LMFDB - Embedded newform 531.6.a.c.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} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 $ x^12 - 2x^11 - 269x^10 + 143x^9 + 25384x^8 + 8539x^7 - 1009736x^6 - 720516x^5 + 15565376x^4 + 6775664x^3 - 75006848x^2 + 21512960*x + 49172480
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			$-0.689340$ 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)

$f(q)$	$=$	$q-2.68934 q^{2} -24.7675 q^{4} +83.5049 q^{5} -48.3401 q^{7} +152.667 q^{8} +O(q^{10})$	\(q-2.68934 q^{2} -24.7675 q^{4} +83.5049 q^{5} -48.3401 q^{7} +152.667 q^{8} -224.573 q^{10} -19.9993 q^{11} +611.920 q^{13} +130.003 q^{14} +381.985 q^{16} -1525.25 q^{17} -1272.41 q^{19} -2068.20 q^{20} +53.7850 q^{22} -3308.46 q^{23} +3848.06 q^{25} -1645.66 q^{26} +1197.26 q^{28} +2171.14 q^{29} +7163.58 q^{31} -5912.63 q^{32} +4101.91 q^{34} -4036.64 q^{35} -5908.05 q^{37} +3421.93 q^{38} +12748.4 q^{40} -2583.15 q^{41} -5333.76 q^{43} +495.332 q^{44} +8897.58 q^{46} +10520.3 q^{47} -14470.2 q^{49} -10348.7 q^{50} -15155.7 q^{52} -6041.06 q^{53} -1670.04 q^{55} -7379.94 q^{56} -5838.93 q^{58} +3481.00 q^{59} +134.207 q^{61} -19265.3 q^{62} +3677.55 q^{64} +51098.3 q^{65} +49806.2 q^{67} +37776.5 q^{68} +10855.9 q^{70} -64703.4 q^{71} +43618.4 q^{73} +15888.8 q^{74} +31514.3 q^{76} +966.770 q^{77} +74841.5 q^{79} +31897.6 q^{80} +6946.96 q^{82} -5270.76 q^{83} -127366. q^{85} +14344.3 q^{86} -3053.23 q^{88} -85402.3 q^{89} -29580.3 q^{91} +81942.2 q^{92} -28292.8 q^{94} -106252. q^{95} +47381.9 q^{97} +38915.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * 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 $

$2$	−2.68934	−0.475413	−0.237706	−	0.971337i	$-0.576396\pi$
$2$	−2.68934	−0.475413	−0.237706	+	0.971337i	$0.576396\pi$
$3$	0	0
$3$	0	0
$4$	−24.7675	−0.773983
$5$	83.5049	1.49378	0.746890	−	0.664948i	$-0.231546\pi$
$5$	83.5049	1.49378	0.746890	+	0.664948i	$0.231546\pi$
$6$	0	0
$7$	−48.3401	−0.372875	−0.186437	−	0.982467i	$-0.559694\pi$
$7$	−48.3401	−0.372875	−0.186437	+	0.982467i	$0.559694\pi$
$8$	152.667	0.843374
$9$	0	0
$10$	−224.573	−0.710162
$11$	−19.9993	−0.0498349	−0.0249174	−	0.999690i	$-0.507932\pi$
$11$	−19.9993	−0.0498349	−0.0249174	+	0.999690i	$0.507932\pi$
$12$	0	0
$13$	611.920	1.00424	0.502119	−	0.864799i	$-0.332554\pi$
$13$	611.920	1.00424	0.502119	+	0.864799i	$0.332554\pi$
$14$	130.003	0.177269
$15$	0	0
$16$	381.985	0.373032
$17$	−1525.25	−1.28002	−0.640012	−	0.768365i	$-0.721071\pi$
$17$	−1525.25	−1.28002	−0.640012	+	0.768365i	$0.721071\pi$
$18$	0	0
$19$	−1272.41	−0.808615	−0.404308	−	0.914623i	$-0.632487\pi$
$19$	−1272.41	−0.808615	−0.404308	+	0.914623i	$0.632487\pi$
$20$	−2068.20	−1.15616
$21$	0	0
$22$	53.7850	0.0236921
$23$	−3308.46	−1.30409	−0.652044	−	0.758181i	$-0.726088\pi$
$23$	−3308.46	−1.30409	−0.652044	+	0.758181i	$0.726088\pi$
$24$	0	0
$25$	3848.06	1.23138
$26$	−1645.66	−0.477427
$27$	0	0
$28$	1197.26	0.288599
$29$	2171.14	0.479394	0.239697	−	0.970848i	$-0.422952\pi$
$29$	2171.14	0.479394	0.239697	+	0.970848i	$0.422952\pi$
$30$	0	0
$31$	7163.58	1.33883	0.669416	−	0.742888i	$-0.266544\pi$
$31$	7163.58	1.33883	0.669416	+	0.742888i	$0.266544\pi$
$32$	−5912.63	−1.02072
$33$	0	0
$34$	4101.91	0.608540
$35$	−4036.64	−0.556993
$36$	0	0
$37$	−5908.05	−0.709480	−0.354740	−	0.934965i	$-0.615431\pi$
$37$	−5908.05	−0.709480	−0.354740	+	0.934965i	$0.615431\pi$
$38$	3421.93	0.384426
$39$	0	0
$40$	12748.4	1.25982
$41$	−2583.15	−0.239988	−0.119994	−	0.992775i	$-0.538288\pi$
$41$	−2583.15	−0.239988	−0.119994	+	0.992775i	$0.538288\pi$
$42$	0	0
$43$	−5333.76	−0.439908	−0.219954	−	0.975510i	$-0.570591\pi$
$43$	−5333.76	−0.439908	−0.219954	+	0.975510i	$0.570591\pi$
$44$	495.332	0.0385713
$45$	0	0
$46$	8897.58	0.619980
$47$	10520.3	0.694680	0.347340	−	0.937739i	$-0.387085\pi$
$47$	10520.3	0.694680	0.347340	+	0.937739i	$0.387085\pi$
$48$	0	0
$49$	−14470.2	−0.860965
$50$	−10348.7	−0.585413
$51$	0	0
$52$	−15155.7	−0.777262
$53$	−6041.06	−0.295409	−0.147705	−	0.989032i	$-0.547188\pi$
$53$	−6041.06	−0.295409	−0.147705	+	0.989032i	$0.547188\pi$
$54$	0	0
$55$	−1670.04	−0.0744424
$56$	−7379.94	−0.314473
$57$	0	0
$58$	−5838.93	−0.227910
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	134.207	0.00461798	0.00230899	−	0.999997i	$-0.499265\pi$
$61$	134.207	0.00461798	0.00230899	+	0.999997i	$0.499265\pi$
$62$	−19265.3	−0.636498
$63$	0	0
$64$	3677.55	0.112230
$65$	51098.3	1.50011
$66$	0	0
$67$	49806.2	1.35549	0.677745	−	0.735297i	$-0.262957\pi$
$67$	49806.2	1.35549	0.677745	+	0.735297i	$0.262957\pi$
$68$	37776.5	0.990717
$69$	0	0
$70$	10855.9	0.264801
$71$	−64703.4	−1.52329	−0.761643	−	0.647997i	$-0.775607\pi$
$71$	−64703.4	−1.52329	−0.761643	+	0.647997i	$0.775607\pi$
$72$	0	0
$73$	43618.4	0.957993	0.478996	−	0.877817i	$-0.341001\pi$
$73$	43618.4	0.957993	0.478996	+	0.877817i	$0.341001\pi$
$74$	15888.8	0.337296
$75$	0	0
$76$	31514.3	0.625854
$77$	966.770	0.0185822
$78$	0	0
$79$	74841.5	1.34920	0.674598	−	0.738185i	$-0.264317\pi$
$79$	74841.5	1.34920	0.674598	+	0.738185i	$0.264317\pi$
$80$	31897.6	0.557228
$81$	0	0
$82$	6946.96	0.114093
$83$	−5270.76	−0.0839804	−0.0419902	−	0.999118i	$-0.513370\pi$
$83$	−5270.76	−0.0839804	−0.0419902	+	0.999118i	$0.513370\pi$
$84$	0	0
$85$	−127366.	−1.91208
$86$	14344.3	0.209138
$87$	0	0
$88$	−3053.23	−0.0420294
$89$	−85402.3	−1.14286	−0.571432	−	0.820649i	$-0.693612\pi$
$89$	−85402.3	−1.14286	−0.571432	+	0.820649i	$0.693612\pi$
$90$	0	0
$91$	−29580.3	−0.374455
$92$	81942.2	1.00934
$93$	0	0
$94$	−28292.8	−0.330260
$95$	−106252.	−1.20789
$96$	0	0
$97$	47381.9	0.511309	0.255654	−	0.966768i	$-0.417709\pi$
$97$	47381.9	0.511309	0.255654	+	0.966768i	$0.417709\pi$
$98$	38915.4	0.409313
$99$	0	0
$100$	−95306.6	−0.953066
$101$	−111845.	−1.09097	−0.545484	−	0.838121i	$-0.683654\pi$
$101$	−111845.	−1.09097	−0.545484	+	0.838121i	$0.683654\pi$
$102$	0	0
$103$	−161601.	−1.50089	−0.750447	−	0.660930i	$-0.770162\pi$
$103$	−161601.	−1.50089	−0.750447	+	0.660930i	$0.770162\pi$
$104$	93420.0	0.846947
$105$	0	0
$106$	16246.5	0.140441
$107$	181209.	1.53010	0.765052	−	0.643969i	$-0.222713\pi$
$107$	181209.	1.53010	0.765052	+	0.643969i	$0.222713\pi$
$108$	0	0
$109$	−69858.3	−0.563186	−0.281593	−	0.959534i	$-0.590863\pi$
$109$	−69858.3	−0.563186	−0.281593	+	0.959534i	$0.590863\pi$
$110$	4491.30	0.0353908
$111$	0	0
$112$	−18465.2	−0.139094
$113$	−152141.	−1.12086	−0.560428	−	0.828203i	$-0.689363\pi$
$113$	−152141.	−1.12086	−0.560428	+	0.828203i	$0.689363\pi$
$114$	0	0
$115$	−276273.	−1.94802
$116$	−53773.6	−0.371043
$117$	0	0
$118$	−9361.59	−0.0618934
$119$	73730.7	0.477289
$120$	0	0
$121$	−160651.	−0.997516
$122$	−360.929	−0.00219545
$123$	0	0
$124$	−177424.	−1.03623
$125$	60379.0	0.345630
$126$	0	0
$127$	275370.	1.51498	0.757491	−	0.652846i	$-0.226425\pi$
$127$	275370.	1.51498	0.757491	+	0.652846i	$0.226425\pi$
$128$	179314.	0.967363
$129$	0	0
$130$	−137421.	−0.713171
$131$	−10248.5	−0.0521775	−0.0260888	−	0.999660i	$-0.508305\pi$
$131$	−10248.5	−0.0521775	−0.0260888	+	0.999660i	$0.508305\pi$
$132$	0	0
$133$	61508.3	0.301512
$134$	−133946.	−0.644417
$135$	0	0
$136$	−232855.	−1.07954
$137$	−391127.	−1.78039	−0.890197	−	0.455576i	$-0.849433\pi$
$137$	−391127.	−1.78039	−0.890197	+	0.455576i	$0.849433\pi$
$138$	0	0
$139$	−45851.0	−0.201285	−0.100643	−	0.994923i	$-0.532090\pi$
$139$	−45851.0	−0.201285	−0.100643	+	0.994923i	$0.532090\pi$
$140$	99977.2	0.431103
$141$	0	0
$142$	174009.	0.724189
$143$	−12238.0	−0.0500460
$144$	0	0
$145$	181301.	0.716109
$146$	−117305.	−0.455442
$147$	0	0
$148$	146327.	0.549125
$149$	−188439.	−0.695354	−0.347677	−	0.937614i	$-0.613029\pi$
$149$	−188439.	−0.695354	−0.347677	+	0.937614i	$0.613029\pi$
$150$	0	0
$151$	−70211.4	−0.250591	−0.125295	−	0.992119i	$-0.539988\pi$
$151$	−70211.4	−0.250591	−0.125295	+	0.992119i	$0.539988\pi$
$152$	−194254.	−0.681965
$153$	0	0
$154$	−2599.97	−0.00883419
$155$	598194.	1.99992
$156$	0	0
$157$	−409601.	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$157$	−409601.	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$158$	−201274.	−0.641425
$159$	0	0
$160$	−493733.	−1.52473
$161$	159932.	0.486261
$162$	0	0
$163$	−447613.	−1.31957	−0.659787	−	0.751452i	$-0.729354\pi$
$163$	−447613.	−1.31957	−0.659787	+	0.751452i	$0.729354\pi$
$164$	63978.0	0.185747
$165$	0	0
$166$	14174.9	0.0399254
$167$	64777.2	0.179734	0.0898672	−	0.995954i	$-0.471356\pi$
$167$	64777.2	0.179734	0.0898672	+	0.995954i	$0.471356\pi$
$168$	0	0
$169$	3153.10	0.00849222
$170$	342529.	0.909025
$171$	0	0
$172$	132104.	0.340481
$173$	−436181.	−1.10803	−0.554014	−	0.832507i	$-0.686905\pi$
$173$	−436181.	−1.10803	−0.554014	+	0.832507i	$0.686905\pi$
$174$	0	0
$175$	−186016.	−0.459150
$176$	−7639.44	−0.0185900
$177$	0	0
$178$	229676.	0.543332
$179$	−248687.	−0.580123	−0.290061	−	0.957008i	$-0.593676\pi$
$179$	−248687.	−0.580123	−0.290061	+	0.957008i	$0.593676\pi$
$180$	0	0
$181$	−509737.	−1.15651	−0.578256	−	0.815856i	$-0.696266\pi$
$181$	−509737.	−1.15651	−0.578256	+	0.815856i	$0.696266\pi$
$182$	79551.5	0.178020
$183$	0	0
$184$	−505093.	−1.09983
$185$	−493351.	−1.05981
$186$	0	0
$187$	30503.9	0.0637899
$188$	−260562.	−0.537671
$189$	0	0
$190$	285748.	0.574248
$191$	−582176.	−1.15471	−0.577353	−	0.816495i	$-0.695914\pi$
$191$	−582176.	−1.15471	−0.577353	+	0.816495i	$0.695914\pi$
$192$	0	0
$193$	33922.3	0.0655529	0.0327765	−	0.999463i	$-0.489565\pi$
$193$	33922.3	0.0655529	0.0327765	+	0.999463i	$0.489565\pi$
$194$	−127426.	−0.243083
$195$	0	0
$196$	358391.	0.666372
$197$	−51381.6	−0.0943283	−0.0471642	−	0.998887i	$-0.515018\pi$
$197$	−51381.6	−0.0943283	−0.0471642	+	0.998887i	$0.515018\pi$
$198$	0	0
$199$	293133.	0.524725	0.262363	−	0.964969i	$-0.415498\pi$
$199$	293133.	0.524725	0.262363	+	0.964969i	$0.415498\pi$
$200$	587472.	1.03851
$201$	0	0
$202$	300789.	0.518660
$203$	−104953.	−0.178754
$204$	0	0
$205$	−215705.	−0.358489
$206$	434599.	0.713544
$207$	0	0
$208$	233744.	0.374613
$209$	25447.3	0.0402972
$210$	0	0
$211$	−257285.	−0.397839	−0.198920	−	0.980016i	$-0.563743\pi$
$211$	−257285.	−0.397839	−0.198920	+	0.980016i	$0.563743\pi$
$212$	149622.	0.228642
$213$	0	0
$214$	−487333.	−0.727430
$215$	−445395.	−0.657126
$216$	0	0
$217$	−346289.	−0.499217
$218$	187873.	0.267746
$219$	0	0
$220$	41362.6	0.0576171
$221$	−933330.	−1.28545
$222$	0	0
$223$	−269708.	−0.363188	−0.181594	−	0.983374i	$-0.558126\pi$
$223$	−269708.	−0.363188	−0.181594	+	0.983374i	$0.558126\pi$
$224$	285817.	0.380600
$225$	0	0
$226$	409159.	0.532869
$227$	−474834.	−0.611614	−0.305807	−	0.952094i	$-0.598926\pi$
$227$	−474834.	−0.611614	−0.305807	+	0.952094i	$0.598926\pi$
$228$	0	0
$229$	735574.	0.926910	0.463455	−	0.886120i	$-0.346610\pi$
$229$	735574.	0.926910	0.463455	+	0.886120i	$0.346610\pi$
$230$	742991.	0.926113
$231$	0	0
$232$	331461.	0.404308
$233$	−147473.	−0.177960	−0.0889800	−	0.996033i	$-0.528361\pi$
$233$	−147473.	−0.177960	−0.0889800	+	0.996033i	$0.528361\pi$
$234$	0	0
$235$	878499.	1.03770
$236$	−86215.5	−0.100764
$237$	0	0
$238$	−198287.	−0.226909
$239$	951198.	1.07715	0.538575	−	0.842577i	$-0.318963\pi$
$239$	951198.	1.07715	0.538575	+	0.842577i	$0.318963\pi$
$240$	0	0
$241$	341256.	0.378475	0.189238	−	0.981931i	$-0.439398\pi$
$241$	341256.	0.378475	0.189238	+	0.981931i	$0.439398\pi$
$242$	432045.	0.474232
$243$	0	0
$244$	−3323.98	−0.00357424
$245$	−1.20833e6	−1.28609
$246$	0	0
$247$	−778611.	−0.812042
$248$	1.09364e6	1.12914
$249$	0	0
$250$	−162380.	−0.164317
$251$	−1.54946e6	−1.55238	−0.776188	−	0.630502i	$-0.782849\pi$
$251$	−1.54946e6	−1.55238	−0.776188	+	0.630502i	$0.782849\pi$
$252$	0	0
$253$	66167.0	0.0649890
$254$	−740564.	−0.720242
$255$	0	0
$256$	−599918.	−0.572126
$257$	819098.	0.773576	0.386788	−	0.922169i	$-0.373584\pi$
$257$	819098.	0.773576	0.386788	+	0.922169i	$0.373584\pi$
$258$	0	0
$259$	285596.	0.264547
$260$	−1.26557e6	−1.16106
$261$	0	0
$262$	27561.8	0.0248058
$263$	−1.22245e6	−1.08979	−0.544894	−	0.838505i	$-0.683430\pi$
$263$	−1.22245e6	−1.08979	−0.544894	+	0.838505i	$0.683430\pi$
$264$	0	0
$265$	−504458.	−0.441276
$266$	−165417.	−0.143343
$267$	0	0
$268$	−1.23357e6	−1.04913
$269$	1.60210e6	1.34992	0.674961	−	0.737853i	$-0.264160\pi$
$269$	1.60210e6	1.34992	0.674961	+	0.737853i	$0.264160\pi$
$270$	0	0
$271$	1.62601e6	1.34493	0.672465	−	0.740129i	$-0.265235\pi$
$271$	1.62601e6	1.34493	0.672465	+	0.740129i	$0.265235\pi$
$272$	−582622.	−0.477490
$273$	0	0
$274$	1.05187e6	0.846422
$275$	−76958.6	−0.0613656
$276$	0	0
$277$	1.36348e6	1.06770	0.533851	−	0.845579i	$-0.320744\pi$
$277$	1.36348e6	1.06770	0.533851	+	0.845579i	$0.320744\pi$
$278$	123309.	0.0956935
$279$	0	0
$280$	−616261.	−0.469753
$281$	−1.74237e6	−1.31636	−0.658179	−	0.752862i	$-0.728673\pi$
$281$	−1.74237e6	−1.31636	−0.658179	+	0.752862i	$0.728673\pi$
$282$	0	0
$283$	−1.23647e6	−0.917735	−0.458868	−	0.888505i	$-0.651745\pi$
$283$	−1.23647e6	−0.917735	−0.458868	+	0.888505i	$0.651745\pi$
$284$	1.60254e6	1.17900
$285$	0	0
$286$	32912.1	0.0237925
$287$	124870.	0.0894854
$288$	0	0
$289$	906525.	0.638462
$290$	−487579.	−0.340447
$291$	0	0
$292$	−1.08032e6	−0.741470
$293$	−2.68981e6	−1.83042	−0.915212	−	0.402972i	$-0.867977\pi$
$293$	−2.68981e6	−1.83042	−0.915212	+	0.402972i	$0.867977\pi$
$294$	0	0
$295$	290680.	0.194474
$296$	−901964.	−0.598357
$297$	0	0
$298$	506777.	0.330580
$299$	−2.02451e6	−1.30961
$300$	0	0
$301$	257835.	0.164031
$302$	188822.	0.119134
$303$	0	0
$304$	−486040.	−0.301640
$305$	11207.0	0.00689825
$306$	0	0
$307$	−1.87388e6	−1.13474	−0.567369	−	0.823463i	$-0.692039\pi$
$307$	−1.87388e6	−1.13474	−0.567369	+	0.823463i	$0.692039\pi$
$308$	−23944.4	−0.0143823
$309$	0	0
$310$	−1.60875e6	−0.950788
$311$	1.66629e6	0.976898	0.488449	−	0.872592i	$-0.337563\pi$
$311$	1.66629e6	0.976898	0.488449	+	0.872592i	$0.337563\pi$
$312$	0	0
$313$	−1.52243e6	−0.878368	−0.439184	−	0.898397i	$-0.644732\pi$
$313$	−1.52243e6	−0.878368	−0.439184	+	0.898397i	$0.644732\pi$
$314$	1.10156e6	0.630497
$315$	0	0
$316$	−1.85363e6	−1.04425
$317$	1.08948e6	0.608933	0.304466	−	0.952523i	$-0.401522\pi$
$317$	1.08948e6	0.608933	0.304466	+	0.952523i	$0.401522\pi$
$318$	0	0
$319$	−43421.3	−0.0238905
$320$	307093.	0.167647
$321$	0	0
$322$	−430110.	−0.231175
$323$	1.94074e6	1.03505
$324$	0	0
$325$	2.35471e6	1.23660
$326$	1.20378e6	0.627342
$327$	0	0
$328$	−394361.	−0.202400
$329$	−508555.	−0.259029
$330$	0	0
$331$	3.01617e6	1.51317	0.756583	−	0.653898i	$-0.226867\pi$
$331$	3.01617e6	1.51317	0.756583	+	0.653898i	$0.226867\pi$
$332$	130543.	0.0649994
$333$	0	0
$334$	−174208.	−0.0854480
$335$	4.15906e6	2.02480
$336$	0	0
$337$	−2.89211e6	−1.38720	−0.693602	−	0.720359i	$-0.743977\pi$
$337$	−2.89211e6	−1.38720	−0.693602	+	0.720359i	$0.743977\pi$
$338$	−8479.77	−0.00403731
$339$	0	0
$340$	3.15452e6	1.47991
$341$	−143267.	−0.0667205
$342$	0	0
$343$	1.51195e6	0.693906
$344$	−814289.	−0.371007
$345$	0	0
$346$	1.17304e6	0.526771
$347$	2.45959e6	1.09658	0.548289	−	0.836289i	$-0.315279\pi$
$347$	2.45959e6	1.09658	0.548289	+	0.836289i	$0.315279\pi$
$348$	0	0
$349$	1.57913e6	0.693992	0.346996	−	0.937867i	$-0.387202\pi$
$349$	1.57913e6	0.693992	0.346996	+	0.937867i	$0.387202\pi$
$350$	500260.	0.218286
$351$	0	0
$352$	118249.	0.0508674
$353$	932705.	0.398389	0.199195	−	0.979960i	$-0.436167\pi$
$353$	932705.	0.398389	0.199195	+	0.979960i	$0.436167\pi$
$354$	0	0
$355$	−5.40305e6	−2.27545
$356$	2.11520e6	0.884557
$357$	0	0
$358$	668803.	0.275798
$359$	94646.8	0.0387587	0.0193794	−	0.999812i	$-0.493831\pi$
$359$	94646.8	0.0387587	0.0193794	+	0.999812i	$0.493831\pi$
$360$	0	0
$361$	−857080.	−0.346141
$362$	1.37086e6	0.549820
$363$	0	0
$364$	732629.	0.289821
$365$	3.64235e6	1.43103
$366$	0	0
$367$	3.17308e6	1.22975	0.614873	−	0.788626i	$-0.289207\pi$
$367$	3.17308e6	1.22975	0.614873	+	0.788626i	$0.289207\pi$
$368$	−1.26378e6	−0.486467
$369$	0	0
$370$	1.32679e6	0.503845
$371$	292026.	0.110151
$372$	0	0
$373$	−2.35041e6	−0.874723	−0.437362	−	0.899286i	$-0.644087\pi$
$373$	−2.35041e6	−0.874723	−0.437362	+	0.899286i	$0.644087\pi$
$374$	−82035.4	−0.0303265
$375$	0	0
$376$	1.60611e6	0.585875
$377$	1.32856e6	0.481425
$378$	0	0
$379$	998577.	0.357095	0.178547	−	0.983931i	$-0.442860\pi$
$379$	998577.	0.357095	0.178547	+	0.983931i	$0.442860\pi$
$380$	2.63159e6	0.934889
$381$	0	0
$382$	1.56567e6	0.548961
$383$	3.24360e6	1.12987	0.564937	−	0.825134i	$-0.308900\pi$
$383$	3.24360e6	1.12987	0.564937	+	0.825134i	$0.308900\pi$
$384$	0	0
$385$	80730.0	0.0277577
$386$	−91228.6	−0.0311647
$387$	0	0
$388$	−1.17353e6	−0.395744
$389$	434588.	0.145614	0.0728070	−	0.997346i	$-0.476804\pi$
$389$	434588.	0.145614	0.0728070	+	0.997346i	$0.476804\pi$
$390$	0	0
$391$	5.04623e6	1.66926
$392$	−2.20913e6	−0.726115
$393$	0	0
$394$	138183.	0.0448449
$395$	6.24963e6	2.01540
$396$	0	0
$397$	2.29030e6	0.729315	0.364658	−	0.931142i	$-0.381186\pi$
$397$	2.29030e6	0.729315	0.364658	+	0.931142i	$0.381186\pi$
$398$	−788334.	−0.249461
$399$	0	0
$400$	1.46990e6	0.459344
$401$	−1.95111e6	−0.605928	−0.302964	−	0.953002i	$-0.597976\pi$
$401$	−1.95111e6	−0.605928	−0.302964	+	0.953002i	$0.597976\pi$
$402$	0	0
$403$	4.38354e6	1.34450
$404$	2.77011e6	0.844391
$405$	0	0
$406$	282255.	0.0849818
$407$	118157.	0.0353568
$408$	0	0
$409$	1.46908e6	0.434247	0.217123	−	0.976144i	$-0.430333\pi$
$409$	1.46908e6	0.434247	0.217123	+	0.976144i	$0.430333\pi$
$410$	580105.	0.170430
$411$	0	0
$412$	4.00244e6	1.16167
$413$	−168272.	−0.0485441
$414$	0	0
$415$	−440134.	−0.125448
$416$	−3.61806e6	−1.02504
$417$	0	0
$418$	−68436.3	−0.0191578
$419$	3.97963e6	1.10741	0.553703	−	0.832714i	$-0.313214\pi$
$419$	3.97963e6	1.10741	0.553703	+	0.832714i	$0.313214\pi$
$420$	0	0
$421$	828909.	0.227930	0.113965	−	0.993485i	$-0.463645\pi$
$421$	828909.	0.227930	0.113965	+	0.993485i	$0.463645\pi$
$422$	691926.	0.189138
$423$	0	0
$424$	−922271.	−0.249140
$425$	−5.86925e6	−1.57620
$426$	0	0
$427$	−6487.61	−0.00172193
$428$	−4.48809e6	−1.18427
$429$	0	0
$430$	1.19782e6	0.312406
$431$	−6.65314e6	−1.72518	−0.862588	−	0.505907i	$-0.831158\pi$
$431$	−6.65314e6	−1.72518	−0.862588	+	0.505907i	$0.831158\pi$
$432$	0	0
$433$	−532462.	−0.136480	−0.0682400	−	0.997669i	$-0.521738\pi$
$433$	−532462.	−0.136480	−0.0682400	+	0.997669i	$0.521738\pi$
$434$	931288.	0.237334
$435$	0	0
$436$	1.73021e6	0.435896
$437$	4.20971e6	1.05450
$438$	0	0
$439$	4.04363e6	1.00141	0.500703	−	0.865619i	$-0.333075\pi$
$439$	4.04363e6	1.00141	0.500703	+	0.865619i	$0.333075\pi$
$440$	−254960.	−0.0627827
$441$	0	0
$442$	2.51004e6	0.611118
$443$	−1.66963e6	−0.404214	−0.202107	−	0.979363i	$-0.564779\pi$
$443$	−1.66963e6	−0.404214	−0.202107	+	0.979363i	$0.564779\pi$
$444$	0	0
$445$	−7.13151e6	−1.70719
$446$	725336.	0.172664
$447$	0	0
$448$	−177773.	−0.0418477
$449$	4.07437e6	0.953772	0.476886	−	0.878965i	$-0.341765\pi$
$449$	4.07437e6	0.953772	0.476886	+	0.878965i	$0.341765\pi$
$450$	0	0
$451$	51661.2	0.0119598
$452$	3.76814e6	0.867524
$453$	0	0
$454$	1.27699e6	0.290769
$455$	−2.47010e6	−0.559353
$456$	0	0
$457$	−6.80572e6	−1.52435	−0.762173	−	0.647373i	$-0.775868\pi$
$457$	−6.80572e6	−1.52435	−0.762173	+	0.647373i	$0.775868\pi$
$458$	−1.97821e6	−0.440665
$459$	0	0
$460$	6.84257e6	1.50773
$461$	−6.87814e6	−1.50737	−0.753683	−	0.657239i	$-0.771724\pi$
$461$	−6.87814e6	−1.50737	−0.753683	+	0.657239i	$0.771724\pi$
$462$	0	0
$463$	−2.39491e6	−0.519202	−0.259601	−	0.965716i	$-0.583591\pi$
$463$	−2.39491e6	−0.519202	−0.259601	+	0.965716i	$0.583591\pi$
$464$	829343.	0.178829
$465$	0	0
$466$	396605.	0.0846044
$467$	−5.32072e6	−1.12896	−0.564480	−	0.825447i	$-0.690923\pi$
$467$	−5.32072e6	−1.12896	−0.564480	+	0.825447i	$0.690923\pi$
$468$	0	0
$469$	−2.40764e6	−0.505428
$470$	−2.36258e6	−0.493336
$471$	0	0
$472$	531434.	0.109798
$473$	106671.	0.0219228
$474$	0	0
$475$	−4.89630e6	−0.995712
$476$	−1.82612e6	−0.369413
$477$	0	0
$478$	−2.55810e6	−0.512091
$479$	476392.	0.0948693	0.0474347	−	0.998874i	$-0.484895\pi$
$479$	476392.	0.0948693	0.0474347	+	0.998874i	$0.484895\pi$
$480$	0	0
$481$	−3.61526e6	−0.712486
$482$	−917753.	−0.179932
$483$	0	0
$484$	3.97892e6	0.772061
$485$	3.95662e6	0.763783
$486$	0	0
$487$	−3.24904e6	−0.620774	−0.310387	−	0.950610i	$-0.600459\pi$
$487$	−3.24904e6	−0.620774	−0.310387	+	0.950610i	$0.600459\pi$
$488$	20489.0	0.00389468
$489$	0	0
$490$	3.24962e6	0.611424
$491$	1.12407e6	0.210421	0.105211	−	0.994450i	$-0.466448\pi$
$491$	1.12407e6	0.210421	0.105211	+	0.994450i	$0.466448\pi$
$492$	0	0
$493$	−3.31153e6	−0.613636
$494$	2.09395e6	0.386055
$495$	0	0
$496$	2.73638e6	0.499428
$497$	3.12777e6	0.567994
$498$	0	0
$499$	876516.	0.157583	0.0787913	−	0.996891i	$-0.474894\pi$
$499$	876516.	0.157583	0.0787913	+	0.996891i	$0.474894\pi$
$500$	−1.49543e6	−0.267512
$501$	0	0
$502$	4.16703e6	0.738019
$503$	1.53970e6	0.271342	0.135671	−	0.990754i	$-0.456681\pi$
$503$	1.53970e6	0.271342	0.135671	+	0.990754i	$0.456681\pi$
$504$	0	0
$505$	−9.33958e6	−1.62967
$506$	−177946.	−0.0308966
$507$	0	0
$508$	−6.82022e6	−1.17257
$509$	−3.23654e6	−0.553715	−0.276858	−	0.960911i	$-0.589293\pi$
$509$	−3.23654e6	−0.553715	−0.276858	+	0.960911i	$0.589293\pi$
$510$	0	0
$511$	−2.10852e6	−0.357211
$512$	−4.12467e6	−0.695367
$513$	0	0
$514$	−2.20283e6	−0.367768
$515$	−1.34944e7	−2.24201
$516$	0	0
$517$	−210400.	−0.0346193
$518$	−768065.	−0.125769
$519$	0	0
$520$	7.80102e6	1.26515
$521$	−8.43608e6	−1.36159	−0.680795	−	0.732474i	$-0.738366\pi$
$521$	−8.43608e6	−1.36159	−0.680795	+	0.732474i	$0.738366\pi$
$522$	0	0
$523$	1.04546e7	1.67129	0.835645	−	0.549270i	$-0.185094\pi$
$523$	1.04546e7	1.67129	0.835645	+	0.549270i	$0.185094\pi$
$524$	253830.	0.0403845
$525$	0	0
$526$	3.28759e6	0.518099
$527$	−1.09262e7	−1.71374
$528$	0	0
$529$	4.50958e6	0.700644
$530$	1.35666e6	0.209788
$531$	0	0
$532$	−1.52340e6	−0.233365
$533$	−1.58068e6	−0.241005
$534$	0	0
$535$	1.51318e7	2.28564
$536$	7.60376e6	1.14318
$537$	0	0
$538$	−4.30859e6	−0.641770
$539$	289395.	0.0429061
$540$	0	0
$541$	−7.11362e6	−1.04496	−0.522478	−	0.852653i	$-0.674992\pi$
$541$	−7.11362e6	−1.04496	−0.522478	+	0.852653i	$0.674992\pi$
$542$	−4.37289e6	−0.639397
$543$	0	0
$544$	9.01823e6	1.30654
$545$	−5.83351e6	−0.841276
$546$	0	0
$547$	−3.99245e6	−0.570520	−0.285260	−	0.958450i	$-0.592080\pi$
$547$	−3.99245e6	−0.570520	−0.285260	+	0.958450i	$0.592080\pi$
$548$	9.68721e6	1.37799
$549$	0	0
$550$	206968.	0.0291740
$551$	−2.76257e6	−0.387645
$552$	0	0
$553$	−3.61785e6	−0.503081
$554$	−3.66686e6	−0.507599
$555$	0	0
$556$	1.13561e6	0.155791
$557$	1.13329e6	0.154776	0.0773881	−	0.997001i	$-0.475342\pi$
$557$	1.13329e6	0.154776	0.0773881	+	0.997001i	$0.475342\pi$
$558$	0	0
$559$	−3.26383e6	−0.441772
$560$	−1.54193e6	−0.207776
$561$	0	0
$562$	4.68582e6	0.625813
$563$	1.33775e7	1.77871	0.889353	−	0.457222i	$-0.151156\pi$
$563$	1.33775e7	1.77871	0.889353	+	0.457222i	$0.151156\pi$
$564$	0	0
$565$	−1.27045e7	−1.67431
$566$	3.32529e6	0.436303
$567$	0	0
$568$	−9.87807e6	−1.28470
$569$	9.96665e6	1.29053	0.645266	−	0.763958i	$-0.276747\pi$
$569$	9.96665e6	1.29053	0.645266	+	0.763958i	$0.276747\pi$
$570$	0	0
$571$	−5.30435e6	−0.680835	−0.340417	−	0.940274i	$-0.610568\pi$
$571$	−5.30435e6	−0.680835	−0.340417	+	0.940274i	$0.610568\pi$
$572$	303104.	0.0387348
$573$	0	0
$574$	−335817.	−0.0425425
$575$	−1.27312e7	−1.60583
$576$	0	0
$577$	1.35522e7	1.69461	0.847307	−	0.531103i	$-0.178222\pi$
$577$	1.35522e7	1.69461	0.847307	+	0.531103i	$0.178222\pi$
$578$	−2.43795e6	−0.303533
$579$	0	0
$580$	−4.49035e6	−0.554256
$581$	254789.	0.0313142
$582$	0	0
$583$	120817.	0.0147217
$584$	6.65909e6	0.807946
$585$	0	0
$586$	7.23380e6	0.870207
$587$	1.53895e7	1.84344	0.921720	−	0.387857i	$-0.126785\pi$
$587$	1.53895e7	1.84344	0.921720	+	0.387857i	$0.126785\pi$
$588$	0	0
$589$	−9.11499e6	−1.08260
$590$	−781738.	−0.0924552
$591$	0	0
$592$	−2.25679e6	−0.264659
$593$	−9.50339e6	−1.10979	−0.554896	−	0.831920i	$-0.687242\pi$
$593$	−9.50339e6	−1.10979	−0.554896	+	0.831920i	$0.687242\pi$
$594$	0	0
$595$	6.15687e6	0.712964
$596$	4.66716e6	0.538192
$597$	0	0
$598$	5.44461e6	0.622606
$599$	−5.58939e6	−0.636498	−0.318249	−	0.948007i	$-0.603095\pi$
$599$	−5.58939e6	−0.636498	−0.318249	+	0.948007i	$0.603095\pi$
$600$	0	0
$601$	−1.67571e7	−1.89240	−0.946200	−	0.323581i	$-0.895113\pi$
$601$	−1.67571e7	−1.89240	−0.946200	+	0.323581i	$0.895113\pi$
$602$	−693405.	−0.0779822
$603$	0	0
$604$	1.73896e6	0.193953
$605$	−1.34151e7	−1.49007
$606$	0	0
$607$	−1.00054e7	−1.10220	−0.551101	−	0.834439i	$-0.685792\pi$
$607$	−1.00054e7	−1.10220	−0.551101	+	0.834439i	$0.685792\pi$
$608$	7.52327e6	0.825368
$609$	0	0
$610$	−30139.4	−0.00327951
$611$	6.43761e6	0.697624
$612$	0	0
$613$	1.22753e7	1.31941	0.659704	−	0.751525i	$-0.270682\pi$
$613$	1.22753e7	1.31941	0.659704	+	0.751525i	$0.270682\pi$
$614$	5.03950e6	0.539469
$615$	0	0
$616$	147594.	0.0156717
$617$	3.39106e6	0.358611	0.179305	−	0.983793i	$-0.442615\pi$
$617$	3.39106e6	0.358611	0.179305	+	0.983793i	$0.442615\pi$
$618$	0	0
$619$	−1.36685e7	−1.43382	−0.716908	−	0.697168i	$-0.754443\pi$
$619$	−1.36685e7	−1.43382	−0.716908	+	0.697168i	$0.754443\pi$
$620$	−1.48157e7	−1.54790
$621$	0	0
$622$	−4.48122e6	−0.464430
$623$	4.12836e6	0.426145
$624$	0	0
$625$	−6.98325e6	−0.715084
$626$	4.09433e6	0.417587
$627$	0	0
$628$	1.01448e7	1.02646
$629$	9.01125e6	0.908151
$630$	0	0
$631$	1.06988e7	1.06969	0.534847	−	0.844949i	$-0.320369\pi$
$631$	1.06988e7	1.06969	0.534847	+	0.844949i	$0.320369\pi$
$632$	1.14258e7	1.13788
$633$	0	0
$634$	−2.92997e6	−0.289494
$635$	2.29947e7	2.26305
$636$	0	0
$637$	−8.85462e6	−0.864613
$638$	116775.	0.0113579
$639$	0	0
$640$	1.49736e7	1.44503
$641$	5.09674e6	0.489945	0.244972	−	0.969530i	$-0.421221\pi$
$641$	5.09674e6	0.489945	0.244972	+	0.969530i	$0.421221\pi$
$642$	0	0
$643$	540230.	0.0515289	0.0257645	−	0.999668i	$-0.491798\pi$
$643$	540230.	0.0515289	0.0257645	+	0.999668i	$0.491798\pi$
$644$	−3.96110e6	−0.376358
$645$	0	0
$646$	−5.21930e6	−0.492075
$647$	−1.70813e6	−0.160421	−0.0802105	−	0.996778i	$-0.525559\pi$
$647$	−1.70813e6	−0.160421	−0.0802105	+	0.996778i	$0.525559\pi$
$648$	0	0
$649$	−69617.6	−0.00648795
$650$	−6.33260e6	−0.587894
$651$	0	0
$652$	1.10862e7	1.02133
$653$	1.69268e6	0.155343	0.0776716	−	0.996979i	$-0.475251\pi$
$653$	1.69268e6	0.155343	0.0776716	+	0.996979i	$0.475251\pi$
$654$	0	0
$655$	−855802.	−0.0779417
$656$	−986724.	−0.0895233
$657$	0	0
$658$	1.36768e6	0.123145
$659$	−1.08698e7	−0.975005	−0.487502	−	0.873122i	$-0.662092\pi$
$659$	−1.08698e7	−0.975005	−0.487502	+	0.873122i	$0.662092\pi$
$660$	0	0
$661$	1.45109e7	1.29178	0.645892	−	0.763429i	$-0.276486\pi$
$661$	1.45109e7	1.29178	0.645892	+	0.763429i	$0.276486\pi$
$662$	−8.11152e6	−0.719378
$663$	0	0
$664$	−804671.	−0.0708269
$665$	5.13624e6	0.450393
$666$	0	0
$667$	−7.18313e6	−0.625172
$668$	−1.60437e6	−0.139111
$669$	0	0
$670$	−1.11851e7	−0.962618
$671$	−2684.06	−0.000230136 0
$672$	0	0
$673$	−1.88349e7	−1.60297	−0.801484	−	0.598016i	$-0.795956\pi$
$673$	−1.88349e7	−1.60297	−0.801484	+	0.598016i	$0.795956\pi$
$674$	7.77787e6	0.659494
$675$	0	0
$676$	−78094.3	−0.00657284
$677$	8.52740e6	0.715064	0.357532	−	0.933901i	$-0.383618\pi$
$677$	8.52740e6	0.715064	0.357532	+	0.933901i	$0.383618\pi$
$678$	0	0
$679$	−2.29045e6	−0.190654
$680$	−1.94445e7	−1.61259
$681$	0	0
$682$	385293.	0.0317198
$683$	−1.61614e7	−1.32564	−0.662821	−	0.748778i	$-0.730641\pi$
$683$	−1.61614e7	−1.32564	−0.662821	+	0.748778i	$0.730641\pi$
$684$	0	0
$685$	−3.26610e7	−2.65952
$686$	−4.06614e6	−0.329892
$687$	0	0
$688$	−2.03742e6	−0.164100
$689$	−3.69665e6	−0.296661
$690$	0	0
$691$	−2.32191e7	−1.84991	−0.924953	−	0.380081i	$-0.875896\pi$
$691$	−2.32191e7	−1.84991	−0.924953	+	0.380081i	$0.875896\pi$
$692$	1.08031e7	0.857595
$693$	0	0
$694$	−6.61468e6	−0.521327
$695$	−3.82878e6	−0.300676
$696$	0	0
$697$	3.93994e6	0.307191
$698$	−4.24682e6	−0.329932
$699$	0	0
$700$	4.60714e6	0.355374
$701$	1.45476e7	1.11814	0.559070	−	0.829121i	$-0.311158\pi$
$701$	1.45476e7	1.11814	0.559070	+	0.829121i	$0.311158\pi$
$702$	0	0
$703$	7.51744e6	0.573696
$704$	−73548.5	−0.00559296
$705$	0	0
$706$	−2.50836e6	−0.189399
$707$	5.40659e6	0.406795
$708$	0	0
$709$	2.04913e7	1.53092	0.765462	−	0.643481i	$-0.222510\pi$
$709$	2.04913e7	1.53092	0.765462	+	0.643481i	$0.222510\pi$
$710$	1.45306e7	1.08178
$711$	0	0
$712$	−1.30381e7	−0.963861
$713$	−2.37005e7	−1.74595
$714$	0	0
$715$	−1.02193e6	−0.0747578
$716$	6.15933e6	0.449005
$717$	0	0
$718$	−254537.	−0.0184264
$719$	−5.74561e6	−0.414490	−0.207245	−	0.978289i	$-0.566450\pi$
$719$	−5.74561e6	−0.414490	−0.207245	+	0.978289i	$0.566450\pi$
$720$	0	0
$721$	7.81180e6	0.559645
$722$	2.30498e6	0.164560
$723$	0	0
$724$	1.26249e7	0.895120
$725$	8.35467e6	0.590316
$726$	0	0
$727$	−2.22394e7	−1.56058	−0.780292	−	0.625416i	$-0.784929\pi$
$727$	−2.22394e7	−1.56058	−0.780292	+	0.625416i	$0.784929\pi$
$728$	−4.51593e6	−0.315805
$729$	0	0
$730$	−9.79551e6	−0.680330
$731$	8.13530e6	0.563093
$732$	0	0
$733$	1.66373e7	1.14373	0.571866	−	0.820347i	$-0.306220\pi$
$733$	1.66373e7	1.14373	0.571866	+	0.820347i	$0.306220\pi$
$734$	−8.53349e6	−0.584637
$735$	0	0
$736$	1.95617e7	1.33111
$737$	−996090.	−0.0675507
$738$	0	0
$739$	−6.69365e6	−0.450871	−0.225435	−	0.974258i	$-0.572380\pi$
$739$	−6.69365e6	−0.450871	−0.225435	+	0.974258i	$0.572380\pi$
$740$	1.22190e7	0.820272
$741$	0	0
$742$	−785357.	−0.0523669
$743$	−2.73131e7	−1.81509	−0.907547	−	0.419951i	$-0.862047\pi$
$743$	−2.73131e7	−1.81509	−0.907547	+	0.419951i	$0.862047\pi$
$744$	0	0
$745$	−1.57356e7	−1.03871
$746$	6.32104e6	0.415854
$747$	0	0
$748$	−755504.	−0.0493723
$749$	−8.75968e6	−0.570537
$750$	0	0
$751$	−3.14794e6	−0.203670	−0.101835	−	0.994801i	$-0.532471\pi$
$751$	−3.14794e6	−0.203670	−0.101835	+	0.994801i	$0.532471\pi$
$752$	4.01861e6	0.259138
$753$	0	0
$754$	−3.57296e6	−0.228876
$755$	−5.86299e6	−0.374328
$756$	0	0
$757$	8.68258e6	0.550692	0.275346	−	0.961345i	$-0.411208\pi$
$757$	8.68258e6	0.550692	0.275346	+	0.961345i	$0.411208\pi$
$758$	−2.68551e6	−0.169767
$759$	0	0
$760$	−1.62212e7	−1.01871
$761$	9.78953e6	0.612774	0.306387	−	0.951907i	$-0.400880\pi$
$761$	9.78953e6	0.612774	0.306387	+	0.951907i	$0.400880\pi$
$762$	0	0
$763$	3.37696e6	0.209998
$764$	1.44190e7	0.893722
$765$	0	0
$766$	−8.72314e6	−0.537157
$767$	2.13009e6	0.130741
$768$	0	0
$769$	−2.96608e7	−1.80870	−0.904351	−	0.426789i	$-0.859645\pi$
$769$	−2.96608e7	−1.80870	−0.904351	+	0.426789i	$0.859645\pi$
$770$	−217110.	−0.0131963
$771$	0	0
$772$	−840169.	−0.0507368
$773$	7.78504e6	0.468611	0.234305	−	0.972163i	$-0.424718\pi$
$773$	7.78504e6	0.468611	0.234305	+	0.972163i	$0.424718\pi$
$774$	0	0
$775$	2.75659e7	1.64861
$776$	7.23365e6	0.431224
$777$	0	0
$778$	−1.16875e6	−0.0692268
$779$	3.28681e6	0.194058
$780$	0	0
$781$	1.29402e6	0.0759127
$782$	−1.35710e7	−0.793589
$783$	0	0
$784$	−5.52741e6	−0.321168
$785$	−3.42037e7	−1.98106
$786$	0	0
$787$	−2.25929e6	−0.130028	−0.0650138	−	0.997884i	$-0.520709\pi$
$787$	−2.25929e6	−0.130028	−0.0650138	+	0.997884i	$0.520709\pi$
$788$	1.27259e6	0.0730085
$789$	0	0
$790$	−1.68074e7	−0.958148
$791$	7.35451e6	0.417939
$792$	0	0
$793$	82124.2	0.00463755
$794$	−6.15938e6	−0.346726
$795$	0	0
$796$	−7.26016e6	−0.406128
$797$	1.95399e7	1.08962	0.544812	−	0.838558i	$-0.316601\pi$
$797$	1.95399e7	1.08962	0.544812	+	0.838558i	$0.316601\pi$
$798$	0	0
$799$	−1.60461e7	−0.889208
$800$	−2.27522e7	−1.25689
$801$	0	0
$802$	5.24720e6	0.288066
$803$	−872338.	−0.0477415
$804$	0	0
$805$	1.33551e7	0.726367
$806$	−1.17888e7	−0.639195
$807$	0	0
$808$	−1.70750e7	−0.920094
$809$	2.92703e7	1.57237	0.786187	−	0.617989i	$-0.212052\pi$
$809$	2.92703e7	1.57237	0.786187	+	0.617989i	$0.212052\pi$
$810$	0	0
$811$	3.63592e6	0.194116	0.0970581	−	0.995279i	$-0.469057\pi$
$811$	3.63592e6	0.194116	0.0970581	+	0.995279i	$0.469057\pi$
$812$	2.59942e6	0.138352
$813$	0	0
$814$	−317764.	−0.0168091
$815$	−3.73779e7	−1.97115
$816$	0	0
$817$	6.78671e6	0.355717
$818$	−3.95085e6	−0.206446
$819$	0	0
$820$	5.34247e6	0.277465
$821$	−1.26564e6	−0.0655318	−0.0327659	−	0.999463i	$-0.510432\pi$
$821$	−1.26564e6	−0.0655318	−0.0327659	+	0.999463i	$0.510432\pi$
$822$	0	0
$823$	2.48803e7	1.28043	0.640216	−	0.768195i	$-0.278845\pi$
$823$	2.48803e7	1.28043	0.640216	+	0.768195i	$0.278845\pi$
$824$	−2.46711e7	−1.26581
$825$	0	0
$826$	452541.	0.0230785
$827$	−1.44783e7	−0.736131	−0.368065	−	0.929800i	$-0.619980\pi$
$827$	−1.44783e7	−0.736131	−0.368065	+	0.929800i	$0.619980\pi$
$828$	0	0
$829$	476856.	0.0240991	0.0120495	−	0.999927i	$-0.496164\pi$
$829$	476856.	0.0240991	0.0120495	+	0.999927i	$0.496164\pi$
$830$	1.18367e6	0.0596397
$831$	0	0
$832$	2.25037e6	0.112705
$833$	2.20707e7	1.10206
$834$	0	0
$835$	5.40921e6	0.268484
$836$	−630264.	−0.0311894
$837$	0	0
$838$	−1.07026e7	−0.526475
$839$	−2.73963e7	−1.34365	−0.671827	−	0.740708i	$-0.734490\pi$
$839$	−2.73963e7	−1.34365	−0.671827	+	0.740708i	$0.734490\pi$
$840$	0	0
$841$	−1.57973e7	−0.770181
$842$	−2.22922e6	−0.108361
$843$	0	0
$844$	6.37228e6	0.307921
$845$	263299.	0.0126855
$846$	0	0
$847$	7.76589e6	0.371949
$848$	−2.30760e6	−0.110197
$849$	0	0
$850$	1.57844e7	0.749343
$851$	1.95466e7	0.925223
$852$	0	0
$853$	2.87120e6	0.135111	0.0675556	−	0.997716i	$-0.478480\pi$
$853$	2.87120e6	0.135111	0.0675556	+	0.997716i	$0.478480\pi$
$854$	17447.4	0.000818626 0
$855$	0	0
$856$	2.76647e7	1.29045
$857$	−2.66310e7	−1.23861	−0.619305	−	0.785150i	$-0.712586\pi$
$857$	−2.66310e7	−1.23861	−0.619305	+	0.785150i	$0.712586\pi$
$858$	0	0
$859$	7.47800e6	0.345782	0.172891	−	0.984941i	$-0.444689\pi$
$859$	7.47800e6	0.345782	0.172891	+	0.984941i	$0.444689\pi$
$860$	1.10313e7	0.508604
$861$	0	0
$862$	1.78926e7	0.820171
$863$	3.33536e7	1.52446	0.762230	−	0.647306i	$-0.224105\pi$
$863$	3.33536e7	1.52446	0.762230	+	0.647306i	$0.224105\pi$
$864$	0	0
$865$	−3.64232e7	−1.65515
$866$	1.43197e6	0.0648843
$867$	0	0
$868$	8.57669e6	0.386385
$869$	−1.49678e6	−0.0672370
$870$	0	0
$871$	3.04774e7	1.36123
$872$	−1.06651e7	−0.474976
$873$	0	0
$874$	−1.13213e7	−0.501325
$875$	−2.91873e6	−0.128877
$876$	0	0
$877$	2.97763e7	1.30729	0.653645	−	0.756802i	$-0.273239\pi$
$877$	2.97763e7	1.30729	0.653645	+	0.756802i	$0.273239\pi$
$878$	−1.08747e7	−0.476081
$879$	0	0
$880$	−637930.	−0.0277694
$881$	−1.74483e7	−0.757380	−0.378690	−	0.925523i	$-0.623625\pi$
$881$	−1.74483e7	−0.757380	−0.378690	+	0.925523i	$0.623625\pi$
$882$	0	0
$883$	3.08101e7	1.32981	0.664907	−	0.746926i	$-0.268471\pi$
$883$	3.08101e7	1.32981	0.664907	+	0.746926i	$0.268471\pi$
$884$	2.31162e7	0.994915
$885$	0	0
$886$	4.49021e6	0.192169
$887$	110509.	0.00471615	0.00235807	−	0.999997i	$-0.499249\pi$
$887$	110509.	0.00471615	0.00235807	+	0.999997i	$0.499249\pi$
$888$	0	0
$889$	−1.33114e7	−0.564898
$890$	1.91790e7	0.811618
$891$	0	0
$892$	6.67997e6	0.281101
$893$	−1.33861e7	−0.561729
$894$	0	0
$895$	−2.07665e7	−0.866576
$896$	−8.66806e6	−0.360705
$897$	0	0
$898$	−1.09574e7	−0.453435
$899$	1.55531e7	0.641828
$900$	0	0
$901$	9.21412e6	0.378131
$902$	−138934.	−0.00568583
$903$	0	0
$904$	−2.32269e7	−0.945301
$905$	−4.25655e7	−1.72757
$906$	0	0
$907$	3.31400e7	1.33763	0.668813	−	0.743431i	$-0.266803\pi$
$907$	3.31400e7	1.33763	0.668813	+	0.743431i	$0.266803\pi$
$908$	1.17604e7	0.473379
$909$	0	0
$910$	6.64293e6	0.265923
$911$	2.55670e7	1.02067	0.510334	−	0.859976i	$-0.329522\pi$
$911$	2.55670e7	1.02067	0.510334	+	0.859976i	$0.329522\pi$
$912$	0	0
$913$	105412.	0.00418515
$914$	1.83029e7	0.724694
$915$	0	0
$916$	−1.82183e7	−0.717412
$917$	495415.	0.0194557
$918$	0	0
$919$	4.08635e7	1.59605	0.798026	−	0.602623i	$-0.205878\pi$
$919$	4.08635e7	1.59605	0.798026	+	0.602623i	$0.205878\pi$
$920$	−4.21777e7	−1.64291
$921$	0	0
$922$	1.84976e7	0.716620
$923$	−3.95933e7	−1.52974
$924$	0	0
$925$	−2.27345e7	−0.873639
$926$	6.44072e6	0.246835
$927$	0	0
$928$	−1.28371e7	−0.489326
$929$	1.44490e7	0.549287	0.274644	−	0.961546i	$-0.411440\pi$
$929$	1.44490e7	0.549287	0.274644	+	0.961546i	$0.411440\pi$
$930$	0	0
$931$	1.84120e7	0.696189
$932$	3.65253e6	0.137738
$933$	0	0
$934$	1.43092e7	0.536722
$935$	2.54723e6	0.0952880
$936$	0	0
$937$	−9.49392e6	−0.353262	−0.176631	−	0.984277i	$-0.556520\pi$
$937$	−9.49392e6	−0.353262	−0.176631	+	0.984277i	$0.556520\pi$
$938$	6.47496e6	0.240287
$939$	0	0
$940$	−2.17582e7	−0.803162
$941$	−4.83451e6	−0.177983	−0.0889915	−	0.996032i	$-0.528364\pi$
$941$	−4.83451e6	−0.177983	−0.0889915	+	0.996032i	$0.528364\pi$
$942$	0	0
$943$	8.54625e6	0.312965
$944$	1.32969e6	0.0485647
$945$	0	0
$946$	−286876.	−0.0104224
$947$	2.50321e7	0.907030	0.453515	−	0.891249i	$-0.350170\pi$
$947$	2.50321e7	0.907030	0.453515	+	0.891249i	$0.350170\pi$
$948$	0	0
$949$	2.66910e7	0.962052
$950$	1.31678e7	0.473374
$951$	0	0
$952$	1.12562e7	0.402533
$953$	1.73815e7	0.619949	0.309975	−	0.950745i	$-0.399679\pi$
$953$	1.73815e7	0.619949	0.309975	+	0.950745i	$0.399679\pi$
$954$	0	0
$955$	−4.86145e7	−1.72488
$956$	−2.35588e7	−0.833696
$957$	0	0
$958$	−1.28118e6	−0.0451021
$959$	1.89071e7	0.663864
$960$	0	0
$961$	2.26878e7	0.792472
$962$	9.72265e6	0.338725
$963$	0	0
$964$	−8.45204e6	−0.292933
$965$	2.83268e6	0.0979216
$966$	0	0
$967$	−5.09719e6	−0.175293	−0.0876465	−	0.996152i	$-0.527935\pi$
$967$	−5.09719e6	−0.175293	−0.0876465	+	0.996152i	$0.527935\pi$
$968$	−2.45261e7	−0.841279
$969$	0	0
$970$	−1.06407e7	−0.363112
$971$	−393529.	−0.0133946	−0.00669729	−	0.999978i	$-0.502132\pi$
$971$	−393529.	−0.0133946	−0.00669729	+	0.999978i	$0.502132\pi$
$972$	0	0
$973$	2.21644e6	0.0750541
$974$	8.73778e6	0.295124
$975$	0	0
$976$	51265.2	0.00172266
$977$	5.44154e7	1.82384	0.911918	−	0.410373i	$-0.134602\pi$
$977$	5.44154e7	1.82384	0.911918	+	0.410373i	$0.134602\pi$
$978$	0	0
$979$	1.70799e6	0.0569545
$980$	2.99274e7	0.995413
$981$	0	0
$982$	−3.02301e6	−0.100037
$983$	6.93627e6	0.228951	0.114475	−	0.993426i	$-0.463481\pi$
$983$	6.93627e6	0.228951	0.114475	+	0.993426i	$0.463481\pi$
$984$	0	0
$985$	−4.29061e6	−0.140906
$986$	8.90582e6	0.291730
$987$	0	0
$988$	1.92842e7	0.628506
$989$	1.76465e7	0.573679
$990$	0	0
$991$	2.32589e7	0.752326	0.376163	−	0.926554i	$-0.377243\pi$
$991$	2.32589e7	0.752326	0.376163	+	0.926554i	$0.377243\pi$
$992$	−4.23556e7	−1.36657
$993$	0	0
$994$	−8.41164e6	−0.270032
$995$	2.44780e7	0.783824
$996$	0	0
$997$	−4.77014e7	−1.51982	−0.759912	−	0.650026i	$-0.774758\pi$
$997$	−4.77014e7	−1.51982	−0.759912	+	0.650026i	$0.774758\pi$
$998$	−2.35725e6	−0.0749168
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.68934 q^{2} -24.7675 q^{4} +83.5049 q^{5} -48.3401 q^{7} +152.667 q^{8} +O(q^{10})\)	\(q-2.68934 q^{2} -24.7675 q^{4} +83.5049 q^{5} -48.3401 q^{7} +152.667 q^{8} -224.573 q^{10} -19.9993 q^{11} +611.920 q^{13} +130.003 q^{14} +381.985 q^{16} -1525.25 q^{17} -1272.41 q^{19} -2068.20 q^{20} +53.7850 q^{22} -3308.46 q^{23} +3848.06 q^{25} -1645.66 q^{26} +1197.26 q^{28} +2171.14 q^{29} +7163.58 q^{31} -5912.63 q^{32} +4101.91 q^{34} -4036.64 q^{35} -5908.05 q^{37} +3421.93 q^{38} +12748.4 q^{40} -2583.15 q^{41} -5333.76 q^{43} +495.332 q^{44} +8897.58 q^{46} +10520.3 q^{47} -14470.2 q^{49} -10348.7 q^{50} -15155.7 q^{52} -6041.06 q^{53} -1670.04 q^{55} -7379.94 q^{56} -5838.93 q^{58} +3481.00 q^{59} +134.207 q^{61} -19265.3 q^{62} +3677.55 q^{64} +51098.3 q^{65} +49806.2 q^{67} +37776.5 q^{68} +10855.9 q^{70} -64703.4 q^{71} +43618.4 q^{73} +15888.8 q^{74} +31514.3 q^{76} +966.770 q^{77} +74841.5 q^{79} +31897.6 q^{80} +6946.96 q^{82} -5270.76 q^{83} -127366. q^{85} +14344.3 q^{86} -3053.23 q^{88} -85402.3 q^{89} -29580.3 q^{91} +81942.2 q^{92} -28292.8 q^{94} -106252. q^{95} +47381.9 q^{97} +38915.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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