LMFDB - Embedded newform 531.6.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.20625$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20625 q^{2} -21.7200 q^{4} -26.7258 q^{5} +39.0273 q^{7} +172.240 q^{8} +O(q^{10})$	\(q-3.20625 q^{2} -21.7200 q^{4} -26.7258 q^{5} +39.0273 q^{7} +172.240 q^{8} +85.6897 q^{10} +606.405 q^{11} -161.707 q^{13} -125.131 q^{14} +142.797 q^{16} +1651.20 q^{17} +882.094 q^{19} +580.485 q^{20} -1944.28 q^{22} -2923.63 q^{23} -2410.73 q^{25} +518.474 q^{26} -847.672 q^{28} +7064.00 q^{29} -2253.17 q^{31} -5969.51 q^{32} -5294.14 q^{34} -1043.04 q^{35} -7561.88 q^{37} -2828.21 q^{38} -4603.25 q^{40} +16708.0 q^{41} -4502.50 q^{43} -13171.1 q^{44} +9373.88 q^{46} -8408.79 q^{47} -15283.9 q^{49} +7729.39 q^{50} +3512.28 q^{52} -11049.2 q^{53} -16206.7 q^{55} +6722.04 q^{56} -22648.9 q^{58} -3481.00 q^{59} +34738.9 q^{61} +7224.23 q^{62} +14570.2 q^{64} +4321.76 q^{65} -57227.0 q^{67} -35863.9 q^{68} +3344.23 q^{70} -26.9044 q^{71} +54481.3 q^{73} +24245.2 q^{74} -19159.1 q^{76} +23666.3 q^{77} +93190.6 q^{79} -3816.37 q^{80} -53569.9 q^{82} +89600.1 q^{83} -44129.6 q^{85} +14436.1 q^{86} +104447. q^{88} +128157. q^{89} -6311.00 q^{91} +63501.2 q^{92} +26960.7 q^{94} -23574.7 q^{95} -26665.5 q^{97} +49003.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

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 $

$2$	−3.20625	−0.566790	−0.283395	−	0.959003i	$-0.591461\pi$
$2$	−3.20625	−0.566790	−0.283395	+	0.959003i	$0.591461\pi$
$3$	0	0
$3$	0	0
$4$	−21.7200	−0.678749
$5$	−26.7258	−0.478086	−0.239043	−	0.971009i	$-0.576834\pi$
$5$	−26.7258	−0.478086	−0.239043	+	0.971009i	$0.576834\pi$
$6$	0	0
$7$	39.0273	0.301039	0.150520	−	0.988607i	$-0.451905\pi$
$7$	39.0273	0.301039	0.150520	+	0.988607i	$0.451905\pi$
$8$	172.240	0.951498
$9$	0	0
$10$	85.6897	0.270974
$11$	606.405	1.51106	0.755528	−	0.655116i	$-0.227380\pi$
$11$	606.405	1.51106	0.755528	+	0.655116i	$0.227380\pi$
$12$	0	0
$13$	−161.707	−0.265382	−0.132691	−	0.991157i	$-0.542362\pi$
$13$	−161.707	−0.265382	−0.132691	+	0.991157i	$0.542362\pi$
$14$	−125.131	−0.170626
$15$	0	0
$16$	142.797	0.139450
$17$	1651.20	1.38572	0.692861	−	0.721071i	$-0.256350\pi$
$17$	1651.20	1.38572	0.692861	+	0.721071i	$0.256350\pi$
$18$	0	0
$19$	882.094	0.560572	0.280286	−	0.959917i	$-0.409571\pi$
$19$	882.094	0.560572	0.280286	+	0.959917i	$0.409571\pi$
$20$	580.485	0.324501
$21$	0	0
$22$	−1944.28	−0.856451
$23$	−2923.63	−1.15240	−0.576199	−	0.817309i	$-0.695465\pi$
$23$	−2923.63	−1.15240	−0.576199	+	0.817309i	$0.695465\pi$
$24$	0	0
$25$	−2410.73	−0.771433
$26$	518.474	0.150416
$27$	0	0
$28$	−847.672	−0.204330
$29$	7064.00	1.55975	0.779876	−	0.625934i	$-0.215282\pi$
$29$	7064.00	1.55975	0.779876	+	0.625934i	$0.215282\pi$
$30$	0	0
$31$	−2253.17	−0.421105	−0.210553	−	0.977583i	$-0.567526\pi$
$31$	−2253.17	−0.421105	−0.210553	+	0.977583i	$0.567526\pi$
$32$	−5969.51	−1.03054
$33$	0	0
$34$	−5294.14	−0.785413
$35$	−1043.04	−0.143923
$36$	0	0
$37$	−7561.88	−0.908082	−0.454041	−	0.890981i	$-0.650018\pi$
$37$	−7561.88	−0.908082	−0.454041	+	0.890981i	$0.650018\pi$
$38$	−2828.21	−0.317726
$39$	0	0
$40$	−4603.25	−0.454898
$41$	16708.0	1.55226	0.776130	−	0.630574i	$-0.217180\pi$
$41$	16708.0	1.55226	0.776130	+	0.630574i	$0.217180\pi$
$42$	0	0
$43$	−4502.50	−0.371349	−0.185675	−	0.982611i	$-0.559447\pi$
$43$	−4502.50	−0.371349	−0.185675	+	0.982611i	$0.559447\pi$
$44$	−13171.1	−1.02563
$45$	0	0
$46$	9373.88	0.653168
$47$	−8408.79	−0.555251	−0.277625	−	0.960689i	$-0.589547\pi$
$47$	−8408.79	−0.555251	−0.277625	+	0.960689i	$0.589547\pi$
$48$	0	0
$49$	−15283.9	−0.909375
$50$	7729.39	0.437241
$51$	0	0
$52$	3512.28	0.180128
$53$	−11049.2	−0.540307	−0.270154	−	0.962817i	$-0.587075\pi$
$53$	−11049.2	−0.540307	−0.270154	+	0.962817i	$0.587075\pi$
$54$	0	0
$55$	−16206.7	−0.722416
$56$	6722.04	0.286438
$57$	0	0
$58$	−22648.9	−0.884052
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	34738.9	1.19534	0.597670	−	0.801742i	$-0.296093\pi$
$61$	34738.9	1.19534	0.597670	+	0.801742i	$0.296093\pi$
$62$	7224.23	0.238678
$63$	0	0
$64$	14570.2	0.444648
$65$	4321.76	0.126875
$66$	0	0
$67$	−57227.0	−1.55745	−0.778724	−	0.627366i	$-0.784133\pi$
$67$	−57227.0	−1.55745	−0.778724	+	0.627366i	$0.784133\pi$
$68$	−35863.9	−0.940558
$69$	0	0
$70$	3344.23	0.0815740
$71$	−26.9044	−0.000633398 0	−0.000316699	−	1.00000i	$-0.500101\pi$
$71$	−26.9044	−0.000633398 0	−0.000316699		1.00000i	$0.500101\pi$
$72$	0	0
$73$	54481.3	1.19658	0.598288	−	0.801281i	$-0.295848\pi$
$73$	54481.3	1.19658	0.598288	+	0.801281i	$0.295848\pi$
$74$	24245.2	0.514692
$75$	0	0
$76$	−19159.1	−0.380488
$77$	23666.3	0.454887
$78$	0	0
$79$	93190.6	1.67998	0.839990	−	0.542601i	$-0.182561\pi$
$79$	93190.6	1.67998	0.839990	+	0.542601i	$0.182561\pi$
$80$	−3816.37	−0.0666692
$81$	0	0
$82$	−53569.9	−0.879805
$83$	89600.1	1.42762	0.713811	−	0.700339i	$-0.246968\pi$
$83$	89600.1	1.42762	0.713811	+	0.700339i	$0.246968\pi$
$84$	0	0
$85$	−44129.6	−0.662495
$86$	14436.1	0.210477
$87$	0	0
$88$	104447.	1.43777
$89$	128157.	1.71502	0.857508	−	0.514471i	$-0.172012\pi$
$89$	128157.	1.71502	0.857508	+	0.514471i	$0.172012\pi$
$90$	0	0
$91$	−6311.00	−0.0798904
$92$	63501.2	0.782190
$93$	0	0
$94$	26960.7	0.314710
$95$	−23574.7	−0.268002
$96$	0	0
$97$	−26665.5	−0.287753	−0.143877	−	0.989596i	$-0.545957\pi$
$97$	−26665.5	−0.287753	−0.143877	+	0.989596i	$0.545957\pi$
$98$	49003.9	0.515425
$99$	0	0
$100$	52361.0	0.523610
$101$	−153961.	−1.50178	−0.750890	−	0.660427i	$-0.770375\pi$
$101$	−153961.	−1.50178	−0.750890	+	0.660427i	$0.770375\pi$
$102$	0	0
$103$	−51991.3	−0.482878	−0.241439	−	0.970416i	$-0.577619\pi$
$103$	−51991.3	−0.482878	−0.241439	+	0.970416i	$0.577619\pi$
$104$	−27852.4	−0.252510
$105$	0	0
$106$	35426.4	0.306241
$107$	−215950.	−1.82345	−0.911725	−	0.410801i	$-0.865249\pi$
$107$	−215950.	−1.82345	−0.911725	+	0.410801i	$0.865249\pi$
$108$	0	0
$109$	62561.7	0.504362	0.252181	−	0.967680i	$-0.418852\pi$
$109$	62561.7	0.504362	0.252181	+	0.967680i	$0.418852\pi$
$110$	51962.6	0.409458
$111$	0	0
$112$	5572.98	0.0419800
$113$	−242539.	−1.78684	−0.893421	−	0.449220i	$-0.851702\pi$
$113$	−242539.	−1.78684	−0.893421	+	0.449220i	$0.851702\pi$
$114$	0	0
$115$	78136.5	0.550946
$116$	−153430.	−1.05868
$117$	0	0
$118$	11160.9	0.0737897
$119$	64441.7	0.417157
$120$	0	0
$121$	206675.	1.28329
$122$	−111381.	−0.677506
$123$	0	0
$124$	48938.9	0.285825
$125$	147947.	0.846898
$126$	0	0
$127$	4520.72	0.0248713	0.0124356	−	0.999923i	$-0.496042\pi$
$127$	4520.72	0.0248713	0.0124356	+	0.999923i	$0.496042\pi$
$128$	144309.	0.778515
$129$	0	0
$130$	−13856.6	−0.0719117
$131$	201847.	1.02765	0.513823	−	0.857896i	$-0.328229\pi$
$131$	201847.	1.02765	0.513823	+	0.857896i	$0.328229\pi$
$132$	0	0
$133$	34425.8	0.168754
$134$	183484.	0.882746
$135$	0	0
$136$	284401.	1.31851
$137$	−128569.	−0.585242	−0.292621	−	0.956229i	$-0.594527\pi$
$137$	−128569.	−0.585242	−0.292621	+	0.956229i	$0.594527\pi$
$138$	0	0
$139$	264506.	1.16118	0.580588	−	0.814198i	$-0.302823\pi$
$139$	264506.	1.16118	0.580588	+	0.814198i	$0.302823\pi$
$140$	22654.7	0.0976875
$141$	0	0
$142$	86.2620	0.000359003 0
$143$	−98060.0	−0.401007
$144$	0	0
$145$	−188791.	−0.745697
$146$	−174680.	−0.678207
$147$	0	0
$148$	164244.	0.616360
$149$	310579.	1.14606	0.573028	−	0.819536i	$-0.305769\pi$
$149$	310579.	1.14606	0.573028	+	0.819536i	$0.305769\pi$
$150$	0	0
$151$	150211.	0.536118	0.268059	−	0.963402i	$-0.413618\pi$
$151$	150211.	0.536118	0.268059	+	0.963402i	$0.413618\pi$
$152$	151932.	0.533383
$153$	0	0
$154$	−75880.1	−0.257826
$155$	60218.0	0.201325
$156$	0	0
$157$	381735.	1.23598	0.617992	−	0.786185i	$-0.287947\pi$
$157$	381735.	1.23598	0.617992	+	0.786185i	$0.287947\pi$
$158$	−298792.	−0.952196
$159$	0	0
$160$	159540.	0.492686
$161$	−114101.	−0.346917
$162$	0	0
$163$	−43367.4	−0.127848	−0.0639240	−	0.997955i	$-0.520362\pi$
$163$	−43367.4	−0.127848	−0.0639240	+	0.997955i	$0.520362\pi$
$164$	−362897.	−1.05359
$165$	0	0
$166$	−287280.	−0.809161
$167$	−227398.	−0.630951	−0.315475	−	0.948934i	$-0.602164\pi$
$167$	−227398.	−0.630951	−0.315475	+	0.948934i	$0.602164\pi$
$168$	0	0
$169$	−345144.	−0.929572
$170$	141490.	0.375495
$171$	0	0
$172$	97794.2	0.252053
$173$	196716.	0.499716	0.249858	−	0.968282i	$-0.419616\pi$
$173$	196716.	0.499716	0.249858	+	0.968282i	$0.419616\pi$
$174$	0	0
$175$	−94084.2	−0.232232
$176$	86592.7	0.210717
$177$	0	0
$178$	−410904.	−0.972053
$179$	51846.3	0.120944	0.0604722	−	0.998170i	$-0.480739\pi$
$179$	51846.3	0.120944	0.0604722	+	0.998170i	$0.480739\pi$
$180$	0	0
$181$	692422.	1.57099	0.785496	−	0.618866i	$-0.212408\pi$
$181$	692422.	1.57099	0.785496	+	0.618866i	$0.212408\pi$
$182$	20234.6	0.0452810
$183$	0	0
$184$	−503565.	−1.09651
$185$	202098.	0.434142
$186$	0	0
$187$	1.00129e6	2.09390
$188$	182639.	0.376876
$189$	0	0
$190$	75586.4	0.151901
$191$	964737.	1.91349	0.956743	−	0.290933i	$-0.0939658\pi$
$191$	964737.	1.91349	0.956743	+	0.290933i	$0.0939658\pi$
$192$	0	0
$193$	−305452.	−0.590268	−0.295134	−	0.955456i	$-0.595364\pi$
$193$	−305452.	−0.590268	−0.295134	+	0.955456i	$0.595364\pi$
$194$	85496.1	0.163096
$195$	0	0
$196$	331965.	0.617238
$197$	31988.0	0.0587249	0.0293624	−	0.999569i	$-0.490652\pi$
$197$	31988.0	0.0587249	0.0293624	+	0.999569i	$0.490652\pi$
$198$	0	0
$199$	428137.	0.766390	0.383195	−	0.923668i	$-0.374824\pi$
$199$	428137.	0.766390	0.383195	+	0.923668i	$0.374824\pi$
$200$	−415223.	−0.734017
$201$	0	0
$202$	493636.	0.851193
$203$	275689.	0.469547
$204$	0	0
$205$	−446535.	−0.742114
$206$	166697.	0.273691
$207$	0	0
$208$	−23091.3	−0.0370075
$209$	534906.	0.847055
$210$	0	0
$211$	−1.07522e6	−1.66261	−0.831306	−	0.555815i	$-0.812406\pi$
$211$	−1.07522e6	−1.66261	−0.831306	+	0.555815i	$0.812406\pi$
$212$	239988.	0.366733
$213$	0	0
$214$	692389.	1.03351
$215$	120333.	0.177537
$216$	0	0
$217$	−87935.3	−0.126769
$218$	−200588.	−0.285867
$219$	0	0
$220$	352009.	0.490339
$221$	−267010.	−0.367746
$222$	0	0
$223$	541635.	0.729365	0.364682	−	0.931132i	$-0.381178\pi$
$223$	541635.	0.729365	0.364682	+	0.931132i	$0.381178\pi$
$224$	−232974.	−0.310232
$225$	0	0
$226$	777641.	1.01276
$227$	503824.	0.648954	0.324477	−	0.945894i	$-0.394812\pi$
$227$	503824.	0.648954	0.324477	+	0.945894i	$0.394812\pi$
$228$	0	0
$229$	236193.	0.297631	0.148815	−	0.988865i	$-0.452454\pi$
$229$	236193.	0.297631	0.148815	+	0.988865i	$0.452454\pi$
$230$	−250525.	−0.312271
$231$	0	0
$232$	1.21670e6	1.48410
$233$	1.35136e6	1.63073	0.815366	−	0.578946i	$-0.196536\pi$
$233$	1.35136e6	1.63073	0.815366	+	0.578946i	$0.196536\pi$
$234$	0	0
$235$	224732.	0.265458
$236$	75607.3	0.0883656
$237$	0	0
$238$	−206616.	−0.236440
$239$	1.00537e6	1.13849	0.569245	−	0.822168i	$-0.307235\pi$
$239$	1.00537e6	1.13849	0.569245	+	0.822168i	$0.307235\pi$
$240$	0	0
$241$	−968910.	−1.07458	−0.537292	−	0.843396i	$-0.680553\pi$
$241$	−968910.	−1.07458	−0.537292	+	0.843396i	$0.680553\pi$
$242$	−662653.	−0.727357
$243$	0	0
$244$	−754528.	−0.811336
$245$	408474.	0.434760
$246$	0	0
$247$	−142641.	−0.148766
$248$	−388086.	−0.400681
$249$	0	0
$250$	−474355.	−0.480013
$251$	1.74198e6	1.74525	0.872626	−	0.488390i	$-0.162416\pi$
$251$	1.74198e6	1.74525	0.872626	+	0.488390i	$0.162416\pi$
$252$	0	0
$253$	−1.77290e6	−1.74134
$254$	−14494.5	−0.0140968
$255$	0	0
$256$	−928936.	−0.885902
$257$	696669.	0.657951	0.328976	−	0.944338i	$-0.393297\pi$
$257$	696669.	0.657951	0.328976	+	0.944338i	$0.393297\pi$
$258$	0	0
$259$	−295120.	−0.273369
$260$	−93868.6	−0.0861166
$261$	0	0
$262$	−647171.	−0.582459
$263$	894241.	0.797196	0.398598	−	0.917126i	$-0.369497\pi$
$263$	894241.	0.797196	0.398598	+	0.917126i	$0.369497\pi$
$264$	0	0
$265$	295299.	0.258314
$266$	−110377.	−0.0956481
$267$	0	0
$268$	1.24297e6	1.05712
$269$	−346103.	−0.291625	−0.145812	−	0.989312i	$-0.546580\pi$
$269$	−346103.	−0.291625	−0.145812	+	0.989312i	$0.546580\pi$
$270$	0	0
$271$	476055.	0.393762	0.196881	−	0.980427i	$-0.436919\pi$
$271$	476055.	0.393762	0.196881	+	0.980427i	$0.436919\pi$
$272$	235786.	0.193239
$273$	0	0
$274$	412224.	0.331709
$275$	−1.46188e6	−1.16568
$276$	0	0
$277$	−1.33986e6	−1.04920	−0.524600	−	0.851349i	$-0.675785\pi$
$277$	−1.33986e6	−1.04920	−0.524600	+	0.851349i	$0.675785\pi$
$278$	−848071.	−0.658143
$279$	0	0
$280$	−179652.	−0.136942
$281$	−767258.	−0.579663	−0.289831	−	0.957078i	$-0.593599\pi$
$281$	−767258.	−0.579663	−0.289831	+	0.957078i	$0.593599\pi$
$282$	0	0
$283$	391631.	0.290677	0.145339	−	0.989382i	$-0.453573\pi$
$283$	391631.	0.290677	0.145339	+	0.989382i	$0.453573\pi$
$284$	584.362	0.000429919 0
$285$	0	0
$286$	314405.	0.227287
$287$	652067.	0.467291
$288$	0	0
$289$	1.30659e6	0.920227
$290$	605312.	0.422653
$291$	0	0
$292$	−1.18333e6	−0.812175
$293$	−1.78575e6	−1.21521	−0.607606	−	0.794239i	$-0.707870\pi$
$293$	−1.78575e6	−1.21521	−0.607606	+	0.794239i	$0.707870\pi$
$294$	0	0
$295$	93032.7	0.0622415
$296$	−1.30245e6	−0.864039
$297$	0	0
$298$	−995791.	−0.649573
$299$	472772.	0.305826
$300$	0	0
$301$	−175720.	−0.111791
$302$	−481615.	−0.303866
$303$	0	0
$304$	125960.	0.0781718
$305$	−928426.	−0.571476
$306$	0	0
$307$	−927343.	−0.561558	−0.280779	−	0.959772i	$-0.590593\pi$
$307$	−927343.	−0.561558	−0.280779	+	0.959772i	$0.590593\pi$
$308$	−514032.	−0.308755
$309$	0	0
$310$	−193074.	−0.114109
$311$	−766407.	−0.449323	−0.224661	−	0.974437i	$-0.572128\pi$
$311$	−766407.	−0.449323	−0.224661	+	0.974437i	$0.572128\pi$
$312$	0	0
$313$	226189.	0.130500	0.0652501	−	0.997869i	$-0.479215\pi$
$313$	226189.	0.130500	0.0652501	+	0.997869i	$0.479215\pi$
$314$	−1.22394e6	−0.700543
$315$	0	0
$316$	−2.02410e6	−1.14029
$317$	−1.08057e6	−0.603955	−0.301978	−	0.953315i	$-0.597647\pi$
$317$	−1.08057e6	−0.603955	−0.301978	+	0.953315i	$0.597647\pi$
$318$	0	0
$319$	4.28364e6	2.35687
$320$	−389401.	−0.212580
$321$	0	0
$322$	365837.	0.196629
$323$	1.45651e6	0.776797
$324$	0	0
$325$	389833.	0.204724
$326$	139047.	0.0724630
$327$	0	0
$328$	2.87777e6	1.47697
$329$	−328172.	−0.167152
$330$	0	0
$331$	−2.89963e6	−1.45470	−0.727350	−	0.686267i	$-0.759248\pi$
$331$	−2.89963e6	−1.45470	−0.727350	+	0.686267i	$0.759248\pi$
$332$	−1.94611e6	−0.968997
$333$	0	0
$334$	729094.	0.357617
$335$	1.52944e6	0.744595
$336$	0	0
$337$	−204495.	−0.0980863	−0.0490432	−	0.998797i	$-0.515617\pi$
$337$	−204495.	−0.0980863	−0.0490432	+	0.998797i	$0.515617\pi$
$338$	1.10662e6	0.526872
$339$	0	0
$340$	958494.	0.449668
$341$	−1.36634e6	−0.636314
$342$	0	0
$343$	−1.25242e6	−0.574797
$344$	−775509.	−0.353338
$345$	0	0
$346$	−630719.	−0.283234
$347$	−1.14557e6	−0.510739	−0.255370	−	0.966843i	$-0.582197\pi$
$347$	−1.14557e6	−0.510739	−0.255370	+	0.966843i	$0.582197\pi$
$348$	0	0
$349$	3.14511e6	1.38220	0.691101	−	0.722758i	$-0.257126\pi$
$349$	3.14511e6	1.38220	0.691101	+	0.722758i	$0.257126\pi$
$350$	301657.	0.131627
$351$	0	0
$352$	−3.61994e6	−1.55720
$353$	2.35284e6	1.00497	0.502487	−	0.864584i	$-0.332418\pi$
$353$	2.35284e6	1.00497	0.502487	+	0.864584i	$0.332418\pi$
$354$	0	0
$355$	719.042	0.000302819 0
$356$	−2.78357e6	−1.16407
$357$	0	0
$358$	−166232.	−0.0685500
$359$	−198560.	−0.0813122	−0.0406561	−	0.999173i	$-0.512945\pi$
$359$	−198560.	−0.0813122	−0.0406561	+	0.999173i	$0.512945\pi$
$360$	0	0
$361$	−1.69801e6	−0.685759
$362$	−2.22007e6	−0.890423
$363$	0	0
$364$	137075.	0.0542255
$365$	−1.45606e6	−0.572067
$366$	0	0
$367$	3.33716e6	1.29334	0.646669	−	0.762771i	$-0.276162\pi$
$367$	3.33716e6	1.29334	0.646669	+	0.762771i	$0.276162\pi$
$368$	−417485.	−0.160702
$369$	0	0
$370$	−647975.	−0.246067
$371$	−431220.	−0.162654
$372$	0	0
$373$	3.49695e6	1.30142	0.650710	−	0.759326i	$-0.274471\pi$
$373$	3.49695e6	1.30142	0.650710	+	0.759326i	$0.274471\pi$
$374$	−3.21039e6	−1.18680
$375$	0	0
$376$	−1.44833e6	−0.528320
$377$	−1.14230e6	−0.413930
$378$	0	0
$379$	−4.62014e6	−1.65218	−0.826089	−	0.563539i	$-0.809439\pi$
$379$	−4.62014e6	−1.65218	−0.826089	+	0.563539i	$0.809439\pi$
$380$	512042.	0.181906
$381$	0	0
$382$	−3.09318e6	−1.08454
$383$	3.77589e6	1.31529	0.657645	−	0.753328i	$-0.271553\pi$
$383$	3.77589e6	1.31529	0.657645	+	0.753328i	$0.271553\pi$
$384$	0	0
$385$	−632502.	−0.217475
$386$	979353.	0.334558
$387$	0	0
$388$	579174.	0.195312
$389$	−1.44066e6	−0.482710	−0.241355	−	0.970437i	$-0.577592\pi$
$389$	−1.44066e6	−0.482710	−0.241355	+	0.970437i	$0.577592\pi$
$390$	0	0
$391$	−4.82749e6	−1.59691
$392$	−2.63249e6	−0.865269
$393$	0	0
$394$	−102562.	−0.0332846
$395$	−2.49060e6	−0.803176
$396$	0	0
$397$	4.04765e6	1.28892	0.644461	−	0.764637i	$-0.277082\pi$
$397$	4.04765e6	1.28892	0.644461	+	0.764637i	$0.277082\pi$
$398$	−1.37271e6	−0.434382
$399$	0	0
$400$	−344245.	−0.107576
$401$	187555.	0.0582462	0.0291231	−	0.999576i	$-0.490729\pi$
$401$	187555.	0.0582462	0.0291231	+	0.999576i	$0.490729\pi$
$402$	0	0
$403$	364355.	0.111754
$404$	3.34402e6	1.01933
$405$	0	0
$406$	−883926.	−0.266134
$407$	−4.58556e6	−1.37216
$408$	0	0
$409$	−3.25079e6	−0.960906	−0.480453	−	0.877021i	$-0.659528\pi$
$409$	−3.25079e6	−0.960906	−0.480453	+	0.877021i	$0.659528\pi$
$410$	1.43170e6	0.420623
$411$	0	0
$412$	1.12925e6	0.327753
$413$	−135854.	−0.0391920
$414$	0	0
$415$	−2.39464e6	−0.682526
$416$	965313.	0.273486
$417$	0	0
$418$	−1.71504e6	−0.480102
$419$	−3.65412e6	−1.01683	−0.508414	−	0.861113i	$-0.669768\pi$
$419$	−3.65412e6	−1.01683	−0.508414	+	0.861113i	$0.669768\pi$
$420$	0	0
$421$	−5.36572e6	−1.47544	−0.737722	−	0.675105i	$-0.764098\pi$
$421$	−5.36572e6	−1.47544	−0.737722	+	0.675105i	$0.764098\pi$
$422$	3.44742e6	0.942351
$423$	0	0
$424$	−1.90311e6	−0.514101
$425$	−3.98059e6	−1.06899
$426$	0	0
$427$	1.35576e6	0.359844
$428$	4.69043e6	1.23767
$429$	0	0
$430$	−385818.	−0.100626
$431$	6.87518e6	1.78275	0.891377	−	0.453264i	$-0.149740\pi$
$431$	6.87518e6	1.78275	0.891377	+	0.453264i	$0.149740\pi$
$432$	0	0
$433$	40489.0	0.0103781	0.00518904	−	0.999987i	$-0.498348\pi$
$433$	40489.0	0.0103781	0.00518904	+	0.999987i	$0.498348\pi$
$434$	281942.	0.0718515
$435$	0	0
$436$	−1.35884e6	−0.342335
$437$	−2.57892e6	−0.646002
$438$	0	0
$439$	−4.44389e6	−1.10053	−0.550265	−	0.834990i	$-0.685473\pi$
$439$	−4.44389e6	−1.10053	−0.550265	+	0.834990i	$0.685473\pi$
$440$	−2.79143e6	−0.687377
$441$	0	0
$442$	856101.	0.208434
$443$	278084.	0.0673236	0.0336618	−	0.999433i	$-0.489283\pi$
$443$	278084.	0.0673236	0.0336618	+	0.999433i	$0.489283\pi$
$444$	0	0
$445$	−3.42511e6	−0.819925
$446$	−1.73662e6	−0.413397
$447$	0	0
$448$	568636.	0.133856
$449$	−1.07874e6	−0.252523	−0.126262	−	0.991997i	$-0.540298\pi$
$449$	−1.07874e6	−0.252523	−0.126262	+	0.991997i	$0.540298\pi$
$450$	0	0
$451$	1.01318e7	2.34555
$452$	5.26795e6	1.21282
$453$	0	0
$454$	−1.61538e6	−0.367821
$455$	168667.	0.0381945
$456$	0	0
$457$	−2.34460e6	−0.525144	−0.262572	−	0.964912i	$-0.584571\pi$
$457$	−2.34460e6	−0.525144	−0.262572	+	0.964912i	$0.584571\pi$
$458$	−757292.	−0.168694
$459$	0	0
$460$	−1.69712e6	−0.373954
$461$	8.15879e6	1.78802	0.894012	−	0.448043i	$-0.147879\pi$
$461$	8.15879e6	1.78802	0.894012	+	0.448043i	$0.147879\pi$
$462$	0	0
$463$	5.42287e6	1.17565	0.587823	−	0.808990i	$-0.299985\pi$
$463$	5.42287e6	1.17565	0.587823	+	0.808990i	$0.299985\pi$
$464$	1.00872e6	0.217508
$465$	0	0
$466$	−4.33281e6	−0.924282
$467$	−8.21137e6	−1.74230	−0.871151	−	0.491015i	$-0.836626\pi$
$467$	−8.21137e6	−1.74230	−0.871151	+	0.491015i	$0.836626\pi$
$468$	0	0
$469$	−2.23341e6	−0.468853
$470$	−720547.	−0.150459
$471$	0	0
$472$	−599566.	−0.123874
$473$	−2.73034e6	−0.561130
$474$	0	0
$475$	−2.12649e6	−0.432444
$476$	−1.39967e6	−0.283145
$477$	0	0
$478$	−3.22345e6	−0.645285
$479$	5.34027e6	1.06347	0.531734	−	0.846911i	$-0.321541\pi$
$479$	5.34027e6	1.06347	0.531734	+	0.846911i	$0.321541\pi$
$480$	0	0
$481$	1.22281e6	0.240989
$482$	3.10656e6	0.609064
$483$	0	0
$484$	−4.48899e6	−0.871034
$485$	712658.	0.137571
$486$	0	0
$487$	2.52006e6	0.481492	0.240746	−	0.970588i	$-0.422608\pi$
$487$	2.52006e6	0.481492	0.240746	+	0.970588i	$0.422608\pi$
$488$	5.98341e6	1.13736
$489$	0	0
$490$	−1.30967e6	−0.246417
$491$	8.33075e6	1.55948	0.779741	−	0.626102i	$-0.215351\pi$
$491$	8.33075e6	1.55948	0.779741	+	0.626102i	$0.215351\pi$
$492$	0	0
$493$	1.16640e7	2.16138
$494$	457343.	0.0843188
$495$	0	0
$496$	−321746.	−0.0587232
$497$	−1050.00	−0.000190678 0
$498$	0	0
$499$	9.76824e6	1.75616	0.878082	−	0.478510i	$-0.158823\pi$
$499$	9.76824e6	1.75616	0.878082	+	0.478510i	$0.158823\pi$
$500$	−3.21341e6	−0.574832
$501$	0	0
$502$	−5.58521e6	−0.989190
$503$	−2.49233e6	−0.439223	−0.219612	−	0.975587i	$-0.570479\pi$
$503$	−2.49233e6	−0.439223	−0.219612	+	0.975587i	$0.570479\pi$
$504$	0	0
$505$	4.11473e6	0.717981
$506$	5.68436e6	0.986974
$507$	0	0
$508$	−98189.9	−0.0168814
$509$	7.02509e6	1.20187	0.600935	−	0.799298i	$-0.294795\pi$
$509$	7.02509e6	1.20187	0.600935	+	0.799298i	$0.294795\pi$
$510$	0	0
$511$	2.12626e6	0.360216
$512$	−1.63948e6	−0.276395
$513$	0	0
$514$	−2.23369e6	−0.372920
$515$	1.38951e6	0.230858
$516$	0	0
$517$	−5.09913e6	−0.839015
$518$	946226.	0.154942
$519$	0	0
$520$	744379.	0.120722
$521$	5.27900e6	0.852034	0.426017	−	0.904715i	$-0.359916\pi$
$521$	5.27900e6	0.852034	0.426017	+	0.904715i	$0.359916\pi$
$522$	0	0
$523$	−291703.	−0.0466322	−0.0233161	−	0.999728i	$-0.507422\pi$
$523$	−291703.	−0.0466322	−0.0233161	+	0.999728i	$0.507422\pi$
$524$	−4.38411e6	−0.697514
$525$	0	0
$526$	−2.86716e6	−0.451843
$527$	−3.72043e6	−0.583535
$528$	0	0
$529$	2.11127e6	0.328023
$530$	−946802.	−0.146410
$531$	0	0
$532$	−747727.	−0.114542
$533$	−2.70180e6	−0.411941
$534$	0	0
$535$	5.77145e6	0.871767
$536$	−9.85674e6	−1.48191
$537$	0	0
$538$	1.10969e6	0.165290
$539$	−9.26821e6	−1.37412
$540$	0	0
$541$	1.81999e6	0.267347	0.133673	−	0.991025i	$-0.457323\pi$
$541$	1.81999e6	0.267347	0.133673	+	0.991025i	$0.457323\pi$
$542$	−1.52635e6	−0.223180
$543$	0	0
$544$	−9.85682e6	−1.42804
$545$	−1.67201e6	−0.241128
$546$	0	0
$547$	−9.72974e6	−1.39038	−0.695189	−	0.718827i	$-0.744679\pi$
$547$	−9.72974e6	−1.39038	−0.695189	+	0.718827i	$0.744679\pi$
$548$	2.79252e6	0.397232
$549$	0	0
$550$	4.68714e6	0.660695
$551$	6.23112e6	0.874353
$552$	0	0
$553$	3.63698e6	0.505740
$554$	4.29591e6	0.594676
$555$	0	0
$556$	−5.74506e6	−0.788147
$557$	3.95131e6	0.539638	0.269819	−	0.962911i	$-0.413036\pi$
$557$	3.95131e6	0.539638	0.269819	+	0.962911i	$0.413036\pi$
$558$	0	0
$559$	728087.	0.0985494
$560$	−148942.	−0.0200701
$561$	0	0
$562$	2.46002e6	0.328547
$563$	907456.	0.120658	0.0603288	−	0.998179i	$-0.480785\pi$
$563$	907456.	0.120658	0.0603288	+	0.998179i	$0.480785\pi$
$564$	0	0
$565$	6.48207e6	0.854265
$566$	−1.25567e6	−0.164753
$567$	0	0
$568$	−4633.99	−0.000602677 0
$569$	4.32375e6	0.559861	0.279931	−	0.960020i	$-0.409689\pi$
$569$	4.32375e6	0.559861	0.279931	+	0.960020i	$0.409689\pi$
$570$	0	0
$571$	−1.76582e6	−0.226650	−0.113325	−	0.993558i	$-0.536150\pi$
$571$	−1.76582e6	−0.226650	−0.113325	+	0.993558i	$0.536150\pi$
$572$	2.12986e6	0.272183
$573$	0	0
$574$	−2.09069e6	−0.264856
$575$	7.04808e6	0.888999
$576$	0	0
$577$	4.97841e6	0.622516	0.311258	−	0.950325i	$-0.399250\pi$
$577$	4.97841e6	0.622516	0.311258	+	0.950325i	$0.399250\pi$
$578$	−4.18925e6	−0.521575
$579$	0	0
$580$	4.10054e6	0.506141
$581$	3.49685e6	0.429770
$582$	0	0
$583$	−6.70028e6	−0.816435
$584$	9.38383e6	1.13854
$585$	0	0
$586$	5.72556e6	0.688769
$587$	−2.55026e6	−0.305484	−0.152742	−	0.988266i	$-0.548810\pi$
$587$	−2.55026e6	−0.305484	−0.152742	+	0.988266i	$0.548810\pi$
$588$	0	0
$589$	−1.98751e6	−0.236060
$590$	−298286.	−0.0352779
$591$	0	0
$592$	−1.07981e6	−0.126632
$593$	−5.95905e6	−0.695889	−0.347945	−	0.937515i	$-0.613120\pi$
$593$	−5.95905e6	−0.695889	−0.347945	+	0.937515i	$0.613120\pi$
$594$	0	0
$595$	−1.72226e6	−0.199437
$596$	−6.74576e6	−0.777885
$597$	0	0
$598$	−1.51582e6	−0.173339
$599$	−7.34100e6	−0.835966	−0.417983	−	0.908455i	$-0.637263\pi$
$599$	−7.34100e6	−0.835966	−0.417983	+	0.908455i	$0.637263\pi$
$600$	0	0
$601$	5.04285e6	0.569495	0.284748	−	0.958603i	$-0.408090\pi$
$601$	5.04285e6	0.569495	0.284748	+	0.958603i	$0.408090\pi$
$602$	563403.	0.0633619
$603$	0	0
$604$	−3.26259e6	−0.363890
$605$	−5.52358e6	−0.613524
$606$	0	0
$607$	1.78269e7	1.96383	0.981916	−	0.189318i	$-0.0606277\pi$
$607$	1.78269e7	1.96383	0.981916	+	0.189318i	$0.0606277\pi$
$608$	−5.26567e6	−0.577690
$609$	0	0
$610$	2.97676e6	0.323906
$611$	1.35976e6	0.147353
$612$	0	0
$613$	1.10977e7	1.19284	0.596421	−	0.802672i	$-0.296589\pi$
$613$	1.10977e7	1.19284	0.596421	+	0.802672i	$0.296589\pi$
$614$	2.97329e6	0.318285
$615$	0	0
$616$	4.07628e6	0.432824
$617$	−6.00790e6	−0.635345	−0.317673	−	0.948200i	$-0.602901\pi$
$617$	−6.00790e6	−0.635345	−0.317673	+	0.948200i	$0.602901\pi$
$618$	0	0
$619$	1.79364e6	0.188151	0.0940757	−	0.995565i	$-0.470010\pi$
$619$	1.79364e6	0.188151	0.0940757	+	0.995565i	$0.470010\pi$
$620$	−1.30793e6	−0.136649
$621$	0	0
$622$	2.45729e6	0.254672
$623$	5.00163e6	0.516287
$624$	0	0
$625$	3.57952e6	0.366543
$626$	−725218.	−0.0739661
$627$	0	0
$628$	−8.29127e6	−0.838923
$629$	−1.24861e7	−1.25835
$630$	0	0
$631$	−1.00827e7	−1.00810	−0.504052	−	0.863673i	$-0.668158\pi$
$631$	−1.00827e7	−1.00810	−0.504052	+	0.863673i	$0.668158\pi$
$632$	1.60511e7	1.59850
$633$	0	0
$634$	3.46457e6	0.342316
$635$	−120820.	−0.0118906
$636$	0	0
$637$	2.47151e6	0.241332
$638$	−1.37344e7	−1.33585
$639$	0	0
$640$	−3.85677e6	−0.372198
$641$	5.75277e6	0.553009	0.276505	−	0.961013i	$-0.410824\pi$
$641$	5.75277e6	0.553009	0.276505	+	0.961013i	$0.410824\pi$
$642$	0	0
$643$	−5.46072e6	−0.520862	−0.260431	−	0.965492i	$-0.583865\pi$
$643$	−5.46072e6	−0.520862	−0.260431	+	0.965492i	$0.583865\pi$
$644$	2.47828e6	0.235470
$645$	0	0
$646$	−4.66993e6	−0.440280
$647$	−2.48477e6	−0.233360	−0.116680	−	0.993170i	$-0.537225\pi$
$647$	−2.48477e6	−0.233360	−0.116680	+	0.993170i	$0.537225\pi$
$648$	0	0
$649$	−2.11089e6	−0.196723
$650$	−1.24990e6	−0.116036
$651$	0	0
$652$	941939.	0.0867768
$653$	9.47172e6	0.869253	0.434626	−	0.900611i	$-0.356880\pi$
$653$	9.47172e6	0.869253	0.434626	+	0.900611i	$0.356880\pi$
$654$	0	0
$655$	−5.39453e6	−0.491304
$656$	2.38585e6	0.216463
$657$	0	0
$658$	1.05220e6	0.0947402
$659$	−9.34926e6	−0.838617	−0.419309	−	0.907844i	$-0.637727\pi$
$659$	−9.34926e6	−0.838617	−0.419309	+	0.907844i	$0.637727\pi$
$660$	0	0
$661$	4.30563e6	0.383295	0.191647	−	0.981464i	$-0.438617\pi$
$661$	4.30563e6	0.383295	0.191647	+	0.981464i	$0.438617\pi$
$662$	9.29694e6	0.824509
$663$	0	0
$664$	1.54327e7	1.35838
$665$	−920057.	−0.0806790
$666$	0	0
$667$	−2.06525e7	−1.79746
$668$	4.93908e6	0.428258
$669$	0	0
$670$	−4.90376e6	−0.422029
$671$	2.10658e7	1.80623
$672$	0	0
$673$	1.43291e7	1.21950	0.609748	−	0.792595i	$-0.291271\pi$
$673$	1.43291e7	1.21950	0.609748	+	0.792595i	$0.291271\pi$
$674$	655662.	0.0555943
$675$	0	0
$676$	7.49652e6	0.630947
$677$	6.86042e6	0.575279	0.287640	−	0.957739i	$-0.407129\pi$
$677$	6.86042e6	0.575279	0.287640	+	0.957739i	$0.407129\pi$
$678$	0	0
$679$	−1.04068e6	−0.0866250
$680$	−7.60086e6	−0.630363
$681$	0	0
$682$	4.38081e6	0.360656
$683$	2.00520e7	1.64478	0.822388	−	0.568927i	$-0.192641\pi$
$683$	2.00520e7	1.64478	0.822388	+	0.568927i	$0.192641\pi$
$684$	0	0
$685$	3.43612e6	0.279796
$686$	4.01557e6	0.325789
$687$	0	0
$688$	−642943.	−0.0517847
$689$	1.78674e6	0.143388
$690$	0	0
$691$	1.21795e7	0.970365	0.485182	−	0.874413i	$-0.338753\pi$
$691$	1.21795e7	0.970365	0.485182	+	0.874413i	$0.338753\pi$
$692$	−4.27266e6	−0.339182
$693$	0	0
$694$	3.67299e6	0.289482
$695$	−7.06914e6	−0.555142
$696$	0	0
$697$	2.75881e7	2.15100
$698$	−1.00840e7	−0.783418
$699$	0	0
$700$	2.04351e6	0.157627
$701$	−1.23445e7	−0.948809	−0.474404	−	0.880307i	$-0.657337\pi$
$701$	−1.23445e7	−0.948809	−0.474404	+	0.880307i	$0.657337\pi$
$702$	0	0
$703$	−6.67029e6	−0.509045
$704$	8.83544e6	0.671888
$705$	0	0
$706$	−7.54378e6	−0.569609
$707$	−6.00867e6	−0.452095
$708$	0	0
$709$	2.21727e7	1.65654	0.828272	−	0.560327i	$-0.189324\pi$
$709$	2.21727e7	1.65654	0.828272	+	0.560327i	$0.189324\pi$
$710$	−2305.42	−0.000171635 0
$711$	0	0
$712$	2.20737e7	1.63183
$713$	6.58745e6	0.485281
$714$	0	0
$715$	2.62074e6	0.191716
$716$	−1.12610e6	−0.0820909
$717$	0	0
$718$	636632.	0.0460869
$719$	2.40050e6	0.173173	0.0865864	−	0.996244i	$-0.472404\pi$
$719$	2.40050e6	0.173173	0.0865864	+	0.996244i	$0.472404\pi$
$720$	0	0
$721$	−2.02908e6	−0.145365
$722$	5.44423e6	0.388681
$723$	0	0
$724$	−1.50394e7	−1.06631
$725$	−1.70294e7	−1.20325
$726$	0	0
$727$	−9.51466e6	−0.667663	−0.333831	−	0.942633i	$-0.608342\pi$
$727$	−9.51466e6	−0.667663	−0.333831	+	0.942633i	$0.608342\pi$
$728$	−1.08700e6	−0.0760155
$729$	0	0
$730$	4.66848e6	0.324242
$731$	−7.43451e6	−0.514587
$732$	0	0
$733$	1.04964e6	0.0721572	0.0360786	−	0.999349i	$-0.488513\pi$
$733$	1.04964e6	0.0721572	0.0360786	+	0.999349i	$0.488513\pi$
$734$	−1.06998e7	−0.733050
$735$	0	0
$736$	1.74526e7	1.18759
$737$	−3.47027e7	−2.35339
$738$	0	0
$739$	−2.68604e6	−0.180926	−0.0904629	−	0.995900i	$-0.528835\pi$
$739$	−2.68604e6	−0.180926	−0.0904629	+	0.995900i	$0.528835\pi$
$740$	−4.38955e6	−0.294674
$741$	0	0
$742$	1.38260e6	0.0921905
$743$	2.09335e7	1.39114	0.695569	−	0.718460i	$-0.255153\pi$
$743$	2.09335e7	1.39114	0.695569	+	0.718460i	$0.255153\pi$
$744$	0	0
$745$	−8.30047e6	−0.547914
$746$	−1.12121e7	−0.737632
$747$	0	0
$748$	−2.17481e7	−1.42124
$749$	−8.42794e6	−0.548930
$750$	0	0
$751$	2.59501e7	1.67895	0.839477	−	0.543395i	$-0.182861\pi$
$751$	2.59501e7	1.67895	0.839477	+	0.543395i	$0.182861\pi$
$752$	−1.20075e6	−0.0774298
$753$	0	0
$754$	3.66250e6	0.234611
$755$	−4.01452e6	−0.256311
$756$	0	0
$757$	−8.97503e6	−0.569241	−0.284620	−	0.958640i	$-0.591868\pi$
$757$	−8.97503e6	−0.569241	−0.284620	+	0.958640i	$0.591868\pi$
$758$	1.48133e7	0.936438
$759$	0	0
$760$	−4.06050e6	−0.255003
$761$	−1.20515e7	−0.754361	−0.377180	−	0.926140i	$-0.623106\pi$
$761$	−1.20515e7	−0.754361	−0.377180	+	0.926140i	$0.623106\pi$
$762$	0	0
$763$	2.44161e6	0.151833
$764$	−2.09541e7	−1.29878
$765$	0	0
$766$	−1.21064e7	−0.745493
$767$	562903.	0.0345498
$768$	0	0
$769$	1.32176e7	0.806001	0.403001	−	0.915200i	$-0.367967\pi$
$769$	1.32176e7	0.806001	0.403001	+	0.915200i	$0.367967\pi$
$770$	2.02796e6	0.123263
$771$	0	0
$772$	6.63440e6	0.400644
$773$	1.10628e7	0.665914	0.332957	−	0.942942i	$-0.391954\pi$
$773$	1.10628e7	0.665914	0.332957	+	0.942942i	$0.391954\pi$
$774$	0	0
$775$	5.43179e6	0.324855
$776$	−4.59285e6	−0.273797
$777$	0	0
$778$	4.61910e6	0.273595
$779$	1.47380e7	0.870152
$780$	0	0
$781$	−16314.9	−0.000957100 0
$782$	1.54781e7	0.905109
$783$	0	0
$784$	−2.18249e6	−0.126813
$785$	−1.02022e7	−0.590907
$786$	0	0
$787$	−1.33119e7	−0.766132	−0.383066	−	0.923721i	$-0.625132\pi$
$787$	−1.33119e7	−0.766132	−0.383066	+	0.923721i	$0.625132\pi$
$788$	−694779.	−0.0398595
$789$	0	0
$790$	7.98547e6	0.455232
$791$	−9.46565e6	−0.537910
$792$	0	0
$793$	−5.61753e6	−0.317221
$794$	−1.29778e7	−0.730548
$795$	0	0
$796$	−9.29912e6	−0.520186
$797$	−1.61590e6	−0.0901089	−0.0450544	−	0.998985i	$-0.514346\pi$
$797$	−1.61590e6	−0.0901089	−0.0450544	+	0.998985i	$0.514346\pi$
$798$	0	0
$799$	−1.38846e7	−0.769423
$800$	1.43909e7	0.794991
$801$	0	0
$802$	−601347.	−0.0330133
$803$	3.30377e7	1.80809
$804$	0	0
$805$	3.04945e6	0.165856
$806$	−1.16821e6	−0.0633408
$807$	0	0
$808$	−2.65181e7	−1.42894
$809$	1.48836e7	0.799535	0.399768	−	0.916617i	$-0.369091\pi$
$809$	1.48836e7	0.799535	0.399768	+	0.916617i	$0.369091\pi$
$810$	0	0
$811$	−3.19912e7	−1.70796	−0.853982	−	0.520303i	$-0.825819\pi$
$811$	−3.19912e7	−1.70796	−0.853982	+	0.520303i	$0.825819\pi$
$812$	−5.98795e6	−0.318705
$813$	0	0
$814$	1.47024e7	0.777728
$815$	1.15903e6	0.0611224
$816$	0	0
$817$	−3.97163e6	−0.208168
$818$	1.04228e7	0.544632
$819$	0	0
$820$	9.69873e6	0.503709
$821$	2.95946e7	1.53234	0.766169	−	0.642640i	$-0.222161\pi$
$821$	2.95946e7	1.53234	0.766169	+	0.642640i	$0.222161\pi$
$822$	0	0
$823$	1.71026e7	0.880163	0.440081	−	0.897958i	$-0.354950\pi$
$823$	1.71026e7	0.880163	0.440081	+	0.897958i	$0.354950\pi$
$824$	−8.95496e6	−0.459458
$825$	0	0
$826$	435581.	0.0222136
$827$	2.24600e7	1.14195	0.570974	−	0.820968i	$-0.306566\pi$
$827$	2.24600e7	1.14195	0.570974	+	0.820968i	$0.306566\pi$
$828$	0	0
$829$	−2.65000e7	−1.33924	−0.669622	−	0.742702i	$-0.733544\pi$
$829$	−2.65000e7	−1.33924	−0.669622	+	0.742702i	$0.733544\pi$
$830$	7.67780e6	0.386849
$831$	0	0
$832$	−2.35611e6	−0.118001
$833$	−2.52367e7	−1.26014
$834$	0	0
$835$	6.07740e6	0.301649
$836$	−1.16182e7	−0.574938
$837$	0	0
$838$	1.17160e7	0.576328
$839$	−3.38841e7	−1.66185	−0.830924	−	0.556386i	$-0.812187\pi$
$839$	−3.38841e7	−1.66185	−0.830924	+	0.556386i	$0.812187\pi$
$840$	0	0
$841$	2.93890e7	1.43283
$842$	1.72038e7	0.836266
$843$	0	0
$844$	2.33537e7	1.12850
$845$	9.22426e6	0.444416
$846$	0	0
$847$	8.06598e6	0.386321
$848$	−1.57779e6	−0.0753459
$849$	0	0
$850$	1.27627e7	0.605894
$851$	2.21081e7	1.04647
$852$	0	0
$853$	−1.88452e6	−0.0886804	−0.0443402	−	0.999016i	$-0.514119\pi$
$853$	−1.88452e6	−0.0886804	−0.0443402	+	0.999016i	$0.514119\pi$
$854$	−4.34691e6	−0.203956
$855$	0	0
$856$	−3.71951e7	−1.73501
$857$	−1.95529e6	−0.0909407	−0.0454704	−	0.998966i	$-0.514479\pi$
$857$	−1.95529e6	−0.0909407	−0.0454704	+	0.998966i	$0.514479\pi$
$858$	0	0
$859$	−1.15147e7	−0.532437	−0.266219	−	0.963913i	$-0.585774\pi$
$859$	−1.15147e7	−0.532437	−0.266219	+	0.963913i	$0.585774\pi$
$860$	−2.61363e6	−0.120503
$861$	0	0
$862$	−2.20435e7	−1.01045
$863$	−2.60655e6	−0.119135	−0.0595674	−	0.998224i	$-0.518972\pi$
$863$	−2.60655e6	−0.119135	−0.0595674	+	0.998224i	$0.518972\pi$
$864$	0	0
$865$	−5.25739e6	−0.238908
$866$	−129818.	−0.00588219
$867$	0	0
$868$	1.90995e6	0.0860445
$869$	5.65112e7	2.53855
$870$	0	0
$871$	9.25402e6	0.413319
$872$	1.07756e7	0.479899
$873$	0	0
$874$	8.26865e6	0.366147
$875$	5.77397e6	0.254950
$876$	0	0
$877$	3.54109e7	1.55467	0.777335	−	0.629087i	$-0.216571\pi$
$877$	3.54109e7	1.55467	0.777335	+	0.629087i	$0.216571\pi$
$878$	1.42482e7	0.623769
$879$	0	0
$880$	−2.31426e6	−0.100741
$881$	−1.78051e7	−0.772867	−0.386434	−	0.922317i	$-0.626293\pi$
$881$	−1.78051e7	−0.772867	−0.386434	+	0.922317i	$0.626293\pi$
$882$	0	0
$883$	−3.70288e7	−1.59822	−0.799111	−	0.601183i	$-0.794696\pi$
$883$	−3.70288e7	−1.59822	−0.799111	+	0.601183i	$0.794696\pi$
$884$	5.79946e6	0.249607
$885$	0	0
$886$	−891607.	−0.0381583
$887$	3.85936e7	1.64705	0.823524	−	0.567281i	$-0.192005\pi$
$887$	3.85936e7	1.64705	0.823524	+	0.567281i	$0.192005\pi$
$888$	0	0
$889$	176431.	0.00748723
$890$	1.09817e7	0.464725
$891$	0	0
$892$	−1.17643e7	−0.495056
$893$	−7.41735e6	−0.311258
$894$	0	0
$895$	−1.38564e6	−0.0578218
$896$	5.63197e6	0.234364
$897$	0	0
$898$	3.45871e6	0.143128
$899$	−1.59164e7	−0.656820
$900$	0	0
$901$	−1.82444e7	−0.748716
$902$	−3.24850e7	−1.32943
$903$	0	0
$904$	−4.17749e7	−1.70018
$905$	−1.85056e7	−0.751070
$906$	0	0
$907$	−9.11662e6	−0.367973	−0.183986	−	0.982929i	$-0.558900\pi$
$907$	−9.11662e6	−0.367973	−0.183986	+	0.982929i	$0.558900\pi$
$908$	−1.09430e7	−0.440477
$909$	0	0
$910$	−540787.	−0.0216483
$911$	4.35218e7	1.73745	0.868723	−	0.495299i	$-0.164941\pi$
$911$	4.35218e7	1.73745	0.868723	+	0.495299i	$0.164941\pi$
$912$	0	0
$913$	5.43339e7	2.15722
$914$	7.51737e6	0.297646
$915$	0	0
$916$	−5.13010e6	−0.202017
$917$	7.87753e6	0.309362
$918$	0	0
$919$	2.99781e6	0.117089	0.0585444	−	0.998285i	$-0.481354\pi$
$919$	2.99781e6	0.117089	0.0585444	+	0.998285i	$0.481354\pi$
$920$	1.34582e7	0.524224
$921$	0	0
$922$	−2.61591e7	−1.01343
$923$	4350.63	0.000168092 0
$924$	0	0
$925$	1.82296e7	0.700525
$926$	−1.73871e7	−0.666344
$927$	0	0
$928$	−4.21686e7	−1.60738
$929$	−2.19993e7	−0.836313	−0.418156	−	0.908375i	$-0.637324\pi$
$929$	−2.19993e7	−0.836313	−0.418156	+	0.908375i	$0.637324\pi$
$930$	0	0
$931$	−1.34818e7	−0.509770
$932$	−2.93516e7	−1.10686
$933$	0	0
$934$	2.63277e7	0.987519
$935$	−2.67604e7	−1.00107
$936$	0	0
$937$	8.01794e6	0.298341	0.149171	−	0.988811i	$-0.452340\pi$
$937$	8.01794e6	0.298341	0.149171	+	0.988811i	$0.452340\pi$
$938$	7.16087e6	0.265741
$939$	0	0
$940$	−4.88118e6	−0.180179
$941$	−9.19445e6	−0.338494	−0.169247	−	0.985574i	$-0.554134\pi$
$941$	−9.19445e6	−0.338494	−0.169247	+	0.985574i	$0.554134\pi$
$942$	0	0
$943$	−4.88479e7	−1.78882
$944$	−497076.	−0.0181549
$945$	0	0
$946$	8.75414e6	0.318043
$947$	4.27420e6	0.154874	0.0774372	−	0.996997i	$-0.475326\pi$
$947$	4.27420e6	0.154874	0.0774372	+	0.996997i	$0.475326\pi$
$948$	0	0
$949$	−8.81002e6	−0.317550
$950$	6.81806e6	0.245105
$951$	0	0
$952$	1.10994e7	0.396924
$953$	5.03679e7	1.79648	0.898239	−	0.439507i	$-0.144847\pi$
$953$	5.03679e7	1.79648	0.898239	+	0.439507i	$0.144847\pi$
$954$	0	0
$955$	−2.57834e7	−0.914812
$956$	−2.18365e7	−0.772750
$957$	0	0
$958$	−1.71222e7	−0.602763
$959$	−5.01770e6	−0.176181
$960$	0	0
$961$	−2.35524e7	−0.822670
$962$	−3.92063e6	−0.136590
$963$	0	0
$964$	2.10447e7	0.729374
$965$	8.16345e6	0.282199
$966$	0	0
$967$	−1.89781e6	−0.0652659	−0.0326329	−	0.999467i	$-0.510389\pi$
$967$	−1.89781e6	−0.0652659	−0.0326329	+	0.999467i	$0.510389\pi$
$968$	3.55977e7	1.22105
$969$	0	0
$970$	−2.28496e6	−0.0779738
$971$	−2.68041e7	−0.912331	−0.456166	−	0.889895i	$-0.650778\pi$
$971$	−2.68041e7	−0.912331	−0.456166	+	0.889895i	$0.650778\pi$
$972$	0	0
$973$	1.03229e7	0.349560
$974$	−8.07994e6	−0.272905
$975$	0	0
$976$	4.96061e6	0.166690
$977$	−4.27764e7	−1.43373	−0.716865	−	0.697212i	$-0.754424\pi$
$977$	−4.27764e7	−1.43373	−0.716865	+	0.697212i	$0.754424\pi$
$978$	0	0
$979$	7.77151e7	2.59148
$980$	−8.87205e6	−0.295093
$981$	0	0
$982$	−2.67104e7	−0.883898
$983$	3.97908e7	1.31340	0.656702	−	0.754150i	$-0.271951\pi$
$983$	3.97908e7	1.31340	0.656702	+	0.754150i	$0.271951\pi$
$984$	0	0
$985$	−854907.	−0.0280756
$986$	−3.73978e7	−1.22505
$987$	0	0
$988$	3.09816e6	0.100975
$989$	1.31636e7	0.427943
$990$	0	0
$991$	2.18022e7	0.705207	0.352604	−	0.935773i	$-0.385296\pi$
$991$	2.18022e7	0.705207	0.352604	+	0.935773i	$0.385296\pi$
$992$	1.34503e7	0.433964
$993$	0	0
$994$	3366.57	0.000108074 0
$995$	−1.14423e7	−0.366400
$996$	0	0
$997$	−2.61869e7	−0.834346	−0.417173	−	0.908827i	$-0.636979\pi$
$997$	−2.61869e7	−0.834346	−0.417173	+	0.908827i	$0.636979\pi$
$998$	−3.13194e7	−0.995376
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.5		11
3.2	odd	2		177.6.a.a.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.7	✓	11	3.2	odd	2
531.6.a.b.1.5		11	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-3.20625 q^{2} -21.7200 q^{4} -26.7258 q^{5} +39.0273 q^{7} +172.240 q^{8} +O(q^{10})\)	\(q-3.20625 q^{2} -21.7200 q^{4} -26.7258 q^{5} +39.0273 q^{7} +172.240 q^{8} +85.6897 q^{10} +606.405 q^{11} -161.707 q^{13} -125.131 q^{14} +142.797 q^{16} +1651.20 q^{17} +882.094 q^{19} +580.485 q^{20} -1944.28 q^{22} -2923.63 q^{23} -2410.73 q^{25} +518.474 q^{26} -847.672 q^{28} +7064.00 q^{29} -2253.17 q^{31} -5969.51 q^{32} -5294.14 q^{34} -1043.04 q^{35} -7561.88 q^{37} -2828.21 q^{38} -4603.25 q^{40} +16708.0 q^{41} -4502.50 q^{43} -13171.1 q^{44} +9373.88 q^{46} -8408.79 q^{47} -15283.9 q^{49} +7729.39 q^{50} +3512.28 q^{52} -11049.2 q^{53} -16206.7 q^{55} +6722.04 q^{56} -22648.9 q^{58} -3481.00 q^{59} +34738.9 q^{61} +7224.23 q^{62} +14570.2 q^{64} +4321.76 q^{65} -57227.0 q^{67} -35863.9 q^{68} +3344.23 q^{70} -26.9044 q^{71} +54481.3 q^{73} +24245.2 q^{74} -19159.1 q^{76} +23666.3 q^{77} +93190.6 q^{79} -3816.37 q^{80} -53569.9 q^{82} +89600.1 q^{83} -44129.6 q^{85} +14436.1 q^{86} +104447. q^{88} +128157. q^{89} -6311.00 q^{91} +63501.2 q^{92} +26960.7 q^{94} -23574.7 q^{95} -26665.5 q^{97} +49003.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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