LMFDB - Embedded newform 832.6.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,6,Mod(1,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("832.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	832.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:	$133.439338084$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	832.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63068 q^{3} +103.462 q^{5} -126.309 q^{7} -240.341 q^{9} -14.8296 q^{11} +169.000 q^{13} -168.714 q^{15} -1051.12 q^{17} -213.723 q^{19} +205.969 q^{21} +4231.16 q^{23} +7579.41 q^{25} +788.176 q^{27} +504.955 q^{29} -4783.58 q^{31} +24.1823 q^{33} -13068.2 q^{35} +4635.74 q^{37} -275.585 q^{39} +7944.15 q^{41} -8516.41 q^{43} -24866.2 q^{45} -24921.2 q^{47} -853.113 q^{49} +1714.05 q^{51} +7808.46 q^{53} -1534.30 q^{55} +348.515 q^{57} -37337.5 q^{59} +18172.2 q^{61} +30357.1 q^{63} +17485.1 q^{65} -34559.9 q^{67} -6899.68 q^{69} -41255.7 q^{71} -1056.42 q^{73} -12359.6 q^{75} +1873.10 q^{77} +47719.3 q^{79} +57117.6 q^{81} -74799.0 q^{83} -108751. q^{85} -823.421 q^{87} +9799.26 q^{89} -21346.2 q^{91} +7800.50 q^{93} -22112.3 q^{95} -138432. q^{97} +3564.15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 28 q^{3} + 42 q^{5} + 36 q^{7} + 212 q^{9} - 376 q^{11} + 338 q^{13} + 1452 q^{15} - 2630 q^{17} - 312 q^{19} - 4074 q^{21} + 2624 q^{23} + 8232 q^{25} - 4732 q^{27} + 812 q^{29} - 7720 q^{31} + 9548 q^{33}+ \cdots - 159808 q^{99}+O(q^{100}) $ 2 * q - 28 * q^3 + 42 * q^5 + 36 * q^7 + 212 * q^9 - 376 * q^11 + 338 * q^13 + 1452 * q^15 - 2630 * q^17 - 312 * q^19 - 4074 * q^21 + 2624 * q^23 + 8232 * q^25 - 4732 * q^27 + 812 * q^29 - 7720 * q^31 + 9548 * q^33 - 23044 * q^35 + 16858 * q^37 - 4732 * q^39 + 7840 * q^41 + 2420 * q^43 - 52668 * q^45 - 9972 * q^47 + 8684 * q^49 + 43348 * q^51 + 43720 * q^53 + 20664 * q^55 + 2940 * q^57 - 38936 * q^59 - 1984 * q^61 + 103776 * q^63 + 7098 * q^65 - 69928 * q^67 + 35480 * q^69 - 67396 * q^71 + 74412 * q^73 - 29568 * q^75 - 56748 * q^77 + 55296 * q^79 + 92762 * q^81 - 75712 * q^83 - 11710 * q^85 - 8920 * q^87 + 116508 * q^89 + 6084 * q^91 + 85232 * q^93 - 16072 * q^95 - 34756 * q^97 - 159808 * 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
$3$	−1.63068	−0.104608	−0.0523042	−	0.998631i	$-0.516657\pi$
$3$	−1.63068	−0.104608	−0.0523042	+	0.998631i	$0.516657\pi$
$4$	0	0
$5$	103.462	1.85079	0.925393	−	0.379008i	$-0.123735\pi$
$5$	103.462	1.85079	0.925393	+	0.379008i	$0.123735\pi$
$6$	0	0
$7$	−126.309	−0.974290	−0.487145	−	0.873321i	$-0.661962\pi$
$7$	−126.309	−0.974290	−0.487145	+	0.873321i	$0.661962\pi$
$8$	0	0
$9$	−240.341	−0.989057
$10$	0	0
$11$	−14.8296	−0.0369527	−0.0184764	−	0.999829i	$-0.505882\pi$
$11$	−14.8296	−0.0369527	−0.0184764	+	0.999829i	$0.505882\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−168.714	−0.193608
$16$	0	0
$17$	−1051.12	−0.882126	−0.441063	−	0.897476i	$-0.645398\pi$
$17$	−1051.12	−0.882126	−0.441063	+	0.897476i	$0.645398\pi$
$18$	0	0
$19$	−213.723	−0.135821	−0.0679107	−	0.997691i	$-0.521633\pi$
$19$	−213.723	−0.135821	−0.0679107	+	0.997691i	$0.521633\pi$
$20$	0	0
$21$	205.969	0.101919
$22$	0	0
$23$	4231.16	1.66778	0.833892	−	0.551928i	$-0.186108\pi$
$23$	4231.16	1.66778	0.833892	+	0.551928i	$0.186108\pi$
$24$	0	0
$25$	7579.41	2.42541
$26$	0	0
$27$	788.176	0.208072
$28$	0	0
$29$	504.955	0.111495	0.0557477	−	0.998445i	$-0.482246\pi$
$29$	504.955	0.111495	0.0557477	+	0.998445i	$0.482246\pi$
$30$	0	0
$31$	−4783.58	−0.894022	−0.447011	−	0.894528i	$-0.647512\pi$
$31$	−4783.58	−0.894022	−0.447011	+	0.894528i	$0.647512\pi$
$32$	0	0
$33$	24.1823	0.00386557
$34$	0	0
$35$	−13068.2	−1.80320
$36$	0	0
$37$	4635.74	0.556692	0.278346	−	0.960481i	$-0.410214\pi$
$37$	4635.74	0.556692	0.278346	+	0.960481i	$0.410214\pi$
$38$	0	0
$39$	−275.585	−0.0290131
$40$	0	0
$41$	7944.15	0.738054	0.369027	−	0.929419i	$-0.379691\pi$
$41$	7944.15	0.738054	0.369027	+	0.929419i	$0.379691\pi$
$42$	0	0
$43$	−8516.41	−0.702401	−0.351201	−	0.936300i	$-0.614226\pi$
$43$	−8516.41	−0.702401	−0.351201	+	0.936300i	$0.614226\pi$
$44$	0	0
$45$	−24866.2	−1.83053
$46$	0	0
$47$	−24921.2	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$47$	−24921.2	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$48$	0	0
$49$	−853.113	−0.0507594
$50$	0	0
$51$	1714.05	0.0922777
$52$	0	0
$53$	7808.46	0.381835	0.190917	−	0.981606i	$-0.438854\pi$
$53$	7808.46	0.381835	0.190917	+	0.981606i	$0.438854\pi$
$54$	0	0
$55$	−1534.30	−0.0683916
$56$	0	0
$57$	348.515	0.0142081
$58$	0	0
$59$	−37337.5	−1.39642	−0.698209	−	0.715894i	$-0.746020\pi$
$59$	−37337.5	−1.39642	−0.698209	+	0.715894i	$0.746020\pi$
$60$	0	0
$61$	18172.2	0.625292	0.312646	−	0.949870i	$-0.398785\pi$
$61$	18172.2	0.625292	0.312646	+	0.949870i	$0.398785\pi$
$62$	0	0
$63$	30357.1	0.963628
$64$	0	0
$65$	17485.1	0.513316
$66$	0	0
$67$	−34559.9	−0.940559	−0.470279	−	0.882518i	$-0.655847\pi$
$67$	−34559.9	−0.940559	−0.470279	+	0.882518i	$0.655847\pi$
$68$	0	0
$69$	−6899.68	−0.174464
$70$	0	0
$71$	−41255.7	−0.971265	−0.485632	−	0.874163i	$-0.661411\pi$
$71$	−41255.7	−0.971265	−0.485632	+	0.874163i	$0.661411\pi$
$72$	0	0
$73$	−1056.42	−0.0232022	−0.0116011	−	0.999933i	$-0.503693\pi$
$73$	−1056.42	−0.0232022	−0.0116011	+	0.999933i	$0.503693\pi$
$74$	0	0
$75$	−12359.6	−0.253718
$76$	0	0
$77$	1873.10	0.0360027
$78$	0	0
$79$	47719.3	0.860253	0.430126	−	0.902769i	$-0.358469\pi$
$79$	47719.3	0.860253	0.430126	+	0.902769i	$0.358469\pi$
$80$	0	0
$81$	57117.6	0.967291
$82$	0	0
$83$	−74799.0	−1.19179	−0.595896	−	0.803061i	$-0.703203\pi$
$83$	−74799.0	−1.19179	−0.595896	+	0.803061i	$0.703203\pi$
$84$	0	0
$85$	−108751.	−1.63263
$86$	0	0
$87$	−823.421	−0.0116634
$88$	0	0
$89$	9799.26	0.131135	0.0655675	−	0.997848i	$-0.479114\pi$
$89$	9799.26	0.131135	0.0655675	+	0.997848i	$0.479114\pi$
$90$	0	0
$91$	−21346.2	−0.270219
$92$	0	0
$93$	7800.50	0.0935222
$94$	0	0
$95$	−22112.3	−0.251376
$96$	0	0
$97$	−138432.	−1.49385	−0.746927	−	0.664906i	$-0.768472\pi$
$97$	−138432.	−1.49385	−0.746927	+	0.664906i	$0.768472\pi$
$98$	0	0
$99$	3564.15	0.0365484
$100$	0	0
$101$	−139151.	−1.35733	−0.678663	−	0.734450i	$-0.737440\pi$
$101$	−139151.	−1.35733	−0.678663	+	0.734450i	$0.737440\pi$
$102$	0	0
$103$	−98512.2	−0.914950	−0.457475	−	0.889223i	$-0.651246\pi$
$103$	−98512.2	−0.914950	−0.457475	+	0.889223i	$0.651246\pi$
$104$	0	0
$105$	21310.0	0.188630
$106$	0	0
$107$	−26848.4	−0.226704	−0.113352	−	0.993555i	$-0.536159\pi$
$107$	−26848.4	−0.226704	−0.113352	+	0.993555i	$0.536159\pi$
$108$	0	0
$109$	63220.9	0.509676	0.254838	−	0.966984i	$-0.417978\pi$
$109$	63220.9	0.509676	0.254838	+	0.966984i	$0.417978\pi$
$110$	0	0
$111$	−7559.43	−0.0582346
$112$	0	0
$113$	114434.	0.843058	0.421529	−	0.906815i	$-0.361494\pi$
$113$	114434.	0.843058	0.421529	+	0.906815i	$0.361494\pi$
$114$	0	0
$115$	437765.	3.08671
$116$	0	0
$117$	−40617.6	−0.274315
$118$	0	0
$119$	132766.	0.859446
$120$	0	0
$121$	−160831.	−0.998634
$122$	0	0
$123$	−12954.4	−0.0772066
$124$	0	0
$125$	460863.	2.63813
$126$	0	0
$127$	248871.	1.36919	0.684596	−	0.728922i	$-0.259978\pi$
$127$	248871.	1.36919	0.684596	+	0.728922i	$0.259978\pi$
$128$	0	0
$129$	13887.6	0.0734770
$130$	0	0
$131$	102963.	0.524205	0.262102	−	0.965040i	$-0.415584\pi$
$131$	102963.	0.524205	0.262102	+	0.965040i	$0.415584\pi$
$132$	0	0
$133$	26995.1	0.132329
$134$	0	0
$135$	81546.3	0.385097
$136$	0	0
$137$	−36037.4	−0.164041	−0.0820204	−	0.996631i	$-0.526137\pi$
$137$	−36037.4	−0.164041	−0.0820204	+	0.996631i	$0.526137\pi$
$138$	0	0
$139$	152655.	0.670151	0.335076	−	0.942191i	$-0.391238\pi$
$139$	152655.	0.670151	0.335076	+	0.942191i	$0.391238\pi$
$140$	0	0
$141$	40638.6	0.172144
$142$	0	0
$143$	−2506.20	−0.0102488
$144$	0	0
$145$	52243.7	0.206354
$146$	0	0
$147$	1391.16	0.00530986
$148$	0	0
$149$	−72547.1	−0.267704	−0.133852	−	0.991001i	$-0.542735\pi$
$149$	−72547.1	−0.267704	−0.133852	+	0.991001i	$0.542735\pi$
$150$	0	0
$151$	−489021.	−1.74536	−0.872681	−	0.488291i	$-0.837621\pi$
$151$	−489021.	−1.74536	−0.872681	+	0.488291i	$0.837621\pi$
$152$	0	0
$153$	252627.	0.872473
$154$	0	0
$155$	−494919.	−1.65464
$156$	0	0
$157$	−89467.9	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$157$	−89467.9	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$158$	0	0
$159$	−12733.1	−0.0399431
$160$	0	0
$161$	−534432.	−1.62490
$162$	0	0
$163$	−225668.	−0.665275	−0.332637	−	0.943055i	$-0.607939\pi$
$163$	−225668.	−0.665275	−0.332637	+	0.943055i	$0.607939\pi$
$164$	0	0
$165$	2501.95	0.00715434
$166$	0	0
$167$	−209528.	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$167$	−209528.	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	51366.5	0.134335
$172$	0	0
$173$	465184.	1.18171	0.590853	−	0.806779i	$-0.298791\pi$
$173$	465184.	1.18171	0.590853	+	0.806779i	$0.298791\pi$
$174$	0	0
$175$	−957345.	−2.36305
$176$	0	0
$177$	60885.7	0.146077
$178$	0	0
$179$	−472573.	−1.10239	−0.551197	−	0.834375i	$-0.685829\pi$
$179$	−472573.	−1.10239	−0.551197	+	0.834375i	$0.685829\pi$
$180$	0	0
$181$	74099.4	0.168120	0.0840598	−	0.996461i	$-0.473211\pi$
$181$	74099.4	0.168120	0.0840598	+	0.996461i	$0.473211\pi$
$182$	0	0
$183$	−29633.1	−0.0654108
$184$	0	0
$185$	479624.	1.03032
$186$	0	0
$187$	15587.7	0.0325970
$188$	0	0
$189$	−99553.5	−0.202722
$190$	0	0
$191$	224128.	0.444543	0.222271	−	0.974985i	$-0.428653\pi$
$191$	224128.	0.444543	0.222271	+	0.974985i	$0.428653\pi$
$192$	0	0
$193$	−662265.	−1.27979	−0.639895	−	0.768463i	$-0.721022\pi$
$193$	−662265.	−1.27979	−0.639895	+	0.768463i	$0.721022\pi$
$194$	0	0
$195$	−28512.7	−0.0536971
$196$	0	0
$197$	−666464.	−1.22352	−0.611760	−	0.791043i	$-0.709538\pi$
$197$	−666464.	−1.22352	−0.611760	+	0.791043i	$0.709538\pi$
$198$	0	0
$199$	−645720.	−1.15588	−0.577938	−	0.816081i	$-0.696142\pi$
$199$	−645720.	−1.15588	−0.577938	+	0.816081i	$0.696142\pi$
$200$	0	0
$201$	56356.3	0.0983903
$202$	0	0
$203$	−63780.1	−0.108629
$204$	0	0
$205$	821919.	1.36598
$206$	0	0
$207$	−1.01692e6	−1.64953
$208$	0	0
$209$	3169.43	0.00501897
$210$	0	0
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	0	0
$213$	67274.9	0.101602
$214$	0	0
$215$	−881125.	−1.29999
$216$	0	0
$217$	604207.	0.871037
$218$	0	0
$219$	1722.69	0.00242715
$220$	0	0
$221$	−177639.	−0.244658
$222$	0	0
$223$	58342.9	0.0785644	0.0392822	−	0.999228i	$-0.487493\pi$
$223$	58342.9	0.0785644	0.0392822	+	0.999228i	$0.487493\pi$
$224$	0	0
$225$	−1.82164e6	−2.39887
$226$	0	0
$227$	111768.	0.143964	0.0719820	−	0.997406i	$-0.477068\pi$
$227$	111768.	0.143964	0.0719820	+	0.997406i	$0.477068\pi$
$228$	0	0
$229$	−1.14984e6	−1.44893	−0.724467	−	0.689309i	$-0.757914\pi$
$229$	−1.14984e6	−1.44893	−0.724467	+	0.689309i	$0.757914\pi$
$230$	0	0
$231$	−3054.44	−0.00376618
$232$	0	0
$233$	630470.	0.760807	0.380404	−	0.924821i	$-0.375785\pi$
$233$	630470.	0.760807	0.380404	+	0.924821i	$0.375785\pi$
$234$	0	0
$235$	−2.57840e6	−3.04566
$236$	0	0
$237$	−77815.0	−0.0899897
$238$	0	0
$239$	165628.	0.187559	0.0937797	−	0.995593i	$-0.470105\pi$
$239$	165628.	0.187559	0.0937797	+	0.995593i	$0.470105\pi$
$240$	0	0
$241$	690968.	0.766329	0.383165	−	0.923680i	$-0.374834\pi$
$241$	690968.	0.766329	0.383165	+	0.923680i	$0.374834\pi$
$242$	0	0
$243$	−284667.	−0.309259
$244$	0	0
$245$	−88264.9	−0.0939448
$246$	0	0
$247$	−36119.3	−0.0376701
$248$	0	0
$249$	121974.	0.124671
$250$	0	0
$251$	−386887.	−0.387615	−0.193807	−	0.981040i	$-0.562084\pi$
$251$	−386887.	−0.387615	−0.193807	+	0.981040i	$0.562084\pi$
$252$	0	0
$253$	−62746.2	−0.0616292
$254$	0	0
$255$	177339.	0.170786
$256$	0	0
$257$	258260.	0.243907	0.121953	−	0.992536i	$-0.461084\pi$
$257$	258260.	0.243907	0.121953	+	0.992536i	$0.461084\pi$
$258$	0	0
$259$	−585535.	−0.542379
$260$	0	0
$261$	−121361.	−0.110275
$262$	0	0
$263$	1.19053e6	1.06133	0.530665	−	0.847582i	$-0.321942\pi$
$263$	1.19053e6	1.06133	0.530665	+	0.847582i	$0.321942\pi$
$264$	0	0
$265$	807879.	0.706695
$266$	0	0
$267$	−15979.5	−0.0137178
$268$	0	0
$269$	−850968.	−0.717022	−0.358511	−	0.933525i	$-0.616715\pi$
$269$	−850968.	−0.717022	−0.358511	+	0.933525i	$0.616715\pi$
$270$	0	0
$271$	40926.1	0.0338514	0.0169257	−	0.999857i	$-0.494612\pi$
$271$	40926.1	0.0338514	0.0169257	+	0.999857i	$0.494612\pi$
$272$	0	0
$273$	34808.8	0.0282672
$274$	0	0
$275$	−112399.	−0.0896256
$276$	0	0
$277$	−1.00054e6	−0.783490	−0.391745	−	0.920074i	$-0.628128\pi$
$277$	−1.00054e6	−0.783490	−0.391745	+	0.920074i	$0.628128\pi$
$278$	0	0
$279$	1.14969e6	0.884239
$280$	0	0
$281$	−1.73596e6	−1.31151	−0.655757	−	0.754972i	$-0.727650\pi$
$281$	−1.73596e6	−1.31151	−0.655757	+	0.754972i	$0.727650\pi$
$282$	0	0
$283$	−1.27363e6	−0.945319	−0.472660	−	0.881245i	$-0.656706\pi$
$283$	−1.27363e6	−0.945319	−0.472660	+	0.881245i	$0.656706\pi$
$284$	0	0
$285$	36058.1	0.0262961
$286$	0	0
$287$	−1.00342e6	−0.719078
$288$	0	0
$289$	−315001.	−0.221854
$290$	0	0
$291$	225739.	0.156270
$292$	0	0
$293$	2043.70	0.00139075	0.000695374	−	1.00000i	$-0.499779\pi$
$293$	2043.70	0.00139075	0.000695374		1.00000i	$0.499779\pi$
$294$	0	0
$295$	−3.86302e6	−2.58447
$296$	0	0
$297$	−11688.3	−0.00768883
$298$	0	0
$299$	715066.	0.462560
$300$	0	0
$301$	1.07570e6	0.684342
$302$	0	0
$303$	226912.	0.141988
$304$	0	0
$305$	1.88013e6	1.15728
$306$	0	0
$307$	−401308.	−0.243014	−0.121507	−	0.992591i	$-0.538773\pi$
$307$	−401308.	−0.243014	−0.121507	+	0.992591i	$0.538773\pi$
$308$	0	0
$309$	160642.	0.0957114
$310$	0	0
$311$	1.92628e6	1.12933	0.564663	−	0.825322i	$-0.309006\pi$
$311$	1.92628e6	1.12933	0.564663	+	0.825322i	$0.309006\pi$
$312$	0	0
$313$	1.64519e6	0.949196	0.474598	−	0.880203i	$-0.342593\pi$
$313$	1.64519e6	0.949196	0.474598	+	0.880203i	$0.342593\pi$
$314$	0	0
$315$	3.14081e6	1.78347
$316$	0	0
$317$	1.99476e6	1.11492	0.557459	−	0.830205i	$-0.311777\pi$
$317$	1.99476e6	1.11492	0.557459	+	0.830205i	$0.311777\pi$
$318$	0	0
$319$	−7488.26	−0.00412006
$320$	0	0
$321$	43781.2	0.0237151
$322$	0	0
$323$	224649.	0.119812
$324$	0	0
$325$	1.28092e6	0.672688
$326$	0	0
$327$	−103093.	−0.0533164
$328$	0	0
$329$	3.14777e6	1.60329
$330$	0	0
$331$	−675924.	−0.339100	−0.169550	−	0.985522i	$-0.554231\pi$
$331$	−675924.	−0.339100	−0.169550	+	0.985522i	$0.554231\pi$
$332$	0	0
$333$	−1.11416e6	−0.550600
$334$	0	0
$335$	−3.57564e6	−1.74077
$336$	0	0
$337$	−2.13552e6	−1.02430	−0.512152	−	0.858895i	$-0.671152\pi$
$337$	−2.13552e6	−1.02430	−0.512152	+	0.858895i	$0.671152\pi$
$338$	0	0
$339$	−186605.	−0.0881909
$340$	0	0
$341$	70938.3	0.0330366
$342$	0	0
$343$	2.23063e6	1.02374
$344$	0	0
$345$	−713855.	−0.322896
$346$	0	0
$347$	−2.57257e6	−1.14695	−0.573473	−	0.819225i	$-0.694404\pi$
$347$	−2.57257e6	−1.14695	−0.573473	+	0.819225i	$0.694404\pi$
$348$	0	0
$349$	−2.02363e6	−0.889339	−0.444670	−	0.895695i	$-0.646679\pi$
$349$	−2.02363e6	−0.889339	−0.444670	+	0.895695i	$0.646679\pi$
$350$	0	0
$351$	133202.	0.0577088
$352$	0	0
$353$	2.04810e6	0.874810	0.437405	−	0.899265i	$-0.355898\pi$
$353$	2.04810e6	0.874810	0.437405	+	0.899265i	$0.355898\pi$
$354$	0	0
$355$	−4.26840e6	−1.79760
$356$	0	0
$357$	−216499.	−0.0899053
$358$	0	0
$359$	1.59901e6	0.654808	0.327404	−	0.944885i	$-0.393826\pi$
$359$	1.59901e6	0.654808	0.327404	+	0.944885i	$0.393826\pi$
$360$	0	0
$361$	−2.43042e6	−0.981553
$362$	0	0
$363$	262265.	0.104466
$364$	0	0
$365$	−109299.	−0.0429424
$366$	0	0
$367$	3.86389e6	1.49747	0.748737	−	0.662867i	$-0.230661\pi$
$367$	3.86389e6	1.49747	0.748737	+	0.662867i	$0.230661\pi$
$368$	0	0
$369$	−1.90930e6	−0.729977
$370$	0	0
$371$	−986276.	−0.372018
$372$	0	0
$373$	1.56702e6	0.583179	0.291589	−	0.956544i	$-0.405816\pi$
$373$	1.56702e6	0.583179	0.291589	+	0.956544i	$0.405816\pi$
$374$	0	0
$375$	−751521.	−0.275971
$376$	0	0
$377$	85337.3	0.0309233
$378$	0	0
$379$	3.19239e6	1.14161	0.570805	−	0.821086i	$-0.306631\pi$
$379$	3.19239e6	1.14161	0.570805	+	0.821086i	$0.306631\pi$
$380$	0	0
$381$	−405829.	−0.143229
$382$	0	0
$383$	−400432.	−0.139486	−0.0697432	−	0.997565i	$-0.522218\pi$
$383$	−400432.	−0.139486	−0.0697432	+	0.997565i	$0.522218\pi$
$384$	0	0
$385$	193795.	0.0666333
$386$	0	0
$387$	2.04684e6	0.694715
$388$	0	0
$389$	−413440.	−0.138528	−0.0692642	−	0.997598i	$-0.522065\pi$
$389$	−413440.	−0.138528	−0.0692642	+	0.997598i	$0.522065\pi$
$390$	0	0
$391$	−4.44746e6	−1.47120
$392$	0	0
$393$	−167899.	−0.0548362
$394$	0	0
$395$	4.93714e6	1.59214
$396$	0	0
$397$	102926.	0.0327753	0.0163877	−	0.999866i	$-0.494783\pi$
$397$	102926.	0.0327753	0.0163877	+	0.999866i	$0.494783\pi$
$398$	0	0
$399$	−44020.5	−0.0138428
$400$	0	0
$401$	2.23365e6	0.693671	0.346836	−	0.937926i	$-0.387256\pi$
$401$	2.23365e6	0.693671	0.346836	+	0.937926i	$0.387256\pi$
$402$	0	0
$403$	−808424.	−0.247957
$404$	0	0
$405$	5.90950e6	1.79025
$406$	0	0
$407$	−68746.0	−0.0205713
$408$	0	0
$409$	−4.46150e6	−1.31878	−0.659390	−	0.751801i	$-0.729185\pi$
$409$	−4.46150e6	−1.31878	−0.659390	+	0.751801i	$0.729185\pi$
$410$	0	0
$411$	58765.6	0.0171600
$412$	0	0
$413$	4.71606e6	1.36052
$414$	0	0
$415$	−7.73887e6	−2.20575
$416$	0	0
$417$	−248931.	−0.0701034
$418$	0	0
$419$	4.22792e6	1.17650	0.588250	−	0.808679i	$-0.299817\pi$
$419$	4.22792e6	1.17650	0.588250	+	0.808679i	$0.299817\pi$
$420$	0	0
$421$	−4.11791e6	−1.13233	−0.566163	−	0.824293i	$-0.691573\pi$
$421$	−4.11791e6	−1.13233	−0.566163	+	0.824293i	$0.691573\pi$
$422$	0	0
$423$	5.98959e6	1.62759
$424$	0	0
$425$	−7.96688e6	−2.13952
$426$	0	0
$427$	−2.29531e6	−0.609216
$428$	0	0
$429$	4086.81	0.00107212
$430$	0	0
$431$	−1.15324e6	−0.299038	−0.149519	−	0.988759i	$-0.547773\pi$
$431$	−1.15324e6	−0.299038	−0.149519	+	0.988759i	$0.547773\pi$
$432$	0	0
$433$	−33734.3	−0.00864673	−0.00432337	−	0.999991i	$-0.501376\pi$
$433$	−33734.3	−0.00864673	−0.00432337	+	0.999991i	$0.501376\pi$
$434$	0	0
$435$	−85192.9	−0.0215864
$436$	0	0
$437$	−904298.	−0.226521
$438$	0	0
$439$	−7.48363e6	−1.85332	−0.926661	−	0.375898i	$-0.877334\pi$
$439$	−7.48363e6	−1.85332	−0.926661	+	0.375898i	$0.877334\pi$
$440$	0	0
$441$	205038.	0.0502039
$442$	0	0
$443$	−3.28028e6	−0.794148	−0.397074	−	0.917786i	$-0.629974\pi$
$443$	−3.28028e6	−0.794148	−0.397074	+	0.917786i	$0.629974\pi$
$444$	0	0
$445$	1.01385e6	0.242703
$446$	0	0
$447$	118301.	0.0280040
$448$	0	0
$449$	7.95356e6	1.86185	0.930927	−	0.365205i	$-0.119001\pi$
$449$	7.95356e6	1.86185	0.930927	+	0.365205i	$0.119001\pi$
$450$	0	0
$451$	−117808.	−0.0272731
$452$	0	0
$453$	797439.	0.182579
$454$	0	0
$455$	−2.20852e6	−0.500118
$456$	0	0
$457$	−3.35187e6	−0.750753	−0.375377	−	0.926872i	$-0.622487\pi$
$457$	−3.35187e6	−0.750753	−0.375377	+	0.926872i	$0.622487\pi$
$458$	0	0
$459$	−828468.	−0.183546
$460$	0	0
$461$	−4.68627e6	−1.02701	−0.513505	−	0.858086i	$-0.671653\pi$
$461$	−4.68627e6	−1.02701	−0.513505	+	0.858086i	$0.671653\pi$
$462$	0	0
$463$	−6.64697e6	−1.44102	−0.720512	−	0.693442i	$-0.756093\pi$
$463$	−6.64697e6	−1.44102	−0.720512	+	0.693442i	$0.756093\pi$
$464$	0	0
$465$	807056.	0.173090
$466$	0	0
$467$	−3.14141e6	−0.666549	−0.333275	−	0.942830i	$-0.608154\pi$
$467$	−3.14141e6	−0.666549	−0.333275	+	0.942830i	$0.608154\pi$
$468$	0	0
$469$	4.36522e6	0.916377
$470$	0	0
$471$	145894.	0.0303029
$472$	0	0
$473$	126295.	0.0259556
$474$	0	0
$475$	−1.61990e6	−0.329423
$476$	0	0
$477$	−1.87669e6	−0.377656
$478$	0	0
$479$	6.68286e6	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$479$	6.68286e6	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$480$	0	0
$481$	783441.	0.154399
$482$	0	0
$483$	871489.	0.169979
$484$	0	0
$485$	−1.43225e7	−2.76481
$486$	0	0
$487$	4.06478e6	0.776631	0.388316	−	0.921526i	$-0.373057\pi$
$487$	4.06478e6	0.776631	0.388316	+	0.921526i	$0.373057\pi$
$488$	0	0
$489$	367993.	0.0695933
$490$	0	0
$491$	−2.10434e6	−0.393923	−0.196962	−	0.980411i	$-0.563107\pi$
$491$	−2.10434e6	−0.393923	−0.196962	+	0.980411i	$0.563107\pi$
$492$	0	0
$493$	−530768.	−0.0983530
$494$	0	0
$495$	368755.	0.0676432
$496$	0	0
$497$	5.21095e6	0.946293
$498$	0	0
$499$	5.96715e6	1.07279	0.536396	−	0.843966i	$-0.319785\pi$
$499$	5.96715e6	1.07279	0.536396	+	0.843966i	$0.319785\pi$
$500$	0	0
$501$	341673.	0.0608158
$502$	0	0
$503$	1.00144e7	1.76483	0.882417	−	0.470467i	$-0.155915\pi$
$503$	1.00144e7	1.76483	0.882417	+	0.470467i	$0.155915\pi$
$504$	0	0
$505$	−1.43969e7	−2.51212
$506$	0	0
$507$	−46573.9	−0.00804680
$508$	0	0
$509$	8.47321e6	1.44962	0.724809	−	0.688950i	$-0.241928\pi$
$509$	8.47321e6	1.44962	0.724809	+	0.688950i	$0.241928\pi$
$510$	0	0
$511$	133435.	0.0226057
$512$	0	0
$513$	−168452.	−0.0282606
$514$	0	0
$515$	−1.01923e7	−1.69338
$516$	0	0
$517$	369571.	0.0608095
$518$	0	0
$519$	−758567.	−0.123616
$520$	0	0
$521$	−197614.	−0.0318951	−0.0159476	−	0.999873i	$-0.505076\pi$
$521$	−197614.	−0.0318951	−0.0159476	+	0.999873i	$0.505076\pi$
$522$	0	0
$523$	8.27263e6	1.32248	0.661240	−	0.750174i	$-0.270030\pi$
$523$	8.27263e6	1.32248	0.661240	+	0.750174i	$0.270030\pi$
$524$	0	0
$525$	1.56113e6	0.247195
$526$	0	0
$527$	5.02812e6	0.788640
$528$	0	0
$529$	1.14664e7	1.78150
$530$	0	0
$531$	8.97374e6	1.38114
$532$	0	0
$533$	1.34256e6	0.204699
$534$	0	0
$535$	−2.77779e6	−0.419580
$536$	0	0
$537$	770618.	0.115320
$538$	0	0
$539$	12651.3	0.00187570
$540$	0	0
$541$	−363216.	−0.0533546	−0.0266773	−	0.999644i	$-0.508493\pi$
$541$	−363216.	−0.0533546	−0.0266773	+	0.999644i	$0.508493\pi$
$542$	0	0
$543$	−120833.	−0.0175867
$544$	0	0
$545$	6.54097e6	0.943302
$546$	0	0
$547$	−620452.	−0.0886624	−0.0443312	−	0.999017i	$-0.514116\pi$
$547$	−620452.	−0.0886624	−0.0443312	+	0.999017i	$0.514116\pi$
$548$	0	0
$549$	−4.36752e6	−0.618449
$550$	0	0
$551$	−107921.	−0.0151435
$552$	0	0
$553$	−6.02736e6	−0.838136
$554$	0	0
$555$	−782114.	−0.107780
$556$	0	0
$557$	−3.89737e6	−0.532272	−0.266136	−	0.963935i	$-0.585747\pi$
$557$	−3.89737e6	−0.532272	−0.266136	+	0.963935i	$0.585747\pi$
$558$	0	0
$559$	−1.43927e6	−0.194811
$560$	0	0
$561$	−25418.5	−0.00340992
$562$	0	0
$563$	510725.	0.0679073	0.0339536	−	0.999423i	$-0.489190\pi$
$563$	510725.	0.0679073	0.0339536	+	0.999423i	$0.489190\pi$
$564$	0	0
$565$	1.18395e7	1.56032
$566$	0	0
$567$	−7.21445e6	−0.942422
$568$	0	0
$569$	9.75625e6	1.26329	0.631644	−	0.775259i	$-0.282381\pi$
$569$	9.75625e6	1.26329	0.631644	+	0.775259i	$0.282381\pi$
$570$	0	0
$571$	−1.41952e7	−1.82201	−0.911006	−	0.412393i	$-0.864693\pi$
$571$	−1.41952e7	−1.82201	−0.911006	+	0.412393i	$0.864693\pi$
$572$	0	0
$573$	−365482.	−0.0465029
$574$	0	0
$575$	3.20697e7	4.04506
$576$	0	0
$577$	−1.16423e6	−0.145579	−0.0727896	−	0.997347i	$-0.523190\pi$
$577$	−1.16423e6	−0.145579	−0.0727896	+	0.997347i	$0.523190\pi$
$578$	0	0
$579$	1.07994e6	0.133877
$580$	0	0
$581$	9.44777e6	1.16115
$582$	0	0
$583$	−115796.	−0.0141098
$584$	0	0
$585$	−4.20238e6	−0.507699
$586$	0	0
$587$	6.58038e6	0.788234	0.394117	−	0.919060i	$-0.371050\pi$
$587$	6.58038e6	0.788234	0.394117	+	0.919060i	$0.371050\pi$
$588$	0	0
$589$	1.02236e6	0.121427
$590$	0	0
$591$	1.08679e6	0.127991
$592$	0	0
$593$	−1.91423e6	−0.223541	−0.111771	−	0.993734i	$-0.535652\pi$
$593$	−1.91423e6	−0.223541	−0.111771	+	0.993734i	$0.535652\pi$
$594$	0	0
$595$	1.37362e7	1.59065
$596$	0	0
$597$	1.05296e6	0.120914
$598$	0	0
$599$	2.33678e6	0.266104	0.133052	−	0.991109i	$-0.457522\pi$
$599$	2.33678e6	0.266104	0.133052	+	0.991109i	$0.457522\pi$
$600$	0	0
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	0	0
$603$	8.30617e6	0.930266
$604$	0	0
$605$	−1.66399e7	−1.84826
$606$	0	0
$607$	120274.	0.0132495	0.00662474	−	0.999978i	$-0.497891\pi$
$607$	120274.	0.0132495	0.00662474	+	0.999978i	$0.497891\pi$
$608$	0	0
$609$	104005.	0.0113635
$610$	0	0
$611$	−4.21169e6	−0.456408
$612$	0	0
$613$	−1.34576e7	−1.44649	−0.723245	−	0.690592i	$-0.757350\pi$
$613$	−1.34576e7	−1.44649	−0.723245	+	0.690592i	$0.757350\pi$
$614$	0	0
$615$	−1.34029e6	−0.142893
$616$	0	0
$617$	−6.84879e6	−0.724270	−0.362135	−	0.932126i	$-0.617952\pi$
$617$	−6.84879e6	−0.724270	−0.362135	+	0.932126i	$0.617952\pi$
$618$	0	0
$619$	−5.40663e6	−0.567153	−0.283577	−	0.958950i	$-0.591521\pi$
$619$	−5.40663e6	−0.567153	−0.283577	+	0.958950i	$0.591521\pi$
$620$	0	0
$621$	3.33490e6	0.347019
$622$	0	0
$623$	−1.23773e6	−0.127763
$624$	0	0
$625$	2.39962e7	2.45721
$626$	0	0
$627$	−5168.33	−0.000525027 0
$628$	0	0
$629$	−4.87273e6	−0.491072
$630$	0	0
$631$	9.62552e6	0.962390	0.481195	−	0.876614i	$-0.340203\pi$
$631$	9.62552e6	0.962390	0.481195	+	0.876614i	$0.340203\pi$
$632$	0	0
$633$	1.41547e6	0.140408
$634$	0	0
$635$	2.57487e7	2.53408
$636$	0	0
$637$	−144176.	−0.0140781
$638$	0	0
$639$	9.91542e6	0.960636
$640$	0	0
$641$	−1.58752e7	−1.52607	−0.763037	−	0.646355i	$-0.776293\pi$
$641$	−1.58752e7	−1.52607	−0.763037	+	0.646355i	$0.776293\pi$
$642$	0	0
$643$	1.57235e7	1.49976	0.749880	−	0.661574i	$-0.230111\pi$
$643$	1.57235e7	1.49976	0.749880	+	0.661574i	$0.230111\pi$
$644$	0	0
$645$	1.43684e6	0.135990
$646$	0	0
$647$	1.58173e7	1.48549	0.742747	−	0.669572i	$-0.233523\pi$
$647$	1.58173e7	1.48549	0.742747	+	0.669572i	$0.233523\pi$
$648$	0	0
$649$	553699.	0.0516015
$650$	0	0
$651$	−985270.	−0.0911178
$652$	0	0
$653$	−5.31229e6	−0.487527	−0.243763	−	0.969835i	$-0.578382\pi$
$653$	−5.31229e6	−0.487527	−0.243763	+	0.969835i	$0.578382\pi$
$654$	0	0
$655$	1.06527e7	0.970192
$656$	0	0
$657$	253901.	0.0229483
$658$	0	0
$659$	9.90554e6	0.888514	0.444257	−	0.895899i	$-0.353468\pi$
$659$	9.90554e6	0.888514	0.444257	+	0.895899i	$0.353468\pi$
$660$	0	0
$661$	1.29988e7	1.15717	0.578587	−	0.815621i	$-0.303604\pi$
$661$	1.29988e7	1.15717	0.578587	+	0.815621i	$0.303604\pi$
$662$	0	0
$663$	289674.	0.0255932
$664$	0	0
$665$	2.79297e6	0.244913
$666$	0	0
$667$	2.13654e6	0.185950
$668$	0	0
$669$	−95138.8	−0.00821850
$670$	0	0
$671$	−269486.	−0.0231062
$672$	0	0
$673$	−1.32503e7	−1.12769	−0.563844	−	0.825881i	$-0.690678\pi$
$673$	−1.32503e7	−1.12769	−0.563844	+	0.825881i	$0.690678\pi$
$674$	0	0
$675$	5.97391e6	0.504660
$676$	0	0
$677$	−2.23310e7	−1.87257	−0.936284	−	0.351245i	$-0.885758\pi$
$677$	−2.23310e7	−1.87257	−0.936284	+	0.351245i	$0.885758\pi$
$678$	0	0
$679$	1.74852e7	1.45545
$680$	0	0
$681$	−182259.	−0.0150598
$682$	0	0
$683$	−1.40049e7	−1.14876	−0.574380	−	0.818588i	$-0.694757\pi$
$683$	−1.40049e7	−1.14876	−0.574380	+	0.818588i	$0.694757\pi$
$684$	0	0
$685$	−3.72851e6	−0.303605
$686$	0	0
$687$	1.87502e6	0.151571
$688$	0	0
$689$	1.31963e6	0.105902
$690$	0	0
$691$	−5.24817e6	−0.418132	−0.209066	−	0.977902i	$-0.567042\pi$
$691$	−5.24817e6	−0.418132	−0.209066	+	0.977902i	$0.567042\pi$
$692$	0	0
$693$	−450183.	−0.0356087
$694$	0	0
$695$	1.57940e7	1.24031
$696$	0	0
$697$	−8.35027e6	−0.651056
$698$	0	0
$699$	−1.02810e6	−0.0795868
$700$	0	0
$701$	2.13994e7	1.64477	0.822386	−	0.568930i	$-0.192642\pi$
$701$	2.13994e7	1.64477	0.822386	+	0.568930i	$0.192642\pi$
$702$	0	0
$703$	−990767.	−0.0756107
$704$	0	0
$705$	4.20456e6	0.318601
$706$	0	0
$707$	1.75760e7	1.32243
$708$	0	0
$709$	−4.35333e6	−0.325242	−0.162621	−	0.986689i	$-0.551995\pi$
$709$	−4.35333e6	−0.325242	−0.162621	+	0.986689i	$0.551995\pi$
$710$	0	0
$711$	−1.14689e7	−0.850839
$712$	0	0
$713$	−2.02401e7	−1.49104
$714$	0	0
$715$	−259296.	−0.0189684
$716$	0	0
$717$	−270087.	−0.0196203
$718$	0	0
$719$	−2.21389e7	−1.59710	−0.798552	−	0.601926i	$-0.794400\pi$
$719$	−2.21389e7	−1.59710	−0.798552	+	0.601926i	$0.794400\pi$
$720$	0	0
$721$	1.24430e7	0.891426
$722$	0	0
$723$	−1.12675e6	−0.0801645
$724$	0	0
$725$	3.82726e6	0.270422
$726$	0	0
$727$	4.83218e6	0.339084	0.169542	−	0.985523i	$-0.445771\pi$
$727$	4.83218e6	0.339084	0.169542	+	0.985523i	$0.445771\pi$
$728$	0	0
$729$	−1.34154e7	−0.934940
$730$	0	0
$731$	8.95178e6	0.619606
$732$	0	0
$733$	2.47827e7	1.70368	0.851840	−	0.523803i	$-0.175487\pi$
$733$	2.47827e7	1.70368	0.851840	+	0.523803i	$0.175487\pi$
$734$	0	0
$735$	143932.	0.00982741
$736$	0	0
$737$	512509.	0.0347562
$738$	0	0
$739$	7.09289e6	0.477762	0.238881	−	0.971049i	$-0.423219\pi$
$739$	7.09289e6	0.477762	0.238881	+	0.971049i	$0.423219\pi$
$740$	0	0
$741$	58899.1	0.00394061
$742$	0	0
$743$	−1.95117e7	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$743$	−1.95117e7	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$744$	0	0
$745$	−7.50588e6	−0.495462
$746$	0	0
$747$	1.79773e7	1.17875
$748$	0	0
$749$	3.39118e6	0.220875
$750$	0	0
$751$	−1.66103e7	−1.07468	−0.537339	−	0.843366i	$-0.680570\pi$
$751$	−1.66103e7	−1.07468	−0.537339	+	0.843366i	$0.680570\pi$
$752$	0	0
$753$	630891.	0.0405477
$754$	0	0
$755$	−5.05952e7	−3.23029
$756$	0	0
$757$	−1.22902e6	−0.0779508	−0.0389754	−	0.999240i	$-0.512409\pi$
$757$	−1.22902e6	−0.0779508	−0.0389754	+	0.999240i	$0.512409\pi$
$758$	0	0
$759$	102319.	0.00644693
$760$	0	0
$761$	1.37482e7	0.860569	0.430284	−	0.902693i	$-0.358413\pi$
$761$	1.37482e7	0.860569	0.430284	+	0.902693i	$0.358413\pi$
$762$	0	0
$763$	−7.98535e6	−0.496572
$764$	0	0
$765$	2.61374e7	1.61476
$766$	0	0
$767$	−6.31004e6	−0.387297
$768$	0	0
$769$	−1.01549e7	−0.619240	−0.309620	−	0.950860i	$-0.600202\pi$
$769$	−1.01549e7	−0.619240	−0.309620	+	0.950860i	$0.600202\pi$
$770$	0	0
$771$	−421140.	−0.0255147
$772$	0	0
$773$	−1.41511e7	−0.851807	−0.425903	−	0.904769i	$-0.640044\pi$
$773$	−1.41511e7	−0.851807	−0.425903	+	0.904769i	$0.640044\pi$
$774$	0	0
$775$	−3.62567e7	−2.16837
$776$	0	0
$777$	954821.	0.0567374
$778$	0	0
$779$	−1.69785e6	−0.100243
$780$	0	0
$781$	611803.	0.0358909
$782$	0	0
$783$	397993.	0.0231991
$784$	0	0
$785$	−9.25654e6	−0.536136
$786$	0	0
$787$	−1.03095e6	−0.0593338	−0.0296669	−	0.999560i	$-0.509445\pi$
$787$	−1.03095e6	−0.0593338	−0.0296669	+	0.999560i	$0.509445\pi$
$788$	0	0
$789$	−1.94137e6	−0.111024
$790$	0	0
$791$	−1.44540e7	−0.821383
$792$	0	0
$793$	3.07110e6	0.173425
$794$	0	0
$795$	−1.31740e6	−0.0739262
$796$	0	0
$797$	1.43337e7	0.799303	0.399651	−	0.916667i	$-0.369131\pi$
$797$	1.43337e7	0.799303	0.399651	+	0.916667i	$0.369131\pi$
$798$	0	0
$799$	2.61952e7	1.45163
$800$	0	0
$801$	−2.35516e6	−0.129700
$802$	0	0
$803$	15666.3	0.000857386 0
$804$	0	0
$805$	−5.52935e7	−3.00735
$806$	0	0
$807$	1.38766e6	0.0750065
$808$	0	0
$809$	1.55020e7	0.832751	0.416376	−	0.909193i	$-0.363300\pi$
$809$	1.55020e7	0.832751	0.416376	+	0.909193i	$0.363300\pi$
$810$	0	0
$811$	2.45861e7	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$811$	2.45861e7	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$812$	0	0
$813$	−66737.5	−0.00354114
$814$	0	0
$815$	−2.33481e7	−1.23128
$816$	0	0
$817$	1.82016e6	0.0954011
$818$	0	0
$819$	5.13036e6	0.267262
$820$	0	0
$821$	3.33396e6	0.172624	0.0863122	−	0.996268i	$-0.472492\pi$
$821$	3.33396e6	0.172624	0.0863122	+	0.996268i	$0.472492\pi$
$822$	0	0
$823$	−3.08787e6	−0.158913	−0.0794564	−	0.996838i	$-0.525318\pi$
$823$	−3.08787e6	−0.158913	−0.0794564	+	0.996838i	$0.525318\pi$
$824$	0	0
$825$	183288.	0.00937559
$826$	0	0
$827$	6.89555e6	0.350595	0.175297	−	0.984516i	$-0.443911\pi$
$827$	6.89555e6	0.350595	0.175297	+	0.984516i	$0.443911\pi$
$828$	0	0
$829$	2.00007e7	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$829$	2.00007e7	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$830$	0	0
$831$	1.63156e6	0.0819596
$832$	0	0
$833$	896725.	0.0447762
$834$	0	0
$835$	−2.16782e7	−1.07599
$836$	0	0
$837$	−3.77030e6	−0.186021
$838$	0	0
$839$	−5.23988e6	−0.256990	−0.128495	−	0.991710i	$-0.541015\pi$
$839$	−5.23988e6	−0.256990	−0.128495	+	0.991710i	$0.541015\pi$
$840$	0	0
$841$	−2.02562e7	−0.987569
$842$	0	0
$843$	2.83079e6	0.137195
$844$	0	0
$845$	2.95498e6	0.142368
$846$	0	0
$847$	2.03144e7	0.972959
$848$	0	0
$849$	2.07689e6	0.0988883
$850$	0	0
$851$	1.96146e7	0.928442
$852$	0	0
$853$	1.32853e7	0.625170	0.312585	−	0.949890i	$-0.398805\pi$
$853$	1.32853e7	0.625170	0.312585	+	0.949890i	$0.398805\pi$
$854$	0	0
$855$	5.31449e6	0.248626
$856$	0	0
$857$	−8.34473e6	−0.388115	−0.194057	−	0.980990i	$-0.562165\pi$
$857$	−8.34473e6	−0.388115	−0.194057	+	0.980990i	$0.562165\pi$
$858$	0	0
$859$	−4.07521e7	−1.88437	−0.942187	−	0.335088i	$-0.891234\pi$
$859$	−4.07521e7	−1.88437	−0.942187	+	0.335088i	$0.891234\pi$
$860$	0	0
$861$	1.63625e6	0.0752216
$862$	0	0
$863$	1.73853e7	0.794614	0.397307	−	0.917686i	$-0.369945\pi$
$863$	1.73853e7	0.794614	0.397307	+	0.917686i	$0.369945\pi$
$864$	0	0
$865$	4.81289e7	2.18708
$866$	0	0
$867$	513667.	0.0232078
$868$	0	0
$869$	−707656.	−0.0317887
$870$	0	0
$871$	−5.84063e6	−0.260864
$872$	0	0
$873$	3.32710e7	1.47751
$874$	0	0
$875$	−5.82109e7	−2.57030
$876$	0	0
$877$	−2.58279e6	−0.113394	−0.0566971	−	0.998391i	$-0.518057\pi$
$877$	−2.58279e6	−0.113394	−0.0566971	+	0.998391i	$0.518057\pi$
$878$	0	0
$879$	−3332.63	−0.000145484 0
$880$	0	0
$881$	−1.66814e7	−0.724090	−0.362045	−	0.932161i	$-0.617921\pi$
$881$	−1.66814e7	−0.724090	−0.362045	+	0.932161i	$0.617921\pi$
$882$	0	0
$883$	2.36384e7	1.02027	0.510137	−	0.860093i	$-0.329595\pi$
$883$	2.36384e7	1.02027	0.510137	+	0.860093i	$0.329595\pi$
$884$	0	0
$885$	6.29936e6	0.270358
$886$	0	0
$887$	5.25660e6	0.224334	0.112167	−	0.993689i	$-0.464221\pi$
$887$	5.25660e6	0.224334	0.112167	+	0.993689i	$0.464221\pi$
$888$	0	0
$889$	−3.14345e7	−1.33399
$890$	0	0
$891$	−847029.	−0.0357441
$892$	0	0
$893$	5.32625e6	0.223508
$894$	0	0
$895$	−4.88935e7	−2.04030
$896$	0	0
$897$	−1.16605e6	−0.0483876
$898$	0	0
$899$	−2.41549e6	−0.0996795
$900$	0	0
$901$	−8.20763e6	−0.336826
$902$	0	0
$903$	−1.75412e6	−0.0715879
$904$	0	0
$905$	7.66648e6	0.311153
$906$	0	0
$907$	3.22789e7	1.30287	0.651435	−	0.758705i	$-0.274167\pi$
$907$	3.22789e7	1.30287	0.651435	+	0.758705i	$0.274167\pi$
$908$	0	0
$909$	3.34438e7	1.34247
$910$	0	0
$911$	−4.20975e7	−1.68058	−0.840292	−	0.542134i	$-0.817617\pi$
$911$	−4.20975e7	−1.68058	−0.840292	+	0.542134i	$0.817617\pi$
$912$	0	0
$913$	1.10924e6	0.0440400
$914$	0	0
$915$	−3.06590e6	−0.121061
$916$	0	0
$917$	−1.30051e7	−0.510728
$918$	0	0
$919$	2.19460e7	0.857168	0.428584	−	0.903502i	$-0.359013\pi$
$919$	2.19460e7	0.857168	0.428584	+	0.903502i	$0.359013\pi$
$920$	0	0
$921$	654407.	0.0254213
$922$	0	0
$923$	−6.97221e6	−0.269380
$924$	0	0
$925$	3.51362e7	1.35021
$926$	0	0
$927$	2.36765e7	0.904937
$928$	0	0
$929$	−1.35921e7	−0.516710	−0.258355	−	0.966050i	$-0.583180\pi$
$929$	−1.35921e7	−0.516710	−0.258355	+	0.966050i	$0.583180\pi$
$930$	0	0
$931$	182330.	0.00689421
$932$	0	0
$933$	−3.14116e6	−0.118137
$934$	0	0
$935$	1.61273e6	0.0603300
$936$	0	0
$937$	3.01018e7	1.12007	0.560033	−	0.828470i	$-0.310788\pi$
$937$	3.01018e7	1.12007	0.560033	+	0.828470i	$0.310788\pi$
$938$	0	0
$939$	−2.68279e6	−0.0992938
$940$	0	0
$941$	1.06275e7	0.391252	0.195626	−	0.980679i	$-0.437326\pi$
$941$	1.06275e7	0.391252	0.195626	+	0.980679i	$0.437326\pi$
$942$	0	0
$943$	3.36130e7	1.23091
$944$	0	0
$945$	−1.03000e7	−0.375196
$946$	0	0
$947$	−2.41709e7	−0.875826	−0.437913	−	0.899017i	$-0.644282\pi$
$947$	−2.41709e7	−0.875826	−0.437913	+	0.899017i	$0.644282\pi$
$948$	0	0
$949$	−178535.	−0.00643514
$950$	0	0
$951$	−3.25282e6	−0.116630
$952$	0	0
$953$	4.27043e7	1.52314	0.761569	−	0.648084i	$-0.224430\pi$
$953$	4.27043e7	1.52314	0.761569	+	0.648084i	$0.224430\pi$
$954$	0	0
$955$	2.31888e7	0.822754
$956$	0	0
$957$	12211.0	0.000430993 0
$958$	0	0
$959$	4.55184e6	0.159823
$960$	0	0
$961$	−5.74655e6	−0.200724
$962$	0	0
$963$	6.45276e6	0.224223
$964$	0	0
$965$	−6.85193e7	−2.36862
$966$	0	0
$967$	4.42692e7	1.52242	0.761211	−	0.648504i	$-0.224605\pi$
$967$	4.42692e7	1.52242	0.761211	+	0.648504i	$0.224605\pi$
$968$	0	0
$969$	−366332.	−0.0125333
$970$	0	0
$971$	3.88962e7	1.32391	0.661956	−	0.749542i	$-0.269726\pi$
$971$	3.88962e7	1.32391	0.661956	+	0.749542i	$0.269726\pi$
$972$	0	0
$973$	−1.92816e7	−0.652922
$974$	0	0
$975$	−2.08877e6	−0.0703688
$976$	0	0
$977$	−2.71611e7	−0.910354	−0.455177	−	0.890401i	$-0.650424\pi$
$977$	−2.71611e7	−0.910354	−0.455177	+	0.890401i	$0.650424\pi$
$978$	0	0
$979$	−145319.	−0.00484580
$980$	0	0
$981$	−1.51946e7	−0.504099
$982$	0	0
$983$	1.98048e7	0.653714	0.326857	−	0.945074i	$-0.394011\pi$
$983$	1.98048e7	0.653714	0.326857	+	0.945074i	$0.394011\pi$
$984$	0	0
$985$	−6.89538e7	−2.26448
$986$	0	0
$987$	−5.13301e6	−0.167718
$988$	0	0
$989$	−3.60343e7	−1.17145
$990$	0	0
$991$	1.44104e7	0.466115	0.233057	−	0.972463i	$-0.425127\pi$
$991$	1.44104e7	0.466115	0.233057	+	0.972463i	$0.425127\pi$
$992$	0	0
$993$	1.10222e6	0.0354727
$994$	0	0
$995$	−6.68075e7	−2.13928
$996$	0	0
$997$	−3.16635e7	−1.00884	−0.504418	−	0.863459i	$-0.668293\pi$
$997$	−3.16635e7	−1.00884	−0.504418	+	0.863459i	$0.668293\pi$
$998$	0	0
$999$	3.65378e6	0.115832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.6.a.i.1.2		2
4.3	odd	2		832.6.a.p.1.1		2
8.3	odd	2		13.6.a.a.1.1	✓	2
8.5	even	2		208.6.a.h.1.1		2
24.11	even	2		117.6.a.c.1.2		2
40.3	even	4		325.6.b.b.274.4		4
40.19	odd	2		325.6.a.b.1.2		2
40.27	even	4		325.6.b.b.274.1		4
56.27	even	2		637.6.a.a.1.1		2
104.51	odd	2		169.6.a.a.1.2		2
104.83	even	4		169.6.b.a.168.4		4
104.99	even	4		169.6.b.a.168.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	8.3	odd	2
117.6.a.c.1.2		2	24.11	even	2
169.6.a.a.1.2		2	104.51	odd	2
169.6.b.a.168.1		4	104.99	even	4
169.6.b.a.168.4		4	104.83	even	4
208.6.a.h.1.1		2	8.5	even	2
325.6.a.b.1.2		2	40.19	odd	2
325.6.b.b.274.1		4	40.27	even	4
325.6.b.b.274.4		4	40.3	even	4
637.6.a.a.1.1		2	56.27	even	2
832.6.a.i.1.2		2	1.1	even	1	trivial
832.6.a.p.1.1		2	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	832.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63068 q^{3} +103.462 q^{5} -126.309 q^{7} -240.341 q^{9} -14.8296 q^{11} +169.000 q^{13} -168.714 q^{15} -1051.12 q^{17} -213.723 q^{19} +205.969 q^{21} +4231.16 q^{23} +7579.41 q^{25} +788.176 q^{27} +504.955 q^{29} -4783.58 q^{31} +24.1823 q^{33} -13068.2 q^{35} +4635.74 q^{37} -275.585 q^{39} +7944.15 q^{41} -8516.41 q^{43} -24866.2 q^{45} -24921.2 q^{47} -853.113 q^{49} +1714.05 q^{51} +7808.46 q^{53} -1534.30 q^{55} +348.515 q^{57} -37337.5 q^{59} +18172.2 q^{61} +30357.1 q^{63} +17485.1 q^{65} -34559.9 q^{67} -6899.68 q^{69} -41255.7 q^{71} -1056.42 q^{73} -12359.6 q^{75} +1873.10 q^{77} +47719.3 q^{79} +57117.6 q^{81} -74799.0 q^{83} -108751. q^{85} -823.421 q^{87} +9799.26 q^{89} -21346.2 q^{91} +7800.50 q^{93} -22112.3 q^{95} -138432. q^{97} +3564.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 28 q^{3} + 42 q^{5} + 36 q^{7} + 212 q^{9} - 376 q^{11} + 338 q^{13} + 1452 q^{15} - 2630 q^{17} - 312 q^{19} - 4074 q^{21} + 2624 q^{23} + 8232 q^{25} - 4732 q^{27} + 812 q^{29} - 7720 q^{31} + 9548 q^{33}+ \cdots - 159808 q^{99}+O(q^{100}) \) 2 * q - 28 * q^3 + 42 * q^5 + 36 * q^7 + 212 * q^9 - 376 * q^11 + 338 * q^13 + 1452 * q^15 - 2630 * q^17 - 312 * q^19 - 4074 * q^21 + 2624 * q^23 + 8232 * q^25 - 4732 * q^27 + 812 * q^29 - 7720 * q^31 + 9548 * q^33 - 23044 * q^35 + 16858 * q^37 - 4732 * q^39 + 7840 * q^41 + 2420 * q^43 - 52668 * q^45 - 9972 * q^47 + 8684 * q^49 + 43348 * q^51 + 43720 * q^53 + 20664 * q^55 + 2940 * q^57 - 38936 * q^59 - 1984 * q^61 + 103776 * q^63 + 7098 * q^65 - 69928 * q^67 + 35480 * q^69 - 67396 * q^71 + 74412 * q^73 - 29568 * q^75 - 56748 * q^77 + 55296 * q^79 + 92762 * q^81 - 75712 * q^83 - 11710 * q^85 - 8920 * q^87 + 116508 * q^89 + 6084 * q^91 + 85232 * q^93 - 16072 * q^95 - 34756 * q^97 - 159808 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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