LMFDB - Embedded newform 784.5.c.b.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,5,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	784.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$81.0420510577$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.b.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.9941i q^{3} -25.3864i q^{5} -143.823 q^{9} +O(q^{10})$	$q+14.9941i q^{3} -25.3864i q^{5} -143.823 q^{9} -43.9706 q^{11} +162.507i q^{13} +380.647 q^{15} +108.236i q^{17} +449.568i q^{19} +435.823 q^{23} -19.4701 q^{25} -941.981i q^{27} +742.118 q^{29} +1039.43i q^{31} -659.300i q^{33} +986.675 q^{37} -2436.65 q^{39} -1143.70i q^{41} -2418.82 q^{43} +3651.16i q^{45} -1345.43i q^{47} -1622.91 q^{51} -3566.20 q^{53} +1116.26i q^{55} -6740.88 q^{57} -4285.01i q^{59} +1396.69i q^{61} +4125.47 q^{65} -7265.79 q^{67} +6534.77i q^{69} -5987.76 q^{71} -577.493i q^{73} -291.937i q^{75} +3146.23 q^{79} +2474.47 q^{81} +4729.96i q^{83} +2747.74 q^{85} +11127.4i q^{87} +837.909i q^{89} -15585.4 q^{93} +11412.9 q^{95} +5622.23i q^{97} +6323.99 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 168 q^{9}+O(q^{10}) $ 4 * q - 168 * q^9	$ 4 q - 168 q^{9} - 108 q^{11} + 708 q^{15} - 972 q^{23} + 1144 q^{25} + 3240 q^{29} + 892 q^{37} - 6624 q^{39} - 2344 q^{43} + 636 q^{51} - 5508 q^{53} - 17460 q^{57} + 6048 q^{65} - 10124 q^{67} - 18792 q^{71} + 1588 q^{79} + 8676 q^{81} + 10380 q^{85} - 37836 q^{93} + 17820 q^{95} + 11448 q^{99}+O(q^{100}) $ 4 * q - 168 * q^9 - 108 * q^11 + 708 * q^15 - 972 * q^23 + 1144 * q^25 + 3240 * q^29 + 892 * q^37 - 6624 * q^39 - 2344 * q^43 + 636 * q^51 - 5508 * q^53 - 17460 * q^57 + 6048 * q^65 - 10124 * q^67 - 18792 * q^71 + 1588 * q^79 + 8676 * q^81 + 10380 * q^85 - 37836 * q^93 + 17820 * q^95 + 11448 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	14.9941i	1.66601i	0.553264	+	0.833006i	$0.313382\pi$
$3$	14.9941i	1.66601i	−0.553264	+	0.833006i	$0.686618\pi$
$4$	0	0
$5$	− 25.3864i	− 1.01546i	−0.861517	−	0.507728i	$-0.830485\pi$
$5$	− 25.3864i	− 1.01546i	0.861517	−	0.507728i	$-0.169515\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−143.823	−1.77560
$10$	0	0
$11$	−43.9706	−0.363393	−0.181697	−	0.983355i	$-0.558159\pi$
$11$	−43.9706	−0.363393	−0.181697	+	0.983355i	$0.558159\pi$
$12$	0	0
$13$	162.507i	0.961579i	0.876836	+	0.480790i	$0.159650\pi$
$13$	162.507i	0.961579i	−0.876836	+	0.480790i	$0.840350\pi$
$14$	0	0
$15$	380.647	1.69176
$16$	0	0
$17$	108.236i	0.374521i	0.982310	+	0.187260i	$0.0599608\pi$
$17$	108.236i	0.374521i	−0.982310	+	0.187260i	$0.940039\pi$
$18$	0	0
$19$	449.568i	1.24534i	0.782484	+	0.622671i	$0.213952\pi$
$19$	449.568i	1.24534i	−0.782484	+	0.622671i	$0.786048\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	435.823	0.823861	0.411931	−	0.911215i	$-0.364855\pi$
$23$	435.823	0.823861	0.411931	+	0.911215i	$0.364855\pi$
$24$	0	0
$25$	−19.4701	−0.0311522
$26$	0	0
$27$	− 941.981i	− 1.29215i
$28$	0	0
$29$	742.118	0.882423	0.441212	−	0.897403i	$-0.354549\pi$
$29$	742.118	0.882423	0.441212	+	0.897403i	$0.354549\pi$
$30$	0	0
$31$	1039.43i	1.08162i	0.841146	+	0.540808i	$0.181881\pi$
$31$	1039.43i	1.08162i	−0.841146	+	0.540808i	$0.818119\pi$
$32$	0	0
$33$	− 659.300i	− 0.605417i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	986.675	0.720727	0.360364	−	0.932812i	$-0.382653\pi$
$37$	986.675	0.720727	0.360364	+	0.932812i	$0.382653\pi$
$38$	0	0
$39$	−2436.65	−1.60200
$40$	0	0
$41$	− 1143.70i	− 0.680366i	−0.940359	−	0.340183i	$-0.889511\pi$
$41$	− 1143.70i	− 0.680366i	0.940359	−	0.340183i	$-0.110489\pi$
$42$	0	0
$43$	−2418.82	−1.30818	−0.654089	−	0.756418i	$-0.726948\pi$
$43$	−2418.82	−1.30818	−0.654089	+	0.756418i	$0.726948\pi$
$44$	0	0
$45$	3651.16i	1.80304i
$46$	0	0
$47$	− 1345.43i	− 0.609066i	−0.952502	−	0.304533i	$-0.901500\pi$
$47$	− 1345.43i	− 0.609066i	0.952502	−	0.304533i	$-0.0985004\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1622.91	−0.623956
$52$	0	0
$53$	−3566.20	−1.26956	−0.634782	−	0.772692i	$-0.718910\pi$
$53$	−3566.20	−1.26956	−0.634782	+	0.772692i	$0.718910\pi$
$54$	0	0
$55$	1116.26i	0.369010i
$56$	0	0
$57$	−6740.88	−2.07475
$58$	0	0
$59$	− 4285.01i	− 1.23097i	−0.788148	−	0.615485i	$-0.788960\pi$
$59$	− 4285.01i	− 1.23097i	0.788148	−	0.615485i	$-0.211040\pi$
$60$	0	0
$61$	1396.69i	0.375353i	0.982231	+	0.187677i	$0.0600957\pi$
$61$	1396.69i	0.375353i	−0.982231	+	0.187677i	$0.939904\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4125.47	0.976442
$66$	0	0
$67$	−7265.79	−1.61858	−0.809288	−	0.587412i	$-0.800147\pi$
$67$	−7265.79	−1.61858	−0.809288	+	0.587412i	$0.800147\pi$
$68$	0	0
$69$	6534.77i	1.37256i
$70$	0	0
$71$	−5987.76	−1.18781	−0.593906	−	0.804534i	$-0.702415\pi$
$71$	−5987.76	−1.18781	−0.593906	+	0.804534i	$0.702415\pi$
$72$	0	0
$73$	− 577.493i	− 0.108368i	−0.998531	−	0.0541840i	$-0.982744\pi$
$73$	− 577.493i	− 0.108368i	0.998531	−	0.0541840i	$-0.0172557\pi$
$74$	0	0
$75$	− 291.937i	− 0.0519000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3146.23	0.504123	0.252061	−	0.967711i	$-0.418891\pi$
$79$	3146.23	0.504123	0.252061	+	0.967711i	$0.418891\pi$
$80$	0	0
$81$	2474.47	0.377148
$82$	0	0
$83$	4729.96i	0.686596i	0.939227	+	0.343298i	$0.111544\pi$
$83$	4729.96i	0.686596i	−0.939227	+	0.343298i	$0.888456\pi$
$84$	0	0
$85$	2747.74	0.380309
$86$	0	0
$87$	11127.4i	1.47013i
$88$	0	0
$89$	837.909i	0.105783i	0.998600	+	0.0528916i	$0.0168438\pi$
$89$	837.909i	0.105783i	−0.998600	+	0.0528916i	$0.983156\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−15585.4	−1.80199
$94$	0	0
$95$	11412.9	1.26459
$96$	0	0
$97$	5622.23i	0.597537i	0.954326	+	0.298769i	$0.0965759\pi$
$97$	5622.23i	0.597537i	−0.954326	+	0.298769i	$0.903424\pi$
$98$	0	0
$99$	6323.99	0.645240
$100$	0	0
$101$	− 6585.50i	− 0.645574i	−0.946472	−	0.322787i	$-0.895380\pi$
$101$	− 6585.50i	− 0.645574i	0.946472	−	0.322787i	$-0.104620\pi$
$102$	0	0
$103$	1524.50i	0.143698i	0.997416	+	0.0718492i	$0.0228900\pi$
$103$	1524.50i	0.143698i	−0.997416	+	0.0718492i	$0.977110\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10606.6	−0.926424	−0.463212	−	0.886248i	$-0.653303\pi$
$107$	−10606.6	−0.926424	−0.463212	+	0.886248i	$0.653303\pi$
$108$	0	0
$109$	−13952.3	−1.17433	−0.587167	−	0.809466i	$-0.699757\pi$
$109$	−13952.3	−1.17433	−0.587167	+	0.809466i	$0.699757\pi$
$110$	0	0
$111$	14794.3i	1.20074i
$112$	0	0
$113$	−8811.30	−0.690054	−0.345027	−	0.938593i	$-0.612130\pi$
$113$	−8811.30	−0.690054	−0.345027	+	0.938593i	$0.612130\pi$
$114$	0	0
$115$	− 11064.0i	− 0.836595i
$116$	0	0
$117$	− 23372.3i	− 1.70738i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−12707.6	−0.867945
$122$	0	0
$123$	17148.7	1.13350
$124$	0	0
$125$	− 15372.2i	− 0.983823i
$126$	0	0
$127$	−3992.70	−0.247548	−0.123774	−	0.992310i	$-0.539500\pi$
$127$	−3992.70	−0.247548	−0.123774	+	0.992310i	$0.539500\pi$
$128$	0	0
$129$	− 36268.1i	− 2.17944i
$130$	0	0
$131$	− 20225.0i	− 1.17854i	−0.807935	−	0.589272i	$-0.799415\pi$
$131$	− 20225.0i	− 1.17854i	0.807935	−	0.589272i	$-0.200585\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−23913.5	−1.31213
$136$	0	0
$137$	25057.7	1.33506	0.667530	−	0.744583i	$-0.267352\pi$
$137$	25057.7	1.33506	0.667530	+	0.744583i	$0.267352\pi$
$138$	0	0
$139$	− 20070.1i	− 1.03877i	−0.854541	−	0.519385i	$-0.826161\pi$
$139$	− 20070.1i	− 1.03877i	0.854541	−	0.519385i	$-0.173839\pi$
$140$	0	0
$141$	20173.5	1.01471
$142$	0	0
$143$	− 7145.52i	− 0.349431i
$144$	0	0
$145$	− 18839.7i	− 0.896062i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−42707.9	−1.92369	−0.961846	−	0.273591i	$-0.911789\pi$
$149$	−42707.9	−1.92369	−0.961846	+	0.273591i	$0.911789\pi$
$150$	0	0
$151$	−29607.4	−1.29852	−0.649258	−	0.760569i	$-0.724920\pi$
$151$	−29607.4	−1.29852	−0.649258	+	0.760569i	$0.724920\pi$
$152$	0	0
$153$	− 15566.9i	− 0.664998i
$154$	0	0
$155$	26387.5	1.09833
$156$	0	0
$157$	12237.1i	0.496452i	0.968702	+	0.248226i	$0.0798476\pi$
$157$	12237.1i	0.496452i	−0.968702	+	0.248226i	$0.920152\pi$
$158$	0	0
$159$	− 53472.0i	− 2.11511i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21531.1	−0.810385	−0.405193	−	0.914231i	$-0.632796\pi$
$163$	−21531.1	−0.810385	−0.405193	+	0.914231i	$0.632796\pi$
$164$	0	0
$165$	−16737.3	−0.614775
$166$	0	0
$167$	− 16578.9i	− 0.594461i	−0.954806	−	0.297230i	$-0.903937\pi$
$167$	− 16578.9i	− 0.594461i	0.954806	−	0.297230i	$-0.0960629\pi$
$168$	0	0
$169$	2152.52	0.0753658
$170$	0	0
$171$	− 64658.4i	− 2.21123i
$172$	0	0
$173$	40383.1i	1.34930i	0.738140	+	0.674648i	$0.235704\pi$
$173$	40383.1i	1.34930i	−0.738140	+	0.674648i	$0.764296\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	64249.9	2.05081
$178$	0	0
$179$	53494.4	1.66956	0.834780	−	0.550583i	$-0.185595\pi$
$179$	53494.4	1.66956	0.834780	+	0.550583i	$0.185595\pi$
$180$	0	0
$181$	54202.1i	1.65447i	0.561857	+	0.827235i	$0.310087\pi$
$181$	54202.1i	1.65447i	−0.561857	+	0.827235i	$0.689913\pi$
$182$	0	0
$183$	−20942.1	−0.625343
$184$	0	0
$185$	− 25048.2i	− 0.731867i
$186$	0	0
$187$	− 4759.22i	− 0.136098i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1203.87	0.0330000	0.0165000	−	0.999864i	$-0.494748\pi$
$191$	1203.87	0.0330000	0.0165000	+	0.999864i	$0.494748\pi$
$192$	0	0
$193$	51312.9	1.37757	0.688783	−	0.724968i	$-0.258145\pi$
$193$	51312.9	1.37757	0.688783	+	0.724968i	$0.258145\pi$
$194$	0	0
$195$	61857.7i	1.62676i
$196$	0	0
$197$	−1456.28	−0.0375242	−0.0187621	−	0.999824i	$-0.505973\pi$
$197$	−1456.28	−0.0375242	−0.0187621	+	0.999824i	$0.505973\pi$
$198$	0	0
$199$	− 68758.2i	− 1.73627i	−0.496325	−	0.868137i	$-0.665317\pi$
$199$	− 68758.2i	− 1.73627i	0.496325	−	0.868137i	$-0.334683\pi$
$200$	0	0
$201$	− 108944.i	− 2.69657i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−29034.3	−0.690882
$206$	0	0
$207$	−62681.5	−1.46285
$208$	0	0
$209$	− 19767.8i	− 0.452549i
$210$	0	0
$211$	622.821	0.0139894	0.00699469	−	0.999976i	$-0.497774\pi$
$211$	622.821	0.0139894	0.00699469	+	0.999976i	$0.497774\pi$
$212$	0	0
$213$	− 89781.2i	− 1.97891i
$214$	0	0
$215$	61405.2i	1.32840i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8658.99	0.180542
$220$	0	0
$221$	−17589.2	−0.360131
$222$	0	0
$223$	− 11509.7i	− 0.231449i	−0.993281	−	0.115725i	$-0.963081\pi$
$223$	− 11509.7i	− 0.231449i	0.993281	−	0.115725i	$-0.0369190\pi$
$224$	0	0
$225$	2800.26	0.0553138
$226$	0	0
$227$	56367.2i	1.09389i	0.837168	+	0.546946i	$0.184210\pi$
$227$	56367.2i	1.09389i	−0.837168	+	0.546946i	$0.815790\pi$
$228$	0	0
$229$	42128.5i	0.803351i	0.915782	+	0.401675i	$0.131572\pi$
$229$	42128.5i	0.803351i	−0.915782	+	0.401675i	$0.868428\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−55101.9	−1.01497	−0.507486	−	0.861660i	$-0.669425\pi$
$233$	−55101.9	−1.01497	−0.507486	+	0.861660i	$0.669425\pi$
$234$	0	0
$235$	−34155.6	−0.618480
$236$	0	0
$237$	47174.9i	0.839875i
$238$	0	0
$239$	−23082.5	−0.404098	−0.202049	−	0.979375i	$-0.564760\pi$
$239$	−23082.5	−0.404098	−0.202049	+	0.979375i	$0.564760\pi$
$240$	0	0
$241$	81572.5i	1.40446i	0.711950	+	0.702231i	$0.247812\pi$
$241$	81572.5i	1.40446i	−0.711950	+	0.702231i	$0.752188\pi$
$242$	0	0
$243$	− 39198.0i	− 0.663821i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−73057.9	−1.19749
$248$	0	0
$249$	−70921.5	−1.14388
$250$	0	0
$251$	27207.6i	0.431859i	0.976409	+	0.215930i	$0.0692782\pi$
$251$	27207.6i	0.431859i	−0.976409	+	0.215930i	$0.930722\pi$
$252$	0	0
$253$	−19163.4	−0.299385
$254$	0	0
$255$	41199.8i	0.633600i
$256$	0	0
$257$	− 107929.i	− 1.63408i	−0.576584	−	0.817038i	$-0.695615\pi$
$257$	− 107929.i	− 1.63408i	0.576584	−	0.817038i	$-0.304385\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−106734.	−1.56683
$262$	0	0
$263$	40144.1	0.580377	0.290188	−	0.956970i	$-0.406282\pi$
$263$	40144.1	0.580377	0.290188	+	0.956970i	$0.406282\pi$
$264$	0	0
$265$	90533.1i	1.28919i
$266$	0	0
$267$	−12563.7	−0.176236
$268$	0	0
$269$	105354.i	1.45595i	0.685606	+	0.727973i	$0.259538\pi$
$269$	105354.i	1.45595i	−0.685606	+	0.727973i	$0.740462\pi$
$270$	0	0
$271$	− 15229.7i	− 0.207373i	−0.994610	−	0.103687i	$-0.966936\pi$
$271$	− 15229.7i	− 0.207373i	0.994610	−	0.103687i	$-0.0330639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	856.113	0.0113205
$276$	0	0
$277$	−9871.62	−0.128656	−0.0643278	−	0.997929i	$-0.520490\pi$
$277$	−9871.62	−0.128656	−0.0643278	+	0.997929i	$0.520490\pi$
$278$	0	0
$279$	− 149495.i	− 1.92051i
$280$	0	0
$281$	−155117.	−1.96448	−0.982240	−	0.187631i	$-0.939919\pi$
$281$	−155117.	−1.96448	−0.982240	+	0.187631i	$0.939919\pi$
$282$	0	0
$283$	109662.i	1.36925i	0.728895	+	0.684626i	$0.240034\pi$
$283$	109662.i	1.36925i	−0.728895	+	0.684626i	$0.759966\pi$
$284$	0	0
$285$	171127.i	2.10682i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	71805.9	0.859734
$290$	0	0
$291$	−84300.3	−0.995504
$292$	0	0
$293$	− 89912.4i	− 1.04733i	−0.851924	−	0.523666i	$-0.824564\pi$
$293$	− 89912.4i	− 1.04733i	0.851924	−	0.523666i	$-0.175436\pi$
$294$	0	0
$295$	−108781.	−1.25000
$296$	0	0
$297$	41419.4i	0.469560i
$298$	0	0
$299$	70824.1i	0.792208i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	98743.7	1.07553
$304$	0	0
$305$	35456.9	0.381155
$306$	0	0
$307$	141681.i	1.50327i	0.659582	+	0.751633i	$0.270733\pi$
$307$	141681.i	1.50327i	−0.659582	+	0.751633i	$0.729267\pi$
$308$	0	0
$309$	−22858.5	−0.239403
$310$	0	0
$311$	85645.6i	0.885491i	0.896647	+	0.442746i	$0.145995\pi$
$311$	85645.6i	0.885491i	−0.896647	+	0.442746i	$0.854005\pi$
$312$	0	0
$313$	− 17831.6i	− 0.182012i	−0.995850	−	0.0910061i	$-0.970992\pi$
$313$	− 17831.6i	− 0.182012i	0.995850	−	0.0910061i	$-0.0290083\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	80805.2	0.804120	0.402060	−	0.915613i	$-0.368294\pi$
$317$	80805.2	0.804120	0.402060	+	0.915613i	$0.368294\pi$
$318$	0	0
$319$	−32631.3	−0.320666
$320$	0	0
$321$	− 159037.i	− 1.54343i
$322$	0	0
$323$	−48659.7	−0.466406
$324$	0	0
$325$	− 3164.03i	− 0.0299553i
$326$	0	0
$327$	− 209202.i	− 1.95646i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−72804.3	−0.664509	−0.332254	−	0.943190i	$-0.607809\pi$
$331$	−72804.3	−0.664509	−0.332254	+	0.943190i	$0.607809\pi$
$332$	0	0
$333$	−141907.	−1.27972
$334$	0	0
$335$	184452.i	1.64359i
$336$	0	0
$337$	126538.	1.11419	0.557095	−	0.830449i	$-0.311916\pi$
$337$	126538.	1.11419	0.557095	+	0.830449i	$0.311916\pi$
$338$	0	0
$339$	− 132118.i	− 1.14964i
$340$	0	0
$341$	− 45704.5i	− 0.393052i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	165894.	1.39378
$346$	0	0
$347$	66949.3	0.556016	0.278008	−	0.960579i	$-0.410326\pi$
$347$	66949.3	0.556016	0.278008	+	0.960579i	$0.410326\pi$
$348$	0	0
$349$	25527.5i	0.209583i	0.994494	+	0.104792i	$0.0334176\pi$
$349$	25527.5i	0.209583i	−0.994494	+	0.104792i	$0.966582\pi$
$350$	0	0
$351$	153078.	1.24251
$352$	0	0
$353$	− 56575.6i	− 0.454025i	−0.973892	−	0.227013i	$-0.927104\pi$
$353$	− 56575.6i	− 0.454025i	0.973892	−	0.227013i	$-0.0728959\pi$
$354$	0	0
$355$	152008.i	1.20617i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37804.8	−0.293331	−0.146665	−	0.989186i	$-0.546854\pi$
$359$	−37804.8	−0.293331	−0.146665	+	0.989186i	$0.546854\pi$
$360$	0	0
$361$	−71790.8	−0.550876
$362$	0	0
$363$	− 190539.i	− 1.44601i
$364$	0	0
$365$	−14660.5	−0.110043
$366$	0	0
$367$	− 10816.4i	− 0.0803065i	−0.999194	−	0.0401532i	$-0.987215\pi$
$367$	− 10816.4i	− 0.0803065i	0.999194	−	0.0401532i	$-0.0127846\pi$
$368$	0	0
$369$	164490.i	1.20806i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−31023.4	−0.222983	−0.111491	−	0.993765i	$-0.535563\pi$
$373$	−31023.4	−0.222983	−0.111491	+	0.993765i	$0.535563\pi$
$374$	0	0
$375$	230493.	1.63906
$376$	0	0
$377$	120599.i	0.848519i
$378$	0	0
$379$	98527.5	0.685929	0.342964	−	0.939348i	$-0.388569\pi$
$379$	98527.5	0.685929	0.342964	+	0.939348i	$0.388569\pi$
$380$	0	0
$381$	− 59867.0i	− 0.412418i
$382$	0	0
$383$	− 5143.33i	− 0.0350628i	−0.999846	−	0.0175314i	$-0.994419\pi$
$383$	− 5143.33i	− 0.0350628i	0.999846	−	0.0175314i	$-0.00558071\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	347883.	2.32280
$388$	0	0
$389$	86282.0	0.570192	0.285096	−	0.958499i	$-0.407975\pi$
$389$	86282.0	0.570192	0.285096	+	0.958499i	$0.407975\pi$
$390$	0	0
$391$	47171.9i	0.308553i
$392$	0	0
$393$	303256.	1.96347
$394$	0	0
$395$	− 79871.5i	− 0.511915i
$396$	0	0
$397$	155926.i	0.989323i	0.869086	+	0.494661i	$0.164708\pi$
$397$	155926.i	0.989323i	−0.869086	+	0.494661i	$0.835292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34614.5	0.215263	0.107631	−	0.994191i	$-0.465673\pi$
$401$	34614.5	0.215263	0.107631	+	0.994191i	$0.465673\pi$
$402$	0	0
$403$	−168915.	−1.04006
$404$	0	0
$405$	− 62817.9i	− 0.382978i
$406$	0	0
$407$	−43384.7	−0.261907
$408$	0	0
$409$	− 313788.i	− 1.87581i	−0.346886	−	0.937907i	$-0.612761\pi$
$409$	− 313788.i	− 1.87581i	0.346886	−	0.937907i	$-0.387239\pi$
$410$	0	0
$411$	375719.i	2.22423i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	120077.	0.697208
$416$	0	0
$417$	300933.	1.73060
$418$	0	0
$419$	7324.74i	0.0417219i	0.999782	+	0.0208609i	$0.00664073\pi$
$419$	7324.74i	0.0417219i	−0.999782	+	0.0208609i	$0.993359\pi$
$420$	0	0
$421$	−25178.3	−0.142057	−0.0710285	−	0.997474i	$-0.522628\pi$
$421$	−25178.3	−0.142057	−0.0710285	+	0.997474i	$0.522628\pi$
$422$	0	0
$423$	193504.i	1.08146i
$424$	0	0
$425$	− 2107.38i	− 0.0116671i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	107141.	0.582157
$430$	0	0
$431$	−13827.2	−0.0744353	−0.0372176	−	0.999307i	$-0.511849\pi$
$431$	−13827.2	−0.0744353	−0.0372176	+	0.999307i	$0.511849\pi$
$432$	0	0
$433$	− 47438.8i	− 0.253022i	−0.991965	−	0.126511i	$-0.959622\pi$
$433$	− 47438.8i	− 0.253022i	0.991965	−	0.126511i	$-0.0403779\pi$
$434$	0	0
$435$	282485.	1.49285
$436$	0	0
$437$	195932.i	1.02599i
$438$	0	0
$439$	109814.i	0.569808i	0.958556	+	0.284904i	$0.0919618\pi$
$439$	109814.i	0.569808i	−0.958556	+	0.284904i	$0.908038\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	82560.7	0.420694	0.210347	−	0.977627i	$-0.432541\pi$
$443$	82560.7	0.420694	0.210347	+	0.977627i	$0.432541\pi$
$444$	0	0
$445$	21271.5	0.107418
$446$	0	0
$447$	− 640367.i	− 3.20490i
$448$	0	0
$449$	−330438.	−1.63907	−0.819534	−	0.573030i	$-0.805768\pi$
$449$	−330438.	−1.63907	−0.819534	+	0.573030i	$0.805768\pi$
$450$	0	0
$451$	50288.9i	0.247240i
$452$	0	0
$453$	− 443937.i	− 2.16334i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−373287.	−1.78735	−0.893677	−	0.448711i	$-0.851883\pi$
$457$	−373287.	−1.78735	−0.893677	+	0.448711i	$0.851883\pi$
$458$	0	0
$459$	101957.	0.483938
$460$	0	0
$461$	− 20355.2i	− 0.0957798i	−0.998853	−	0.0478899i	$-0.984750\pi$
$461$	− 20355.2i	− 0.0957798i	0.998853	−	0.0478899i	$-0.0152497\pi$
$462$	0	0
$463$	31552.7	0.147189	0.0735944	−	0.997288i	$-0.476553\pi$
$463$	31552.7	0.147189	0.0735944	+	0.997288i	$0.476553\pi$
$464$	0	0
$465$	395657.i	1.82984i
$466$	0	0
$467$	− 241854.i	− 1.10897i	−0.832193	−	0.554486i	$-0.812915\pi$
$467$	− 241854.i	− 1.10897i	0.832193	−	0.554486i	$-0.187085\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−183484.	−0.827096
$472$	0	0
$473$	106357.	0.475383
$474$	0	0
$475$	− 8753.16i	− 0.0387951i
$476$	0	0
$477$	512903.	2.25423
$478$	0	0
$479$	− 272083.i	− 1.18585i	−0.805257	−	0.592926i	$-0.797973\pi$
$479$	− 272083.i	− 1.18585i	0.805257	−	0.592926i	$-0.202027\pi$
$480$	0	0
$481$	160342.i	0.693036i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	142728.	0.606773
$486$	0	0
$487$	407746.	1.71922	0.859611	−	0.510949i	$-0.170706\pi$
$487$	407746.	1.71922	0.859611	+	0.510949i	$0.170706\pi$
$488$	0	0
$489$	− 322840.i	− 1.35011i
$490$	0	0
$491$	286772.	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$491$	286772.	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$492$	0	0
$493$	80324.2i	0.330486i
$494$	0	0
$495$	− 160544.i	− 0.655213i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−351435.	−1.41138	−0.705690	−	0.708520i	$-0.749363\pi$
$499$	−351435.	−1.41138	−0.705690	+	0.708520i	$0.749363\pi$
$500$	0	0
$501$	248586.	0.990379
$502$	0	0
$503$	116045.i	0.458660i	0.973349	+	0.229330i	$0.0736535\pi$
$503$	116045.i	0.458660i	−0.973349	+	0.229330i	$0.926347\pi$
$504$	0	0
$505$	−167182.	−0.655552
$506$	0	0
$507$	32275.2i	0.125560i
$508$	0	0
$509$	− 83173.8i	− 0.321034i	−0.987033	−	0.160517i	$-0.948684\pi$
$509$	− 83173.8i	− 0.321034i	0.987033	−	0.160517i	$-0.0513162\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	423485.	1.60917
$514$	0	0
$515$	38701.5	0.145920
$516$	0	0
$517$	59159.2i	0.221330i
$518$	0	0
$519$	−605508.	−2.24794
$520$	0	0
$521$	202013.i	0.744224i	0.928188	+	0.372112i	$0.121366\pi$
$521$	202013.i	0.744224i	−0.928188	+	0.372112i	$0.878634\pi$
$522$	0	0
$523$	− 27867.6i	− 0.101882i	−0.998702	−	0.0509408i	$-0.983778\pi$
$523$	− 27867.6i	− 0.101882i	0.998702	−	0.0509408i	$-0.0162220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−112505.	−0.405087
$528$	0	0
$529$	−89899.7	−0.321253
$530$	0	0
$531$	616284.i	2.18571i
$532$	0	0
$533$	185858.	0.654226
$534$	0	0
$535$	269264.i	0.940743i
$536$	0	0
$537$	802101.i	2.78151i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−37993.6	−0.129812	−0.0649062	−	0.997891i	$-0.520675\pi$
$541$	−37993.6	−0.129812	−0.0649062	+	0.997891i	$0.520675\pi$
$542$	0	0
$543$	−812712.	−2.75637
$544$	0	0
$545$	354198.i	1.19249i
$546$	0	0
$547$	−360160.	−1.20371	−0.601854	−	0.798606i	$-0.705571\pi$
$547$	−360160.	−1.20371	−0.601854	+	0.798606i	$0.705571\pi$
$548$	0	0
$549$	− 200877.i	− 0.666476i
$550$	0	0
$551$	333633.i	1.09892i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	375575.	1.21930
$556$	0	0
$557$	310543.	1.00095	0.500474	−	0.865751i	$-0.333159\pi$
$557$	310543.	1.00095	0.500474	+	0.865751i	$0.333159\pi$
$558$	0	0
$559$	− 393075.i	− 1.25792i
$560$	0	0
$561$	71360.2	0.226741
$562$	0	0
$563$	− 16689.4i	− 0.0526530i	−0.999653	−	0.0263265i	$-0.991619\pi$
$563$	− 16689.4i	− 0.0526530i	0.999653	−	0.0263265i	$-0.00838095\pi$
$564$	0	0
$565$	223687.i	0.700720i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−130262.	−0.402341	−0.201170	−	0.979556i	$-0.564474\pi$
$569$	−130262.	−0.402341	−0.201170	+	0.979556i	$0.564474\pi$
$570$	0	0
$571$	498239.	1.52815	0.764073	−	0.645129i	$-0.223196\pi$
$571$	498239.	1.52815	0.764073	+	0.645129i	$0.223196\pi$
$572$	0	0
$573$	18051.0i	0.0549784i
$574$	0	0
$575$	−8485.52	−0.0256651
$576$	0	0
$577$	− 33716.2i	− 0.101271i	−0.998717	−	0.0506357i	$-0.983875\pi$
$577$	− 33716.2i	− 0.101271i	0.998717	−	0.0506357i	$-0.0161247\pi$
$578$	0	0
$579$	769392.i	2.29504i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	156808.	0.461350
$584$	0	0
$585$	−593339.	−1.73377
$586$	0	0
$587$	356809.i	1.03552i	0.855525	+	0.517762i	$0.173235\pi$
$587$	356809.i	1.03552i	−0.855525	+	0.517762i	$0.826765\pi$
$588$	0	0
$589$	−467296.	−1.34698
$590$	0	0
$591$	− 21835.6i	− 0.0625157i
$592$	0	0
$593$	33339.2i	0.0948082i	0.998876	+	0.0474041i	$0.0150949\pi$
$593$	33339.2i	0.0948082i	−0.998876	+	0.0474041i	$0.984905\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.03097e6	2.89265
$598$	0	0
$599$	−324330.	−0.903927	−0.451963	−	0.892036i	$-0.649276\pi$
$599$	−324330.	−0.903927	−0.451963	+	0.892036i	$0.649276\pi$
$600$	0	0
$601$	− 66322.0i	− 0.183615i	−0.995777	−	0.0918076i	$-0.970736\pi$
$601$	− 66322.0i	− 0.183615i	0.995777	−	0.0918076i	$-0.0292645\pi$
$602$	0	0
$603$	1.04499e6	2.87394
$604$	0	0
$605$	322600.i	0.881361i
$606$	0	0
$607$	− 487655.i	− 1.32354i	−0.749709	−	0.661768i	$-0.769806\pi$
$607$	− 487655.i	− 1.32354i	0.749709	−	0.661768i	$-0.230194\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	218641.	0.585665
$612$	0	0
$613$	−341510.	−0.908831	−0.454415	−	0.890790i	$-0.650152\pi$
$613$	−341510.	−0.908831	−0.454415	+	0.890790i	$0.650152\pi$
$614$	0	0
$615$	− 435344.i	− 1.15102i
$616$	0	0
$617$	240873.	0.632728	0.316364	−	0.948638i	$-0.397538\pi$
$617$	240873.	0.632728	0.316364	+	0.948638i	$0.397538\pi$
$618$	0	0
$619$	154335.i	0.402794i	0.979510	+	0.201397i	$0.0645481\pi$
$619$	154335.i	0.402794i	−0.979510	+	0.201397i	$0.935452\pi$
$620$	0	0
$621$	− 410536.i	− 1.06456i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−402415.	−1.03018
$626$	0	0
$627$	296400.	0.753952
$628$	0	0
$629$	106794.i	0.269927i
$630$	0	0
$631$	−220248.	−0.553164	−0.276582	−	0.960990i	$-0.589202\pi$
$631$	−220248.	−0.553164	−0.276582	+	0.960990i	$0.589202\pi$
$632$	0	0
$633$	9338.64i	0.0233065i
$634$	0	0
$635$	101360.i	0.251374i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	861180.	2.10908
$640$	0	0
$641$	−681208.	−1.65792	−0.828960	−	0.559308i	$-0.811067\pi$
$641$	−681208.	−1.65792	−0.828960	+	0.559308i	$0.811067\pi$
$642$	0	0
$643$	572102.i	1.38373i	0.722027	+	0.691865i	$0.243211\pi$
$643$	572102.i	1.38373i	−0.722027	+	0.691865i	$0.756789\pi$
$644$	0	0
$645$	−920716.	−2.21313
$646$	0	0
$647$	607275.i	1.45070i	0.688382	+	0.725348i	$0.258321\pi$
$647$	607275.i	1.45070i	−0.688382	+	0.725348i	$0.741679\pi$
$648$	0	0
$649$	188414.i	0.447326i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−484670.	−1.13663	−0.568316	−	0.822810i	$-0.692405\pi$
$653$	−484670.	−1.13663	−0.568316	+	0.822810i	$0.692405\pi$
$654$	0	0
$655$	−513440.	−1.19676
$656$	0	0
$657$	83057.0i	0.192418i
$658$	0	0
$659$	−329627.	−0.759017	−0.379509	−	0.925188i	$-0.623907\pi$
$659$	−329627.	−0.759017	−0.379509	+	0.925188i	$0.623907\pi$
$660$	0	0
$661$	− 210608.i	− 0.482028i	−0.970522	−	0.241014i	$-0.922520\pi$
$661$	− 210608.i	− 0.482028i	0.970522	−	0.241014i	$-0.0774800\pi$
$662$	0	0
$663$	− 263734.i	− 0.599983i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	323432.	0.726994
$668$	0	0
$669$	172578.	0.385597
$670$	0	0
$671$	− 61413.2i	− 0.136401i
$672$	0	0
$673$	94709.8	0.209105	0.104553	−	0.994519i	$-0.466659\pi$
$673$	94709.8	0.209105	0.104553	+	0.994519i	$0.466659\pi$
$674$	0	0
$675$	18340.5i	0.0402535i
$676$	0	0
$677$	− 538552.i	− 1.17503i	−0.809212	−	0.587517i	$-0.800106\pi$
$677$	− 538552.i	− 1.17503i	0.809212	−	0.587517i	$-0.199894\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−845176.	−1.82244
$682$	0	0
$683$	129072.	0.276688	0.138344	−	0.990384i	$-0.455822\pi$
$683$	129072.	0.276688	0.138344	+	0.990384i	$0.455822\pi$
$684$	0	0
$685$	− 636126.i	− 1.35570i
$686$	0	0
$687$	−631680.	−1.33839
$688$	0	0
$689$	− 579532.i	− 1.22079i
$690$	0	0
$691$	130713.i	0.273756i	0.990588	+	0.136878i	$0.0437068\pi$
$691$	130713.i	0.273756i	−0.990588	+	0.136878i	$0.956293\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−509507.	−1.05483
$696$	0	0
$697$	123790.	0.254811
$698$	0	0
$699$	− 826204.i	− 1.69096i
$700$	0	0
$701$	−182501.	−0.371389	−0.185694	−	0.982608i	$-0.559453\pi$
$701$	−182501.	−0.371389	−0.185694	+	0.982608i	$0.559453\pi$
$702$	0	0
$703$	443578.i	0.897552i
$704$	0	0
$705$	− 512133.i	− 1.03040i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	813798.	1.61892	0.809458	−	0.587178i	$-0.199761\pi$
$709$	813798.	1.61892	0.809458	+	0.587178i	$0.199761\pi$
$710$	0	0
$711$	−452502.	−0.895119
$712$	0	0
$713$	453008.i	0.891101i
$714$	0	0
$715$	−181399.	−0.354832
$716$	0	0
$717$	− 346101.i	− 0.673232i
$718$	0	0
$719$	− 420567.i	− 0.813537i	−0.913531	−	0.406769i	$-0.866656\pi$
$719$	− 420567.i	− 0.813537i	0.913531	−	0.406769i	$-0.133344\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.22311e6	−2.33985
$724$	0	0
$725$	−14449.1	−0.0274894
$726$	0	0
$727$	172948.i	0.327225i	0.986525	+	0.163613i	$0.0523147\pi$
$727$	172948.i	0.327225i	−0.986525	+	0.163613i	$0.947685\pi$
$728$	0	0
$729$	788171.	1.48308
$730$	0	0
$731$	− 261805.i	− 0.489939i
$732$	0	0
$733$	− 131562.i	− 0.244862i	−0.992477	−	0.122431i	$-0.960931\pi$
$733$	− 131562.i	− 0.244862i	0.992477	−	0.122431i	$-0.0390690\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	319481.	0.588179
$738$	0	0
$739$	489768.	0.896812	0.448406	−	0.893830i	$-0.351992\pi$
$739$	489768.	0.896812	0.448406	+	0.893830i	$0.351992\pi$
$740$	0	0
$741$	− 1.09544e6i	− 1.99504i
$742$	0	0
$743$	−580258.	−1.05110	−0.525549	−	0.850763i	$-0.676140\pi$
$743$	−580258.	−1.05110	−0.525549	+	0.850763i	$0.676140\pi$
$744$	0	0
$745$	1.08420e6i	1.95343i
$746$	0	0
$747$	− 680279.i	− 1.21912i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−437397.	−0.775525	−0.387763	−	0.921759i	$-0.626752\pi$
$751$	−437397.	−0.775525	−0.387763	+	0.921759i	$0.626752\pi$
$752$	0	0
$753$	−407953.	−0.719483
$754$	0	0
$755$	751627.i	1.31859i
$756$	0	0
$757$	−24171.7	−0.0421809	−0.0210905	−	0.999778i	$-0.506714\pi$
$757$	−24171.7	−0.0421809	−0.0210905	+	0.999778i	$0.506714\pi$
$758$	0	0
$759$	− 287338.i	− 0.498780i
$760$	0	0
$761$	648221.i	1.11932i	0.828723	+	0.559659i	$0.189068\pi$
$761$	648221.i	1.11932i	−0.828723	+	0.559659i	$0.810932\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−395189.	−0.675276
$766$	0	0
$767$	696343.	1.18368
$768$	0	0
$769$	179564.i	0.303646i	0.988408	+	0.151823i	$0.0485143\pi$
$769$	179564.i	0.303646i	−0.988408	+	0.151823i	$0.951486\pi$
$770$	0	0
$771$	1.61830e6	2.72239
$772$	0	0
$773$	610915.i	1.02240i	0.859461	+	0.511201i	$0.170799\pi$
$773$	610915.i	1.02240i	−0.859461	+	0.511201i	$0.829201\pi$
$774$	0	0
$775$	− 20237.9i	− 0.0336947i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	514169.	0.847288
$780$	0	0
$781$	263285.	0.431643
$782$	0	0
$783$	− 699061.i	− 1.14023i
$784$	0	0
$785$	310655.	0.504126
$786$	0	0
$787$	− 1.19155e6i	− 1.92381i	−0.273384	−	0.961905i	$-0.588143\pi$
$787$	− 1.19155e6i	− 1.92381i	0.273384	−	0.961905i	$-0.411857\pi$
$788$	0	0
$789$	601925.i	0.966915i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−226972.	−0.360932
$794$	0	0
$795$	−1.35746e6	−2.14780
$796$	0	0
$797$	900495.i	1.41764i	0.705392	+	0.708818i	$0.250771\pi$
$797$	900495.i	1.41764i	−0.705392	+	0.708818i	$0.749229\pi$
$798$	0	0
$799$	145624.	0.228108
$800$	0	0
$801$	− 120511.i	− 0.187828i
$802$	0	0
$803$	25392.7i	0.0393802i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.57969e6	−2.42562
$808$	0	0
$809$	555764.	0.849168	0.424584	−	0.905389i	$-0.360420\pi$
$809$	555764.	0.849168	0.424584	+	0.905389i	$0.360420\pi$
$810$	0	0
$811$	70954.2i	0.107879i	0.998544	+	0.0539395i	$0.0171778\pi$
$811$	70954.2i	0.107879i	−0.998544	+	0.0539395i	$0.982822\pi$
$812$	0	0
$813$	228356.	0.345486
$814$	0	0
$815$	546598.i	0.822911i
$816$	0	0
$817$	− 1.08743e6i	− 1.62913i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−624712.	−0.926816	−0.463408	−	0.886145i	$-0.653374\pi$
$821$	−624712.	−0.926816	−0.463408	+	0.886145i	$0.653374\pi$
$822$	0	0
$823$	210059.	0.310128	0.155064	−	0.987904i	$-0.450442\pi$
$823$	210059.	0.310128	0.155064	+	0.987904i	$0.450442\pi$
$824$	0	0
$825$	12836.6i	0.0188601i
$826$	0	0
$827$	880910.	1.28801	0.644007	−	0.765020i	$-0.277271\pi$
$827$	880910.	1.28801	0.644007	+	0.765020i	$0.277271\pi$
$828$	0	0
$829$	− 1.29270e6i	− 1.88100i	−0.339799	−	0.940498i	$-0.610359\pi$
$829$	− 1.29270e6i	− 1.88100i	0.339799	−	0.940498i	$-0.389641\pi$
$830$	0	0
$831$	− 148016.i	− 0.214342i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−420879.	−0.603649
$836$	0	0
$837$	979126.	1.39762
$838$	0	0
$839$	1.11586e6i	1.58520i	0.609741	+	0.792601i	$0.291274\pi$
$839$	1.11586e6i	1.58520i	−0.609741	+	0.792601i	$0.708726\pi$
$840$	0	0
$841$	−156542.	−0.221330
$842$	0	0
$843$	− 2.32584e6i	− 3.27285i
$844$	0	0
$845$	− 54644.8i	− 0.0765307i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.64428e6	−2.28119
$850$	0	0
$851$	430015.	0.593779
$852$	0	0
$853$	1.28959e6i	1.77236i	0.463341	+	0.886180i	$0.346650\pi$
$853$	1.28959e6i	1.77236i	−0.463341	+	0.886180i	$0.653350\pi$
$854$	0	0
$855$	−1.64145e6	−2.24540
$856$	0	0
$857$	− 654547.i	− 0.891208i	−0.895230	−	0.445604i	$-0.852989\pi$
$857$	− 654547.i	− 0.891208i	0.895230	−	0.445604i	$-0.147011\pi$
$858$	0	0
$859$	244561.i	0.331437i	0.986173	+	0.165718i	$0.0529943\pi$
$859$	244561.i	0.331437i	−0.986173	+	0.165718i	$0.947006\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	465277.	0.624726	0.312363	−	0.949963i	$-0.398879\pi$
$863$	465277.	0.624726	0.312363	+	0.949963i	$0.398879\pi$
$864$	0	0
$865$	1.02518e6	1.37015
$866$	0	0
$867$	1.07667e6i	1.43233i
$868$	0	0
$869$	−138342.	−0.183195
$870$	0	0
$871$	− 1.18074e6i	− 1.55639i
$872$	0	0
$873$	− 808608.i	− 1.06099i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	304172.	0.395475	0.197738	−	0.980255i	$-0.436641\pi$
$877$	304172.	0.395475	0.197738	+	0.980255i	$0.436641\pi$
$878$	0	0
$879$	1.34816e6	1.74487
$880$	0	0
$881$	697876.i	0.899138i	0.893246	+	0.449569i	$0.148422\pi$
$881$	697876.i	0.899138i	−0.893246	+	0.449569i	$0.851578\pi$
$882$	0	0
$883$	−891773.	−1.14375	−0.571877	−	0.820339i	$-0.693785\pi$
$883$	−891773.	−1.14375	−0.571877	+	0.820339i	$0.693785\pi$
$884$	0	0
$885$	− 1.63107e6i	− 2.08251i
$886$	0	0
$887$	− 1.26256e6i	− 1.60473i	−0.596831	−	0.802367i	$-0.703574\pi$
$887$	− 1.26256e6i	− 1.60473i	0.596831	−	0.802367i	$-0.296426\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−108804.	−0.137053
$892$	0	0
$893$	604862.	0.758496
$894$	0	0
$895$	− 1.35803e6i	− 1.69537i
$896$	0	0
$897$	−1.06195e6	−1.31983
$898$	0	0
$899$	771382.i	0.954443i
$900$	0	0
$901$	− 385993.i	− 0.475477i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.37600e6	1.68004
$906$	0	0
$907$	−951981.	−1.15721	−0.578607	−	0.815607i	$-0.696403\pi$
$907$	−951981.	−1.15721	−0.578607	+	0.815607i	$0.696403\pi$
$908$	0	0
$909$	947149.i	1.14628i
$910$	0	0
$911$	743243.	0.895559	0.447779	−	0.894144i	$-0.352215\pi$
$911$	743243.	0.895559	0.447779	+	0.894144i	$0.352215\pi$
$912$	0	0
$913$	− 207979.i	− 0.249504i
$914$	0	0
$915$	531645.i	0.635009i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	524300.	0.620796	0.310398	−	0.950607i	$-0.399538\pi$
$919$	524300.	0.620796	0.310398	+	0.950607i	$0.399538\pi$
$920$	0	0
$921$	−2.12438e6	−2.50446
$922$	0	0
$923$	− 973052.i	− 1.14218i
$924$	0	0
$925$	−19210.7	−0.0224522
$926$	0	0
$927$	− 219258.i	− 0.255151i
$928$	0	0
$929$	− 979392.i	− 1.13482i	−0.823437	−	0.567408i	$-0.807946\pi$
$929$	− 979392.i	− 1.13482i	0.823437	−	0.567408i	$-0.192054\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.28418e6	−1.47524
$934$	0	0
$935$	−120819.	−0.138202
$936$	0	0
$937$	− 1.13988e6i	− 1.29831i	−0.760656	−	0.649155i	$-0.775123\pi$
$937$	− 1.13988e6i	− 1.29831i	0.760656	−	0.649155i	$-0.224877\pi$
$938$	0	0
$939$	267368.	0.303235
$940$	0	0
$941$	801121.i	0.904730i	0.891833	+	0.452365i	$0.149420\pi$
$941$	801121.i	0.904730i	−0.891833	+	0.452365i	$0.850580\pi$
$942$	0	0
$943$	− 498448.i	− 0.560527i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−680286.	−0.758563	−0.379282	−	0.925281i	$-0.623829\pi$
$947$	−680286.	−0.758563	−0.379282	+	0.925281i	$0.623829\pi$
$948$	0	0
$949$	93846.5	0.104204
$950$	0	0
$951$	1.21160e6i	1.33967i
$952$	0	0
$953$	−394774.	−0.434673	−0.217337	−	0.976097i	$-0.569737\pi$
$953$	−394774.	−0.434673	−0.217337	+	0.976097i	$0.569737\pi$
$954$	0	0
$955$	− 30562.0i	− 0.0335101i
$956$	0	0
$957$	− 489278.i	− 0.534234i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−156900.	−0.169893
$962$	0	0
$963$	1.52548e6	1.64496
$964$	0	0
$965$	− 1.30265e6i	− 1.39886i
$966$	0	0
$967$	604328.	0.646279	0.323140	−	0.946351i	$-0.395262\pi$
$967$	604328.	0.646279	0.323140	+	0.946351i	$0.395262\pi$
$968$	0	0
$969$	− 729609.i	− 0.777038i
$970$	0	0
$971$	758986.i	0.804999i	0.915420	+	0.402499i	$0.131858\pi$
$971$	758986.i	0.804999i	−0.915420	+	0.402499i	$0.868142\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	47441.8	0.0499059
$976$	0	0
$977$	1.22518e6	1.28355	0.641773	−	0.766894i	$-0.278199\pi$
$977$	1.22518e6	1.28355	0.641773	+	0.766894i	$0.278199\pi$
$978$	0	0
$979$	− 36843.3i	− 0.0384409i
$980$	0	0
$981$	2.00666e6	2.08514
$982$	0	0
$983$	1.20047e6i	1.24235i	0.783670	+	0.621177i	$0.213345\pi$
$983$	1.20047e6i	1.24235i	−0.783670	+	0.621177i	$0.786655\pi$
$984$	0	0
$985$	36969.6i	0.0381042i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.05418e6	−1.07776
$990$	0	0
$991$	−1.80993e6	−1.84295	−0.921475	−	0.388438i	$-0.873015\pi$
$991$	−1.80993e6	−1.84295	−0.921475	+	0.388438i	$0.873015\pi$
$992$	0	0
$993$	− 1.09164e6i	− 1.10708i
$994$	0	0
$995$	−1.74552e6	−1.76311
$996$	0	0
$997$	749289.i	0.753805i	0.926253	+	0.376902i	$0.123011\pi$
$997$	749289.i	0.753805i	−0.926253	+	0.376902i	$0.876989\pi$
$998$	0	0
$999$	− 929429.i	− 0.931291i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.b.97.4		4
4.3	odd	2		98.5.b.b.97.3		4
7.4	even	3		112.5.s.b.33.2		4
7.5	odd	6		112.5.s.b.17.2		4
7.6	odd	2	inner	784.5.c.b.97.1		4
12.11	even	2		882.5.c.b.685.2		4
28.3	even	6		98.5.d.a.19.1		4
28.11	odd	6		14.5.d.a.5.1	yes	4
28.19	even	6		14.5.d.a.3.1	✓	4
28.23	odd	6		98.5.d.a.31.1		4
28.27	even	2		98.5.b.b.97.4		4
84.11	even	6		126.5.n.a.19.2		4
84.47	odd	6		126.5.n.a.73.2		4
84.83	odd	2		882.5.c.b.685.1		4
140.19	even	6		350.5.k.a.101.2		4
140.39	odd	6		350.5.k.a.201.2		4
140.47	odd	12		350.5.i.a.199.3		8
140.67	even	12		350.5.i.a.299.2		8
140.103	odd	12		350.5.i.a.199.2		8
140.123	even	12		350.5.i.a.299.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.d.a.3.1	✓	4	28.19	even	6
14.5.d.a.5.1	yes	4	28.11	odd	6
98.5.b.b.97.3		4	4.3	odd	2
98.5.b.b.97.4		4	28.27	even	2
98.5.d.a.19.1		4	28.3	even	6
98.5.d.a.31.1		4	28.23	odd	6
112.5.s.b.17.2		4	7.5	odd	6
112.5.s.b.33.2		4	7.4	even	3
126.5.n.a.19.2		4	84.11	even	6
126.5.n.a.73.2		4	84.47	odd	6
350.5.i.a.199.2		8	140.103	odd	12
350.5.i.a.199.3		8	140.47	odd	12
350.5.i.a.299.2		8	140.67	even	12
350.5.i.a.299.3		8	140.123	even	12
350.5.k.a.101.2		4	140.19	even	6
350.5.k.a.201.2		4	140.39	odd	6
784.5.c.b.97.1		4	7.6	odd	2	inner
784.5.c.b.97.4		4	1.1	even	1	trivial
882.5.c.b.685.1		4	84.83	odd	2
882.5.c.b.685.2		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+14.9941i q^{3} -25.3864i q^{5} -143.823 q^{9} +O(q^{10})\)	\(q+14.9941i q^{3} -25.3864i q^{5} -143.823 q^{9} -43.9706 q^{11} +162.507i q^{13} +380.647 q^{15} +108.236i q^{17} +449.568i q^{19} +435.823 q^{23} -19.4701 q^{25} -941.981i q^{27} +742.118 q^{29} +1039.43i q^{31} -659.300i q^{33} +986.675 q^{37} -2436.65 q^{39} -1143.70i q^{41} -2418.82 q^{43} +3651.16i q^{45} -1345.43i q^{47} -1622.91 q^{51} -3566.20 q^{53} +1116.26i q^{55} -6740.88 q^{57} -4285.01i q^{59} +1396.69i q^{61} +4125.47 q^{65} -7265.79 q^{67} +6534.77i q^{69} -5987.76 q^{71} -577.493i q^{73} -291.937i q^{75} +3146.23 q^{79} +2474.47 q^{81} +4729.96i q^{83} +2747.74 q^{85} +11127.4i q^{87} +837.909i q^{89} -15585.4 q^{93} +11412.9 q^{95} +5622.23i q^{97} +6323.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 168 q^{9}+O(q^{10}) \) 4 * q - 168 * q^9	\( 4 q - 168 q^{9} - 108 q^{11} + 708 q^{15} - 972 q^{23} + 1144 q^{25} + 3240 q^{29} + 892 q^{37} - 6624 q^{39} - 2344 q^{43} + 636 q^{51} - 5508 q^{53} - 17460 q^{57} + 6048 q^{65} - 10124 q^{67} - 18792 q^{71} + 1588 q^{79} + 8676 q^{81} + 10380 q^{85} - 37836 q^{93} + 17820 q^{95} + 11448 q^{99}+O(q^{100}) \) 4 * q - 168 * q^9 - 108 * q^11 + 708 * q^15 - 972 * q^23 + 1144 * q^25 + 3240 * q^29 + 892 * q^37 - 6624 * q^39 - 2344 * q^43 + 636 * q^51 - 5508 * q^53 - 17460 * q^57 + 6048 * q^65 - 10124 * q^67 - 18792 * q^71 + 1588 * q^79 + 8676 * q^81 + 10380 * q^85 - 37836 * q^93 + 17820 * q^95 + 11448 * q^99

Introduction

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