LMFDB - Embedded newform 784.3.c.e.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,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, 3, names="a")

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
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:	$21.3624527258$
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^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.3.c.e.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.717439i q^{3} -6.63103i q^{5} +8.48528 q^{9} +O(q^{10})$	$q+0.717439i q^{3} -6.63103i q^{5} +8.48528 q^{9} +4.75736 q^{11} +15.2913i q^{13} +4.75736 q^{15} +3.76127i q^{17} -4.18154i q^{19} +27.7279 q^{23} -18.9706 q^{25} +12.5446i q^{27} +3.51472 q^{29} -48.8667i q^{31} +3.41311i q^{33} -2.94113 q^{37} -10.9706 q^{39} -27.9590i q^{41} +10.4853 q^{43} -56.2662i q^{45} -52.6790i q^{47} -2.69848 q^{51} +55.9706 q^{53} -31.5462i q^{55} +3.00000 q^{57} +38.7206i q^{59} -90.5080i q^{61} +101.397 q^{65} +34.6396 q^{67} +19.8931i q^{69} -36.4264 q^{71} -52.6069i q^{73} -13.6102i q^{75} +33.7868 q^{79} +67.3675 q^{81} +127.577i q^{83} +24.9411 q^{85} +2.52160i q^{87} -50.3314i q^{89} +35.0589 q^{93} -27.7279 q^{95} +101.792i q^{97} +40.3675 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 36 q^{11} + 36 q^{15} + 60 q^{23} - 8 q^{25} + 48 q^{29} + 124 q^{37} + 24 q^{39} + 8 q^{43} + 108 q^{51} + 156 q^{53} + 12 q^{57} + 168 q^{65} - 116 q^{67} + 24 q^{71} + 220 q^{79} - 36 q^{81} - 36 q^{85} + 276 q^{93} - 60 q^{95} - 144 q^{99}+O(q^{100}) $ 4 * q + 36 * q^11 + 36 * q^15 + 60 * q^23 - 8 * q^25 + 48 * q^29 + 124 * q^37 + 24 * q^39 + 8 * q^43 + 108 * q^51 + 156 * q^53 + 12 * q^57 + 168 * q^65 - 116 * q^67 + 24 * q^71 + 220 * q^79 - 36 * q^81 - 36 * q^85 + 276 * q^93 - 60 * q^95 - 144 * 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
$3$	0.717439i	0.239146i	0.992825	+	0.119573i	$0.0381526\pi$
$3$	0.717439i	0.239146i	−0.992825	+	0.119573i	$0.961847\pi$
$4$	0	0
$5$	− 6.63103i	− 1.32621i	−0.748528	−	0.663103i	$-0.769239\pi$
$5$	− 6.63103i	− 1.32621i	0.748528	−	0.663103i	$-0.230761\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.48528	0.942809
$10$	0	0
$11$	4.75736	0.432487	0.216244	−	0.976339i	$-0.430619\pi$
$11$	4.75736	0.432487	0.216244	+	0.976339i	$0.430619\pi$
$12$	0	0
$13$	15.2913i	1.17625i	0.808769	+	0.588126i	$0.200134\pi$
$13$	15.2913i	1.17625i	−0.808769	+	0.588126i	$0.799866\pi$
$14$	0	0
$15$	4.75736	0.317157
$16$	0	0
$17$	3.76127i	0.221251i	0.993862	+	0.110626i	$0.0352855\pi$
$17$	3.76127i	0.221251i	−0.993862	+	0.110626i	$0.964715\pi$
$18$	0	0
$19$	− 4.18154i	− 0.220081i	−0.993927	−	0.110041i	$-0.964902\pi$
$19$	− 4.18154i	− 0.220081i	0.993927	−	0.110041i	$-0.0350981\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	27.7279	1.20556	0.602781	−	0.797907i	$-0.294059\pi$
$23$	27.7279	1.20556	0.602781	+	0.797907i	$0.294059\pi$
$24$	0	0
$25$	−18.9706	−0.758823
$26$	0	0
$27$	12.5446i	0.464616i
$28$	0	0
$29$	3.51472	0.121197	0.0605986	−	0.998162i	$-0.480699\pi$
$29$	3.51472	0.121197	0.0605986	+	0.998162i	$0.480699\pi$
$30$	0	0
$31$	− 48.8667i	− 1.57635i	−0.615454	−	0.788173i	$-0.711027\pi$
$31$	− 48.8667i	− 1.57635i	0.615454	−	0.788173i	$-0.288973\pi$
$32$	0	0
$33$	3.41311i	0.103428i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.94113	−0.0794899	−0.0397449	−	0.999210i	$-0.512655\pi$
$37$	−2.94113	−0.0794899	−0.0397449	+	0.999210i	$0.512655\pi$
$38$	0	0
$39$	−10.9706	−0.281296
$40$	0	0
$41$	− 27.9590i	− 0.681927i	−0.940077	−	0.340963i	$-0.889247\pi$
$41$	− 27.9590i	− 0.681927i	0.940077	−	0.340963i	$-0.110753\pi$
$42$	0	0
$43$	10.4853	0.243844	0.121922	−	0.992540i	$-0.461094\pi$
$43$	10.4853	0.243844	0.121922	+	0.992540i	$0.461094\pi$
$44$	0	0
$45$	− 56.2662i	− 1.25036i
$46$	0	0
$47$	− 52.6790i	− 1.12083i	−0.828212	−	0.560415i	$-0.810642\pi$
$47$	− 52.6790i	− 1.12083i	0.828212	−	0.560415i	$-0.189358\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.69848	−0.0529115
$52$	0	0
$53$	55.9706	1.05605	0.528024	−	0.849229i	$-0.322933\pi$
$53$	55.9706	1.05605	0.528024	+	0.849229i	$0.322933\pi$
$54$	0	0
$55$	− 31.5462i	− 0.573567i
$56$	0	0
$57$	3.00000	0.0526316
$58$	0	0
$59$	38.7206i	0.656281i	0.944629	+	0.328141i	$0.106422\pi$
$59$	38.7206i	0.656281i	−0.944629	+	0.328141i	$0.893578\pi$
$60$	0	0
$61$	− 90.5080i	− 1.48374i	−0.670545	−	0.741869i	$-0.733940\pi$
$61$	− 90.5080i	− 1.48374i	0.670545	−	0.741869i	$-0.266060\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	101.397	1.55995
$66$	0	0
$67$	34.6396	0.517009	0.258505	−	0.966010i	$-0.416770\pi$
$67$	34.6396	0.517009	0.258505	+	0.966010i	$0.416770\pi$
$68$	0	0
$69$	19.8931i	0.288306i
$70$	0	0
$71$	−36.4264	−0.513048	−0.256524	−	0.966538i	$-0.582577\pi$
$71$	−36.4264	−0.513048	−0.256524	+	0.966538i	$0.582577\pi$
$72$	0	0
$73$	− 52.6069i	− 0.720642i	−0.932828	−	0.360321i	$-0.882667\pi$
$73$	− 52.6069i	− 0.720642i	0.932828	−	0.360321i	$-0.117333\pi$
$74$	0	0
$75$	− 13.6102i	− 0.181470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	33.7868	0.427681	0.213840	−	0.976869i	$-0.431403\pi$
$79$	33.7868	0.427681	0.213840	+	0.976869i	$0.431403\pi$
$80$	0	0
$81$	67.3675	0.831698
$82$	0	0
$83$	127.577i	1.53708i	0.639803	+	0.768539i	$0.279016\pi$
$83$	127.577i	1.53708i	−0.639803	+	0.768539i	$0.720984\pi$
$84$	0	0
$85$	24.9411	0.293425
$86$	0	0
$87$	2.52160i	0.0289839i
$88$	0	0
$89$	− 50.3314i	− 0.565522i	−0.959190	−	0.282761i	$-0.908750\pi$
$89$	− 50.3314i	− 0.565522i	0.959190	−	0.282761i	$-0.0912503\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	35.0589	0.376977
$94$	0	0
$95$	−27.7279	−0.291873
$96$	0	0
$97$	101.792i	1.04940i	0.851287	+	0.524700i	$0.175823\pi$
$97$	101.792i	1.04940i	−0.851287	+	0.524700i	$0.824177\pi$
$98$	0	0
$99$	40.3675	0.407753
$100$	0	0
$101$	59.6793i	0.590884i	0.955361	+	0.295442i	$0.0954669\pi$
$101$	59.6793i	0.590884i	−0.955361	+	0.295442i	$0.904533\pi$
$102$	0	0
$103$	− 120.178i	− 1.16678i	−0.812193	−	0.583388i	$-0.801727\pi$
$103$	− 120.178i	− 1.16678i	0.812193	−	0.583388i	$-0.198273\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	113.610	1.06178	0.530889	−	0.847442i	$-0.321858\pi$
$107$	113.610	1.06178	0.530889	+	0.847442i	$0.321858\pi$
$108$	0	0
$109$	−145.309	−1.33311	−0.666553	−	0.745457i	$-0.732231\pi$
$109$	−145.309	−1.33311	−0.666553	+	0.745457i	$0.732231\pi$
$110$	0	0
$111$	− 2.11008i	− 0.0190097i
$112$	0	0
$113$	34.5442	0.305700	0.152850	−	0.988249i	$-0.451155\pi$
$113$	34.5442	0.305700	0.152850	+	0.988249i	$0.451155\pi$
$114$	0	0
$115$	− 183.865i	− 1.59882i
$116$	0	0
$117$	129.751i	1.10898i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−98.3675	−0.812955
$122$	0	0
$123$	20.0589	0.163080
$124$	0	0
$125$	− 39.9814i	− 0.319851i
$126$	0	0
$127$	247.338	1.94754	0.973772	−	0.227526i	$-0.0730636\pi$
$127$	247.338	1.94754	0.973772	+	0.227526i	$0.0730636\pi$
$128$	0	0
$129$	7.52255i	0.0583143i
$130$	0	0
$131$	147.645i	1.12706i	0.826096	+	0.563529i	$0.190557\pi$
$131$	147.645i	1.12706i	−0.826096	+	0.563529i	$0.809443\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	83.1838	0.616176
$136$	0	0
$137$	−32.5736	−0.237763	−0.118882	−	0.992908i	$-0.537931\pi$
$137$	−32.5736	−0.237763	−0.118882	+	0.992908i	$0.537931\pi$
$138$	0	0
$139$	− 68.5857i	− 0.493422i	−0.969089	−	0.246711i	$-0.920650\pi$
$139$	− 68.5857i	− 0.493422i	0.969089	−	0.246711i	$-0.0793499\pi$
$140$	0	0
$141$	37.7939	0.268042
$142$	0	0
$143$	72.7461i	0.508714i
$144$	0	0
$145$	− 23.3062i	− 0.160732i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	92.3970	0.620114	0.310057	−	0.950718i	$-0.399652\pi$
$149$	92.3970	0.620114	0.310057	+	0.950718i	$0.399652\pi$
$150$	0	0
$151$	91.7868	0.607860	0.303930	−	0.952694i	$-0.401701\pi$
$151$	91.7868	0.607860	0.303930	+	0.952694i	$0.401701\pi$
$152$	0	0
$153$	31.9155i	0.208598i
$154$	0	0
$155$	−324.037	−2.09056
$156$	0	0
$157$	− 8.45631i	− 0.0538618i	−0.999637	−	0.0269309i	$-0.991427\pi$
$157$	− 8.45631i	− 0.0538618i	0.999637	−	0.0269309i	$-0.00857341\pi$
$158$	0	0
$159$	40.1555i	0.252550i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−221.978	−1.36183	−0.680913	−	0.732364i	$-0.738417\pi$
$163$	−221.978	−1.36183	−0.680913	+	0.732364i	$0.738417\pi$
$164$	0	0
$165$	22.6325	0.137166
$166$	0	0
$167$	− 168.841i	− 1.01102i	−0.862820	−	0.505511i	$-0.831304\pi$
$167$	− 168.841i	− 1.01102i	0.862820	−	0.505511i	$-0.168696\pi$
$168$	0	0
$169$	−64.8234	−0.383570
$170$	0	0
$171$	− 35.4815i	− 0.207494i
$172$	0	0
$173$	− 164.341i	− 0.949947i	−0.880000	−	0.474974i	$-0.842458\pi$
$173$	− 164.341i	− 0.949947i	0.880000	−	0.474974i	$-0.157542\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−27.7797	−0.156947
$178$	0	0
$179$	−185.184	−1.03455	−0.517273	−	0.855820i	$-0.673053\pi$
$179$	−185.184	−1.03455	−0.517273	+	0.855820i	$0.673053\pi$
$180$	0	0
$181$	− 155.086i	− 0.856830i	−0.903582	−	0.428415i	$-0.859072\pi$
$181$	− 155.086i	− 0.856830i	0.903582	−	0.428415i	$-0.140928\pi$
$182$	0	0
$183$	64.9340	0.354831
$184$	0	0
$185$	19.5027i	0.105420i
$186$	0	0
$187$	17.8937i	0.0956884i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−248.095	−1.29893	−0.649465	−	0.760392i	$-0.725007\pi$
$191$	−248.095	−1.29893	−0.649465	+	0.760392i	$0.725007\pi$
$192$	0	0
$193$	154.338	0.799679	0.399840	−	0.916585i	$-0.369066\pi$
$193$	154.338	0.799679	0.399840	+	0.916585i	$0.369066\pi$
$194$	0	0
$195$	72.7461i	0.373057i
$196$	0	0
$197$	−181.103	−0.919303	−0.459651	−	0.888099i	$-0.652026\pi$
$197$	−181.103	−0.919303	−0.459651	+	0.888099i	$0.652026\pi$
$198$	0	0
$199$	348.707i	1.75229i	0.482043	+	0.876147i	$0.339895\pi$
$199$	348.707i	1.75229i	−0.482043	+	0.876147i	$0.660105\pi$
$200$	0	0
$201$	24.8518i	0.123641i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−185.397	−0.904375
$206$	0	0
$207$	235.279	1.13661
$208$	0	0
$209$	− 19.8931i	− 0.0951823i
$210$	0	0
$211$	−364.073	−1.72547	−0.862733	−	0.505660i	$-0.831249\pi$
$211$	−364.073	−1.72547	−0.862733	+	0.505660i	$0.831249\pi$
$212$	0	0
$213$	− 26.1337i	− 0.122694i
$214$	0	0
$215$	− 69.5282i	− 0.323387i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	37.7422	0.172339
$220$	0	0
$221$	−57.5147	−0.260248
$222$	0	0
$223$	− 123.231i	− 0.552603i	−0.961071	−	0.276302i	$-0.910891\pi$
$223$	− 123.231i	− 0.552603i	0.961071	−	0.276302i	$-0.0891089\pi$
$224$	0	0
$225$	−160.971	−0.715425
$226$	0	0
$227$	− 76.3756i	− 0.336456i	−0.985748	−	0.168228i	$-0.946195\pi$
$227$	− 76.3756i	− 0.336456i	0.985748	−	0.168228i	$-0.0538045\pi$
$228$	0	0
$229$	357.286i	1.56020i	0.625654	+	0.780101i	$0.284832\pi$
$229$	357.286i	1.56020i	−0.625654	+	0.780101i	$0.715168\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−273.073	−1.17199	−0.585994	−	0.810315i	$-0.699296\pi$
$233$	−273.073	−1.17199	−0.585994	+	0.810315i	$0.699296\pi$
$234$	0	0
$235$	−349.316	−1.48645
$236$	0	0
$237$	24.2400i	0.102278i
$238$	0	0
$239$	265.103	1.10922	0.554608	−	0.832112i	$-0.312868\pi$
$239$	265.103	1.10922	0.554608	+	0.832112i	$0.312868\pi$
$240$	0	0
$241$	87.6383i	0.363644i	0.983331	+	0.181822i	$0.0581995\pi$
$241$	87.6383i	0.363644i	−0.983331	+	0.181822i	$0.941800\pi$
$242$	0	0
$243$	161.234i	0.663513i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	63.9411	0.258871
$248$	0	0
$249$	−91.5290	−0.367586
$250$	0	0
$251$	495.655i	1.97472i	0.158491	+	0.987360i	$0.449337\pi$
$251$	495.655i	1.97472i	−0.158491	+	0.987360i	$0.550663\pi$
$252$	0	0
$253$	131.912	0.521390
$254$	0	0
$255$	17.8937i	0.0701715i
$256$	0	0
$257$	400.536i	1.55851i	0.626709	+	0.779254i	$0.284402\pi$
$257$	400.536i	1.55851i	−0.626709	+	0.779254i	$0.715598\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	29.8234	0.114266
$262$	0	0
$263$	32.3452	0.122986	0.0614928	−	0.998108i	$-0.480414\pi$
$263$	32.3452	0.122986	0.0614928	+	0.998108i	$0.480414\pi$
$264$	0	0
$265$	− 371.142i	− 1.40054i
$266$	0	0
$267$	36.1097	0.135242
$268$	0	0
$269$	306.963i	1.14113i	0.821253	+	0.570564i	$0.193275\pi$
$269$	306.963i	1.14113i	−0.821253	+	0.570564i	$0.806725\pi$
$270$	0	0
$271$	75.9852i	0.280388i	0.990124	+	0.140194i	$0.0447726\pi$
$271$	75.9852i	0.280388i	−0.990124	+	0.140194i	$0.955227\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−90.2498	−0.328181
$276$	0	0
$277$	278.411	1.00509	0.502547	−	0.864550i	$-0.332396\pi$
$277$	278.411	1.00509	0.502547	+	0.864550i	$0.332396\pi$
$278$	0	0
$279$	− 414.648i	− 1.48619i
$280$	0	0
$281$	394.690	1.40459	0.702296	−	0.711885i	$-0.252158\pi$
$281$	394.690	1.40459	0.702296	+	0.711885i	$0.252158\pi$
$282$	0	0
$283$	146.396i	0.517301i	0.965971	+	0.258650i	$0.0832778\pi$
$283$	146.396i	0.517301i	−0.965971	+	0.258650i	$0.916722\pi$
$284$	0	0
$285$	− 19.8931i	− 0.0698003i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	274.853	0.951048
$290$	0	0
$291$	−73.0294	−0.250960
$292$	0	0
$293$	299.678i	1.02279i	0.859345	+	0.511396i	$0.170872\pi$
$293$	299.678i	1.02279i	−0.859345	+	0.511396i	$0.829128\pi$
$294$	0	0
$295$	256.757	0.870364
$296$	0	0
$297$	59.6793i	0.200940i
$298$	0	0
$299$	423.996i	1.41805i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−42.8162	−0.141308
$304$	0	0
$305$	−600.161	−1.96774
$306$	0	0
$307$	20.9886i	0.0683666i	0.999416	+	0.0341833i	$0.0108830\pi$
$307$	20.9886i	0.0683666i	−0.999416	+	0.0341833i	$0.989117\pi$
$308$	0	0
$309$	86.2203	0.279030
$310$	0	0
$311$	182.039i	0.585336i	0.956214	+	0.292668i	$0.0945430\pi$
$311$	182.039i	0.585336i	−0.956214	+	0.292668i	$0.905457\pi$
$312$	0	0
$313$	− 97.9286i	− 0.312871i	−0.987688	−	0.156435i	$-0.950000\pi$
$313$	− 97.9286i	− 0.312871i	0.987688	−	0.156435i	$-0.0500003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−481.971	−1.52041	−0.760206	−	0.649682i	$-0.774902\pi$
$317$	−481.971	−1.52041	−0.760206	+	0.649682i	$0.774902\pi$
$318$	0	0
$319$	16.7208	0.0524162
$320$	0	0
$321$	81.5084i	0.253920i
$322$	0	0
$323$	15.7279	0.0486933
$324$	0	0
$325$	− 290.084i	− 0.892567i
$326$	0	0
$327$	− 104.250i	− 0.318808i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−225.007	−0.679780	−0.339890	−	0.940465i	$-0.610390\pi$
$331$	−225.007	−0.679780	−0.339890	+	0.940465i	$0.610390\pi$
$332$	0	0
$333$	−24.9563	−0.0749438
$334$	0	0
$335$	− 229.696i	− 0.685661i
$336$	0	0
$337$	−264.368	−0.784473	−0.392237	−	0.919864i	$-0.628299\pi$
$337$	−264.368	−0.784473	−0.392237	+	0.919864i	$0.628299\pi$
$338$	0	0
$339$	24.7833i	0.0731071i
$340$	0	0
$341$	− 232.476i	− 0.681749i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	131.912	0.382353
$346$	0	0
$347$	191.257	0.551173	0.275586	−	0.961276i	$-0.411128\pi$
$347$	191.257	0.551173	0.275586	+	0.961276i	$0.411128\pi$
$348$	0	0
$349$	135.448i	0.388104i	0.980991	+	0.194052i	$0.0621630\pi$
$349$	135.448i	0.388104i	−0.980991	+	0.194052i	$0.937837\pi$
$350$	0	0
$351$	−191.823	−0.546505
$352$	0	0
$353$	348.490i	0.987225i	0.869682	+	0.493612i	$0.164324\pi$
$353$	348.490i	0.987225i	−0.869682	+	0.493612i	$0.835676\pi$
$354$	0	0
$355$	241.545i	0.680407i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−304.831	−0.849110	−0.424555	−	0.905402i	$-0.639569\pi$
$359$	−304.831	−0.849110	−0.424555	+	0.905402i	$0.639569\pi$
$360$	0	0
$361$	343.515	0.951564
$362$	0	0
$363$	− 70.5727i	− 0.194415i
$364$	0	0
$365$	−348.838	−0.955720
$366$	0	0
$367$	− 95.0042i	− 0.258867i	−0.991588	−	0.129434i	$-0.958684\pi$
$367$	− 95.0042i	− 0.258867i	0.991588	−	0.129434i	$-0.0413159\pi$
$368$	0	0
$369$	− 237.240i	− 0.642927i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	253.558	0.679781	0.339891	−	0.940465i	$-0.389610\pi$
$373$	253.558	0.679781	0.339891	+	0.940465i	$0.389610\pi$
$374$	0	0
$375$	28.6842	0.0764912
$376$	0	0
$377$	53.7446i	0.142559i
$378$	0	0
$379$	−508.250	−1.34103	−0.670514	−	0.741897i	$-0.733926\pi$
$379$	−508.250	−1.34103	−0.670514	+	0.741897i	$0.733926\pi$
$380$	0	0
$381$	177.450i	0.465748i
$382$	0	0
$383$	− 477.761i	− 1.24742i	−0.781656	−	0.623709i	$-0.785625\pi$
$383$	− 477.761i	− 1.24742i	0.781656	−	0.623709i	$-0.214375\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	88.9706	0.229898
$388$	0	0
$389$	170.220	0.437584	0.218792	−	0.975771i	$-0.429788\pi$
$389$	170.220	0.437584	0.218792	+	0.975771i	$0.429788\pi$
$390$	0	0
$391$	104.292i	0.266732i
$392$	0	0
$393$	−105.926	−0.269532
$394$	0	0
$395$	− 224.041i	− 0.567193i
$396$	0	0
$397$	244.550i	0.615995i	0.951387	+	0.307997i	$0.0996588\pi$
$397$	244.550i	0.615995i	−0.951387	+	0.307997i	$0.900341\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−417.573	−1.04133	−0.520664	−	0.853762i	$-0.674316\pi$
$401$	−417.573	−1.04133	−0.520664	+	0.853762i	$0.674316\pi$
$402$	0	0
$403$	747.235	1.85418
$404$	0	0
$405$	− 446.716i	− 1.10300i
$406$	0	0
$407$	−13.9920	−0.0343784
$408$	0	0
$409$	− 308.212i	− 0.753574i	−0.926300	−	0.376787i	$-0.877029\pi$
$409$	− 308.212i	− 0.753574i	0.926300	−	0.376787i	$-0.122971\pi$
$410$	0	0
$411$	− 23.3696i	− 0.0568603i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	845.970	2.03848
$416$	0	0
$417$	49.2061	0.118000
$418$	0	0
$419$	103.142i	0.246163i	0.992397	+	0.123081i	$0.0392776\pi$
$419$	103.142i	0.246163i	−0.992397	+	0.123081i	$0.960722\pi$
$420$	0	0
$421$	−165.220	−0.392447	−0.196224	−	0.980559i	$-0.562868\pi$
$421$	−165.220	−0.392447	−0.196224	+	0.980559i	$0.562868\pi$
$422$	0	0
$423$	− 446.996i	− 1.05673i
$424$	0	0
$425$	− 71.3535i	− 0.167891i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−52.1909	−0.121657
$430$	0	0
$431$	−594.536	−1.37943	−0.689717	−	0.724079i	$-0.742265\pi$
$431$	−594.536	−1.37943	−0.689717	+	0.724079i	$0.742265\pi$
$432$	0	0
$433$	− 40.6267i	− 0.0938261i	−0.998899	−	0.0469131i	$-0.985062\pi$
$433$	− 40.6267i	− 0.0938261i	0.998899	−	0.0469131i	$-0.0149384\pi$
$434$	0	0
$435$	16.7208	0.0384386
$436$	0	0
$437$	− 115.945i	− 0.265321i
$438$	0	0
$439$	− 146.600i	− 0.333941i	−0.985962	−	0.166971i	$-0.946602\pi$
$439$	− 146.600i	− 0.333941i	0.985962	−	0.166971i	$-0.0533985\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−107.360	−0.242349	−0.121174	−	0.992631i	$-0.538666\pi$
$443$	−107.360	−0.242349	−0.121174	+	0.992631i	$0.538666\pi$
$444$	0	0
$445$	−333.749	−0.749999
$446$	0	0
$447$	66.2892i	0.148298i
$448$	0	0
$449$	135.161	0.301028	0.150514	−	0.988608i	$-0.451907\pi$
$449$	135.161	0.301028	0.150514	+	0.988608i	$0.451907\pi$
$450$	0	0
$451$	− 133.011i	− 0.294925i
$452$	0	0
$453$	65.8514i	0.145367i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	159.735	0.349530	0.174765	−	0.984610i	$-0.444083\pi$
$457$	159.735	0.349530	0.174765	+	0.984610i	$0.444083\pi$
$458$	0	0
$459$	−47.1838	−0.102797
$460$	0	0
$461$	310.250i	0.672993i	0.941685	+	0.336497i	$0.109242\pi$
$461$	310.250i	0.672993i	−0.941685	+	0.336497i	$0.890758\pi$
$462$	0	0
$463$	326.014	0.704135	0.352067	−	0.935975i	$-0.385479\pi$
$463$	326.014	0.704135	0.352067	+	0.935975i	$0.385479\pi$
$464$	0	0
$465$	− 232.476i	− 0.499949i
$466$	0	0
$467$	− 595.558i	− 1.27529i	−0.770332	−	0.637643i	$-0.779910\pi$
$467$	− 595.558i	− 1.27529i	0.770332	−	0.637643i	$-0.220090\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.06688	0.0128809
$472$	0	0
$473$	49.8823	0.105459
$474$	0	0
$475$	79.3262i	0.167002i
$476$	0	0
$477$	474.926	0.995652
$478$	0	0
$479$	506.680i	1.05779i	0.848688	+	0.528894i	$0.177393\pi$
$479$	506.680i	1.05779i	−0.848688	+	0.528894i	$0.822607\pi$
$480$	0	0
$481$	− 44.9736i	− 0.0935002i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	674.985	1.39172
$486$	0	0
$487$	−211.302	−0.433884	−0.216942	−	0.976184i	$-0.569608\pi$
$487$	−211.302	−0.433884	−0.216942	+	0.976184i	$0.569608\pi$
$488$	0	0
$489$	− 159.255i	− 0.325676i
$490$	0	0
$491$	784.161	1.59707	0.798534	−	0.601949i	$-0.205609\pi$
$491$	784.161	1.59707	0.798534	+	0.601949i	$0.205609\pi$
$492$	0	0
$493$	13.2198i	0.0268151i
$494$	0	0
$495$	− 267.678i	− 0.540764i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	171.492	0.343672	0.171836	−	0.985126i	$-0.445030\pi$
$499$	171.492	0.343672	0.171836	+	0.985126i	$0.445030\pi$
$500$	0	0
$501$	121.133	0.241782
$502$	0	0
$503$	20.0883i	0.0399370i	0.999801	+	0.0199685i	$0.00635659\pi$
$503$	20.0883i	0.0399370i	−0.999801	+	0.0199685i	$0.993643\pi$
$504$	0	0
$505$	395.735	0.783634
$506$	0	0
$507$	− 46.5068i	− 0.0917294i
$508$	0	0
$509$	− 476.764i	− 0.936668i	−0.883551	−	0.468334i	$-0.844854\pi$
$509$	− 476.764i	− 0.936668i	0.883551	−	0.468334i	$-0.155146\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	52.4558	0.102253
$514$	0	0
$515$	−796.904	−1.54739
$516$	0	0
$517$	− 250.613i	− 0.484744i
$518$	0	0
$519$	117.905	0.227176
$520$	0	0
$521$	− 854.274i	− 1.63968i	−0.572592	−	0.819841i	$-0.694062\pi$
$521$	− 854.274i	− 1.63968i	0.572592	−	0.819841i	$-0.305938\pi$
$522$	0	0
$523$	593.002i	1.13385i	0.823771	+	0.566923i	$0.191866\pi$
$523$	593.002i	1.13385i	−0.823771	+	0.566923i	$0.808134\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	183.801	0.348769
$528$	0	0
$529$	239.838	0.453379
$530$	0	0
$531$	328.555i	0.618748i
$532$	0	0
$533$	427.529	0.802118
$534$	0	0
$535$	− 753.352i	− 1.40814i
$536$	0	0
$537$	− 132.858i	− 0.247408i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	855.191	1.58076	0.790380	−	0.612617i	$-0.209883\pi$
$541$	855.191	1.58076	0.790380	+	0.612617i	$0.209883\pi$
$542$	0	0
$543$	111.265	0.204908
$544$	0	0
$545$	963.546i	1.76797i
$546$	0	0
$547$	−415.897	−0.760323	−0.380161	−	0.924920i	$-0.624132\pi$
$547$	−415.897	−0.760323	−0.380161	+	0.924920i	$0.624132\pi$
$548$	0	0
$549$	− 767.986i	− 1.39888i
$550$	0	0
$551$	− 14.6969i	− 0.0266732i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13.9920	−0.0252108
$556$	0	0
$557$	−584.220	−1.04887	−0.524435	−	0.851451i	$-0.675723\pi$
$557$	−584.220	−1.04887	−0.524435	+	0.851451i	$0.675723\pi$
$558$	0	0
$559$	160.333i	0.286822i
$560$	0	0
$561$	−12.8377	−0.0228835
$562$	0	0
$563$	911.147i	1.61838i	0.587548	+	0.809189i	$0.300093\pi$
$563$	911.147i	1.61838i	−0.587548	+	0.809189i	$0.699907\pi$
$564$	0	0
$565$	− 229.063i	− 0.405422i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	699.999	1.23023	0.615113	−	0.788439i	$-0.289110\pi$
$569$	699.999	1.23023	0.615113	+	0.788439i	$0.289110\pi$
$570$	0	0
$571$	562.463	0.985049	0.492525	−	0.870299i	$-0.336074\pi$
$571$	562.463	0.985049	0.492525	+	0.870299i	$0.336074\pi$
$572$	0	0
$573$	− 177.993i	− 0.310634i
$574$	0	0
$575$	−526.014	−0.914807
$576$	0	0
$577$	661.659i	1.14672i	0.819302	+	0.573362i	$0.194361\pi$
$577$	661.659i	1.14672i	−0.819302	+	0.573362i	$0.805639\pi$
$578$	0	0
$579$	110.728i	0.191240i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	266.272	0.456727
$584$	0	0
$585$	860.382	1.47074
$586$	0	0
$587$	823.029i	1.40209i	0.713116	+	0.701046i	$0.247283\pi$
$587$	823.029i	1.40209i	−0.713116	+	0.701046i	$0.752717\pi$
$588$	0	0
$589$	−204.338	−0.346924
$590$	0	0
$591$	− 129.930i	− 0.219848i
$592$	0	0
$593$	− 622.256i	− 1.04934i	−0.851307	−	0.524668i	$-0.824189\pi$
$593$	− 622.256i	− 1.04934i	0.851307	−	0.524668i	$-0.175811\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−250.176	−0.419055
$598$	0	0
$599$	512.845	0.856168	0.428084	−	0.903739i	$-0.359189\pi$
$599$	512.845	0.856168	0.428084	+	0.903739i	$0.359189\pi$
$600$	0	0
$601$	680.160i	1.13171i	0.824504	+	0.565857i	$0.191454\pi$
$601$	680.160i	1.13171i	−0.824504	+	0.565857i	$0.808546\pi$
$602$	0	0
$603$	293.927	0.487441
$604$	0	0
$605$	652.278i	1.07815i
$606$	0	0
$607$	− 38.7381i	− 0.0638189i	−0.999491	−	0.0319095i	$-0.989841\pi$
$607$	− 38.7381i	− 0.0638189i	0.999491	−	0.0319095i	$-0.0101588\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	805.529	1.31838
$612$	0	0
$613$	401.103	0.654329	0.327164	−	0.944967i	$-0.393907\pi$
$613$	401.103	0.654329	0.327164	+	0.944967i	$0.393907\pi$
$614$	0	0
$615$	− 133.011i	− 0.216278i
$616$	0	0
$617$	−959.044	−1.55437	−0.777183	−	0.629275i	$-0.783352\pi$
$617$	−959.044	−1.55437	−0.777183	+	0.629275i	$0.783352\pi$
$618$	0	0
$619$	− 1004.53i	− 1.62283i	−0.584469	−	0.811416i	$-0.698697\pi$
$619$	− 1004.53i	− 1.62283i	0.584469	−	0.811416i	$-0.301303\pi$
$620$	0	0
$621$	347.836i	0.560123i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−739.382	−1.18301
$626$	0	0
$627$	14.2721	0.0227625
$628$	0	0
$629$	− 11.0624i	− 0.0175873i
$630$	0	0
$631$	386.514	0.612542	0.306271	−	0.951944i	$-0.400919\pi$
$631$	386.514	0.612542	0.306271	+	0.951944i	$0.400919\pi$
$632$	0	0
$633$	− 261.200i	− 0.412639i
$634$	0	0
$635$	− 1640.11i	− 2.58284i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−309.088	−0.483706
$640$	0	0
$641$	−992.147	−1.54781	−0.773906	−	0.633301i	$-0.781700\pi$
$641$	−992.147	−1.54781	−0.773906	+	0.633301i	$0.781700\pi$
$642$	0	0
$643$	− 944.986i	− 1.46965i	−0.678256	−	0.734826i	$-0.737264\pi$
$643$	− 944.986i	− 1.46965i	0.678256	−	0.734826i	$-0.262736\pi$
$644$	0	0
$645$	49.8823	0.0773368
$646$	0	0
$647$	2.89088i	0.00446812i	0.999998	+	0.00223406i	$0.000711124\pi$
$647$	2.89088i	0.00446812i	−0.999998	+	0.00223406i	$0.999289\pi$
$648$	0	0
$649$	184.208i	0.283833i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−323.059	−0.494730	−0.247365	−	0.968922i	$-0.579565\pi$
$653$	−323.059	−0.494730	−0.247365	+	0.968922i	$0.579565\pi$
$654$	0	0
$655$	979.036	1.49471
$656$	0	0
$657$	− 446.384i	− 0.679428i
$658$	0	0
$659$	−295.955	−0.449098	−0.224549	−	0.974463i	$-0.572091\pi$
$659$	−295.955	−0.449098	−0.224549	+	0.974463i	$0.572091\pi$
$660$	0	0
$661$	− 20.7511i	− 0.0313935i	−0.999877	−	0.0156968i	$-0.995003\pi$
$661$	− 20.7511i	− 0.0313935i	0.999877	−	0.0156968i	$-0.00499664\pi$
$662$	0	0
$663$	− 41.2633i	− 0.0622373i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	97.4558	0.146111
$668$	0	0
$669$	88.4104	0.132153
$670$	0	0
$671$	− 430.579i	− 0.641698i
$672$	0	0
$673$	627.044	0.931714	0.465857	−	0.884860i	$-0.345746\pi$
$673$	627.044	0.931714	0.465857	+	0.884860i	$0.345746\pi$
$674$	0	0
$675$	− 237.979i	− 0.352561i
$676$	0	0
$677$	− 109.246i	− 0.161368i	−0.996740	−	0.0806838i	$-0.974290\pi$
$677$	− 109.246i	− 0.161368i	0.996740	−	0.0806838i	$-0.0257104\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	54.7948	0.0804623
$682$	0	0
$683$	−793.566	−1.16188	−0.580941	−	0.813946i	$-0.697315\pi$
$683$	−793.566	−1.16188	−0.580941	+	0.813946i	$0.697315\pi$
$684$	0	0
$685$	215.996i	0.315323i
$686$	0	0
$687$	−256.331	−0.373116
$688$	0	0
$689$	855.862i	1.24218i
$690$	0	0
$691$	− 183.889i	− 0.266121i	−0.991108	−	0.133060i	$-0.957520\pi$
$691$	− 183.889i	− 0.266121i	0.991108	−	0.133060i	$-0.0424804\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−454.794	−0.654380
$696$	0	0
$697$	105.161	0.150877
$698$	0	0
$699$	− 195.913i	− 0.280277i
$700$	0	0
$701$	−1043.82	−1.48905	−0.744525	−	0.667595i	$-0.767324\pi$
$701$	−1043.82	−1.48905	−0.744525	+	0.667595i	$0.767324\pi$
$702$	0	0
$703$	12.2984i	0.0174942i
$704$	0	0
$705$	− 250.613i	− 0.355479i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−980.558	−1.38301	−0.691507	−	0.722369i	$-0.743053\pi$
$709$	−980.558	−1.38301	−0.691507	+	0.722369i	$0.743053\pi$
$710$	0	0
$711$	286.690	0.403221
$712$	0	0
$713$	− 1354.97i	− 1.90038i
$714$	0	0
$715$	482.382	0.674660
$716$	0	0
$717$	190.195i	0.265265i
$718$	0	0
$719$	778.484i	1.08273i	0.840787	+	0.541366i	$0.182093\pi$
$719$	778.484i	1.08273i	−0.840787	+	0.541366i	$0.817907\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−62.8751	−0.0869642
$724$	0	0
$725$	−66.6762	−0.0919672
$726$	0	0
$727$	735.255i	1.01135i	0.862723	+	0.505677i	$0.168757\pi$
$727$	735.255i	1.01135i	−0.862723	+	0.505677i	$0.831243\pi$
$728$	0	0
$729$	490.632	0.673021
$730$	0	0
$731$	39.4380i	0.0539508i
$732$	0	0
$733$	− 478.860i	− 0.653288i	−0.945147	−	0.326644i	$-0.894082\pi$
$733$	− 478.860i	− 0.653288i	0.945147	−	0.326644i	$-0.105918\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	164.793	0.223600
$738$	0	0
$739$	19.9045	0.0269344	0.0134672	−	0.999909i	$-0.495713\pi$
$739$	19.9045	0.0269344	0.0134672	+	0.999909i	$0.495713\pi$
$740$	0	0
$741$	45.8739i	0.0619080i
$742$	0	0
$743$	−43.3095	−0.0582901	−0.0291450	−	0.999575i	$-0.509278\pi$
$743$	−43.3095	−0.0582901	−0.0291450	+	0.999575i	$0.509278\pi$
$744$	0	0
$745$	− 612.687i	− 0.822399i
$746$	0	0
$747$	1082.53i	1.44917i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−225.330	−0.300040	−0.150020	−	0.988683i	$-0.547934\pi$
$751$	−225.330	−0.300040	−0.150020	+	0.988683i	$0.547934\pi$
$752$	0	0
$753$	−355.602	−0.472247
$754$	0	0
$755$	− 608.641i	− 0.806147i
$756$	0	0
$757$	935.779	1.23617	0.618084	−	0.786112i	$-0.287909\pi$
$757$	935.779	1.23617	0.618084	+	0.786112i	$0.287909\pi$
$758$	0	0
$759$	94.6386i	0.124689i
$760$	0	0
$761$	1402.71i	1.84325i	0.388081	+	0.921625i	$0.373138\pi$
$761$	1402.71i	1.84325i	−0.388081	+	0.921625i	$0.626862\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	211.632	0.276644
$766$	0	0
$767$	−592.087	−0.771952
$768$	0	0
$769$	1.72330i	0.00224097i	0.999999	+	0.00112048i	$0.000356661\pi$
$769$	1.72330i	0.00224097i	−0.999999	+	0.00112048i	$0.999643\pi$
$770$	0	0
$771$	−287.360	−0.372711
$772$	0	0
$773$	− 224.258i	− 0.290113i	−0.989423	−	0.145057i	$-0.953664\pi$
$773$	− 224.258i	− 0.290113i	0.989423	−	0.145057i	$-0.0463364\pi$
$774$	0	0
$775$	927.029i	1.19617i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−116.912	−0.150079
$780$	0	0
$781$	−173.294	−0.221887
$782$	0	0
$783$	44.0908i	0.0563101i
$784$	0	0
$785$	−56.0740	−0.0714319
$786$	0	0
$787$	70.2034i	0.0892038i	0.999005	+	0.0446019i	$0.0142019\pi$
$787$	70.2034i	0.0892038i	−0.999005	+	0.0446019i	$0.985798\pi$
$788$	0	0
$789$	23.2057i	0.0294116i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1383.98	1.74525
$794$	0	0
$795$	266.272	0.334933
$796$	0	0
$797$	1305.38i	1.63787i	0.573889	+	0.818933i	$0.305434\pi$
$797$	1305.38i	1.63787i	−0.573889	+	0.818933i	$0.694566\pi$
$798$	0	0
$799$	198.140	0.247985
$800$	0	0
$801$	− 427.076i	− 0.533179i
$802$	0	0
$803$	− 250.270i	− 0.311668i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−220.227	−0.272897
$808$	0	0
$809$	762.765	0.942849	0.471424	−	0.881907i	$-0.343740\pi$
$809$	762.765	0.942849	0.471424	+	0.881907i	$0.343740\pi$
$810$	0	0
$811$	− 1214.98i	− 1.49813i	−0.662498	−	0.749064i	$-0.730504\pi$
$811$	− 1214.98i	− 1.49813i	0.662498	−	0.749064i	$-0.269496\pi$
$812$	0	0
$813$	−54.5147	−0.0670538
$814$	0	0
$815$	1471.94i	1.80606i
$816$	0	0
$817$	− 43.8446i	− 0.0536654i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−583.368	−0.710557	−0.355279	−	0.934760i	$-0.615614\pi$
$821$	−583.368	−0.710557	−0.355279	+	0.934760i	$0.615614\pi$
$822$	0	0
$823$	−1030.74	−1.25242	−0.626210	−	0.779655i	$-0.715395\pi$
$823$	−1030.74	−1.25242	−0.626210	+	0.779655i	$0.715395\pi$
$824$	0	0
$825$	− 64.7487i	− 0.0784833i
$826$	0	0
$827$	−152.102	−0.183920	−0.0919599	−	0.995763i	$-0.529313\pi$
$827$	−152.102	−0.183920	−0.0919599	+	0.995763i	$0.529313\pi$
$828$	0	0
$829$	− 614.410i	− 0.741146i	−0.928803	−	0.370573i	$-0.879161\pi$
$829$	− 614.410i	− 0.741146i	0.928803	−	0.370573i	$-0.120839\pi$
$830$	0	0
$831$	199.743i	0.240365i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1119.59	−1.34082
$836$	0	0
$837$	613.014	0.732395
$838$	0	0
$839$	1546.14i	1.84284i	0.388568	+	0.921420i	$0.372970\pi$
$839$	1546.14i	1.84284i	−0.388568	+	0.921420i	$0.627030\pi$
$840$	0	0
$841$	−828.647	−0.985311
$842$	0	0
$843$	283.166i	0.335903i
$844$	0	0
$845$	429.846i	0.508693i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−105.030	−0.123711
$850$	0	0
$851$	−81.5513	−0.0958300
$852$	0	0
$853$	− 1235.15i	− 1.44800i	−0.689798	−	0.724002i	$-0.742301\pi$
$853$	− 1235.15i	− 1.44800i	0.689798	−	0.724002i	$-0.257699\pi$
$854$	0	0
$855$	−235.279	−0.275180
$856$	0	0
$857$	1100.68i	1.28434i	0.766561	+	0.642172i	$0.221966\pi$
$857$	1100.68i	1.28434i	−0.766561	+	0.642172i	$0.778034\pi$
$858$	0	0
$859$	− 591.771i	− 0.688906i	−0.938803	−	0.344453i	$-0.888064\pi$
$859$	− 591.771i	− 0.688906i	0.938803	−	0.344453i	$-0.111936\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	64.7271	0.0750024	0.0375012	−	0.999297i	$-0.488060\pi$
$863$	64.7271	0.0750024	0.0375012	+	0.999297i	$0.488060\pi$
$864$	0	0
$865$	−1089.75	−1.25983
$866$	0	0
$867$	197.190i	0.227440i
$868$	0	0
$869$	160.736	0.184967
$870$	0	0
$871$	529.684i	0.608133i
$872$	0	0
$873$	863.732i	0.989384i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−304.192	−0.346855	−0.173427	−	0.984847i	$-0.555484\pi$
$877$	−304.192	−0.346855	−0.173427	+	0.984847i	$0.555484\pi$
$878$	0	0
$879$	−215.001	−0.244597
$880$	0	0
$881$	863.732i	0.980400i	0.871610	+	0.490200i	$0.163076\pi$
$881$	863.732i	0.980400i	−0.871610	+	0.490200i	$0.836924\pi$
$882$	0	0
$883$	567.456	0.642645	0.321323	−	0.946970i	$-0.395873\pi$
$883$	567.456	0.642645	0.321323	+	0.946970i	$0.395873\pi$
$884$	0	0
$885$	184.208i	0.208144i
$886$	0	0
$887$	889.761i	1.00311i	0.865125	+	0.501556i	$0.167239\pi$
$887$	889.761i	1.00311i	−0.865125	+	0.501556i	$0.832761\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	320.492	0.359699
$892$	0	0
$893$	−220.279	−0.246673
$894$	0	0
$895$	1227.96i	1.37202i
$896$	0	0
$897$	−304.191	−0.339120
$898$	0	0
$899$	− 171.753i	− 0.191049i
$900$	0	0
$901$	210.521i	0.233652i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1028.38	−1.13633
$906$	0	0
$907$	−373.978	−0.412324	−0.206162	−	0.978518i	$-0.566097\pi$
$907$	−373.978	−0.412324	−0.206162	+	0.978518i	$0.566097\pi$
$908$	0	0
$909$	506.395i	0.557091i
$910$	0	0
$911$	−1133.75	−1.24451	−0.622256	−	0.782814i	$-0.713784\pi$
$911$	−1133.75	−1.24451	−0.622256	+	0.782814i	$0.713784\pi$
$912$	0	0
$913$	606.932i	0.664766i
$914$	0	0
$915$	− 430.579i	− 0.470578i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−456.302	−0.496521	−0.248260	−	0.968693i	$-0.579859\pi$
$919$	−456.302	−0.496521	−0.248260	+	0.968693i	$0.579859\pi$
$920$	0	0
$921$	−15.0580	−0.0163496
$922$	0	0
$923$	− 557.007i	− 0.603474i
$924$	0	0
$925$	55.7948	0.0603187
$926$	0	0
$927$	− 1019.74i	− 1.10005i
$928$	0	0
$929$	− 951.540i	− 1.02426i	−0.858907	−	0.512131i	$-0.828856\pi$
$929$	− 951.540i	− 1.02426i	0.858907	−	0.512131i	$-0.171144\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−130.602	−0.139981
$934$	0	0
$935$	118.654	0.126903
$936$	0	0
$937$	− 1295.71i	− 1.38283i	−0.722460	−	0.691413i	$-0.756989\pi$
$937$	− 1295.71i	− 1.38283i	0.722460	−	0.691413i	$-0.243011\pi$
$938$	0	0
$939$	70.2578	0.0748219
$940$	0	0
$941$	1356.89i	1.44197i	0.692952	+	0.720984i	$0.256310\pi$
$941$	1356.89i	1.44197i	−0.692952	+	0.720984i	$0.743690\pi$
$942$	0	0
$943$	− 775.245i	− 0.822105i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	709.462	0.749168	0.374584	−	0.927193i	$-0.377786\pi$
$947$	709.462	0.749168	0.374584	+	0.927193i	$0.377786\pi$
$948$	0	0
$949$	804.426	0.847657
$950$	0	0
$951$	− 345.784i	− 0.363601i
$952$	0	0
$953$	936.603	0.982794	0.491397	−	0.870936i	$-0.336486\pi$
$953$	936.603	0.982794	0.491397	+	0.870936i	$0.336486\pi$
$954$	0	0
$955$	1645.13i	1.72265i
$956$	0	0
$957$	11.9961i	0.0125351i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1426.95	−1.48486
$962$	0	0
$963$	964.014	1.00105
$964$	0	0
$965$	− 1023.42i	− 1.06054i
$966$	0	0
$967$	−1374.37	−1.42127	−0.710635	−	0.703561i	$-0.751592\pi$
$967$	−1374.37	−1.42127	−0.710635	+	0.703561i	$0.751592\pi$
$968$	0	0
$969$	11.2838i	0.0116448i
$970$	0	0
$971$	31.4617i	0.0324014i	0.999869	+	0.0162007i	$0.00515706\pi$
$971$	31.4617i	0.0324014i	−0.999869	+	0.0162007i	$0.994843\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	208.118	0.213454
$976$	0	0
$977$	−541.897	−0.554654	−0.277327	−	0.960776i	$-0.589449\pi$
$977$	−541.897	−0.554654	−0.277327	+	0.960776i	$0.589449\pi$
$978$	0	0
$979$	− 239.445i	− 0.244581i
$980$	0	0
$981$	−1232.98	−1.25687
$982$	0	0
$983$	22.4198i	0.0228075i	0.999935	+	0.0114038i	$0.00363001\pi$
$983$	22.4198i	0.0228075i	−0.999935	+	0.0114038i	$0.996370\pi$
$984$	0	0
$985$	1200.90i	1.21918i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	290.735	0.293969
$990$	0	0
$991$	−678.035	−0.684193	−0.342096	−	0.939665i	$-0.611137\pi$
$991$	−678.035	−0.684193	−0.342096	+	0.939665i	$0.611137\pi$
$992$	0	0
$993$	− 161.429i	− 0.162567i
$994$	0	0
$995$	2312.28	2.32390
$996$	0	0
$997$	− 876.163i	− 0.878799i	−0.898292	−	0.439400i	$-0.855191\pi$
$997$	− 876.163i	− 0.878799i	0.898292	−	0.439400i	$-0.144809\pi$
$998$	0	0
$999$	− 36.8953i	− 0.0369322i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.c.e.97.3		4
4.3	odd	2		98.3.b.b.97.3		4
7.2	even	3		112.3.s.b.17.1		4
7.3	odd	6		112.3.s.b.33.1		4
7.4	even	3		784.3.s.c.705.2		4
7.5	odd	6		784.3.s.c.129.2		4
7.6	odd	2	inner	784.3.c.e.97.2		4
12.11	even	2		882.3.c.f.685.2		4
21.2	odd	6		1008.3.cg.l.577.2		4
21.17	even	6		1008.3.cg.l.145.2		4
28.3	even	6		14.3.d.a.5.1	yes	4
28.11	odd	6		98.3.d.a.19.1		4
28.19	even	6		98.3.d.a.31.1		4
28.23	odd	6		14.3.d.a.3.1	✓	4
28.27	even	2		98.3.b.b.97.4		4
56.3	even	6		448.3.s.d.257.1		4
56.37	even	6		448.3.s.c.129.2		4
56.45	odd	6		448.3.s.c.257.2		4
56.51	odd	6		448.3.s.d.129.1		4
84.11	even	6		882.3.n.b.19.2		4
84.23	even	6		126.3.n.c.73.2		4
84.47	odd	6		882.3.n.b.325.2		4
84.59	odd	6		126.3.n.c.19.2		4
84.83	odd	2		882.3.c.f.685.1		4
140.3	odd	12		350.3.i.a.299.3		8
140.23	even	12		350.3.i.a.199.2		8
140.59	even	6		350.3.k.a.201.2		4
140.79	odd	6		350.3.k.a.101.2		4
140.87	odd	12		350.3.i.a.299.2		8
140.107	even	12		350.3.i.a.199.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.1	✓	4	28.23	odd	6
14.3.d.a.5.1	yes	4	28.3	even	6
98.3.b.b.97.3		4	4.3	odd	2
98.3.b.b.97.4		4	28.27	even	2
98.3.d.a.19.1		4	28.11	odd	6
98.3.d.a.31.1		4	28.19	even	6
112.3.s.b.17.1		4	7.2	even	3
112.3.s.b.33.1		4	7.3	odd	6
126.3.n.c.19.2		4	84.59	odd	6
126.3.n.c.73.2		4	84.23	even	6
350.3.i.a.199.2		8	140.23	even	12
350.3.i.a.199.3		8	140.107	even	12
350.3.i.a.299.2		8	140.87	odd	12
350.3.i.a.299.3		8	140.3	odd	12
350.3.k.a.101.2		4	140.79	odd	6
350.3.k.a.201.2		4	140.59	even	6
448.3.s.c.129.2		4	56.37	even	6
448.3.s.c.257.2		4	56.45	odd	6
448.3.s.d.129.1		4	56.51	odd	6
448.3.s.d.257.1		4	56.3	even	6
784.3.c.e.97.2		4	7.6	odd	2	inner
784.3.c.e.97.3		4	1.1	even	1	trivial
784.3.s.c.129.2		4	7.5	odd	6
784.3.s.c.705.2		4	7.4	even	3
882.3.c.f.685.1		4	84.83	odd	2
882.3.c.f.685.2		4	12.11	even	2
882.3.n.b.19.2		4	84.11	even	6
882.3.n.b.325.2		4	84.47	odd	6
1008.3.cg.l.145.2		4	21.17	even	6
1008.3.cg.l.577.2		4	21.2	odd	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.717439i q^{3} -6.63103i q^{5} +8.48528 q^{9} +O(q^{10})\)	\(q+0.717439i q^{3} -6.63103i q^{5} +8.48528 q^{9} +4.75736 q^{11} +15.2913i q^{13} +4.75736 q^{15} +3.76127i q^{17} -4.18154i q^{19} +27.7279 q^{23} -18.9706 q^{25} +12.5446i q^{27} +3.51472 q^{29} -48.8667i q^{31} +3.41311i q^{33} -2.94113 q^{37} -10.9706 q^{39} -27.9590i q^{41} +10.4853 q^{43} -56.2662i q^{45} -52.6790i q^{47} -2.69848 q^{51} +55.9706 q^{53} -31.5462i q^{55} +3.00000 q^{57} +38.7206i q^{59} -90.5080i q^{61} +101.397 q^{65} +34.6396 q^{67} +19.8931i q^{69} -36.4264 q^{71} -52.6069i q^{73} -13.6102i q^{75} +33.7868 q^{79} +67.3675 q^{81} +127.577i q^{83} +24.9411 q^{85} +2.52160i q^{87} -50.3314i q^{89} +35.0589 q^{93} -27.7279 q^{95} +101.792i q^{97} +40.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 36 q^{11} + 36 q^{15} + 60 q^{23} - 8 q^{25} + 48 q^{29} + 124 q^{37} + 24 q^{39} + 8 q^{43} + 108 q^{51} + 156 q^{53} + 12 q^{57} + 168 q^{65} - 116 q^{67} + 24 q^{71} + 220 q^{79} - 36 q^{81} - 36 q^{85} + 276 q^{93} - 60 q^{95} - 144 q^{99}+O(q^{100}) \) 4 * q + 36 * q^11 + 36 * q^15 + 60 * q^23 - 8 * q^25 + 48 * q^29 + 124 * q^37 + 24 * q^39 + 8 * q^43 + 108 * q^51 + 156 * q^53 + 12 * q^57 + 168 * q^65 - 116 * q^67 + 24 * q^71 + 220 * q^79 - 36 * q^81 - 36 * q^85 + 276 * q^93 - 60 * q^95 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more