LMFDB - Embedded newform 531.6.a.d.1.6

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 $ x^12 - 4x^11 - 283x^10 + 1045x^9 + 27968x^8 - 94393x^7 - 1130486x^6 + 3566264x^5 + 15496192x^4 - 53008480x^3 - 16576192x^2 + 120303168*x - 50564480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.501674$ 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)

$f(q)$	$=$	$q+0.501674 q^{2} -31.7483 q^{4} +14.0506 q^{5} +117.234 q^{7} -31.9809 q^{8} +O(q^{10})$	\(q+0.501674 q^{2} -31.7483 q^{4} +14.0506 q^{5} +117.234 q^{7} -31.9809 q^{8} +7.04881 q^{10} -53.5537 q^{11} -471.768 q^{13} +58.8130 q^{14} +999.902 q^{16} -1164.87 q^{17} +1826.16 q^{19} -446.082 q^{20} -26.8665 q^{22} -254.207 q^{23} -2927.58 q^{25} -236.673 q^{26} -3721.97 q^{28} +6598.12 q^{29} +1104.44 q^{31} +1525.01 q^{32} -584.382 q^{34} +1647.20 q^{35} -7748.51 q^{37} +916.136 q^{38} -449.350 q^{40} +5824.87 q^{41} -6404.30 q^{43} +1700.24 q^{44} -127.529 q^{46} +2856.66 q^{47} -3063.28 q^{49} -1468.69 q^{50} +14977.8 q^{52} +16675.9 q^{53} -752.461 q^{55} -3749.23 q^{56} +3310.10 q^{58} +3481.00 q^{59} +37099.5 q^{61} +554.070 q^{62} -31231.8 q^{64} -6628.61 q^{65} +10974.2 q^{67} +36982.5 q^{68} +826.357 q^{70} +15478.9 q^{71} -42171.4 q^{73} -3887.22 q^{74} -57977.5 q^{76} -6278.29 q^{77} -89645.3 q^{79} +14049.2 q^{80} +2922.18 q^{82} -58689.8 q^{83} -16367.0 q^{85} -3212.87 q^{86} +1712.69 q^{88} -61752.2 q^{89} -55307.0 q^{91} +8070.64 q^{92} +1433.11 q^{94} +25658.6 q^{95} -13874.8 q^{97} -1536.77 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * 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 $

$2$	0.501674	0.0886842	0.0443421	−	0.999016i	$-0.485881\pi$
$2$	0.501674	0.0886842	0.0443421	+	0.999016i	$0.485881\pi$
$3$	0	0
$3$	0	0
$4$	−31.7483	−0.992135
$5$	14.0506	0.251344	0.125672	−	0.992072i	$-0.459891\pi$
$5$	14.0506	0.251344	0.125672	+	0.992072i	$0.459891\pi$
$6$	0	0
$7$	117.234	0.904288	0.452144	−	0.891945i	$-0.350659\pi$
$7$	117.234	0.904288	0.452144	+	0.891945i	$0.350659\pi$
$8$	−31.9809	−0.176671
$9$	0	0
$10$	7.04881	0.0222903
$11$	−53.5537	−0.133447	−0.0667233	−	0.997772i	$-0.521254\pi$
$11$	−53.5537	−0.133447	−0.0667233	+	0.997772i	$0.521254\pi$
$12$	0	0
$13$	−471.768	−0.774230	−0.387115	−	0.922031i	$-0.626528\pi$
$13$	−471.768	−0.774230	−0.387115	+	0.922031i	$0.626528\pi$
$14$	58.8130	0.0801961
$15$	0	0
$16$	999.902	0.976467
$17$	−1164.87	−0.977582	−0.488791	−	0.872401i	$-0.662562\pi$
$17$	−1164.87	−0.977582	−0.488791	+	0.872401i	$0.662562\pi$
$18$	0	0
$19$	1826.16	1.16053	0.580263	−	0.814429i	$-0.302950\pi$
$19$	1826.16	1.16053	0.580263	+	0.814429i	$0.302950\pi$
$20$	−446.082	−0.249368
$21$	0	0
$22$	−26.8665	−0.0118346
$23$	−254.207	−0.100200	−0.0501000	−	0.998744i	$-0.515954\pi$
$23$	−254.207	−0.100200	−0.0501000	+	0.998744i	$0.515954\pi$
$24$	0	0
$25$	−2927.58	−0.936826
$26$	−236.673	−0.0686620
$27$	0	0
$28$	−3721.97	−0.897176
$29$	6598.12	1.45689	0.728443	−	0.685107i	$-0.240245\pi$
$29$	6598.12	1.45689	0.728443	+	0.685107i	$0.240245\pi$
$30$	0	0
$31$	1104.44	0.206414	0.103207	−	0.994660i	$-0.467090\pi$
$31$	1104.44	0.206414	0.103207	+	0.994660i	$0.467090\pi$
$32$	1525.01	0.263268
$33$	0	0
$34$	−584.382	−0.0866961
$35$	1647.20	0.227288
$36$	0	0
$37$	−7748.51	−0.930494	−0.465247	−	0.885181i	$-0.654035\pi$
$37$	−7748.51	−0.930494	−0.465247	+	0.885181i	$0.654035\pi$
$38$	916.136	0.102920
$39$	0	0
$40$	−449.350	−0.0444053
$41$	5824.87	0.541161	0.270581	−	0.962697i	$-0.412784\pi$
$41$	5824.87	0.541161	0.270581	+	0.962697i	$0.412784\pi$
$42$	0	0
$43$	−6404.30	−0.528203	−0.264101	−	0.964495i	$-0.585075\pi$
$43$	−6404.30	−0.528203	−0.264101	+	0.964495i	$0.585075\pi$
$44$	1700.24	0.132397
$45$	0	0
$46$	−127.529	−0.00888616
$47$	2856.66	0.188632	0.0943158	−	0.995542i	$-0.469934\pi$
$47$	2856.66	0.188632	0.0943158	+	0.995542i	$0.469934\pi$
$48$	0	0
$49$	−3063.28	−0.182262
$50$	−1468.69	−0.0830817
$51$	0	0
$52$	14977.8	0.768141
$53$	16675.9	0.815452	0.407726	−	0.913104i	$-0.366322\pi$
$53$	16675.9	0.815452	0.407726	+	0.913104i	$0.366322\pi$
$54$	0	0
$55$	−752.461	−0.0335411
$56$	−3749.23	−0.159761
$57$	0	0
$58$	3310.10	0.129203
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	37099.5	1.27657	0.638284	−	0.769801i	$-0.279645\pi$
$61$	37099.5	1.27657	0.638284	+	0.769801i	$0.279645\pi$
$62$	554.070	0.0183057
$63$	0	0
$64$	−31231.8	−0.953119
$65$	−6628.61	−0.194598
$66$	0	0
$67$	10974.2	0.298665	0.149332	−	0.988787i	$-0.452288\pi$
$67$	10974.2	0.298665	0.149332	+	0.988787i	$0.452288\pi$
$68$	36982.5	0.969894
$69$	0	0
$70$	826.357	0.0201568
$71$	15478.9	0.364414	0.182207	−	0.983260i	$-0.441676\pi$
$71$	15478.9	0.364414	0.182207	+	0.983260i	$0.441676\pi$
$72$	0	0
$73$	−42171.4	−0.926214	−0.463107	−	0.886302i	$-0.653265\pi$
$73$	−42171.4	−0.926214	−0.463107	+	0.886302i	$0.653265\pi$
$74$	−3887.22	−0.0825202
$75$	0	0
$76$	−57977.5	−1.15140
$77$	−6278.29	−0.120674
$78$	0	0
$79$	−89645.3	−1.61607	−0.808034	−	0.589136i	$-0.799468\pi$
$79$	−89645.3	−1.61607	−0.808034	+	0.589136i	$0.799468\pi$
$80$	14049.2	0.245430
$81$	0	0
$82$	2922.18	0.0479924
$83$	−58689.8	−0.935120	−0.467560	−	0.883961i	$-0.654867\pi$
$83$	−58689.8	−0.935120	−0.467560	+	0.883961i	$0.654867\pi$
$84$	0	0
$85$	−16367.0	−0.245710
$86$	−3212.87	−0.0468433
$87$	0	0
$88$	1712.69	0.0235761
$89$	−61752.2	−0.826376	−0.413188	−	0.910646i	$-0.635585\pi$
$89$	−61752.2	−0.826376	−0.413188	+	0.910646i	$0.635585\pi$
$90$	0	0
$91$	−55307.0	−0.700127
$92$	8070.64	0.0994119
$93$	0	0
$94$	1433.11	0.0167286
$95$	25658.6	0.291692
$96$	0	0
$97$	−13874.8	−0.149726	−0.0748630	−	0.997194i	$-0.523852\pi$
$97$	−13874.8	−0.149726	−0.0748630	+	0.997194i	$0.523852\pi$
$98$	−1536.77	−0.0161638
$99$	0	0
$100$	92945.8	0.929458
$101$	−101934.	−0.994293	−0.497146	−	0.867667i	$-0.665619\pi$
$101$	−101934.	−0.994293	−0.497146	+	0.867667i	$0.665619\pi$
$102$	0	0
$103$	−176131.	−1.63585	−0.817925	−	0.575325i	$-0.804875\pi$
$103$	−176131.	−1.63585	−0.817925	+	0.575325i	$0.804875\pi$
$104$	15087.5	0.136784
$105$	0	0
$106$	8365.84	0.0723177
$107$	−140675.	−1.18783	−0.593917	−	0.804526i	$-0.702419\pi$
$107$	−140675.	−1.18783	−0.593917	+	0.804526i	$0.702419\pi$
$108$	0	0
$109$	−87074.8	−0.701982	−0.350991	−	0.936379i	$-0.614155\pi$
$109$	−87074.8	−0.701982	−0.350991	+	0.936379i	$0.614155\pi$
$110$	−377.490	−0.00297456
$111$	0	0
$112$	117222.	0.883008
$113$	257288.	1.89550	0.947748	−	0.319019i	$-0.103353\pi$
$113$	257288.	1.89550	0.947748	+	0.319019i	$0.103353\pi$
$114$	0	0
$115$	−3571.75	−0.0251847
$116$	−209479.	−1.44543
$117$	0	0
$118$	1746.33	0.0115457
$119$	−136561.	−0.884016
$120$	0	0
$121$	−158183.	−0.982192
$122$	18611.9	0.113211
$123$	0	0
$124$	−35064.2	−0.204791
$125$	−85042.3	−0.486810
$126$	0	0
$127$	−60712.0	−0.334014	−0.167007	−	0.985956i	$-0.553410\pi$
$127$	−60712.0	−0.334014	−0.167007	+	0.985956i	$0.553410\pi$
$128$	−64468.6	−0.347795
$129$	0	0
$130$	−3325.40	−0.0172578
$131$	−191296.	−0.973928	−0.486964	−	0.873422i	$-0.661896\pi$
$131$	−191296.	−0.973928	−0.486964	+	0.873422i	$0.661896\pi$
$132$	0	0
$133$	214087.	1.04945
$134$	5505.44	0.0264869
$135$	0	0
$136$	37253.4	0.172710
$137$	−262613.	−1.19541	−0.597703	−	0.801718i	$-0.703920\pi$
$137$	−262613.	−1.19541	−0.597703	+	0.801718i	$0.703920\pi$
$138$	0	0
$139$	−149545.	−0.656502	−0.328251	−	0.944590i	$-0.606459\pi$
$139$	−149545.	−0.656502	−0.328251	+	0.944590i	$0.606459\pi$
$140$	−52295.8	−0.225500
$141$	0	0
$142$	7765.38	0.0323178
$143$	25264.9	0.103318
$144$	0	0
$145$	92707.4	0.366180
$146$	−21156.3	−0.0821406
$147$	0	0
$148$	246002.	0.923176
$149$	356966.	1.31723	0.658614	−	0.752481i	$-0.271143\pi$
$149$	356966.	1.31723	0.658614	+	0.752481i	$0.271143\pi$
$150$	0	0
$151$	−236793.	−0.845135	−0.422568	−	0.906331i	$-0.638871\pi$
$151$	−236793.	−0.845135	−0.422568	+	0.906331i	$0.638871\pi$
$152$	−58402.2	−0.205031
$153$	0	0
$154$	−3149.65	−0.0107019
$155$	15518.1	0.0518810
$156$	0	0
$157$	13948.1	0.0451613	0.0225807	−	0.999745i	$-0.492812\pi$
$157$	13948.1	0.0451613	0.0225807	+	0.999745i	$0.492812\pi$
$158$	−44972.7	−0.143320
$159$	0	0
$160$	21427.3	0.0661710
$161$	−29801.6	−0.0906097
$162$	0	0
$163$	119894.	0.353450	0.176725	−	0.984260i	$-0.443450\pi$
$163$	119894.	0.353450	0.176725	+	0.984260i	$0.443450\pi$
$164$	−184930.	−0.536905
$165$	0	0
$166$	−29443.1	−0.0829304
$167$	6906.10	0.0191620	0.00958102	−	0.999954i	$-0.496950\pi$
$167$	6906.10	0.0191620	0.00958102	+	0.999954i	$0.496950\pi$
$168$	0	0
$169$	−148728.	−0.400568
$170$	−8210.91	−0.0217906
$171$	0	0
$172$	203326.	0.524049
$173$	394688.	1.00263	0.501313	−	0.865266i	$-0.332851\pi$
$173$	394688.	1.00263	0.501313	+	0.865266i	$0.332851\pi$
$174$	0	0
$175$	−343211.	−0.847161
$176$	−53548.5	−0.130306
$177$	0	0
$178$	−30979.5	−0.0732865
$179$	−728867.	−1.70026	−0.850130	−	0.526572i	$-0.823477\pi$
$179$	−728867.	−1.70026	−0.850130	+	0.526572i	$0.823477\pi$
$180$	0	0
$181$	−409033.	−0.928030	−0.464015	−	0.885827i	$-0.653592\pi$
$181$	−409033.	−0.928030	−0.464015	+	0.885827i	$0.653592\pi$
$182$	−27746.1	−0.0620902
$183$	0	0
$184$	8129.75	0.0177024
$185$	−108871.	−0.233875
$186$	0	0
$187$	62382.8	0.130455
$188$	−90694.3	−0.187148
$189$	0	0
$190$	12872.2	0.0258685
$191$	549926.	1.09074	0.545369	−	0.838196i	$-0.316389\pi$
$191$	549926.	1.09074	0.545369	+	0.838196i	$0.316389\pi$
$192$	0	0
$193$	272527.	0.526644	0.263322	−	0.964708i	$-0.415182\pi$
$193$	272527.	0.526644	0.263322	+	0.964708i	$0.415182\pi$
$194$	−6960.61	−0.0132783
$195$	0	0
$196$	97254.1	0.180829
$197$	196950.	0.361569	0.180784	−	0.983523i	$-0.442136\pi$
$197$	196950.	0.361569	0.180784	+	0.983523i	$0.442136\pi$
$198$	0	0
$199$	−573962.	−1.02743	−0.513713	−	0.857962i	$-0.671730\pi$
$199$	−573962.	−1.02743	−0.513713	+	0.857962i	$0.671730\pi$
$200$	93626.6	0.165510
$201$	0	0
$202$	−51137.4	−0.0881781
$203$	773522.	1.31744
$204$	0	0
$205$	81842.8	0.136018
$206$	−88360.4	−0.145074
$207$	0	0
$208$	−471722.	−0.756010
$209$	−97797.6	−0.154868
$210$	0	0
$211$	−812952.	−1.25707	−0.628534	−	0.777782i	$-0.716345\pi$
$211$	−812952.	−1.25707	−0.628534	+	0.777782i	$0.716345\pi$
$212$	−529431.	−0.809039
$213$	0	0
$214$	−70572.7	−0.105342
$215$	−89984.2	−0.132761
$216$	0	0
$217$	129478.	0.186658
$218$	−43683.1	−0.0622548
$219$	0	0
$220$	23889.4	0.0332773
$221$	549546.	0.756873
$222$	0	0
$223$	−211386.	−0.284652	−0.142326	−	0.989820i	$-0.545458\pi$
$223$	−211386.	−0.284652	−0.142326	+	0.989820i	$0.545458\pi$
$224$	178783.	0.238070
$225$	0	0
$226$	129074.	0.168101
$227$	383490.	0.493958	0.246979	−	0.969021i	$-0.420562\pi$
$227$	383490.	0.493958	0.246979	+	0.969021i	$0.420562\pi$
$228$	0	0
$229$	−67560.6	−0.0851344	−0.0425672	−	0.999094i	$-0.513554\pi$
$229$	−67560.6	−0.0851344	−0.0425672	+	0.999094i	$0.513554\pi$
$230$	−1791.86	−0.00223349
$231$	0	0
$232$	−211014.	−0.257389
$233$	−592445.	−0.714922	−0.357461	−	0.933928i	$-0.616357\pi$
$233$	−592445.	−0.714922	−0.357461	+	0.933928i	$0.616357\pi$
$234$	0	0
$235$	40137.8	0.0474115
$236$	−110516.	−0.129165
$237$	0	0
$238$	−68509.2	−0.0783983
$239$	−45816.0	−0.0518827	−0.0259413	−	0.999663i	$-0.508258\pi$
$239$	−45816.0	−0.0518827	−0.0259413	+	0.999663i	$0.508258\pi$
$240$	0	0
$241$	−1.56926e6	−1.74041	−0.870204	−	0.492691i	$-0.836013\pi$
$241$	−1.56926e6	−1.74041	−0.870204	+	0.492691i	$0.836013\pi$
$242$	−79356.3	−0.0871049
$243$	0	0
$244$	−1.17785e6	−1.26653
$245$	−43040.9	−0.0458106
$246$	0	0
$247$	−861523.	−0.898514
$248$	−35321.0	−0.0364673
$249$	0	0
$250$	−42663.5	−0.0431724
$251$	36553.8	0.0366225	0.0183113	−	0.999832i	$-0.494171\pi$
$251$	36553.8	0.0366225	0.0183113	+	0.999832i	$0.494171\pi$
$252$	0	0
$253$	13613.7	0.0133714
$254$	−30457.6	−0.0296218
$255$	0	0
$256$	967076.	0.922276
$257$	821929.	0.776250	0.388125	−	0.921607i	$-0.373123\pi$
$257$	821929.	0.776250	0.388125	+	0.921607i	$0.373123\pi$
$258$	0	0
$259$	−908385.	−0.841435
$260$	210447.	0.193068
$261$	0	0
$262$	−95968.0	−0.0863721
$263$	242612.	0.216283	0.108141	−	0.994136i	$-0.465510\pi$
$263$	242612.	0.216283	0.108141	+	0.994136i	$0.465510\pi$
$264$	0	0
$265$	234306.	0.204959
$266$	107402.	0.0930697
$267$	0	0
$268$	−348411.	−0.296316
$269$	1.06646e6	0.898596	0.449298	−	0.893382i	$-0.351674\pi$
$269$	1.06646e6	0.898596	0.449298	+	0.893382i	$0.351674\pi$
$270$	0	0
$271$	202507.	0.167501	0.0837505	−	0.996487i	$-0.473310\pi$
$271$	202507.	0.167501	0.0837505	+	0.996487i	$0.473310\pi$
$272$	−1.16475e6	−0.954577
$273$	0	0
$274$	−131746.	−0.106014
$275$	156783.	0.125016
$276$	0	0
$277$	1.26102e6	0.987469	0.493735	−	0.869613i	$-0.335631\pi$
$277$	1.26102e6	0.987469	0.493735	+	0.869613i	$0.335631\pi$
$278$	−75023.0	−0.0582214
$279$	0	0
$280$	−52678.9	−0.0401552
$281$	−1.93346e6	−1.46072	−0.730362	−	0.683060i	$-0.760649\pi$
$281$	−1.93346e6	−1.46072	−0.730362	+	0.683060i	$0.760649\pi$
$282$	0	0
$283$	−502495.	−0.372963	−0.186482	−	0.982458i	$-0.559708\pi$
$283$	−502495.	−0.372963	−0.186482	+	0.982458i	$0.559708\pi$
$284$	−491431.	−0.361548
$285$	0	0
$286$	12674.7	0.00916271
$287$	682870.	0.489366
$288$	0	0
$289$	−62946.4	−0.0443329
$290$	46508.9	0.0324744
$291$	0	0
$292$	1.33887e6	0.918929
$293$	86625.1	0.0589487	0.0294744	−	0.999566i	$-0.490617\pi$
$293$	86625.1	0.0589487	0.0294744	+	0.999566i	$0.490617\pi$
$294$	0	0
$295$	48910.1	0.0327223
$296$	247804.	0.164391
$297$	0	0
$298$	179080.	0.116817
$299$	119927.	0.0775778
$300$	0	0
$301$	−750800.	−0.477648
$302$	−118793.	−0.0749502
$303$	0	0
$304$	1.82598e6	1.13322
$305$	521270.	0.320858
$306$	0	0
$307$	630671.	0.381907	0.190953	−	0.981599i	$-0.438842\pi$
$307$	630671.	0.381907	0.190953	+	0.981599i	$0.438842\pi$
$308$	199325.	0.119725
$309$	0	0
$310$	7785.01	0.00460103
$311$	828845.	0.485929	0.242964	−	0.970035i	$-0.421880\pi$
$311$	828845.	0.485929	0.242964	+	0.970035i	$0.421880\pi$
$312$	0	0
$313$	1.64437e6	0.948724	0.474362	−	0.880330i	$-0.342679\pi$
$313$	1.64437e6	0.948724	0.474362	+	0.880330i	$0.342679\pi$
$314$	6997.40	0.00400510
$315$	0	0
$316$	2.84609e6	1.60336
$317$	−2.66630e6	−1.49025	−0.745127	−	0.666923i	$-0.767611\pi$
$317$	−2.66630e6	−1.49025	−0.745127	+	0.666923i	$0.767611\pi$
$318$	0	0
$319$	−353354.	−0.194416
$320$	−438825.	−0.239561
$321$	0	0
$322$	−14950.7	−0.00803565
$323$	−2.12723e6	−1.13451
$324$	0	0
$325$	1.38114e6	0.725319
$326$	60147.6	0.0313454
$327$	0	0
$328$	−186284.	−0.0956074
$329$	334897.	0.170577
$330$	0	0
$331$	1.09024e6	0.546955	0.273477	−	0.961878i	$-0.411826\pi$
$331$	1.09024e6	0.546955	0.273477	+	0.961878i	$0.411826\pi$
$332$	1.86330e6	0.927765
$333$	0	0
$334$	3464.61	0.00169937
$335$	154193.	0.0750677
$336$	0	0
$337$	−1.37111e6	−0.657654	−0.328827	−	0.944390i	$-0.606653\pi$
$337$	−1.37111e6	−0.657654	−0.328827	+	0.944390i	$0.606653\pi$
$338$	−74613.0	−0.0355241
$339$	0	0
$340$	519626.	0.243777
$341$	−59147.0	−0.0275453
$342$	0	0
$343$	−2.32946e6	−1.06911
$344$	204815.	0.0933181
$345$	0	0
$346$	198005.	0.0889170
$347$	2.35316e6	1.04913	0.524564	−	0.851371i	$-0.324228\pi$
$347$	2.35316e6	1.04913	0.524564	+	0.851371i	$0.324228\pi$
$348$	0	0
$349$	844331.	0.371064	0.185532	−	0.982638i	$-0.440599\pi$
$349$	844331.	0.371064	0.185532	+	0.982638i	$0.440599\pi$
$350$	−172180.	−0.0751298
$351$	0	0
$352$	−81670.0	−0.0351323
$353$	−1.07476e6	−0.459066	−0.229533	−	0.973301i	$-0.573720\pi$
$353$	−1.07476e6	−0.459066	−0.229533	+	0.973301i	$0.573720\pi$
$354$	0	0
$355$	217488.	0.0915935
$356$	1.96053e6	0.819877
$357$	0	0
$358$	−365653.	−0.150786
$359$	−1.95175e6	−0.799259	−0.399629	−	0.916677i	$-0.630861\pi$
$359$	−1.95175e6	−0.799259	−0.399629	+	0.916677i	$0.630861\pi$
$360$	0	0
$361$	858761.	0.346820
$362$	−205201.	−0.0823016
$363$	0	0
$364$	1.75591e6	0.694621
$365$	−592533.	−0.232799
$366$	0	0
$367$	1.56632e6	0.607036	0.303518	−	0.952826i	$-0.401839\pi$
$367$	1.56632e6	0.607036	0.303518	+	0.952826i	$0.401839\pi$
$368$	−254182.	−0.0978420
$369$	0	0
$370$	−54617.7	−0.0207410
$371$	1.95497e6	0.737404
$372$	0	0
$373$	2.85853e6	1.06382	0.531912	−	0.846799i	$-0.321474\pi$
$373$	2.85853e6	1.06382	0.531912	+	0.846799i	$0.321474\pi$
$374$	31295.8	0.0115693
$375$	0	0
$376$	−91358.5	−0.0333257
$377$	−3.11278e6	−1.12796
$378$	0	0
$379$	1.66138e6	0.594116	0.297058	−	0.954859i	$-0.403995\pi$
$379$	1.66138e6	0.594116	0.297058	+	0.954859i	$0.403995\pi$
$380$	−814618.	−0.289398
$381$	0	0
$382$	275883.	0.0967313
$383$	4.91425e6	1.71183	0.855914	−	0.517118i	$-0.172995\pi$
$383$	4.91425e6	1.71183	0.855914	+	0.517118i	$0.172995\pi$
$384$	0	0
$385$	−88213.7	−0.0303308
$386$	136720.	0.0467050
$387$	0	0
$388$	440501.	0.148548
$389$	−4.04295e6	−1.35464	−0.677320	−	0.735688i	$-0.736859\pi$
$389$	−4.04295e6	−1.35464	−0.677320	+	0.735688i	$0.736859\pi$
$390$	0	0
$391$	296117.	0.0979538
$392$	97966.4	0.0322005
$393$	0	0
$394$	98804.7	0.0320654
$395$	−1.25957e6	−0.406190
$396$	0	0
$397$	2.31593e6	0.737477	0.368738	−	0.929533i	$-0.379790\pi$
$397$	2.31593e6	0.737477	0.368738	+	0.929533i	$0.379790\pi$
$398$	−287942.	−0.0911164
$399$	0	0
$400$	−2.92730e6	−0.914780
$401$	3.24006e6	1.00622	0.503110	−	0.864223i	$-0.332189\pi$
$401$	3.24006e6	1.00622	0.503110	+	0.864223i	$0.332189\pi$
$402$	0	0
$403$	−521041.	−0.159812
$404$	3.23622e6	0.986473
$405$	0	0
$406$	388055.	0.116837
$407$	414961.	0.124171
$408$	0	0
$409$	2.30082e6	0.680101	0.340051	−	0.940407i	$-0.389556\pi$
$409$	2.30082e6	0.680101	0.340051	+	0.940407i	$0.389556\pi$
$410$	41058.4	0.0120626
$411$	0	0
$412$	5.59187e6	1.62298
$413$	408090.	0.117728
$414$	0	0
$415$	−824626.	−0.235037
$416$	−719452.	−0.203830
$417$	0	0
$418$	−49062.5	−0.0137344
$419$	−4.71896e6	−1.31314	−0.656570	−	0.754265i	$-0.727993\pi$
$419$	−4.71896e6	−1.31314	−0.656570	+	0.754265i	$0.727993\pi$
$420$	0	0
$421$	1.59180e6	0.437708	0.218854	−	0.975758i	$-0.429768\pi$
$421$	1.59180e6	0.437708	0.218854	+	0.975758i	$0.429768\pi$
$422$	−407837.	−0.111482
$423$	0	0
$424$	−533308.	−0.144067
$425$	3.41024e6	0.915824
$426$	0	0
$427$	4.34931e6	1.15438
$428$	4.46618e6	1.17849
$429$	0	0
$430$	−45142.7	−0.0117738
$431$	3.84179e6	0.996186	0.498093	−	0.867124i	$-0.334034\pi$
$431$	3.84179e6	0.996186	0.498093	+	0.867124i	$0.334034\pi$
$432$	0	0
$433$	−97620.0	−0.0250218	−0.0125109	−	0.999922i	$-0.503982\pi$
$433$	−97620.0	−0.0250218	−0.0125109	+	0.999922i	$0.503982\pi$
$434$	64955.6	0.0165536
$435$	0	0
$436$	2.76448e6	0.696461
$437$	−464222.	−0.116285
$438$	0	0
$439$	4.43043e6	1.09720	0.548599	−	0.836086i	$-0.315161\pi$
$439$	4.43043e6	1.09720	0.548599	+	0.836086i	$0.315161\pi$
$440$	24064.3	0.00592573
$441$	0	0
$442$	275693.	0.0671227
$443$	6.14299e6	1.48720	0.743602	−	0.668623i	$-0.233116\pi$
$443$	6.14299e6	1.48720	0.743602	+	0.668623i	$0.233116\pi$
$444$	0	0
$445$	−867655.	−0.207705
$446$	−106047.	−0.0252441
$447$	0	0
$448$	−3.66142e6	−0.861895
$449$	−5.39476e6	−1.26286	−0.631431	−	0.775432i	$-0.717532\pi$
$449$	−5.39476e6	−1.26286	−0.631431	+	0.775432i	$0.717532\pi$
$450$	0	0
$451$	−311943.	−0.0722161
$452$	−8.16845e6	−1.88059
$453$	0	0
$454$	192387.	0.0438063
$455$	−777096.	−0.175973
$456$	0	0
$457$	−1.53312e6	−0.343388	−0.171694	−	0.985150i	$-0.554924\pi$
$457$	−1.53312e6	−0.343388	−0.171694	+	0.985150i	$0.554924\pi$
$458$	−33893.4	−0.00755008
$459$	0	0
$460$	113397.	0.0249866
$461$	−2.69249e6	−0.590067	−0.295033	−	0.955487i	$-0.595331\pi$
$461$	−2.69249e6	−0.590067	−0.295033	+	0.955487i	$0.595331\pi$
$462$	0	0
$463$	−2.40113e6	−0.520550	−0.260275	−	0.965535i	$-0.583813\pi$
$463$	−2.40113e6	−0.520550	−0.260275	+	0.965535i	$0.583813\pi$
$464$	6.59748e6	1.42260
$465$	0	0
$466$	−297214.	−0.0634023
$467$	7.70828e6	1.63556	0.817778	−	0.575534i	$-0.195206\pi$
$467$	7.70828e6	1.63556	0.817778	+	0.575534i	$0.195206\pi$
$468$	0	0
$469$	1.28654e6	0.270079
$470$	20136.1	0.00420465
$471$	0	0
$472$	−111325.	−0.0230006
$473$	342974.	0.0704869
$474$	0	0
$475$	−5.34623e6	−1.08721
$476$	4.33559e6	0.877064
$477$	0	0
$478$	−22984.7	−0.00460118
$479$	−4.29567e6	−0.855445	−0.427723	−	0.903910i	$-0.640684\pi$
$479$	−4.29567e6	−0.855445	−0.427723	+	0.903910i	$0.640684\pi$
$480$	0	0
$481$	3.65550e6	0.720416
$482$	−787254.	−0.154347
$483$	0	0
$484$	5.02205e6	0.974467
$485$	−194949.	−0.0376328
$486$	0	0
$487$	−9.17052e6	−1.75215	−0.876076	−	0.482173i	$-0.839848\pi$
$487$	−9.17052e6	−1.75215	−0.876076	+	0.482173i	$0.839848\pi$
$488$	−1.18647e6	−0.225532
$489$	0	0
$490$	−21592.5	−0.00406268
$491$	1.90208e6	0.356062	0.178031	−	0.984025i	$-0.443027\pi$
$491$	1.90208e6	0.356062	0.178031	+	0.984025i	$0.443027\pi$
$492$	0	0
$493$	−7.68592e6	−1.42423
$494$	−432204.	−0.0796840
$495$	0	0
$496$	1.10434e6	0.201556
$497$	1.81465e6	0.329536
$498$	0	0
$499$	−5.54825e6	−0.997482	−0.498741	−	0.866751i	$-0.666204\pi$
$499$	−5.54825e6	−0.997482	−0.498741	+	0.866751i	$0.666204\pi$
$500$	2.69995e6	0.482982
$501$	0	0
$502$	18338.1	0.00324784
$503$	−5.18973e6	−0.914587	−0.457294	−	0.889316i	$-0.651181\pi$
$503$	−5.18973e6	−0.914587	−0.457294	+	0.889316i	$0.651181\pi$
$504$	0	0
$505$	−1.43223e6	−0.249910
$506$	6829.65	0.00118583
$507$	0	0
$508$	1.92750e6	0.331387
$509$	−2.94232e6	−0.503379	−0.251689	−	0.967808i	$-0.580986\pi$
$509$	−2.94232e6	−0.503379	−0.251689	+	0.967808i	$0.580986\pi$
$510$	0	0
$511$	−4.94391e6	−0.837565
$512$	2.54815e6	0.429586
$513$	0	0
$514$	412340.	0.0688411
$515$	−2.47475e6	−0.411162
$516$	0	0
$517$	−152985.	−0.0251723
$518$	−455713.	−0.0746220
$519$	0	0
$520$	211989.	0.0343799
$521$	3.27821e6	0.529106	0.264553	−	0.964371i	$-0.414776\pi$
$521$	3.27821e6	0.529106	0.264553	+	0.964371i	$0.414776\pi$
$522$	0	0
$523$	−890700.	−0.142389	−0.0711946	−	0.997462i	$-0.522681\pi$
$523$	−890700.	−0.142389	−0.0711946	+	0.997462i	$0.522681\pi$
$524$	6.07332e6	0.966268
$525$	0	0
$526$	121712.	0.0191809
$527$	−1.28653e6	−0.201787
$528$	0	0
$529$	−6.37172e6	−0.989960
$530$	117545.	0.0181767
$531$	0	0
$532$	−6.79691e6	−1.04120
$533$	−2.74799e6	−0.418983
$534$	0	0
$535$	−1.97656e6	−0.298556
$536$	−350963.	−0.0527654
$537$	0	0
$538$	535016.	0.0796913
$539$	164050.	0.0243223
$540$	0	0
$541$	5.79897e6	0.851839	0.425920	−	0.904761i	$-0.359951\pi$
$541$	5.79897e6	0.851839	0.425920	+	0.904761i	$0.359951\pi$
$542$	101593.	0.0148547
$543$	0	0
$544$	−1.77643e6	−0.257366
$545$	−1.22345e6	−0.176439
$546$	0	0
$547$	2.82959e6	0.404348	0.202174	−	0.979350i	$-0.435199\pi$
$547$	2.82959e6	0.404348	0.202174	+	0.979350i	$0.435199\pi$
$548$	8.33753e6	1.18600
$549$	0	0
$550$	78653.8	0.0110870
$551$	1.20492e7	1.69075
$552$	0	0
$553$	−1.05094e7	−1.46139
$554$	632622.	0.0875729
$555$	0	0
$556$	4.74782e6	0.651339
$557$	9.20725e6	1.25745	0.628726	−	0.777627i	$-0.283577\pi$
$557$	9.20725e6	1.25745	0.628726	+	0.777627i	$0.283577\pi$
$558$	0	0
$559$	3.02134e6	0.408950
$560$	1.64704e6	0.221939
$561$	0	0
$562$	−969964.	−0.129543
$563$	−4.99478e6	−0.664118	−0.332059	−	0.943259i	$-0.607743\pi$
$563$	−4.99478e6	−0.664118	−0.332059	+	0.943259i	$0.607743\pi$
$564$	0	0
$565$	3.61504e6	0.476423
$566$	−252089.	−0.0330759
$567$	0	0
$568$	−495030.	−0.0643814
$569$	−1.02729e6	−0.133018	−0.0665092	−	0.997786i	$-0.521186\pi$
$569$	−1.02729e6	−0.133018	−0.0665092	+	0.997786i	$0.521186\pi$
$570$	0	0
$571$	7.08898e6	0.909900	0.454950	−	0.890517i	$-0.349657\pi$
$571$	7.08898e6	0.909900	0.454950	+	0.890517i	$0.349657\pi$
$572$	−802118.	−0.102506
$573$	0	0
$574$	342578.	0.0433990
$575$	744211.	0.0938700
$576$	0	0
$577$	−4.15153e6	−0.519121	−0.259561	−	0.965727i	$-0.583578\pi$
$577$	−4.15153e6	−0.519121	−0.259561	+	0.965727i	$0.583578\pi$
$578$	−31578.6	−0.00393163
$579$	0	0
$580$	−2.94331e6	−0.363300
$581$	−6.88042e6	−0.845618
$582$	0	0
$583$	−893054.	−0.108819
$584$	1.34868e6	0.163635
$585$	0	0
$586$	43457.5	0.00522782
$587$	−1.65158e7	−1.97835	−0.989176	−	0.146737i	$-0.953123\pi$
$587$	−1.65158e7	−1.97835	−0.989176	+	0.146737i	$0.953123\pi$
$588$	0	0
$589$	2.01689e6	0.239549
$590$	24536.9	0.00290195
$591$	0	0
$592$	−7.74775e6	−0.908597
$593$	−7.04831e6	−0.823092	−0.411546	−	0.911389i	$-0.635011\pi$
$593$	−7.04831e6	−0.823092	−0.411546	+	0.911389i	$0.635011\pi$
$594$	0	0
$595$	−1.91877e6	−0.222193
$596$	−1.13331e7	−1.30687
$597$	0	0
$598$	60164.0	0.00687993
$599$	−1.26382e7	−1.43919	−0.719594	−	0.694395i	$-0.755672\pi$
$599$	−1.26382e7	−1.43919	−0.719594	+	0.694395i	$0.755672\pi$
$600$	0	0
$601$	6.61706e6	0.747272	0.373636	−	0.927575i	$-0.378111\pi$
$601$	6.61706e6	0.747272	0.373636	+	0.927575i	$0.378111\pi$
$602$	−376656.	−0.0423598
$603$	0	0
$604$	7.51778e6	0.838489
$605$	−2.22256e6	−0.246868
$606$	0	0
$607$	−2.10068e6	−0.231413	−0.115707	−	0.993283i	$-0.536913\pi$
$607$	−2.10068e6	−0.231413	−0.115707	+	0.993283i	$0.536913\pi$
$608$	2.78492e6	0.305529
$609$	0	0
$610$	261507.	0.0284550
$611$	−1.34768e6	−0.146044
$612$	0	0
$613$	1.51680e7	1.63034	0.815168	−	0.579225i	$-0.196645\pi$
$613$	1.51680e7	1.63034	0.815168	+	0.579225i	$0.196645\pi$
$614$	316391.	0.0338691
$615$	0	0
$616$	200785.	0.0213196
$617$	1.74862e6	0.184920	0.0924600	−	0.995716i	$-0.470527\pi$
$617$	1.74862e6	0.184920	0.0924600	+	0.995716i	$0.470527\pi$
$618$	0	0
$619$	−1.06999e7	−1.12241	−0.561206	−	0.827676i	$-0.689662\pi$
$619$	−1.06999e7	−1.12241	−0.561206	+	0.827676i	$0.689662\pi$
$620$	−492673.	−0.0514730
$621$	0	0
$622$	415810.	0.0430942
$623$	−7.23944e6	−0.747282
$624$	0	0
$625$	7.95380e6	0.814469
$626$	824939.	0.0841368
$627$	0	0
$628$	−442829.	−0.0448061
$629$	9.02597e6	0.909635
$630$	0	0
$631$	1.06482e7	1.06464	0.532322	−	0.846542i	$-0.321319\pi$
$631$	1.06482e7	1.06464	0.532322	+	0.846542i	$0.321319\pi$
$632$	2.86693e6	0.285512
$633$	0	0
$634$	−1.33761e6	−0.132162
$635$	−853038.	−0.0839526
$636$	0	0
$637$	1.44516e6	0.141113
$638$	−177268.	−0.0172417
$639$	0	0
$640$	−905821.	−0.0874163
$641$	−1.71884e7	−1.65231	−0.826155	−	0.563443i	$-0.809477\pi$
$641$	−1.71884e7	−1.65231	−0.826155	+	0.563443i	$0.809477\pi$
$642$	0	0
$643$	9.90174e6	0.944461	0.472230	−	0.881475i	$-0.343449\pi$
$643$	9.90174e6	0.944461	0.472230	+	0.881475i	$0.343449\pi$
$644$	946151.	0.0898971
$645$	0	0
$646$	−1.06718e6	−0.100613
$647$	−4.53741e6	−0.426136	−0.213068	−	0.977037i	$-0.568346\pi$
$647$	−4.53741e6	−0.426136	−0.213068	+	0.977037i	$0.568346\pi$
$648$	0	0
$649$	−186420.	−0.0173733
$650$	692881.	0.0643243
$651$	0	0
$652$	−3.80643e6	−0.350670
$653$	5.46204e6	0.501270	0.250635	−	0.968082i	$-0.419361\pi$
$653$	5.46204e6	0.501270	0.250635	+	0.968082i	$0.419361\pi$
$654$	0	0
$655$	−2.68782e6	−0.244791
$656$	5.82430e6	0.528426
$657$	0	0
$658$	168009.	0.0151275
$659$	−1.36246e6	−0.122211	−0.0611056	−	0.998131i	$-0.519463\pi$
$659$	−1.36246e6	−0.122211	−0.0611056	+	0.998131i	$0.519463\pi$
$660$	0	0
$661$	1.11826e7	0.995497	0.497748	−	0.867321i	$-0.334160\pi$
$661$	1.11826e7	0.995497	0.497748	+	0.867321i	$0.334160\pi$
$662$	546944.	0.0485062
$663$	0	0
$664$	1.87695e6	0.165209
$665$	3.00805e6	0.263773
$666$	0	0
$667$	−1.67729e6	−0.145980
$668$	−219257.	−0.0190113
$669$	0	0
$670$	77354.7	0.00665732
$671$	−1.98682e6	−0.170354
$672$	0	0
$673$	−2.22259e7	−1.89157	−0.945785	−	0.324794i	$-0.894705\pi$
$673$	−2.22259e7	−1.89157	−0.945785	+	0.324794i	$0.894705\pi$
$674$	−687850.	−0.0583235
$675$	0	0
$676$	4.72187e6	0.397418
$677$	7.09861e6	0.595253	0.297627	−	0.954682i	$-0.403805\pi$
$677$	7.09861e6	0.595253	0.297627	+	0.954682i	$0.403805\pi$
$678$	0	0
$679$	−1.62659e6	−0.135395
$680$	523432.	0.0434098
$681$	0	0
$682$	−29672.5	−0.00244283
$683$	2.04470e7	1.67717	0.838585	−	0.544771i	$-0.183384\pi$
$683$	2.04470e7	1.67717	0.838585	+	0.544771i	$0.183384\pi$
$684$	0	0
$685$	−3.68987e6	−0.300458
$686$	−1.16863e6	−0.0948128
$687$	0	0
$688$	−6.40368e6	−0.515773
$689$	−7.86713e6	−0.631347
$690$	0	0
$691$	7.78175e6	0.619987	0.309993	−	0.950739i	$-0.399673\pi$
$691$	7.78175e6	0.619987	0.309993	+	0.950739i	$0.399673\pi$
$692$	−1.25307e7	−0.994740
$693$	0	0
$694$	1.18052e6	0.0930411
$695$	−2.10120e6	−0.165008
$696$	0	0
$697$	−6.78519e6	−0.529029
$698$	423579.	0.0329075
$699$	0	0
$700$	1.08964e7	0.840498
$701$	1.01109e7	0.777136	0.388568	−	0.921420i	$-0.372970\pi$
$701$	1.01109e7	0.777136	0.388568	+	0.921420i	$0.372970\pi$
$702$	0	0
$703$	−1.41500e7	−1.07986
$704$	1.67258e6	0.127191
$705$	0	0
$706$	−539179.	−0.0407119
$707$	−1.19500e7	−0.899127
$708$	0	0
$709$	5.55179e6	0.414780	0.207390	−	0.978258i	$-0.433503\pi$
$709$	5.55179e6	0.414780	0.207390	+	0.978258i	$0.433503\pi$
$710$	109108.	0.00812290
$711$	0	0
$712$	1.97489e6	0.145997
$713$	−280757.	−0.0206827
$714$	0	0
$715$	354987.	0.0259685
$716$	2.31403e7	1.68689
$717$	0	0
$718$	−979140.	−0.0708816
$719$	−1.36153e7	−0.982215	−0.491107	−	0.871099i	$-0.663408\pi$
$719$	−1.36153e7	−0.982215	−0.491107	+	0.871099i	$0.663408\pi$
$720$	0	0
$721$	−2.06485e7	−1.47928
$722$	430818.	0.0307575
$723$	0	0
$724$	1.29861e7	0.920731
$725$	−1.93165e7	−1.36485
$726$	0	0
$727$	7.71813e6	0.541597	0.270798	−	0.962636i	$-0.412712\pi$
$727$	7.71813e6	0.541597	0.270798	+	0.962636i	$0.412712\pi$
$728$	1.76877e6	0.123692
$729$	0	0
$730$	−297258.	−0.0206456
$731$	7.46015e6	0.516362
$732$	0	0
$733$	−2.00055e7	−1.37527	−0.687636	−	0.726056i	$-0.741351\pi$
$733$	−2.00055e7	−1.37527	−0.687636	+	0.726056i	$0.741351\pi$
$734$	785779.	0.0538345
$735$	0	0
$736$	−387669.	−0.0263795
$737$	−587706.	−0.0398558
$738$	0	0
$739$	−1.40919e7	−0.949204	−0.474602	−	0.880201i	$-0.657408\pi$
$739$	−1.40919e7	−0.949204	−0.474602	+	0.880201i	$0.657408\pi$
$740$	3.45647e6	0.232035
$741$	0	0
$742$	980758.	0.0653961
$743$	−1.01829e7	−0.676706	−0.338353	−	0.941019i	$-0.609870\pi$
$743$	−1.01829e7	−0.676706	−0.338353	+	0.941019i	$0.609870\pi$
$744$	0	0
$745$	5.01558e6	0.331078
$746$	1.43405e6	0.0943444
$747$	0	0
$748$	−1.98055e6	−0.129429
$749$	−1.64918e7	−1.07415
$750$	0	0
$751$	−2.24603e7	−1.45317	−0.726583	−	0.687079i	$-0.758893\pi$
$751$	−2.24603e7	−1.45317	−0.726583	+	0.687079i	$0.758893\pi$
$752$	2.85638e6	0.184193
$753$	0	0
$754$	−1.56160e6	−0.100033
$755$	−3.32708e6	−0.212420
$756$	0	0
$757$	−2.42736e7	−1.53955	−0.769777	−	0.638313i	$-0.779633\pi$
$757$	−2.42736e7	−1.53955	−0.769777	+	0.638313i	$0.779633\pi$
$758$	833471.	0.0526887
$759$	0	0
$760$	−820584.	−0.0515334
$761$	−2.19171e7	−1.37190	−0.685948	−	0.727651i	$-0.740612\pi$
$761$	−2.19171e7	−1.37190	−0.685948	+	0.727651i	$0.740612\pi$
$762$	0	0
$763$	−1.02081e7	−0.634795
$764$	−1.74592e7	−1.08216
$765$	0	0
$766$	2.46535e6	0.151812
$767$	−1.64222e6	−0.100796
$768$	0	0
$769$	1.95692e7	1.19332	0.596660	−	0.802494i	$-0.296494\pi$
$769$	1.95692e7	1.19332	0.596660	+	0.802494i	$0.296494\pi$
$770$	−44254.5	−0.00268986
$771$	0	0
$772$	−8.65229e6	−0.522502
$773$	−2.41099e7	−1.45127	−0.725634	−	0.688081i	$-0.758453\pi$
$773$	−2.41099e7	−1.45127	−0.725634	+	0.688081i	$0.758453\pi$
$774$	0	0
$775$	−3.23335e6	−0.193374
$776$	443728.	0.0264522
$777$	0	0
$778$	−2.02824e6	−0.120135
$779$	1.06371e7	0.628031
$780$	0	0
$781$	−828955.	−0.0486299
$782$	148554.	0.00868695
$783$	0	0
$784$	−3.06298e6	−0.177973
$785$	195979.	0.0113510
$786$	0	0
$787$	−3.33270e7	−1.91805	−0.959025	−	0.283321i	$-0.908564\pi$
$787$	−3.33270e7	−1.91805	−0.959025	+	0.283321i	$0.908564\pi$
$788$	−6.25284e6	−0.358725
$789$	0	0
$790$	−631892.	−0.0360226
$791$	3.01628e7	1.71408
$792$	0	0
$793$	−1.75024e7	−0.988356
$794$	1.16184e6	0.0654026
$795$	0	0
$796$	1.82223e7	1.01935
$797$	864232.	0.0481930	0.0240965	−	0.999710i	$-0.492329\pi$
$797$	864232.	0.0481930	0.0240965	+	0.999710i	$0.492329\pi$
$798$	0	0
$799$	−3.32763e6	−0.184403
$800$	−4.46460e6	−0.246636
$801$	0	0
$802$	1.62545e6	0.0892358
$803$	2.25844e6	0.123600
$804$	0	0
$805$	−418730.	−0.0227742
$806$	−261392.	−0.0141728
$807$	0	0
$808$	3.25993e6	0.175663
$809$	−1.13424e7	−0.609303	−0.304651	−	0.952464i	$-0.598540\pi$
$809$	−1.13424e7	−0.609303	−0.304651	+	0.952464i	$0.598540\pi$
$810$	0	0
$811$	3.43446e6	0.183361	0.0916805	−	0.995788i	$-0.470776\pi$
$811$	3.43446e6	0.183361	0.0916805	+	0.995788i	$0.470776\pi$
$812$	−2.45580e7	−1.30708
$813$	0	0
$814$	208175.	0.0110120
$815$	1.68458e6	0.0888376
$816$	0	0
$817$	−1.16953e7	−0.612993
$818$	1.15426e6	0.0603142
$819$	0	0
$820$	−2.59837e6	−0.134948
$821$	−1.81504e7	−0.939782	−0.469891	−	0.882724i	$-0.655707\pi$
$821$	−1.81504e7	−0.939782	−0.469891	+	0.882724i	$0.655707\pi$
$822$	0	0
$823$	−1.63665e6	−0.0842281	−0.0421141	−	0.999113i	$-0.513409\pi$
$823$	−1.63665e6	−0.0842281	−0.0421141	+	0.999113i	$0.513409\pi$
$824$	5.63283e6	0.289007
$825$	0	0
$826$	204728.	0.0104406
$827$	1.28654e7	0.654125	0.327063	−	0.945003i	$-0.393941\pi$
$827$	1.28654e7	0.654125	0.327063	+	0.945003i	$0.393941\pi$
$828$	0	0
$829$	−8.03655e6	−0.406147	−0.203073	−	0.979164i	$-0.565093\pi$
$829$	−8.03655e6	−0.406147	−0.203073	+	0.979164i	$0.565093\pi$
$830$	−413693.	−0.0208441
$831$	0	0
$832$	1.47342e7	0.737933
$833$	3.56831e6	0.178176
$834$	0	0
$835$	97034.7	0.00481627
$836$	3.10491e6	0.153650
$837$	0	0
$838$	−2.36738e6	−0.116455
$839$	3.59741e7	1.76435	0.882176	−	0.470920i	$-0.156078\pi$
$839$	3.59741e7	1.76435	0.882176	+	0.470920i	$0.156078\pi$
$840$	0	0
$841$	2.30241e7	1.12251
$842$	798566.	0.0388178
$843$	0	0
$844$	2.58099e7	1.24718
$845$	−2.08972e6	−0.100681
$846$	0	0
$847$	−1.85444e7	−0.888185
$848$	1.66742e7	0.796262
$849$	0	0
$850$	1.71083e6	0.0812192
$851$	1.96972e6	0.0932355
$852$	0	0
$853$	−2.61300e7	−1.22961	−0.614803	−	0.788680i	$-0.710765\pi$
$853$	−2.61300e7	−1.22961	−0.614803	+	0.788680i	$0.710765\pi$
$854$	2.18193e6	0.102376
$855$	0	0
$856$	4.49889e6	0.209856
$857$	2.58354e7	1.20161	0.600804	−	0.799396i	$-0.294847\pi$
$857$	2.58354e7	1.20161	0.600804	+	0.799396i	$0.294847\pi$
$858$	0	0
$859$	−2.85214e7	−1.31883	−0.659415	−	0.751779i	$-0.729196\pi$
$859$	−2.85214e7	−1.31883	−0.659415	+	0.751779i	$0.729196\pi$
$860$	2.85685e6	0.131717
$861$	0	0
$862$	1.92732e6	0.0883460
$863$	2.80905e7	1.28390	0.641952	−	0.766745i	$-0.278125\pi$
$863$	2.80905e7	1.28390	0.641952	+	0.766745i	$0.278125\pi$
$864$	0	0
$865$	5.54560e6	0.252004
$866$	−48973.4	−0.00221904
$867$	0	0
$868$	−4.11070e6	−0.185190
$869$	4.80083e6	0.215659
$870$	0	0
$871$	−5.17725e6	−0.231235
$872$	2.78473e6	0.124020
$873$	0	0
$874$	−232888.	−0.0103126
$875$	−9.96981e6	−0.440217
$876$	0	0
$877$	1.23614e7	0.542712	0.271356	−	0.962479i	$-0.412528\pi$
$877$	1.23614e7	0.542712	0.271356	+	0.962479i	$0.412528\pi$
$878$	2.22263e6	0.0973041
$879$	0	0
$880$	−752387.	−0.0327518
$881$	1.44013e7	0.625119	0.312560	−	0.949898i	$-0.398814\pi$
$881$	1.44013e7	0.625119	0.312560	+	0.949898i	$0.398814\pi$
$882$	0	0
$883$	76484.3	0.00330119	0.00165059	−	0.999999i	$-0.499475\pi$
$883$	76484.3	0.00330119	0.00165059	+	0.999999i	$0.499475\pi$
$884$	−1.74472e7	−0.750921
$885$	0	0
$886$	3.08178e6	0.131891
$887$	4.12317e7	1.75963	0.879817	−	0.475312i	$-0.157665\pi$
$887$	4.12317e7	1.75963	0.879817	+	0.475312i	$0.157665\pi$
$888$	0	0
$889$	−7.11748e6	−0.302045
$890$	−435280.	−0.0184202
$891$	0	0
$892$	6.71115e6	0.282413
$893$	5.21672e6	0.218912
$894$	0	0
$895$	−1.02410e7	−0.427351
$896$	−7.55788e6	−0.314507
$897$	0	0
$898$	−2.70641e6	−0.111996
$899$	7.28725e6	0.300721
$900$	0	0
$901$	−1.94251e7	−0.797172
$902$	−156494.	−0.00640443
$903$	0	0
$904$	−8.22828e6	−0.334879
$905$	−5.74715e6	−0.233255
$906$	0	0
$907$	3.00251e7	1.21190	0.605950	−	0.795503i	$-0.292793\pi$
$907$	3.00251e7	1.21190	0.605950	+	0.795503i	$0.292793\pi$
$908$	−1.21752e7	−0.490073
$909$	0	0
$910$	−389849.	−0.0156060
$911$	7.61632e6	0.304053	0.152026	−	0.988376i	$-0.451420\pi$
$911$	7.61632e6	0.304053	0.152026	+	0.988376i	$0.451420\pi$
$912$	0	0
$913$	3.14306e6	0.124789
$914$	−769124.	−0.0304531
$915$	0	0
$916$	2.14494e6	0.0844648
$917$	−2.24263e7	−0.880712
$918$	0	0
$919$	1.41234e7	0.551634	0.275817	−	0.961210i	$-0.411052\pi$
$919$	1.41234e7	0.551634	0.275817	+	0.961210i	$0.411052\pi$
$920$	114228.	0.00444941
$921$	0	0
$922$	−1.35075e6	−0.0523296
$923$	−7.30247e6	−0.282140
$924$	0	0
$925$	2.26844e7	0.871711
$926$	−1.20458e6	−0.0461646
$927$	0	0
$928$	1.00622e7	0.383551
$929$	2.95615e7	1.12380	0.561898	−	0.827206i	$-0.310071\pi$
$929$	2.95615e7	1.12380	0.561898	+	0.827206i	$0.310071\pi$
$930$	0	0
$931$	−5.59405e6	−0.211520
$932$	1.88091e7	0.709299
$933$	0	0
$934$	3.86704e6	0.145048
$935$	876515.	0.0327892
$936$	0	0
$937$	3.40344e7	1.26639	0.633197	−	0.773990i	$-0.281742\pi$
$937$	3.40344e7	1.26639	0.633197	+	0.773990i	$0.281742\pi$
$938$	645423.	0.0239518
$939$	0	0
$940$	−1.27431e6	−0.0470386
$941$	−2.51116e7	−0.924485	−0.462242	−	0.886754i	$-0.652955\pi$
$941$	−2.51116e7	−0.924485	−0.462242	+	0.886754i	$0.652955\pi$
$942$	0	0
$943$	−1.48072e6	−0.0542243
$944$	3.48066e6	0.127125
$945$	0	0
$946$	172061.	0.00625108
$947$	2.02733e7	0.734599	0.367300	−	0.930103i	$-0.380282\pi$
$947$	2.02733e7	0.734599	0.367300	+	0.930103i	$0.380282\pi$
$948$	0	0
$949$	1.98951e7	0.717102
$950$	−2.68206e6	−0.0964184
$951$	0	0
$952$	4.36735e6	0.156180
$953$	−1.24755e7	−0.444965	−0.222483	−	0.974937i	$-0.571416\pi$
$953$	−1.24755e7	−0.444965	−0.222483	+	0.974937i	$0.571416\pi$
$954$	0	0
$955$	7.72678e6	0.274151
$956$	1.45458e6	0.0514746
$957$	0	0
$958$	−2.15503e6	−0.0758645
$959$	−3.07871e7	−1.08099
$960$	0	0
$961$	−2.74094e7	−0.957393
$962$	1.83387e6	0.0638896
$963$	0	0
$964$	4.98212e7	1.72672
$965$	3.82917e6	0.132369
$966$	0	0
$967$	3.79545e7	1.30526	0.652630	−	0.757677i	$-0.273666\pi$
$967$	3.79545e7	1.30526	0.652630	+	0.757677i	$0.273666\pi$
$968$	5.05883e6	0.173525
$969$	0	0
$970$	−97800.7	−0.00333743
$971$	5.56592e6	0.189447	0.0947237	−	0.995504i	$-0.469803\pi$
$971$	5.56592e6	0.189447	0.0947237	+	0.995504i	$0.469803\pi$
$972$	0	0
$973$	−1.75318e7	−0.593667
$974$	−4.60061e6	−0.155388
$975$	0	0
$976$	3.70959e7	1.24653
$977$	4.95886e7	1.66206	0.831028	−	0.556230i	$-0.187753\pi$
$977$	4.95886e7	1.66206	0.831028	+	0.556230i	$0.187753\pi$
$978$	0	0
$979$	3.30706e6	0.110277
$980$	1.36648e6	0.0454503
$981$	0	0
$982$	954224.	0.0315771
$983$	−1.16536e7	−0.384660	−0.192330	−	0.981330i	$-0.561604\pi$
$983$	−1.16536e7	−0.384660	−0.192330	+	0.981330i	$0.561604\pi$
$984$	0	0
$985$	2.76726e6	0.0908783
$986$	−3.85582e6	−0.126306
$987$	0	0
$988$	2.73519e7	0.891447
$989$	1.62802e6	0.0529259
$990$	0	0
$991$	−1.62828e7	−0.526679	−0.263340	−	0.964703i	$-0.584824\pi$
$991$	−1.62828e7	−0.526679	−0.263340	+	0.964703i	$0.584824\pi$
$992$	1.68429e6	0.0543422
$993$	0	0
$994$	910363.	0.0292246
$995$	−8.06450e6	−0.258238
$996$	0	0
$997$	1.47322e7	0.469386	0.234693	−	0.972070i	$-0.424592\pi$
$997$	1.47322e7	0.469386	0.234693	+	0.972070i	$0.424592\pi$
$998$	−2.78341e6	−0.0884609
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.d.1.6		12
3.2	odd	2		177.6.a.b.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.7	✓	12	3.2	odd	2
531.6.a.d.1.6		12	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)

\(f(q)\)	\(=\)	\(q+0.501674 q^{2} -31.7483 q^{4} +14.0506 q^{5} +117.234 q^{7} -31.9809 q^{8} +O(q^{10})\)	\(q+0.501674 q^{2} -31.7483 q^{4} +14.0506 q^{5} +117.234 q^{7} -31.9809 q^{8} +7.04881 q^{10} -53.5537 q^{11} -471.768 q^{13} +58.8130 q^{14} +999.902 q^{16} -1164.87 q^{17} +1826.16 q^{19} -446.082 q^{20} -26.8665 q^{22} -254.207 q^{23} -2927.58 q^{25} -236.673 q^{26} -3721.97 q^{28} +6598.12 q^{29} +1104.44 q^{31} +1525.01 q^{32} -584.382 q^{34} +1647.20 q^{35} -7748.51 q^{37} +916.136 q^{38} -449.350 q^{40} +5824.87 q^{41} -6404.30 q^{43} +1700.24 q^{44} -127.529 q^{46} +2856.66 q^{47} -3063.28 q^{49} -1468.69 q^{50} +14977.8 q^{52} +16675.9 q^{53} -752.461 q^{55} -3749.23 q^{56} +3310.10 q^{58} +3481.00 q^{59} +37099.5 q^{61} +554.070 q^{62} -31231.8 q^{64} -6628.61 q^{65} +10974.2 q^{67} +36982.5 q^{68} +826.357 q^{70} +15478.9 q^{71} -42171.4 q^{73} -3887.22 q^{74} -57977.5 q^{76} -6278.29 q^{77} -89645.3 q^{79} +14049.2 q^{80} +2922.18 q^{82} -58689.8 q^{83} -16367.0 q^{85} -3212.87 q^{86} +1712.69 q^{88} -61752.2 q^{89} -55307.0 q^{91} +8070.64 q^{92} +1433.11 q^{94} +25658.6 q^{95} -13874.8 q^{97} -1536.77 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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