LMFDB - Embedded newform 585.6.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [585,6,Mod(1,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	585.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,134] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$93.8245345906$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.75663$ of defining polynomial
Character	$\chi$	$=$	585.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.75663 q^{2} -24.4010 q^{4} +25.0000 q^{5} +85.7300 q^{7} -155.477 q^{8} +68.9158 q^{10} -431.046 q^{11} +169.000 q^{13} +236.326 q^{14} +352.239 q^{16} +438.894 q^{17} +1617.28 q^{19} -610.025 q^{20} -1188.23 q^{22} -2173.44 q^{23} +625.000 q^{25} +465.871 q^{26} -2091.90 q^{28} -8289.72 q^{29} -2745.88 q^{31} +5946.25 q^{32} +1209.87 q^{34} +2143.25 q^{35} -2137.03 q^{37} +4458.24 q^{38} -3886.92 q^{40} +19520.9 q^{41} +8153.29 q^{43} +10517.9 q^{44} -5991.39 q^{46} +13235.5 q^{47} -9457.36 q^{49} +1722.89 q^{50} -4123.77 q^{52} -1753.17 q^{53} -10776.1 q^{55} -13329.0 q^{56} -22851.7 q^{58} +1976.46 q^{59} +45578.3 q^{61} -7569.39 q^{62} +5119.96 q^{64} +4225.00 q^{65} -19457.7 q^{67} -10709.5 q^{68} +5908.15 q^{70} +64224.9 q^{71} +1029.22 q^{73} -5891.00 q^{74} -39463.2 q^{76} -36953.6 q^{77} -107661. q^{79} +8805.99 q^{80} +53812.1 q^{82} +46473.4 q^{83} +10972.4 q^{85} +22475.6 q^{86} +67017.6 q^{88} +3410.51 q^{89} +14488.4 q^{91} +53034.2 q^{92} +36485.3 q^{94} +40431.9 q^{95} +133264. q^{97} -26070.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) $ 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

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 $

$2$	2.75663	0.487308	0.243654	−	0.969862i	$-0.421654\pi$
$2$	2.75663	0.487308	0.243654	+	0.969862i	$0.421654\pi$
$3$	0	0
$3$	0	0
$4$	−24.4010	−0.762531
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	85.7300	0.661284	0.330642	−	0.943756i	$-0.392735\pi$
$7$	85.7300	0.661284	0.330642	+	0.943756i	$0.392735\pi$
$8$	−155.477	−0.858896
$9$	0	0
$10$	68.9158	0.217931
$11$	−431.046	−1.07409	−0.537046	−	0.843553i	$-0.680460\pi$
$11$	−431.046	−1.07409	−0.537046	+	0.843553i	$0.680460\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	236.326	0.322249
$15$	0	0
$16$	352.239	0.343984
$17$	438.894	0.368331	0.184165	−	0.982895i	$-0.441042\pi$
$17$	438.894	0.368331	0.184165	+	0.982895i	$0.441042\pi$
$18$	0	0
$19$	1617.28	1.02778	0.513891	−	0.857856i	$-0.328204\pi$
$19$	1617.28	1.02778	0.513891	+	0.857856i	$0.328204\pi$
$20$	−610.025	−0.341014
$21$	0	0
$22$	−1188.23	−0.523414
$23$	−2173.44	−0.856700	−0.428350	−	0.903613i	$-0.640905\pi$
$23$	−2173.44	−0.856700	−0.428350	+	0.903613i	$0.640905\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	465.871	0.135155
$27$	0	0
$28$	−2091.90	−0.504249
$29$	−8289.72	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$29$	−8289.72	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$30$	0	0
$31$	−2745.88	−0.513190	−0.256595	−	0.966519i	$-0.582601\pi$
$31$	−2745.88	−0.513190	−0.256595	+	0.966519i	$0.582601\pi$
$32$	5946.25	1.02652
$33$	0	0
$34$	1209.87	0.179491
$35$	2143.25	0.295735
$36$	0	0
$37$	−2137.03	−0.256629	−0.128315	−	0.991734i	$-0.540957\pi$
$37$	−2137.03	−0.256629	−0.128315	+	0.991734i	$0.540957\pi$
$38$	4458.24	0.500846
$39$	0	0
$40$	−3886.92	−0.384110
$41$	19520.9	1.81360	0.906799	−	0.421562i	$-0.138518\pi$
$41$	19520.9	1.81360	0.906799	+	0.421562i	$0.138518\pi$
$42$	0	0
$43$	8153.29	0.672452	0.336226	−	0.941781i	$-0.390849\pi$
$43$	8153.29	0.672452	0.336226	+	0.941781i	$0.390849\pi$
$44$	10517.9	0.819029
$45$	0	0
$46$	−5991.39	−0.417477
$47$	13235.5	0.873966	0.436983	−	0.899470i	$-0.356047\pi$
$47$	13235.5	0.873966	0.436983	+	0.899470i	$0.356047\pi$
$48$	0	0
$49$	−9457.36	−0.562704
$50$	1722.89	0.0974616
$51$	0	0
$52$	−4123.77	−0.211488
$53$	−1753.17	−0.0857303	−0.0428651	−	0.999081i	$-0.513649\pi$
$53$	−1753.17	−0.0857303	−0.0428651	+	0.999081i	$0.513649\pi$
$54$	0	0
$55$	−10776.1	−0.480349
$56$	−13329.0	−0.567974
$57$	0	0
$58$	−22851.7	−0.891967
$59$	1976.46	0.0739192	0.0369596	−	0.999317i	$-0.488233\pi$
$59$	1976.46	0.0739192	0.0369596	+	0.999317i	$0.488233\pi$
$60$	0	0
$61$	45578.3	1.56832	0.784158	−	0.620561i	$-0.213095\pi$
$61$	45578.3	1.56832	0.784158	+	0.620561i	$0.213095\pi$
$62$	−7569.39	−0.250082
$63$	0	0
$64$	5119.96	0.156249
$65$	4225.00	0.124035
$66$	0	0
$67$	−19457.7	−0.529546	−0.264773	−	0.964311i	$-0.585297\pi$
$67$	−19457.7	−0.529546	−0.264773	+	0.964311i	$0.585297\pi$
$68$	−10709.5	−0.280863
$69$	0	0
$70$	5908.15	0.144114
$71$	64224.9	1.51202	0.756010	−	0.654561i	$-0.227146\pi$
$71$	64224.9	1.51202	0.756010	+	0.654561i	$0.227146\pi$
$72$	0	0
$73$	1029.22	0.0226048	0.0113024	−	0.999936i	$-0.496402\pi$
$73$	1029.22	0.0226048	0.0113024	+	0.999936i	$0.496402\pi$
$74$	−5891.00	−0.125058
$75$	0	0
$76$	−39463.2	−0.783715
$77$	−36953.6	−0.710280
$78$	0	0
$79$	−107661.	−1.94085	−0.970424	−	0.241407i	$-0.922391\pi$
$79$	−107661.	−1.94085	−0.970424	+	0.241407i	$0.922391\pi$
$80$	8805.99	0.153834
$81$	0	0
$82$	53812.1	0.883782
$83$	46473.4	0.740473	0.370237	−	0.928938i	$-0.379277\pi$
$83$	46473.4	0.740473	0.370237	+	0.928938i	$0.379277\pi$
$84$	0	0
$85$	10972.4	0.164722
$86$	22475.6	0.327692
$87$	0	0
$88$	67017.6	0.922534
$89$	3410.51	0.0456398	0.0228199	−	0.999740i	$-0.492736\pi$
$89$	3410.51	0.0456398	0.0228199	+	0.999740i	$0.492736\pi$
$90$	0	0
$91$	14488.4	0.183407
$92$	53034.2	0.653260
$93$	0	0
$94$	36485.3	0.425891
$95$	40431.9	0.459638
$96$	0	0
$97$	133264.	1.43808	0.719041	−	0.694967i	$-0.244581\pi$
$97$	133264.	1.43808	0.719041	+	0.694967i	$0.244581\pi$
$98$	−26070.5	−0.274210
$99$	0	0
$100$	−15250.6	−0.152506
$101$	−112628.	−1.09860	−0.549302	−	0.835624i	$-0.685106\pi$
$101$	−112628.	−1.09860	−0.549302	+	0.835624i	$0.685106\pi$
$102$	0	0
$103$	102456.	0.951581	0.475791	−	0.879559i	$-0.342162\pi$
$103$	102456.	0.951581	0.475791	+	0.879559i	$0.342162\pi$
$104$	−26275.6	−0.238215
$105$	0	0
$106$	−4832.84	−0.0417771
$107$	90417.9	0.763475	0.381738	−	0.924271i	$-0.375326\pi$
$107$	90417.9	0.763475	0.381738	+	0.924271i	$0.375326\pi$
$108$	0	0
$109$	164916.	1.32953	0.664764	−	0.747053i	$-0.268532\pi$
$109$	164916.	1.32953	0.664764	+	0.747053i	$0.268532\pi$
$110$	−29705.9	−0.234078
$111$	0	0
$112$	30197.5	0.227471
$113$	36453.6	0.268562	0.134281	−	0.990943i	$-0.457128\pi$
$113$	36453.6	0.268562	0.134281	+	0.990943i	$0.457128\pi$
$114$	0	0
$115$	−54336.1	−0.383128
$116$	202277.	1.39573
$117$	0	0
$118$	5448.36	0.0360214
$119$	37626.4	0.243571
$120$	0	0
$121$	24749.6	0.153676
$122$	125643.	0.764253
$123$	0	0
$124$	67002.3	0.391323
$125$	15625.0	0.0894427
$126$	0	0
$127$	178579.	0.982476	0.491238	−	0.871025i	$-0.336544\pi$
$127$	178579.	0.982476	0.491238	+	0.871025i	$0.336544\pi$
$128$	−176166.	−0.950381
$129$	0	0
$130$	11646.8	0.0604431
$131$	−43555.7	−0.221752	−0.110876	−	0.993834i	$-0.535366\pi$
$131$	−43555.7	−0.221752	−0.110876	+	0.993834i	$0.535366\pi$
$132$	0	0
$133$	138649.	0.679655
$134$	−53637.6	−0.258052
$135$	0	0
$136$	−68237.9	−0.316358
$137$	347518.	1.58189	0.790944	−	0.611888i	$-0.209590\pi$
$137$	347518.	1.58189	0.790944	+	0.611888i	$0.209590\pi$
$138$	0	0
$139$	54594.1	0.239667	0.119834	−	0.992794i	$-0.461764\pi$
$139$	54594.1	0.239667	0.119834	+	0.992794i	$0.461764\pi$
$140$	−52297.4	−0.225507
$141$	0	0
$142$	177044.	0.736819
$143$	−72846.8	−0.297900
$144$	0	0
$145$	−207243.	−0.818578
$146$	2837.18	0.0110155
$147$	0	0
$148$	52145.6	0.195688
$149$	249528.	0.920777	0.460388	−	0.887718i	$-0.347710\pi$
$149$	249528.	0.920777	0.460388	+	0.887718i	$0.347710\pi$
$150$	0	0
$151$	398788.	1.42331	0.711655	−	0.702529i	$-0.247946\pi$
$151$	398788.	1.42331	0.711655	+	0.702529i	$0.247946\pi$
$152$	−251449.	−0.882757
$153$	0	0
$154$	−101867.	−0.346125
$155$	−68647.1	−0.229505
$156$	0	0
$157$	432099.	1.39905	0.699527	−	0.714606i	$-0.253394\pi$
$157$	432099.	1.39905	0.699527	+	0.714606i	$0.253394\pi$
$158$	−296782.	−0.945791
$159$	0	0
$160$	148656.	0.459075
$161$	−186329.	−0.566522
$162$	0	0
$163$	56309.9	0.166003	0.0830014	−	0.996549i	$-0.473549\pi$
$163$	56309.9	0.166003	0.0830014	+	0.996549i	$0.473549\pi$
$164$	−476330.	−1.38292
$165$	0	0
$166$	128110.	0.360839
$167$	512611.	1.42232	0.711160	−	0.703031i	$-0.248170\pi$
$167$	512611.	1.42232	0.711160	+	0.703031i	$0.248170\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	30246.8	0.0802706
$171$	0	0
$172$	−198948.	−0.512766
$173$	363030.	0.922204	0.461102	−	0.887347i	$-0.347454\pi$
$173$	363030.	0.922204	0.461102	+	0.887347i	$0.347454\pi$
$174$	0	0
$175$	53581.3	0.132257
$176$	−151831.	−0.369471
$177$	0	0
$178$	9401.51	0.0222406
$179$	812489.	1.89533	0.947665	−	0.319265i	$-0.103436\pi$
$179$	812489.	1.89533	0.947665	+	0.319265i	$0.103436\pi$
$180$	0	0
$181$	−464974.	−1.05495	−0.527475	−	0.849570i	$-0.676861\pi$
$181$	−464974.	−1.05495	−0.527475	+	0.849570i	$0.676861\pi$
$182$	39939.1	0.0893758
$183$	0	0
$184$	337920.	0.735816
$185$	−53425.7	−0.114768
$186$	0	0
$187$	−189184.	−0.395621
$188$	−322959.	−0.666426
$189$	0	0
$190$	111456.	0.223985
$191$	15537.6	0.0308177	0.0154088	−	0.999881i	$-0.495095\pi$
$191$	15537.6	0.0308177	0.0154088	+	0.999881i	$0.495095\pi$
$192$	0	0
$193$	−319408.	−0.617238	−0.308619	−	0.951186i	$-0.599867\pi$
$193$	−319408.	−0.617238	−0.308619	+	0.951186i	$0.599867\pi$
$194$	367360.	0.700789
$195$	0	0
$196$	230769.	0.429079
$197$	−635878.	−1.16737	−0.583685	−	0.811980i	$-0.698390\pi$
$197$	−635878.	−1.16737	−0.583685	+	0.811980i	$0.698390\pi$
$198$	0	0
$199$	57426.3	0.102797	0.0513983	−	0.998678i	$-0.483632\pi$
$199$	57426.3	0.102797	0.0513983	+	0.998678i	$0.483632\pi$
$200$	−97173.0	−0.171779
$201$	0	0
$202$	−310473.	−0.535359
$203$	−710678.	−1.21041
$204$	0	0
$205$	488024.	0.811066
$206$	282434.	0.463713
$207$	0	0
$208$	59528.5	0.0954039
$209$	−697121.	−1.10393
$210$	0	0
$211$	−407094.	−0.629489	−0.314745	−	0.949176i	$-0.601919\pi$
$211$	−407094.	−0.629489	−0.314745	+	0.949176i	$0.601919\pi$
$212$	42779.1	0.0653720
$213$	0	0
$214$	249249.	0.372048
$215$	203832.	0.300730
$216$	0	0
$217$	−235405.	−0.339364
$218$	454614.	0.647890
$219$	0	0
$220$	262949.	0.366281
$221$	74173.2	0.102157
$222$	0	0
$223$	94338.9	0.127037	0.0635183	−	0.997981i	$-0.479768\pi$
$223$	94338.9	0.127037	0.0635183	+	0.997981i	$0.479768\pi$
$224$	509772.	0.678822
$225$	0	0
$226$	100489.	0.130872
$227$	−251896.	−0.324456	−0.162228	−	0.986753i	$-0.551868\pi$
$227$	−251896.	−0.324456	−0.162228	+	0.986753i	$0.551868\pi$
$228$	0	0
$229$	−428033.	−0.539372	−0.269686	−	0.962948i	$-0.586920\pi$
$229$	−428033.	−0.539372	−0.269686	+	0.962948i	$0.586920\pi$
$230$	−149785.	−0.186701
$231$	0	0
$232$	1.28886e6	1.57212
$233$	−47973.8	−0.0578914	−0.0289457	−	0.999581i	$-0.509215\pi$
$233$	−47973.8	−0.0578914	−0.0289457	+	0.999581i	$0.509215\pi$
$234$	0	0
$235$	330887.	0.390850
$236$	−48227.5	−0.0563657
$237$	0	0
$238$	103722.	0.118694
$239$	1.39917e6	1.58444	0.792220	−	0.610236i	$-0.208925\pi$
$239$	1.39917e6	1.58444	0.792220	+	0.610236i	$0.208925\pi$
$240$	0	0
$241$	−576792.	−0.639700	−0.319850	−	0.947468i	$-0.603633\pi$
$241$	−576792.	−0.639700	−0.319850	+	0.947468i	$0.603633\pi$
$242$	68225.6	0.0748875
$243$	0	0
$244$	−1.11216e6	−1.19589
$245$	−236434.	−0.251649
$246$	0	0
$247$	273320.	0.285055
$248$	426921.	0.440776
$249$	0	0
$250$	43072.4	0.0435862
$251$	−676974.	−0.678247	−0.339124	−	0.940742i	$-0.610130\pi$
$251$	−676974.	−0.678247	−0.339124	+	0.940742i	$0.610130\pi$
$252$	0	0
$253$	936855.	0.920176
$254$	492278.	0.478769
$255$	0	0
$256$	−649464.	−0.619377
$257$	2.05124e6	1.93724	0.968620	−	0.248545i	$-0.0799525\pi$
$257$	2.05124e6	1.93724	0.968620	+	0.248545i	$0.0799525\pi$
$258$	0	0
$259$	−183208.	−0.169705
$260$	−103094.	−0.0945803
$261$	0	0
$262$	−120067.	−0.108061
$263$	−1.51723e6	−1.35258	−0.676289	−	0.736636i	$-0.736413\pi$
$263$	−1.51723e6	−1.35258	−0.676289	+	0.736636i	$0.736413\pi$
$264$	0	0
$265$	−43829.2	−0.0383397
$266$	382205.	0.331201
$267$	0	0
$268$	474786.	0.403795
$269$	−180817.	−0.152355	−0.0761777	−	0.997094i	$-0.524272\pi$
$269$	−180817.	−0.152355	−0.0761777	+	0.997094i	$0.524272\pi$
$270$	0	0
$271$	−1.58203e6	−1.30856	−0.654278	−	0.756254i	$-0.727028\pi$
$271$	−1.58203e6	−1.30856	−0.654278	+	0.756254i	$0.727028\pi$
$272$	154596.	0.126700
$273$	0	0
$274$	957979.	0.770867
$275$	−269404.	−0.214819
$276$	0	0
$277$	−1.53514e6	−1.20212	−0.601062	−	0.799202i	$-0.705256\pi$
$277$	−1.53514e6	−1.20212	−0.601062	+	0.799202i	$0.705256\pi$
$278$	150496.	0.116792
$279$	0	0
$280$	−333226.	−0.254006
$281$	−959201.	−0.724676	−0.362338	−	0.932047i	$-0.618021\pi$
$281$	−959201.	−0.724676	−0.362338	+	0.932047i	$0.618021\pi$
$282$	0	0
$283$	2.63997e6	1.95945	0.979724	−	0.200353i	$-0.0642089\pi$
$283$	2.63997e6	1.95945	0.979724	+	0.200353i	$0.0642089\pi$
$284$	−1.56715e6	−1.15296
$285$	0	0
$286$	−200812.	−0.145169
$287$	1.67353e6	1.19930
$288$	0	0
$289$	−1.22723e6	−0.864333
$290$	−571293.	−0.398900
$291$	0	0
$292$	−25114.0	−0.0172369
$293$	−286709.	−0.195106	−0.0975532	−	0.995230i	$-0.531102\pi$
$293$	−286709.	−0.195106	−0.0975532	+	0.995230i	$0.531102\pi$
$294$	0	0
$295$	49411.4	0.0330577
$296$	332258.	0.220418
$297$	0	0
$298$	687858.	0.448702
$299$	−367312.	−0.237606
$300$	0	0
$301$	698981.	0.444682
$302$	1.09931e6	0.693591
$303$	0	0
$304$	569669.	0.353540
$305$	1.13946e6	0.701372
$306$	0	0
$307$	1.89896e6	1.14993	0.574963	−	0.818179i	$-0.305016\pi$
$307$	1.89896e6	1.14993	0.574963	+	0.818179i	$0.305016\pi$
$308$	901704.	0.541610
$309$	0	0
$310$	−189235.	−0.111840
$311$	−1.74416e6	−1.02255	−0.511277	−	0.859416i	$-0.670827\pi$
$311$	−1.74416e6	−1.02255	−0.511277	+	0.859416i	$0.670827\pi$
$312$	0	0
$313$	605507.	0.349348	0.174674	−	0.984626i	$-0.444113\pi$
$313$	605507.	0.349348	0.174674	+	0.984626i	$0.444113\pi$
$314$	1.19114e6	0.681770
$315$	0	0
$316$	2.62704e6	1.47996
$317$	2.19697e6	1.22793	0.613967	−	0.789332i	$-0.289573\pi$
$317$	2.19697e6	1.22793	0.613967	+	0.789332i	$0.289573\pi$
$318$	0	0
$319$	3.57325e6	1.96601
$320$	127999.	0.0698765
$321$	0	0
$322$	−513642.	−0.276071
$323$	709814.	0.378563
$324$	0	0
$325$	105625.	0.0554700
$326$	155226.	0.0808945
$327$	0	0
$328$	−3.03505e6	−1.55769
$329$	1.13468e6	0.577940
$330$	0	0
$331$	−3.68205e6	−1.84722	−0.923612	−	0.383329i	$-0.874777\pi$
$331$	−3.68205e6	−1.84722	−0.923612	+	0.383329i	$0.874777\pi$
$332$	−1.13400e6	−0.564633
$333$	0	0
$334$	1.41308e6	0.693108
$335$	−486442.	−0.236820
$336$	0	0
$337$	−3.46729e6	−1.66309	−0.831543	−	0.555460i	$-0.812542\pi$
$337$	−3.46729e6	−1.66309	−0.831543	+	0.555460i	$0.812542\pi$
$338$	78732.2	0.0374852
$339$	0	0
$340$	−267736.	−0.125606
$341$	1.18360e6	0.551214
$342$	0	0
$343$	−2.25164e6	−1.03339
$344$	−1.26765e6	−0.577566
$345$	0	0
$346$	1.00074e6	0.449398
$347$	−3.91532e6	−1.74559	−0.872797	−	0.488083i	$-0.837696\pi$
$347$	−3.91532e6	−1.74559	−0.872797	+	0.488083i	$0.837696\pi$
$348$	0	0
$349$	−3.32051e6	−1.45929	−0.729644	−	0.683827i	$-0.760314\pi$
$349$	−3.32051e6	−1.45929	−0.729644	+	0.683827i	$0.760314\pi$
$350$	147704.	0.0644498
$351$	0	0
$352$	−2.56311e6	−1.10258
$353$	−3.09575e6	−1.32230	−0.661148	−	0.750255i	$-0.729931\pi$
$353$	−3.09575e6	−1.32230	−0.661148	+	0.750255i	$0.729931\pi$
$354$	0	0
$355$	1.60562e6	0.676196
$356$	−83219.7	−0.0348017
$357$	0	0
$358$	2.23973e6	0.923610
$359$	2.71429e6	1.11153	0.555764	−	0.831340i	$-0.312426\pi$
$359$	2.71429e6	1.11153	0.555764	+	0.831340i	$0.312426\pi$
$360$	0	0
$361$	139488.	0.0563340
$362$	−1.28176e6	−0.514086
$363$	0	0
$364$	−353531.	−0.139854
$365$	25730.5	0.0101092
$366$	0	0
$367$	2.54964e6	0.988130	0.494065	−	0.869425i	$-0.335511\pi$
$367$	2.54964e6	0.988130	0.494065	+	0.869425i	$0.335511\pi$
$368$	−765573.	−0.294691
$369$	0	0
$370$	−147275.	−0.0559274
$371$	−150299.	−0.0566920
$372$	0	0
$373$	3.67946e6	1.36934	0.684671	−	0.728852i	$-0.259946\pi$
$373$	3.67946e6	1.36934	0.684671	+	0.728852i	$0.259946\pi$
$374$	−521510.	−0.192790
$375$	0	0
$376$	−2.05781e6	−0.750646
$377$	−1.40096e6	−0.507660
$378$	0	0
$379$	−1.06822e6	−0.382000	−0.191000	−	0.981590i	$-0.561173\pi$
$379$	−1.06822e6	−0.382000	−0.191000	+	0.981590i	$0.561173\pi$
$380$	−986579.	−0.350488
$381$	0	0
$382$	42831.4	0.0150177
$383$	601784.	0.209625	0.104813	−	0.994492i	$-0.466576\pi$
$383$	601784.	0.209625	0.104813	+	0.994492i	$0.466576\pi$
$384$	0	0
$385$	−923839.	−0.317647
$386$	−880490.	−0.300785
$387$	0	0
$388$	−3.25178e6	−1.09658
$389$	−5.00396e6	−1.67664	−0.838319	−	0.545180i	$-0.816461\pi$
$389$	−5.00396e6	−1.67664	−0.838319	+	0.545180i	$0.816461\pi$
$390$	0	0
$391$	−953913.	−0.315549
$392$	1.47040e6	0.483304
$393$	0	0
$394$	−1.75288e6	−0.568869
$395$	−2.69153e6	−0.867973
$396$	0	0
$397$	3.47583e6	1.10683	0.553417	−	0.832904i	$-0.313323\pi$
$397$	3.47583e6	1.10683	0.553417	+	0.832904i	$0.313323\pi$
$398$	158303.	0.0500936
$399$	0	0
$400$	220150.	0.0687968
$401$	5.15091e6	1.59964	0.799822	−	0.600238i	$-0.204927\pi$
$401$	5.15091e6	1.59964	0.799822	+	0.600238i	$0.204927\pi$
$402$	0	0
$403$	−464054.	−0.142333
$404$	2.74822e6	0.837719
$405$	0	0
$406$	−1.95908e6	−0.589843
$407$	921158.	0.275644
$408$	0	0
$409$	−1.11646e6	−0.330015	−0.165008	−	0.986292i	$-0.552765\pi$
$409$	−1.11646e6	−0.330015	−0.165008	+	0.986292i	$0.552765\pi$
$410$	1.34530e6	0.395239
$411$	0	0
$412$	−2.50004e6	−0.725610
$413$	169442.	0.0488816
$414$	0	0
$415$	1.16184e6	0.331150
$416$	1.00492e6	0.284706
$417$	0	0
$418$	−1.92171e6	−0.537955
$419$	−1.79740e6	−0.500160	−0.250080	−	0.968225i	$-0.580457\pi$
$419$	−1.79740e6	−0.500160	−0.250080	+	0.968225i	$0.580457\pi$
$420$	0	0
$421$	5.31302e6	1.46095	0.730477	−	0.682937i	$-0.239298\pi$
$421$	5.31302e6	1.46095	0.730477	+	0.682937i	$0.239298\pi$
$422$	−1.12221e6	−0.306755
$423$	0	0
$424$	272577.	0.0736334
$425$	274309.	0.0736661
$426$	0	0
$427$	3.90743e6	1.03710
$428$	−2.20629e6	−0.582173
$429$	0	0
$430$	561890.	0.146548
$431$	4.63630e6	1.20221	0.601103	−	0.799172i	$-0.294728\pi$
$431$	4.63630e6	1.20221	0.601103	+	0.799172i	$0.294728\pi$
$432$	0	0
$433$	−3.68664e6	−0.944955	−0.472478	−	0.881343i	$-0.656640\pi$
$433$	−3.68664e6	−0.944955	−0.472478	+	0.881343i	$0.656640\pi$
$434$	−648924.	−0.165375
$435$	0	0
$436$	−4.02412e6	−1.01381
$437$	−3.51506e6	−0.880501
$438$	0	0
$439$	−1.67032e6	−0.413656	−0.206828	−	0.978377i	$-0.566314\pi$
$439$	−1.67032e6	−0.413656	−0.206828	+	0.978377i	$0.566314\pi$
$440$	1.67544e6	0.412570
$441$	0	0
$442$	204468.	0.0497817
$443$	−3.59209e6	−0.869637	−0.434819	−	0.900518i	$-0.643187\pi$
$443$	−3.59209e6	−0.869637	−0.434819	+	0.900518i	$0.643187\pi$
$444$	0	0
$445$	85262.6	0.0204107
$446$	260058.	0.0619060
$447$	0	0
$448$	438934.	0.103325
$449$	−80455.2	−0.0188338	−0.00941690	−	0.999956i	$-0.502998\pi$
$449$	−80455.2	−0.0188338	−0.00941690	+	0.999956i	$0.502998\pi$
$450$	0	0
$451$	−8.41443e6	−1.94797
$452$	−889503.	−0.204786
$453$	0	0
$454$	−694384.	−0.158110
$455$	362209.	0.0820221
$456$	0	0
$457$	2.87687e6	0.644362	0.322181	−	0.946678i	$-0.395584\pi$
$457$	2.87687e6	0.644362	0.322181	+	0.946678i	$0.395584\pi$
$458$	−1.17993e6	−0.262840
$459$	0	0
$460$	1.32585e6	0.292147
$461$	−81644.7	−0.0178927	−0.00894635	−	0.999960i	$-0.502848\pi$
$461$	−81644.7	−0.0178927	−0.00894635	+	0.999960i	$0.502848\pi$
$462$	0	0
$463$	−2.50214e6	−0.542450	−0.271225	−	0.962516i	$-0.587429\pi$
$463$	−2.50214e6	−0.542450	−0.271225	+	0.962516i	$0.587429\pi$
$464$	−2.91997e6	−0.629626
$465$	0	0
$466$	−132246.	−0.0282109
$467$	−1.38773e6	−0.294452	−0.147226	−	0.989103i	$-0.547034\pi$
$467$	−1.38773e6	−0.294452	−0.147226	+	0.989103i	$0.547034\pi$
$468$	0	0
$469$	−1.66811e6	−0.350180
$470$	912133.	0.190464
$471$	0	0
$472$	−307293.	−0.0634889
$473$	−3.51444e6	−0.722276
$474$	0	0
$475$	1.01080e6	0.205556
$476$	−918122.	−0.185730
$477$	0	0
$478$	3.85700e6	0.772111
$479$	4.96374e6	0.988485	0.494243	−	0.869324i	$-0.335445\pi$
$479$	4.96374e6	0.988485	0.494243	+	0.869324i	$0.335445\pi$
$480$	0	0
$481$	−361158.	−0.0711761
$482$	−1.59000e6	−0.311731
$483$	0	0
$484$	−603915.	−0.117182
$485$	3.33160e6	0.643130
$486$	0	0
$487$	8.57371e6	1.63812	0.819061	−	0.573706i	$-0.194495\pi$
$487$	8.57371e6	1.63812	0.819061	+	0.573706i	$0.194495\pi$
$488$	−7.08637e6	−1.34702
$489$	0	0
$490$	−651762.	−0.122631
$491$	−2.55302e6	−0.477915	−0.238958	−	0.971030i	$-0.576806\pi$
$491$	−2.55302e6	−0.477915	−0.238958	+	0.971030i	$0.576806\pi$
$492$	0	0
$493$	−3.63831e6	−0.674191
$494$	753442.	0.138910
$495$	0	0
$496$	−967209.	−0.176529
$497$	5.50600e6	0.999874
$498$	0	0
$499$	161861.	0.0290999	0.0145500	−	0.999894i	$-0.495368\pi$
$499$	161861.	0.0290999	0.0145500	+	0.999894i	$0.495368\pi$
$500$	−381265.	−0.0682028
$501$	0	0
$502$	−1.86617e6	−0.330515
$503$	−5.40474e6	−0.952477	−0.476239	−	0.879316i	$-0.658000\pi$
$503$	−5.40474e6	−0.952477	−0.476239	+	0.879316i	$0.658000\pi$
$504$	0	0
$505$	−2.81569e6	−0.491311
$506$	2.58256e6	0.448409
$507$	0	0
$508$	−4.35751e6	−0.749169
$509$	−1.72999e6	−0.295971	−0.147985	−	0.988990i	$-0.547279\pi$
$509$	−1.72999e6	−0.295971	−0.147985	+	0.988990i	$0.547279\pi$
$510$	0	0
$511$	88235.1	0.0149482
$512$	3.84698e6	0.648553
$513$	0	0
$514$	5.65451e6	0.944033
$515$	2.56141e6	0.425560
$516$	0	0
$517$	−5.70510e6	−0.938721
$518$	−505036.	−0.0826985
$519$	0	0
$520$	−656889.	−0.106533
$521$	−8.15642e6	−1.31645	−0.658226	−	0.752820i	$-0.728693\pi$
$521$	−8.15642e6	−1.31645	−0.658226	+	0.752820i	$0.728693\pi$
$522$	0	0
$523$	−7.57602e6	−1.21112	−0.605559	−	0.795800i	$-0.707051\pi$
$523$	−7.57602e6	−1.21112	−0.605559	+	0.795800i	$0.707051\pi$
$524$	1.06280e6	0.169093
$525$	0	0
$526$	−4.18245e6	−0.659123
$527$	−1.20515e6	−0.189023
$528$	0	0
$529$	−1.71248e6	−0.266064
$530$	−120821.	−0.0186833
$531$	0	0
$532$	−3.38318e6	−0.518258
$533$	3.29904e6	0.503002
$534$	0	0
$535$	2.26045e6	0.341436
$536$	3.02521e6	0.454825
$537$	0	0
$538$	−498445.	−0.0742440
$539$	4.07656e6	0.604396
$540$	0	0
$541$	1.29055e7	1.89575	0.947876	−	0.318640i	$-0.103226\pi$
$541$	1.29055e7	1.89575	0.947876	+	0.318640i	$0.103226\pi$
$542$	−4.36108e6	−0.637670
$543$	0	0
$544$	2.60978e6	0.378099
$545$	4.12291e6	0.594583
$546$	0	0
$547$	−7.61965e6	−1.08885	−0.544423	−	0.838811i	$-0.683252\pi$
$547$	−7.61965e6	−1.08885	−0.544423	+	0.838811i	$0.683252\pi$
$548$	−8.47978e6	−1.20624
$549$	0	0
$550$	−742647.	−0.104683
$551$	−1.34068e7	−1.88125
$552$	0	0
$553$	−9.22980e6	−1.28345
$554$	−4.23182e6	−0.585805
$555$	0	0
$556$	−1.33215e6	−0.182754
$557$	7.35984e6	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$557$	7.35984e6	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$558$	0	0
$559$	1.37791e6	0.186505
$560$	754937.	0.101728
$561$	0	0
$562$	−2.64416e6	−0.353140
$563$	5.68526e6	0.755927	0.377963	−	0.925821i	$-0.376625\pi$
$563$	5.68526e6	0.755927	0.377963	+	0.925821i	$0.376625\pi$
$564$	0	0
$565$	911339.	0.120104
$566$	7.27744e6	0.954855
$567$	0	0
$568$	−9.98547e6	−1.29867
$569$	1.43108e7	1.85304	0.926519	−	0.376249i	$-0.122786\pi$
$569$	1.43108e7	1.85304	0.926519	+	0.376249i	$0.122786\pi$
$570$	0	0
$571$	−1.95049e6	−0.250354	−0.125177	−	0.992134i	$-0.539950\pi$
$571$	−1.95049e6	−0.250354	−0.125177	+	0.992134i	$0.539950\pi$
$572$	1.77753e6	0.227158
$573$	0	0
$574$	4.61331e6	0.584430
$575$	−1.35840e6	−0.171340
$576$	0	0
$577$	−987263.	−0.123451	−0.0617253	−	0.998093i	$-0.519660\pi$
$577$	−987263.	−0.123451	−0.0617253	+	0.998093i	$0.519660\pi$
$578$	−3.38302e6	−0.421196
$579$	0	0
$580$	5.05693e6	0.624191
$581$	3.98417e6	0.489663
$582$	0	0
$583$	755697.	0.0920823
$584$	−160020.	−0.0194152
$585$	0	0
$586$	−790350.	−0.0950770
$587$	−1.11813e6	−0.133936	−0.0669679	−	0.997755i	$-0.521332\pi$
$587$	−1.11813e6	−0.133936	−0.0669679	+	0.997755i	$0.521332\pi$
$588$	0	0
$589$	−4.44086e6	−0.527447
$590$	136209.	0.0161093
$591$	0	0
$592$	−752746.	−0.0882763
$593$	905001.	0.105685	0.0528424	−	0.998603i	$-0.483172\pi$
$593$	905001.	0.105685	0.0528424	+	0.998603i	$0.483172\pi$
$594$	0	0
$595$	940661.	0.108928
$596$	−6.08874e6	−0.702121
$597$	0	0
$598$	−1.01254e6	−0.115787
$599$	5.11859e6	0.582885	0.291443	−	0.956588i	$-0.405865\pi$
$599$	5.11859e6	0.582885	0.291443	+	0.956588i	$0.405865\pi$
$600$	0	0
$601$	8.84991e6	0.999431	0.499716	−	0.866190i	$-0.333438\pi$
$601$	8.84991e6	0.999431	0.499716	+	0.866190i	$0.333438\pi$
$602$	1.92683e6	0.216697
$603$	0	0
$604$	−9.73082e6	−1.08532
$605$	618741.	0.0687259
$606$	0	0
$607$	1.37995e7	1.52017	0.760085	−	0.649824i	$-0.225157\pi$
$607$	1.37995e7	1.52017	0.760085	+	0.649824i	$0.225157\pi$
$608$	9.61674e6	1.05504
$609$	0	0
$610$	3.14107e6	0.341785
$611$	2.23679e6	0.242395
$612$	0	0
$613$	1.55631e7	1.67280	0.836401	−	0.548118i	$-0.184655\pi$
$613$	1.55631e7	1.67280	0.836401	+	0.548118i	$0.184655\pi$
$614$	5.23473e6	0.560368
$615$	0	0
$616$	5.74542e6	0.610057
$617$	−1.57277e7	−1.66323	−0.831615	−	0.555353i	$-0.812583\pi$
$617$	−1.57277e7	−1.66323	−0.831615	+	0.555353i	$0.812583\pi$
$618$	0	0
$619$	−5.68261e6	−0.596103	−0.298051	−	0.954550i	$-0.596337\pi$
$619$	−5.68261e6	−0.596103	−0.298051	+	0.954550i	$0.596337\pi$
$620$	1.67506e6	0.175005
$621$	0	0
$622$	−4.80802e6	−0.498299
$623$	292383.	0.0301809
$624$	0	0
$625$	390625.	0.0400000
$626$	1.66916e6	0.170240
$627$	0	0
$628$	−1.05436e7	−1.06682
$629$	−937930.	−0.0945244
$630$	0	0
$631$	7.92146e6	0.792012	0.396006	−	0.918248i	$-0.370396\pi$
$631$	7.92146e6	0.792012	0.396006	+	0.918248i	$0.370396\pi$
$632$	1.67388e7	1.66699
$633$	0	0
$634$	6.05622e6	0.598382
$635$	4.46449e6	0.439377
$636$	0	0
$637$	−1.59829e6	−0.156066
$638$	9.85014e6	0.958055
$639$	0	0
$640$	−4.40415e6	−0.425023
$641$	−8.63330e6	−0.829911	−0.414956	−	0.909842i	$-0.636203\pi$
$641$	−8.63330e6	−0.829911	−0.414956	+	0.909842i	$0.636203\pi$
$642$	0	0
$643$	5.41901e6	0.516884	0.258442	−	0.966027i	$-0.416791\pi$
$643$	5.41901e6	0.516884	0.258442	+	0.966027i	$0.416791\pi$
$644$	4.54662e6	0.431990
$645$	0	0
$646$	1.95670e6	0.184477
$647$	−2.85189e6	−0.267838	−0.133919	−	0.990992i	$-0.542756\pi$
$647$	−2.85189e6	−0.267838	−0.133919	+	0.990992i	$0.542756\pi$
$648$	0	0
$649$	−851944.	−0.0793961
$650$	291169.	0.0270310
$651$	0	0
$652$	−1.37402e6	−0.126582
$653$	−1.31018e7	−1.20239	−0.601197	−	0.799101i	$-0.705309\pi$
$653$	−1.31018e7	−1.20239	−0.601197	+	0.799101i	$0.705309\pi$
$654$	0	0
$655$	−1.08889e6	−0.0991704
$656$	6.87605e6	0.623849
$657$	0	0
$658$	3.12789e6	0.281635
$659$	987681.	0.0885937	0.0442969	−	0.999018i	$-0.485895\pi$
$659$	987681.	0.0885937	0.0442969	+	0.999018i	$0.485895\pi$
$660$	0	0
$661$	2.30367e6	0.205077	0.102538	−	0.994729i	$-0.467304\pi$
$661$	2.30367e6	0.205077	0.102538	+	0.994729i	$0.467304\pi$
$662$	−1.01500e7	−0.900167
$663$	0	0
$664$	−7.22553e6	−0.635989
$665$	3.46623e6	0.303951
$666$	0	0
$667$	1.80172e7	1.56810
$668$	−1.25082e7	−1.08456
$669$	0	0
$670$	−1.34094e6	−0.115404
$671$	−1.96464e7	−1.68452
$672$	0	0
$673$	−3.89370e6	−0.331379	−0.165689	−	0.986178i	$-0.552985\pi$
$673$	−3.89370e6	−0.331379	−0.165689	+	0.986178i	$0.552985\pi$
$674$	−9.55803e6	−0.810436
$675$	0	0
$676$	−696916.	−0.0586562
$677$	−2.83801e6	−0.237981	−0.118991	−	0.992895i	$-0.537966\pi$
$677$	−2.83801e6	−0.237981	−0.118991	+	0.992895i	$0.537966\pi$
$678$	0	0
$679$	1.14247e7	0.950981
$680$	−1.70595e6	−0.141479
$681$	0	0
$682$	3.26276e6	0.268611
$683$	6.98711e6	0.573121	0.286560	−	0.958062i	$-0.407488\pi$
$683$	6.98711e6	0.573121	0.286560	+	0.958062i	$0.407488\pi$
$684$	0	0
$685$	8.68795e6	0.707442
$686$	−6.20695e6	−0.503580
$687$	0	0
$688$	2.87191e6	0.231313
$689$	−296286.	−0.0237773
$690$	0	0
$691$	3.58854e6	0.285905	0.142953	−	0.989730i	$-0.454340\pi$
$691$	3.58854e6	0.285905	0.142953	+	0.989730i	$0.454340\pi$
$692$	−8.85829e6	−0.703209
$693$	0	0
$694$	−1.07931e7	−0.850643
$695$	1.36485e6	0.107182
$696$	0	0
$697$	8.56764e6	0.668004
$698$	−9.15342e6	−0.711123
$699$	0	0
$700$	−1.30744e6	−0.100850
$701$	1.07723e7	0.827971	0.413986	−	0.910283i	$-0.364136\pi$
$701$	1.07723e7	0.827971	0.413986	+	0.910283i	$0.364136\pi$
$702$	0	0
$703$	−3.45617e6	−0.263759
$704$	−2.20694e6	−0.167826
$705$	0	0
$706$	−8.53384e6	−0.644366
$707$	−9.65556e6	−0.726489
$708$	0	0
$709$	1.68465e7	1.25862	0.629309	−	0.777155i	$-0.283338\pi$
$709$	1.68465e7	1.25862	0.629309	+	0.777155i	$0.283338\pi$
$710$	4.42611e6	0.329516
$711$	0	0
$712$	−530254.	−0.0391998
$713$	5.96803e6	0.439650
$714$	0	0
$715$	−1.82117e6	−0.133225
$716$	−1.98255e7	−1.44525
$717$	0	0
$718$	7.48230e6	0.541656
$719$	−7.10379e6	−0.512470	−0.256235	−	0.966615i	$-0.582482\pi$
$719$	−7.10379e6	−0.512470	−0.256235	+	0.966615i	$0.582482\pi$
$720$	0	0
$721$	8.78359e6	0.629265
$722$	384518.	0.0274520
$723$	0	0
$724$	1.13458e7	0.804432
$725$	−5.18108e6	−0.366079
$726$	0	0
$727$	−2.50048e7	−1.75464	−0.877319	−	0.479908i	$-0.840670\pi$
$727$	−2.50048e7	−1.75464	−0.877319	+	0.479908i	$0.840670\pi$
$728$	−2.25260e6	−0.157528
$729$	0	0
$730$	70929.5	0.00492629
$731$	3.57843e6	0.247685
$732$	0	0
$733$	−7.77802e6	−0.534699	−0.267349	−	0.963600i	$-0.586148\pi$
$733$	−7.77802e6	−0.534699	−0.267349	+	0.963600i	$0.586148\pi$
$734$	7.02842e6	0.481524
$735$	0	0
$736$	−1.29238e7	−0.879422
$737$	8.38715e6	0.568782
$738$	0	0
$739$	1.00573e7	0.677439	0.338720	−	0.940887i	$-0.390006\pi$
$739$	1.00573e7	0.677439	0.338720	+	0.940887i	$0.390006\pi$
$740$	1.30364e6	0.0875142
$741$	0	0
$742$	−414320.	−0.0276265
$743$	3.40838e6	0.226504	0.113252	−	0.993566i	$-0.463873\pi$
$743$	3.40838e6	0.226504	0.113252	+	0.993566i	$0.463873\pi$
$744$	0	0
$745$	6.23821e6	0.411784
$746$	1.01429e7	0.667292
$747$	0	0
$748$	4.61627e6	0.301673
$749$	7.75153e6	0.504874
$750$	0	0
$751$	1.57084e7	1.01633	0.508163	−	0.861261i	$-0.330325\pi$
$751$	1.57084e7	1.01633	0.508163	+	0.861261i	$0.330325\pi$
$752$	4.66205e6	0.300630
$753$	0	0
$754$	−3.86194e6	−0.247387
$755$	9.96970e6	0.636524
$756$	0	0
$757$	−2.52846e7	−1.60367	−0.801836	−	0.597544i	$-0.796143\pi$
$757$	−2.52846e7	−1.60367	−0.801836	+	0.597544i	$0.796143\pi$
$758$	−2.94469e6	−0.186152
$759$	0	0
$760$	−6.28623e6	−0.394781
$761$	−1.34808e7	−0.843825	−0.421913	−	0.906636i	$-0.638641\pi$
$761$	−1.34808e7	−0.843825	−0.421913	+	0.906636i	$0.638641\pi$
$762$	0	0
$763$	1.41383e7	0.879195
$764$	−379132.	−0.0234994
$765$	0	0
$766$	1.65890e6	0.102152
$767$	334021.	0.0205015
$768$	0	0
$769$	8.09020e6	0.493337	0.246668	−	0.969100i	$-0.420664\pi$
$769$	8.09020e6	0.493337	0.246668	+	0.969100i	$0.420664\pi$
$770$	−2.54668e6	−0.154792
$771$	0	0
$772$	7.79387e6	0.470663
$773$	−2.75295e6	−0.165710	−0.0828551	−	0.996562i	$-0.526404\pi$
$773$	−2.75295e6	−0.165710	−0.0828551	+	0.996562i	$0.526404\pi$
$774$	0	0
$775$	−1.71618e6	−0.102638
$776$	−2.07195e7	−1.23516
$777$	0	0
$778$	−1.37941e7	−0.817040
$779$	3.15708e7	1.86398
$780$	0	0
$781$	−2.76839e7	−1.62405
$782$	−2.62959e6	−0.153770
$783$	0	0
$784$	−3.33126e6	−0.193561
$785$	1.08025e7	0.625676
$786$	0	0
$787$	2.06536e7	1.18867	0.594333	−	0.804219i	$-0.297416\pi$
$787$	2.06536e7	1.18867	0.594333	+	0.804219i	$0.297416\pi$
$788$	1.55161e7	0.890156
$789$	0	0
$790$	−7.41956e6	−0.422971
$791$	3.12516e6	0.177595
$792$	0	0
$793$	7.70274e6	0.434973
$794$	9.58160e6	0.539370
$795$	0	0
$796$	−1.40126e6	−0.0783855
$797$	−5.27093e6	−0.293928	−0.146964	−	0.989142i	$-0.546950\pi$
$797$	−5.27093e6	−0.293928	−0.146964	+	0.989142i	$0.546950\pi$
$798$	0	0
$799$	5.80897e6	0.321909
$800$	3.71641e6	0.205304
$801$	0	0
$802$	1.41992e7	0.779519
$803$	−443641.	−0.0242797
$804$	0	0
$805$	−4.65824e6	−0.253356
$806$	−1.27923e6	−0.0693602
$807$	0	0
$808$	1.75110e7	0.943586
$809$	2.62432e7	1.40976	0.704882	−	0.709325i	$-0.251000\pi$
$809$	2.62432e7	1.40976	0.704882	+	0.709325i	$0.251000\pi$
$810$	0	0
$811$	1.50454e7	0.803251	0.401626	−	0.915804i	$-0.368445\pi$
$811$	1.50454e7	0.803251	0.401626	+	0.915804i	$0.368445\pi$
$812$	1.73412e7	0.922975
$813$	0	0
$814$	2.53929e6	0.134323
$815$	1.40775e6	0.0742387
$816$	0	0
$817$	1.31861e7	0.691134
$818$	−3.07766e6	−0.160819
$819$	0	0
$820$	−1.19083e7	−0.618463
$821$	−1.82770e6	−0.0946338	−0.0473169	−	0.998880i	$-0.515067\pi$
$821$	−1.82770e6	−0.0946338	−0.0473169	+	0.998880i	$0.515067\pi$
$822$	0	0
$823$	2.45074e7	1.26124	0.630619	−	0.776093i	$-0.282801\pi$
$823$	2.45074e7	1.26124	0.630619	+	0.776093i	$0.282801\pi$
$824$	−1.59296e7	−0.817309
$825$	0	0
$826$	467088.	0.0238204
$827$	7.03772e6	0.357823	0.178911	−	0.983865i	$-0.442742\pi$
$827$	7.03772e6	0.357823	0.178911	+	0.983865i	$0.442742\pi$
$828$	0	0
$829$	3.44967e7	1.74338	0.871689	−	0.490059i	$-0.163025\pi$
$829$	3.44967e7	1.74338	0.871689	+	0.490059i	$0.163025\pi$
$830$	3.20275e6	0.161372
$831$	0	0
$832$	865273.	0.0433356
$833$	−4.15078e6	−0.207261
$834$	0	0
$835$	1.28153e7	0.636080
$836$	1.70104e7	0.841782
$837$	0	0
$838$	−4.95476e6	−0.243732
$839$	−2.79128e6	−0.136899	−0.0684493	−	0.997655i	$-0.521805\pi$
$839$	−2.79128e6	−0.136899	−0.0684493	+	0.997655i	$0.521805\pi$
$840$	0	0
$841$	4.82083e7	2.35035
$842$	1.46460e7	0.711935
$843$	0	0
$844$	9.93349e6	0.480005
$845$	714025.	0.0344010
$846$	0	0
$847$	2.12179e6	0.101623
$848$	−617535.	−0.0294898
$849$	0	0
$850$	756169.	0.0358981
$851$	4.64471e6	0.219854
$852$	0	0
$853$	−2.48047e7	−1.16724	−0.583621	−	0.812026i	$-0.698365\pi$
$853$	−2.48047e7	−1.16724	−0.583621	+	0.812026i	$0.698365\pi$
$854$	1.07713e7	0.505388
$855$	0	0
$856$	−1.40579e7	−0.655746
$857$	1.83304e7	0.852550	0.426275	−	0.904594i	$-0.359826\pi$
$857$	1.83304e7	0.852550	0.426275	+	0.904594i	$0.359826\pi$
$858$	0	0
$859$	1.67092e7	0.772632	0.386316	−	0.922366i	$-0.373747\pi$
$859$	1.67092e7	0.772632	0.386316	+	0.922366i	$0.373747\pi$
$860$	−4.97370e6	−0.229316
$861$	0	0
$862$	1.27806e7	0.585844
$863$	−8.29285e6	−0.379033	−0.189516	−	0.981878i	$-0.560692\pi$
$863$	−8.29285e6	−0.379033	−0.189516	+	0.981878i	$0.560692\pi$
$864$	0	0
$865$	9.07575e6	0.412422
$866$	−1.01627e7	−0.460484
$867$	0	0
$868$	5.74411e6	0.258775
$869$	4.64069e7	2.08465
$870$	0	0
$871$	−3.28834e6	−0.146870
$872$	−2.56407e7	−1.14193
$873$	0	0
$874$	−9.68974e6	−0.429075
$875$	1.33953e6	0.0591470
$876$	0	0
$877$	−1.44569e7	−0.634711	−0.317356	−	0.948307i	$-0.602795\pi$
$877$	−1.44569e7	−0.634711	−0.317356	+	0.948307i	$0.602795\pi$
$878$	−4.60447e6	−0.201578
$879$	0	0
$880$	−3.79578e6	−0.165232
$881$	3.42081e7	1.48487	0.742437	−	0.669916i	$-0.233670\pi$
$881$	3.42081e7	1.48487	0.742437	+	0.669916i	$0.233670\pi$
$882$	0	0
$883$	−1.41341e7	−0.610051	−0.305026	−	0.952344i	$-0.598665\pi$
$883$	−1.41341e7	−0.610051	−0.305026	+	0.952344i	$0.598665\pi$
$884$	−1.80990e6	−0.0778975
$885$	0	0
$886$	−9.90207e6	−0.423781
$887$	−2.15650e7	−0.920325	−0.460163	−	0.887835i	$-0.652209\pi$
$887$	−2.15650e7	−0.920325	−0.460163	+	0.887835i	$0.652209\pi$
$888$	0	0
$889$	1.53096e7	0.649696
$890$	235038.	0.00994632
$891$	0	0
$892$	−2.30196e6	−0.0968693
$893$	2.14054e7	0.898246
$894$	0	0
$895$	2.03122e7	0.847618
$896$	−1.51027e7	−0.628471
$897$	0	0
$898$	−221785.	−0.00917787
$899$	2.27626e7	0.939340
$900$	0	0
$901$	−769456.	−0.0315771
$902$	−2.31955e7	−0.949264
$903$	0	0
$904$	−5.66768e6	−0.230666
$905$	−1.16243e7	−0.471788
$906$	0	0
$907$	1.53494e7	0.619547	0.309773	−	0.950810i	$-0.399747\pi$
$907$	1.53494e7	0.619547	0.309773	+	0.950810i	$0.399747\pi$
$908$	6.14650e6	0.247408
$909$	0	0
$910$	998478.	0.0399701
$911$	8.00925e6	0.319739	0.159870	−	0.987138i	$-0.448893\pi$
$911$	8.00925e6	0.319739	0.159870	+	0.987138i	$0.448893\pi$
$912$	0	0
$913$	−2.00322e7	−0.795337
$914$	7.93047e6	0.314003
$915$	0	0
$916$	1.04444e7	0.411287
$917$	−3.73403e6	−0.146641
$918$	0	0
$919$	−1.00787e7	−0.393655	−0.196828	−	0.980438i	$-0.563064\pi$
$919$	−1.00787e7	−0.393655	−0.196828	+	0.980438i	$0.563064\pi$
$920$	8.44800e6	0.329067
$921$	0	0
$922$	−225064.	−0.00871926
$923$	1.08540e7	0.419359
$924$	0	0
$925$	−1.33564e6	−0.0513258
$926$	−6.89748e6	−0.264340
$927$	0	0
$928$	−4.92927e7	−1.87894
$929$	1.20808e7	0.459257	0.229629	−	0.973278i	$-0.426249\pi$
$929$	1.20808e7	0.459257	0.229629	+	0.973278i	$0.426249\pi$
$930$	0	0
$931$	−1.52952e7	−0.578336
$932$	1.17061e6	0.0441440
$933$	0	0
$934$	−3.82547e6	−0.143489
$935$	−4.72959e6	−0.176927
$936$	0	0
$937$	1.85069e7	0.688628	0.344314	−	0.938854i	$-0.388111\pi$
$937$	1.85069e7	0.688628	0.344314	+	0.938854i	$0.388111\pi$
$938$	−4.59835e6	−0.170646
$939$	0	0
$940$	−8.07396e6	−0.298035
$941$	1.31161e7	0.482871	0.241436	−	0.970417i	$-0.422382\pi$
$941$	1.31161e7	0.482871	0.241436	+	0.970417i	$0.422382\pi$
$942$	0	0
$943$	−4.24277e7	−1.55371
$944$	696186.	0.0254270
$945$	0	0
$946$	−9.68802e6	−0.351971
$947$	2.46835e7	0.894399	0.447199	−	0.894434i	$-0.352421\pi$
$947$	2.46835e7	0.894399	0.447199	+	0.894434i	$0.352421\pi$
$948$	0	0
$949$	173938.	0.00626945
$950$	2.78640e6	0.100169
$951$	0	0
$952$	−5.85003e6	−0.209202
$953$	−3.21978e7	−1.14840	−0.574201	−	0.818715i	$-0.694687\pi$
$953$	−3.21978e7	−1.14840	−0.574201	+	0.818715i	$0.694687\pi$
$954$	0	0
$955$	388439.	0.0137821
$956$	−3.41411e7	−1.20818
$957$	0	0
$958$	1.36832e7	0.481697
$959$	2.97927e7	1.04608
$960$	0	0
$961$	−2.10893e7	−0.736636
$962$	−995579.	−0.0346847
$963$	0	0
$964$	1.40743e7	0.487791
$965$	−7.98520e6	−0.276037
$966$	0	0
$967$	2.28108e7	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$967$	2.28108e7	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$968$	−3.84799e6	−0.131991
$969$	0	0
$970$	9.18400e6	0.313403
$971$	−4.02182e7	−1.36891	−0.684455	−	0.729055i	$-0.739960\pi$
$971$	−4.02182e7	−1.36891	−0.684455	+	0.729055i	$0.739960\pi$
$972$	0	0
$973$	4.68035e6	0.158488
$974$	2.36346e7	0.798271
$975$	0	0
$976$	1.60545e7	0.539475
$977$	−3.17409e7	−1.06386	−0.531928	−	0.846790i	$-0.678532\pi$
$977$	−3.17409e7	−1.06386	−0.531928	+	0.846790i	$0.678532\pi$
$978$	0	0
$979$	−1.47008e6	−0.0490214
$980$	5.76922e6	0.191890
$981$	0	0
$982$	−7.03774e6	−0.232892
$983$	−4.52276e6	−0.149286	−0.0746431	−	0.997210i	$-0.523782\pi$
$983$	−4.52276e6	−0.149286	−0.0746431	+	0.997210i	$0.523782\pi$
$984$	0	0
$985$	−1.58970e7	−0.522064
$986$	−1.00295e7	−0.328539
$987$	0	0
$988$	−6.66928e6	−0.217363
$989$	−1.77207e7	−0.576090
$990$	0	0
$991$	−2.04730e7	−0.662213	−0.331107	−	0.943593i	$-0.607422\pi$
$991$	−2.04730e7	−0.662213	−0.331107	+	0.943593i	$0.607422\pi$
$992$	−1.63277e7	−0.526801
$993$	0	0
$994$	1.51780e7	0.487247
$995$	1.43566e6	0.0459720
$996$	0	0
$997$	−3.49736e7	−1.11430	−0.557150	−	0.830412i	$-0.688105\pi$
$997$	−3.49736e7	−1.11430	−0.557150	+	0.830412i	$0.688105\pi$
$998$	446192.	0.0141806
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.6.a.m.1.4		6
3.2	odd	2		65.6.a.d.1.3	✓	6
12.11	even	2		1040.6.a.q.1.2		6
15.2	even	4		325.6.b.g.274.5		12
15.8	even	4		325.6.b.g.274.8		12
15.14	odd	2		325.6.a.g.1.4		6
39.38	odd	2		845.6.a.h.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.3	✓	6	3.2	odd	2
325.6.a.g.1.4		6	15.14	odd	2
325.6.b.g.274.5		12	15.2	even	4
325.6.b.g.274.8		12	15.8	even	4
585.6.a.m.1.4		6	1.1	even	1	trivial
845.6.a.h.1.4		6	39.38	odd	2
1040.6.a.q.1.2		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q+2.75663 q^{2} -24.4010 q^{4} +25.0000 q^{5} +85.7300 q^{7} -155.477 q^{8} +68.9158 q^{10} -431.046 q^{11} +169.000 q^{13} +236.326 q^{14} +352.239 q^{16} +438.894 q^{17} +1617.28 q^{19} -610.025 q^{20} -1188.23 q^{22} -2173.44 q^{23} +625.000 q^{25} +465.871 q^{26} -2091.90 q^{28} -8289.72 q^{29} -2745.88 q^{31} +5946.25 q^{32} +1209.87 q^{34} +2143.25 q^{35} -2137.03 q^{37} +4458.24 q^{38} -3886.92 q^{40} +19520.9 q^{41} +8153.29 q^{43} +10517.9 q^{44} -5991.39 q^{46} +13235.5 q^{47} -9457.36 q^{49} +1722.89 q^{50} -4123.77 q^{52} -1753.17 q^{53} -10776.1 q^{55} -13329.0 q^{56} -22851.7 q^{58} +1976.46 q^{59} +45578.3 q^{61} -7569.39 q^{62} +5119.96 q^{64} +4225.00 q^{65} -19457.7 q^{67} -10709.5 q^{68} +5908.15 q^{70} +64224.9 q^{71} +1029.22 q^{73} -5891.00 q^{74} -39463.2 q^{76} -36953.6 q^{77} -107661. q^{79} +8805.99 q^{80} +53812.1 q^{82} +46473.4 q^{83} +10972.4 q^{85} +22475.6 q^{86} +67017.6 q^{88} +3410.51 q^{89} +14488.4 q^{91} +53034.2 q^{92} +36485.3 q^{94} +40431.9 q^{95} +133264. q^{97} -26070.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) \) 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

Introduction

L-functions

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Database

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