	0.987315i	$-0.550755\pi$
$61$	−2.48017	−0.317553	−0.158777	+	0.987315i	$0.550755\pi$
$62$	−9.25603	−1.17552
$63$	0	0
$64$	6.38833	0.798541
$65$	0	0
$66$	−4.91740	−0.605290
$67$	−7.57284	−0.925169	−0.462585	−	0.886575i	$-0.653078\pi$
$67$	−7.57284	−0.925169	−0.462585	+	0.886575i	$0.653078\pi$
$68$	−1.49958	−0.181851
$69$	3.77178	0.454069
$70$	0	0
$71$	6.60471	0.783834	0.391917	−	0.920000i	$-0.371812\pi$
$71$	6.60471	0.783834	0.391917	+	0.920000i	$0.371812\pi$
$72$	3.90219	0.459877
$73$	16.3712	1.91610	0.958049	−	0.286604i	$-0.0925263\pi$
$73$	16.3712	1.91610	0.958049	+	0.286604i	$0.0925263\pi$
$74$	−2.14296	−0.249114
$75$	−9.89245	−1.14228
$76$	−0.526647	−0.0604105
$77$	0	0
$78$	0	0
$79$	−14.9623	−1.68339	−0.841696	−	0.539951i	$-0.818443\pi$
$79$	−14.9623	−1.68339	−0.841696	+	0.539951i	$0.818443\pi$
$80$	2.66858	0.298356
$81$	−11.2496	−1.24995
$82$	2.06134	0.227637
$83$	−10.1222	−1.11105	−0.555526	−	0.831499i	$-0.687483\pi$
$83$	−10.1222	−1.11105	−0.555526	+	0.831499i	$0.687483\pi$
$84$	0	0
$85$	2.82601	0.306524
$86$	17.5795	1.89564
$87$	14.6549	1.57117
$88$	3.90219	0.415974
$89$	−16.4850	−1.74741	−0.873703	−	0.486460i	$-0.838288\pi$
$89$	−16.4850	−1.74741	−0.873703	+	0.486460i	$0.838288\pi$
$90$	1.36312	0.143685
$91$	0	0
$92$	0.554804	0.0578423
$93$	12.9410	1.34192
$94$	−0.706904	−0.0729115
$95$	0.992484	0.101827
$96$	3.72883	0.380573
$97$	0.973869	0.0988814	0.0494407	−	0.998777i	$-0.484256\pi$
$97$	0.973869	0.0988814	0.0494407	+	0.998777i	$0.484256\pi$
$98$	0	0
$99$	2.31275	0.232440
$100$	−1.45511	−0.145511
$101$	2.94023	0.292564	0.146282	−	0.989243i	$-0.453269\pi$
$101$	2.94023	0.292564	0.146282	+	0.989243i	$0.453269\pi$
$102$	15.5039	1.53511
$103$	−0.528682	−0.0520926	−0.0260463	−	0.999661i	$-0.508292\pi$
$103$	−0.528682	−0.0520926	−0.0260463	+	0.999661i	$0.508292\pi$
$104$	0	0
$105$	0	0
$106$	−12.5415	−1.21814
$107$	−19.3224	−1.86797	−0.933983	−	0.357318i	$-0.883691\pi$
$107$	−19.3224	−1.86797	−0.933983	+	0.357318i	$0.883691\pi$
$108$	−0.983651	−0.0946518
$109$	−5.73240	−0.549064	−0.274532	−	0.961578i	$-0.588523\pi$
$109$	−5.73240	−0.549064	−0.274532	+	0.961578i	$0.588523\pi$
$110$	1.36312	0.129968
$111$	2.99610	0.284377
$112$	0	0
$113$	5.14959	0.484433	0.242216	−	0.970222i	$-0.422126\pi$
$113$	5.14959	0.484433	0.242216	+	0.970222i	$0.422126\pi$
$114$	5.44491	0.509962
$115$	−1.04555	−0.0974978
$116$	2.15564	0.200147
$117$	0	0
$118$	−18.0602	−1.66257
$119$	0	0
$120$	−3.21554	−0.293537
$121$	−8.68725	−0.789750
$122$	3.77178	0.341481
$123$	−2.88199	−0.259860
$124$	1.90353	0.170942
$125$	5.68917	0.508855
$126$	0	0
$127$	−9.00331	−0.798915	−0.399457	−	0.916752i	$-0.630801\pi$
$127$	−9.00331	−0.798915	−0.399457	+	0.916752i	$0.630801\pi$
$128$	−13.2227	−1.16873
$129$	−24.5781	−2.16398
$130$	0	0
$131$	−6.79840	−0.593979	−0.296989	−	0.954881i	$-0.595983\pi$
$131$	−6.79840	−0.593979	−0.296989	+	0.954881i	$0.595983\pi$
$132$	1.01128	0.0880205
$133$	0	0
$134$	11.5166	0.994880
$135$	1.85372	0.159543
$136$	−12.3031	−1.05498
$137$	−14.0756	−1.20256	−0.601279	−	0.799039i	$-0.705342\pi$
$137$	−14.0756	−1.20256	−0.601279	+	0.799039i	$0.705342\pi$
$138$	−5.73602	−0.488283
$139$	−17.2863	−1.46620	−0.733101	−	0.680120i	$-0.761928\pi$
$139$	−17.2863	−1.46620	−0.733101	+	0.680120i	$0.761928\pi$
$140$	0	0
$141$	0.988330	0.0832324
$142$	−10.0443	−0.842896
$143$	0	0
$144$	−6.88559	−0.573799
$145$	−4.06238	−0.337363
$146$	−24.8968	−2.06048
$147$	0	0
$148$	0.440707	0.0362259
$149$	−17.1252	−1.40295	−0.701475	−	0.712694i	$-0.747475\pi$
$149$	−17.1252	−1.40295	−0.701475	+	0.712694i	$0.747475\pi$
$150$	15.0442	1.22835
$151$	15.7975	1.28558	0.642789	−	0.766043i	$-0.277777\pi$
$151$	15.7975	1.28558	0.642789	+	0.766043i	$0.277777\pi$
$152$	−4.32079	−0.350462
$153$	−7.29179	−0.589507
$154$	0	0
$155$	−3.58727	−0.288137
$156$	0	0
$157$	3.78351	0.301957	0.150979	−	0.988537i	$-0.451758\pi$
$157$	3.78351	0.301957	0.150979	+	0.988537i	$0.451758\pi$
$158$	22.7543	1.81023
$159$	17.5344	1.39057
$160$	−1.03364	−0.0817166
$161$	0	0
$162$	17.1080	1.34413
$163$	1.71551	0.134369	0.0671847	−	0.997741i	$-0.478598\pi$
$163$	1.71551	0.134369	0.0671847	+	0.997741i	$0.478598\pi$
$164$	−0.423922	−0.0331027
$165$	−1.90579	−0.148365
$166$	15.3935	1.19477
$167$	−12.6521	−0.979049	−0.489524	−	0.871990i	$-0.662830\pi$
$167$	−12.6521	−0.979049	−0.489524	+	0.871990i	$0.662830\pi$
$168$	0	0
$169$	0	0
$170$	−4.29772	−0.329620
$171$	−2.56085	−0.195833
$172$	−3.61527	−0.275662
$173$	11.4874	0.873372	0.436686	−	0.899614i	$-0.356152\pi$
$173$	11.4874	0.873372	0.436686	+	0.899614i	$0.356152\pi$
$174$	−22.2868	−1.68956
$175$	0	0
$176$	−6.88559	−0.519021
$177$	25.2501	1.89792
$178$	25.0699	1.87907
$179$	−2.18451	−0.163278	−0.0816389	−	0.996662i	$-0.526015\pi$
$179$	−2.18451	−0.163278	−0.0816389	+	0.996662i	$0.526015\pi$
$180$	−0.280329	−0.0208945
$181$	11.5981	0.862081	0.431041	−	0.902333i	$-0.358147\pi$
$181$	11.5981	0.862081	0.431041	+	0.902333i	$0.358147\pi$
$182$	0	0
$183$	−5.27337	−0.389819
$184$	4.55180	0.335563
$185$	−0.830527	−0.0610616
$186$	−19.6803	−1.44303
$187$	−7.29179	−0.533229
$188$	0.145377	0.0106027
$189$	0	0
$190$	−1.50934	−0.109499
$191$	17.7592	1.28501	0.642506	−	0.766281i	$-0.277895\pi$
$191$	17.7592	1.28501	0.642506	+	0.766281i	$0.277895\pi$
$192$	13.5829	0.980264
$193$	22.6379	1.62951	0.814756	−	0.579804i	$-0.196871\pi$
$193$	22.6379	1.62951	0.814756	+	0.579804i	$0.196871\pi$
$194$	−1.48103	−0.106332
$195$	0	0
$196$	0	0
$197$	20.0063	1.42539	0.712696	−	0.701474i	$-0.247474\pi$
$197$	20.0063	1.42539	0.712696	+	0.701474i	$0.247474\pi$
$198$	−3.51717	−0.249954
$199$	1.84885	0.131062	0.0655309	−	0.997851i	$-0.479126\pi$
$199$	1.84885	0.131062	0.0655309	+	0.997851i	$0.479126\pi$
$200$	−11.9383	−0.844162
$201$	−16.1015	−1.13571
$202$	−4.47143	−0.314609
$203$	0	0
$204$	−3.18842	−0.223234
$205$	0.798895	0.0557972
$206$	0.804005	0.0560177
$207$	2.69777	0.187508
$208$	0	0
$209$	−2.56085	−0.177138
$210$	0	0
$211$	−16.1695	−1.11315	−0.556576	−	0.830796i	$-0.687885\pi$
$211$	−16.1695	−1.11315	−0.556576	+	0.830796i	$0.687885\pi$
$212$	2.57920	0.177140
$213$	14.0430	0.962210
$214$	29.3850	2.00871
$215$	6.81310	0.464650
$216$	−8.07020	−0.549108
$217$	0	0
$218$	8.71768	0.590436
$219$	34.8085	2.35214
$220$	−0.280329	−0.0188998
$221$	0	0
$222$	−4.55639	−0.305805
$223$	12.4318	0.832494	0.416247	−	0.909252i	$-0.363345\pi$
$223$	12.4318	0.832494	0.416247	+	0.909252i	$0.363345\pi$
$224$	0	0
$225$	−7.07558	−0.471705
$226$	−7.83136	−0.520934
$227$	−1.23497	−0.0819681	−0.0409841	−	0.999160i	$-0.513049\pi$
$227$	−1.23497	−0.0819681	−0.0409841	+	0.999160i	$0.513049\pi$
$228$	−1.11976	−0.0741581
$229$	−6.55514	−0.433176	−0.216588	−	0.976263i	$-0.569493\pi$
$229$	−6.55514	−0.433176	−0.216588	+	0.976263i	$0.569493\pi$
$230$	1.59004	0.104844
$231$	0	0
$232$	17.6856	1.16112
$233$	6.29887	0.412653	0.206326	−	0.978483i	$-0.433849\pi$
$233$	6.29887	0.412653	0.206326	+	0.978483i	$0.433849\pi$
$234$	0	0
$235$	−0.273968	−0.0178717
$236$	3.71413	0.241769
$237$	−31.8130	−2.06648
$238$	0	0
$239$	−18.9193	−1.22379	−0.611895	−	0.790939i	$-0.709592\pi$
$239$	−18.9193	−1.22379	−0.611895	+	0.790939i	$0.709592\pi$
$240$	5.67397	0.366253
$241$	−22.2968	−1.43626	−0.718131	−	0.695908i	$-0.755002\pi$
$241$	−22.2968	−1.43626	−0.718131	+	0.695908i	$0.755002\pi$
$242$	13.2113	0.849257
$243$	−14.4835	−0.929118
$244$	−0.775678	−0.0496577
$245$	0	0
$246$	4.38285	0.279440
$247$	0	0
$248$	15.6172	0.991695
$249$	−21.5219	−1.36389
$250$	−8.65194	−0.547197
$251$	6.95315	0.438879	0.219439	−	0.975626i	$-0.429577\pi$
$251$	6.95315	0.438879	0.219439	+	0.975626i	$0.429577\pi$
$252$	0	0
$253$	2.69777	0.169607
$254$	13.6920	0.859113
$255$	6.00869	0.376279
$256$	7.33206	0.458254
$257$	−21.1551	−1.31962	−0.659811	−	0.751432i	$-0.729364\pi$
$257$	−21.1551	−1.31962	−0.659811	+	0.751432i	$0.729364\pi$
$258$	37.3777	2.32703
$259$	0	0
$260$	0	0
$261$	10.4819	0.648816
$262$	10.3388	0.638735
$263$	−8.42992	−0.519811	−0.259906	−	0.965634i	$-0.583691\pi$
$263$	−8.42992	−0.519811	−0.259906	+	0.965634i	$0.583691\pi$
$264$	8.29687	0.510637
$265$	−4.86060	−0.298584
$266$	0	0
$267$	−35.0506	−2.14506
$268$	−2.36842	−0.144674
$269$	5.83039	0.355485	0.177743	−	0.984077i	$-0.443121\pi$
$269$	5.83039	0.355485	0.177743	+	0.984077i	$0.443121\pi$
$270$	−2.81909	−0.171565
$271$	−18.4299	−1.11954	−0.559769	−	0.828648i	$-0.689110\pi$
$271$	−18.4299	−1.11954	−0.559769	+	0.828648i	$0.689110\pi$
$272$	21.7093	1.31632
$273$	0	0
$274$	21.4058	1.29317
$275$	−7.07558	−0.426673
$276$	1.17963	0.0710054
$277$	−6.18307	−0.371505	−0.185752	−	0.982597i	$-0.559472\pi$
$277$	−6.18307	−0.371505	−0.185752	+	0.982597i	$0.559472\pi$
$278$	26.2885	1.57668
$279$	9.25603	0.554144
$280$	0	0
$281$	−5.64049	−0.336483	−0.168242	−	0.985746i	$-0.553809\pi$
$281$	−5.64049	−0.336483	−0.168242	+	0.985746i	$0.553809\pi$
$282$	−1.50303	−0.0895039
$283$	16.4554	0.978173	0.489086	−	0.872235i	$-0.337330\pi$
$283$	16.4554	0.978173	0.489086	+	0.872235i	$0.337330\pi$
$284$	2.06563	0.122573
$285$	2.11023	0.124999
$286$	0	0
$287$	0	0
$288$	2.66705	0.157157
$289$	5.99003	0.352355
$290$	6.17797	0.362783
$291$	2.07065	0.121384
$292$	5.12011	0.299632
$293$	−30.6171	−1.78867	−0.894335	−	0.447397i	$-0.852351\pi$
$293$	−30.6171	−1.78867	−0.894335	+	0.447397i	$0.852351\pi$
$294$	0	0
$295$	−6.99940	−0.407521
$296$	3.61571	0.210159
$297$	−4.78306	−0.277541
$298$	26.0435	1.50866
$299$	0	0
$300$	−3.09388	−0.178625
$301$	0	0
$302$	−24.0244	−1.38245
$303$	6.25156	0.359143
$304$	7.62424	0.437280
$305$	1.46179	0.0837019
$306$	11.0892	0.633925
$307$	9.96020	0.568459	0.284229	−	0.958756i	$-0.408262\pi$
$307$	9.96020	0.568459	0.284229	+	0.958756i	$0.408262\pi$
$308$	0	0
$309$	−1.12409	−0.0639472
$310$	5.45543	0.309847
$311$	−27.7468	−1.57337	−0.786687	−	0.617352i	$-0.788205\pi$
$311$	−27.7468	−1.57337	−0.786687	+	0.617352i	$0.788205\pi$
$312$	0	0
$313$	16.5227	0.933920	0.466960	−	0.884278i	$-0.345349\pi$
$313$	16.5227	0.933920	0.466960	+	0.884278i	$0.345349\pi$
$314$	−5.75387	−0.324710
$315$	0	0
$316$	−4.67949	−0.263242
$317$	23.6793	1.32996	0.664980	−	0.746861i	$-0.268440\pi$
$317$	23.6793	1.32996	0.664980	+	0.746861i	$0.268440\pi$
$318$	−26.6659	−1.49535
$319$	10.4819	0.586876
$320$	−3.76523	−0.210483
$321$	−41.0835	−2.29306
$322$	0	0
$323$	8.07401	0.449250
$324$	−3.51832	−0.195462
$325$	0	0
$326$	−2.60891	−0.144494
$327$	−12.1883	−0.674014
$328$	−3.47800	−0.192040
$329$	0	0
$330$	2.89827	0.159545
$331$	7.95209	0.437086	0.218543	−	0.975827i	$-0.429870\pi$
$331$	7.95209	0.437086	0.218543	+	0.975827i	$0.429870\pi$
$332$	−3.16573	−0.173742
$333$	2.14296	0.117434
$334$	19.2410	1.05282
$335$	4.46337	0.243860
$336$	0	0
$337$	−7.91326	−0.431063	−0.215531	−	0.976497i	$-0.569148\pi$
$337$	−7.91326	−0.431063	−0.215531	+	0.976497i	$0.569148\pi$
$338$	0	0
$339$	10.9491	0.594674
$340$	0.883839	0.0479329
$341$	9.25603	0.501242
$342$	3.89447	0.210589
$343$	0	0
$344$	−29.6609	−1.59921
$345$	−2.22305	−0.119685
$346$	−17.4698	−0.939180
$347$	7.13571	0.383065	0.191533	−	0.981486i	$-0.438654\pi$
$347$	7.13571	0.383065	0.191533	+	0.981486i	$0.438654\pi$
$348$	4.58335	0.245694
$349$	1.37680	0.0736987	0.0368493	−	0.999321i	$-0.488268\pi$
$349$	1.37680	0.0736987	0.0368493	+	0.999321i	$0.488268\pi$
$350$	0	0
$351$	0	0
$352$	2.66705	0.142154
$353$	0.693215	0.0368961	0.0184481	−	0.999830i	$-0.494127\pi$
$353$	0.693215	0.0368961	0.0184481	+	0.999830i	$0.494127\pi$
$354$	−38.3997	−2.04092
$355$	−3.89276	−0.206606
$356$	−5.15571	−0.273252
$357$	0	0
$358$	3.32214	0.175581
$359$	−5.80365	−0.306305	−0.153152	−	0.988203i	$-0.548943\pi$
$359$	−5.80365	−0.306305	−0.153152	+	0.988203i	$0.548943\pi$
$360$	−2.29991	−0.121216
$361$	−16.1644	−0.850760
$362$	−17.6381	−0.927038
$363$	−18.4709	−0.969472
$364$	0	0
$365$	−9.64902	−0.505053
$366$	8.01960	0.419191
$367$	7.35157	0.383749	0.191874	−	0.981420i	$-0.438543\pi$
$367$	7.35157	0.383749	0.191874	+	0.981420i	$0.438543\pi$
$368$	−8.03187	−0.418690
$369$	−2.06134	−0.107309
$370$	1.26304	0.0656625
$371$	0	0
$372$	4.04731	0.209843
$373$	18.3988	0.952656	0.476328	−	0.879268i	$-0.341967\pi$
$373$	18.3988	0.952656	0.476328	+	0.879268i	$0.341967\pi$
$374$	11.0892	0.573407
$375$	12.0964	0.624654
$376$	1.19272	0.0615099
$377$	0	0
$378$	0	0
$379$	−4.85101	−0.249180	−0.124590	−	0.992208i	$-0.539761\pi$
$379$	−4.85101	−0.249180	−0.124590	+	0.992208i	$0.539761\pi$
$380$	0.310401	0.0159232
$381$	−19.1429	−0.980723
$382$	−27.0078	−1.38184
$383$	−22.8205	−1.16607	−0.583037	−	0.812446i	$-0.698136\pi$
$383$	−22.8205	−1.16607	−0.583037	+	0.812446i	$0.698136\pi$
$384$	−28.1142	−1.43470
$385$	0	0
$386$	−34.4271	−1.75229
$387$	−17.5795	−0.893615
$388$	0.304579	0.0154627
$389$	−20.3122	−1.02987	−0.514933	−	0.857230i	$-0.672183\pi$
$389$	−20.3122	−1.02987	−0.514933	+	0.857230i	$0.672183\pi$
$390$	0	0
$391$	−8.50569	−0.430151
$392$	0	0
$393$	−14.4548	−0.729150
$394$	−30.4251	−1.53279
$395$	8.81866	0.443715
$396$	0.723317	0.0363481
$397$	−34.1377	−1.71332	−0.856662	−	0.515878i	$-0.827466\pi$
$397$	−34.1377	−1.71332	−0.856662	+	0.515878i	$0.827466\pi$
$398$	−2.81169	−0.140937
$399$	0	0
$400$	21.0656	1.05328
$401$	−3.02596	−0.151109	−0.0755547	−	0.997142i	$-0.524073\pi$
$401$	−3.02596	−0.151109	−0.0755547	+	0.997142i	$0.524073\pi$
$402$	24.4867	1.22128
$403$	0	0
$404$	0.919564	0.0457500
$405$	6.63040	0.329467
$406$	0	0
$407$	2.14296	0.106223
$408$	−26.1589	−1.29506
$409$	5.38325	0.266184	0.133092	−	0.991104i	$-0.457509\pi$
$409$	5.38325	0.266184	0.133092	+	0.991104i	$0.457509\pi$
$410$	−1.21494	−0.0600015
$411$	−29.9276	−1.47622
$412$	−0.165346	−0.00814602
$413$	0	0
$414$	−4.10269	−0.201636
$415$	5.96592	0.292856
$416$	0	0
$417$	−36.7543	−1.79986
$418$	3.89447	0.190485
$419$	5.88235	0.287371	0.143686	−	0.989623i	$-0.454105\pi$
$419$	5.88235	0.287371	0.143686	+	0.989623i	$0.454105\pi$
$420$	0	0
$421$	−28.7614	−1.40174	−0.700872	−	0.713287i	$-0.747206\pi$
$421$	−28.7614	−1.40174	−0.700872	+	0.713287i	$0.747206\pi$
$422$	24.5901	1.19703
$423$	0.706904	0.0343708
$424$	21.1607	1.02765
$425$	22.3083	1.08211
$426$	−21.3562	−1.03471
$427$	0	0
$428$	−6.04311	−0.292105
$429$	0	0
$430$	−10.3612	−0.499661
$431$	8.38588	0.403934	0.201967	−	0.979392i	$-0.435267\pi$
$431$	8.38588	0.403934	0.201967	+	0.979392i	$0.435267\pi$
$432$	14.2403	0.685135
$433$	−27.5993	−1.32634	−0.663168	−	0.748471i	$-0.730788\pi$
$433$	−27.5993	−1.32634	−0.663168	+	0.748471i	$0.730788\pi$
$434$	0	0
$435$	−8.63749	−0.414136
$436$	−1.79282	−0.0858604
$437$	−2.98717	−0.142896
$438$	−52.9359	−2.52937
$439$	31.1737	1.48784	0.743921	−	0.668268i	$-0.232964\pi$
$439$	31.1737	1.48784	0.743921	+	0.668268i	$0.232964\pi$
$440$	−2.29991	−0.109644
$441$	0	0
$442$	0	0
$443$	−23.5883	−1.12071	−0.560357	−	0.828251i	$-0.689336\pi$
$443$	−23.5883	−1.12071	−0.560357	+	0.828251i	$0.689336\pi$
$444$	0.937036	0.0444698
$445$	9.71611	0.460588
$446$	−18.9059	−0.895221
$447$	−36.4118	−1.72222
$448$	0	0
$449$	2.82656	0.133394	0.0666968	−	0.997773i	$-0.478754\pi$
$449$	2.82656	0.133394	0.0666968	+	0.997773i	$0.478754\pi$
$450$	10.7604	0.507248
$451$	−2.06134	−0.0970649
$452$	1.61054	0.0757536
$453$	33.5887	1.57814
$454$	1.87812	0.0881444
$455$	0	0
$456$	−9.18691	−0.430217
$457$	−37.7432	−1.76555	−0.882776	−	0.469795i	$-0.844328\pi$
$457$	−37.7432	−1.76555	−0.882776	+	0.469795i	$0.844328\pi$
$458$	9.96888	0.465815
$459$	15.0803	0.703890
$460$	−0.326997	−0.0152463
$461$	34.6586	1.61421	0.807106	−	0.590407i	$-0.201033\pi$
$461$	34.6586	1.61421	0.807106	+	0.590407i	$0.201033\pi$
$462$	0	0
$463$	18.5114	0.860296	0.430148	−	0.902758i	$-0.358461\pi$
$463$	18.5114	0.860296	0.430148	+	0.902758i	$0.358461\pi$
$464$	−31.2071	−1.44875
$465$	−7.62730	−0.353707
$466$	−9.57916	−0.443746
$467$	6.62783	0.306700	0.153350	−	0.988172i	$-0.450994\pi$
$467$	6.62783	0.306700	0.153350	+	0.988172i	$0.450994\pi$
$468$	0	0
$469$	0	0
$470$	0.416643	0.0192183
$471$	8.04455	0.370673
$472$	30.4720	1.40259
$473$	−17.5795	−0.808305
$474$	48.3804	2.22219
$475$	7.83460	0.359476
$476$	0	0
$477$	12.5415	0.574237
$478$	28.7720	1.31600
$479$	−17.4526	−0.797430	−0.398715	−	0.917075i	$-0.630544\pi$
$479$	−17.4526	−0.797430	−0.398715	+	0.917075i	$0.630544\pi$
$480$	−2.19774	−0.100313
$481$	0	0
$482$	33.9083	1.54448
$483$	0	0
$484$	−2.71695	−0.123498
$485$	−0.573990	−0.0260635
$486$	22.0261	0.999126
$487$	−35.5138	−1.60928	−0.804641	−	0.593762i	$-0.797642\pi$
$487$	−35.5138	−1.60928	−0.804641	+	0.593762i	$0.797642\pi$
$488$	−6.36393	−0.288081
$489$	3.64754	0.164948
$490$	0	0
$491$	−29.8118	−1.34539	−0.672695	−	0.739920i	$-0.734863\pi$
$491$	−29.8118	−1.34539	−0.672695	+	0.739920i	$0.734863\pi$
$492$	−0.901347	−0.0406359
$493$	−33.0481	−1.48841
$494$	0	0
$495$	−1.36312	−0.0612675
$496$	−27.5573	−1.23736
$497$	0	0
$498$	32.7299	1.46666
$499$	−7.50966	−0.336178	−0.168089	−	0.985772i	$-0.553760\pi$
$499$	−7.50966	−0.336178	−0.168089	+	0.985772i	$0.553760\pi$
$500$	1.77930	0.0795726
$501$	−26.9010	−1.20185
$502$	−10.5742	−0.471948
$503$	0.984343	0.0438897	0.0219448	−	0.999759i	$-0.493014\pi$
$503$	0.984343	0.0438897	0.0219448	+	0.999759i	$0.493014\pi$
$504$	0	0
$505$	−1.73295	−0.0771152
$506$	−4.10269	−0.182387
$507$	0	0
$508$	−2.81580	−0.124931
$509$	12.9792	0.575291	0.287646	−	0.957737i	$-0.407127\pi$
$509$	12.9792	0.575291	0.287646	+	0.957737i	$0.407127\pi$
$510$	−9.13786	−0.404631
$511$	0	0
$512$	15.2950	0.675949
$513$	5.29616	0.233831
$514$	32.1722	1.41905
$515$	0.311600	0.0137308
$516$	−7.68683	−0.338394
$517$	0.706904	0.0310896
$518$	0	0
$519$	24.4247	1.07212
$520$	0	0
$521$	−19.4146	−0.850569	−0.425285	−	0.905060i	$-0.639826\pi$
$521$	−19.4146	−0.850569	−0.425285	+	0.905060i	$0.639826\pi$
$522$	−15.9407	−0.697704
$523$	27.2719	1.19252	0.596259	−	0.802792i	$-0.296653\pi$
$523$	27.2719	1.19252	0.596259	+	0.802792i	$0.296653\pi$
$524$	−2.12621	−0.0928840
$525$	0	0
$526$	12.8200	0.558979
$527$	−29.1830	−1.27123
$528$	−14.6402	−0.637134
$529$	−19.8531	−0.863179
$530$	7.39187	0.321082
$531$	18.0602	0.783744
$532$	0	0
$533$	0	0
$534$	53.3040	2.30669
$535$	11.3884	0.492365
$536$	−19.4313	−0.839305
$537$	−4.64473	−0.200435
$538$	−8.86670	−0.382271
$539$	0	0
$540$	0.579755	0.0249487
$541$	−30.0990	−1.29406	−0.647029	−	0.762465i	$-0.723989\pi$
$541$	−30.0990	−1.29406	−0.647029	+	0.762465i	$0.723989\pi$
$542$	28.0278	1.20389
$543$	24.6600	1.05826
$544$	−8.40885	−0.360526
$545$	3.37863	0.144724
$546$	0	0
$547$	−26.1451	−1.11788	−0.558942	−	0.829207i	$-0.688793\pi$
$547$	−26.1451	−1.11788	−0.558942	+	0.829207i	$0.688793\pi$
$548$	−4.40216	−0.188051
$549$	−3.77178	−0.160976
$550$	10.7604	0.458823
$551$	−11.6064	−0.494449
$552$	9.67809	0.411927
$553$	0	0
$554$	9.40305	0.399497
$555$	−1.76588	−0.0749573
$556$	−5.40631	−0.229279
$557$	17.9063	0.758716	0.379358	−	0.925250i	$-0.376145\pi$
$557$	17.9063	0.758716	0.379358	+	0.925250i	$0.376145\pi$
$558$	−14.0763	−0.595899
$559$	0	0
$560$	0	0
$561$	−15.5039	−0.654575
$562$	8.57790	0.361837
$563$	−31.6549	−1.33410	−0.667048	−	0.745015i	$-0.732442\pi$
$563$	−31.6549	−1.33410	−0.667048	+	0.745015i	$0.732442\pi$
$564$	0.309102	0.0130155
$565$	−3.03512	−0.127689
$566$	−25.0250	−1.05188
$567$	0	0
$568$	16.9472	0.711087
$569$	−26.1111	−1.09463	−0.547317	−	0.836925i	$-0.684351\pi$
$569$	−26.1111	−1.09463	−0.547317	+	0.836925i	$0.684351\pi$
$570$	−3.20918	−0.134418
$571$	13.3041	0.556760	0.278380	−	0.960471i	$-0.410203\pi$
$571$	13.3041	0.556760	0.278380	+	0.960471i	$0.410203\pi$
$572$	0	0
$573$	37.7599	1.57744
$574$	0	0
$575$	−8.25348	−0.344194
$576$	9.71520	0.404800
$577$	−16.7713	−0.698198	−0.349099	−	0.937086i	$-0.613512\pi$
$577$	−16.7713	−0.698198	−0.349099	+	0.937086i	$0.613512\pi$
$578$	−9.10948	−0.378905
$579$	48.1329	2.00034
$580$	−1.27052	−0.0527554
$581$	0	0
$582$	−3.14899	−0.130530
$583$	12.5415	0.519417
$584$	42.0071	1.73827
$585$	0	0
$586$	46.5617	1.92345
$587$	−10.0652	−0.415436	−0.207718	−	0.978189i	$-0.566604\pi$
$587$	−10.0652	−0.415436	−0.207718	+	0.978189i	$0.566604\pi$
$588$	0	0
$589$	−10.2490	−0.422301
$590$	10.6445	0.438227
$591$	42.5377	1.74977
$592$	−6.38009	−0.262220
$593$	39.4322	1.61929	0.809643	−	0.586923i	$-0.199661\pi$
$593$	39.4322	1.61929	0.809643	+	0.586923i	$0.199661\pi$
$594$	7.27395	0.298454
$595$	0	0
$596$	−5.35593	−0.219388
$597$	3.93105	0.160887
$598$	0	0
$599$	13.7720	0.562709	0.281355	−	0.959604i	$-0.409216\pi$
$599$	13.7720	0.562709	0.281355	+	0.959604i	$0.409216\pi$
$600$	−25.3833	−1.03627
$601$	33.2623	1.35680	0.678399	−	0.734694i	$-0.262674\pi$
$601$	33.2623	1.35680	0.678399	+	0.734694i	$0.262674\pi$
$602$	0	0
$603$	−11.5166	−0.468991
$604$	4.94068	0.201034
$605$	5.12019	0.208165
$606$	−9.50720	−0.386204
$607$	43.9649	1.78448	0.892240	−	0.451562i	$-0.149133\pi$
$607$	43.9649	1.78448	0.892240	+	0.451562i	$0.149133\pi$
$608$	−2.95316	−0.119766
$609$	0	0
$610$	−2.22305	−0.0900088
$611$	0	0
$612$	−2.28052	−0.0921846
$613$	2.70091	0.109089	0.0545443	−	0.998511i	$-0.482629\pi$
$613$	2.70091	0.109089	0.0545443	+	0.998511i	$0.482629\pi$
$614$	−15.1472	−0.611292
$615$	1.69862	0.0684949
$616$	0	0
$617$	6.00415	0.241718	0.120859	−	0.992670i	$-0.461435\pi$
$617$	6.00415	0.241718	0.120859	+	0.992670i	$0.461435\pi$
$618$	1.70948	0.0687655
$619$	−13.3641	−0.537148	−0.268574	−	0.963259i	$-0.586552\pi$
$619$	−13.3641	−0.537148	−0.268574	+	0.963259i	$0.586552\pi$
$620$	−1.12193	−0.0450576
$621$	−5.57932	−0.223890
$622$	42.1965	1.69193
$623$	0	0
$624$	0	0
$625$	19.9099	0.796398
$626$	−25.1273	−1.00429
$627$	−5.44491	−0.217449
$628$	1.18330	0.0472188
$629$	−6.75647	−0.269398
$630$	0	0
$631$	40.4457	1.61012	0.805059	−	0.593195i	$-0.202134\pi$
$631$	40.4457	1.61012	0.805059	+	0.593195i	$0.202134\pi$
$632$	−38.3921	−1.52716
$633$	−34.3797	−1.36647
$634$	−36.0108	−1.43017
$635$	5.30648	0.210581
$636$	5.48393	0.217452
$637$	0	0
$638$	−15.9407	−0.631097
$639$	10.0443	0.397345
$640$	7.79334	0.308059
$641$	10.2198	0.403658	0.201829	−	0.979421i	$-0.435311\pi$
$641$	10.2198	0.403658	0.201829	+	0.979421i	$0.435311\pi$
$642$	62.4786	2.46584
$643$	31.8027	1.25418	0.627088	−	0.778948i	$-0.284247\pi$
$643$	31.8027	1.25418	0.627088	+	0.778948i	$0.284247\pi$
$644$	0	0
$645$	14.4861	0.570389
$646$	−12.2787	−0.483101
$647$	−34.5233	−1.35725	−0.678626	−	0.734484i	$-0.737424\pi$
$647$	−34.5233	−1.35725	−0.678626	+	0.734484i	$0.737424\pi$
$648$	−28.8655	−1.13394
$649$	18.0602	0.708923
$650$	0	0
$651$	0	0
$652$	0.536530	0.0210121
$653$	10.1180	0.395947	0.197974	−	0.980207i	$-0.436564\pi$
$653$	10.1180	0.395947	0.197974	+	0.980207i	$0.436564\pi$
$654$	18.5356	0.724800
$655$	4.00692	0.156563
$656$	6.13709	0.239613
$657$	24.8968	0.971317
$658$	0	0
$659$	−34.7681	−1.35437	−0.677187	−	0.735811i	$-0.736801\pi$
$659$	−34.7681	−1.35437	−0.677187	+	0.735811i	$0.736801\pi$
$660$	−0.596039	−0.0232008
$661$	−16.5615	−0.644168	−0.322084	−	0.946711i	$-0.604383\pi$
$661$	−16.5615	−0.644168	−0.322084	+	0.946711i	$0.604383\pi$
$662$	−12.0933	−0.470020
$663$	0	0
$664$	−25.9727	−1.00794
$665$	0	0
$666$	−3.25896	−0.126282
$667$	12.2269	0.473428
$668$	−3.95697	−0.153100
$669$	26.4326	1.02194
$670$	−6.78777	−0.262234
$671$	−3.77178	−0.145608
$672$	0	0
$673$	−41.8874	−1.61464	−0.807321	−	0.590113i	$-0.799083\pi$
$673$	−41.8874	−1.61464	−0.807321	+	0.590113i	$0.799083\pi$
$674$	12.0343	0.463543
$675$	14.6332	0.563231
$676$	0	0
$677$	−31.5715	−1.21339	−0.606696	−	0.794934i	$-0.707505\pi$
$677$	−31.5715	−1.21339	−0.606696	+	0.794934i	$0.707505\pi$
$678$	−16.6511	−0.639483
$679$	0	0
$680$	7.25132	0.278075
$681$	−2.62582	−0.100621
$682$	−14.0763	−0.539011
$683$	−26.2676	−1.00510	−0.502551	−	0.864548i	$-0.667605\pi$
$683$	−26.2676	−1.00510	−0.502551	+	0.864548i	$0.667605\pi$
$684$	−0.800911	−0.0306236
$685$	8.29602	0.316975
$686$	0	0
$687$	−13.9376	−0.531753
$688$	52.3381	1.99537
$689$	0	0
$690$	3.38076	0.128703
$691$	6.10095	0.232091	0.116046	−	0.993244i	$-0.462978\pi$
$691$	6.10095	0.232091	0.116046	+	0.993244i	$0.462978\pi$
$692$	3.59271	0.136574
$693$	0	0
$694$	−10.8518	−0.411929
$695$	10.1884	0.386467
$696$	37.6034	1.42535
$697$	6.49914	0.246172
$698$	−2.09381	−0.0792518
$699$	13.3927	0.506560
$700$	0	0
$701$	−20.5701	−0.776921	−0.388461	−	0.921465i	$-0.626993\pi$
$701$	−20.5701	−0.776921	−0.388461	+	0.921465i	$0.626993\pi$
$702$	0	0
$703$	−2.37285	−0.0894936
$704$	9.71520	0.366155
$705$	−0.582513	−0.0219387
$706$	−1.05422	−0.0396762
$707$	0	0
$708$	7.89702	0.296788
$709$	−43.1529	−1.62064	−0.810320	−	0.585988i	$-0.800707\pi$
$709$	−43.1529	−1.62064	−0.810320	+	0.585988i	$0.800707\pi$
$710$	5.92000	0.222174
$711$	−22.7543	−0.853353
$712$	−42.2992	−1.58523
$713$	10.7969	0.404348
$714$	0	0
$715$	0	0
$716$	−0.683209	−0.0255327
$717$	−40.2265	−1.50229
$718$	8.82603	0.329385
$719$	28.2084	1.05200	0.525999	−	0.850485i	$-0.323692\pi$
$719$	28.2084	1.05200	0.525999	+	0.850485i	$0.323692\pi$
$720$	4.05831	0.151244
$721$	0	0
$722$	24.5824	0.914864
$723$	−47.4076	−1.76311
$724$	3.62733	0.134809
$725$	−32.0682	−1.19098
$726$	28.0901	1.04252
$727$	19.5116	0.723646	0.361823	−	0.932247i	$-0.382155\pi$
$727$	19.5116	0.723646	0.361823	+	0.932247i	$0.382155\pi$
$728$	0	0
$729$	2.95370	0.109396
$730$	14.6740	0.543108
$731$	55.4257	2.04999
$732$	−1.64926	−0.0609582
$733$	−20.3666	−0.752259	−0.376129	−	0.926567i	$-0.622745\pi$
$733$	−20.3666	−0.752259	−0.376129	+	0.926567i	$0.622745\pi$
$734$	−11.1801	−0.412664
$735$	0	0
$736$	3.11105	0.114675
$737$	−11.5166	−0.424218
$738$	3.13484	0.115395
$739$	−10.4199	−0.383303	−0.191651	−	0.981463i	$-0.561384\pi$
$739$	−10.4199	−0.383303	−0.191651	+	0.981463i	$0.561384\pi$
$740$	−0.259749	−0.00954856
$741$	0	0
$742$	0	0
$743$	−17.4094	−0.638689	−0.319344	−	0.947639i	$-0.603463\pi$
$743$	−17.4094	−0.638689	−0.319344	+	0.947639i	$0.603463\pi$
$744$	33.2055	1.21737
$745$	10.0934	0.369795
$746$	−27.9805	−1.02444
$747$	−15.3935	−0.563220
$748$	−2.28052	−0.0833841
$749$	0	0
$750$	−18.3958	−0.671721
$751$	−1.81525	−0.0662395	−0.0331197	−	0.999451i	$-0.510544\pi$
$751$	−1.81525	−0.0662395	−0.0331197	+	0.999451i	$0.510544\pi$
$752$	−2.10461	−0.0767474
$753$	14.7839	0.538754
$754$	0	0
$755$	−9.31088	−0.338858
$756$	0	0
$757$	53.5627	1.94677	0.973385	−	0.229176i	$-0.0736033\pi$
$757$	53.5627	1.94677	0.973385	+	0.229176i	$0.0736033\pi$
$758$	7.37728	0.267955
$759$	5.73602	0.208204
$760$	2.54664	0.0923762
$761$	−3.68166	−0.133460	−0.0667300	−	0.997771i	$-0.521257\pi$
$761$	−3.68166	−0.133460	−0.0667300	+	0.997771i	$0.521257\pi$
$762$	29.1121	1.05462
$763$	0	0
$764$	5.55423	0.200945
$765$	4.29772	0.155384
$766$	34.7048	1.25394
$767$	0	0
$768$	15.5895	0.562538
$769$	−5.23794	−0.188885	−0.0944424	−	0.995530i	$-0.530107\pi$
$769$	−5.23794	−0.188885	−0.0944424	+	0.995530i	$0.530107\pi$
$770$	0	0
$771$	−44.9803	−1.61993
$772$	7.08004	0.254816
$773$	40.6138	1.46078	0.730388	−	0.683032i	$-0.239339\pi$
$773$	40.6138	1.46078	0.730388	+	0.683032i	$0.239339\pi$
$774$	26.7344	0.960948
$775$	−28.3177	−1.01720
$776$	2.49887	0.0897043
$777$	0	0
$778$	30.8902	1.10747
$779$	2.28247	0.0817781
$780$	0	0
$781$	10.0443	0.359412
$782$	12.9352	0.462563
$783$	−21.6780	−0.774707
$784$	0	0
$785$	−2.22997	−0.0795911
$786$	21.9825	0.784090
$787$	−34.2000	−1.21910	−0.609550	−	0.792748i	$-0.708650\pi$
$787$	−34.2000	−1.21910	−0.609550	+	0.792748i	$0.708650\pi$
$788$	6.25701	0.222897
$789$	−17.9238	−0.638104
$790$	−13.4112	−0.477149
$791$	0	0
$792$	5.93434	0.210868
$793$	0	0
$794$	51.9158	1.84242
$795$	−10.3347	−0.366532
$796$	0.578232	0.0204949
$797$	−31.3805	−1.11156	−0.555778	−	0.831331i	$-0.687579\pi$
$797$	−31.3805	−1.11156	−0.555778	+	0.831331i	$0.687579\pi$
$798$	0	0
$799$	−2.22877	−0.0788483
$800$	−8.15951	−0.288482
$801$	−25.0699	−0.885803
$802$	4.60180	0.162495
$803$	24.8968	0.878590
$804$	−5.03576	−0.177598
$805$	0	0
$806$	0	0
$807$	12.3966	0.436382
$808$	7.54441	0.265412
$809$	−37.9418	−1.33396	−0.666981	−	0.745075i	$-0.732414\pi$
$809$	−37.9418	−1.33396	−0.666981	+	0.745075i	$0.732414\pi$
$810$	−10.0833	−0.354292
$811$	32.1023	1.12726	0.563632	−	0.826026i	$-0.309403\pi$
$811$	32.1023	1.12726	0.563632	+	0.826026i	$0.309403\pi$
$812$	0	0
$813$	−39.1859	−1.37431
$814$	−3.25896	−0.114227
$815$	−1.01111	−0.0354176
$816$	46.1586	1.61588
$817$	19.4653	0.681004
$818$	−8.18670	−0.286241
$819$	0	0
$820$	0.249856	0.00872534
$821$	−16.9223	−0.590594	−0.295297	−	0.955406i	$-0.595419\pi$
$821$	−16.9223	−0.590594	−0.295297	+	0.955406i	$0.595419\pi$
$822$	45.5132	1.58745
$823$	−14.4733	−0.504508	−0.252254	−	0.967661i	$-0.581172\pi$
$823$	−14.4733	−0.504508	−0.252254	+	0.967661i	$0.581172\pi$
$824$	−1.35656	−0.0472579
$825$	−15.0442	−0.523771
$826$	0	0
$827$	27.5792	0.959021	0.479511	−	0.877536i	$-0.340814\pi$
$827$	27.5792	0.959021	0.479511	+	0.877536i	$0.340814\pi$
$828$	0.843731	0.0293217
$829$	−21.0930	−0.732592	−0.366296	−	0.930498i	$-0.619374\pi$
$829$	−21.0930	−0.732592	−0.366296	+	0.930498i	$0.619374\pi$
$830$	−9.07281	−0.314922
$831$	−13.1465	−0.456047
$832$	0	0
$833$	0	0
$834$	55.8949	1.93548
$835$	7.45704	0.258061
$836$	−0.800911	−0.0277001
$837$	−19.1426	−0.661666
$838$	−8.94572	−0.309025
$839$	12.7098	0.438789	0.219395	−	0.975636i	$-0.429592\pi$
$839$	12.7098	0.438789	0.219395	+	0.975636i	$0.429592\pi$
$840$	0	0
$841$	18.5067	0.638160
$842$	43.7396	1.50736
$843$	−11.9929	−0.413056
$844$	−5.05703	−0.174070
$845$	0	0
$846$	−1.07504	−0.0369606
$847$	0	0
$848$	−37.3390	−1.28223
$849$	34.9877	1.20077
$850$	−33.9259	−1.16365
$851$	2.49971	0.0856890
$852$	4.39197	0.150467
$853$	35.0471	1.19999	0.599996	−	0.800003i	$-0.295169\pi$
$853$	35.0471	1.19999	0.599996	+	0.800003i	$0.295169\pi$
$854$	0	0
$855$	1.50934	0.0516184
$856$	−49.5797	−1.69460
$857$	25.5940	0.874275	0.437138	−	0.899395i	$-0.355992\pi$
$857$	25.5940	0.874275	0.437138	+	0.899395i	$0.355992\pi$
$858$	0	0
$859$	−0.745486	−0.0254357	−0.0127178	−	0.999919i	$-0.504048\pi$
$859$	−0.745486	−0.0254357	−0.0127178	+	0.999919i	$0.504048\pi$
$860$	2.13081	0.0726600
$861$	0	0
$862$	−12.7530	−0.434370
$863$	−21.9816	−0.748263	−0.374132	−	0.927376i	$-0.622059\pi$
$863$	−21.9816	−0.748263	−0.374132	+	0.927376i	$0.622059\pi$
$864$	−5.51579	−0.187651
$865$	−6.77058	−0.230207
$866$	41.9722	1.42627
$867$	12.7361	0.432540
$868$	0	0
$869$	−22.7543	−0.771887
$870$	13.1357	0.445341
$871$	0	0
$872$	−14.7089	−0.498106
$873$	1.48103	0.0501254
$874$	4.54280	0.153663
$875$	0	0
$876$	10.8864	0.367818
$877$	29.4190	0.993411	0.496705	−	0.867919i	$-0.334543\pi$
$877$	29.4190	0.993411	0.496705	+	0.867919i	$0.334543\pi$
$878$	−47.4082	−1.59995
$879$	−65.0984	−2.19572
$880$	4.05831	0.136806
$881$	48.2828	1.62669	0.813345	−	0.581781i	$-0.197644\pi$
$881$	48.2828	1.62669	0.813345	+	0.581781i	$0.197644\pi$
$882$	0	0
$883$	16.4465	0.553469	0.276735	−	0.960946i	$-0.410748\pi$
$883$	16.4465	0.553469	0.276735	+	0.960946i	$0.410748\pi$
$884$	0	0
$885$	−14.8822	−0.500260
$886$	35.8724	1.20516
$887$	29.1653	0.979276	0.489638	−	0.871926i	$-0.337129\pi$
$887$	29.1653	0.979276	0.489638	+	0.871926i	$0.337129\pi$
$888$	7.68776	0.257984
$889$	0	0
$890$	−14.7760	−0.495293
$891$	−17.1080	−0.573141
$892$	3.88806	0.130182
$893$	−0.782736	−0.0261933
$894$	55.3741	1.85199
$895$	1.28753	0.0430374
$896$	0	0
$897$	0	0
$898$	−4.29856	−0.143445
$899$	41.9506	1.39913
$900$	−2.21290	−0.0737633
$901$	−39.5417	−1.31733
$902$	3.13484	0.104379
$903$	0	0
$904$	13.2135	0.439473
$905$	−6.83583	−0.227231
$906$	−51.0808	−1.69705
$907$	−35.3215	−1.17283	−0.586416	−	0.810010i	$-0.699461\pi$
$907$	−35.3215	−1.17283	−0.586416	+	0.810010i	$0.699461\pi$
$908$	−0.386240	−0.0128178
$909$	4.47143	0.148308
$910$	0	0
$911$	−43.6496	−1.44617	−0.723087	−	0.690757i	$-0.757278\pi$
$911$	−43.6496	−1.44617	−0.723087	+	0.690757i	$0.757278\pi$
$912$	16.2107	0.536791
$913$	−15.3935	−0.509452
$914$	57.3988	1.89858
$915$	3.10808	0.102750
$916$	−2.05013	−0.0677382
$917$	0	0
$918$	−22.9338	−0.756927
$919$	−17.7878	−0.586765	−0.293382	−	0.955995i	$-0.594781\pi$
$919$	−17.7878	−0.586765	−0.293382	+	0.955995i	$0.594781\pi$
$920$	−2.68279	−0.0884491
$921$	21.1775	0.697822
$922$	−52.7079	−1.73584
$923$	0	0
$924$	0	0
$925$	−6.55613	−0.215564
$926$	−28.1516	−0.925118
$927$	−0.804005	−0.0264070
$928$	12.0877	0.396798
$929$	3.91420	0.128421	0.0642103	−	0.997936i	$-0.479547\pi$
$929$	3.91420	0.128421	0.0642103	+	0.997936i	$0.479547\pi$
$930$	11.5994	0.380359
$931$	0	0
$932$	1.96998	0.0645289
$933$	−58.9955	−1.93142
$934$	−10.0794	−0.329809
$935$	4.29772	0.140550
$936$	0	0
$937$	−0.207362	−0.00677421	−0.00338710	−	0.999994i	$-0.501078\pi$
$937$	−0.207362	−0.00677421	−0.00338710	+	0.999994i	$0.501078\pi$
$938$	0	0
$939$	35.1308	1.14645
$940$	−0.0856839	−0.00279470
$941$	27.4656	0.895355	0.447677	−	0.894195i	$-0.352251\pi$
$941$	27.4656	0.895355	0.447677	+	0.894195i	$0.352251\pi$
$942$	−12.2339	−0.398603
$943$	−2.40451	−0.0783015
$944$	−53.7692	−1.75004
$945$	0	0
$946$	26.7344	0.869210
$947$	−23.0363	−0.748580	−0.374290	−	0.927312i	$-0.622114\pi$
$947$	−23.0363	−0.748580	−0.374290	+	0.927312i	$0.622114\pi$
$948$	−9.94959	−0.323148
$949$	0	0
$950$	−11.9147	−0.386563
$951$	50.3471	1.63262
$952$	0	0
$953$	14.4562	0.468281	0.234140	−	0.972203i	$-0.424772\pi$
$953$	14.4562	0.468281	0.234140	+	0.972203i	$0.424772\pi$
$954$	−19.0728	−0.617505
$955$	−10.4671	−0.338708
$956$	−5.91706	−0.191371
$957$	22.2868	0.720431
$958$	26.5414	0.857515
$959$	0	0
$960$	−8.00567	−0.258382
$961$	6.04426	0.194976
$962$	0	0
$963$	−29.3850	−0.946917
$964$	−6.97336	−0.224597
$965$	−13.3426	−0.429513
$966$	0	0
$967$	32.3357	1.03984	0.519922	−	0.854213i	$-0.325961\pi$
$967$	32.3357	1.03984	0.519922	+	0.854213i	$0.325961\pi$
$968$	−22.2908	−0.716454
$969$	17.1671	0.551485
$970$	0.872909	0.0280274
$971$	24.6235	0.790206	0.395103	−	0.918637i	$-0.370709\pi$
$971$	24.6235	0.790206	0.395103	+	0.918637i	$0.370709\pi$
$972$	−4.52974	−0.145292
$973$	0	0
$974$	54.0084	1.73054
$975$	0	0
$976$	11.2294	0.359446
$977$	6.00992	0.192274	0.0961372	−	0.995368i	$-0.469351\pi$
$977$	6.00992	0.192274	0.0961372	+	0.995368i	$0.469351\pi$
$978$	−5.54709	−0.177376
$979$	−25.0699	−0.801239
$980$	0	0
$981$	−8.71768	−0.278334
$982$	45.3371	1.44676
$983$	−35.9093	−1.14533	−0.572664	−	0.819790i	$-0.694090\pi$
$983$	−35.9093	−1.14533	−0.572664	+	0.819790i	$0.694090\pi$
$984$	−7.39496	−0.235743
$985$	−11.7916	−0.375710
$986$	50.2587	1.60056
$987$	0	0
$988$	0	0
$989$	−20.5060	−0.652053
$990$	2.07299	0.0658840
$991$	28.0000	0.889449	0.444724	−	0.895668i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444724	+	0.895668i	$0.353302\pi$
$992$	10.6740	0.338900
$993$	16.9078	0.536553
$994$	0	0
$995$	−1.08970	−0.0345457
$996$	−6.73101	−0.213280
$997$	−33.0081	−1.04538	−0.522688	−	0.852524i	$-0.675071\pi$
$997$	−33.0081	−1.04538	−0.522688	+	0.852524i	$0.675071\pi$
$998$	11.4205	0.361509
$999$	−4.43191	−0.140219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ci.1.4		8
7.6	odd	2	inner	8281.2.a.ci.1.3		8
13.3	even	3		637.2.f.l.295.5	✓	16
13.9	even	3		637.2.f.l.393.5	yes	16
13.12	even	2		8281.2.a.cl.1.6		8
91.3	odd	6		637.2.g.m.373.5		16
91.9	even	3		637.2.g.m.263.6		16
91.16	even	3		637.2.h.m.165.3		16
91.48	odd	6		637.2.f.l.393.6	yes	16
91.55	odd	6		637.2.f.l.295.6	yes	16
91.61	odd	6		637.2.g.m.263.5		16
91.68	odd	6		637.2.h.m.165.4		16
91.74	even	3		637.2.h.m.471.3		16
91.81	even	3		637.2.g.m.373.6		16
91.87	odd	6		637.2.h.m.471.4		16
91.90	odd	2		8281.2.a.cl.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.l.295.5	✓	16	13.3	even	3
637.2.f.l.295.6	yes	16	91.55	odd	6
637.2.f.l.393.5	yes	16	13.9	even	3
637.2.f.l.393.6	yes	16	91.48	odd	6
637.2.g.m.263.5		16	91.61	odd	6
637.2.g.m.263.6		16	91.9	even	3
637.2.g.m.373.5		16	91.3	odd	6
637.2.g.m.373.6		16	91.81	even	3
637.2.h.m.165.3		16	91.16	even	3
637.2.h.m.165.4		16	91.68	odd	6
637.2.h.m.471.3		16	91.74	even	3
637.2.h.m.471.4		16	91.87	odd	6
8281.2.a.ci.1.3		8	7.6	odd	2	inner
8281.2.a.ci.1.4		8	1.1	even	1	trivial
8281.2.a.cl.1.5		8	91.90	odd	2
8281.2.a.cl.1.6		8	13.12	even	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52077 q^{2} +2.12621 q^{3} +0.312752 q^{4} -0.589391 q^{5} -3.23349 q^{6} +2.56592 q^{8} +1.52077 q^{9} +O(q^{10})\)	\(q-1.52077 q^{2} +2.12621 q^{3} +0.312752 q^{4} -0.589391 q^{5} -3.23349 q^{6} +2.56592 q^{8} +1.52077 q^{9} +0.896331 q^{10} +1.52077 q^{11} +0.664976 q^{12} -1.25317 q^{15} -4.52769 q^{16} -4.79479 q^{17} -2.31275 q^{18} -1.68391 q^{19} -0.184333 q^{20} -2.31275 q^{22} +1.77394 q^{23} +5.45569 q^{24} -4.65262 q^{25} -3.14515 q^{27} +6.89251 q^{29} +1.90579 q^{30} +6.08640 q^{31} +1.75375 q^{32} +3.23349 q^{33} +7.29179 q^{34} +0.475625 q^{36} +1.40913 q^{37} +2.56085 q^{38} -1.51233 q^{40} -1.35546 q^{41} -11.5596 q^{43} +0.475625 q^{44} -0.896331 q^{45} -2.69777 q^{46} +0.464832 q^{47} -9.62682 q^{48} +7.07558 q^{50} -10.1947 q^{51} +8.24681 q^{53} +4.78306 q^{54} -0.896331 q^{55} -3.58035 q^{57} -10.4819 q^{58} +11.8756 q^{59} -0.391931 q^{60} -2.48017 q^{61} -9.25603 q^{62} +6.38833 q^{64} -4.91740 q^{66} -7.57284 q^{67} -1.49958 q^{68} +3.77178 q^{69} +6.60471 q^{71} +3.90219 q^{72} +16.3712 q^{73} -2.14296 q^{74} -9.89245 q^{75} -0.526647 q^{76} -14.9623 q^{79} +2.66858 q^{80} -11.2496 q^{81} +2.06134 q^{82} -10.1222 q^{83} +2.82601 q^{85} +17.5795 q^{86} +14.6549 q^{87} +3.90219 q^{88} -16.4850 q^{89} +1.36312 q^{90} +0.554804 q^{92} +12.9410 q^{93} -0.706904 q^{94} +0.992484 q^{95} +3.72883 q^{96} +0.973869 q^{97} +2.31275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9	\( 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{15} + 4 q^{16} - 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} + 8 q^{37} - 32 q^{43} - 4 q^{44} + 4 q^{46} - 36 q^{50} - 44 q^{51} - 4 q^{53} - 48 q^{57} + 48 q^{58} + 64 q^{60} - 32 q^{64} - 20 q^{67} - 8 q^{71} - 28 q^{72} - 76 q^{74} - 4 q^{79} - 56 q^{81} - 36 q^{85} + 4 q^{86} - 28 q^{88} - 80 q^{92} - 8 q^{93} - 52 q^{95} + 28 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^15 + 4 * q^16 - 28 * q^18 - 28 * q^22 - 12 * q^23 - 12 * q^25 - 8 * q^29 - 28 * q^30 - 4 * q^36 + 8 * q^37 - 32 * q^43 - 4 * q^44 + 4 * q^46 - 36 * q^50 - 44 * q^51 - 4 * q^53 - 48 * q^57 + 48 * q^58 + 64 * q^60 - 32 * q^64 - 20 * q^67 - 8 * q^71 - 28 * q^72 - 76 * q^74 - 4 * q^79 - 56 * q^81 - 36 * q^85 + 4 * q^86 - 28 * q^88 - 80 * q^92 - 8 * q^93 - 52 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more