LMFDB - Embedded newform 8281.2.a.cl.1.8

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			1.8
Root			$-1.11758$ 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)

$f(q)$	$=$	\(q+2.33152 q^{2} +2.30901 q^{3} +3.43596 q^{4} -3.37112 q^{5} +5.38349 q^{6} +3.34797 q^{8} +2.33152 q^{9} -7.85981 q^{10} -2.33152 q^{11} +7.93366 q^{12} -7.78393 q^{15} +0.933914 q^{16} -5.45734 q^{17} +5.43596 q^{18} +7.16819 q^{19} -11.5830 q^{20} -5.43596 q^{22} -6.45242 q^{23} +7.73048 q^{24} +6.36442 q^{25} -1.54354 q^{27} -8.44287 q^{29} -18.1484 q^{30} +3.05121 q^{31} -4.51850 q^{32} -5.38349 q^{33} -12.7239 q^{34} +8.01100 q^{36} -3.05653 q^{37} +16.7127 q^{38} -11.2864 q^{40} +0.937666 q^{41} -4.09209 q^{43} -8.01100 q^{44} -7.85981 q^{45} -15.0439 q^{46} +3.46336 q^{47} +2.15642 q^{48} +14.8387 q^{50} -12.6010 q^{51} -2.34387 q^{53} -3.59878 q^{54} +7.85981 q^{55} +16.5514 q^{57} -19.6847 q^{58} -7.24693 q^{59} -26.7453 q^{60} +6.39013 q^{61} +7.11394 q^{62} -12.4028 q^{64} -12.5517 q^{66} -4.61340 q^{67} -18.7512 q^{68} -14.8987 q^{69} +7.58739 q^{71} +7.80584 q^{72} +2.06996 q^{73} -7.12636 q^{74} +14.6955 q^{75} +24.6296 q^{76} -7.58868 q^{79} -3.14833 q^{80} -10.5586 q^{81} +2.18618 q^{82} +2.89335 q^{83} +18.3973 q^{85} -9.54077 q^{86} -19.4946 q^{87} -7.80584 q^{88} -13.1597 q^{89} -18.3253 q^{90} -22.1703 q^{92} +7.04526 q^{93} +8.07488 q^{94} -24.1648 q^{95} -10.4333 q^{96} +3.55712 q^{97} -5.43596 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	2.33152	1.64863	0.824315	−	0.566131i	$-0.191560\pi$
$2$	2.33152	1.64863	0.824315	+	0.566131i	$0.191560\pi$
$3$	2.30901	1.33311	0.666553	−	0.745458i	$-0.267769\pi$
$3$	2.30901	1.33311	0.666553	+	0.745458i	$0.267769\pi$
$4$	3.43596	1.71798
$5$	−3.37112	−1.50761	−0.753804	−	0.657099i	$-0.771783\pi$
$5$	−3.37112	−1.50761	−0.753804	+	0.657099i	$0.771783\pi$
$6$	5.38349	2.19780
$7$	0	0
$7$	0	0
$8$	3.34797	1.18369
$9$	2.33152	0.777172
$10$	−7.85981	−2.48549
$11$	−2.33152	−0.702978	−0.351489	−	0.936192i	$-0.614325\pi$
$11$	−2.33152	−0.702978	−0.351489	+	0.936192i	$0.614325\pi$
$12$	7.93366	2.29025
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−7.78393	−2.00980
$16$	0.933914	0.233479
$17$	−5.45734	−1.32360	−0.661800	−	0.749681i	$-0.730207\pi$
$17$	−5.45734	−1.32360	−0.661800	+	0.749681i	$0.730207\pi$
$18$	5.43596	1.28127
$19$	7.16819	1.64450	0.822248	−	0.569129i	$-0.192720\pi$
$19$	7.16819	1.64450	0.822248	+	0.569129i	$0.192720\pi$
$20$	−11.5830	−2.59004
$21$	0	0
$22$	−5.43596	−1.15895
$23$	−6.45242	−1.34542	−0.672711	−	0.739905i	$-0.734870\pi$
$23$	−6.45242	−1.34542	−0.672711	+	0.739905i	$0.734870\pi$
$24$	7.73048	1.57798
$25$	6.36442	1.27288
$26$	0	0
$27$	−1.54354	−0.297054
$28$	0	0
$29$	−8.44287	−1.56780	−0.783900	−	0.620887i	$-0.786773\pi$
$29$	−8.44287	−1.56780	−0.783900	+	0.620887i	$0.786773\pi$
$30$	−18.1484	−3.31342
$31$	3.05121	0.548013	0.274007	−	0.961728i	$-0.411651\pi$
$31$	3.05121	0.548013	0.274007	+	0.961728i	$0.411651\pi$
$32$	−4.51850	−0.798766
$33$	−5.38349	−0.937145
$34$	−12.7239	−2.18213
$35$	0	0
$36$	8.01100	1.33517
$37$	−3.05653	−0.502491	−0.251246	−	0.967923i	$-0.580840\pi$
$37$	−3.05653	−0.502491	−0.251246	+	0.967923i	$0.580840\pi$
$38$	16.7127	2.71117
$39$	0	0
$40$	−11.2864	−1.78454
$41$	0.937666	0.146439	0.0732194	−	0.997316i	$-0.476673\pi$
$41$	0.937666	0.146439	0.0732194	+	0.997316i	$0.476673\pi$
$42$	0	0
$43$	−4.09209	−0.624038	−0.312019	−	0.950076i	$-0.601005\pi$
$43$	−4.09209	−0.624038	−0.312019	+	0.950076i	$0.601005\pi$
$44$	−8.01100	−1.20770
$45$	−7.85981	−1.17167
$46$	−15.0439	−2.21810
$47$	3.46336	0.505183	0.252591	−	0.967573i	$-0.418717\pi$
$47$	3.46336	0.505183	0.252591	+	0.967573i	$0.418717\pi$
$48$	2.15642	0.311252
$49$	0	0
$50$	14.8387	2.09852
$51$	−12.6010	−1.76450
$52$	0	0
$53$	−2.34387	−0.321956	−0.160978	−	0.986958i	$-0.551465\pi$
$53$	−2.34387	−0.321956	−0.160978	+	0.986958i	$0.551465\pi$
$54$	−3.59878	−0.489732
$55$	7.85981	1.05982
$56$	0	0
$57$	16.5514	2.19229
$58$	−19.6847	−2.58472
$59$	−7.24693	−0.943470	−0.471735	−	0.881740i	$-0.656372\pi$
$59$	−7.24693	−0.943470	−0.471735	+	0.881740i	$0.656372\pi$
$60$	−26.7453	−3.45280
$61$	6.39013	0.818172	0.409086	−	0.912496i	$-0.365848\pi$
$61$	6.39013	0.818172	0.409086	+	0.912496i	$0.365848\pi$
$62$	7.11394	0.903471
$63$	0	0
$64$	−12.4028	−1.55035
$65$	0	0
$66$	−12.5517	−1.54500
$67$	−4.61340	−0.563616	−0.281808	−	0.959471i	$-0.590934\pi$
$67$	−4.61340	−0.563616	−0.281808	+	0.959471i	$0.590934\pi$
$68$	−18.7512	−2.27392
$69$	−14.8987	−1.79359
$70$	0	0
$71$	7.58739	0.900458	0.450229	−	0.892913i	$-0.351342\pi$
$71$	7.58739	0.900458	0.450229	+	0.892913i	$0.351342\pi$
$72$	7.80584	0.919927
$73$	2.06996	0.242271	0.121136	−	0.992636i	$-0.461346\pi$
$73$	2.06996	0.242271	0.121136	+	0.992636i	$0.461346\pi$
$74$	−7.12636	−0.828422
$75$	14.6955	1.69689
$76$	24.6296	2.82521
$77$	0	0
$78$	0	0
$79$	−7.58868	−0.853794	−0.426897	−	0.904300i	$-0.640393\pi$
$79$	−7.58868	−0.853794	−0.426897	+	0.904300i	$0.640393\pi$
$80$	−3.14833	−0.351994
$81$	−10.5586	−1.17318
$82$	2.18618	0.241424
$83$	2.89335	0.317587	0.158793	−	0.987312i	$-0.449240\pi$
$83$	2.89335	0.317587	0.158793	+	0.987312i	$0.449240\pi$
$84$	0	0
$85$	18.3973	1.99547
$86$	−9.54077	−1.02881
$87$	−19.4946	−2.09004
$88$	−7.80584	−0.832105
$89$	−13.1597	−1.39492	−0.697461	−	0.716622i	$-0.745687\pi$
$89$	−13.1597	−1.39492	−0.697461	+	0.716622i	$0.745687\pi$
$90$	−18.3253	−1.93165
$91$	0	0
$92$	−22.1703	−2.31141
$93$	7.04526	0.730560
$94$	8.07488	0.832860
$95$	−24.1648	−2.47926
$96$	−10.4333	−1.06484
$97$	3.55712	0.361171	0.180585	−	0.983559i	$-0.442201\pi$
$97$	3.55712	0.361171	0.180585	+	0.983559i	$0.442201\pi$
$98$	0	0
$99$	−5.43596	−0.546335
$100$	21.8679	2.18679
$101$	4.72865	0.470518	0.235259	−	0.971933i	$-0.424406\pi$
$101$	4.72865	0.470518	0.235259	+	0.971933i	$0.424406\pi$
$102$	−29.3795	−2.90901
$103$	−5.98286	−0.589508	−0.294754	−	0.955573i	$-0.595238\pi$
$103$	−5.98286	−0.589508	−0.294754	+	0.955573i	$0.595238\pi$
$104$	0	0
$105$	0	0
$106$	−5.46477	−0.530786
$107$	13.1826	1.27441	0.637206	−	0.770694i	$-0.280090\pi$
$107$	13.1826	1.27441	0.637206	+	0.770694i	$0.280090\pi$
$108$	−5.30353	−0.510333
$109$	−4.11545	−0.394188	−0.197094	−	0.980385i	$-0.563150\pi$
$109$	−4.11545	−0.394188	−0.197094	+	0.980385i	$0.563150\pi$
$110$	18.3253	1.74725
$111$	−7.05756	−0.669874
$112$	0	0
$113$	14.2805	1.34340	0.671699	−	0.740824i	$-0.265565\pi$
$113$	14.2805	1.34340	0.671699	+	0.740824i	$0.265565\pi$
$114$	38.5899	3.61427
$115$	21.7518	2.02837
$116$	−29.0094	−2.69345
$117$	0	0
$118$	−16.8963	−1.55543
$119$	0	0
$120$	−26.0604	−2.37897
$121$	−5.56404	−0.505822
$122$	14.8987	1.34886
$123$	2.16508	0.195219
$124$	10.4838	0.941477
$125$	−4.59963	−0.411403
$126$	0	0
$127$	−11.0771	−0.982932	−0.491466	−	0.870897i	$-0.663539\pi$
$127$	−11.0771	−0.982932	−0.491466	+	0.870897i	$0.663539\pi$
$128$	−19.8803	−1.75718
$129$	−9.44867	−0.831909
$130$	0	0
$131$	−0.672012	−0.0587139	−0.0293570	−	0.999569i	$-0.509346\pi$
$131$	−0.672012	−0.0587139	−0.0293570	+	0.999569i	$0.509346\pi$
$132$	−18.4975	−1.61000
$133$	0	0
$134$	−10.7562	−0.929195
$135$	5.20344	0.447841
$136$	−18.2710	−1.56673
$137$	−7.83875	−0.669709	−0.334855	−	0.942270i	$-0.608687\pi$
$137$	−7.83875	−0.669709	−0.334855	+	0.942270i	$0.608687\pi$
$138$	−34.7365	−2.95697
$139$	3.27521	0.277799	0.138900	−	0.990306i	$-0.455643\pi$
$139$	3.27521	0.277799	0.138900	+	0.990306i	$0.455643\pi$
$140$	0	0
$141$	7.99692	0.673462
$142$	17.6901	1.48452
$143$	0	0
$144$	2.17744	0.181453
$145$	28.4619	2.36363
$146$	4.82615	0.399415
$147$	0	0
$148$	−10.5021	−0.863270
$149$	14.1637	1.16034	0.580169	−	0.814496i	$-0.302987\pi$
$149$	14.1637	1.16034	0.580169	+	0.814496i	$0.302987\pi$
$150$	34.2628	2.79754
$151$	1.34625	0.109556	0.0547781	−	0.998499i	$-0.482555\pi$
$151$	1.34625	0.109556	0.0547781	+	0.998499i	$0.482555\pi$
$152$	23.9989	1.94657
$153$	−12.7239	−1.02866
$154$	0	0
$155$	−10.2860	−0.826190
$156$	0	0
$157$	−13.0401	−1.04071	−0.520357	−	0.853949i	$-0.674201\pi$
$157$	−13.0401	−1.04071	−0.520357	+	0.853949i	$0.674201\pi$
$158$	−17.6931	−1.40759
$159$	−5.41202	−0.429201
$160$	15.2324	1.20423
$161$	0	0
$162$	−24.6175	−1.93413
$163$	−4.93256	−0.386348	−0.193174	−	0.981165i	$-0.561878\pi$
$163$	−4.93256	−0.386348	−0.193174	+	0.981165i	$0.561878\pi$
$164$	3.22179	0.251579
$165$	18.1484	1.41285
$166$	6.74590	0.523583
$167$	−3.64256	−0.281870	−0.140935	−	0.990019i	$-0.545011\pi$
$167$	−3.64256	−0.281870	−0.140935	+	0.990019i	$0.545011\pi$
$168$	0	0
$169$	0	0
$170$	42.8937	3.28979
$171$	16.7127	1.27806
$172$	−14.0603	−1.07209
$173$	12.6917	0.964930	0.482465	−	0.875915i	$-0.339742\pi$
$173$	12.6917	0.964930	0.482465	+	0.875915i	$0.339742\pi$
$174$	−45.4520	−3.44571
$175$	0	0
$176$	−2.17744	−0.164130
$177$	−16.7332	−1.25775
$178$	−30.6820	−2.29971
$179$	−8.78939	−0.656950	−0.328475	−	0.944513i	$-0.606535\pi$
$179$	−8.78939	−0.656950	−0.328475	+	0.944513i	$0.606535\pi$
$180$	−27.0060	−2.01291
$181$	−17.1982	−1.27833	−0.639167	−	0.769068i	$-0.720721\pi$
$181$	−17.1982	−1.27833	−0.639167	+	0.769068i	$0.720721\pi$
$182$	0	0
$183$	14.7548	1.09071
$184$	−21.6025	−1.59256
$185$	10.3039	0.757560
$186$	16.4261	1.20442
$187$	12.7239	0.930462
$188$	11.9000	0.867895
$189$	0	0
$190$	−56.3406	−4.08738
$191$	1.06780	0.0772636	0.0386318	−	0.999254i	$-0.487700\pi$
$191$	1.06780	0.0772636	0.0386318	+	0.999254i	$0.487700\pi$
$192$	−28.6381	−2.06678
$193$	−3.15580	−0.227159	−0.113580	−	0.993529i	$-0.536232\pi$
$193$	−3.15580	−0.227159	−0.113580	+	0.993529i	$0.536232\pi$
$194$	8.29348	0.595437
$195$	0	0
$196$	0	0
$197$	17.6957	1.26076	0.630382	−	0.776285i	$-0.282898\pi$
$197$	17.6957	1.26076	0.630382	+	0.776285i	$0.282898\pi$
$198$	−12.6740	−0.900704
$199$	12.9895	0.920803	0.460402	−	0.887711i	$-0.347705\pi$
$199$	12.9895	0.920803	0.460402	+	0.887711i	$0.347705\pi$
$200$	21.3079	1.50670
$201$	−10.6524	−0.751360
$202$	11.0249	0.775710
$203$	0	0
$204$	−43.2967	−3.03138
$205$	−3.16098	−0.220773
$206$	−13.9491	−0.971881
$207$	−15.0439	−1.04562
$208$	0	0
$209$	−16.7127	−1.15605
$210$	0	0
$211$	27.5402	1.89595	0.947974	−	0.318348i	$-0.103128\pi$
$211$	27.5402	1.89595	0.947974	+	0.318348i	$0.103128\pi$
$212$	−8.05346	−0.553114
$213$	17.5193	1.20041
$214$	30.7355	2.10103
$215$	13.7949	0.940805
$216$	−5.16771	−0.351618
$217$	0	0
$218$	−9.59523	−0.649871
$219$	4.77956	0.322973
$220$	27.0060	1.82074
$221$	0	0
$222$	−16.4548	−1.10437
$223$	8.70197	0.582727	0.291363	−	0.956612i	$-0.405891\pi$
$223$	8.70197	0.582727	0.291363	+	0.956612i	$0.405891\pi$
$224$	0	0
$225$	14.8387	0.989250
$226$	33.2953	2.21477
$227$	21.9669	1.45800	0.728998	−	0.684516i	$-0.239987\pi$
$227$	21.9669	1.45800	0.728998	+	0.684516i	$0.239987\pi$
$228$	56.8700	3.76631
$229$	−20.5188	−1.35592	−0.677959	−	0.735100i	$-0.737135\pi$
$229$	−20.5188	−1.35592	−0.677959	+	0.735100i	$0.737135\pi$
$230$	50.7148	3.34403
$231$	0	0
$232$	−28.2664	−1.85578
$233$	−17.2888	−1.13263	−0.566313	−	0.824190i	$-0.691631\pi$
$233$	−17.2888	−1.13263	−0.566313	+	0.824190i	$0.691631\pi$
$234$	0	0
$235$	−11.6754	−0.761618
$236$	−24.9002	−1.62086
$237$	−17.5223	−1.13820
$238$	0	0
$239$	−3.25961	−0.210847	−0.105423	−	0.994427i	$-0.533620\pi$
$239$	−3.25961	−0.210847	−0.105423	+	0.994427i	$0.533620\pi$
$240$	−7.26953	−0.469246
$241$	−1.46571	−0.0944149	−0.0472074	−	0.998885i	$-0.515032\pi$
$241$	−1.46571	−0.0944149	−0.0472074	+	0.998885i	$0.515032\pi$
$242$	−12.9726	−0.833913
$243$	−19.7492	−1.26691
$244$	21.9562	1.40560
$245$	0	0
$246$	5.04791	0.321843
$247$	0	0
$248$	10.2154	0.648676
$249$	6.68077	0.423377
$250$	−10.7241	−0.678252
$251$	−17.1028	−1.07952	−0.539761	−	0.841818i	$-0.681485\pi$
$251$	−17.1028	−1.07952	−0.539761	+	0.841818i	$0.681485\pi$
$252$	0	0
$253$	15.0439	0.945802
$254$	−25.8264	−1.62049
$255$	42.4796	2.66017
$256$	−21.5456	−1.34660
$257$	−3.85011	−0.240163	−0.120082	−	0.992764i	$-0.538316\pi$
$257$	−3.85011	−0.240163	−0.120082	+	0.992764i	$0.538316\pi$
$258$	−22.0297	−1.37151
$259$	0	0
$260$	0	0
$261$	−19.6847	−1.21845
$262$	−1.56681	−0.0967976
$263$	−30.5159	−1.88169	−0.940844	−	0.338840i	$-0.889966\pi$
$263$	−30.5159	−1.88169	−0.940844	+	0.338840i	$0.889966\pi$
$264$	−18.0237	−1.10928
$265$	7.90147	0.485383
$266$	0	0
$267$	−30.3858	−1.85958
$268$	−15.8515	−0.968282
$269$	−16.7771	−1.02292	−0.511460	−	0.859307i	$-0.670895\pi$
$269$	−16.7771	−1.02292	−0.511460	+	0.859307i	$0.670895\pi$
$270$	12.1319	0.738324
$271$	31.1772	1.89388	0.946941	−	0.321407i	$-0.104156\pi$
$271$	31.1772	1.89388	0.946941	+	0.321407i	$0.104156\pi$
$272$	−5.09669	−0.309032
$273$	0	0
$274$	−18.2762	−1.10410
$275$	−14.8387	−0.894810
$276$	−51.1913	−3.08135
$277$	0.395882	0.0237863	0.0118931	−	0.999929i	$-0.496214\pi$
$277$	0.395882	0.0237863	0.0118931	+	0.999929i	$0.496214\pi$
$278$	7.63619	0.457988
$279$	7.11394	0.425900
$280$	0	0
$281$	22.9459	1.36884	0.684420	−	0.729088i	$-0.260056\pi$
$281$	22.9459	1.36884	0.684420	+	0.729088i	$0.260056\pi$
$282$	18.6449	1.11029
$283$	−19.3515	−1.15033	−0.575164	−	0.818038i	$-0.695062\pi$
$283$	−19.3515	−1.15033	−0.575164	+	0.818038i	$0.695062\pi$
$284$	26.0700	1.54697
$285$	−55.7967	−3.30511
$286$	0	0
$287$	0	0
$288$	−10.5350	−0.620778
$289$	12.7826	0.751916
$290$	66.3593	3.89675
$291$	8.21342	0.481479
$292$	7.11232	0.416217
$293$	−15.7723	−0.921429	−0.460715	−	0.887548i	$-0.652407\pi$
$293$	−15.7723	−0.921429	−0.460715	+	0.887548i	$0.652407\pi$
$294$	0	0
$295$	24.4302	1.42238
$296$	−10.2332	−0.594792
$297$	3.59878	0.208822
$298$	33.0229	1.91297
$299$	0	0
$300$	50.4932	2.91523
$301$	0	0
$302$	3.13880	0.180618
$303$	10.9185	0.627250
$304$	6.69448	0.383955
$305$	−21.5419	−1.23348
$306$	−29.6659	−1.69589
$307$	19.2535	1.09885	0.549427	−	0.835542i	$-0.314846\pi$
$307$	19.2535	1.09885	0.549427	+	0.835542i	$0.314846\pi$
$308$	0	0
$309$	−13.8145	−0.785877
$310$	−23.9819	−1.36208
$311$	−3.06464	−0.173780	−0.0868898	−	0.996218i	$-0.527693\pi$
$311$	−3.06464	−0.173780	−0.0868898	+	0.996218i	$0.527693\pi$
$312$	0	0
$313$	35.2751	1.99387	0.996934	−	0.0782450i	$-0.0249316\pi$
$313$	35.2751	1.99387	0.996934	+	0.0782450i	$0.0249316\pi$
$314$	−30.4032	−1.71575
$315$	0	0
$316$	−26.0744	−1.46680
$317$	−32.7065	−1.83698	−0.918490	−	0.395444i	$-0.870591\pi$
$317$	−32.7065	−1.83698	−0.918490	+	0.395444i	$0.870591\pi$
$318$	−12.6182	−0.707594
$319$	19.6847	1.10213
$320$	41.8112	2.33732
$321$	30.4388	1.69893
$322$	0	0
$323$	−39.1193	−2.17665
$324$	−36.2789	−2.01549
$325$	0	0
$326$	−11.5003	−0.636944
$327$	−9.50260	−0.525495
$328$	3.13928	0.173338
$329$	0	0
$330$	42.3132	2.32926
$331$	−4.77703	−0.262569	−0.131285	−	0.991345i	$-0.541910\pi$
$331$	−4.77703	−0.262569	−0.131285	+	0.991345i	$0.541910\pi$
$332$	9.94145	0.545608
$333$	−7.12636	−0.390522
$334$	−8.49269	−0.464699
$335$	15.5523	0.849713
$336$	0	0
$337$	26.2392	1.42934	0.714669	−	0.699463i	$-0.246577\pi$
$337$	26.2392	1.42934	0.714669	+	0.699463i	$0.246577\pi$
$338$	0	0
$339$	32.9738	1.79089
$340$	63.2125	3.42818
$341$	−7.11394	−0.385241
$342$	38.9660	2.10704
$343$	0	0
$344$	−13.7002	−0.738665
$345$	50.2252	2.70403
$346$	29.5908	1.59081
$347$	−10.4442	−0.560675	−0.280338	−	0.959901i	$-0.590446\pi$
$347$	−10.4442	−0.560675	−0.280338	+	0.959901i	$0.590446\pi$
$348$	−66.9829	−3.59066
$349$	23.4688	1.25626	0.628128	−	0.778110i	$-0.283822\pi$
$349$	23.4688	1.25626	0.628128	+	0.778110i	$0.283822\pi$
$350$	0	0
$351$	0	0
$352$	10.5350	0.561515
$353$	2.93135	0.156020	0.0780099	−	0.996953i	$-0.475143\pi$
$353$	2.93135	0.156020	0.0780099	+	0.996953i	$0.475143\pi$
$354$	−39.0137	−2.07356
$355$	−25.5780	−1.35754
$356$	−45.2162	−2.39645
$357$	0	0
$358$	−20.4926	−1.08307
$359$	17.1014	0.902576	0.451288	−	0.892378i	$-0.350965\pi$
$359$	17.1014	0.902576	0.451288	+	0.892378i	$0.350965\pi$
$360$	−26.3144	−1.38689
$361$	32.3830	1.70437
$362$	−40.0979	−2.10750
$363$	−12.8474	−0.674314
$364$	0	0
$365$	−6.97809	−0.365250
$366$	34.4012	1.79818
$367$	1.04860	0.0547365	0.0273683	−	0.999625i	$-0.491287\pi$
$367$	1.04860	0.0547365	0.0273683	+	0.999625i	$0.491287\pi$
$368$	−6.02600	−0.314127
$369$	2.18618	0.113808
$370$	24.0238	1.24894
$371$	0	0
$372$	24.2073	1.25509
$373$	−15.0107	−0.777226	−0.388613	−	0.921401i	$-0.627046\pi$
$373$	−15.0107	−0.777226	−0.388613	+	0.921401i	$0.627046\pi$
$374$	29.6659	1.53399
$375$	−10.6206	−0.548444
$376$	11.5952	0.597978
$377$	0	0
$378$	0	0
$379$	27.1497	1.39459	0.697294	−	0.716786i	$-0.254387\pi$
$379$	27.1497	1.39459	0.697294	+	0.716786i	$0.254387\pi$
$380$	−83.0294	−4.25932
$381$	−25.5771	−1.31035
$382$	2.48960	0.127379
$383$	−16.0264	−0.818911	−0.409455	−	0.912330i	$-0.634281\pi$
$383$	−16.0264	−0.818911	−0.409455	+	0.912330i	$0.634281\pi$
$384$	−45.9037	−2.34251
$385$	0	0
$386$	−7.35779	−0.374502
$387$	−9.54077	−0.484985
$388$	12.2221	0.620485
$389$	7.99427	0.405326	0.202663	−	0.979249i	$-0.435040\pi$
$389$	7.99427	0.405326	0.202663	+	0.979249i	$0.435040\pi$
$390$	0	0
$391$	35.2130	1.78080
$392$	0	0
$393$	−1.55168	−0.0782719
$394$	41.2577	2.07853
$395$	25.5823	1.28719
$396$	−18.6778	−0.938593
$397$	−12.2087	−0.612738	−0.306369	−	0.951913i	$-0.599114\pi$
$397$	−12.2087	−0.612738	−0.306369	+	0.951913i	$0.599114\pi$
$398$	30.2853	1.51806
$399$	0	0
$400$	5.94383	0.297191
$401$	−37.8412	−1.88970	−0.944850	−	0.327503i	$-0.893793\pi$
$401$	−37.8412	−1.88970	−0.944850	+	0.327503i	$0.893793\pi$
$402$	−24.8362	−1.23872
$403$	0	0
$404$	16.2475	0.808341
$405$	35.5942	1.76869
$406$	0	0
$407$	7.12636	0.353240
$408$	−42.1879	−2.08861
$409$	0.235074	0.0116237	0.00581184	−	0.999983i	$-0.498150\pi$
$409$	0.235074	0.0116237	0.00581184	+	0.999983i	$0.498150\pi$
$410$	−7.36988	−0.363972
$411$	−18.0997	−0.892794
$412$	−20.5569	−1.01276
$413$	0	0
$414$	−35.0751	−1.72385
$415$	−9.75383	−0.478797
$416$	0	0
$417$	7.56248	0.370336
$418$	−38.9660	−1.90589
$419$	−0.444046	−0.0216931	−0.0108465	−	0.999941i	$-0.503453\pi$
$419$	−0.444046	−0.0216931	−0.0108465	+	0.999941i	$0.503453\pi$
$420$	0	0
$421$	−9.45998	−0.461051	−0.230526	−	0.973066i	$-0.574045\pi$
$421$	−9.45998	−0.461051	−0.230526	+	0.973066i	$0.574045\pi$
$422$	64.2105	3.12572
$423$	8.07488	0.392614
$424$	−7.84721	−0.381094
$425$	−34.7328	−1.68479
$426$	40.8466	1.97903
$427$	0	0
$428$	45.2950	2.18942
$429$	0	0
$430$	32.1630	1.55104
$431$	−21.5446	−1.03777	−0.518883	−	0.854845i	$-0.673652\pi$
$431$	−21.5446	−1.03777	−0.518883	+	0.854845i	$0.673652\pi$
$432$	−1.44153	−0.0693557
$433$	13.1590	0.632380	0.316190	−	0.948696i	$-0.397596\pi$
$433$	13.1590	0.632380	0.316190	+	0.948696i	$0.397596\pi$
$434$	0	0
$435$	65.7187	3.15097
$436$	−14.1405	−0.677208
$437$	−46.2522	−2.21254
$438$	11.1436	0.532463
$439$	29.7847	1.42155	0.710773	−	0.703422i	$-0.248345\pi$
$439$	29.7847	1.42155	0.710773	+	0.703422i	$0.248345\pi$
$440$	26.3144	1.25449
$441$	0	0
$442$	0	0
$443$	14.8467	0.705387	0.352693	−	0.935739i	$-0.385266\pi$
$443$	14.8467	0.705387	0.352693	+	0.935739i	$0.385266\pi$
$444$	−24.2495	−1.15083
$445$	44.3628	2.10300
$446$	20.2888	0.960701
$447$	32.7041	1.54685
$448$	0	0
$449$	26.2227	1.23753	0.618763	−	0.785578i	$-0.287634\pi$
$449$	26.2227	1.23753	0.618763	+	0.785578i	$0.287634\pi$
$450$	34.5968	1.63091
$451$	−2.18618	−0.102943
$452$	49.0674	2.30793
$453$	3.10850	0.146050
$454$	51.2162	2.40369
$455$	0	0
$456$	55.4136	2.59498
$457$	15.8708	0.742406	0.371203	−	0.928552i	$-0.378945\pi$
$457$	15.8708	0.742406	0.371203	+	0.928552i	$0.378945\pi$
$458$	−47.8398	−2.23541
$459$	8.42361	0.393180
$460$	74.7385	3.48470
$461$	22.0886	1.02877	0.514384	−	0.857560i	$-0.328020\pi$
$461$	22.0886	1.02877	0.514384	+	0.857560i	$0.328020\pi$
$462$	0	0
$463$	−18.2887	−0.849949	−0.424974	−	0.905205i	$-0.639717\pi$
$463$	−18.2887	−0.849949	−0.424974	+	0.905205i	$0.639717\pi$
$464$	−7.88491	−0.366048
$465$	−23.7504	−1.10140
$466$	−40.3091	−1.86728
$467$	−4.53318	−0.209771	−0.104885	−	0.994484i	$-0.533448\pi$
$467$	−4.53318	−0.209771	−0.104885	+	0.994484i	$0.533448\pi$
$468$	0	0
$469$	0	0
$470$	−27.2213	−1.25563
$471$	−30.1097	−1.38738
$472$	−24.2625	−1.11677
$473$	9.54077	0.438685
$474$	−40.8536	−1.87647
$475$	45.6214	2.09325
$476$	0	0
$477$	−5.46477	−0.250215
$478$	−7.59984	−0.347609
$479$	−10.4461	−0.477293	−0.238646	−	0.971107i	$-0.576704\pi$
$479$	−10.4461	−0.477293	−0.238646	+	0.971107i	$0.576704\pi$
$480$	35.1717	1.60536
$481$	0	0
$482$	−3.41733	−0.155655
$483$	0	0
$484$	−19.1178	−0.868992
$485$	−11.9915	−0.544505
$486$	−46.0456	−2.08867
$487$	−35.9143	−1.62743	−0.813715	−	0.581263i	$-0.802559\pi$
$487$	−35.9143	−1.62743	−0.813715	+	0.581263i	$0.802559\pi$
$488$	21.3939	0.968458
$489$	−11.3893	−0.515042
$490$	0	0
$491$	7.70248	0.347608	0.173804	−	0.984780i	$-0.444394\pi$
$491$	7.70248	0.347608	0.173804	+	0.984780i	$0.444394\pi$
$492$	7.43913	0.335382
$493$	46.0756	2.07514
$494$	0	0
$495$	18.3253	0.823659
$496$	2.84957	0.127949
$497$	0	0
$498$	15.5763	0.697992
$499$	−8.49078	−0.380099	−0.190050	−	0.981774i	$-0.560865\pi$
$499$	−8.49078	−0.380099	−0.190050	+	0.981774i	$0.560865\pi$
$500$	−15.8041	−0.706783
$501$	−8.41071	−0.375763
$502$	−39.8755	−1.77973
$503$	−30.4001	−1.35547	−0.677736	−	0.735305i	$-0.737039\pi$
$503$	−30.4001	−1.35547	−0.677736	+	0.735305i	$0.737039\pi$
$504$	0	0
$505$	−15.9408	−0.709357
$506$	35.0751	1.55928
$507$	0	0
$508$	−38.0604	−1.68866
$509$	−15.5801	−0.690578	−0.345289	−	0.938496i	$-0.612219\pi$
$509$	−15.5801	−0.690578	−0.345289	+	0.938496i	$0.612219\pi$
$510$	99.0418	4.38564
$511$	0	0
$512$	−10.4733	−0.462860
$513$	−11.0644	−0.488504
$514$	−8.97660	−0.395941
$515$	20.1689	0.888748
$516$	−32.4653	−1.42920
$517$	−8.07488	−0.355133
$518$	0	0
$519$	29.3052	1.28635
$520$	0	0
$521$	−27.1574	−1.18979	−0.594893	−	0.803805i	$-0.702806\pi$
$521$	−27.1574	−1.18979	−0.594893	+	0.803805i	$0.702806\pi$
$522$	−45.8951	−2.00877
$523$	18.0575	0.789598	0.394799	−	0.918767i	$-0.370814\pi$
$523$	18.0575	0.789598	0.394799	+	0.918767i	$0.370814\pi$
$524$	−2.30901	−0.100869
$525$	0	0
$526$	−71.1482	−3.10221
$527$	−16.6515	−0.725350
$528$	−5.02772	−0.218803
$529$	18.6337	0.810160
$530$	18.4224	0.800217
$531$	−16.8963	−0.733238
$532$	0	0
$533$	0	0
$534$	−70.8449	−3.06576
$535$	−44.4401	−1.92131
$536$	−15.4455	−0.667145
$537$	−20.2948	−0.875783
$538$	−39.1162	−1.68642
$539$	0	0
$540$	17.8788	0.769382
$541$	−39.9882	−1.71923	−0.859613	−	0.510945i	$-0.829295\pi$
$541$	−39.9882	−1.71923	−0.859613	+	0.510945i	$0.829295\pi$
$542$	72.6902	3.12231
$543$	−39.7108	−1.70415
$544$	24.6590	1.05725
$545$	13.8736	0.594282
$546$	0	0
$547$	−22.6124	−0.966836	−0.483418	−	0.875390i	$-0.660605\pi$
$547$	−22.6124	−0.966836	−0.483418	+	0.875390i	$0.660605\pi$
$548$	−26.9336	−1.15055
$549$	14.8987	0.635860
$550$	−34.5968	−1.47521
$551$	−60.5201	−2.57824
$552$	−49.8803	−2.12305
$553$	0	0
$554$	0.923005	0.0392147
$555$	23.7919	1.00991
$556$	11.2535	0.477254
$557$	9.97372	0.422600	0.211300	−	0.977421i	$-0.432230\pi$
$557$	9.97372	0.422600	0.211300	+	0.977421i	$0.432230\pi$
$558$	16.5863	0.702152
$559$	0	0
$560$	0	0
$561$	29.3795	1.24040
$562$	53.4988	2.25671
$563$	21.5279	0.907294	0.453647	−	0.891181i	$-0.350123\pi$
$563$	21.5279	0.907294	0.453647	+	0.891181i	$0.350123\pi$
$564$	27.4771	1.15700
$565$	−48.1413	−2.02532
$566$	−45.1183	−1.89646
$567$	0	0
$568$	25.4024	1.06586
$569$	−33.1004	−1.38764	−0.693819	−	0.720149i	$-0.744073\pi$
$569$	−33.1004	−1.38764	−0.693819	+	0.720149i	$0.744073\pi$
$570$	−130.091	−5.44891
$571$	19.9357	0.834284	0.417142	−	0.908841i	$-0.363032\pi$
$571$	19.9357	0.834284	0.417142	+	0.908841i	$0.363032\pi$
$572$	0	0
$573$	2.46557	0.103001
$574$	0	0
$575$	−41.0659	−1.71257
$576$	−28.9173	−1.20489
$577$	29.4695	1.22683	0.613416	−	0.789760i	$-0.289795\pi$
$577$	29.4695	1.22683	0.613416	+	0.789760i	$0.289795\pi$
$578$	29.8028	1.23963
$579$	−7.28676	−0.302827
$580$	97.7939	4.06067
$581$	0	0
$582$	19.1497	0.793781
$583$	5.46477	0.226328
$584$	6.93018	0.286773
$585$	0	0
$586$	−36.7734	−1.51910
$587$	6.98859	0.288450	0.144225	−	0.989545i	$-0.453931\pi$
$587$	6.98859	0.288450	0.144225	+	0.989545i	$0.453931\pi$
$588$	0	0
$589$	21.8717	0.901206
$590$	56.9595	2.34498
$591$	40.8594	1.68073
$592$	−2.85454	−0.117321
$593$	0.970248	0.0398433	0.0199216	−	0.999802i	$-0.493658\pi$
$593$	0.970248	0.0398433	0.0199216	+	0.999802i	$0.493658\pi$
$594$	8.39061	0.344271
$595$	0	0
$596$	48.6660	1.99344
$597$	29.9929	1.22753
$598$	0	0
$599$	−22.5998	−0.923404	−0.461702	−	0.887035i	$-0.652761\pi$
$599$	−22.5998	−0.923404	−0.461702	+	0.887035i	$0.652761\pi$
$600$	49.2001	2.00858
$601$	30.4292	1.24123	0.620617	−	0.784114i	$-0.286882\pi$
$601$	30.4292	1.24123	0.620617	+	0.784114i	$0.286882\pi$
$602$	0	0
$603$	−10.7562	−0.438027
$604$	4.62567	0.188216
$605$	18.7570	0.762581
$606$	25.4566	1.03410
$607$	32.1576	1.30524	0.652618	−	0.757687i	$-0.273671\pi$
$607$	32.1576	1.30524	0.652618	+	0.757687i	$0.273671\pi$
$608$	−32.3895	−1.31357
$609$	0	0
$610$	−50.2252	−2.03356
$611$	0	0
$612$	−43.7188	−1.76723
$613$	−13.9209	−0.562258	−0.281129	−	0.959670i	$-0.590709\pi$
$613$	−13.9209	−0.562258	−0.281129	+	0.959670i	$0.590709\pi$
$614$	44.8898	1.81160
$615$	−7.29873	−0.294313
$616$	0	0
$617$	10.1679	0.409344	0.204672	−	0.978831i	$-0.434387\pi$
$617$	10.1679	0.409344	0.204672	+	0.978831i	$0.434387\pi$
$618$	−32.2086	−1.29562
$619$	−43.7905	−1.76009	−0.880044	−	0.474893i	$-0.842487\pi$
$619$	−43.7905	−1.76009	−0.880044	+	0.474893i	$0.842487\pi$
$620$	−35.3422	−1.41938
$621$	9.95954	0.399663
$622$	−7.14525	−0.286498
$623$	0	0
$624$	0	0
$625$	−16.3162	−0.652650
$626$	82.2445	3.28715
$627$	−38.5899	−1.54113
$628$	−44.8053	−1.78793
$629$	16.6806	0.665097
$630$	0	0
$631$	−16.0893	−0.640504	−0.320252	−	0.947332i	$-0.603768\pi$
$631$	−16.0893	−0.640504	−0.320252	+	0.947332i	$0.603768\pi$
$632$	−25.4067	−1.01062
$633$	63.5906	2.52750
$634$	−76.2557	−3.02850
$635$	37.3421	1.48188
$636$	−18.5955	−0.737359
$637$	0	0
$638$	45.8951	1.81700
$639$	17.6901	0.699810
$640$	67.0187	2.64915
$641$	−49.2508	−1.94529	−0.972645	−	0.232296i	$-0.925376\pi$
$641$	−49.2508	−1.94529	−0.972645	+	0.232296i	$0.925376\pi$
$642$	70.9684	2.80090
$643$	−2.67158	−0.105357	−0.0526784	−	0.998612i	$-0.516776\pi$
$643$	−2.67158	−0.105357	−0.0526784	+	0.998612i	$0.516776\pi$
$644$	0	0
$645$	31.8526	1.25419
$646$	−91.2072	−3.58850
$647$	33.3628	1.31163	0.655814	−	0.754923i	$-0.272326\pi$
$647$	33.3628	1.31163	0.655814	+	0.754923i	$0.272326\pi$
$648$	−35.3498	−1.38867
$649$	16.8963	0.663239
$650$	0	0
$651$	0	0
$652$	−16.9481	−0.663738
$653$	23.8487	0.933274	0.466637	−	0.884449i	$-0.345466\pi$
$653$	23.8487	0.933274	0.466637	+	0.884449i	$0.345466\pi$
$654$	−22.1554	−0.866347
$655$	2.26543	0.0885177
$656$	0.875700	0.0341903
$657$	4.82615	0.188286
$658$	0	0
$659$	−17.1623	−0.668547	−0.334274	−	0.942476i	$-0.608491\pi$
$659$	−17.1623	−0.668547	−0.334274	+	0.942476i	$0.608491\pi$
$660$	62.3571	2.42725
$661$	0.466403	0.0181410	0.00907048	−	0.999959i	$-0.497113\pi$
$661$	0.466403	0.0181410	0.00907048	+	0.999959i	$0.497113\pi$
$662$	−11.1377	−0.432880
$663$	0	0
$664$	9.68686	0.375923
$665$	0	0
$666$	−16.6152	−0.643826
$667$	54.4769	2.10935
$668$	−12.5157	−0.484248
$669$	20.0929	0.776837
$670$	36.2604	1.40086
$671$	−14.8987	−0.575157
$672$	0	0
$673$	17.5412	0.676165	0.338082	−	0.941117i	$-0.390222\pi$
$673$	17.5412	0.676165	0.338082	+	0.941117i	$0.390222\pi$
$674$	61.1770	2.35645
$675$	−9.82372	−0.378115
$676$	0	0
$677$	−9.71961	−0.373555	−0.186777	−	0.982402i	$-0.559804\pi$
$677$	−9.71961	−0.373555	−0.186777	+	0.982402i	$0.559804\pi$
$678$	76.8790	2.95252
$679$	0	0
$680$	61.5937	2.36201
$681$	50.7218	1.94366
$682$	−16.5863	−0.635121
$683$	36.1154	1.38192	0.690960	−	0.722893i	$-0.257188\pi$
$683$	36.1154	1.38192	0.690960	+	0.722893i	$0.257188\pi$
$684$	57.4244	2.19568
$685$	26.4253	1.00966
$686$	0	0
$687$	−47.3780	−1.80758
$688$	−3.82166	−0.145700
$689$	0	0
$690$	117.101	4.45795
$691$	8.77269	0.333729	0.166864	−	0.985980i	$-0.446636\pi$
$691$	8.77269	0.333729	0.166864	+	0.985980i	$0.446636\pi$
$692$	43.6081	1.65773
$693$	0	0
$694$	−24.3509	−0.924346
$695$	−11.0411	−0.418813
$696$	−65.2674	−2.47396
$697$	−5.11717	−0.193826
$698$	54.7178	2.07110
$699$	−39.9199	−1.50991
$700$	0	0
$701$	1.51585	0.0572530	0.0286265	−	0.999590i	$-0.490887\pi$
$701$	1.51585	0.0572530	0.0286265	+	0.999590i	$0.490887\pi$
$702$	0	0
$703$	−21.9098	−0.826345
$704$	28.9173	1.08986
$705$	−26.9586	−1.01532
$706$	6.83448	0.257219
$707$	0	0
$708$	−57.4947	−2.16078
$709$	15.1020	0.567168	0.283584	−	0.958947i	$-0.408477\pi$
$709$	15.1020	0.567168	0.283584	+	0.958947i	$0.408477\pi$
$710$	−59.6355	−2.23808
$711$	−17.6931	−0.663544
$712$	−44.0582	−1.65115
$713$	−19.6877	−0.737309
$714$	0	0
$715$	0	0
$716$	−30.2000	−1.12863
$717$	−7.52647	−0.281081
$718$	39.8721	1.48801
$719$	−19.1989	−0.715999	−0.357999	−	0.933722i	$-0.616541\pi$
$719$	−19.1989	−0.715999	−0.357999	+	0.933722i	$0.616541\pi$
$720$	−7.34039	−0.273560
$721$	0	0
$722$	75.5014	2.80987
$723$	−3.38434	−0.125865
$724$	−59.0925	−2.19615
$725$	−53.7340	−1.99563
$726$	−29.9539	−1.11169
$727$	2.41101	0.0894195	0.0447098	−	0.999000i	$-0.485764\pi$
$727$	2.41101	0.0894195	0.0447098	+	0.999000i	$0.485764\pi$
$728$	0	0
$729$	−13.9254	−0.515755
$730$	−16.2695	−0.602162
$731$	22.3319	0.825976
$732$	50.6971	1.87382
$733$	−13.4992	−0.498604	−0.249302	−	0.968426i	$-0.580201\pi$
$733$	−13.4992	−0.498604	−0.249302	+	0.968426i	$0.580201\pi$
$734$	2.44483	0.0902403
$735$	0	0
$736$	29.1553	1.07468
$737$	10.7562	0.396210
$738$	5.09712	0.187628
$739$	0.0428191	0.00157513	0.000787563	−	1.00000i	$-0.499749\pi$
$739$	0.0428191	0.00157513	0.000787563		1.00000i	$0.499749\pi$
$740$	35.4039	1.30147
$741$	0	0
$742$	0	0
$743$	21.6508	0.794289	0.397145	−	0.917756i	$-0.370001\pi$
$743$	21.6508	0.794289	0.397145	+	0.917756i	$0.370001\pi$
$744$	23.5873	0.864753
$745$	−47.7475	−1.74933
$746$	−34.9977	−1.28136
$747$	6.74590	0.246819
$748$	43.7188	1.59852
$749$	0	0
$750$	−24.7620	−0.904181
$751$	−44.8858	−1.63791	−0.818953	−	0.573860i	$-0.805445\pi$
$751$	−44.8858	−1.63791	−0.818953	+	0.573860i	$0.805445\pi$
$752$	3.23448	0.117949
$753$	−39.4906	−1.43912
$754$	0	0
$755$	−4.53837	−0.165168
$756$	0	0
$757$	−7.55451	−0.274573	−0.137287	−	0.990531i	$-0.543838\pi$
$757$	−7.55451	−0.274573	−0.137287	+	0.990531i	$0.543838\pi$
$758$	63.3000	2.29916
$759$	34.7365	1.26085
$760$	−80.9030	−2.93466
$761$	23.3891	0.847856	0.423928	−	0.905696i	$-0.360651\pi$
$761$	23.3891	0.847856	0.423928	+	0.905696i	$0.360651\pi$
$762$	−59.6333	−2.16029
$763$	0	0
$764$	3.66894	0.132737
$765$	42.8937	1.55082
$766$	−37.3658	−1.35008
$767$	0	0
$768$	−49.7489	−1.79516
$769$	−43.8509	−1.58130	−0.790652	−	0.612266i	$-0.790258\pi$
$769$	−43.8509	−1.58130	−0.790652	+	0.612266i	$0.790258\pi$
$770$	0	0
$771$	−8.88994	−0.320163
$772$	−10.8432	−0.390255
$773$	−43.1225	−1.55101	−0.775505	−	0.631341i	$-0.782505\pi$
$773$	−43.1225	−1.55101	−0.775505	+	0.631341i	$0.782505\pi$
$774$	−22.2445	−0.799560
$775$	19.4192	0.697558
$776$	11.9091	0.427513
$777$	0	0
$778$	18.6388	0.668232
$779$	6.72137	0.240818
$780$	0	0
$781$	−17.6901	−0.633002
$782$	82.0997	2.93588
$783$	13.0319	0.465721
$784$	0	0
$785$	43.9597	1.56899
$786$	−3.61777	−0.129041
$787$	−24.8960	−0.887446	−0.443723	−	0.896164i	$-0.646343\pi$
$787$	−24.8960	−0.887446	−0.443723	+	0.896164i	$0.646343\pi$
$788$	60.8017	2.16597
$789$	−70.4613	−2.50849
$790$	59.6456	2.12210
$791$	0	0
$792$	−18.1994	−0.646689
$793$	0	0
$794$	−28.4648	−1.01018
$795$	18.2445	0.647067
$796$	44.6315	1.58192
$797$	−31.2422	−1.10665	−0.553327	−	0.832964i	$-0.686642\pi$
$797$	−31.2422	−1.10665	−0.553327	+	0.832964i	$0.686642\pi$
$798$	0	0
$799$	−18.9007	−0.668660
$800$	−28.7577	−1.01674
$801$	−30.6820	−1.08409
$802$	−88.2274	−3.11542
$803$	−4.82615	−0.170311
$804$	−36.6011	−1.29082
$805$	0	0
$806$	0	0
$807$	−38.7386	−1.36366
$808$	15.8314	0.556945
$809$	39.5458	1.39036	0.695179	−	0.718837i	$-0.255325\pi$
$809$	39.5458	1.39036	0.695179	+	0.718837i	$0.255325\pi$
$810$	82.9884	2.91592
$811$	−43.8260	−1.53894	−0.769470	−	0.638683i	$-0.779480\pi$
$811$	−43.8260	−1.53894	−0.769470	+	0.638683i	$0.779480\pi$
$812$	0	0
$813$	71.9885	2.52475
$814$	16.6152	0.582363
$815$	16.6282	0.582461
$816$	−11.7683	−0.411973
$817$	−29.3329	−1.02623
$818$	0.548079	0.0191631
$819$	0	0
$820$	−10.8610	−0.379283
$821$	−45.0324	−1.57164	−0.785820	−	0.618455i	$-0.787759\pi$
$821$	−45.0324	−1.57164	−0.785820	+	0.618455i	$0.787759\pi$
$822$	−42.1998	−1.47189
$823$	25.1987	0.878373	0.439186	−	0.898396i	$-0.355267\pi$
$823$	25.1987	0.878373	0.439186	+	0.898396i	$0.355267\pi$
$824$	−20.0304	−0.697793
$825$	−34.2628	−1.19288
$826$	0	0
$827$	22.3091	0.775762	0.387881	−	0.921709i	$-0.373207\pi$
$827$	22.3091	0.775762	0.387881	+	0.921709i	$0.373207\pi$
$828$	−51.6903	−1.79636
$829$	−26.1963	−0.909834	−0.454917	−	0.890534i	$-0.650331\pi$
$829$	−26.1963	−0.909834	−0.454917	+	0.890534i	$0.650331\pi$
$830$	−22.7412	−0.789359
$831$	0.914095	0.0317096
$832$	0	0
$833$	0	0
$834$	17.6320	0.610547
$835$	12.2795	0.424950
$836$	−57.4244	−1.98606
$837$	−4.70965	−0.162789
$838$	−1.03530	−0.0357639
$839$	43.4414	1.49976	0.749882	−	0.661571i	$-0.230110\pi$
$839$	43.4414	1.49976	0.749882	+	0.661571i	$0.230110\pi$
$840$	0	0
$841$	42.2820	1.45800
$842$	−22.0561	−0.760103
$843$	52.9823	1.82481
$844$	94.6272	3.25720
$845$	0	0
$846$	18.8267	0.647275
$847$	0	0
$848$	−2.18898	−0.0751697
$849$	−44.6828	−1.53351
$850$	−80.9801	−2.77759
$851$	19.7220	0.676063
$852$	60.1958	2.06227
$853$	1.15312	0.0394820	0.0197410	−	0.999805i	$-0.493716\pi$
$853$	1.15312	0.0394820	0.0197410	+	0.999805i	$0.493716\pi$
$854$	0	0
$855$	−56.3406	−1.92681
$856$	44.1350	1.50850
$857$	25.0087	0.854280	0.427140	−	0.904185i	$-0.359521\pi$
$857$	25.0087	0.854280	0.427140	+	0.904185i	$0.359521\pi$
$858$	0	0
$859$	4.08914	0.139520	0.0697598	−	0.997564i	$-0.477777\pi$
$859$	4.08914	0.139520	0.0697598	+	0.997564i	$0.477777\pi$
$860$	47.3988	1.61629
$861$	0	0
$862$	−50.2316	−1.71089
$863$	−17.3929	−0.592060	−0.296030	−	0.955179i	$-0.595663\pi$
$863$	−17.3929	−0.592060	−0.296030	+	0.955179i	$0.595663\pi$
$864$	6.97447	0.237276
$865$	−42.7851	−1.45474
$866$	30.6803	1.04256
$867$	29.5151	1.00238
$868$	0	0
$869$	17.6931	0.600198
$870$	153.224	5.19478
$871$	0	0
$872$	−13.7784	−0.466595
$873$	8.29348	0.280692
$874$	−107.838	−3.64766
$875$	0	0
$876$	16.4224	0.554862
$877$	14.9816	0.505894	0.252947	−	0.967480i	$-0.418600\pi$
$877$	14.9816	0.505894	0.252947	+	0.967480i	$0.418600\pi$
$878$	69.4434	2.34360
$879$	−36.4184	−1.22836
$880$	7.34039	0.247444
$881$	−42.6763	−1.43780	−0.718901	−	0.695112i	$-0.755355\pi$
$881$	−42.6763	−1.43780	−0.718901	+	0.695112i	$0.755355\pi$
$882$	0	0
$883$	35.8874	1.20771	0.603854	−	0.797095i	$-0.293631\pi$
$883$	35.8874	1.20771	0.603854	+	0.797095i	$0.293631\pi$
$884$	0	0
$885$	56.4096	1.89619
$886$	34.6152	1.16292
$887$	−27.6672	−0.928975	−0.464488	−	0.885580i	$-0.653761\pi$
$887$	−27.6672	−0.928975	−0.464488	+	0.885580i	$0.653761\pi$
$888$	−23.6285	−0.792920
$889$	0	0
$890$	103.433	3.46707
$891$	24.6175	0.824717
$892$	29.8996	1.00111
$893$	24.8260	0.830771
$894$	76.2502	2.55019
$895$	29.6300	0.990423
$896$	0	0
$897$	0	0
$898$	61.1386	2.04022
$899$	−25.7609	−0.859176
$900$	50.9854	1.69951
$901$	12.7913	0.426140
$902$	−5.09712	−0.169716
$903$	0	0
$904$	47.8108	1.59016
$905$	57.9772	1.92723
$906$	7.24752	0.240783
$907$	18.1933	0.604097	0.302049	−	0.953293i	$-0.402330\pi$
$907$	18.1933	0.604097	0.302049	+	0.953293i	$0.402330\pi$
$908$	75.4775	2.50481
$909$	11.0249	0.365673
$910$	0	0
$911$	−33.1527	−1.09840	−0.549199	−	0.835691i	$-0.685067\pi$
$911$	−33.1527	−1.09840	−0.549199	+	0.835691i	$0.685067\pi$
$912$	15.4576	0.511852
$913$	−6.74590	−0.223257
$914$	37.0031	1.22395
$915$	−49.7403	−1.64436
$916$	−70.5017	−2.32944
$917$	0	0
$918$	19.6398	0.648209
$919$	2.98328	0.0984092	0.0492046	−	0.998789i	$-0.484331\pi$
$919$	2.98328	0.0984092	0.0492046	+	0.998789i	$0.484331\pi$
$920$	72.8245	2.40095
$921$	44.4564	1.46489
$922$	51.4999	1.69606
$923$	0	0
$924$	0	0
$925$	−19.4531	−0.639613
$926$	−42.6404	−1.40125
$927$	−13.9491	−0.458149
$928$	38.1491	1.25231
$929$	−33.3980	−1.09575	−0.547876	−	0.836560i	$-0.684563\pi$
$929$	−33.3980	−1.09575	−0.547876	+	0.836560i	$0.684563\pi$
$930$	−55.3744	−1.81580
$931$	0	0
$932$	−59.4036	−1.94583
$933$	−7.07627	−0.231667
$934$	−10.5692	−0.345834
$935$	−42.8937	−1.40277
$936$	0	0
$937$	−51.3187	−1.67651	−0.838254	−	0.545279i	$-0.816424\pi$
$937$	−51.3187	−1.67651	−0.838254	+	0.545279i	$0.816424\pi$
$938$	0	0
$939$	81.4505	2.65804
$940$	−40.1162	−1.30845
$941$	−19.0039	−0.619509	−0.309755	−	0.950817i	$-0.600247\pi$
$941$	−19.0039	−0.619509	−0.309755	+	0.950817i	$0.600247\pi$
$942$	−70.2013	−2.28728
$943$	−6.05021	−0.197022
$944$	−6.76801	−0.220280
$945$	0	0
$946$	22.2445	0.723229
$947$	46.7080	1.51780	0.758902	−	0.651204i	$-0.225736\pi$
$947$	46.7080	1.51780	0.758902	+	0.651204i	$0.225736\pi$
$948$	−60.2061	−1.95540
$949$	0	0
$950$	106.367	3.45100
$951$	−75.5196	−2.44889
$952$	0	0
$953$	−1.73113	−0.0560769	−0.0280385	−	0.999607i	$-0.508926\pi$
$953$	−1.73113	−0.0560769	−0.0280385	+	0.999607i	$0.508926\pi$
$954$	−12.7412	−0.412512
$955$	−3.59969	−0.116483
$956$	−11.1999	−0.362231
$957$	45.4520	1.46926
$958$	−24.3551	−0.786879
$959$	0	0
$960$	96.5424	3.11589
$961$	−21.6901	−0.699681
$962$	0	0
$963$	30.7355	0.990437
$964$	−5.03613	−0.162203
$965$	10.6386	0.342467
$966$	0	0
$967$	4.88811	0.157191	0.0785955	−	0.996907i	$-0.474956\pi$
$967$	4.88811	0.157191	0.0785955	+	0.996907i	$0.474956\pi$
$968$	−18.6282	−0.598734
$969$	−90.3267	−2.90171
$970$	−27.9583	−0.897687
$971$	8.57152	0.275073	0.137537	−	0.990497i	$-0.456082\pi$
$971$	8.57152	0.275073	0.137537	+	0.990497i	$0.456082\pi$
$972$	−67.8576	−2.17653
$973$	0	0
$974$	−83.7347	−2.68303
$975$	0	0
$976$	5.96783	0.191026
$977$	40.0382	1.28093	0.640467	−	0.767985i	$-0.278741\pi$
$977$	40.0382	1.28093	0.640467	+	0.767985i	$0.278741\pi$
$978$	−26.5543	−0.849114
$979$	30.6820	0.980600
$980$	0	0
$981$	−9.59523	−0.306352
$982$	17.9584	0.573077
$983$	−23.9120	−0.762675	−0.381337	−	0.924436i	$-0.624536\pi$
$983$	−23.9120	−0.762675	−0.381337	+	0.924436i	$0.624536\pi$
$984$	7.24862	0.231077
$985$	−59.6541	−1.90074
$986$	107.426	3.42114
$987$	0	0
$988$	0	0
$989$	26.4039	0.839594
$990$	42.7256	1.35791
$991$	41.0138	1.30285	0.651424	−	0.758714i	$-0.274172\pi$
$991$	41.0138	1.30285	0.651424	+	0.758714i	$0.274172\pi$
$992$	−13.7869	−0.437734
$993$	−11.0302	−0.350033
$994$	0	0
$995$	−43.7892	−1.38821
$996$	22.9549	0.727354
$997$	25.4245	0.805201	0.402600	−	0.915376i	$-0.368106\pi$
$997$	25.4245	0.805201	0.402600	+	0.915376i	$0.368106\pi$
$998$	−19.7964	−0.626643
$999$	4.71787	0.149267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cl.1.8		8
7.6	odd	2	inner	8281.2.a.cl.1.7		8
13.4	even	6		637.2.f.l.393.7	yes	16
13.10	even	6		637.2.f.l.295.7	✓	16
13.12	even	2		8281.2.a.ci.1.2		8
91.4	even	6		637.2.h.m.471.1		16
91.10	odd	6		637.2.g.m.373.7		16
91.17	odd	6		637.2.h.m.471.2		16
91.23	even	6		637.2.h.m.165.1		16
91.30	even	6		637.2.g.m.263.8		16
91.62	odd	6		637.2.f.l.295.8	yes	16
91.69	odd	6		637.2.f.l.393.8	yes	16
91.75	odd	6		637.2.h.m.165.2		16
91.82	odd	6		637.2.g.m.263.7		16
91.88	even	6		637.2.g.m.373.8		16
91.90	odd	2		8281.2.a.ci.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.l.295.7	✓	16	13.10	even	6
637.2.f.l.295.8	yes	16	91.62	odd	6
637.2.f.l.393.7	yes	16	13.4	even	6
637.2.f.l.393.8	yes	16	91.69	odd	6
637.2.g.m.263.7		16	91.82	odd	6
637.2.g.m.263.8		16	91.30	even	6
637.2.g.m.373.7		16	91.10	odd	6
637.2.g.m.373.8		16	91.88	even	6
637.2.h.m.165.1		16	91.23	even	6
637.2.h.m.165.2		16	91.75	odd	6
637.2.h.m.471.1		16	91.4	even	6
637.2.h.m.471.2		16	91.17	odd	6
8281.2.a.ci.1.1		8	91.90	odd	2
8281.2.a.ci.1.2		8	13.12	even	2
8281.2.a.cl.1.7		8	7.6	odd	2	inner
8281.2.a.cl.1.8		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.33152 q^{2} +2.30901 q^{3} +3.43596 q^{4} -3.37112 q^{5} +5.38349 q^{6} +3.34797 q^{8} +2.33152 q^{9} -7.85981 q^{10} -2.33152 q^{11} +7.93366 q^{12} -7.78393 q^{15} +0.933914 q^{16} -5.45734 q^{17} +5.43596 q^{18} +7.16819 q^{19} -11.5830 q^{20} -5.43596 q^{22} -6.45242 q^{23} +7.73048 q^{24} +6.36442 q^{25} -1.54354 q^{27} -8.44287 q^{29} -18.1484 q^{30} +3.05121 q^{31} -4.51850 q^{32} -5.38349 q^{33} -12.7239 q^{34} +8.01100 q^{36} -3.05653 q^{37} +16.7127 q^{38} -11.2864 q^{40} +0.937666 q^{41} -4.09209 q^{43} -8.01100 q^{44} -7.85981 q^{45} -15.0439 q^{46} +3.46336 q^{47} +2.15642 q^{48} +14.8387 q^{50} -12.6010 q^{51} -2.34387 q^{53} -3.59878 q^{54} +7.85981 q^{55} +16.5514 q^{57} -19.6847 q^{58} -7.24693 q^{59} -26.7453 q^{60} +6.39013 q^{61} +7.11394 q^{62} -12.4028 q^{64} -12.5517 q^{66} -4.61340 q^{67} -18.7512 q^{68} -14.8987 q^{69} +7.58739 q^{71} +7.80584 q^{72} +2.06996 q^{73} -7.12636 q^{74} +14.6955 q^{75} +24.6296 q^{76} -7.58868 q^{79} -3.14833 q^{80} -10.5586 q^{81} +2.18618 q^{82} +2.89335 q^{83} +18.3973 q^{85} -9.54077 q^{86} -19.4946 q^{87} -7.80584 q^{88} -13.1597 q^{89} -18.3253 q^{90} -22.1703 q^{92} +7.04526 q^{93} +8.07488 q^{94} -24.1648 q^{95} -10.4333 q^{96} +3.55712 q^{97} -5.43596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{9} - 4 q^{11} - 8 q^{15} + 4 q^{16} + 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} - 8 q^{37} - 32 q^{43} + 4 q^{44} - 4 q^{46}+ \cdots - 28 q^{99}+O(q^{100}) \) 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

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Database

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