LMFDB - Embedded newform 784.6.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.48528 q^{3} +9.89949 q^{5} -171.000 q^{9} +O(q^{10})$	$q+8.48528 q^{3} +9.89949 q^{5} -171.000 q^{9} -308.000 q^{11} +912.168 q^{13} +84.0000 q^{15} +451.134 q^{17} +2616.30 q^{19} +2324.00 q^{23} -3027.00 q^{25} -3512.91 q^{27} -6488.00 q^{29} -814.587 q^{31} -2613.47 q^{33} -12200.0 q^{37} +7740.00 q^{39} +527.502 q^{41} +15028.0 q^{43} -1692.81 q^{45} -1600.89 q^{47} +3828.00 q^{51} +10778.0 q^{53} -3049.04 q^{55} +22200.0 q^{57} +52136.4 q^{59} +36765.3 q^{61} +9030.00 q^{65} +23808.0 q^{67} +19719.8 q^{69} -56448.0 q^{71} -51379.8 q^{73} -25684.9 q^{75} +11224.0 q^{79} +11745.0 q^{81} +38690.1 q^{83} +4466.00 q^{85} -55052.5 q^{87} +128177. q^{89} -6912.00 q^{93} +25900.0 q^{95} -56883.9 q^{97} +52668.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 342 q^{9}+O(q^{10}) $ 2 * q - 342 * q^9	$ 2 q - 342 q^{9} - 616 q^{11} + 168 q^{15} + 4648 q^{23} - 6054 q^{25} - 12976 q^{29} - 24400 q^{37} + 15480 q^{39} + 30056 q^{43} + 7656 q^{51} + 21556 q^{53} + 44400 q^{57} + 18060 q^{65} + 47616 q^{67} - 112896 q^{71} + 22448 q^{79} + 23490 q^{81} + 8932 q^{85} - 13824 q^{93} + 51800 q^{95} + 105336 q^{99}+O(q^{100}) $ 2 * q - 342 * q^9 - 616 * q^11 + 168 * q^15 + 4648 * q^23 - 6054 * q^25 - 12976 * q^29 - 24400 * q^37 + 15480 * q^39 + 30056 * q^43 + 7656 * q^51 + 21556 * q^53 + 44400 * q^57 + 18060 * q^65 + 47616 * q^67 - 112896 * q^71 + 22448 * q^79 + 23490 * q^81 + 8932 * q^85 - 13824 * q^93 + 51800 * q^95 + 105336 * q^99

Display 100 coefficients 10 coefficients

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
$3$	8.48528	0.544331	0.272166	−	0.962250i	$-0.412260\pi$
$3$	8.48528	0.544331	0.272166	+	0.962250i	$0.412260\pi$
$4$	0	0
$5$	9.89949	0.177088	0.0885438	−	0.996072i	$-0.471779\pi$
$5$	9.89949	0.177088	0.0885438	+	0.996072i	$0.471779\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−171.000	−0.703704
$10$	0	0
$11$	−308.000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−308.000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	912.168	1.49698	0.748491	−	0.663145i	$-0.230779\pi$
$13$	912.168	1.49698	0.748491	+	0.663145i	$0.230779\pi$
$14$	0	0
$15$	84.0000	0.0963943
$16$	0	0
$17$	451.134	0.378602	0.189301	−	0.981919i	$-0.439378\pi$
$17$	451.134	0.378602	0.189301	+	0.981919i	$0.439378\pi$
$18$	0	0
$19$	2616.30	1.66266	0.831329	−	0.555781i	$-0.187581\pi$
$19$	2616.30	1.66266	0.831329	+	0.555781i	$0.187581\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2324.00	0.916044	0.458022	−	0.888941i	$-0.348558\pi$
$23$	2324.00	0.916044	0.458022	+	0.888941i	$0.348558\pi$
$24$	0	0
$25$	−3027.00	−0.968640
$26$	0	0
$27$	−3512.91	−0.927379
$28$	0	0
$29$	−6488.00	−1.43257	−0.716285	−	0.697808i	$-0.754159\pi$
$29$	−6488.00	−1.43257	−0.716285	+	0.697808i	$0.754159\pi$
$30$	0	0
$31$	−814.587	−0.152242	−0.0761208	−	0.997099i	$-0.524253\pi$
$31$	−814.587	−0.152242	−0.0761208	+	0.997099i	$0.524253\pi$
$32$	0	0
$33$	−2613.47	−0.417765
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12200.0	−1.46506	−0.732530	−	0.680735i	$-0.761661\pi$
$37$	−12200.0	−1.46506	−0.732530	+	0.680735i	$0.761661\pi$
$38$	0	0
$39$	7740.00	0.814853
$40$	0	0
$41$	527.502	0.0490077	0.0245038	−	0.999700i	$-0.492199\pi$
$41$	527.502	0.0490077	0.0245038	+	0.999700i	$0.492199\pi$
$42$	0	0
$43$	15028.0	1.23945	0.619726	−	0.784818i	$-0.287244\pi$
$43$	15028.0	1.23945	0.619726	+	0.784818i	$0.287244\pi$
$44$	0	0
$45$	−1692.81	−0.124617
$46$	0	0
$47$	−1600.89	−0.105710	−0.0528551	−	0.998602i	$-0.516832\pi$
$47$	−1600.89	−0.105710	−0.0528551	+	0.998602i	$0.516832\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3828.00	0.206085
$52$	0	0
$53$	10778.0	0.527046	0.263523	−	0.964653i	$-0.415116\pi$
$53$	10778.0	0.527046	0.263523	+	0.964653i	$0.415116\pi$
$54$	0	0
$55$	−3049.04	−0.135912
$56$	0	0
$57$	22200.0	0.905036
$58$	0	0
$59$	52136.4	1.94989	0.974947	−	0.222437i	$-0.0714013\pi$
$59$	52136.4	1.94989	0.974947	+	0.222437i	$0.0714013\pi$
$60$	0	0
$61$	36765.3	1.26507	0.632534	−	0.774533i	$-0.282015\pi$
$61$	36765.3	1.26507	0.632534	+	0.774533i	$0.282015\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9030.00	0.265097
$66$	0	0
$67$	23808.0	0.647942	0.323971	−	0.946067i	$-0.394982\pi$
$67$	23808.0	0.647942	0.323971	+	0.946067i	$0.394982\pi$
$68$	0	0
$69$	19719.8	0.498631
$70$	0	0
$71$	−56448.0	−1.32893	−0.664466	−	0.747319i	$-0.731341\pi$
$71$	−56448.0	−1.32893	−0.664466	+	0.747319i	$0.731341\pi$
$72$	0	0
$73$	−51379.8	−1.12846	−0.564229	−	0.825619i	$-0.690826\pi$
$73$	−51379.8	−1.12846	−0.564229	+	0.825619i	$0.690826\pi$
$74$	0	0
$75$	−25684.9	−0.527261
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11224.0	0.202339	0.101170	−	0.994869i	$-0.467742\pi$
$79$	11224.0	0.202339	0.101170	+	0.994869i	$0.467742\pi$
$80$	0	0
$81$	11745.0	0.198903
$82$	0	0
$83$	38690.1	0.616459	0.308229	−	0.951312i	$-0.400264\pi$
$83$	38690.1	0.616459	0.308229	+	0.951312i	$0.400264\pi$
$84$	0	0
$85$	4466.00	0.0670458
$86$	0	0
$87$	−55052.5	−0.779792
$88$	0	0
$89$	128177.	1.71528	0.857642	−	0.514248i	$-0.171929\pi$
$89$	128177.	1.71528	0.857642	+	0.514248i	$0.171929\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6912.00	−0.0828698
$94$	0	0
$95$	25900.0	0.294436
$96$	0	0
$97$	−56883.9	−0.613847	−0.306923	−	0.951734i	$-0.599300\pi$
$97$	−56883.9	−0.613847	−0.306923	+	0.951734i	$0.599300\pi$
$98$	0	0
$99$	52668.0	0.540081
$100$	0	0
$101$	39825.7	0.388472	0.194236	−	0.980955i	$-0.437777\pi$
$101$	39825.7	0.388472	0.194236	+	0.980955i	$0.437777\pi$
$102$	0	0
$103$	112809.	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$103$	112809.	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6376.00	−0.0538380	−0.0269190	−	0.999638i	$-0.508570\pi$
$107$	−6376.00	−0.0538380	−0.0269190	+	0.999638i	$0.508570\pi$
$108$	0	0
$109$	−33512.0	−0.270168	−0.135084	−	0.990834i	$-0.543130\pi$
$109$	−33512.0	−0.270168	−0.135084	+	0.990834i	$0.543130\pi$
$110$	0	0
$111$	−103520.	−0.797478
$112$	0	0
$113$	−2682.00	−0.0197589	−0.00987945	−	0.999951i	$-0.503145\pi$
$113$	−2682.00	−0.0197589	−0.00987945	+	0.999951i	$0.503145\pi$
$114$	0	0
$115$	23006.4	0.162220
$116$	0	0
$117$	−155981.	−1.05343
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−66187.0	−0.410969
$122$	0	0
$123$	4476.00	0.0266764
$124$	0	0
$125$	−60901.7	−0.348622
$126$	0	0
$127$	234404.	1.28960	0.644801	−	0.764350i	$-0.276940\pi$
$127$	234404.	1.28960	0.644801	+	0.764350i	$0.276940\pi$
$128$	0	0
$129$	127517.	0.674673
$130$	0	0
$131$	326946.	1.66456	0.832278	−	0.554359i	$-0.187036\pi$
$131$	326946.	1.66456	0.832278	+	0.554359i	$0.187036\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−34776.0	−0.164227
$136$	0	0
$137$	253632.	1.15452	0.577262	−	0.816559i	$-0.304121\pi$
$137$	253632.	1.15452	0.577262	+	0.816559i	$0.304121\pi$
$138$	0	0
$139$	−295658.	−1.29794	−0.648968	−	0.760816i	$-0.724799\pi$
$139$	−295658.	−1.29794	−0.648968	+	0.760816i	$0.724799\pi$
$140$	0	0
$141$	−13584.0	−0.0575413
$142$	0	0
$143$	−280948.	−1.14891
$144$	0	0
$145$	−64227.9	−0.253690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	438170.	1.61688	0.808438	−	0.588581i	$-0.200313\pi$
$149$	438170.	1.61688	0.808438	+	0.588581i	$0.200313\pi$
$150$	0	0
$151$	−141164.	−0.503827	−0.251914	−	0.967750i	$-0.581060\pi$
$151$	−141164.	−0.503827	−0.251914	+	0.967750i	$0.581060\pi$
$152$	0	0
$153$	−77143.9	−0.266424
$154$	0	0
$155$	−8064.00	−0.0269601
$156$	0	0
$157$	−152301.	−0.493121	−0.246560	−	0.969127i	$-0.579300\pi$
$157$	−152301.	−0.493121	−0.246560	+	0.969127i	$0.579300\pi$
$158$	0	0
$159$	91454.4	0.286887
$160$	0	0
$161$	0	0
$162$	0	0
$163$	437900.	1.29094	0.645470	−	0.763786i	$-0.276662\pi$
$163$	437900.	1.29094	0.645470	+	0.763786i	$0.276662\pi$
$164$	0	0
$165$	−25872.0	−0.0739810
$166$	0	0
$167$	471618.	1.30858	0.654288	−	0.756246i	$-0.272969\pi$
$167$	471618.	1.30858	0.654288	+	0.756246i	$0.272969\pi$
$168$	0	0
$169$	460757.	1.24095
$170$	0	0
$171$	−447386.	−1.17002
$172$	0	0
$173$	666778.	1.69381	0.846907	−	0.531741i	$-0.178462\pi$
$173$	666778.	1.69381	0.846907	+	0.531741i	$0.178462\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	442392.	1.06139
$178$	0	0
$179$	146688.	0.342186	0.171093	−	0.985255i	$-0.445270\pi$
$179$	146688.	0.342186	0.171093	+	0.985255i	$0.445270\pi$
$180$	0	0
$181$	−120979.	−0.274482	−0.137241	−	0.990538i	$-0.543823\pi$
$181$	−120979.	−0.274482	−0.137241	+	0.990538i	$0.543823\pi$
$182$	0	0
$183$	311964.	0.688615
$184$	0	0
$185$	−120774.	−0.259444
$186$	0	0
$187$	−138949.	−0.290571
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−187696.	−0.372282	−0.186141	−	0.982523i	$-0.559598\pi$
$191$	−187696.	−0.372282	−0.186141	+	0.982523i	$0.559598\pi$
$192$	0	0
$193$	−513226.	−0.991780	−0.495890	−	0.868385i	$-0.665158\pi$
$193$	−513226.	−0.991780	−0.495890	+	0.868385i	$0.665158\pi$
$194$	0	0
$195$	76622.1	0.144300
$196$	0	0
$197$	971302.	1.78315	0.891577	−	0.452870i	$-0.149600\pi$
$197$	971302.	1.78315	0.891577	+	0.452870i	$0.149600\pi$
$198$	0	0
$199$	433632.	0.776226	0.388113	−	0.921612i	$-0.373127\pi$
$199$	433632.	0.776226	0.388113	+	0.921612i	$0.373127\pi$
$200$	0	0
$201$	202018.	0.352695
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5222.00	0.00867865
$206$	0	0
$207$	−397404.	−0.644624
$208$	0	0
$209$	−805819.	−1.27606
$210$	0	0
$211$	96072.0	0.148556	0.0742781	−	0.997238i	$-0.476335\pi$
$211$	96072.0	0.148556	0.0742781	+	0.997238i	$0.476335\pi$
$212$	0	0
$213$	−478977.	−0.723379
$214$	0	0
$215$	148770.	0.219492
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−435972.	−0.614254
$220$	0	0
$221$	411510.	0.566761
$222$	0	0
$223$	−1.26660e6	−1.70560	−0.852799	−	0.522239i	$-0.825097\pi$
$223$	−1.26660e6	−1.70560	−0.852799	+	0.522239i	$0.825097\pi$
$224$	0	0
$225$	517617.	0.681636
$226$	0	0
$227$	−444247.	−0.572216	−0.286108	−	0.958197i	$-0.592362\pi$
$227$	−444247.	−0.572216	−0.286108	+	0.958197i	$0.592362\pi$
$228$	0	0
$229$	1.23979e6	1.56228	0.781140	−	0.624356i	$-0.214638\pi$
$229$	1.23979e6	1.56228	0.781140	+	0.624356i	$0.214638\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−391296.	−0.472189	−0.236094	−	0.971730i	$-0.575867\pi$
$233$	−391296.	−0.472189	−0.236094	+	0.971730i	$0.575867\pi$
$234$	0	0
$235$	−15848.0	−0.0187200
$236$	0	0
$237$	95238.8	0.110139
$238$	0	0
$239$	1.15993e6	1.31352	0.656762	−	0.754098i	$-0.271926\pi$
$239$	1.15993e6	1.31352	0.656762	+	0.754098i	$0.271926\pi$
$240$	0	0
$241$	−1.49016e6	−1.65269	−0.826345	−	0.563164i	$-0.809584\pi$
$241$	−1.49016e6	−1.65269	−0.826345	+	0.563164i	$0.809584\pi$
$242$	0	0
$243$	953296.	1.03565
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.38650e6	2.48897
$248$	0	0
$249$	328296.	0.335558
$250$	0	0
$251$	−1.09274e6	−1.09479	−0.547396	−	0.836874i	$-0.684381\pi$
$251$	−1.09274e6	−1.09479	−0.547396	+	0.836874i	$0.684381\pi$
$252$	0	0
$253$	−715792.	−0.703049
$254$	0	0
$255$	37895.3	0.0364951
$256$	0	0
$257$	−610418.	−0.576494	−0.288247	−	0.957556i	$-0.593072\pi$
$257$	−610418.	−0.576494	−0.288247	+	0.957556i	$0.593072\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.10945e6	1.00810
$262$	0	0
$263$	1.36601e6	1.21777	0.608883	−	0.793260i	$-0.291618\pi$
$263$	1.36601e6	1.21777	0.608883	+	0.793260i	$0.291618\pi$
$264$	0	0
$265$	106697.	0.0933333
$266$	0	0
$267$	1.08762e6	0.933682
$268$	0	0
$269$	893270.	0.752665	0.376333	−	0.926485i	$-0.377185\pi$
$269$	893270.	0.752665	0.376333	+	0.926485i	$0.377185\pi$
$270$	0	0
$271$	858264.	0.709900	0.354950	−	0.934885i	$-0.384498\pi$
$271$	858264.	0.709900	0.354950	+	0.934885i	$0.384498\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	932316.	0.743415
$276$	0	0
$277$	−1.55532e6	−1.21793	−0.608963	−	0.793199i	$-0.708414\pi$
$277$	−1.55532e6	−1.21793	−0.608963	+	0.793199i	$0.708414\pi$
$278$	0	0
$279$	139294.	0.107133
$280$	0	0
$281$	−110704.	−0.0836368	−0.0418184	−	0.999125i	$-0.513315\pi$
$281$	−110704.	−0.0836368	−0.0418184	+	0.999125i	$0.513315\pi$
$282$	0	0
$283$	1.12844e6	0.837552	0.418776	−	0.908090i	$-0.362459\pi$
$283$	1.12844e6	0.837552	0.418776	+	0.908090i	$0.362459\pi$
$284$	0	0
$285$	219769.	0.160271
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.21634e6	−0.856660
$290$	0	0
$291$	−482676.	−0.334136
$292$	0	0
$293$	−159974.	−0.108863	−0.0544317	−	0.998517i	$-0.517335\pi$
$293$	−159974.	−0.108863	−0.0544317	+	0.998517i	$0.517335\pi$
$294$	0	0
$295$	516124.	0.345302
$296$	0	0
$297$	1.08198e6	0.711748
$298$	0	0
$299$	2.11988e6	1.37130
$300$	0	0
$301$	0	0
$302$	0	0
$303$	337932.	0.211457
$304$	0	0
$305$	363958.	0.224028
$306$	0	0
$307$	116364.	0.0704651	0.0352325	−	0.999379i	$-0.488783\pi$
$307$	116364.	0.0704651	0.0352325	+	0.999379i	$0.488783\pi$
$308$	0	0
$309$	957216.	0.570314
$310$	0	0
$311$	−2.64100e6	−1.54835	−0.774173	−	0.632974i	$-0.781834\pi$
$311$	−2.64100e6	−1.54835	−0.774173	+	0.632974i	$0.781834\pi$
$312$	0	0
$313$	3.23791e6	1.86812	0.934059	−	0.357120i	$-0.116241\pi$
$313$	3.23791e6	1.86812	0.934059	+	0.357120i	$0.116241\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	861710.	0.481629	0.240815	−	0.970571i	$-0.422585\pi$
$317$	861710.	0.481629	0.240815	+	0.970571i	$0.422585\pi$
$318$	0	0
$319$	1.99830e6	1.09947
$320$	0	0
$321$	−54102.2	−0.0293057
$322$	0	0
$323$	1.18030e6	0.629486
$324$	0	0
$325$	−2.76113e6	−1.45004
$326$	0	0
$327$	−284359.	−0.147061
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−777340.	−0.389979	−0.194989	−	0.980805i	$-0.562467\pi$
$331$	−777340.	−0.389979	−0.194989	+	0.980805i	$0.562467\pi$
$332$	0	0
$333$	2.08620e6	1.03097
$334$	0	0
$335$	235687.	0.114742
$336$	0	0
$337$	2.15751e6	1.03485	0.517426	−	0.855728i	$-0.326890\pi$
$337$	2.15751e6	1.03485	0.517426	+	0.855728i	$0.326890\pi$
$338$	0	0
$339$	−22757.5	−0.0107554
$340$	0	0
$341$	250893.	0.116843
$342$	0	0
$343$	0	0
$344$	0	0
$345$	195216.	0.0883014
$346$	0	0
$347$	−969060.	−0.432043	−0.216022	−	0.976389i	$-0.569308\pi$
$347$	−969060.	−0.432043	−0.216022	+	0.976389i	$0.569308\pi$
$348$	0	0
$349$	1.00307e6	0.440825	0.220412	−	0.975407i	$-0.429260\pi$
$349$	1.00307e6	0.440825	0.220412	+	0.975407i	$0.429260\pi$
$350$	0	0
$351$	−3.20436e6	−1.38827
$352$	0	0
$353$	−372095.	−0.158934	−0.0794671	−	0.996837i	$-0.525322\pi$
$353$	−372095.	−0.158934	−0.0794671	+	0.996837i	$0.525322\pi$
$354$	0	0
$355$	−558807.	−0.235337
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.86915e6	−1.17494	−0.587472	−	0.809245i	$-0.699877\pi$
$359$	−2.86915e6	−1.17494	−0.587472	+	0.809245i	$0.699877\pi$
$360$	0	0
$361$	4.36890e6	1.76443
$362$	0	0
$363$	−561615.	−0.223703
$364$	0	0
$365$	−508634.	−0.199836
$366$	0	0
$367$	684734.	0.265373	0.132686	−	0.991158i	$-0.457640\pi$
$367$	684734.	0.265373	0.132686	+	0.991158i	$0.457640\pi$
$368$	0	0
$369$	−90202.8	−0.0344869
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.79305e6	0.667299	0.333649	−	0.942697i	$-0.391720\pi$
$373$	1.79305e6	0.667299	0.333649	+	0.942697i	$0.391720\pi$
$374$	0	0
$375$	−516768.	−0.189766
$376$	0	0
$377$	−5.91814e6	−2.14453
$378$	0	0
$379$	−2.02956e6	−0.725780	−0.362890	−	0.931832i	$-0.618210\pi$
$379$	−2.02956e6	−0.725780	−0.362890	+	0.931832i	$0.618210\pi$
$380$	0	0
$381$	1.98898e6	0.701970
$382$	0	0
$383$	−733700.	−0.255577	−0.127788	−	0.991801i	$-0.540788\pi$
$383$	−733700.	−0.255577	−0.127788	+	0.991801i	$0.540788\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.56979e6	−0.872208
$388$	0	0
$389$	−2.78722e6	−0.933895	−0.466947	−	0.884285i	$-0.654646\pi$
$389$	−2.78722e6	−0.933895	−0.466947	+	0.884285i	$0.654646\pi$
$390$	0	0
$391$	1.04844e6	0.346817
$392$	0	0
$393$	2.77423e6	0.906069
$394$	0	0
$395$	111112.	0.0358317
$396$	0	0
$397$	697916.	0.222242	0.111121	−	0.993807i	$-0.464556\pi$
$397$	697916.	0.222242	0.111121	+	0.993807i	$0.464556\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.31631e6	0.719343	0.359672	−	0.933079i	$-0.382889\pi$
$401$	2.31631e6	0.719343	0.359672	+	0.933079i	$0.382889\pi$
$402$	0	0
$403$	−743040.	−0.227903
$404$	0	0
$405$	116270.	0.0352232
$406$	0	0
$407$	3.75760e6	1.12441
$408$	0	0
$409$	1.10529e6	0.326714	0.163357	−	0.986567i	$-0.447768\pi$
$409$	1.10529e6	0.326714	0.163357	+	0.986567i	$0.447768\pi$
$410$	0	0
$411$	2.15214e6	0.628443
$412$	0	0
$413$	0	0
$414$	0	0
$415$	383012.	0.109167
$416$	0	0
$417$	−2.50874e6	−0.706507
$418$	0	0
$419$	1.14925e6	0.319801	0.159900	−	0.987133i	$-0.448883\pi$
$419$	1.14925e6	0.319801	0.159900	+	0.987133i	$0.448883\pi$
$420$	0	0
$421$	−2.56929e6	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$421$	−2.56929e6	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$422$	0	0
$423$	273752.	0.0743886
$424$	0	0
$425$	−1.36558e6	−0.366729
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.38392e6	−0.625386
$430$	0	0
$431$	−1.16353e6	−0.301707	−0.150853	−	0.988556i	$-0.548202\pi$
$431$	−1.16353e6	−0.301707	−0.150853	+	0.988556i	$0.548202\pi$
$432$	0	0
$433$	3.23855e6	0.830100	0.415050	−	0.909799i	$-0.363764\pi$
$433$	3.23855e6	0.830100	0.415050	+	0.909799i	$0.363764\pi$
$434$	0	0
$435$	−544992.	−0.138092
$436$	0	0
$437$	6.08027e6	1.52307
$438$	0	0
$439$	2.23623e6	0.553802	0.276901	−	0.960898i	$-0.410693\pi$
$439$	2.23623e6	0.553802	0.276901	+	0.960898i	$0.410693\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.24046e6	1.51080	0.755401	−	0.655263i	$-0.227442\pi$
$443$	6.24046e6	1.51080	0.755401	+	0.655263i	$0.227442\pi$
$444$	0	0
$445$	1.26889e6	0.303755
$446$	0	0
$447$	3.71800e6	0.880116
$448$	0	0
$449$	−5.76885e6	−1.35043	−0.675217	−	0.737619i	$-0.735950\pi$
$449$	−5.76885e6	−1.35043	−0.675217	+	0.737619i	$0.735950\pi$
$450$	0	0
$451$	−162471.	−0.0376126
$452$	0	0
$453$	−1.19782e6	−0.274249
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−929346.	−0.208155	−0.104078	−	0.994569i	$-0.533189\pi$
$457$	−929346.	−0.208155	−0.104078	+	0.994569i	$0.533189\pi$
$458$	0	0
$459$	−1.58479e6	−0.351108
$460$	0	0
$461$	3.81306e6	0.835645	0.417822	−	0.908529i	$-0.362793\pi$
$461$	3.81306e6	0.835645	0.417822	+	0.908529i	$0.362793\pi$
$462$	0	0
$463$	1.63858e6	0.355235	0.177618	−	0.984100i	$-0.443161\pi$
$463$	1.63858e6	0.355235	0.177618	+	0.984100i	$0.443161\pi$
$464$	0	0
$465$	−68425.3	−0.0146752
$466$	0	0
$467$	−5.07980e6	−1.07784	−0.538920	−	0.842357i	$-0.681167\pi$
$467$	−5.07980e6	−1.07784	−0.538920	+	0.842357i	$0.681167\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.29232e6	−0.268421
$472$	0	0
$473$	−4.62862e6	−0.951260
$474$	0	0
$475$	−7.91953e6	−1.61052
$476$	0	0
$477$	−1.84304e6	−0.370884
$478$	0	0
$479$	−3.46178e6	−0.689383	−0.344692	−	0.938716i	$-0.612016\pi$
$479$	−3.46178e6	−0.689383	−0.344692	+	0.938716i	$0.612016\pi$
$480$	0	0
$481$	−1.11284e7	−2.19317
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−563122.	−0.108705
$486$	0	0
$487$	2.71763e6	0.519239	0.259620	−	0.965711i	$-0.416403\pi$
$487$	2.71763e6	0.519239	0.259620	+	0.965711i	$0.416403\pi$
$488$	0	0
$489$	3.71570e6	0.702699
$490$	0	0
$491$	−7.65113e6	−1.43226	−0.716130	−	0.697967i	$-0.754088\pi$
$491$	−7.65113e6	−1.43226	−0.716130	+	0.697967i	$0.754088\pi$
$492$	0	0
$493$	−2.92696e6	−0.542374
$494$	0	0
$495$	521387.	0.0956416
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.12042e6	−0.560998	−0.280499	−	0.959854i	$-0.590500\pi$
$499$	−3.12042e6	−0.560998	−0.280499	+	0.959854i	$0.590500\pi$
$500$	0	0
$501$	4.00181e6	0.712298
$502$	0	0
$503$	5.18492e6	0.913739	0.456869	−	0.889534i	$-0.348971\pi$
$503$	5.18492e6	0.913739	0.456869	+	0.889534i	$0.348971\pi$
$504$	0	0
$505$	394254.	0.0687935
$506$	0	0
$507$	3.90965e6	0.675489
$508$	0	0
$509$	−3.10106e6	−0.530537	−0.265268	−	0.964175i	$-0.585461\pi$
$509$	−3.10106e6	−0.530537	−0.265268	+	0.964175i	$0.585461\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.19080e6	−1.54191
$514$	0	0
$515$	1.11675e6	0.185540
$516$	0	0
$517$	493074.	0.0811308
$518$	0	0
$519$	5.65780e6	0.921996
$520$	0	0
$521$	−7.34131e6	−1.18489	−0.592447	−	0.805610i	$-0.701838\pi$
$521$	−7.34131e6	−1.18489	−0.592447	+	0.805610i	$0.701838\pi$
$522$	0	0
$523$	−4.23449e6	−0.676934	−0.338467	−	0.940978i	$-0.609908\pi$
$523$	−4.23449e6	−0.676934	−0.338467	+	0.940978i	$0.609908\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−367488.	−0.0576390
$528$	0	0
$529$	−1.03537e6	−0.160863
$530$	0	0
$531$	−8.91532e6	−1.37215
$532$	0	0
$533$	481170.	0.0733636
$534$	0	0
$535$	−63119.2	−0.00953404
$536$	0	0
$537$	1.24469e6	0.186262
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.07075e6	−0.157288	−0.0786439	−	0.996903i	$-0.525059\pi$
$541$	−1.07075e6	−0.157288	−0.0786439	+	0.996903i	$0.525059\pi$
$542$	0	0
$543$	−1.02654e6	−0.149409
$544$	0	0
$545$	−331752.	−0.0478434
$546$	0	0
$547$	−3.65868e6	−0.522825	−0.261413	−	0.965227i	$-0.584188\pi$
$547$	−3.65868e6	−0.522825	−0.261413	+	0.965227i	$0.584188\pi$
$548$	0	0
$549$	−6.28687e6	−0.890233
$550$	0	0
$551$	−1.69745e7	−2.38187
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.02480e6	−0.141223
$556$	0	0
$557$	−1.02367e7	−1.39805	−0.699023	−	0.715099i	$-0.746382\pi$
$557$	−1.02367e7	−1.39805	−0.699023	+	0.715099i	$0.746382\pi$
$558$	0	0
$559$	1.37081e7	1.85544
$560$	0	0
$561$	−1.17902e6	−0.158167
$562$	0	0
$563$	8.72952e6	1.16070	0.580349	−	0.814368i	$-0.302916\pi$
$563$	8.72952e6	1.16070	0.580349	+	0.814368i	$0.302916\pi$
$564$	0	0
$565$	−26550.4	−0.00349905
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.02684e6	−0.262445	−0.131223	−	0.991353i	$-0.541890\pi$
$569$	−2.02684e6	−0.262445	−0.131223	+	0.991353i	$0.541890\pi$
$570$	0	0
$571$	−1.25842e7	−1.61523	−0.807616	−	0.589709i	$-0.799243\pi$
$571$	−1.25842e7	−1.61523	−0.807616	+	0.589709i	$0.799243\pi$
$572$	0	0
$573$	−1.59265e6	−0.202644
$574$	0	0
$575$	−7.03475e6	−0.887317
$576$	0	0
$577$	1.15215e7	1.44069	0.720344	−	0.693617i	$-0.243984\pi$
$577$	1.15215e7	1.44069	0.720344	+	0.693617i	$0.243984\pi$
$578$	0	0
$579$	−4.35487e6	−0.539857
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.31962e6	−0.404499
$584$	0	0
$585$	−1.54413e6	−0.186550
$586$	0	0
$587$	1.05984e7	1.26953	0.634767	−	0.772704i	$-0.281096\pi$
$587$	1.05984e7	1.26953	0.634767	+	0.772704i	$0.281096\pi$
$588$	0	0
$589$	−2.13120e6	−0.253126
$590$	0	0
$591$	8.24177e6	0.970626
$592$	0	0
$593$	9.41252e6	1.09918	0.549590	−	0.835434i	$-0.314784\pi$
$593$	9.41252e6	1.09918	0.549590	+	0.835434i	$0.314784\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.67949e6	0.422524
$598$	0	0
$599$	1.44437e7	1.64480	0.822399	−	0.568911i	$-0.192635\pi$
$599$	1.44437e7	1.64480	0.822399	+	0.568911i	$0.192635\pi$
$600$	0	0
$601$	5.72420e6	0.646441	0.323220	−	0.946324i	$-0.395235\pi$
$601$	5.72420e6	0.646441	0.323220	+	0.946324i	$0.395235\pi$
$602$	0	0
$603$	−4.07117e6	−0.455959
$604$	0	0
$605$	−655218.	−0.0727775
$606$	0	0
$607$	−1.09737e7	−1.20888	−0.604440	−	0.796651i	$-0.706603\pi$
$607$	−1.09737e7	−1.20888	−0.604440	+	0.796651i	$0.706603\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.46028e6	−0.158246
$612$	0	0
$613$	2.78508e6	0.299355	0.149677	−	0.988735i	$-0.452176\pi$
$613$	2.78508e6	0.299355	0.149677	+	0.988735i	$0.452176\pi$
$614$	0	0
$615$	44310.1	0.00472406
$616$	0	0
$617$	−1.14021e7	−1.20579	−0.602893	−	0.797822i	$-0.705985\pi$
$617$	−1.14021e7	−1.20579	−0.602893	+	0.797822i	$0.705985\pi$
$618$	0	0
$619$	−2.70782e6	−0.284049	−0.142024	−	0.989863i	$-0.545361\pi$
$619$	−2.70782e6	−0.284049	−0.142024	+	0.989863i	$0.545361\pi$
$620$	0	0
$621$	−8.16399e6	−0.849520
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.85648e6	0.906903
$626$	0	0
$627$	−6.83760e6	−0.694600
$628$	0	0
$629$	−5.50384e6	−0.554675
$630$	0	0
$631$	−1.73174e7	−1.73145	−0.865725	−	0.500520i	$-0.833142\pi$
$631$	−1.73174e7	−1.73145	−0.865725	+	0.500520i	$0.833142\pi$
$632$	0	0
$633$	815198.	0.0808637
$634$	0	0
$635$	2.32048e6	0.228372
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.65261e6	0.935174
$640$	0	0
$641$	1.01435e7	0.975085	0.487543	−	0.873099i	$-0.337893\pi$
$641$	1.01435e7	0.975085	0.487543	+	0.873099i	$0.337893\pi$
$642$	0	0
$643$	−4.96664e6	−0.473734	−0.236867	−	0.971542i	$-0.576121\pi$
$643$	−4.96664e6	−0.473734	−0.236867	+	0.971542i	$0.576121\pi$
$644$	0	0
$645$	1.26235e6	0.119476
$646$	0	0
$647$	1.93247e6	0.181490	0.0907450	−	0.995874i	$-0.471075\pi$
$647$	1.93247e6	0.181490	0.0907450	+	0.995874i	$0.471075\pi$
$648$	0	0
$649$	−1.60580e7	−1.49651
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.04428e7	−0.958372	−0.479186	−	0.877713i	$-0.659068\pi$
$653$	−1.04428e7	−0.958372	−0.479186	+	0.877713i	$0.659068\pi$
$654$	0	0
$655$	3.23660e6	0.294772
$656$	0	0
$657$	8.78594e6	0.794100
$658$	0	0
$659$	−3.99510e6	−0.358356	−0.179178	−	0.983817i	$-0.557344\pi$
$659$	−3.99510e6	−0.358356	−0.179178	+	0.983817i	$0.557344\pi$
$660$	0	0
$661$	4.87665e6	0.434128	0.217064	−	0.976157i	$-0.430352\pi$
$661$	4.87665e6	0.434128	0.217064	+	0.976157i	$0.430352\pi$
$662$	0	0
$663$	3.49178e6	0.308505
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.50781e7	−1.31230
$668$	0	0
$669$	−1.07474e7	−0.928410
$670$	0	0
$671$	−1.13237e7	−0.970918
$672$	0	0
$673$	−1.04085e7	−0.885831	−0.442916	−	0.896563i	$-0.646056\pi$
$673$	−1.04085e7	−0.885831	−0.442916	+	0.896563i	$0.646056\pi$
$674$	0	0
$675$	1.06336e7	0.898296
$676$	0	0
$677$	−748652.	−0.0627781	−0.0313891	−	0.999507i	$-0.509993\pi$
$677$	−748652.	−0.0627781	−0.0313891	+	0.999507i	$0.509993\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.76956e6	−0.311475
$682$	0	0
$683$	−2.15155e6	−0.176482	−0.0882409	−	0.996099i	$-0.528125\pi$
$683$	−2.15155e6	−0.176482	−0.0882409	+	0.996099i	$0.528125\pi$
$684$	0	0
$685$	2.51083e6	0.204452
$686$	0	0
$687$	1.05200e7	0.850398
$688$	0	0
$689$	9.83134e6	0.788978
$690$	0	0
$691$	2.26790e7	1.80688	0.903441	−	0.428713i	$-0.141033\pi$
$691$	2.26790e7	1.80688	0.903441	+	0.428713i	$0.141033\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.92687e6	−0.229848
$696$	0	0
$697$	237974.	0.0185544
$698$	0	0
$699$	−3.32026e6	−0.257027
$700$	0	0
$701$	−1.97940e6	−0.152138	−0.0760691	−	0.997103i	$-0.524237\pi$
$701$	−1.97940e6	−0.152138	−0.0760691	+	0.997103i	$0.524237\pi$
$702$	0	0
$703$	−3.19188e7	−2.43589
$704$	0	0
$705$	−134475.	−0.0101899
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.28570e7	−0.960559	−0.480280	−	0.877115i	$-0.659465\pi$
$709$	−1.28570e7	−0.960559	−0.480280	+	0.877115i	$0.659465\pi$
$710$	0	0
$711$	−1.91930e6	−0.142387
$712$	0	0
$713$	−1.89310e6	−0.139460
$714$	0	0
$715$	−2.78124e6	−0.203457
$716$	0	0
$717$	9.84235e6	0.714992
$718$	0	0
$719$	−5.07359e6	−0.366010	−0.183005	−	0.983112i	$-0.558583\pi$
$719$	−5.07359e6	−0.366010	−0.183005	+	0.983112i	$0.558583\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.26445e7	−0.899611
$724$	0	0
$725$	1.96392e7	1.38764
$726$	0	0
$727$	2.54045e7	1.78268	0.891342	−	0.453331i	$-0.149765\pi$
$727$	2.54045e7	1.78268	0.891342	+	0.453331i	$0.149765\pi$
$728$	0	0
$729$	5.23495e6	0.364833
$730$	0	0
$731$	6.77964e6	0.469260
$732$	0	0
$733$	−1.85116e7	−1.27257	−0.636287	−	0.771453i	$-0.719531\pi$
$733$	−1.85116e7	−1.27257	−0.636287	+	0.771453i	$0.719531\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.33286e6	−0.497284
$738$	0	0
$739$	−2.31134e7	−1.55687	−0.778435	−	0.627725i	$-0.783986\pi$
$739$	−2.31134e7	−1.55687	−0.778435	+	0.627725i	$0.783986\pi$
$740$	0	0
$741$	2.02501e7	1.35482
$742$	0	0
$743$	2.47666e7	1.64586	0.822932	−	0.568140i	$-0.192337\pi$
$743$	2.47666e7	1.64586	0.822932	+	0.568140i	$0.192337\pi$
$744$	0	0
$745$	4.33766e6	0.286329
$746$	0	0
$747$	−6.61600e6	−0.433804
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5.23128e6	−0.338460	−0.169230	−	0.985577i	$-0.554128\pi$
$751$	−5.23128e6	−0.338460	−0.169230	+	0.985577i	$0.554128\pi$
$752$	0	0
$753$	−9.27218e6	−0.595929
$754$	0	0
$755$	−1.39745e6	−0.0892215
$756$	0	0
$757$	−1.00525e7	−0.637579	−0.318789	−	0.947826i	$-0.603276\pi$
$757$	−1.00525e7	−0.637579	−0.318789	+	0.947826i	$0.603276\pi$
$758$	0	0
$759$	−6.07370e6	−0.382691
$760$	0	0
$761$	−8.28396e6	−0.518533	−0.259267	−	0.965806i	$-0.583481\pi$
$761$	−8.28396e6	−0.518533	−0.259267	+	0.965806i	$0.583481\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−763686.	−0.0471804
$766$	0	0
$767$	4.75571e7	2.91895
$768$	0	0
$769$	−8.53977e6	−0.520751	−0.260376	−	0.965507i	$-0.583846\pi$
$769$	−8.53977e6	−0.520751	−0.260376	+	0.965507i	$0.583846\pi$
$770$	0	0
$771$	−5.17957e6	−0.313804
$772$	0	0
$773$	1.39445e7	0.839374	0.419687	−	0.907669i	$-0.362140\pi$
$773$	1.39445e7	0.839374	0.419687	+	0.907669i	$0.362140\pi$
$774$	0	0
$775$	2.46575e6	0.147467
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.38010e6	0.0814830
$780$	0	0
$781$	1.73860e7	1.01993
$782$	0	0
$783$	2.27917e7	1.32854
$784$	0	0
$785$	−1.50770e6	−0.0873256
$786$	0	0
$787$	7.28018e6	0.418992	0.209496	−	0.977810i	$-0.432818\pi$
$787$	7.28018e6	0.418992	0.209496	+	0.977810i	$0.432818\pi$
$788$	0	0
$789$	1.15910e7	0.662868
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.35361e7	1.89378
$794$	0	0
$795$	905352.	0.0508042
$796$	0	0
$797$	−2.56951e7	−1.43286	−0.716430	−	0.697659i	$-0.754225\pi$
$797$	−2.56951e7	−1.43286	−0.716430	+	0.697659i	$0.754225\pi$
$798$	0	0
$799$	−722216.	−0.0400221
$800$	0	0
$801$	−2.19183e7	−1.20705
$802$	0	0
$803$	1.58250e7	0.866072
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.57964e6	0.409699
$808$	0	0
$809$	−1.44220e7	−0.774739	−0.387370	−	0.921924i	$-0.626616\pi$
$809$	−1.44220e7	−0.774739	−0.387370	+	0.921924i	$0.626616\pi$
$810$	0	0
$811$	−3.39576e7	−1.81294	−0.906472	−	0.422265i	$-0.861235\pi$
$811$	−3.39576e7	−1.81294	−0.906472	+	0.422265i	$0.861235\pi$
$812$	0	0
$813$	7.28261e6	0.386421
$814$	0	0
$815$	4.33499e6	0.228609
$816$	0	0
$817$	3.93177e7	2.06079
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.58002e6	0.0818098	0.0409049	−	0.999163i	$-0.486976\pi$
$821$	1.58002e6	0.0818098	0.0409049	+	0.999163i	$0.486976\pi$
$822$	0	0
$823$	−1.36104e7	−0.700443	−0.350221	−	0.936667i	$-0.613894\pi$
$823$	−1.36104e7	−0.700443	−0.350221	+	0.936667i	$0.613894\pi$
$824$	0	0
$825$	7.91096e6	0.404664
$826$	0	0
$827$	−3.31632e7	−1.68614	−0.843069	−	0.537805i	$-0.819254\pi$
$827$	−3.31632e7	−1.68614	−0.843069	+	0.537805i	$0.819254\pi$
$828$	0	0
$829$	−1.50934e7	−0.762781	−0.381391	−	0.924414i	$-0.624555\pi$
$829$	−1.50934e7	−0.762781	−0.381391	+	0.924414i	$0.624555\pi$
$830$	0	0
$831$	−1.31973e7	−0.662955
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.66878e6	0.231732
$836$	0	0
$837$	2.86157e6	0.141186
$838$	0	0
$839$	1.85218e7	0.908400	0.454200	−	0.890900i	$-0.349925\pi$
$839$	1.85218e7	0.908400	0.454200	+	0.890900i	$0.349925\pi$
$840$	0	0
$841$	2.15830e7	1.05226
$842$	0	0
$843$	−939355.	−0.0455261
$844$	0	0
$845$	4.56126e6	0.219757
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.57511e6	0.455905
$850$	0	0
$851$	−2.83528e7	−1.34206
$852$	0	0
$853$	−1.26298e7	−0.594325	−0.297163	−	0.954827i	$-0.596040\pi$
$853$	−1.26298e7	−0.594325	−0.297163	+	0.954827i	$0.596040\pi$
$854$	0	0
$855$	−4.42890e6	−0.207196
$856$	0	0
$857$	939099.	0.0436776	0.0218388	−	0.999762i	$-0.493048\pi$
$857$	939099.	0.0436776	0.0218388	+	0.999762i	$0.493048\pi$
$858$	0	0
$859$	538354.	0.0248935	0.0124467	−	0.999923i	$-0.496038\pi$
$859$	538354.	0.0248935	0.0124467	+	0.999923i	$0.496038\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.38362e6	0.428888	0.214444	−	0.976736i	$-0.431206\pi$
$863$	9.38362e6	0.428888	0.214444	+	0.976736i	$0.431206\pi$
$864$	0	0
$865$	6.60076e6	0.299953
$866$	0	0
$867$	−1.03209e7	−0.466307
$868$	0	0
$869$	−3.45699e6	−0.155292
$870$	0	0
$871$	2.17169e7	0.969956
$872$	0	0
$873$	9.72715e6	0.431966
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.80202e6	0.210827	0.105413	−	0.994428i	$-0.466383\pi$
$877$	4.80202e6	0.210827	0.105413	+	0.994428i	$0.466383\pi$
$878$	0	0
$879$	−1.35743e6	−0.0592577
$880$	0	0
$881$	2.17116e7	0.942437	0.471219	−	0.882016i	$-0.343814\pi$
$881$	2.17116e7	0.942437	0.471219	+	0.882016i	$0.343814\pi$
$882$	0	0
$883$	−1.00011e7	−0.431666	−0.215833	−	0.976430i	$-0.569247\pi$
$883$	−1.00011e7	−0.431666	−0.215833	+	0.976430i	$0.569247\pi$
$884$	0	0
$885$	4.37946e6	0.187959
$886$	0	0
$887$	−1.53920e7	−0.656881	−0.328440	−	0.944525i	$-0.606523\pi$
$887$	−1.53920e7	−0.656881	−0.328440	+	0.944525i	$0.606523\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.61746e6	−0.152654
$892$	0	0
$893$	−4.18840e6	−0.175760
$894$	0	0
$895$	1.45214e6	0.0605968
$896$	0	0
$897$	1.79878e7	0.746442
$898$	0	0
$899$	5.28504e6	0.218097
$900$	0	0
$901$	4.86232e6	0.199541
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.19763e6	−0.0486073
$906$	0	0
$907$	2.65614e7	1.07209	0.536047	−	0.844188i	$-0.319917\pi$
$907$	2.65614e7	1.07209	0.536047	+	0.844188i	$0.319917\pi$
$908$	0	0
$909$	−6.81019e6	−0.273369
$910$	0	0
$911$	4.46773e7	1.78357	0.891787	−	0.452455i	$-0.149452\pi$
$911$	4.46773e7	1.78357	0.891787	+	0.452455i	$0.149452\pi$
$912$	0	0
$913$	−1.19165e7	−0.473122
$914$	0	0
$915$	3.08829e6	0.121945
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.33228e7	0.520363	0.260182	−	0.965560i	$-0.416218\pi$
$919$	1.33228e7	0.520363	0.260182	+	0.965560i	$0.416218\pi$
$920$	0	0
$921$	987384.	0.0383563
$922$	0	0
$923$	−5.14900e7	−1.98939
$924$	0	0
$925$	3.69294e7	1.41912
$926$	0	0
$927$	−1.92903e7	−0.737294
$928$	0	0
$929$	−1.38805e6	−0.0527673	−0.0263836	−	0.999652i	$-0.508399\pi$
$929$	−1.38805e6	−0.0527673	−0.0263836	+	0.999652i	$0.508399\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.24097e7	−0.842813
$934$	0	0
$935$	−1.37553e6	−0.0514565
$936$	0	0
$937$	−3.59646e7	−1.33822	−0.669109	−	0.743164i	$-0.733324\pi$
$937$	−3.59646e7	−1.33822	−0.669109	+	0.743164i	$0.733324\pi$
$938$	0	0
$939$	2.74746e7	1.01687
$940$	0	0
$941$	−3.63095e7	−1.33674	−0.668369	−	0.743830i	$-0.733007\pi$
$941$	−3.63095e7	−1.33674	−0.668369	+	0.743830i	$0.733007\pi$
$942$	0	0
$943$	1.22591e6	0.0448932
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.20605e7	1.52405	0.762025	−	0.647548i	$-0.224205\pi$
$947$	4.20605e7	1.52405	0.762025	+	0.647548i	$0.224205\pi$
$948$	0	0
$949$	−4.68670e7	−1.68928
$950$	0	0
$951$	7.31185e6	0.262166
$952$	0	0
$953$	−1.16644e7	−0.416035	−0.208018	−	0.978125i	$-0.566701\pi$
$953$	−1.16644e7	−0.416035	−0.208018	+	0.978125i	$0.566701\pi$
$954$	0	0
$955$	−1.85810e6	−0.0659264
$956$	0	0
$957$	1.69562e7	0.598478
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.79656e7	−0.976823
$962$	0	0
$963$	1.09030e6	0.0378860
$964$	0	0
$965$	−5.08068e6	−0.175632
$966$	0	0
$967$	4.43761e7	1.52610	0.763050	−	0.646339i	$-0.223701\pi$
$967$	4.43761e7	1.52610	0.763050	+	0.646339i	$0.223701\pi$
$968$	0	0
$969$	1.00152e7	0.342649
$970$	0	0
$971$	3.06950e7	1.04477	0.522384	−	0.852711i	$-0.325043\pi$
$971$	3.06950e7	1.04477	0.522384	+	0.852711i	$0.325043\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.34290e7	−0.789300
$976$	0	0
$977$	1.75819e6	0.0589291	0.0294646	−	0.999566i	$-0.490620\pi$
$977$	1.75819e6	0.0589291	0.0294646	+	0.999566i	$0.490620\pi$
$978$	0	0
$979$	−3.94786e7	−1.31645
$980$	0	0
$981$	5.73055e6	0.190118
$982$	0	0
$983$	−4.55941e7	−1.50496	−0.752480	−	0.658615i	$-0.771143\pi$
$983$	−4.55941e7	−1.50496	−0.752480	+	0.658615i	$0.771143\pi$
$984$	0	0
$985$	9.61540e6	0.315774
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.49251e7	1.13539
$990$	0	0
$991$	4.13157e7	1.33638	0.668191	−	0.743990i	$-0.267069\pi$
$991$	4.13157e7	1.33638	0.668191	+	0.743990i	$0.267069\pi$
$992$	0	0
$993$	−6.59595e6	−0.212278
$994$	0	0
$995$	4.29274e6	0.137460
$996$	0	0
$997$	−3.97352e7	−1.26601	−0.633005	−	0.774147i	$-0.718179\pi$
$997$	−3.97352e7	−1.26601	−0.633005	+	0.774147i	$0.718179\pi$
$998$	0	0
$999$	4.28575e7	1.35867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.x.1.2		2
4.3	odd	2		98.6.a.d.1.1	✓	2
7.6	odd	2	inner	784.6.a.x.1.1		2
12.11	even	2		882.6.a.bp.1.1		2
28.3	even	6		98.6.c.h.79.1		4
28.11	odd	6		98.6.c.h.79.2		4
28.19	even	6		98.6.c.h.67.1		4
28.23	odd	6		98.6.c.h.67.2		4
28.27	even	2		98.6.a.d.1.2	yes	2
84.83	odd	2		882.6.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.6.a.d.1.1	✓	2	4.3	odd	2
98.6.a.d.1.2	yes	2	28.27	even	2
98.6.c.h.67.1		4	28.19	even	6
98.6.c.h.67.2		4	28.23	odd	6
98.6.c.h.79.1		4	28.3	even	6
98.6.c.h.79.2		4	28.11	odd	6
784.6.a.x.1.1		2	7.6	odd	2	inner
784.6.a.x.1.2		2	1.1	even	1	trivial
882.6.a.bp.1.1		2	12.11	even	2
882.6.a.bp.1.2		2	84.83	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+8.48528 q^{3} +9.89949 q^{5} -171.000 q^{9} +O(q^{10})\)	\(q+8.48528 q^{3} +9.89949 q^{5} -171.000 q^{9} -308.000 q^{11} +912.168 q^{13} +84.0000 q^{15} +451.134 q^{17} +2616.30 q^{19} +2324.00 q^{23} -3027.00 q^{25} -3512.91 q^{27} -6488.00 q^{29} -814.587 q^{31} -2613.47 q^{33} -12200.0 q^{37} +7740.00 q^{39} +527.502 q^{41} +15028.0 q^{43} -1692.81 q^{45} -1600.89 q^{47} +3828.00 q^{51} +10778.0 q^{53} -3049.04 q^{55} +22200.0 q^{57} +52136.4 q^{59} +36765.3 q^{61} +9030.00 q^{65} +23808.0 q^{67} +19719.8 q^{69} -56448.0 q^{71} -51379.8 q^{73} -25684.9 q^{75} +11224.0 q^{79} +11745.0 q^{81} +38690.1 q^{83} +4466.00 q^{85} -55052.5 q^{87} +128177. q^{89} -6912.00 q^{93} +25900.0 q^{95} -56883.9 q^{97} +52668.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 342 q^{9}+O(q^{10}) \) 2 * q - 342 * q^9	\( 2 q - 342 q^{9} - 616 q^{11} + 168 q^{15} + 4648 q^{23} - 6054 q^{25} - 12976 q^{29} - 24400 q^{37} + 15480 q^{39} + 30056 q^{43} + 7656 q^{51} + 21556 q^{53} + 44400 q^{57} + 18060 q^{65} + 47616 q^{67} - 112896 q^{71} + 22448 q^{79} + 23490 q^{81} + 8932 q^{85} - 13824 q^{93} + 51800 q^{95} + 105336 q^{99}+O(q^{100}) \) 2 * q - 342 * q^9 - 616 * q^11 + 168 * q^15 + 4648 * q^23 - 6054 * q^25 - 12976 * q^29 - 24400 * q^37 + 15480 * q^39 + 30056 * q^43 + 7656 * q^51 + 21556 * q^53 + 44400 * q^57 + 18060 * q^65 + 47616 * q^67 - 112896 * q^71 + 22448 * q^79 + 23490 * q^81 + 8932 * q^85 - 13824 * q^93 + 51800 * q^95 + 105336 * q^99

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