LMFDB - Embedded newform 1040.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.2250 q^{3} +25.0000 q^{5} +51.1669 q^{7} +126.600 q^{9} -494.607 q^{11} -169.000 q^{13} +480.624 q^{15} -2364.88 q^{17} +699.408 q^{19} +983.681 q^{21} +3809.57 q^{23} +625.000 q^{25} -2237.80 q^{27} +2263.52 q^{29} +9088.51 q^{31} -9508.81 q^{33} +1279.17 q^{35} -4487.29 q^{37} -3249.02 q^{39} -12430.1 q^{41} +12572.6 q^{43} +3164.99 q^{45} -3601.31 q^{47} -14189.0 q^{49} -45464.8 q^{51} +7547.01 q^{53} -12365.2 q^{55} +13446.1 q^{57} -50069.9 q^{59} -24756.7 q^{61} +6477.70 q^{63} -4225.00 q^{65} -43724.6 q^{67} +73238.8 q^{69} -7603.34 q^{71} -21623.2 q^{73} +12015.6 q^{75} -25307.5 q^{77} -103537. q^{79} -73785.2 q^{81} +80394.0 q^{83} -59122.0 q^{85} +43516.1 q^{87} +59301.0 q^{89} -8647.20 q^{91} +174726. q^{93} +17485.2 q^{95} -73477.5 q^{97} -62617.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{3} + 50 q^{5} + 252 q^{7} - 106 q^{9} + 36 q^{11} - 338 q^{13} + 400 q^{15} - 2380 q^{17} + 1092 q^{19} + 336 q^{21} + 1176 q^{23} + 1250 q^{25} - 704 q^{27} - 4872 q^{29} + 2260 q^{31} - 11220 q^{33}+ \cdots - 186036 q^{99}+O(q^{100}) $ 2 * q + 16 * q^3 + 50 * q^5 + 252 * q^7 - 106 * q^9 + 36 * q^11 - 338 * q^13 + 400 * q^15 - 2380 * q^17 + 1092 * q^19 + 336 * q^21 + 1176 * q^23 + 1250 * q^25 - 704 * q^27 - 4872 * q^29 + 2260 * q^31 - 11220 * q^33 + 6300 * q^35 - 17760 * q^37 - 2704 * q^39 - 9220 * q^41 + 23656 * q^43 - 2650 * q^45 - 12860 * q^47 + 9338 * q^49 - 45416 * q^51 + 11068 * q^53 + 900 * q^55 + 12180 * q^57 - 90404 * q^59 - 69000 * q^61 - 40236 * q^63 - 8450 * q^65 - 50796 * q^67 + 81732 * q^69 + 28668 * q^71 - 62688 * q^73 + 10000 * q^75 + 81256 * q^77 - 84064 * q^79 - 22210 * q^81 + 12828 * q^83 - 59500 * q^85 + 66528 * q^87 + 92620 * q^89 - 42588 * q^91 + 196748 * q^93 + 27300 * q^95 - 75684 * q^97 - 186036 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	19.2250	1.23328	0.616641	−	0.787244i	$-0.288493\pi$
$3$	19.2250	1.23328	0.616641	+	0.787244i	$0.288493\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	51.1669	0.394679	0.197339	−	0.980335i	$-0.436770\pi$
$7$	51.1669	0.394679	0.197339	+	0.980335i	$0.436770\pi$
$8$	0	0
$9$	126.600	0.520986
$10$	0	0
$11$	−494.607	−1.23248	−0.616238	−	0.787560i	$-0.711344\pi$
$11$	−494.607	−1.23248	−0.616238	+	0.787560i	$0.711344\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	480.624	0.551541
$16$	0	0
$17$	−2364.88	−1.98466	−0.992332	−	0.123603i	$-0.960555\pi$
$17$	−2364.88	−1.98466	−0.992332	+	0.123603i	$0.960555\pi$
$18$	0	0
$19$	699.408	0.444474	0.222237	−	0.974993i	$-0.428664\pi$
$19$	699.408	0.444474	0.222237	+	0.974993i	$0.428664\pi$
$20$	0	0
$21$	983.681	0.486750
$22$	0	0
$23$	3809.57	1.50161	0.750803	−	0.660526i	$-0.229667\pi$
$23$	3809.57	1.50161	0.750803	+	0.660526i	$0.229667\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2237.80	−0.590760
$28$	0	0
$29$	2263.52	0.499792	0.249896	−	0.968273i	$-0.419603\pi$
$29$	2263.52	0.499792	0.249896	+	0.968273i	$0.419603\pi$
$30$	0	0
$31$	9088.51	1.69859	0.849294	−	0.527920i	$-0.177028\pi$
$31$	9088.51	1.69859	0.849294	+	0.527920i	$0.177028\pi$
$32$	0	0
$33$	−9508.81	−1.51999
$34$	0	0
$35$	1279.17	0.176506
$36$	0	0
$37$	−4487.29	−0.538865	−0.269433	−	0.963019i	$-0.586836\pi$
$37$	−4487.29	−0.538865	−0.269433	+	0.963019i	$0.586836\pi$
$38$	0	0
$39$	−3249.02	−0.342051
$40$	0	0
$41$	−12430.1	−1.15482	−0.577409	−	0.816455i	$-0.695936\pi$
$41$	−12430.1	−1.15482	−0.577409	+	0.816455i	$0.695936\pi$
$42$	0	0
$43$	12572.6	1.03694	0.518470	−	0.855096i	$-0.326502\pi$
$43$	12572.6	1.03694	0.518470	+	0.855096i	$0.326502\pi$
$44$	0	0
$45$	3164.99	0.232992
$46$	0	0
$47$	−3601.31	−0.237802	−0.118901	−	0.992906i	$-0.537937\pi$
$47$	−3601.31	−0.237802	−0.118901	+	0.992906i	$0.537937\pi$
$48$	0	0
$49$	−14189.0	−0.844229
$50$	0	0
$51$	−45464.8	−2.44765
$52$	0	0
$53$	7547.01	0.369050	0.184525	−	0.982828i	$-0.440925\pi$
$53$	7547.01	0.369050	0.184525	+	0.982828i	$0.440925\pi$
$54$	0	0
$55$	−12365.2	−0.551180
$56$	0	0
$57$	13446.1	0.548162
$58$	0	0
$59$	−50069.9	−1.87261	−0.936304	−	0.351192i	$-0.885777\pi$
$59$	−50069.9	−1.87261	−0.936304	+	0.351192i	$0.885777\pi$
$60$	0	0
$61$	−24756.7	−0.851861	−0.425930	−	0.904756i	$-0.640053\pi$
$61$	−24756.7	−0.851861	−0.425930	+	0.904756i	$0.640053\pi$
$62$	0	0
$63$	6477.70	0.205622
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−43724.6	−1.18998	−0.594989	−	0.803734i	$-0.702844\pi$
$67$	−43724.6	−1.18998	−0.594989	+	0.803734i	$0.702844\pi$
$68$	0	0
$69$	73238.8	1.85190
$70$	0	0
$71$	−7603.34	−0.179002	−0.0895011	−	0.995987i	$-0.528527\pi$
$71$	−7603.34	−0.179002	−0.0895011	+	0.995987i	$0.528527\pi$
$72$	0	0
$73$	−21623.2	−0.474911	−0.237456	−	0.971398i	$-0.576313\pi$
$73$	−21623.2	−0.474911	−0.237456	+	0.971398i	$0.576313\pi$
$74$	0	0
$75$	12015.6	0.246657
$76$	0	0
$77$	−25307.5	−0.486432
$78$	0	0
$79$	−103537.	−1.86651	−0.933253	−	0.359220i	$-0.883043\pi$
$79$	−103537.	−1.86651	−0.933253	+	0.359220i	$0.883043\pi$
$80$	0	0
$81$	−73785.2	−1.24956
$82$	0	0
$83$	80394.0	1.28094	0.640470	−	0.767983i	$-0.278740\pi$
$83$	80394.0	1.28094	0.640470	+	0.767983i	$0.278740\pi$
$84$	0	0
$85$	−59122.0	−0.887568
$86$	0	0
$87$	43516.1	0.616385
$88$	0	0
$89$	59301.0	0.793574	0.396787	−	0.917911i	$-0.370125\pi$
$89$	59301.0	0.793574	0.396787	+	0.917911i	$0.370125\pi$
$90$	0	0
$91$	−8647.20	−0.109464
$92$	0	0
$93$	174726.	2.09484
$94$	0	0
$95$	17485.2	0.198775
$96$	0	0
$97$	−73477.5	−0.792913	−0.396456	−	0.918054i	$-0.629760\pi$
$97$	−73477.5	−0.792913	−0.396456	+	0.918054i	$0.629760\pi$
$98$	0	0
$99$	−62617.0	−0.642103
$100$	0	0
$101$	97567.8	0.951707	0.475853	−	0.879525i	$-0.342139\pi$
$101$	97567.8	0.951707	0.475853	+	0.879525i	$0.342139\pi$
$102$	0	0
$103$	−30991.0	−0.287835	−0.143917	−	0.989590i	$-0.545970\pi$
$103$	−30991.0	−0.287835	−0.143917	+	0.989590i	$0.545970\pi$
$104$	0	0
$105$	24592.0	0.217681
$106$	0	0
$107$	−202039.	−1.70599	−0.852993	−	0.521923i	$-0.825215\pi$
$107$	−202039.	−1.70599	−0.852993	+	0.521923i	$0.825215\pi$
$108$	0	0
$109$	−171150.	−1.37978	−0.689890	−	0.723914i	$-0.742341\pi$
$109$	−171150.	−1.37978	−0.689890	+	0.723914i	$0.742341\pi$
$110$	0	0
$111$	−86268.1	−0.664573
$112$	0	0
$113$	−138118.	−1.01755	−0.508774	−	0.860900i	$-0.669901\pi$
$113$	−138118.	−1.01755	−0.508774	+	0.860900i	$0.669901\pi$
$114$	0	0
$115$	95239.2	0.671539
$116$	0	0
$117$	−21395.3	−0.144495
$118$	0	0
$119$	−121003.	−0.783304
$120$	0	0
$121$	83585.1	0.518998
$122$	0	0
$123$	−238968.	−1.42422
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−86030.1	−0.473305	−0.236653	−	0.971594i	$-0.576050\pi$
$127$	−86030.1	−0.473305	−0.236653	+	0.971594i	$0.576050\pi$
$128$	0	0
$129$	241708.	1.27884
$130$	0	0
$131$	297647.	1.51538	0.757692	−	0.652612i	$-0.226327\pi$
$131$	297647.	1.51538	0.757692	+	0.652612i	$0.226327\pi$
$132$	0	0
$133$	35786.5	0.175424
$134$	0	0
$135$	−55944.9	−0.264196
$136$	0	0
$137$	−75631.0	−0.344269	−0.172135	−	0.985073i	$-0.555066\pi$
$137$	−75631.0	−0.344269	−0.172135	+	0.985073i	$0.555066\pi$
$138$	0	0
$139$	−58569.2	−0.257118	−0.128559	−	0.991702i	$-0.541035\pi$
$139$	−58569.2	−0.257118	−0.128559	+	0.991702i	$0.541035\pi$
$140$	0	0
$141$	−69235.0	−0.293277
$142$	0	0
$143$	83588.6	0.341827
$144$	0	0
$145$	56588.0	0.223514
$146$	0	0
$147$	−272782.	−1.04117
$148$	0	0
$149$	−344984.	−1.27301	−0.636507	−	0.771271i	$-0.719621\pi$
$149$	−344984.	−1.27301	−0.636507	+	0.771271i	$0.719621\pi$
$150$	0	0
$151$	−12433.7	−0.0443770	−0.0221885	−	0.999754i	$-0.507063\pi$
$151$	−12433.7	−0.0443770	−0.0221885	+	0.999754i	$0.507063\pi$
$152$	0	0
$153$	−299393.	−1.03398
$154$	0	0
$155$	227213.	0.759632
$156$	0	0
$157$	−416674.	−1.34911	−0.674555	−	0.738225i	$-0.735664\pi$
$157$	−416674.	−1.34911	−0.674555	+	0.738225i	$0.735664\pi$
$158$	0	0
$159$	145091.	0.455143
$160$	0	0
$161$	194924.	0.592652
$162$	0	0
$163$	225121.	0.663661	0.331831	−	0.943339i	$-0.392334\pi$
$163$	225121.	0.663661	0.331831	+	0.943339i	$0.392334\pi$
$164$	0	0
$165$	−237720.	−0.679761
$166$	0	0
$167$	94547.0	0.262335	0.131168	−	0.991360i	$-0.458127\pi$
$167$	94547.0	0.262335	0.131168	+	0.991360i	$0.458127\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	88544.7	0.231565
$172$	0	0
$173$	−92745.6	−0.235602	−0.117801	−	0.993037i	$-0.537584\pi$
$173$	−92745.6	−0.235602	−0.117801	+	0.993037i	$0.537584\pi$
$174$	0	0
$175$	31979.3	0.0789357
$176$	0	0
$177$	−962592.	−2.30945
$178$	0	0
$179$	−226852.	−0.529187	−0.264594	−	0.964360i	$-0.585238\pi$
$179$	−226852.	−0.529187	−0.264594	+	0.964360i	$0.585238\pi$
$180$	0	0
$181$	691487.	1.56887	0.784436	−	0.620210i	$-0.212953\pi$
$181$	691487.	1.56887	0.784436	+	0.620210i	$0.212953\pi$
$182$	0	0
$183$	−475947.	−1.05058
$184$	0	0
$185$	−112182.	−0.240988
$186$	0	0
$187$	1.16969e6	2.44605
$188$	0	0
$189$	−114501.	−0.233160
$190$	0	0
$191$	340612.	0.675579	0.337790	−	0.941222i	$-0.390321\pi$
$191$	340612.	0.675579	0.337790	+	0.941222i	$0.390321\pi$
$192$	0	0
$193$	−479839.	−0.927262	−0.463631	−	0.886028i	$-0.653454\pi$
$193$	−479839.	−0.927262	−0.463631	+	0.886028i	$0.653454\pi$
$194$	0	0
$195$	−81225.5	−0.152970
$196$	0	0
$197$	775437.	1.42358	0.711789	−	0.702394i	$-0.247885\pi$
$197$	775437.	1.42358	0.711789	+	0.702394i	$0.247885\pi$
$198$	0	0
$199$	−960804.	−1.71990	−0.859948	−	0.510382i	$-0.829504\pi$
$199$	−960804.	−1.71990	−0.859948	+	0.510382i	$0.829504\pi$
$200$	0	0
$201$	−840605.	−1.46758
$202$	0	0
$203$	115817.	0.197257
$204$	0	0
$205$	−310752.	−0.516451
$206$	0	0
$207$	482289.	0.782315
$208$	0	0
$209$	−345932.	−0.547804
$210$	0	0
$211$	−1.00406e6	−1.55258	−0.776289	−	0.630377i	$-0.782900\pi$
$211$	−1.00406e6	−1.55258	−0.776289	+	0.630377i	$0.782900\pi$
$212$	0	0
$213$	−146174.	−0.220760
$214$	0	0
$215$	314315.	0.463734
$216$	0	0
$217$	465030.	0.670397
$218$	0	0
$219$	−415705.	−0.585700
$220$	0	0
$221$	399665.	0.550447
$222$	0	0
$223$	−369742.	−0.497893	−0.248947	−	0.968517i	$-0.580084\pi$
$223$	−369742.	−0.497893	−0.248947	+	0.968517i	$0.580084\pi$
$224$	0	0
$225$	79124.7	0.104197
$226$	0	0
$227$	622238.	0.801479	0.400740	−	0.916192i	$-0.368753\pi$
$227$	622238.	0.801479	0.400740	+	0.916192i	$0.368753\pi$
$228$	0	0
$229$	328149.	0.413506	0.206753	−	0.978393i	$-0.433710\pi$
$229$	328149.	0.413506	0.206753	+	0.978393i	$0.433710\pi$
$230$	0	0
$231$	−486536.	−0.599908
$232$	0	0
$233$	1.25086e6	1.50945	0.754723	−	0.656044i	$-0.227771\pi$
$233$	1.25086e6	1.50945	0.754723	+	0.656044i	$0.227771\pi$
$234$	0	0
$235$	−90032.7	−0.106348
$236$	0	0
$237$	−1.99050e6	−2.30193
$238$	0	0
$239$	−14131.5	−0.0160027	−0.00800134	−	0.999968i	$-0.502547\pi$
$239$	−14131.5	−0.0160027	−0.00800134	+	0.999968i	$0.502547\pi$
$240$	0	0
$241$	61638.0	0.0683606	0.0341803	−	0.999416i	$-0.489118\pi$
$241$	61638.0	0.0683606	0.0341803	+	0.999416i	$0.489118\pi$
$242$	0	0
$243$	−874735.	−0.950300
$244$	0	0
$245$	−354724.	−0.377551
$246$	0	0
$247$	−118200.	−0.123275
$248$	0	0
$249$	1.54557e6	1.57976
$250$	0	0
$251$	66537.0	0.0666621	0.0333310	−	0.999444i	$-0.489388\pi$
$251$	66537.0	0.0666621	0.0333310	+	0.999444i	$0.489388\pi$
$252$	0	0
$253$	−1.88424e6	−1.85069
$254$	0	0
$255$	−1.13662e6	−1.09462
$256$	0	0
$257$	−270539.	−0.255504	−0.127752	−	0.991806i	$-0.540776\pi$
$257$	−270539.	−0.255504	−0.127752	+	0.991806i	$0.540776\pi$
$258$	0	0
$259$	−229601.	−0.212679
$260$	0	0
$261$	286561.	0.260385
$262$	0	0
$263$	1.11803e6	0.996698	0.498349	−	0.866976i	$-0.333940\pi$
$263$	1.11803e6	0.996698	0.498349	+	0.866976i	$0.333940\pi$
$264$	0	0
$265$	188675.	0.165044
$266$	0	0
$267$	1.14006e6	0.978701
$268$	0	0
$269$	947186.	0.798095	0.399047	−	0.916930i	$-0.369341\pi$
$269$	947186.	0.798095	0.399047	+	0.916930i	$0.369341\pi$
$270$	0	0
$271$	2.13848e6	1.76881	0.884405	−	0.466720i	$-0.154564\pi$
$271$	2.13848e6	1.76881	0.884405	+	0.466720i	$0.154564\pi$
$272$	0	0
$273$	−166242.	−0.135000
$274$	0	0
$275$	−309129.	−0.246495
$276$	0	0
$277$	−1.34126e6	−1.05030	−0.525151	−	0.851009i	$-0.675991\pi$
$277$	−1.34126e6	−1.05030	−0.525151	+	0.851009i	$0.675991\pi$
$278$	0	0
$279$	1.15060e6	0.884941
$280$	0	0
$281$	1.64581e6	1.24341	0.621705	−	0.783252i	$-0.286440\pi$
$281$	1.64581e6	1.24341	0.621705	+	0.783252i	$0.286440\pi$
$282$	0	0
$283$	−444740.	−0.330096	−0.165048	−	0.986286i	$-0.552778\pi$
$283$	−444740.	−0.330096	−0.165048	+	0.986286i	$0.552778\pi$
$284$	0	0
$285$	336152.	0.245146
$286$	0	0
$287$	−636007.	−0.455782
$288$	0	0
$289$	4.17280e6	2.93889
$290$	0	0
$291$	−1.41260e6	−0.977885
$292$	0	0
$293$	458582.	0.312067	0.156033	−	0.987752i	$-0.450129\pi$
$293$	458582.	0.312067	0.156033	+	0.987752i	$0.450129\pi$
$294$	0	0
$295$	−1.25175e6	−0.837455
$296$	0	0
$297$	1.10683e6	0.728098
$298$	0	0
$299$	−643817.	−0.416471
$300$	0	0
$301$	643300.	0.409258
$302$	0	0
$303$	1.87574e6	1.17372
$304$	0	0
$305$	−618918.	−0.380964
$306$	0	0
$307$	−2.49374e6	−1.51010	−0.755050	−	0.655668i	$-0.772387\pi$
$307$	−2.49374e6	−1.51010	−0.755050	+	0.655668i	$0.772387\pi$
$308$	0	0
$309$	−595802.	−0.354982
$310$	0	0
$311$	1.74516e6	1.02314	0.511569	−	0.859242i	$-0.329065\pi$
$311$	1.74516e6	1.02314	0.511569	+	0.859242i	$0.329065\pi$
$312$	0	0
$313$	791926.	0.456903	0.228451	−	0.973555i	$-0.426634\pi$
$313$	791926.	0.456903	0.228451	+	0.973555i	$0.426634\pi$
$314$	0	0
$315$	161943.	0.0919569
$316$	0	0
$317$	−2.73806e6	−1.53037	−0.765183	−	0.643813i	$-0.777352\pi$
$317$	−2.73806e6	−1.53037	−0.765183	+	0.643813i	$0.777352\pi$
$318$	0	0
$319$	−1.11955e6	−0.615982
$320$	0	0
$321$	−3.88419e6	−2.10396
$322$	0	0
$323$	−1.65402e6	−0.882132
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	−3.29035e6	−1.70166
$328$	0	0
$329$	−184268.	−0.0938554
$330$	0	0
$331$	−134538.	−0.0674954	−0.0337477	−	0.999430i	$-0.510744\pi$
$331$	−134538.	−0.0674954	−0.0337477	+	0.999430i	$0.510744\pi$
$332$	0	0
$333$	−568089.	−0.280741
$334$	0	0
$335$	−1.09312e6	−0.532175
$336$	0	0
$337$	−393396.	−0.188693	−0.0943463	−	0.995539i	$-0.530076\pi$
$337$	−393396.	−0.188693	−0.0943463	+	0.995539i	$0.530076\pi$
$338$	0	0
$339$	−2.65532e6	−1.25492
$340$	0	0
$341$	−4.49524e6	−2.09347
$342$	0	0
$343$	−1.58597e6	−0.727878
$344$	0	0
$345$	1.83097e6	0.828197
$346$	0	0
$347$	409410.	0.182530	0.0912651	−	0.995827i	$-0.470909\pi$
$347$	409410.	0.182530	0.0912651	+	0.995827i	$0.470909\pi$
$348$	0	0
$349$	862372.	0.378993	0.189497	−	0.981881i	$-0.439314\pi$
$349$	862372.	0.378993	0.189497	+	0.981881i	$0.439314\pi$
$350$	0	0
$351$	378187.	0.163847
$352$	0	0
$353$	−967902.	−0.413423	−0.206711	−	0.978402i	$-0.566276\pi$
$353$	−967902.	−0.413423	−0.206711	+	0.978402i	$0.566276\pi$
$354$	0	0
$355$	−190083.	−0.0800522
$356$	0	0
$357$	−2.32629e6	−0.966035
$358$	0	0
$359$	2.87414e6	1.17699	0.588493	−	0.808502i	$-0.299721\pi$
$359$	2.87414e6	1.17699	0.588493	+	0.808502i	$0.299721\pi$
$360$	0	0
$361$	−1.98693e6	−0.802443
$362$	0	0
$363$	1.60692e6	0.640071
$364$	0	0
$365$	−540579.	−0.212387
$366$	0	0
$367$	−2.80197e6	−1.08592	−0.542961	−	0.839758i	$-0.682697\pi$
$367$	−2.80197e6	−1.08592	−0.542961	+	0.839758i	$0.682697\pi$
$368$	0	0
$369$	−1.57364e6	−0.601644
$370$	0	0
$371$	386157.	0.145656
$372$	0	0
$373$	−508739.	−0.189331	−0.0946657	−	0.995509i	$-0.530178\pi$
$373$	−508739.	−0.189331	−0.0946657	+	0.995509i	$0.530178\pi$
$374$	0	0
$375$	300390.	0.110308
$376$	0	0
$377$	−382535.	−0.138617
$378$	0	0
$379$	−26753.7	−0.00956722	−0.00478361	−	0.999989i	$-0.501523\pi$
$379$	−26753.7	−0.00956722	−0.00478361	+	0.999989i	$0.501523\pi$
$380$	0	0
$381$	−1.65393e6	−0.583719
$382$	0	0
$383$	4.57628e6	1.59410	0.797049	−	0.603914i	$-0.206393\pi$
$383$	4.57628e6	1.59410	0.797049	+	0.603914i	$0.206393\pi$
$384$	0	0
$385$	−632687.	−0.217539
$386$	0	0
$387$	1.59168e6	0.540231
$388$	0	0
$389$	−905452.	−0.303383	−0.151692	−	0.988428i	$-0.548472\pi$
$389$	−905452.	−0.303383	−0.151692	+	0.988428i	$0.548472\pi$
$390$	0	0
$391$	−9.00917e6	−2.98018
$392$	0	0
$393$	5.72225e6	1.86890
$394$	0	0
$395$	−2.58843e6	−0.834727
$396$	0	0
$397$	2.20323e6	0.701590	0.350795	−	0.936452i	$-0.385911\pi$
$397$	2.20323e6	0.701590	0.350795	+	0.936452i	$0.385911\pi$
$398$	0	0
$399$	687995.	0.216348
$400$	0	0
$401$	−1.12383e6	−0.349011	−0.174505	−	0.984656i	$-0.555833\pi$
$401$	−1.12383e6	−0.349011	−0.174505	+	0.984656i	$0.555833\pi$
$402$	0	0
$403$	−1.53596e6	−0.471104
$404$	0	0
$405$	−1.84463e6	−0.558820
$406$	0	0
$407$	2.21945e6	0.664139
$408$	0	0
$409$	1.00525e6	0.297143	0.148572	−	0.988902i	$-0.452532\pi$
$409$	1.00525e6	0.297143	0.148572	+	0.988902i	$0.452532\pi$
$410$	0	0
$411$	−1.45400e6	−0.424581
$412$	0	0
$413$	−2.56192e6	−0.739078
$414$	0	0
$415$	2.00985e6	0.572854
$416$	0	0
$417$	−1.12599e6	−0.317099
$418$	0	0
$419$	−1.52964e6	−0.425653	−0.212826	−	0.977090i	$-0.568267\pi$
$419$	−1.52964e6	−0.425653	−0.212826	+	0.977090i	$0.568267\pi$
$420$	0	0
$421$	2.29967e6	0.632354	0.316177	−	0.948700i	$-0.397601\pi$
$421$	2.29967e6	0.632354	0.316177	+	0.948700i	$0.397601\pi$
$422$	0	0
$423$	−455924.	−0.123891
$424$	0	0
$425$	−1.47805e6	−0.396933
$426$	0	0
$427$	−1.26672e6	−0.336211
$428$	0	0
$429$	1.60699e6	0.421570
$430$	0	0
$431$	2.62082e6	0.679585	0.339793	−	0.940500i	$-0.389643\pi$
$431$	2.62082e6	0.679585	0.339793	+	0.940500i	$0.389643\pi$
$432$	0	0
$433$	−591237.	−0.151545	−0.0757726	−	0.997125i	$-0.524142\pi$
$433$	−591237.	−0.151545	−0.0757726	+	0.997125i	$0.524142\pi$
$434$	0	0
$435$	1.08790e6	0.275656
$436$	0	0
$437$	2.66444e6	0.667425
$438$	0	0
$439$	5.86023e6	1.45129	0.725644	−	0.688070i	$-0.241542\pi$
$439$	5.86023e6	1.45129	0.725644	+	0.688070i	$0.241542\pi$
$440$	0	0
$441$	−1.79632e6	−0.439831
$442$	0	0
$443$	−1.49526e6	−0.361998	−0.180999	−	0.983483i	$-0.557933\pi$
$443$	−1.49526e6	−0.361998	−0.180999	+	0.983483i	$0.557933\pi$
$444$	0	0
$445$	1.48253e6	0.354897
$446$	0	0
$447$	−6.63231e6	−1.56999
$448$	0	0
$449$	−3.87629e6	−0.907404	−0.453702	−	0.891154i	$-0.649897\pi$
$449$	−3.87629e6	−0.907404	−0.453702	+	0.891154i	$0.649897\pi$
$450$	0	0
$451$	6.14800e6	1.42329
$452$	0	0
$453$	−239038.	−0.0547294
$454$	0	0
$455$	−216180.	−0.0489539
$456$	0	0
$457$	6.46622e6	1.44830	0.724152	−	0.689640i	$-0.242231\pi$
$457$	6.46622e6	1.44830	0.724152	+	0.689640i	$0.242231\pi$
$458$	0	0
$459$	5.29212e6	1.17246
$460$	0	0
$461$	−7.26166e6	−1.59141	−0.795707	−	0.605681i	$-0.792901\pi$
$461$	−7.26166e6	−1.59141	−0.795707	+	0.605681i	$0.792901\pi$
$462$	0	0
$463$	−4.81594e6	−1.04407	−0.522034	−	0.852925i	$-0.674827\pi$
$463$	−4.81594e6	−1.04407	−0.522034	+	0.852925i	$0.674827\pi$
$464$	0	0
$465$	4.36816e6	0.936841
$466$	0	0
$467$	−231914.	−0.0492078	−0.0246039	−	0.999697i	$-0.507832\pi$
$467$	−231914.	−0.0492078	−0.0246039	+	0.999697i	$0.507832\pi$
$468$	0	0
$469$	−2.23725e6	−0.469659
$470$	0	0
$471$	−8.01055e6	−1.66383
$472$	0	0
$473$	−6.21849e6	−1.27800
$474$	0	0
$475$	437130.	0.0888948
$476$	0	0
$477$	955448.	0.192270
$478$	0	0
$479$	−3.25748e6	−0.648700	−0.324350	−	0.945937i	$-0.605145\pi$
$479$	−3.25748e6	−0.648700	−0.324350	+	0.945937i	$0.605145\pi$
$480$	0	0
$481$	758353.	0.149454
$482$	0	0
$483$	3.74740e6	0.730907
$484$	0	0
$485$	−1.83694e6	−0.354601
$486$	0	0
$487$	−7.51626e6	−1.43608	−0.718042	−	0.696000i	$-0.754961\pi$
$487$	−7.51626e6	−1.43608	−0.718042	+	0.696000i	$0.754961\pi$
$488$	0	0
$489$	4.32794e6	0.818482
$490$	0	0
$491$	456011.	0.0853634	0.0426817	−	0.999089i	$-0.486410\pi$
$491$	456011.	0.0853634	0.0426817	+	0.999089i	$0.486410\pi$
$492$	0	0
$493$	−5.35296e6	−0.991920
$494$	0	0
$495$	−1.56543e6	−0.287157
$496$	0	0
$497$	−389039.	−0.0706483
$498$	0	0
$499$	−3.33321e6	−0.599255	−0.299627	−	0.954056i	$-0.596862\pi$
$499$	−3.33321e6	−0.599255	−0.299627	+	0.954056i	$0.596862\pi$
$500$	0	0
$501$	1.81766e6	0.323534
$502$	0	0
$503$	3.46235e6	0.610171	0.305085	−	0.952325i	$-0.401315\pi$
$503$	3.46235e6	0.610171	0.305085	+	0.952325i	$0.401315\pi$
$504$	0	0
$505$	2.43919e6	0.425616
$506$	0	0
$507$	549084.	0.0948679
$508$	0	0
$509$	−7.66998e6	−1.31220	−0.656100	−	0.754674i	$-0.727795\pi$
$509$	−7.66998e6	−1.31220	−0.656100	+	0.754674i	$0.727795\pi$
$510$	0	0
$511$	−1.10639e6	−0.187437
$512$	0	0
$513$	−1.56513e6	−0.262578
$514$	0	0
$515$	−774776.	−0.128724
$516$	0	0
$517$	1.78123e6	0.293085
$518$	0	0
$519$	−1.78303e6	−0.290563
$520$	0	0
$521$	−7.45075e6	−1.20256	−0.601278	−	0.799040i	$-0.705342\pi$
$521$	−7.45075e6	−1.20256	−0.601278	+	0.799040i	$0.705342\pi$
$522$	0	0
$523$	3.50008e6	0.559530	0.279765	−	0.960068i	$-0.409743\pi$
$523$	3.50008e6	0.559530	0.279765	+	0.960068i	$0.409743\pi$
$524$	0	0
$525$	614801.	0.0973500
$526$	0	0
$527$	−2.14932e7	−3.37113
$528$	0	0
$529$	8.07646e6	1.25482
$530$	0	0
$531$	−6.33883e6	−0.975602
$532$	0	0
$533$	2.10068e6	0.320289
$534$	0	0
$535$	−5.05097e6	−0.762940
$536$	0	0
$537$	−4.36122e6	−0.652638
$538$	0	0
$539$	7.01796e6	1.04049
$540$	0	0
$541$	−6.76874e6	−0.994294	−0.497147	−	0.867666i	$-0.665619\pi$
$541$	−6.76874e6	−0.994294	−0.497147	+	0.867666i	$0.665619\pi$
$542$	0	0
$543$	1.32938e7	1.93486
$544$	0	0
$545$	−4.27874e6	−0.617056
$546$	0	0
$547$	1.19664e7	1.71000	0.855001	−	0.518627i	$-0.173557\pi$
$547$	1.19664e7	1.71000	0.855001	+	0.518627i	$0.173557\pi$
$548$	0	0
$549$	−3.13419e6	−0.443807
$550$	0	0
$551$	1.58313e6	0.222145
$552$	0	0
$553$	−5.29768e6	−0.736670
$554$	0	0
$555$	−2.15670e6	−0.297206
$556$	0	0
$557$	445095.	0.0607875	0.0303938	−	0.999538i	$-0.490324\pi$
$557$	445095.	0.0607875	0.0303938	+	0.999538i	$0.490324\pi$
$558$	0	0
$559$	−2.12477e6	−0.287595
$560$	0	0
$561$	2.24872e7	3.01667
$562$	0	0
$563$	9.94384e6	1.32216	0.661078	−	0.750317i	$-0.270099\pi$
$563$	9.94384e6	1.32216	0.661078	+	0.750317i	$0.270099\pi$
$564$	0	0
$565$	−3.45296e6	−0.455061
$566$	0	0
$567$	−3.77536e6	−0.493174
$568$	0	0
$569$	6.04270e6	0.782439	0.391220	−	0.920297i	$-0.372053\pi$
$569$	6.04270e6	0.782439	0.391220	+	0.920297i	$0.372053\pi$
$570$	0	0
$571$	5.12993e6	0.658447	0.329224	−	0.944252i	$-0.393213\pi$
$571$	5.12993e6	0.658447	0.329224	+	0.944252i	$0.393213\pi$
$572$	0	0
$573$	6.54825e6	0.833180
$574$	0	0
$575$	2.38098e6	0.300321
$576$	0	0
$577$	−5.30151e6	−0.662918	−0.331459	−	0.943470i	$-0.607541\pi$
$577$	−5.30151e6	−0.662918	−0.331459	+	0.943470i	$0.607541\pi$
$578$	0	0
$579$	−9.22489e6	−1.14358
$580$	0	0
$581$	4.11351e6	0.505560
$582$	0	0
$583$	−3.73281e6	−0.454846
$584$	0	0
$585$	−534883.	−0.0646203
$586$	0	0
$587$	−9.26443e6	−1.10975	−0.554873	−	0.831935i	$-0.687233\pi$
$587$	−9.26443e6	−1.10975	−0.554873	+	0.831935i	$0.687233\pi$
$588$	0	0
$589$	6.35657e6	0.754979
$590$	0	0
$591$	1.49078e7	1.75567
$592$	0	0
$593$	−4.97831e6	−0.581360	−0.290680	−	0.956820i	$-0.593882\pi$
$593$	−4.97831e6	−0.581360	−0.290680	+	0.956820i	$0.593882\pi$
$594$	0	0
$595$	−3.02509e6	−0.350304
$596$	0	0
$597$	−1.84714e7	−2.12112
$598$	0	0
$599$	9.54062e6	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$599$	9.54062e6	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$600$	0	0
$601$	−3.23751e6	−0.365615	−0.182808	−	0.983149i	$-0.558519\pi$
$601$	−3.23751e6	−0.365615	−0.182808	+	0.983149i	$0.558519\pi$
$602$	0	0
$603$	−5.53552e6	−0.619962
$604$	0	0
$605$	2.08963e6	0.232103
$606$	0	0
$607$	1.01617e7	1.11942	0.559711	−	0.828688i	$-0.310912\pi$
$607$	1.01617e7	1.11942	0.559711	+	0.828688i	$0.310912\pi$
$608$	0	0
$609$	2.22658e6	0.243274
$610$	0	0
$611$	608621.	0.0659544
$612$	0	0
$613$	2.31456e6	0.248781	0.124391	−	0.992233i	$-0.460302\pi$
$613$	2.31456e6	0.248781	0.124391	+	0.992233i	$0.460302\pi$
$614$	0	0
$615$	−5.97419e6	−0.636929
$616$	0	0
$617$	−6.41200e6	−0.678079	−0.339040	−	0.940772i	$-0.610102\pi$
$617$	−6.41200e6	−0.678079	−0.339040	+	0.940772i	$0.610102\pi$
$618$	0	0
$619$	1.37209e7	1.43932	0.719659	−	0.694327i	$-0.244298\pi$
$619$	1.37209e7	1.43932	0.719659	+	0.694327i	$0.244298\pi$
$620$	0	0
$621$	−8.52503e6	−0.887089
$622$	0	0
$623$	3.03425e6	0.313207
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−6.65054e6	−0.675597
$628$	0	0
$629$	1.06119e7	1.06947
$630$	0	0
$631$	3.42813e6	0.342755	0.171378	−	0.985205i	$-0.445178\pi$
$631$	3.42813e6	0.342755	0.171378	+	0.985205i	$0.445178\pi$
$632$	0	0
$633$	−1.93030e7	−1.91477
$634$	0	0
$635$	−2.15075e6	−0.211669
$636$	0	0
$637$	2.39793e6	0.234147
$638$	0	0
$639$	−962579.	−0.0932576
$640$	0	0
$641$	1.19999e7	1.15354	0.576770	−	0.816906i	$-0.304313\pi$
$641$	1.19999e7	1.15354	0.576770	+	0.816906i	$0.304313\pi$
$642$	0	0
$643$	−39448.9	−0.00376277	−0.00188139	−	0.999998i	$-0.500599\pi$
$643$	−39448.9	−0.00376277	−0.00188139	+	0.999998i	$0.500599\pi$
$644$	0	0
$645$	6.04269e6	0.571915
$646$	0	0
$647$	3.43534e6	0.322633	0.161317	−	0.986903i	$-0.448426\pi$
$647$	3.43534e6	0.322633	0.161317	+	0.986903i	$0.448426\pi$
$648$	0	0
$649$	2.47649e7	2.30794
$650$	0	0
$651$	8.94019e6	0.826788
$652$	0	0
$653$	1.35146e7	1.24028	0.620142	−	0.784490i	$-0.287075\pi$
$653$	1.35146e7	1.24028	0.620142	+	0.784490i	$0.287075\pi$
$654$	0	0
$655$	7.44117e6	0.677700
$656$	0	0
$657$	−2.73748e6	−0.247422
$658$	0	0
$659$	2.17250e7	1.94871	0.974355	−	0.225018i	$-0.0722442\pi$
$659$	2.17250e7	1.94871	0.974355	+	0.225018i	$0.0722442\pi$
$660$	0	0
$661$	3.12575e6	0.278260	0.139130	−	0.990274i	$-0.455569\pi$
$661$	3.12575e6	0.278260	0.139130	+	0.990274i	$0.455569\pi$
$662$	0	0
$663$	7.68354e6	0.678856
$664$	0	0
$665$	894663.	0.0784522
$666$	0	0
$667$	8.62304e6	0.750491
$668$	0	0
$669$	−7.10827e6	−0.614043
$670$	0	0
$671$	1.22449e7	1.04990
$672$	0	0
$673$	−1.26422e7	−1.07594	−0.537968	−	0.842965i	$-0.680808\pi$
$673$	−1.26422e7	−1.07594	−0.537968	+	0.842965i	$0.680808\pi$
$674$	0	0
$675$	−1.39862e6	−0.118152
$676$	0	0
$677$	−1.82316e7	−1.52881	−0.764404	−	0.644737i	$-0.776967\pi$
$677$	−1.82316e7	−1.52881	−0.764404	+	0.644737i	$0.776967\pi$
$678$	0	0
$679$	−3.75961e6	−0.312946
$680$	0	0
$681$	1.19625e7	0.988450
$682$	0	0
$683$	1.60919e7	1.31994	0.659972	−	0.751290i	$-0.270568\pi$
$683$	1.60919e7	1.31994	0.659972	+	0.751290i	$0.270568\pi$
$684$	0	0
$685$	−1.89077e6	−0.153962
$686$	0	0
$687$	6.30865e6	0.509970
$688$	0	0
$689$	−1.27544e6	−0.102356
$690$	0	0
$691$	−1.96555e6	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$691$	−1.96555e6	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$692$	0	0
$693$	−3.20392e6	−0.253424
$694$	0	0
$695$	−1.46423e6	−0.114987
$696$	0	0
$697$	2.93956e7	2.29193
$698$	0	0
$699$	2.40477e7	1.86157
$700$	0	0
$701$	2.38464e7	1.83285	0.916426	−	0.400204i	$-0.131061\pi$
$701$	2.38464e7	1.83285	0.916426	+	0.400204i	$0.131061\pi$
$702$	0	0
$703$	−3.13845e6	−0.239512
$704$	0	0
$705$	−1.73088e6	−0.131157
$706$	0	0
$707$	4.99224e6	0.375618
$708$	0	0
$709$	−6.90141e6	−0.515611	−0.257805	−	0.966197i	$-0.582999\pi$
$709$	−6.90141e6	−0.515611	−0.257805	+	0.966197i	$0.582999\pi$
$710$	0	0
$711$	−1.31078e7	−0.972423
$712$	0	0
$713$	3.46233e7	2.55061
$714$	0	0
$715$	2.08971e6	0.152870
$716$	0	0
$717$	−271677.	−0.0197358
$718$	0	0
$719$	1.72590e6	0.124507	0.0622533	−	0.998060i	$-0.480171\pi$
$719$	1.72590e6	0.124507	0.0622533	+	0.998060i	$0.480171\pi$
$720$	0	0
$721$	−1.58571e6	−0.113602
$722$	0	0
$723$	1.18499e6	0.0843079
$724$	0	0
$725$	1.41470e6	0.0999585
$726$	0	0
$727$	−4.93378e6	−0.346213	−0.173107	−	0.984903i	$-0.555381\pi$
$727$	−4.93378e6	−0.346213	−0.173107	+	0.984903i	$0.555381\pi$
$728$	0	0
$729$	1.11306e6	0.0775709
$730$	0	0
$731$	−2.97327e7	−2.05798
$732$	0	0
$733$	1.95099e7	1.34120	0.670601	−	0.741818i	$-0.266036\pi$
$733$	1.95099e7	1.34120	0.670601	+	0.741818i	$0.266036\pi$
$734$	0	0
$735$	−6.81956e6	−0.465627
$736$	0	0
$737$	2.16265e7	1.46662
$738$	0	0
$739$	−1.62743e7	−1.09621	−0.548103	−	0.836411i	$-0.684650\pi$
$739$	−1.62743e7	−1.09621	−0.548103	+	0.836411i	$0.684650\pi$
$740$	0	0
$741$	−2.27239e6	−0.152033
$742$	0	0
$743$	−1.79935e7	−1.19576	−0.597878	−	0.801587i	$-0.703989\pi$
$743$	−1.79935e7	−1.19576	−0.597878	+	0.801587i	$0.703989\pi$
$744$	0	0
$745$	−8.62460e6	−0.569309
$746$	0	0
$747$	1.01779e7	0.667351
$748$	0	0
$749$	−1.03377e7	−0.673316
$750$	0	0
$751$	−1.05276e7	−0.681130	−0.340565	−	0.940221i	$-0.610618\pi$
$751$	−1.05276e7	−0.681130	−0.340565	+	0.940221i	$0.610618\pi$
$752$	0	0
$753$	1.27917e6	0.0822132
$754$	0	0
$755$	−310843.	−0.0198460
$756$	0	0
$757$	7.81642e6	0.495756	0.247878	−	0.968791i	$-0.420267\pi$
$757$	7.81642e6	0.495756	0.247878	+	0.968791i	$0.420267\pi$
$758$	0	0
$759$	−3.62244e7	−2.28243
$760$	0	0
$761$	−6.86800e6	−0.429901	−0.214951	−	0.976625i	$-0.568959\pi$
$761$	−6.86800e6	−0.429901	−0.214951	+	0.976625i	$0.568959\pi$
$762$	0	0
$763$	−8.75719e6	−0.544570
$764$	0	0
$765$	−7.48482e6	−0.462411
$766$	0	0
$767$	8.46181e6	0.519368
$768$	0	0
$769$	1.29868e7	0.791928	0.395964	−	0.918266i	$-0.370410\pi$
$769$	1.29868e7	0.791928	0.395964	+	0.918266i	$0.370410\pi$
$770$	0	0
$771$	−5.20111e6	−0.315108
$772$	0	0
$773$	2.61945e7	1.57674	0.788371	−	0.615200i	$-0.210925\pi$
$773$	2.61945e7	1.57674	0.788371	+	0.615200i	$0.210925\pi$
$774$	0	0
$775$	5.68032e6	0.339718
$776$	0	0
$777$	−4.41407e6	−0.262293
$778$	0	0
$779$	−8.69369e6	−0.513287
$780$	0	0
$781$	3.76066e6	0.220616
$782$	0	0
$783$	−5.06530e6	−0.295257
$784$	0	0
$785$	−1.04169e7	−0.603340
$786$	0	0
$787$	−2.55783e6	−0.147209	−0.0736046	−	0.997288i	$-0.523450\pi$
$787$	−2.55783e6	−0.147209	−0.0736046	+	0.997288i	$0.523450\pi$
$788$	0	0
$789$	2.14941e7	1.22921
$790$	0	0
$791$	−7.06708e6	−0.401605
$792$	0	0
$793$	4.18389e6	0.236264
$794$	0	0
$795$	3.62728e6	0.203546
$796$	0	0
$797$	2.70410e7	1.50792	0.753958	−	0.656923i	$-0.228142\pi$
$797$	2.70410e7	1.50792	0.753958	+	0.656923i	$0.228142\pi$
$798$	0	0
$799$	8.51666e6	0.471957
$800$	0	0
$801$	7.50748e6	0.413441
$802$	0	0
$803$	1.06950e7	0.585317
$804$	0	0
$805$	4.87309e6	0.265042
$806$	0	0
$807$	1.82096e7	0.984276
$808$	0	0
$809$	−7.14004e6	−0.383557	−0.191778	−	0.981438i	$-0.561425\pi$
$809$	−7.14004e6	−0.383557	−0.191778	+	0.981438i	$0.561425\pi$
$810$	0	0
$811$	3.49368e7	1.86522	0.932612	−	0.360882i	$-0.117524\pi$
$811$	3.49368e7	1.86522	0.932612	+	0.360882i	$0.117524\pi$
$812$	0	0
$813$	4.11122e7	2.18144
$814$	0	0
$815$	5.62802e6	0.296798
$816$	0	0
$817$	8.79337e6	0.460893
$818$	0	0
$819$	−1.09473e6	−0.0570293
$820$	0	0
$821$	−5.14344e6	−0.266315	−0.133158	−	0.991095i	$-0.542512\pi$
$821$	−5.14344e6	−0.266315	−0.133158	+	0.991095i	$0.542512\pi$
$822$	0	0
$823$	−4.85712e6	−0.249965	−0.124982	−	0.992159i	$-0.539887\pi$
$823$	−4.85712e6	−0.249965	−0.124982	+	0.992159i	$0.539887\pi$
$824$	0	0
$825$	−5.94300e6	−0.303998
$826$	0	0
$827$	−1.70403e7	−0.866391	−0.433196	−	0.901300i	$-0.642614\pi$
$827$	−1.70403e7	−0.866391	−0.433196	+	0.901300i	$0.642614\pi$
$828$	0	0
$829$	−2.27045e7	−1.14743	−0.573715	−	0.819055i	$-0.694498\pi$
$829$	−2.27045e7	−1.14743	−0.573715	+	0.819055i	$0.694498\pi$
$830$	0	0
$831$	−2.57857e7	−1.29532
$832$	0	0
$833$	3.35552e7	1.67551
$834$	0	0
$835$	2.36368e6	0.117320
$836$	0	0
$837$	−2.03382e7	−1.00346
$838$	0	0
$839$	1.63556e7	0.802159	0.401080	−	0.916043i	$-0.368635\pi$
$839$	1.63556e7	0.802159	0.401080	+	0.916043i	$0.368635\pi$
$840$	0	0
$841$	−1.53876e7	−0.750208
$842$	0	0
$843$	3.16407e7	1.53347
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	4.27679e6	0.204837
$848$	0	0
$849$	−8.55012e6	−0.407102
$850$	0	0
$851$	−1.70946e7	−0.809163
$852$	0	0
$853$	−4.52879e6	−0.213113	−0.106556	−	0.994307i	$-0.533982\pi$
$853$	−4.52879e6	−0.213113	−0.106556	+	0.994307i	$0.533982\pi$
$854$	0	0
$855$	2.21362e6	0.103559
$856$	0	0
$857$	−3.52177e7	−1.63798	−0.818990	−	0.573807i	$-0.805466\pi$
$857$	−3.52177e7	−1.63798	−0.818990	+	0.573807i	$0.805466\pi$
$858$	0	0
$859$	3.55901e7	1.64568	0.822841	−	0.568271i	$-0.192388\pi$
$859$	3.55901e7	1.64568	0.822841	+	0.568271i	$0.192388\pi$
$860$	0	0
$861$	−1.22272e7	−0.562108
$862$	0	0
$863$	−2.78981e7	−1.27511	−0.637555	−	0.770405i	$-0.720054\pi$
$863$	−2.78981e7	−1.27511	−0.637555	+	0.770405i	$0.720054\pi$
$864$	0	0
$865$	−2.31864e6	−0.105364
$866$	0	0
$867$	8.02220e7	3.62448
$868$	0	0
$869$	5.12103e7	2.30042
$870$	0	0
$871$	7.38946e6	0.330041
$872$	0	0
$873$	−9.30222e6	−0.413096
$874$	0	0
$875$	799482.	0.0353011
$876$	0	0
$877$	6.79883e6	0.298494	0.149247	−	0.988800i	$-0.452315\pi$
$877$	6.79883e6	0.298494	0.149247	+	0.988800i	$0.452315\pi$
$878$	0	0
$879$	8.81622e6	0.384867
$880$	0	0
$881$	−4.02331e7	−1.74640	−0.873201	−	0.487360i	$-0.837960\pi$
$881$	−4.02331e7	−1.74640	−0.873201	+	0.487360i	$0.837960\pi$
$882$	0	0
$883$	−7.81137e6	−0.337152	−0.168576	−	0.985689i	$-0.553917\pi$
$883$	−7.81137e6	−0.337152	−0.168576	+	0.985689i	$0.553917\pi$
$884$	0	0
$885$	−2.40648e7	−1.03282
$886$	0	0
$887$	2.69396e7	1.14970	0.574848	−	0.818260i	$-0.305061\pi$
$887$	2.69396e7	1.14970	0.574848	+	0.818260i	$0.305061\pi$
$888$	0	0
$889$	−4.40189e6	−0.186803
$890$	0	0
$891$	3.64947e7	1.54005
$892$	0	0
$893$	−2.51878e6	−0.105697
$894$	0	0
$895$	−5.67129e6	−0.236660
$896$	0	0
$897$	−1.23774e7	−0.513626
$898$	0	0
$899$	2.05720e7	0.848942
$900$	0	0
$901$	−1.78478e7	−0.732440
$902$	0	0
$903$	1.23674e7	0.504731
$904$	0	0
$905$	1.72872e7	0.701621
$906$	0	0
$907$	−3.66946e7	−1.48110	−0.740549	−	0.672002i	$-0.765435\pi$
$907$	−3.66946e7	−1.48110	−0.740549	+	0.672002i	$0.765435\pi$
$908$	0	0
$909$	1.23520e7	0.495826
$910$	0	0
$911$	1.68568e7	0.672945	0.336473	−	0.941693i	$-0.390766\pi$
$911$	1.68568e7	0.672945	0.336473	+	0.941693i	$0.390766\pi$
$912$	0	0
$913$	−3.97635e7	−1.57873
$914$	0	0
$915$	−1.18987e7	−0.469836
$916$	0	0
$917$	1.52296e7	0.598090
$918$	0	0
$919$	−3.12525e7	−1.22066	−0.610332	−	0.792146i	$-0.708964\pi$
$919$	−3.12525e7	−1.22066	−0.610332	+	0.792146i	$0.708964\pi$
$920$	0	0
$921$	−4.79421e7	−1.86238
$922$	0	0
$923$	1.28496e6	0.0496463
$924$	0	0
$925$	−2.80456e6	−0.107773
$926$	0	0
$927$	−3.92345e6	−0.149958
$928$	0	0
$929$	−3.27128e7	−1.24359	−0.621797	−	0.783178i	$-0.713597\pi$
$929$	−3.27128e7	−1.24359	−0.621797	+	0.783178i	$0.713597\pi$
$930$	0	0
$931$	−9.92387e6	−0.375238
$932$	0	0
$933$	3.35506e7	1.26182
$934$	0	0
$935$	2.92422e7	1.09391
$936$	0	0
$937$	4.53506e7	1.68746	0.843731	−	0.536767i	$-0.180355\pi$
$937$	4.53506e7	1.68746	0.843731	+	0.536767i	$0.180355\pi$
$938$	0	0
$939$	1.52247e7	0.563490
$940$	0	0
$941$	−4.45649e7	−1.64066	−0.820331	−	0.571890i	$-0.806211\pi$
$941$	−4.45649e7	−1.64066	−0.820331	+	0.571890i	$0.806211\pi$
$942$	0	0
$943$	−4.73532e7	−1.73408
$944$	0	0
$945$	−2.86252e6	−0.104272
$946$	0	0
$947$	−1.90646e7	−0.690800	−0.345400	−	0.938456i	$-0.612257\pi$
$947$	−1.90646e7	−0.690800	−0.345400	+	0.938456i	$0.612257\pi$
$948$	0	0
$949$	3.65432e6	0.131717
$950$	0	0
$951$	−5.26392e7	−1.88737
$952$	0	0
$953$	4.22576e6	0.150721	0.0753603	−	0.997156i	$-0.475989\pi$
$953$	4.22576e6	0.150721	0.0753603	+	0.997156i	$0.475989\pi$
$954$	0	0
$955$	8.51529e6	0.302128
$956$	0	0
$957$	−2.15234e7	−0.759680
$958$	0	0
$959$	−3.86980e6	−0.135876
$960$	0	0
$961$	5.39718e7	1.88520
$962$	0	0
$963$	−2.55780e7	−0.888794
$964$	0	0
$965$	−1.19960e7	−0.414684
$966$	0	0
$967$	3.50435e7	1.20515	0.602575	−	0.798062i	$-0.294141\pi$
$967$	3.50435e7	1.20515	0.602575	+	0.798062i	$0.294141\pi$
$968$	0	0
$969$	−3.17984e7	−1.08792
$970$	0	0
$971$	−7.67176e6	−0.261124	−0.130562	−	0.991440i	$-0.541678\pi$
$971$	−7.67176e6	−0.261124	−0.130562	+	0.991440i	$0.541678\pi$
$972$	0	0
$973$	−2.99680e6	−0.101479
$974$	0	0
$975$	−2.03064e6	−0.0684102
$976$	0	0
$977$	−4.64539e7	−1.55699	−0.778494	−	0.627652i	$-0.784016\pi$
$977$	−4.64539e7	−1.55699	−0.778494	+	0.627652i	$0.784016\pi$
$978$	0	0
$979$	−2.93307e7	−0.978061
$980$	0	0
$981$	−2.16675e7	−0.718846
$982$	0	0
$983$	−1.52982e7	−0.504959	−0.252480	−	0.967602i	$-0.581246\pi$
$983$	−1.52982e7	−0.504959	−0.252480	+	0.967602i	$0.581246\pi$
$984$	0	0
$985$	1.93859e7	0.636643
$986$	0	0
$987$	−3.54254e6	−0.115750
$988$	0	0
$989$	4.78961e7	1.55708
$990$	0	0
$991$	8.70500e6	0.281569	0.140784	−	0.990040i	$-0.455038\pi$
$991$	8.70500e6	0.281569	0.140784	+	0.990040i	$0.455038\pi$
$992$	0	0
$993$	−2.58649e6	−0.0832409
$994$	0	0
$995$	−2.40201e7	−0.769161
$996$	0	0
$997$	−4.01903e7	−1.28051	−0.640256	−	0.768162i	$-0.721172\pi$
$997$	−4.01903e7	−1.28051	−0.640256	+	0.768162i	$0.721172\pi$
$998$	0	0
$999$	1.00416e7	0.318340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.h.1.2		2
4.3	odd	2		130.6.a.e.1.1	✓	2
20.3	even	4		650.6.b.c.599.1		4
20.7	even	4		650.6.b.c.599.4		4
20.19	odd	2		650.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.e.1.1	✓	2	4.3	odd	2
650.6.a.c.1.2		2	20.19	odd	2
650.6.b.c.599.1		4	20.3	even	4
650.6.b.c.599.4		4	20.7	even	4
1040.6.a.h.1.2		2	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+19.2250 q^{3} +25.0000 q^{5} +51.1669 q^{7} +126.600 q^{9} -494.607 q^{11} -169.000 q^{13} +480.624 q^{15} -2364.88 q^{17} +699.408 q^{19} +983.681 q^{21} +3809.57 q^{23} +625.000 q^{25} -2237.80 q^{27} +2263.52 q^{29} +9088.51 q^{31} -9508.81 q^{33} +1279.17 q^{35} -4487.29 q^{37} -3249.02 q^{39} -12430.1 q^{41} +12572.6 q^{43} +3164.99 q^{45} -3601.31 q^{47} -14189.0 q^{49} -45464.8 q^{51} +7547.01 q^{53} -12365.2 q^{55} +13446.1 q^{57} -50069.9 q^{59} -24756.7 q^{61} +6477.70 q^{63} -4225.00 q^{65} -43724.6 q^{67} +73238.8 q^{69} -7603.34 q^{71} -21623.2 q^{73} +12015.6 q^{75} -25307.5 q^{77} -103537. q^{79} -73785.2 q^{81} +80394.0 q^{83} -59122.0 q^{85} +43516.1 q^{87} +59301.0 q^{89} -8647.20 q^{91} +174726. q^{93} +17485.2 q^{95} -73477.5 q^{97} -62617.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{3} + 50 q^{5} + 252 q^{7} - 106 q^{9} + 36 q^{11} - 338 q^{13} + 400 q^{15} - 2380 q^{17} + 1092 q^{19} + 336 q^{21} + 1176 q^{23} + 1250 q^{25} - 704 q^{27} - 4872 q^{29} + 2260 q^{31} - 11220 q^{33}+ \cdots - 186036 q^{99}+O(q^{100}) \) 2 * q + 16 * q^3 + 50 * q^5 + 252 * q^7 - 106 * q^9 + 36 * q^11 - 338 * q^13 + 400 * q^15 - 2380 * q^17 + 1092 * q^19 + 336 * q^21 + 1176 * q^23 + 1250 * q^25 - 704 * q^27 - 4872 * q^29 + 2260 * q^31 - 11220 * q^33 + 6300 * q^35 - 17760 * q^37 - 2704 * q^39 - 9220 * q^41 + 23656 * q^43 - 2650 * q^45 - 12860 * q^47 + 9338 * q^49 - 45416 * q^51 + 11068 * q^53 + 900 * q^55 + 12180 * q^57 - 90404 * q^59 - 69000 * q^61 - 40236 * q^63 - 8450 * q^65 - 50796 * q^67 + 81732 * q^69 + 28668 * q^71 - 62688 * q^73 + 10000 * q^75 + 81256 * q^77 - 84064 * q^79 - 22210 * q^81 + 12828 * q^83 - 59500 * q^85 + 66528 * q^87 + 92620 * q^89 - 42588 * q^91 + 196748 * q^93 + 27300 * q^95 - 75684 * q^97 - 186036 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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