LMFDB - Embedded newform 8281.2.a.ce.1.6

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:	$6$
Coefficient field:	6.6.6995813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1$ x^6 - x^5 - 6x^4 + 4x^3 + 7*x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.363441$ 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)

$f(q)$	$=$	$q+2.38804 q^{2} -2.75148 q^{3} +3.70272 q^{4} -0.982280 q^{5} -6.57063 q^{6} +4.06616 q^{8} +4.57063 q^{9} -2.34572 q^{10} +0.587802 q^{11} -10.1880 q^{12} +2.70272 q^{15} +2.30470 q^{16} -6.45420 q^{17} +10.9148 q^{18} +3.82689 q^{19} -3.63711 q^{20} +1.40369 q^{22} +8.26001 q^{23} -11.1880 q^{24} -4.03513 q^{25} -4.32156 q^{27} -3.96018 q^{29} +6.45420 q^{30} +2.98872 q^{31} -2.62861 q^{32} -1.61733 q^{33} -15.4129 q^{34} +16.9238 q^{36} -1.75588 q^{37} +9.13877 q^{38} -3.99411 q^{40} -3.67169 q^{41} +6.38085 q^{43} +2.17647 q^{44} -4.48964 q^{45} +19.7252 q^{46} +4.34059 q^{47} -6.34134 q^{48} -9.63603 q^{50} +17.7586 q^{51} +0.425541 q^{53} -10.3200 q^{54} -0.577387 q^{55} -10.5296 q^{57} -9.45706 q^{58} -6.00863 q^{59} +10.0074 q^{60} +2.20674 q^{61} +7.13717 q^{62} -10.8866 q^{64} -3.86223 q^{66} -7.01303 q^{67} -23.8981 q^{68} -22.7272 q^{69} -3.60253 q^{71} +18.5849 q^{72} -4.93427 q^{73} -4.19311 q^{74} +11.1026 q^{75} +14.1699 q^{76} +2.78541 q^{79} -2.26386 q^{80} -1.82122 q^{81} -8.76812 q^{82} +2.86819 q^{83} +6.33983 q^{85} +15.2377 q^{86} +10.8964 q^{87} +2.39010 q^{88} +2.09311 q^{89} -10.7214 q^{90} +30.5845 q^{92} -8.22340 q^{93} +10.3655 q^{94} -3.75908 q^{95} +7.23255 q^{96} -7.69704 q^{97} +2.68663 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 2 q^{2} - q^{3} + 4 q^{4} + q^{5} - 9 q^{6} + 3 q^{8} - 3 q^{9} - 4 q^{10} + 4 q^{11} - 5 q^{12} - 2 q^{15} - 8 q^{16} - 5 q^{17} + 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots + 10 q^{99}+O(q^{100})$ 6 * q + 2 * q^2 - q^3 + 4 * q^4 + q^5 - 9 * q^6 + 3 * q^8 - 3 * q^9 - 4 * q^10 + 4 * q^11 - 5 * q^12 - 2 * q^15 - 8 * q^16 - 5 * q^17 + 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 - 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 + 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 - 13 * q^37 + 17 * q^38 + 5 * q^40 - 8 * q^41 + 11 * q^43 + 21 * q^44 - 7 * q^45 + 16 * q^46 - q^47 - 21 * q^48 + 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 - 9 * q^55 - 21 * q^57 - 8 * q^58 + 13 * q^59 + 20 * q^60 + 5 * q^61 - 5 * q^62 - 15 * q^64 - 18 * q^66 - 11 * q^67 - 29 * q^68 - 23 * q^69 + 6 * q^71 + 25 * q^72 - 30 * q^73 + 3 * q^74 + 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 - q^82 + 27 * q^83 - q^85 - 7 * q^86 - 16 * q^87 + 4 * q^89 - 8 * q^90 + 27 * q^92 - 7 * q^93 - 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 + 10 * 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

)

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

$2$	2.38804	1.68860	0.844299	−	0.535873i	$-0.180017\pi$
$2$	2.38804	1.68860	0.844299	+	0.535873i	$0.180017\pi$
$3$	−2.75148	−1.58857	−0.794283	−	0.607548i	$-0.792153\pi$
$3$	−2.75148	−1.58857	−0.794283	+	0.607548i	$0.792153\pi$
$4$	3.70272	1.85136
$5$	−0.982280	−0.439289	−0.219644	−	0.975580i	$-0.570490\pi$
$5$	−0.982280	−0.439289	−0.219644	+	0.975580i	$0.570490\pi$
$6$	−6.57063	−2.68245
$7$	0	0
$7$	0	0
$8$	4.06616	1.43761
$9$	4.57063	1.52354
$10$	−2.34572	−0.741782
$11$	0.587802	0.177229	0.0886146	−	0.996066i	$-0.471756\pi$
$11$	0.587802	0.177229	0.0886146	+	0.996066i	$0.471756\pi$
$12$	−10.1880	−2.94101
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.70272	0.697840
$16$	2.30470	0.576176
$17$	−6.45420	−1.56537	−0.782687	−	0.622416i	$-0.786151\pi$
$17$	−6.45420	−1.56537	−0.782687	+	0.622416i	$0.786151\pi$
$18$	10.9148	2.57265
$19$	3.82689	0.877950	0.438975	−	0.898499i	$-0.355342\pi$
$19$	3.82689	0.877950	0.438975	+	0.898499i	$0.355342\pi$
$20$	−3.63711	−0.813282
$21$	0	0
$22$	1.40369	0.299269
$23$	8.26001	1.72233	0.861166	−	0.508324i	$-0.169735\pi$
$23$	8.26001	1.72233	0.861166	+	0.508324i	$0.169735\pi$
$24$	−11.1880	−2.28373
$25$	−4.03513	−0.807025
$26$	0	0
$27$	−4.32156	−0.831685
$28$	0	0
$29$	−3.96018	−0.735387	−0.367694	−	0.929947i	$-0.619853\pi$
$29$	−3.96018	−0.735387	−0.367694	+	0.929947i	$0.619853\pi$
$30$	6.45420	1.17837
$31$	2.98872	0.536790	0.268395	−	0.963309i	$-0.413507\pi$
$31$	2.98872	0.536790	0.268395	+	0.963309i	$0.413507\pi$
$32$	−2.62861	−0.464676
$33$	−1.61733	−0.281540
$34$	−15.4129	−2.64329
$35$	0	0
$36$	16.9238	2.82063
$37$	−1.75588	−0.288665	−0.144333	−	0.989529i	$-0.546104\pi$
$37$	−1.75588	−0.288665	−0.144333	+	0.989529i	$0.546104\pi$
$38$	9.13877	1.48250
$39$	0	0
$40$	−3.99411	−0.631524
$41$	−3.67169	−0.573421	−0.286710	−	0.958017i	$-0.592562\pi$
$41$	−3.67169	−0.573421	−0.286710	+	0.958017i	$0.592562\pi$
$42$	0	0
$43$	6.38085	0.973070	0.486535	−	0.873661i	$-0.338261\pi$
$43$	6.38085	0.973070	0.486535	+	0.873661i	$0.338261\pi$
$44$	2.17647	0.328115
$45$	−4.48964	−0.669276
$46$	19.7252	2.90832
$47$	4.34059	0.633141	0.316570	−	0.948569i	$-0.397469\pi$
$47$	4.34059	0.633141	0.316570	+	0.948569i	$0.397469\pi$
$48$	−6.34134	−0.915294
$49$	0	0
$50$	−9.63603	−1.36274
$51$	17.7586	2.48670
$52$	0	0
$53$	0.425541	0.0584525	0.0292263	−	0.999573i	$-0.490696\pi$
$53$	0.425541	0.0584525	0.0292263	+	0.999573i	$0.490696\pi$
$54$	−10.3200	−1.40438
$55$	−0.577387	−0.0778548
$56$	0	0
$57$	−10.5296	−1.39468
$58$	−9.45706	−1.24177
$59$	−6.00863	−0.782256	−0.391128	−	0.920336i	$-0.627915\pi$
$59$	−6.00863	−0.782256	−0.391128	+	0.920336i	$0.627915\pi$
$60$	10.0074	1.29195
$61$	2.20674	0.282544	0.141272	−	0.989971i	$-0.454881\pi$
$61$	2.20674	0.282544	0.141272	+	0.989971i	$0.454881\pi$
$62$	7.13717	0.906422
$63$	0	0
$64$	−10.8866	−1.36083
$65$	0	0
$66$	−3.86223	−0.475408
$67$	−7.01303	−0.856778	−0.428389	−	0.903594i	$-0.640919\pi$
$67$	−7.01303	−0.856778	−0.428389	+	0.903594i	$0.640919\pi$
$68$	−23.8981	−2.89807
$69$	−22.7272	−2.73604
$70$	0	0
$71$	−3.60253	−0.427542	−0.213771	−	0.976884i	$-0.568575\pi$
$71$	−3.60253	−0.427542	−0.213771	+	0.976884i	$0.568575\pi$
$72$	18.5849	2.19026
$73$	−4.93427	−0.577513	−0.288756	−	0.957403i	$-0.593242\pi$
$73$	−4.93427	−0.577513	−0.288756	+	0.957403i	$0.593242\pi$
$74$	−4.19311	−0.487439
$75$	11.1026	1.28201
$76$	14.1699	1.62540
$77$	0	0
$78$	0	0
$79$	2.78541	0.313383	0.156691	−	0.987648i	$-0.449917\pi$
$79$	2.78541	0.313383	0.156691	+	0.987648i	$0.449917\pi$
$80$	−2.26386	−0.253108
$81$	−1.82122	−0.202357
$82$	−8.76812	−0.968277
$83$	2.86819	0.314825	0.157412	−	0.987533i	$-0.449685\pi$
$83$	2.86819	0.314825	0.157412	+	0.987533i	$0.449685\pi$
$84$	0	0
$85$	6.33983	0.687651
$86$	15.2377	1.64312
$87$	10.8964	1.16821
$88$	2.39010	0.254786
$89$	2.09311	0.221870	0.110935	−	0.993828i	$-0.464616\pi$
$89$	2.09311	0.221870	0.110935	+	0.993828i	$0.464616\pi$
$90$	−10.7214	−1.13014
$91$	0	0
$92$	30.5845	3.18866
$93$	−8.22340	−0.852727
$94$	10.3655	1.06912
$95$	−3.75908	−0.385674
$96$	7.23255	0.738169
$97$	−7.69704	−0.781516	−0.390758	−	0.920493i	$-0.627787\pi$
$97$	−7.69704	−0.781516	−0.390758	+	0.920493i	$0.627787\pi$
$98$	0	0
$99$	2.68663	0.270016
$100$	−14.9409	−1.49409
$101$	−2.63732	−0.262423	−0.131212	−	0.991354i	$-0.541887\pi$
$101$	−2.63732	−0.262423	−0.131212	+	0.991354i	$0.541887\pi$
$102$	42.4082	4.19903
$103$	−10.8619	−1.07026	−0.535128	−	0.844771i	$-0.679737\pi$
$103$	−10.8619	−1.07026	−0.535128	+	0.844771i	$0.679737\pi$
$104$	0	0
$105$	0	0
$106$	1.01621	0.0987027
$107$	−15.9805	−1.54489	−0.772446	−	0.635080i	$-0.780967\pi$
$107$	−15.9805	−1.54489	−0.772446	+	0.635080i	$0.780967\pi$
$108$	−16.0015	−1.53975
$109$	−9.23477	−0.884530	−0.442265	−	0.896884i	$-0.645825\pi$
$109$	−9.23477	−0.884530	−0.442265	+	0.896884i	$0.645825\pi$
$110$	−1.37882	−0.131465
$111$	4.83127	0.458564
$112$	0	0
$113$	10.1802	0.957677	0.478838	−	0.877903i	$-0.341058\pi$
$113$	10.1802	0.957677	0.478838	+	0.877903i	$0.341058\pi$
$114$	−25.1451	−2.35506
$115$	−8.11364	−0.756601
$116$	−14.6635	−1.36147
$117$	0	0
$118$	−14.3488	−1.32092
$119$	0	0
$120$	10.9897	1.00322
$121$	−10.6545	−0.968590
$122$	5.26978	0.477104
$123$	10.1026	0.910917
$124$	11.0664	0.993792
$125$	8.87502	0.793806
$126$	0	0
$127$	4.25026	0.377149	0.188575	−	0.982059i	$-0.439613\pi$
$127$	4.25026	0.377149	0.188575	+	0.982059i	$0.439613\pi$
$128$	−20.7404	−1.83321
$129$	−17.5568	−1.54579
$130$	0	0
$131$	−2.16957	−0.189556	−0.0947779	−	0.995498i	$-0.530214\pi$
$131$	−2.16957	−0.189556	−0.0947779	+	0.995498i	$0.530214\pi$
$132$	−5.98851	−0.521233
$133$	0	0
$134$	−16.7474	−1.44675
$135$	4.24498	0.365350
$136$	−26.2438	−2.25039
$137$	−8.36316	−0.714513	−0.357257	−	0.934006i	$-0.616288\pi$
$137$	−8.36316	−0.714513	−0.357257	+	0.934006i	$0.616288\pi$
$138$	−54.2735	−4.62007
$139$	−0.576914	−0.0489332	−0.0244666	−	0.999701i	$-0.507789\pi$
$139$	−0.576914	−0.0489332	−0.0244666	+	0.999701i	$0.507789\pi$
$140$	0	0
$141$	−11.9430	−1.00579
$142$	−8.60298	−0.721946
$143$	0	0
$144$	10.5340	0.877830
$145$	3.89001	0.323048
$146$	−11.7832	−0.975187
$147$	0	0
$148$	−6.50154	−0.534423
$149$	−2.80662	−0.229928	−0.114964	−	0.993370i	$-0.536675\pi$
$149$	−2.80662	−0.229928	−0.114964	+	0.993370i	$0.536675\pi$
$150$	26.5133	2.16480
$151$	23.0109	1.87260	0.936300	−	0.351202i	$-0.114227\pi$
$151$	23.0109	1.87260	0.936300	+	0.351202i	$0.114227\pi$
$152$	15.5608	1.26215
$153$	−29.4998	−2.38492
$154$	0	0
$155$	−2.93576	−0.235806
$156$	0	0
$157$	22.5760	1.80176	0.900879	−	0.434071i	$-0.142923\pi$
$157$	22.5760	1.80176	0.900879	+	0.434071i	$0.142923\pi$
$158$	6.65165	0.529177
$159$	−1.17087	−0.0928557
$160$	2.58203	0.204127
$161$	0	0
$162$	−4.34913	−0.341700
$163$	−8.17714	−0.640483	−0.320242	−	0.947336i	$-0.603764\pi$
$163$	−8.17714	−0.640483	−0.320242	+	0.947336i	$0.603764\pi$
$164$	−13.5952	−1.06161
$165$	1.58867	0.123678
$166$	6.84934	0.531612
$167$	2.32771	0.180124	0.0900619	−	0.995936i	$-0.471293\pi$
$167$	2.32771	0.180124	0.0900619	+	0.995936i	$0.471293\pi$
$168$	0	0
$169$	0	0
$170$	15.1398	1.16117
$171$	17.4913	1.33760
$172$	23.6265	1.80150
$173$	−8.13372	−0.618396	−0.309198	−	0.950998i	$-0.600061\pi$
$173$	−8.13372	−0.618396	−0.309198	+	0.950998i	$0.600061\pi$
$174$	26.0209	1.97264
$175$	0	0
$176$	1.35471	0.102115
$177$	16.5326	1.24267
$178$	4.99843	0.374648
$179$	−20.9925	−1.56906	−0.784528	−	0.620093i	$-0.787095\pi$
$179$	−20.9925	−1.56906	−0.784528	+	0.620093i	$0.787095\pi$
$180$	−16.6239	−1.23907
$181$	−1.60807	−0.119527	−0.0597635	−	0.998213i	$-0.519035\pi$
$181$	−1.60807	−0.119527	−0.0597635	+	0.998213i	$0.519035\pi$
$182$	0	0
$183$	−6.07180	−0.448841
$184$	33.5865	2.47603
$185$	1.72477	0.126807
$186$	−19.6378	−1.43991
$187$	−3.79379	−0.277430
$188$	16.0720	1.17217
$189$	0	0
$190$	−8.97683	−0.651247
$191$	−11.5622	−0.836614	−0.418307	−	0.908306i	$-0.637376\pi$
$191$	−11.5622	−0.836614	−0.418307	+	0.908306i	$0.637376\pi$
$192$	29.9543	2.16176
$193$	−23.5788	−1.69724	−0.848621	−	0.529001i	$-0.822567\pi$
$193$	−23.5788	−1.69724	−0.848621	+	0.529001i	$0.822567\pi$
$194$	−18.3808	−1.31967
$195$	0	0
$196$	0	0
$197$	1.47094	0.104800	0.0524002	−	0.998626i	$-0.483313\pi$
$197$	1.47094	0.104800	0.0524002	+	0.998626i	$0.483313\pi$
$198$	6.41577	0.455949
$199$	9.39399	0.665922	0.332961	−	0.942941i	$-0.391952\pi$
$199$	9.39399	0.665922	0.332961	+	0.942941i	$0.391952\pi$
$200$	−16.4075	−1.16018
$201$	19.2962	1.36105
$202$	−6.29802	−0.443127
$203$	0	0
$204$	65.7551	4.60378
$205$	3.60662	0.251897
$206$	−25.9386	−1.80723
$207$	37.7535	2.62405
$208$	0	0
$209$	2.24946	0.155598
$210$	0	0
$211$	−8.94219	−0.615605	−0.307803	−	0.951450i	$-0.599594\pi$
$211$	−8.94219	−0.615605	−0.307803	+	0.951450i	$0.599594\pi$
$212$	1.57566	0.108217
$213$	9.91229	0.679179
$214$	−38.1620	−2.60870
$215$	−6.26778	−0.427459
$216$	−17.5722	−1.19563
$217$	0	0
$218$	−22.0530	−1.49362
$219$	13.5765	0.917418
$220$	−2.13790	−0.144137
$221$	0	0
$222$	11.5373	0.774330
$223$	−21.8196	−1.46115	−0.730574	−	0.682833i	$-0.760748\pi$
$223$	−21.8196	−1.46115	−0.730574	+	0.682833i	$0.760748\pi$
$224$	0	0
$225$	−18.4431	−1.22954
$226$	24.3108	1.61713
$227$	18.5525	1.23137	0.615687	−	0.787990i	$-0.288878\pi$
$227$	18.5525	1.23137	0.615687	+	0.787990i	$0.288878\pi$
$228$	−38.9882	−2.58206
$229$	−19.3505	−1.27872	−0.639359	−	0.768909i	$-0.720800\pi$
$229$	−19.3505	−1.27872	−0.639359	+	0.768909i	$0.720800\pi$
$230$	−19.3757	−1.27759
$231$	0	0
$232$	−16.1027	−1.05720
$233$	16.1634	1.05890	0.529450	−	0.848341i	$-0.322398\pi$
$233$	16.1634	1.05890	0.529450	+	0.848341i	$0.322398\pi$
$234$	0	0
$235$	−4.26368	−0.278132
$236$	−22.2483	−1.44824
$237$	−7.66398	−0.497829
$238$	0	0
$239$	−16.1037	−1.04166	−0.520831	−	0.853660i	$-0.674378\pi$
$239$	−16.1037	−1.04166	−0.520831	+	0.853660i	$0.674378\pi$
$240$	6.22897	0.402079
$241$	4.00600	0.258049	0.129025	−	0.991641i	$-0.458815\pi$
$241$	4.00600	0.258049	0.129025	+	0.991641i	$0.458815\pi$
$242$	−25.4433	−1.63556
$243$	17.9757	1.15314
$244$	8.17095	0.523092
$245$	0	0
$246$	24.1253	1.53817
$247$	0	0
$248$	12.1526	0.771692
$249$	−7.89176	−0.500120
$250$	21.1939	1.34042
$251$	3.24688	0.204941	0.102471	−	0.994736i	$-0.467325\pi$
$251$	3.24688	0.204941	0.102471	+	0.994736i	$0.467325\pi$
$252$	0	0
$253$	4.85525	0.305247
$254$	10.1498	0.636853
$255$	−17.4439	−1.09238
$256$	−27.7557	−1.73473
$257$	−26.8924	−1.67750	−0.838751	−	0.544516i	$-0.816713\pi$
$257$	−26.8924	−1.67750	−0.838751	+	0.544516i	$0.816713\pi$
$258$	−41.9262	−2.61021
$259$	0	0
$260$	0	0
$261$	−18.1005	−1.12040
$262$	−5.18100	−0.320084
$263$	−3.80706	−0.234753	−0.117377	−	0.993087i	$-0.537448\pi$
$263$	−3.80706	−0.234753	−0.117377	+	0.993087i	$0.537448\pi$
$264$	−6.57631	−0.404744
$265$	−0.418000	−0.0256775
$266$	0	0
$267$	−5.75915	−0.352455
$268$	−25.9673	−1.58620
$269$	−23.8381	−1.45343	−0.726716	−	0.686938i	$-0.758954\pi$
$269$	−23.8381	−1.45343	−0.726716	+	0.686938i	$0.758954\pi$
$270$	10.1372	0.616929
$271$	−9.90135	−0.601464	−0.300732	−	0.953709i	$-0.597231\pi$
$271$	−9.90135	−0.601464	−0.300732	+	0.953709i	$0.597231\pi$
$272$	−14.8750	−0.901931
$273$	0	0
$274$	−19.9715	−1.20653
$275$	−2.37186	−0.143028
$276$	−84.1526	−5.06539
$277$	11.7858	0.708139	0.354069	−	0.935219i	$-0.384798\pi$
$277$	11.7858	0.708139	0.354069	+	0.935219i	$0.384798\pi$
$278$	−1.37769	−0.0826285
$279$	13.6603	0.817823
$280$	0	0
$281$	−12.9976	−0.775372	−0.387686	−	0.921791i	$-0.626726\pi$
$281$	−12.9976	−0.775372	−0.387686	+	0.921791i	$0.626726\pi$
$282$	−28.5204	−1.69837
$283$	−16.8050	−0.998952	−0.499476	−	0.866328i	$-0.666474\pi$
$283$	−16.8050	−0.998952	−0.499476	+	0.866328i	$0.666474\pi$
$284$	−13.3392	−0.791534
$285$	10.3430	0.612668
$286$	0	0
$287$	0	0
$288$	−12.0144	−0.707955
$289$	24.6567	1.45039
$290$	9.28948	0.545497
$291$	21.1783	1.24149
$292$	−18.2702	−1.06918
$293$	14.0956	0.823476	0.411738	−	0.911302i	$-0.364922\pi$
$293$	14.0956	0.823476	0.411738	+	0.911302i	$0.364922\pi$
$294$	0	0
$295$	5.90215	0.343637
$296$	−7.13970	−0.414987
$297$	−2.54022	−0.147399
$298$	−6.70232	−0.388255
$299$	0	0
$300$	41.1097	2.37347
$301$	0	0
$302$	54.9508	3.16207
$303$	7.25654	0.416877
$304$	8.81986	0.505854
$305$	−2.16764	−0.124119
$306$	−70.4466	−4.02716
$307$	−15.8786	−0.906240	−0.453120	−	0.891450i	$-0.649689\pi$
$307$	−15.8786	−0.906240	−0.453120	+	0.891450i	$0.649689\pi$
$308$	0	0
$309$	29.8863	1.70017
$310$	−7.01070	−0.398181
$311$	−28.6034	−1.62195	−0.810975	−	0.585081i	$-0.801063\pi$
$311$	−28.6034	−1.62195	−0.810975	+	0.585081i	$0.801063\pi$
$312$	0	0
$313$	−18.5792	−1.05016	−0.525080	−	0.851053i	$-0.675965\pi$
$313$	−18.5792	−1.05016	−0.525080	+	0.851053i	$0.675965\pi$
$314$	53.9122	3.04244
$315$	0	0
$316$	10.3136	0.580184
$317$	−30.6445	−1.72117	−0.860584	−	0.509309i	$-0.829901\pi$
$317$	−30.6445	−1.72117	−0.860584	+	0.509309i	$0.829901\pi$
$318$	−2.79607	−0.156796
$319$	−2.32781	−0.130332
$320$	10.6937	0.597796
$321$	43.9700	2.45416
$322$	0	0
$323$	−24.6995	−1.37432
$324$	−6.74346	−0.374637
$325$	0	0
$326$	−19.5273	−1.08152
$327$	25.4093	1.40514
$328$	−14.9297	−0.824353
$329$	0	0
$330$	3.79379	0.208842
$331$	−27.2277	−1.49657	−0.748284	−	0.663378i	$-0.769122\pi$
$331$	−27.2277	−1.49657	−0.748284	+	0.663378i	$0.769122\pi$
$332$	10.6201	0.582854
$333$	−8.02549	−0.439794
$334$	5.55867	0.304157
$335$	6.88876	0.376373
$336$	0	0
$337$	−12.3160	−0.670898	−0.335449	−	0.942058i	$-0.608888\pi$
$337$	−12.3160	−0.670898	−0.335449	+	0.942058i	$0.608888\pi$
$338$	0	0
$339$	−28.0107	−1.52133
$340$	23.4746	1.27309
$341$	1.75678	0.0951348
$342$	41.7699	2.25866
$343$	0	0
$344$	25.9456	1.39889
$345$	22.3245	1.20191
$346$	−19.4236	−1.04422
$347$	6.14506	0.329884	0.164942	−	0.986303i	$-0.447256\pi$
$347$	6.14506	0.329884	0.164942	+	0.986303i	$0.447256\pi$
$348$	40.3462	2.16278
$349$	−13.0313	−0.697547	−0.348774	−	0.937207i	$-0.613402\pi$
$349$	−13.0313	−0.697547	−0.348774	+	0.937207i	$0.613402\pi$
$350$	0	0
$351$	0	0
$352$	−1.54510	−0.0823541
$353$	−31.6665	−1.68544	−0.842718	−	0.538356i	$-0.819046\pi$
$353$	−31.6665	−1.68544	−0.842718	+	0.538356i	$0.819046\pi$
$354$	39.4805	2.09836
$355$	3.53870	0.187814
$356$	7.75021	0.410761
$357$	0	0
$358$	−50.1310	−2.64950
$359$	−19.9322	−1.05198	−0.525991	−	0.850490i	$-0.676305\pi$
$359$	−19.9322	−1.05198	−0.525991	+	0.850490i	$0.676305\pi$
$360$	−18.2556	−0.962155
$361$	−4.35488	−0.229204
$362$	−3.84014	−0.201833
$363$	29.3156	1.53867
$364$	0	0
$365$	4.84684	0.253695
$366$	−14.4997	−0.757911
$367$	19.7190	1.02932	0.514662	−	0.857393i	$-0.327918\pi$
$367$	19.7190	1.02932	0.514662	+	0.857393i	$0.327918\pi$
$368$	19.0369	0.992366
$369$	−16.7819	−0.873632
$370$	4.11881	0.214127
$371$	0	0
$372$	−30.4490	−1.57870
$373$	17.5469	0.908544	0.454272	−	0.890863i	$-0.349899\pi$
$373$	17.5469	0.908544	0.454272	+	0.890863i	$0.349899\pi$
$374$	−9.05972	−0.468467
$375$	−24.4194	−1.26101
$376$	17.6496	0.910207
$377$	0	0
$378$	0	0
$379$	11.7014	0.601058	0.300529	−	0.953773i	$-0.402837\pi$
$379$	11.7014	0.601058	0.300529	+	0.953773i	$0.402837\pi$
$380$	−13.9188	−0.714021
$381$	−11.6945	−0.599127
$382$	−27.6110	−1.41270
$383$	21.5288	1.10007	0.550036	−	0.835141i	$-0.314614\pi$
$383$	21.5288	1.10007	0.550036	+	0.835141i	$0.314614\pi$
$384$	57.0669	2.91218
$385$	0	0
$386$	−56.3072	−2.86596
$387$	29.1645	1.48252
$388$	−28.5000	−1.44687
$389$	26.4910	1.34315	0.671574	−	0.740938i	$-0.265619\pi$
$389$	26.4910	1.34315	0.671574	+	0.740938i	$0.265619\pi$
$390$	0	0
$391$	−53.3118	−2.69609
$392$	0	0
$393$	5.96951	0.301122
$394$	3.51267	0.176966
$395$	−2.73605	−0.137665
$396$	9.94784	0.499898
$397$	33.7989	1.69632	0.848160	−	0.529740i	$-0.177711\pi$
$397$	33.7989	1.69632	0.848160	+	0.529740i	$0.177711\pi$
$398$	22.4332	1.12447
$399$	0	0
$400$	−9.29977	−0.464989
$401$	−21.6119	−1.07925	−0.539623	−	0.841907i	$-0.681433\pi$
$401$	−21.6119	−1.07925	−0.539623	+	0.841907i	$0.681433\pi$
$402$	46.0800	2.29826
$403$	0	0
$404$	−9.76527	−0.485840
$405$	1.78895	0.0888934
$406$	0	0
$407$	−1.03211	−0.0511599
$408$	72.2093	3.57489
$409$	−7.74217	−0.382826	−0.191413	−	0.981510i	$-0.561307\pi$
$409$	−7.74217	−0.382826	−0.191413	+	0.981510i	$0.561307\pi$
$410$	8.61275	0.425353
$411$	23.0111	1.13505
$412$	−40.2186	−1.98143
$413$	0	0
$414$	90.1567	4.43096
$415$	−2.81736	−0.138299
$416$	0	0
$417$	1.58737	0.0777337
$418$	5.37179	0.262743
$419$	−8.10194	−0.395806	−0.197903	−	0.980222i	$-0.563413\pi$
$419$	−8.10194	−0.395806	−0.197903	+	0.980222i	$0.563413\pi$
$420$	0	0
$421$	32.1124	1.56506	0.782530	−	0.622612i	$-0.213929\pi$
$421$	32.1124	1.56506	0.782530	+	0.622612i	$0.213929\pi$
$422$	−21.3543	−1.03951
$423$	19.8393	0.964618
$424$	1.73032	0.0840316
$425$	26.0435	1.26330
$426$	23.6709	1.14686
$427$	0	0
$428$	−59.1713	−2.86015
$429$	0	0
$430$	−14.9677	−0.721806
$431$	29.5281	1.42232	0.711159	−	0.703031i	$-0.248171\pi$
$431$	29.5281	1.42232	0.711159	+	0.703031i	$0.248171\pi$
$432$	−9.95992	−0.479197
$433$	22.0910	1.06163	0.530813	−	0.847489i	$-0.321887\pi$
$433$	22.0910	1.06163	0.530813	+	0.847489i	$0.321887\pi$
$434$	0	0
$435$	−10.7033	−0.513183
$436$	−34.1938	−1.63758
$437$	31.6102	1.51212
$438$	32.4213	1.54915
$439$	−6.35580	−0.303346	−0.151673	−	0.988431i	$-0.548466\pi$
$439$	−6.35580	−0.303346	−0.151673	+	0.988431i	$0.548466\pi$
$440$	−2.34775	−0.111924
$441$	0	0
$442$	0	0
$443$	−13.5627	−0.644383	−0.322192	−	0.946675i	$-0.604420\pi$
$443$	−13.5627	−0.644383	−0.322192	+	0.946675i	$0.604420\pi$
$444$	17.8889	0.848967
$445$	−2.05602	−0.0974648
$446$	−52.1060	−2.46729
$447$	7.72237	0.365255
$448$	0	0
$449$	−21.9118	−1.03408	−0.517041	−	0.855961i	$-0.672966\pi$
$449$	−21.9118	−1.03408	−0.517041	+	0.855961i	$0.672966\pi$
$450$	−44.0428	−2.07620
$451$	−2.15823	−0.101627
$452$	37.6946	1.77300
$453$	−63.3139	−2.97475
$454$	44.3041	2.07930
$455$	0	0
$456$	−42.8151	−2.00500
$457$	−15.2146	−0.711710	−0.355855	−	0.934541i	$-0.615810\pi$
$457$	−15.2146	−0.711710	−0.355855	+	0.934541i	$0.615810\pi$
$458$	−46.2097	−2.15924
$459$	27.8922	1.30190
$460$	−30.0426	−1.40074
$461$	16.2163	0.755267	0.377633	−	0.925955i	$-0.376738\pi$
$461$	16.2163	0.755267	0.377633	+	0.925955i	$0.376738\pi$
$462$	0	0
$463$	1.44769	0.0672799	0.0336400	−	0.999434i	$-0.489290\pi$
$463$	1.44769	0.0672799	0.0336400	+	0.999434i	$0.489290\pi$
$464$	−9.12705	−0.423713
$465$	8.07768	0.374593
$466$	38.5988	1.78805
$467$	14.0067	0.648155	0.324078	−	0.946031i	$-0.394946\pi$
$467$	14.0067	0.648155	0.324078	+	0.946031i	$0.394946\pi$
$468$	0	0
$469$	0	0
$470$	−10.1818	−0.469652
$471$	−62.1172	−2.86221
$472$	−24.4320	−1.12458
$473$	3.75068	0.172456
$474$	−18.3019	−0.840633
$475$	−15.4420	−0.708528
$476$	0	0
$477$	1.94499	0.0890550
$478$	−38.4562	−1.75895
$479$	30.0243	1.37185	0.685923	−	0.727674i	$-0.259399\pi$
$479$	30.0243	1.37185	0.685923	+	0.727674i	$0.259399\pi$
$480$	−7.10439	−0.324270
$481$	0	0
$482$	9.56649	0.435742
$483$	0	0
$484$	−39.4506	−1.79321
$485$	7.56065	0.343312
$486$	42.9267	1.94719
$487$	28.4903	1.29102	0.645510	−	0.763752i	$-0.276645\pi$
$487$	28.4903	1.29102	0.645510	+	0.763752i	$0.276645\pi$
$488$	8.97297	0.406187
$489$	22.4992	1.01745
$490$	0	0
$491$	−28.4677	−1.28473	−0.642365	−	0.766399i	$-0.722047\pi$
$491$	−28.4677	−1.28473	−0.642365	+	0.766399i	$0.722047\pi$
$492$	37.4070	1.68644
$493$	25.5598	1.15116
$494$	0	0
$495$	−2.63902	−0.118615
$496$	6.88812	0.309286
$497$	0	0
$498$	−18.8458	−0.844501
$499$	26.2329	1.17434	0.587172	−	0.809462i	$-0.300241\pi$
$499$	26.2329	1.17434	0.587172	+	0.809462i	$0.300241\pi$
$500$	32.8617	1.46962
$501$	−6.40465	−0.286139
$502$	7.75367	0.346063
$503$	8.53175	0.380412	0.190206	−	0.981744i	$-0.439084\pi$
$503$	8.53175	0.380412	0.190206	+	0.981744i	$0.439084\pi$
$504$	0	0
$505$	2.59059	0.115280
$506$	11.5945	0.515440
$507$	0	0
$508$	15.7375	0.698240
$509$	13.0260	0.577366	0.288683	−	0.957425i	$-0.406783\pi$
$509$	13.0260	0.577366	0.288683	+	0.957425i	$0.406783\pi$
$510$	−41.6567	−1.84459
$511$	0	0
$512$	−24.8008	−1.09605
$513$	−16.5382	−0.730177
$514$	−64.2200	−2.83262
$515$	10.6694	0.470151
$516$	−65.0078	−2.86181
$517$	2.55141	0.112211
$518$	0	0
$519$	22.3798	0.982363
$520$	0	0
$521$	4.46570	0.195646	0.0978230	−	0.995204i	$-0.468812\pi$
$521$	4.46570	0.195646	0.0978230	+	0.995204i	$0.468812\pi$
$522$	−43.2248	−1.89190
$523$	−2.90811	−0.127163	−0.0635815	−	0.997977i	$-0.520252\pi$
$523$	−2.90811	−0.127163	−0.0635815	+	0.997977i	$0.520252\pi$
$524$	−8.03330	−0.350936
$525$	0	0
$526$	−9.09140	−0.396404
$527$	−19.2898	−0.840277
$528$	−3.72746	−0.162217
$529$	45.2278	1.96643
$530$	−0.998200	−0.0433590
$531$	−27.4632	−1.19180
$532$	0	0
$533$	0	0
$534$	−13.7531	−0.595154
$535$	15.6973	0.678654
$536$	−28.5161	−1.23171
$537$	57.7605	2.49255
$538$	−56.9262	−2.45426
$539$	0	0
$540$	15.7180	0.676394
$541$	18.4639	0.793824	0.396912	−	0.917857i	$-0.370082\pi$
$541$	18.4639	0.793824	0.396912	+	0.917857i	$0.370082\pi$
$542$	−23.6448	−1.01563
$543$	4.42458	0.189877
$544$	16.9655	0.727392
$545$	9.07112	0.388564
$546$	0	0
$547$	34.9817	1.49571	0.747856	−	0.663861i	$-0.231083\pi$
$547$	34.9817	1.49571	0.747856	+	0.663861i	$0.231083\pi$
$548$	−30.9665	−1.32282
$549$	10.0862	0.430469
$550$	−5.66408	−0.241517
$551$	−15.1552	−0.645633
$552$	−92.4127	−3.93334
$553$	0	0
$554$	28.1449	1.19576
$555$	−4.74566	−0.201442
$556$	−2.13615	−0.0905930
$557$	−0.0531413	−0.00225167	−0.00112583	−	0.999999i	$-0.500358\pi$
$557$	−0.0531413	−0.00225167	−0.00112583	+	0.999999i	$0.500358\pi$
$558$	32.6214	1.38097
$559$	0	0
$560$	0	0
$561$	10.4385	0.440716
$562$	−31.0388	−1.30929
$563$	7.98506	0.336530	0.168265	−	0.985742i	$-0.446184\pi$
$563$	7.98506	0.336530	0.168265	+	0.985742i	$0.446184\pi$
$564$	−44.2218	−1.86207
$565$	−9.99985	−0.420697
$566$	−40.1309	−1.68683
$567$	0	0
$568$	−14.6485	−0.614637
$569$	−26.7241	−1.12033	−0.560167	−	0.828380i	$-0.689263\pi$
$569$	−26.7241	−1.12033	−0.560167	+	0.828380i	$0.689263\pi$
$570$	24.6995	1.03455
$571$	13.4929	0.564662	0.282331	−	0.959317i	$-0.408892\pi$
$571$	13.4929	0.564662	0.282331	+	0.959317i	$0.408892\pi$
$572$	0	0
$573$	31.8132	1.32902
$574$	0	0
$575$	−33.3302	−1.38996
$576$	−49.7587	−2.07328
$577$	−12.0132	−0.500118	−0.250059	−	0.968231i	$-0.580450\pi$
$577$	−12.0132	−0.500118	−0.250059	+	0.968231i	$0.580450\pi$
$578$	58.8811	2.44913
$579$	64.8767	2.69618
$580$	14.4036	0.598078
$581$	0	0
$582$	50.5745	2.09638
$583$	0.250134	0.0103595
$584$	−20.0636	−0.830236
$585$	0	0
$586$	33.6609	1.39052
$587$	−10.4235	−0.430225	−0.215113	−	0.976589i	$-0.569012\pi$
$587$	−10.4235	−0.430225	−0.215113	+	0.976589i	$0.569012\pi$
$588$	0	0
$589$	11.4375	0.471275
$590$	14.0946	0.580264
$591$	−4.04727	−0.166482
$592$	−4.04679	−0.166322
$593$	22.3501	0.917810	0.458905	−	0.888485i	$-0.348242\pi$
$593$	22.3501	0.917810	0.458905	+	0.888485i	$0.348242\pi$
$594$	−6.06615	−0.248897
$595$	0	0
$596$	−10.3921	−0.425679
$597$	−25.8474	−1.05786
$598$	0	0
$599$	1.15893	0.0473524	0.0236762	−	0.999720i	$-0.492463\pi$
$599$	1.15893	0.0473524	0.0236762	+	0.999720i	$0.492463\pi$
$600$	45.1448	1.84303
$601$	42.1813	1.72061	0.860306	−	0.509778i	$-0.170272\pi$
$601$	42.1813	1.72061	0.860306	+	0.509778i	$0.170272\pi$
$602$	0	0
$603$	−32.0540	−1.30534
$604$	85.2029	3.46686
$605$	10.4657	0.425491
$606$	17.3289	0.703937
$607$	−18.1569	−0.736965	−0.368482	−	0.929635i	$-0.620123\pi$
$607$	−18.1569	−0.736965	−0.368482	+	0.929635i	$0.620123\pi$
$608$	−10.0594	−0.407962
$609$	0	0
$610$	−5.17640	−0.209586
$611$	0	0
$612$	−109.229	−4.41534
$613$	0.902645	0.0364575	0.0182288	−	0.999834i	$-0.494197\pi$
$613$	0.902645	0.0364575	0.0182288	+	0.999834i	$0.494197\pi$
$614$	−37.9187	−1.53027
$615$	−9.92354	−0.400156
$616$	0	0
$617$	26.0436	1.04848	0.524238	−	0.851572i	$-0.324350\pi$
$617$	26.0436	1.04848	0.524238	+	0.851572i	$0.324350\pi$
$618$	71.3696	2.87091
$619$	−26.8341	−1.07855	−0.539277	−	0.842128i	$-0.681303\pi$
$619$	−26.8341	−1.07855	−0.539277	+	0.842128i	$0.681303\pi$
$620$	−10.8703	−0.436562
$621$	−35.6961	−1.43244
$622$	−68.3060	−2.73882
$623$	0	0
$624$	0	0
$625$	11.4579	0.458315
$626$	−44.3679	−1.77330
$627$	−6.18933	−0.247178
$628$	83.5925	3.33570
$629$	11.3328	0.451869
$630$	0	0
$631$	33.6121	1.33808	0.669039	−	0.743228i	$-0.266706\pi$
$631$	33.6121	1.33808	0.669039	+	0.743228i	$0.266706\pi$
$632$	11.3259	0.450521
$633$	24.6042	0.977930
$634$	−73.1802	−2.90636
$635$	−4.17494	−0.165678
$636$	−4.33539	−0.171909
$637$	0	0
$638$	−5.55889	−0.220078
$639$	−16.4659	−0.651379
$640$	20.3729	0.805310
$641$	21.1841	0.836722	0.418361	−	0.908281i	$-0.362605\pi$
$641$	21.1841	0.836722	0.418361	+	0.908281i	$0.362605\pi$
$642$	105.002	4.14410
$643$	−0.661539	−0.0260886	−0.0130443	−	0.999915i	$-0.504152\pi$
$643$	−0.661539	−0.0260886	−0.0130443	+	0.999915i	$0.504152\pi$
$644$	0	0
$645$	17.2457	0.679047
$646$	−58.9834	−2.32067
$647$	−40.0323	−1.57383	−0.786916	−	0.617060i	$-0.788324\pi$
$647$	−40.0323	−1.57383	−0.786916	+	0.617060i	$0.788324\pi$
$648$	−7.40537	−0.290910
$649$	−3.53188	−0.138639
$650$	0	0
$651$	0	0
$652$	−30.2777	−1.18577
$653$	12.7120	0.497460	0.248730	−	0.968573i	$-0.419987\pi$
$653$	12.7120	0.497460	0.248730	+	0.968573i	$0.419987\pi$
$654$	60.6782	2.37271
$655$	2.13112	0.0832698
$656$	−8.46215	−0.330391
$657$	−22.5527	−0.879867
$658$	0	0
$659$	−14.1904	−0.552781	−0.276391	−	0.961045i	$-0.589138\pi$
$659$	−14.1904	−0.552781	−0.276391	+	0.961045i	$0.589138\pi$
$660$	5.88239	0.228972
$661$	−50.1780	−1.95170	−0.975848	−	0.218449i	$-0.929900\pi$
$661$	−50.1780	−1.95170	−0.975848	+	0.218449i	$0.929900\pi$
$662$	−65.0207	−2.52710
$663$	0	0
$664$	11.6625	0.452594
$665$	0	0
$666$	−19.1652	−0.742635
$667$	−32.7112	−1.26658
$668$	8.61888	0.333474
$669$	60.0362	2.32113
$670$	16.4506	0.635542
$671$	1.29713	0.0500751
$672$	0	0
$673$	−1.87427	−0.0722479	−0.0361240	−	0.999347i	$-0.511501\pi$
$673$	−1.87427	−0.0722479	−0.0361240	+	0.999347i	$0.511501\pi$
$674$	−29.4112	−1.13288
$675$	17.4380	0.671191
$676$	0	0
$677$	−2.00879	−0.0772041	−0.0386020	−	0.999255i	$-0.512290\pi$
$677$	−2.00879	−0.0772041	−0.0386020	+	0.999255i	$0.512290\pi$
$678$	−66.8906	−2.56892
$679$	0	0
$680$	25.7788	0.988571
$681$	−51.0469	−1.95612
$682$	4.19525	0.160644
$683$	14.1012	0.539568	0.269784	−	0.962921i	$-0.413048\pi$
$683$	14.1012	0.539568	0.269784	+	0.962921i	$0.413048\pi$
$684$	64.7655	2.47637
$685$	8.21497	0.313878
$686$	0	0
$687$	53.2425	2.03133
$688$	14.7060	0.560660
$689$	0	0
$690$	53.3118	2.02954
$691$	35.6920	1.35779	0.678895	−	0.734236i	$-0.262459\pi$
$691$	35.6920	1.35779	0.678895	+	0.734236i	$0.262459\pi$
$692$	−30.1169	−1.14487
$693$	0	0
$694$	14.6746	0.557041
$695$	0.566691	0.0214958
$696$	44.3064	1.67943
$697$	23.6978	0.897618
$698$	−31.1191	−1.17788
$699$	−44.4732	−1.68213
$700$	0	0
$701$	−6.15865	−0.232609	−0.116305	−	0.993214i	$-0.537105\pi$
$701$	−6.15865	−0.232609	−0.116305	+	0.993214i	$0.537105\pi$
$702$	0	0
$703$	−6.71957	−0.253434
$704$	−6.39918	−0.241178
$705$	11.7314	0.441831
$706$	−75.6207	−2.84602
$707$	0	0
$708$	61.2156	2.30062
$709$	−34.0371	−1.27829	−0.639144	−	0.769087i	$-0.720711\pi$
$709$	−34.0371	−1.27829	−0.639144	+	0.769087i	$0.720711\pi$
$710$	8.45054	0.317143
$711$	12.7311	0.477452
$712$	8.51094	0.318961
$713$	24.6869	0.924530
$714$	0	0
$715$	0	0
$716$	−77.7295	−2.90489
$717$	44.3090	1.65475
$718$	−47.5989	−1.77637
$719$	−22.9648	−0.856444	−0.428222	−	0.903674i	$-0.640860\pi$
$719$	−22.9648	−0.856444	−0.428222	+	0.903674i	$0.640860\pi$
$720$	−10.3473	−0.385621
$721$	0	0
$722$	−10.3996	−0.387034
$723$	−11.0224	−0.409929
$724$	−5.95424	−0.221288
$725$	15.9798	0.593476
$726$	70.0067	2.59819
$727$	1.06558	0.0395203	0.0197601	−	0.999805i	$-0.493710\pi$
$727$	1.06558	0.0395203	0.0197601	+	0.999805i	$0.493710\pi$
$728$	0	0
$729$	−43.9962	−1.62949
$730$	11.5744	0.428389
$731$	−41.1833	−1.52322
$732$	−22.4822	−0.830966
$733$	26.3378	0.972808	0.486404	−	0.873734i	$-0.338308\pi$
$733$	26.3378	0.972808	0.486404	+	0.873734i	$0.338308\pi$
$734$	47.0897	1.73811
$735$	0	0
$736$	−21.7123	−0.800326
$737$	−4.12228	−0.151846
$738$	−40.0759	−1.47521
$739$	−34.2149	−1.25862	−0.629308	−	0.777156i	$-0.716662\pi$
$739$	−34.2149	−1.25862	−0.629308	+	0.777156i	$0.716662\pi$
$740$	6.38633	0.234766
$741$	0	0
$742$	0	0
$743$	−22.4782	−0.824644	−0.412322	−	0.911038i	$-0.635282\pi$
$743$	−22.4782	−0.824644	−0.412322	+	0.911038i	$0.635282\pi$
$744$	−33.4377	−1.22588
$745$	2.75689	0.101005
$746$	41.9027	1.53417
$747$	13.1094	0.479649
$748$	−14.0474	−0.513623
$749$	0	0
$750$	−58.3145	−2.12934
$751$	−42.5424	−1.55239	−0.776197	−	0.630491i	$-0.782854\pi$
$751$	−42.5424	−1.55239	−0.776197	+	0.630491i	$0.782854\pi$
$752$	10.0038	0.364801
$753$	−8.93372	−0.325563
$754$	0	0
$755$	−22.6031	−0.822612
$756$	0	0
$757$	−11.2380	−0.408454	−0.204227	−	0.978924i	$-0.565468\pi$
$757$	−11.2380	−0.408454	−0.204227	+	0.978924i	$0.565468\pi$
$758$	27.9433	1.01495
$759$	−13.3591	−0.484906
$760$	−15.2850	−0.554447
$761$	−12.8084	−0.464306	−0.232153	−	0.972679i	$-0.574577\pi$
$761$	−12.8084	−0.464306	−0.232153	+	0.972679i	$0.574577\pi$
$762$	−27.9269	−1.01168
$763$	0	0
$764$	−42.8117	−1.54887
$765$	28.9770	1.04767
$766$	51.4117	1.85758
$767$	0	0
$768$	76.3692	2.75574
$769$	51.3517	1.85179	0.925895	−	0.377781i	$-0.123313\pi$
$769$	51.3517	1.85179	0.925895	+	0.377781i	$0.123313\pi$
$770$	0	0
$771$	73.9938	2.66482
$772$	−87.3059	−3.14221
$773$	−20.0046	−0.719517	−0.359759	−	0.933045i	$-0.617141\pi$
$773$	−20.0046	−0.719517	−0.359759	+	0.933045i	$0.617141\pi$
$774$	69.6459	2.50337
$775$	−12.0599	−0.433203
$776$	−31.2974	−1.12351
$777$	0	0
$778$	63.2615	2.26804
$779$	−14.0512	−0.503435
$780$	0	0
$781$	−2.11758	−0.0757729
$782$	−127.310	−4.55261
$783$	17.1142	0.611611
$784$	0	0
$785$	−22.1759	−0.791492
$786$	14.2554	0.508474
$787$	−29.3192	−1.04512	−0.522558	−	0.852604i	$-0.675022\pi$
$787$	−29.3192	−1.04512	−0.522558	+	0.852604i	$0.675022\pi$
$788$	5.44650	0.194023
$789$	10.4750	0.372921
$790$	−6.53378	−0.232462
$791$	0	0
$792$	10.9243	0.388177
$793$	0	0
$794$	80.7131	2.86440
$795$	1.15012	0.0407905
$796$	34.7833	1.23286
$797$	3.10100	0.109843	0.0549215	−	0.998491i	$-0.482509\pi$
$797$	3.10100	0.109843	0.0549215	+	0.998491i	$0.482509\pi$
$798$	0	0
$799$	−28.0151	−0.991102
$800$	10.6068	0.375005
$801$	9.56685	0.338028
$802$	−51.6100	−1.82241
$803$	−2.90038	−0.102352
$804$	71.4484	2.51979
$805$	0	0
$806$	0	0
$807$	65.5899	2.30887
$808$	−10.7238	−0.377261
$809$	7.99003	0.280914	0.140457	−	0.990087i	$-0.455143\pi$
$809$	7.99003	0.280914	0.140457	+	0.990087i	$0.455143\pi$
$810$	4.27207	0.150105
$811$	48.2554	1.69448	0.847239	−	0.531213i	$-0.178263\pi$
$811$	48.2554	1.69448	0.847239	+	0.531213i	$0.178263\pi$
$812$	0	0
$813$	27.2434	0.955466
$814$	−2.46472	−0.0863884
$815$	8.03224	0.281357
$816$	40.9283	1.43278
$817$	24.4188	0.854307
$818$	−18.4886	−0.646439
$819$	0	0
$820$	13.3543	0.466353
$821$	27.5519	0.961569	0.480785	−	0.876839i	$-0.340352\pi$
$821$	27.5519	0.961569	0.480785	+	0.876839i	$0.340352\pi$
$822$	54.9513	1.91665
$823$	20.4274	0.712056	0.356028	−	0.934475i	$-0.384131\pi$
$823$	20.4274	0.712056	0.356028	+	0.934475i	$0.384131\pi$
$824$	−44.1663	−1.53861
$825$	6.52611	0.227210
$826$	0	0
$827$	27.7142	0.963719	0.481859	−	0.876249i	$-0.339962\pi$
$827$	27.7142	0.963719	0.481859	+	0.876249i	$0.339962\pi$
$828$	139.791	4.85806
$829$	9.25664	0.321496	0.160748	−	0.986995i	$-0.448609\pi$
$829$	9.25664	0.321496	0.160748	+	0.986995i	$0.448609\pi$
$830$	−6.72797	−0.233531
$831$	−32.4283	−1.12493
$832$	0	0
$833$	0	0
$834$	3.79069	0.131261
$835$	−2.28647	−0.0791264
$836$	8.32912	0.288069
$837$	−12.9159	−0.446440
$838$	−19.3477	−0.668357
$839$	30.3739	1.04862	0.524312	−	0.851526i	$-0.324323\pi$
$839$	30.3739	1.04862	0.524312	+	0.851526i	$0.324323\pi$
$840$	0	0
$841$	−13.3170	−0.459205
$842$	76.6855	2.64276
$843$	35.7626	1.23173
$844$	−33.1104	−1.13971
$845$	0	0
$846$	47.3769	1.62885
$847$	0	0
$848$	0.980745	0.0336789
$849$	46.2385	1.58690
$850$	62.1929	2.13320
$851$	−14.5036	−0.497177
$852$	36.7025	1.25741
$853$	5.30773	0.181733	0.0908666	−	0.995863i	$-0.471036\pi$
$853$	5.30773	0.181733	0.0908666	+	0.995863i	$0.471036\pi$
$854$	0	0
$855$	−17.1814	−0.587591
$856$	−64.9793	−2.22095
$857$	−16.6371	−0.568314	−0.284157	−	0.958778i	$-0.591714\pi$
$857$	−16.6371	−0.568314	−0.284157	+	0.958778i	$0.591714\pi$
$858$	0	0
$859$	−10.5885	−0.361276	−0.180638	−	0.983550i	$-0.557816\pi$
$859$	−10.5885	−0.361276	−0.180638	+	0.983550i	$0.557816\pi$
$860$	−23.2078	−0.791381
$861$	0	0
$862$	70.5142	2.40172
$863$	56.0632	1.90841	0.954207	−	0.299148i	$-0.0967025\pi$
$863$	56.0632	1.90841	0.954207	+	0.299148i	$0.0967025\pi$
$864$	11.3597	0.386464
$865$	7.98959	0.271654
$866$	52.7541	1.79266
$867$	−67.8424	−2.30405
$868$	0	0
$869$	1.63727	0.0555405
$870$	−25.5598	−0.866559
$871$	0	0
$872$	−37.5501	−1.27161
$873$	−35.1804	−1.19067
$874$	75.4863	2.55336
$875$	0	0
$876$	50.2702	1.69847
$877$	3.66051	0.123607	0.0618033	−	0.998088i	$-0.480315\pi$
$877$	3.66051	0.123607	0.0618033	+	0.998088i	$0.480315\pi$
$878$	−15.1779	−0.512229
$879$	−38.7838	−1.30815
$880$	−1.33071	−0.0448581
$881$	10.2299	0.344653	0.172326	−	0.985040i	$-0.444872\pi$
$881$	10.2299	0.344653	0.172326	+	0.985040i	$0.444872\pi$
$882$	0	0
$883$	−3.98979	−0.134267	−0.0671335	−	0.997744i	$-0.521385\pi$
$883$	−3.98979	−0.134267	−0.0671335	+	0.997744i	$0.521385\pi$
$884$	0	0
$885$	−16.2396	−0.545890
$886$	−32.3882	−1.08810
$887$	−14.2208	−0.477487	−0.238743	−	0.971083i	$-0.576735\pi$
$887$	−14.2208	−0.477487	−0.238743	+	0.971083i	$0.576735\pi$
$888$	19.6447	0.659234
$889$	0	0
$890$	−4.90986	−0.164579
$891$	−1.07052	−0.0358636
$892$	−80.7919	−2.70511
$893$	16.6110	0.555866
$894$	18.4413	0.616769
$895$	20.6206	0.689269
$896$	0	0
$897$	0	0
$898$	−52.3262	−1.74615
$899$	−11.8359	−0.394749
$900$	−68.2896	−2.27632
$901$	−2.74652	−0.0915000
$902$	−5.15392	−0.171607
$903$	0	0
$904$	41.3945	1.37676
$905$	1.57958	0.0525069
$906$	−151.196	−5.02315
$907$	43.4253	1.44191	0.720956	−	0.692981i	$-0.243703\pi$
$907$	43.4253	1.44191	0.720956	+	0.692981i	$0.243703\pi$
$908$	68.6949	2.27972
$909$	−12.0542	−0.399814
$910$	0	0
$911$	24.8617	0.823706	0.411853	−	0.911250i	$-0.364882\pi$
$911$	24.8617	0.823706	0.411853	+	0.911250i	$0.364882\pi$
$912$	−24.2677	−0.803582
$913$	1.68593	0.0557961
$914$	−36.3331	−1.20179
$915$	5.96421	0.197171
$916$	−71.6495	−2.36737
$917$	0	0
$918$	66.6076	2.19838
$919$	−1.66327	−0.0548664	−0.0274332	−	0.999624i	$-0.508733\pi$
$919$	−1.66327	−0.0548664	−0.0274332	+	0.999624i	$0.508733\pi$
$920$	−32.9914	−1.08769
$921$	43.6896	1.43962
$922$	38.7250	1.27534
$923$	0	0
$924$	0	0
$925$	7.08521	0.232960
$926$	3.45714	0.113609
$927$	−49.6458	−1.63058
$928$	10.4098	0.341717
$929$	−9.49521	−0.311528	−0.155764	−	0.987794i	$-0.549784\pi$
$929$	−9.49521	−0.311528	−0.155764	+	0.987794i	$0.549784\pi$
$930$	19.2898	0.632537
$931$	0	0
$932$	59.8486	1.96040
$933$	78.7016	2.57657
$934$	33.4486	1.09447
$935$	3.72657	0.121872
$936$	0	0
$937$	−6.41678	−0.209627	−0.104813	−	0.994492i	$-0.533425\pi$
$937$	−6.41678	−0.209627	−0.104813	+	0.994492i	$0.533425\pi$
$938$	0	0
$939$	51.1203	1.66825
$940$	−15.7872	−0.514922
$941$	51.5186	1.67946	0.839730	−	0.543005i	$-0.182713\pi$
$941$	51.5186	1.67946	0.839730	+	0.543005i	$0.182713\pi$
$942$	−148.338	−4.83312
$943$	−30.3282	−0.987621
$944$	−13.8481	−0.450717
$945$	0	0
$946$	8.95676	0.291209
$947$	8.40219	0.273034	0.136517	−	0.990638i	$-0.456409\pi$
$947$	8.40219	0.273034	0.136517	+	0.990638i	$0.456409\pi$
$948$	−28.3776	−0.921661
$949$	0	0
$950$	−36.8761	−1.19642
$951$	84.3177	2.73419
$952$	0	0
$953$	−36.0911	−1.16910	−0.584552	−	0.811356i	$-0.698730\pi$
$953$	−36.0911	−1.16910	−0.584552	+	0.811356i	$0.698730\pi$
$954$	4.64471	0.150378
$955$	11.3573	0.367515
$956$	−59.6275	−1.92849
$957$	6.40491	0.207041
$958$	71.6991	2.31649
$959$	0	0
$960$	−29.4235	−0.949639
$961$	−22.0676	−0.711857
$962$	0	0
$963$	−73.0409	−2.35371
$964$	14.8331	0.477743
$965$	23.1610	0.745580
$966$	0	0
$967$	−3.18338	−0.102371	−0.0511853	−	0.998689i	$-0.516300\pi$
$967$	−3.18338	−0.102371	−0.0511853	+	0.998689i	$0.516300\pi$
$968$	−43.3229	−1.39245
$969$	67.9602	2.18320
$970$	18.0551	0.579715
$971$	37.7476	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$971$	37.7476	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$972$	66.5591	2.13488
$973$	0	0
$974$	68.0359	2.18001
$975$	0	0
$976$	5.08589	0.162795
$977$	21.3076	0.681692	0.340846	−	0.940119i	$-0.389287\pi$
$977$	21.3076	0.681692	0.340846	+	0.940119i	$0.389287\pi$
$978$	53.7290	1.71806
$979$	1.23034	0.0393217
$980$	0	0
$981$	−42.2087	−1.34762
$982$	−67.9819	−2.16939
$983$	22.0316	0.702700	0.351350	−	0.936244i	$-0.385723\pi$
$983$	22.0316	0.702700	0.351350	+	0.936244i	$0.385723\pi$
$984$	41.0787	1.30954
$985$	−1.44488	−0.0460377
$986$	61.0378	1.94384
$987$	0	0
$988$	0	0
$989$	52.7059	1.67595
$990$	−6.30208	−0.200293
$991$	−22.0259	−0.699676	−0.349838	−	0.936810i	$-0.613763\pi$
$991$	−22.0259	−0.699676	−0.349838	+	0.936810i	$0.613763\pi$
$992$	−7.85617	−0.249434
$993$	74.9164	2.37740
$994$	0	0
$995$	−9.22753	−0.292532
$996$	−29.2210	−0.925902
$997$	10.0820	0.319301	0.159651	−	0.987174i	$-0.448963\pi$
$997$	10.0820	0.319301	0.159651	+	0.987174i	$0.448963\pi$
$998$	62.6450	1.98299
$999$	7.58815	0.240078

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	8281.2.a.ce.1.6		6
7.2	even	3		1183.2.e.g.508.1		12
7.4	even	3		1183.2.e.g.170.1		12
7.6	odd	2		8281.2.a.cf.1.6		6
13.4	even	6		637.2.f.k.393.6		12
13.10	even	6		637.2.f.k.295.6		12
13.12	even	2		8281.2.a.bz.1.1		6
91.4	even	6		91.2.h.b.16.1	yes	12
91.10	odd	6		637.2.g.l.373.6		12
91.17	odd	6		637.2.h.l.471.1		12
91.23	even	6		91.2.h.b.74.1	yes	12
91.25	even	6		1183.2.e.h.170.6		12
91.30	even	6		91.2.g.b.81.6	yes	12
91.51	even	6		1183.2.e.h.508.6		12
91.62	odd	6		637.2.f.j.295.6		12
91.69	odd	6		637.2.f.j.393.6		12
91.75	odd	6		637.2.h.l.165.1		12
91.82	odd	6		637.2.g.l.263.6		12
91.88	even	6		91.2.g.b.9.6	✓	12
91.90	odd	2		8281.2.a.ca.1.1		6
273.23	odd	6		819.2.s.d.802.6		12
273.95	odd	6		819.2.s.d.289.6		12
273.179	odd	6		819.2.n.d.100.1		12
273.212	odd	6		819.2.n.d.172.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.6	✓	12	91.88	even	6
91.2.g.b.81.6	yes	12	91.30	even	6
91.2.h.b.16.1	yes	12	91.4	even	6
91.2.h.b.74.1	yes	12	91.23	even	6
637.2.f.j.295.6		12	91.62	odd	6
637.2.f.j.393.6		12	91.69	odd	6
637.2.f.k.295.6		12	13.10	even	6
637.2.f.k.393.6		12	13.4	even	6
637.2.g.l.263.6		12	91.82	odd	6
637.2.g.l.373.6		12	91.10	odd	6
637.2.h.l.165.1		12	91.75	odd	6
637.2.h.l.471.1		12	91.17	odd	6
819.2.n.d.100.1		12	273.179	odd	6
819.2.n.d.172.1		12	273.212	odd	6
819.2.s.d.289.6		12	273.95	odd	6
819.2.s.d.802.6		12	273.23	odd	6
1183.2.e.g.170.1		12	7.4	even	3
1183.2.e.g.508.1		12	7.2	even	3
1183.2.e.h.170.6		12	91.25	even	6
1183.2.e.h.508.6		12	91.51	even	6
8281.2.a.bz.1.1		6	13.12	even	2
8281.2.a.ca.1.1		6	91.90	odd	2
8281.2.a.ce.1.6		6	1.1	even	1	trivial
8281.2.a.cf.1.6		6	7.6	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 $\F_{q}$
Belyi maps

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	8281.2.a.ce.1.6

Level	$8281$
Weight	$2$
Character	8281.1
Self dual	yes
Analytic conductor	$66.124$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$