LMFDB - Embedded newform 784.3.c.e.97.4

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			97.4
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.3.c.e.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.18154i q^{3} -3.16693i q^{5} -8.48528 q^{9} +O(q^{10})$	$q+4.18154i q^{3} -3.16693i q^{5} -8.48528 q^{9} +13.2426 q^{11} -5.49333i q^{13} +13.2426 q^{15} -13.5592i q^{17} -0.717439i q^{19} +2.27208 q^{23} +14.9706 q^{25} +2.15232i q^{27} +20.4853 q^{29} -24.6180i q^{31} +55.3746i q^{33} +64.9411 q^{37} +22.9706 q^{39} -21.0308i q^{41} -6.48528 q^{43} +26.8723i q^{45} +47.7800i q^{47} +56.6985 q^{51} +22.0294 q^{53} -41.9385i q^{55} +3.00000 q^{57} +83.7539i q^{59} -66.2593i q^{61} -17.3970 q^{65} -92.6396 q^{67} +9.50079i q^{69} +48.4264 q^{71} +130.991i q^{73} +62.6000i q^{75} +76.2132 q^{79} -85.3675 q^{81} -107.981i q^{83} -42.9411 q^{85} +85.6600i q^{87} +167.907i q^{89} +102.941 q^{93} -2.27208 q^{95} +25.5816i q^{97} -112.368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.18154i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	4.18154i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	0	0
$5$	− 3.16693i	− 0.633386i	−0.948528	−	0.316693i	$-0.897428\pi$
$5$	− 3.16693i	− 0.633386i	0.948528	−	0.316693i	$-0.102572\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.48528	−0.942809
$10$	0	0
$11$	13.2426	1.20388	0.601938	−	0.798543i	$-0.294395\pi$
$11$	13.2426	1.20388	0.601938	+	0.798543i	$0.294395\pi$
$12$	0	0
$13$	− 5.49333i	− 0.422563i	−0.977425	−	0.211282i	$-0.932236\pi$
$13$	− 5.49333i	− 0.422563i	0.977425	−	0.211282i	$-0.0677638\pi$
$14$	0	0
$15$	13.2426	0.882843
$16$	0	0
$17$	− 13.5592i	− 0.797602i	−0.917037	−	0.398801i	$-0.869426\pi$
$17$	− 13.5592i	− 0.797602i	0.917037	−	0.398801i	$-0.130574\pi$
$18$	0	0
$19$	− 0.717439i	− 0.0377599i	−0.999822	−	0.0188800i	$-0.993990\pi$
$19$	− 0.717439i	− 0.0377599i	0.999822	−	0.0188800i	$-0.00601004\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.27208	0.0987860	0.0493930	−	0.998779i	$-0.484271\pi$
$23$	2.27208	0.0987860	0.0493930	+	0.998779i	$0.484271\pi$
$24$	0	0
$25$	14.9706	0.598823
$26$	0	0
$27$	2.15232i	0.0797154i
$28$	0	0
$29$	20.4853	0.706389	0.353195	−	0.935550i	$-0.385095\pi$
$29$	20.4853	0.706389	0.353195	+	0.935550i	$0.385095\pi$
$30$	0	0
$31$	− 24.6180i	− 0.794129i	−0.917791	−	0.397064i	$-0.870029\pi$
$31$	− 24.6180i	− 0.794129i	0.917791	−	0.397064i	$-0.129971\pi$
$32$	0	0
$33$	55.3746i	1.67802i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	64.9411	1.75517	0.877583	−	0.479425i	$-0.159155\pi$
$37$	64.9411	1.75517	0.877583	+	0.479425i	$0.159155\pi$
$38$	0	0
$39$	22.9706	0.588989
$40$	0	0
$41$	− 21.0308i	− 0.512946i	−0.966551	−	0.256473i	$-0.917439\pi$
$41$	− 21.0308i	− 0.512946i	0.966551	−	0.256473i	$-0.0825605\pi$
$42$	0	0
$43$	−6.48528	−0.150820	−0.0754102	−	0.997153i	$-0.524027\pi$
$43$	−6.48528	−0.150820	−0.0754102	+	0.997153i	$0.524027\pi$
$44$	0	0
$45$	26.8723i	0.597162i
$46$	0	0
$47$	47.7800i	1.01660i	0.861181	+	0.508298i	$0.169725\pi$
$47$	47.7800i	1.01660i	−0.861181	+	0.508298i	$0.830275\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	56.6985	1.11173
$52$	0	0
$53$	22.0294	0.415650	0.207825	−	0.978166i	$-0.433362\pi$
$53$	22.0294	0.415650	0.207825	+	0.978166i	$0.433362\pi$
$54$	0	0
$55$	− 41.9385i	− 0.762518i
$56$	0	0
$57$	3.00000	0.0526316
$58$	0	0
$59$	83.7539i	1.41956i	0.704425	+	0.709779i	$0.251205\pi$
$59$	83.7539i	1.41956i	−0.704425	+	0.709779i	$0.748795\pi$
$60$	0	0
$61$	− 66.2593i	− 1.08622i	−0.839662	−	0.543109i	$-0.817247\pi$
$61$	− 66.2593i	− 1.08622i	0.839662	−	0.543109i	$-0.182753\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.3970	−0.267646
$66$	0	0
$67$	−92.6396	−1.38268	−0.691340	−	0.722529i	$-0.742979\pi$
$67$	−92.6396	−1.38268	−0.691340	+	0.722529i	$0.742979\pi$
$68$	0	0
$69$	9.50079i	0.137693i
$70$	0	0
$71$	48.4264	0.682062	0.341031	−	0.940052i	$-0.389224\pi$
$71$	48.4264	0.682062	0.341031	+	0.940052i	$0.389224\pi$
$72$	0	0
$73$	130.991i	1.79439i	0.441634	+	0.897195i	$0.354399\pi$
$73$	130.991i	1.79439i	−0.441634	+	0.897195i	$0.645601\pi$
$74$	0	0
$75$	62.6000i	0.834667i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	76.2132	0.964724	0.482362	−	0.875972i	$-0.339779\pi$
$79$	76.2132	0.964724	0.482362	+	0.875972i	$0.339779\pi$
$80$	0	0
$81$	−85.3675	−1.05392
$82$	0	0
$83$	− 107.981i	− 1.30098i	−0.759514	−	0.650491i	$-0.774563\pi$
$83$	− 107.981i	− 1.30098i	0.759514	−	0.650491i	$-0.225437\pi$
$84$	0	0
$85$	−42.9411	−0.505190
$86$	0	0
$87$	85.6600i	0.984598i
$88$	0	0
$89$	167.907i	1.88659i	0.331949	+	0.943297i	$0.392294\pi$
$89$	167.907i	1.88659i	−0.331949	+	0.943297i	$0.607706\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	102.941	1.10689
$94$	0	0
$95$	−2.27208	−0.0239166
$96$	0	0
$97$	25.5816i	0.263728i	0.991268	+	0.131864i	$0.0420962\pi$
$97$	25.5816i	0.263728i	−0.991268	+	0.131864i	$0.957904\pi$
$98$	0	0
$99$	−112.368	−1.13503
$100$	0	0
$101$	28.5024i	0.282202i	0.989995	+	0.141101i	$0.0450642\pi$
$101$	28.5024i	0.282202i	−0.989995	+	0.141101i	$0.954936\pi$
$102$	0	0
$103$	56.4912i	0.548458i	0.961664	+	0.274229i	$0.0884227\pi$
$103$	56.4912i	0.548458i	−0.961664	+	0.274229i	$0.911577\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−47.6102	−0.444955	−0.222477	−	0.974938i	$-0.571414\pi$
$107$	−47.6102	−0.444955	−0.222477	+	0.974938i	$0.571414\pi$
$108$	0	0
$109$	75.3087	0.690905	0.345453	−	0.938436i	$-0.387725\pi$
$109$	75.3087	0.690905	0.345453	+	0.938436i	$0.387725\pi$
$110$	0	0
$111$	271.554i	2.44643i
$112$	0	0
$113$	85.4558	0.756246	0.378123	−	0.925755i	$-0.376570\pi$
$113$	85.4558	0.756246	0.378123	+	0.925755i	$0.376570\pi$
$114$	0	0
$115$	− 7.19551i	− 0.0625696i
$116$	0	0
$117$	46.6124i	0.398397i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	54.3675	0.449318
$122$	0	0
$123$	87.9411	0.714968
$124$	0	0
$125$	− 126.584i	− 1.01267i
$126$	0	0
$127$	60.6619	0.477653	0.238826	−	0.971062i	$-0.423237\pi$
$127$	60.6619	0.477653	0.238826	+	0.971062i	$0.423237\pi$
$128$	0	0
$129$	− 27.1185i	− 0.210221i
$130$	0	0
$131$	− 132.948i	− 1.01487i	−0.861691	−	0.507434i	$-0.830594\pi$
$131$	− 132.948i	− 1.01487i	0.861691	−	0.507434i	$-0.169406\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.81623	0.0504906
$136$	0	0
$137$	−117.426	−0.857127	−0.428564	−	0.903512i	$-0.640980\pi$
$137$	−117.426	−0.857127	−0.428564	+	0.903512i	$0.640980\pi$
$138$	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
$141$	−199.794	−1.41698
$142$	0	0
$143$	− 72.7461i	− 0.508714i
$144$	0	0
$145$	− 64.8754i	− 0.447417i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−26.3970	−0.177161	−0.0885804	−	0.996069i	$-0.528233\pi$
$149$	−26.3970	−0.177161	−0.0885804	+	0.996069i	$0.528233\pi$
$150$	0	0
$151$	134.213	0.888829	0.444415	−	0.895821i	$-0.353412\pi$
$151$	134.213	0.888829	0.444415	+	0.895821i	$0.353412\pi$
$152$	0	0
$153$	115.054i	0.751986i
$154$	0	0
$155$	−77.9634	−0.502990
$156$	0	0
$157$	− 226.695i	− 1.44392i	−0.691937	−	0.721958i	$-0.743243\pi$
$157$	− 226.695i	− 1.44392i	0.691937	−	0.721958i	$-0.256757\pi$
$158$	0	0
$159$	92.1170i	0.579352i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	91.9777	0.564280	0.282140	−	0.959373i	$-0.408956\pi$
$163$	91.9777	0.564280	0.282140	+	0.959373i	$0.408956\pi$
$164$	0	0
$165$	175.368	1.06283
$166$	0	0
$167$	− 203.482i	− 1.21845i	−0.792996	−	0.609227i	$-0.791480\pi$
$167$	− 203.482i	− 1.21845i	0.792996	−	0.609227i	$-0.208520\pi$
$168$	0	0
$169$	138.823	0.821440
$170$	0	0
$171$	6.08767i	0.0356004i
$172$	0	0
$173$	− 70.8101i	− 0.409307i	−0.978834	−	0.204654i	$-0.934393\pi$
$173$	− 70.8101i	− 0.409307i	0.978834	−	0.204654i	$-0.0656068\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−350.220	−1.97865
$178$	0	0
$179$	−108.816	−0.607912	−0.303956	−	0.952686i	$-0.598308\pi$
$179$	−108.816	−0.607912	−0.303956	+	0.952686i	$0.598308\pi$
$180$	0	0
$181$	− 99.6607i	− 0.550611i	−0.961357	−	0.275306i	$-0.911221\pi$
$181$	− 99.6607i	− 0.550611i	0.961357	−	0.275306i	$-0.0887791\pi$
$182$	0	0
$183$	277.066	1.51402
$184$	0	0
$185$	− 205.664i	− 1.11170i
$186$	0	0
$187$	− 179.560i	− 0.960214i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−69.9045	−0.365992	−0.182996	−	0.983114i	$-0.558580\pi$
$191$	−69.9045	−0.365992	−0.182996	+	0.983114i	$0.558580\pi$
$192$	0	0
$193$	−32.3381	−0.167555	−0.0837774	−	0.996484i	$-0.526698\pi$
$193$	−32.3381	−0.167555	−0.0837774	+	0.996484i	$0.526698\pi$
$194$	0	0
$195$	− 72.7461i	− 0.373057i
$196$	0	0
$197$	277.103	1.40661	0.703306	−	0.710887i	$-0.251706\pi$
$197$	277.103	1.40661	0.703306	+	0.710887i	$0.251706\pi$
$198$	0	0
$199$	− 167.444i	− 0.841429i	−0.907193	−	0.420715i	$-0.861779\pi$
$199$	− 167.444i	− 0.841429i	0.907193	−	0.420715i	$-0.138221\pi$
$200$	0	0
$201$	− 387.376i	− 1.92725i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−66.6030	−0.324893
$206$	0	0
$207$	−19.2792	−0.0931363
$208$	0	0
$209$	− 9.50079i	− 0.0454583i
$210$	0	0
$211$	128.073	0.606982	0.303491	−	0.952834i	$-0.401848\pi$
$211$	128.073	0.606982	0.303491	+	0.952834i	$0.401848\pi$
$212$	0	0
$213$	202.497i	0.950690i
$214$	0	0
$215$	20.5384i	0.0955276i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−547.742	−2.50111
$220$	0	0
$221$	−74.4853	−0.337037
$222$	0	0
$223$	417.169i	1.87071i	0.353705	+	0.935357i	$0.384922\pi$
$223$	417.169i	1.87071i	−0.353705	+	0.935357i	$0.615078\pi$
$224$	0	0
$225$	−127.029	−0.564575
$226$	0	0
$227$	− 232.260i	− 1.02317i	−0.859232	−	0.511586i	$-0.829058\pi$
$227$	− 232.260i	− 1.02317i	0.859232	−	0.511586i	$-0.170942\pi$
$228$	0	0
$229$	83.6221i	0.365162i	0.983191	+	0.182581i	$0.0584452\pi$
$229$	83.6221i	0.365162i	−0.983191	+	0.182581i	$0.941555\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	219.073	0.940228	0.470114	−	0.882606i	$-0.344213\pi$
$233$	219.073	0.940228	0.470114	+	0.882606i	$0.344213\pi$
$234$	0	0
$235$	151.316	0.643897
$236$	0	0
$237$	318.689i	1.34468i
$238$	0	0
$239$	−193.103	−0.807961	−0.403980	−	0.914768i	$-0.632374\pi$
$239$	−193.103	−0.807961	−0.403980	+	0.914768i	$0.632374\pi$
$240$	0	0
$241$	49.5332i	0.205532i	0.994706	+	0.102766i	$0.0327692\pi$
$241$	49.5332i	0.205532i	−0.994706	+	0.102766i	$0.967231\pi$
$242$	0	0
$243$	− 337.597i	− 1.38929i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.94113	−0.0159560
$248$	0	0
$249$	451.529	1.81337
$250$	0	0
$251$	− 162.524i	− 0.647507i	−0.946141	−	0.323754i	$-0.895055\pi$
$251$	− 162.524i	− 0.647507i	0.946141	−	0.323754i	$-0.104945\pi$
$252$	0	0
$253$	30.0883	0.118926
$254$	0	0
$255$	− 179.560i	− 0.704157i
$256$	0	0
$257$	99.1595i	0.385835i	0.981215	+	0.192917i	$0.0617949\pi$
$257$	99.1595i	0.385835i	−0.981215	+	0.192917i	$0.938205\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−173.823	−0.665990
$262$	0	0
$263$	−434.345	−1.65150	−0.825751	−	0.564034i	$-0.809249\pi$
$263$	−434.345	−1.65150	−0.825751	+	0.564034i	$0.809249\pi$
$264$	0	0
$265$	− 69.7657i	− 0.263267i
$266$	0	0
$267$	−702.110	−2.62962
$268$	0	0
$269$	− 91.4083i	− 0.339808i	−0.985461	−	0.169904i	$-0.945654\pi$
$269$	− 91.4083i	− 0.339808i	0.985461	−	0.169904i	$-0.0543457\pi$
$270$	0	0
$271$	17.0954i	0.0630828i	0.999502	+	0.0315414i	$0.0100416\pi$
$271$	17.0954i	0.0630828i	−0.999502	+	0.0315414i	$0.989958\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	198.250	0.720908
$276$	0	0
$277$	−400.411	−1.44553	−0.722764	−	0.691095i	$-0.757129\pi$
$277$	−400.411	−1.44553	−0.722764	+	0.691095i	$0.757129\pi$
$278$	0	0
$279$	208.891i	0.748712i
$280$	0	0
$281$	−538.690	−1.91705	−0.958524	−	0.285012i	$-0.908002\pi$
$281$	−538.690	−1.91705	−0.958524	+	0.285012i	$0.908002\pi$
$282$	0	0
$283$	309.209i	1.09261i	0.837586	+	0.546306i	$0.183966\pi$
$283$	309.209i	1.09261i	−0.837586	+	0.546306i	$0.816034\pi$
$284$	0	0
$285$	− 9.50079i	− 0.0333361i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	105.147	0.363831
$290$	0	0
$291$	−106.971	−0.367596
$292$	0	0
$293$	327.391i	1.11738i	0.829378	+	0.558688i	$0.188695\pi$
$293$	327.391i	1.11738i	−0.829378	+	0.558688i	$0.811305\pi$
$294$	0	0
$295$	265.243	0.899128
$296$	0	0
$297$	28.5024i	0.0959675i
$298$	0	0
$299$	− 12.4813i	− 0.0417434i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−119.184	−0.393346
$304$	0	0
$305$	−209.839	−0.687995
$306$	0	0
$307$	− 256.140i	− 0.834331i	−0.908831	−	0.417165i	$-0.863024\pi$
$307$	− 256.140i	− 0.834331i	0.908831	−	0.417165i	$-0.136976\pi$
$308$	0	0
$309$	−236.220	−0.764467
$310$	0	0
$311$	− 216.332i	− 0.695602i	−0.937568	−	0.347801i	$-0.886928\pi$
$311$	− 216.332i	− 0.695602i	0.937568	−	0.347801i	$-0.113072\pi$
$312$	0	0
$313$	− 156.818i	− 0.501017i	−0.968114	−	0.250509i	$-0.919402\pi$
$313$	− 156.818i	− 0.501017i	0.968114	−	0.250509i	$-0.0805978\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−448.029	−1.41334	−0.706671	−	0.707542i	$-0.749804\pi$
$317$	−448.029	−1.41334	−0.706671	+	0.707542i	$0.749804\pi$
$318$	0	0
$319$	271.279	0.850405
$320$	0	0
$321$	− 199.084i	− 0.620199i
$322$	0	0
$323$	−9.72792	−0.0301174
$324$	0	0
$325$	− 82.2382i	− 0.253041i
$326$	0	0
$327$	314.906i	0.963016i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	55.0071	0.166185	0.0830924	−	0.996542i	$-0.473520\pi$
$331$	55.0071	0.166185	0.0830924	+	0.996542i	$0.473520\pi$
$332$	0	0
$333$	−551.044	−1.65479
$334$	0	0
$335$	293.383i	0.875770i
$336$	0	0
$337$	−111.632	−0.331254	−0.165627	−	0.986189i	$-0.552965\pi$
$337$	−111.632	−0.331254	−0.165627	+	0.986189i	$0.552965\pi$
$338$	0	0
$339$	357.337i	1.05409i
$340$	0	0
$341$	− 326.007i	− 0.956033i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	30.0883	0.0872125
$346$	0	0
$347$	−377.257	−1.08720	−0.543598	−	0.839346i	$-0.682938\pi$
$347$	−377.257	−1.08720	−0.543598	+	0.839346i	$0.682938\pi$
$348$	0	0
$349$	− 204.034i	− 0.584624i	−0.956323	−	0.292312i	$-0.905575\pi$
$349$	− 204.034i	− 0.584624i	0.956323	−	0.292312i	$-0.0944246\pi$
$350$	0	0
$351$	11.8234	0.0336848
$352$	0	0
$353$	− 417.076i	− 1.18152i	−0.806848	−	0.590759i	$-0.798828\pi$
$353$	− 417.076i	− 1.18152i	0.806848	−	0.590759i	$-0.201172\pi$
$354$	0	0
$355$	− 153.363i	− 0.432008i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	178.831	0.498135	0.249068	−	0.968486i	$-0.419876\pi$
$359$	178.831	0.498135	0.249068	+	0.968486i	$0.419876\pi$
$360$	0	0
$361$	360.485	0.998574
$362$	0	0
$363$	227.340i	0.626281i
$364$	0	0
$365$	414.838	1.13654
$366$	0	0
$367$	628.993i	1.71388i	0.515418	+	0.856939i	$0.327637\pi$
$367$	628.993i	1.71388i	−0.515418	+	0.856939i	$0.672363\pi$
$368$	0	0
$369$	178.452i	0.483610i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−255.558	−0.685143	−0.342572	−	0.939492i	$-0.611298\pi$
$373$	−255.558	−0.685143	−0.342572	+	0.939492i	$0.611298\pi$
$374$	0	0
$375$	529.316	1.41151
$376$	0	0
$377$	− 112.532i	− 0.298494i
$378$	0	0
$379$	−219.750	−0.579816	−0.289908	−	0.957055i	$-0.593625\pi$
$379$	−219.750	−0.579816	−0.289908	+	0.957055i	$0.593625\pi$
$380$	0	0
$381$	253.660i	0.665775i
$382$	0	0
$383$	− 17.0357i	− 0.0444797i	−0.999753	−	0.0222398i	$-0.992920\pi$
$383$	− 17.0357i	− 0.0444797i	0.999753	−	0.0222398i	$-0.00707974\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	55.0294	0.142195
$388$	0	0
$389$	−152.220	−0.391312	−0.195656	−	0.980673i	$-0.562684\pi$
$389$	−152.220	−0.391312	−0.195656	+	0.980673i	$0.562684\pi$
$390$	0	0
$391$	− 30.8076i	− 0.0787919i
$392$	0	0
$393$	555.926	1.41457
$394$	0	0
$395$	− 241.362i	− 0.611042i
$396$	0	0
$397$	372.722i	0.938845i	0.882974	+	0.469423i	$0.155538\pi$
$397$	372.722i	0.938845i	−0.882974	+	0.469423i	$0.844462\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	651.573	1.62487	0.812435	−	0.583052i	$-0.198142\pi$
$401$	651.573	1.62487	0.812435	+	0.583052i	$0.198142\pi$
$402$	0	0
$403$	−135.235	−0.335570
$404$	0	0
$405$	270.353i	0.667538i
$406$	0	0
$407$	859.992	2.11300
$408$	0	0
$409$	533.565i	1.30456i	0.757978	+	0.652280i	$0.226187\pi$
$409$	533.565i	1.30456i	−0.757978	+	0.652280i	$0.773813\pi$
$410$	0	0
$411$	− 491.023i	− 1.19470i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−341.970	−0.824023
$416$	0	0
$417$	286.794	0.687755
$418$	0	0
$419$	− 534.252i	− 1.27507i	−0.770423	−	0.637533i	$-0.779955\pi$
$419$	− 534.252i	− 1.27507i	0.770423	−	0.637533i	$-0.220045\pi$
$420$	0	0
$421$	157.220	0.373445	0.186723	−	0.982413i	$-0.440213\pi$
$421$	157.220	0.373445	0.186723	+	0.982413i	$0.440213\pi$
$422$	0	0
$423$	− 405.427i	− 0.958455i
$424$	0	0
$425$	− 202.989i	− 0.477622i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	304.191	0.709070
$430$	0	0
$431$	228.536	0.530246	0.265123	−	0.964215i	$-0.414587\pi$
$431$	228.536	0.530246	0.265123	+	0.964215i	$0.414587\pi$
$432$	0	0
$433$	− 47.5549i	− 0.109827i	−0.998491	−	0.0549133i	$-0.982512\pi$
$433$	− 47.5549i	− 0.109827i	0.998491	−	0.0549133i	$-0.0174882\pi$
$434$	0	0
$435$	271.279	0.623630
$436$	0	0
$437$	− 1.63008i	− 0.00373015i
$438$	0	0
$439$	− 73.8540i	− 0.168232i	−0.996456	−	0.0841161i	$-0.973193\pi$
$439$	− 73.8540i	− 0.168232i	0.996456	−	0.0841161i	$-0.0268067\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−234.640	−0.529661	−0.264830	−	0.964295i	$-0.585316\pi$
$443$	−234.640	−0.529661	−0.264830	+	0.964295i	$0.585316\pi$
$444$	0	0
$445$	531.749	1.19494
$446$	0	0
$447$	− 110.380i	− 0.246935i
$448$	0	0
$449$	−255.161	−0.568288	−0.284144	−	0.958782i	$-0.591709\pi$
$449$	−255.161	−0.568288	−0.284144	+	0.958782i	$0.591709\pi$
$450$	0	0
$451$	− 278.503i	− 0.617524i
$452$	0	0
$453$	561.218i	1.23889i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−145.735	−0.318895	−0.159448	−	0.987206i	$-0.550971\pi$
$457$	−145.735	−0.318895	−0.159448	+	0.987206i	$0.550971\pi$
$458$	0	0
$459$	29.1838	0.0635812
$460$	0	0
$461$	− 888.329i	− 1.92696i	−0.267777	−	0.963481i	$-0.586289\pi$
$461$	− 888.329i	− 1.92696i	0.267777	−	0.963481i	$-0.413711\pi$
$462$	0	0
$463$	−234.014	−0.505430	−0.252715	−	0.967541i	$-0.581324\pi$
$463$	−234.014	−0.505430	−0.252715	+	0.967541i	$0.581324\pi$
$464$	0	0
$465$	− 326.007i	− 0.701091i
$466$	0	0
$467$	786.618i	1.68441i	0.539159	+	0.842204i	$0.318742\pi$
$467$	786.618i	1.68441i	−0.539159	+	0.842204i	$0.681258\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	947.933	2.01260
$472$	0	0
$473$	−85.8823	−0.181569
$474$	0	0
$475$	− 10.7405i	− 0.0226115i
$476$	0	0
$477$	−186.926	−0.391878
$478$	0	0
$479$	− 736.932i	− 1.53848i	−0.638960	−	0.769240i	$-0.720635\pi$
$479$	− 736.932i	− 1.53848i	0.638960	−	0.769240i	$-0.279365\pi$
$480$	0	0
$481$	− 356.743i	− 0.741669i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	81.0152	0.167042
$486$	0	0
$487$	−270.698	−0.555849	−0.277925	−	0.960603i	$-0.589647\pi$
$487$	−270.698	−0.555849	−0.277925	+	0.960603i	$0.589647\pi$
$488$	0	0
$489$	384.609i	0.786520i
$490$	0	0
$491$	−760.161	−1.54819	−0.774094	−	0.633070i	$-0.781794\pi$
$491$	−760.161	−1.54819	−0.774094	+	0.633070i	$0.781794\pi$
$492$	0	0
$493$	− 277.765i	− 0.563417i
$494$	0	0
$495$	355.860i	0.718909i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−125.492	−0.251488	−0.125744	−	0.992063i	$-0.540132\pi$
$499$	−125.492	−0.251488	−0.125744	+	0.992063i	$0.540132\pi$
$500$	0	0
$501$	850.867	1.69834
$502$	0	0
$503$	117.083i	0.232770i	0.993204	+	0.116385i	$0.0371306\pi$
$503$	117.083i	0.232770i	−0.993204	+	0.116385i	$0.962869\pi$
$504$	0	0
$505$	90.2649	0.178742
$506$	0	0
$507$	580.496i	1.14496i
$508$	0	0
$509$	662.925i	1.30241i	0.758903	+	0.651204i	$0.225736\pi$
$509$	662.925i	1.30241i	−0.758903	+	0.651204i	$0.774264\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.54416	0.00301005
$514$	0	0
$515$	178.904	0.347386
$516$	0	0
$517$	632.733i	1.22386i
$518$	0	0
$519$	296.095	0.570511
$520$	0	0
$521$	− 47.1383i	− 0.0904765i	−0.998976	−	0.0452383i	$-0.985595\pi$
$521$	− 47.1383i	− 0.0904765i	0.998976	−	0.0452383i	$-0.0144047\pi$
$522$	0	0
$523$	499.471i	0.955011i	0.878629	+	0.477506i	$0.158459\pi$
$523$	499.471i	0.955011i	−0.878629	+	0.477506i	$0.841541\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−333.801	−0.633399
$528$	0	0
$529$	−523.838	−0.990241
$530$	0	0
$531$	− 710.675i	− 1.33837i
$532$	0	0
$533$	−115.529	−0.216752
$534$	0	0
$535$	150.778i	0.281828i
$536$	0	0
$537$	− 455.019i	− 0.847336i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	498.809	0.922013	0.461007	−	0.887397i	$-0.347488\pi$
$541$	498.809	0.922013	0.461007	+	0.887397i	$0.347488\pi$
$542$	0	0
$543$	416.735	0.767468
$544$	0	0
$545$	− 238.497i	− 0.437609i
$546$	0	0
$547$	279.897	0.511694	0.255847	−	0.966717i	$-0.417646\pi$
$547$	279.897	0.511694	0.255847	+	0.966717i	$0.417646\pi$
$548$	0	0
$549$	562.229i	1.02410i
$550$	0	0
$551$	− 14.6969i	− 0.0266732i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	859.992	1.54954
$556$	0	0
$557$	−261.780	−0.469981	−0.234991	−	0.971998i	$-0.575506\pi$
$557$	−261.780	−0.469981	−0.234991	+	0.971998i	$0.575506\pi$
$558$	0	0
$559$	35.6258i	0.0637312i
$560$	0	0
$561$	750.838	1.33839
$562$	0	0
$563$	485.062i	0.861567i	0.902455	+	0.430784i	$0.141763\pi$
$563$	485.062i	0.861567i	−0.902455	+	0.430784i	$0.858237\pi$
$564$	0	0
$565$	− 270.633i	− 0.478996i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−453.999	−0.797890	−0.398945	−	0.916975i	$-0.630623\pi$
$569$	−453.999	−0.797890	−0.398945	+	0.916975i	$0.630623\pi$
$570$	0	0
$571$	231.537	0.405494	0.202747	−	0.979231i	$-0.435013\pi$
$571$	231.537	0.405494	0.202747	+	0.979231i	$0.435013\pi$
$572$	0	0
$573$	− 292.309i	− 0.510137i
$574$	0	0
$575$	34.0143	0.0591553
$576$	0	0
$577$	651.267i	1.12871i	0.825531	+	0.564356i	$0.190876\pi$
$577$	651.267i	1.12871i	−0.825531	+	0.564356i	$0.809124\pi$
$578$	0	0
$579$	− 135.223i	− 0.233546i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	291.728	0.500391
$584$	0	0
$585$	147.618	0.252339
$586$	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
$589$	−17.6619	−0.0299863
$590$	0	0
$591$	1158.72i	1.96060i
$592$	0	0
$593$	808.418i	1.36327i	0.731694	+	0.681634i	$0.238730\pi$
$593$	808.418i	1.36327i	−0.731694	+	0.681634i	$0.761270\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	700.176	1.17282
$598$	0	0
$599$	−530.845	−0.886218	−0.443109	−	0.896468i	$-0.646125\pi$
$599$	−530.845	−0.886218	−0.443109	+	0.896468i	$0.646125\pi$
$600$	0	0
$601$	936.503i	1.55824i	0.626874	+	0.779121i	$0.284334\pi$
$601$	936.503i	1.55824i	−0.626874	+	0.779121i	$0.715666\pi$
$602$	0	0
$603$	786.073	1.30360
$604$	0	0
$605$	− 172.178i	− 0.284592i
$606$	0	0
$607$	602.121i	0.991962i	0.868333	+	0.495981i	$0.165191\pi$
$607$	602.121i	0.991962i	−0.868333	+	0.495981i	$0.834809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	262.471	0.429576
$612$	0	0
$613$	1096.90	1.78939	0.894695	−	0.446677i	$-0.147393\pi$
$613$	1096.90	1.78939	0.894695	+	0.446677i	$0.147393\pi$
$614$	0	0
$615$	− 278.503i	− 0.452851i
$616$	0	0
$617$	−432.956	−0.701712	−0.350856	−	0.936429i	$-0.614109\pi$
$617$	−432.956	−0.701712	−0.350856	+	0.936429i	$0.614109\pi$
$618$	0	0
$619$	− 225.110i	− 0.363668i	−0.983329	−	0.181834i	$-0.941797\pi$
$619$	− 225.110i	− 0.363668i	0.983329	−	0.181834i	$-0.0582034\pi$
$620$	0	0
$621$	4.89023i	0.00787477i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.6182	−0.0425891
$626$	0	0
$627$	39.7279	0.0633619
$628$	0	0
$629$	− 880.552i	− 1.39992i
$630$	0	0
$631$	−750.514	−1.18940	−0.594702	−	0.803946i	$-0.702730\pi$
$631$	−750.514	−1.18940	−0.594702	+	0.803946i	$0.702730\pi$
$632$	0	0
$633$	535.543i	0.846040i
$634$	0	0
$635$	− 192.112i	− 0.302538i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−410.912	−0.643054
$640$	0	0
$641$	−1161.85	−1.81256	−0.906281	−	0.422675i	$-0.861091\pi$
$641$	−1161.85	−1.81256	−0.906281	+	0.422675i	$0.861091\pi$
$642$	0	0
$643$	121.957i	0.189669i	0.995493	+	0.0948347i	$0.0302322\pi$
$643$	121.957i	0.189669i	−0.995493	+	0.0948347i	$0.969768\pi$
$644$	0	0
$645$	−85.8823	−0.133151
$646$	0	0
$647$	158.775i	0.245403i	0.992444	+	0.122701i	$0.0391557\pi$
$647$	158.775i	0.245403i	−0.992444	+	0.122701i	$0.960844\pi$
$648$	0	0
$649$	1109.12i	1.70897i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−390.941	−0.598685	−0.299342	−	0.954146i	$-0.596767\pi$
$653$	−390.941	−0.598685	−0.299342	+	0.954146i	$0.596767\pi$
$654$	0	0
$655$	−421.036	−0.642803
$656$	0	0
$657$	− 1111.49i	− 1.69177i
$658$	0	0
$659$	331.955	0.503726	0.251863	−	0.967763i	$-0.418957\pi$
$659$	331.955	0.503726	0.251863	+	0.967763i	$0.418957\pi$
$660$	0	0
$661$	647.820i	0.980061i	0.871705	+	0.490031i	$0.163014\pi$
$661$	647.820i	0.980061i	−0.871705	+	0.490031i	$0.836986\pi$
$662$	0	0
$663$	− 311.463i	− 0.469779i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	46.5442	0.0697813
$668$	0	0
$669$	−1744.41	−2.60749
$670$	0	0
$671$	− 877.448i	− 1.30767i
$672$	0	0
$673$	100.956	0.150009	0.0750047	−	0.997183i	$-0.476103\pi$
$673$	100.956	0.150009	0.0750047	+	0.997183i	$0.476103\pi$
$674$	0	0
$675$	32.2214i	0.0477354i
$676$	0	0
$677$	− 743.177i	− 1.09775i	−0.835905	−	0.548875i	$-0.815056\pi$
$677$	− 743.177i	− 1.09775i	0.835905	−	0.548875i	$-0.184944\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	971.205	1.42615
$682$	0	0
$683$	−4.43442	−0.00649256	−0.00324628	−	0.999995i	$-0.501033\pi$
$683$	−4.43442	−0.00649256	−0.00324628	+	0.999995i	$0.501033\pi$
$684$	0	0
$685$	371.881i	0.542892i
$686$	0	0
$687$	−349.669	−0.508980
$688$	0	0
$689$	− 121.015i	− 0.175638i
$690$	0	0
$691$	− 977.169i	− 1.41414i	−0.707145	−	0.707069i	$-0.750017\pi$
$691$	− 977.169i	− 1.41414i	0.707145	−	0.707069i	$-0.249983\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−217.206	−0.312527
$696$	0	0
$697$	−285.161	−0.409127
$698$	0	0
$699$	916.063i	1.31053i
$700$	0	0
$701$	−840.177	−1.19854	−0.599270	−	0.800547i	$-0.704542\pi$
$701$	−840.177	−1.19854	−0.599270	+	0.800547i	$0.704542\pi$
$702$	0	0
$703$	− 46.5913i	− 0.0662750i
$704$	0	0
$705$	632.733i	0.897494i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	682.558	0.962705	0.481352	−	0.876527i	$-0.340146\pi$
$709$	682.558	0.962705	0.481352	+	0.876527i	$0.340146\pi$
$710$	0	0
$711$	−646.690	−0.909551
$712$	0	0
$713$	− 55.9340i	− 0.0784488i
$714$	0	0
$715$	−230.382	−0.322212
$716$	0	0
$717$	− 807.466i	− 1.12617i
$718$	0	0
$719$	137.625i	0.191412i	0.995410	+	0.0957060i	$0.0305109\pi$
$719$	137.625i	0.191412i	−0.995410	+	0.0957060i	$0.969489\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−207.125	−0.286480
$724$	0	0
$725$	306.676	0.423002
$726$	0	0
$727$	264.137i	0.363325i	0.983361	+	0.181662i	$0.0581478\pi$
$727$	264.137i	0.363325i	−0.983361	+	0.181662i	$0.941852\pi$
$728$	0	0
$729$	643.368	0.882534
$730$	0	0
$731$	87.9354i	0.120295i
$732$	0	0
$733$	− 579.319i	− 0.790340i	−0.918608	−	0.395170i	$-0.870686\pi$
$733$	− 579.319i	− 0.790340i	0.918608	−	0.395170i	$-0.129314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1226.79	−1.66458
$738$	0	0
$739$	198.095	0.268059	0.134029	−	0.990977i	$-0.457208\pi$
$739$	198.095	0.268059	0.134029	+	0.990977i	$0.457208\pi$
$740$	0	0
$741$	− 16.4800i	− 0.0222402i
$742$	0	0
$743$	−976.690	−1.31452	−0.657261	−	0.753663i	$-0.728285\pi$
$743$	−976.690	−1.31452	−0.657261	+	0.753663i	$0.728285\pi$
$744$	0	0
$745$	83.5973i	0.112211i
$746$	0	0
$747$	916.253i	1.22658i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	835.330	1.11229	0.556145	−	0.831085i	$-0.312280\pi$
$751$	835.330	1.11229	0.556145	+	0.831085i	$0.312280\pi$
$752$	0	0
$753$	679.602	0.902526
$754$	0	0
$755$	− 425.044i	− 0.562972i
$756$	0	0
$757$	104.221	0.137677	0.0688383	−	0.997628i	$-0.478071\pi$
$757$	104.221	0.137677	0.0688383	+	0.997628i	$0.478071\pi$
$758$	0	0
$759$	125.815i	0.165765i
$760$	0	0
$761$	547.080i	0.718897i	0.933165	+	0.359448i	$0.117035\pi$
$761$	547.080i	0.718897i	−0.933165	+	0.359448i	$0.882965\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	364.368	0.476297
$766$	0	0
$767$	460.087	0.599853
$768$	0	0
$769$	341.205i	0.443700i	0.975081	+	0.221850i	$0.0712095\pi$
$769$	341.205i	0.443700i	−0.975081	+	0.221850i	$0.928790\pi$
$770$	0	0
$771$	−414.640	−0.537795
$772$	0	0
$773$	− 490.993i	− 0.635179i	−0.948228	−	0.317590i	$-0.897127\pi$
$773$	− 490.993i	− 0.635179i	0.948228	−	0.317590i	$-0.102873\pi$
$774$	0	0
$775$	− 368.545i	− 0.475542i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15.0883	−0.0193688
$780$	0	0
$781$	641.294	0.821118
$782$	0	0
$783$	44.0908i	0.0563101i
$784$	0	0
$785$	−717.926	−0.914555
$786$	0	0
$787$	− 300.455i	− 0.381773i	−0.981612	−	0.190887i	$-0.938864\pi$
$787$	− 300.455i	− 0.381773i	0.981612	−	0.190887i	$-0.0611363\pi$
$788$	0	0
$789$	− 1816.23i	− 2.30194i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−363.984	−0.458996
$794$	0	0
$795$	291.728	0.366953
$796$	0	0
$797$	370.072i	0.464331i	0.972676	+	0.232165i	$0.0745811\pi$
$797$	370.072i	0.464331i	−0.972676	+	0.232165i	$0.925419\pi$
$798$	0	0
$799$	647.860	0.810838
$800$	0	0
$801$	− 1424.74i	− 1.77870i
$802$	0	0
$803$	1734.66i	2.16022i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	382.227	0.473640
$808$	0	0
$809$	491.235	0.607213	0.303607	−	0.952797i	$-0.401809\pi$
$809$	491.235	0.607213	0.303607	+	0.952797i	$0.401809\pi$
$810$	0	0
$811$	156.802i	0.193344i	0.995316	+	0.0966722i	$0.0308199\pi$
$811$	156.802i	0.193344i	−0.995316	+	0.0966722i	$0.969180\pi$
$812$	0	0
$813$	−71.4853	−0.0879278
$814$	0	0
$815$	− 291.287i	− 0.357407i
$816$	0	0
$817$	4.65279i	0.00569497i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−430.632	−0.524522	−0.262261	−	0.964997i	$-0.584468\pi$
$821$	−430.632	−0.524522	−0.262261	+	0.964997i	$0.584468\pi$
$822$	0	0
$823$	708.741	0.861168	0.430584	−	0.902550i	$-0.358308\pi$
$823$	708.741	0.861168	0.430584	+	0.902550i	$0.358308\pi$
$824$	0	0
$825$	828.990i	1.00484i
$826$	0	0
$827$	1460.10	1.76554	0.882770	−	0.469805i	$-0.155676\pi$
$827$	1460.10	1.76554	0.882770	+	0.469805i	$0.155676\pi$
$828$	0	0
$829$	− 257.608i	− 0.310745i	−0.987856	−	0.155373i	$-0.950342\pi$
$829$	− 257.608i	− 0.310745i	0.987856	−	0.155373i	$-0.0496578\pi$
$830$	0	0
$831$	− 1674.34i	− 2.01484i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−644.412	−0.771751
$836$	0	0
$837$	52.9857	0.0633043
$838$	0	0
$839$	− 213.621i	− 0.254613i	−0.991863	−	0.127307i	$-0.959367\pi$
$839$	− 213.621i	− 0.254613i	0.991863	−	0.127307i	$-0.0406332\pi$
$840$	0	0
$841$	−421.353	−0.501015
$842$	0	0
$843$	− 2252.56i	− 2.67207i
$844$	0	0
$845$	− 439.644i	− 0.520288i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1292.97	−1.52293
$850$	0	0
$851$	147.551	0.173386
$852$	0	0
$853$	1127.37i	1.32165i	0.750539	+	0.660826i	$0.229794\pi$
$853$	1127.37i	1.32165i	−0.750539	+	0.660826i	$0.770206\pi$
$854$	0	0
$855$	19.2792	0.0225488
$856$	0	0
$857$	1270.42i	1.48241i	0.671280	+	0.741204i	$0.265745\pi$
$857$	1270.42i	1.48241i	−0.671280	+	0.741204i	$0.734255\pi$
$858$	0	0
$859$	− 255.753i	− 0.297733i	−0.988857	−	0.148867i	$-0.952438\pi$
$859$	− 255.753i	− 0.297733i	0.988857	−	0.148867i	$-0.0475625\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1114.73	−1.29169	−0.645844	−	0.763469i	$-0.723494\pi$
$863$	−1114.73	−1.29169	−0.645844	+	0.763469i	$0.723494\pi$
$864$	0	0
$865$	−224.251	−0.259249
$866$	0	0
$867$	439.677i	0.507125i
$868$	0	0
$869$	1009.26	1.16141
$870$	0	0
$871$	508.900i	0.584270i
$872$	0	0
$873$	− 217.067i	− 0.248645i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1101.81	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$877$	−1101.81	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$878$	0	0
$879$	−1369.00	−1.55745
$880$	0	0
$881$	− 217.067i	− 0.246387i	−0.992383	−	0.123194i	$-0.960686\pi$
$881$	− 217.067i	− 0.246387i	0.992383	−	0.123194i	$-0.0393136\pi$
$882$	0	0
$883$	516.544	0.584988	0.292494	−	0.956267i	$-0.405515\pi$
$883$	516.544	0.584988	0.292494	+	0.956267i	$0.405515\pi$
$884$	0	0
$885$	1109.12i	1.25325i
$886$	0	0
$887$	− 1129.81i	− 1.27374i	−0.770970	−	0.636872i	$-0.780228\pi$
$887$	− 1129.81i	− 1.27374i	0.770970	−	0.636872i	$-0.219772\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1130.49	−1.26879
$892$	0	0
$893$	34.2792	0.0383866
$894$	0	0
$895$	344.613i	0.385043i
$896$	0	0
$897$	52.1909	0.0581838
$898$	0	0
$899$	− 504.306i	− 0.560964i
$900$	0	0
$901$	− 298.702i	− 0.331523i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−315.618	−0.348749
$906$	0	0
$907$	−60.0223	−0.0661767	−0.0330884	−	0.999452i	$-0.510534\pi$
$907$	−60.0223	−0.0661767	−0.0330884	+	0.999452i	$0.510534\pi$
$908$	0	0
$909$	− 241.851i	− 0.266062i
$910$	0	0
$911$	−1422.25	−1.56120	−0.780598	−	0.625033i	$-0.785085\pi$
$911$	−1422.25	−1.56120	−0.780598	+	0.625033i	$0.785085\pi$
$912$	0	0
$913$	− 1429.96i	− 1.56622i
$914$	0	0
$915$	− 877.448i	− 0.958960i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1669.70	−1.81686	−0.908432	−	0.418033i	$-0.862720\pi$
$919$	−1669.70	−1.81686	−0.908432	+	0.418033i	$0.862720\pi$
$920$	0	0
$921$	1071.06	1.16293
$922$	0	0
$923$	− 266.022i	− 0.288215i
$924$	0	0
$925$	972.205	1.05103
$926$	0	0
$927$	− 479.344i	− 0.517092i
$928$	0	0
$929$	− 968.860i	− 1.04291i	−0.853280	−	0.521453i	$-0.825390\pi$
$929$	− 968.860i	− 1.04291i	0.853280	−	0.521453i	$-0.174610\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	904.602	0.969563
$934$	0	0
$935$	−568.654	−0.608186
$936$	0	0
$937$	− 1212.57i	− 1.29410i	−0.762449	−	0.647049i	$-0.776003\pi$
$937$	− 1212.57i	− 1.29410i	0.762449	−	0.647049i	$-0.223997\pi$
$938$	0	0
$939$	655.742	0.698341
$940$	0	0
$941$	− 1494.06i	− 1.58774i	−0.608087	−	0.793870i	$-0.708063\pi$
$941$	− 1494.06i	− 1.58774i	0.608087	−	0.793870i	$-0.291937\pi$
$942$	0	0
$943$	− 47.7836i	− 0.0506719i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−775.462	−0.818862	−0.409431	−	0.912341i	$-0.634273\pi$
$947$	−775.462	−0.818862	−0.409431	+	0.912341i	$0.634273\pi$
$948$	0	0
$949$	719.574	0.758244
$950$	0	0
$951$	− 1873.45i	− 1.96998i
$952$	0	0
$953$	1055.40	1.10745	0.553723	−	0.832701i	$-0.313206\pi$
$953$	1055.40	1.10745	0.553723	+	0.832701i	$0.313206\pi$
$954$	0	0
$955$	221.383i	0.231814i
$956$	0	0
$957$	1134.37i	1.18533i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	354.955	0.369360
$962$	0	0
$963$	403.986	0.419507
$964$	0	0
$965$	102.412i	0.106127i
$966$	0	0
$967$	−1221.63	−1.26332	−0.631661	−	0.775245i	$-0.717627\pi$
$967$	−1221.63	−1.26332	−0.631661	+	0.775245i	$0.717627\pi$
$968$	0	0
$969$	− 40.6777i	− 0.0419791i
$970$	0	0
$971$	− 526.259i	− 0.541976i	−0.962583	−	0.270988i	$-0.912650\pi$
$971$	− 526.259i	− 0.541976i	0.962583	−	0.270988i	$-0.0873504\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	343.882	0.352700
$976$	0	0
$977$	−1000.10	−1.02365	−0.511823	−	0.859091i	$-0.671030\pi$
$977$	−1000.10	−1.02365	−0.511823	+	0.859091i	$0.671030\pi$
$978$	0	0
$979$	2223.53i	2.27123i
$980$	0	0
$981$	−639.015	−0.651392
$982$	0	0
$983$	− 1075.70i	− 1.09430i	−0.837033	−	0.547152i	$-0.815712\pi$
$983$	− 1075.70i	− 1.09430i	0.837033	−	0.547152i	$-0.184288\pi$
$984$	0	0
$985$	− 877.564i	− 0.890928i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.7351	−0.0148990
$990$	0	0
$991$	1876.03	1.89307	0.946536	−	0.322597i	$-0.104556\pi$
$991$	1876.03	1.89307	0.946536	+	0.322597i	$0.104556\pi$
$992$	0	0
$993$	230.015i	0.231636i
$994$	0	0
$995$	−530.285	−0.532949
$996$	0	0
$997$	582.224i	0.583976i	0.956422	+	0.291988i	$0.0943167\pi$
$997$	582.224i	0.583976i	−0.956422	+	0.291988i	$0.905683\pi$
$998$	0	0
$999$	139.774i	0.139914i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.c.e.97.4		4
4.3	odd	2		98.3.b.b.97.1		4
7.2	even	3		784.3.s.c.129.1		4
7.3	odd	6		784.3.s.c.705.1		4
7.4	even	3		112.3.s.b.33.2		4
7.5	odd	6		112.3.s.b.17.2		4
7.6	odd	2	inner	784.3.c.e.97.1		4
12.11	even	2		882.3.c.f.685.4		4
21.5	even	6		1008.3.cg.l.577.1		4
21.11	odd	6		1008.3.cg.l.145.1		4
28.3	even	6		98.3.d.a.19.2		4
28.11	odd	6		14.3.d.a.5.2	yes	4
28.19	even	6		14.3.d.a.3.2	✓	4
28.23	odd	6		98.3.d.a.31.2		4
28.27	even	2		98.3.b.b.97.2		4
56.5	odd	6		448.3.s.c.129.1		4
56.11	odd	6		448.3.s.d.257.2		4
56.19	even	6		448.3.s.d.129.2		4
56.53	even	6		448.3.s.c.257.1		4
84.11	even	6		126.3.n.c.19.1		4
84.23	even	6		882.3.n.b.325.1		4
84.47	odd	6		126.3.n.c.73.1		4
84.59	odd	6		882.3.n.b.19.1		4
84.83	odd	2		882.3.c.f.685.3		4
140.19	even	6		350.3.k.a.101.1		4
140.39	odd	6		350.3.k.a.201.1		4
140.47	odd	12		350.3.i.a.199.1		8
140.67	even	12		350.3.i.a.299.4		8
140.103	odd	12		350.3.i.a.199.4		8
140.123	even	12		350.3.i.a.299.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.2	✓	4	28.19	even	6
14.3.d.a.5.2	yes	4	28.11	odd	6
98.3.b.b.97.1		4	4.3	odd	2
98.3.b.b.97.2		4	28.27	even	2
98.3.d.a.19.2		4	28.3	even	6
98.3.d.a.31.2		4	28.23	odd	6
112.3.s.b.17.2		4	7.5	odd	6
112.3.s.b.33.2		4	7.4	even	3
126.3.n.c.19.1		4	84.11	even	6
126.3.n.c.73.1		4	84.47	odd	6
350.3.i.a.199.1		8	140.47	odd	12
350.3.i.a.199.4		8	140.103	odd	12
350.3.i.a.299.1		8	140.123	even	12
350.3.i.a.299.4		8	140.67	even	12
350.3.k.a.101.1		4	140.19	even	6
350.3.k.a.201.1		4	140.39	odd	6
448.3.s.c.129.1		4	56.5	odd	6
448.3.s.c.257.1		4	56.53	even	6
448.3.s.d.129.2		4	56.19	even	6
448.3.s.d.257.2		4	56.11	odd	6
784.3.c.e.97.1		4	7.6	odd	2	inner
784.3.c.e.97.4		4	1.1	even	1	trivial
784.3.s.c.129.1		4	7.2	even	3
784.3.s.c.705.1		4	7.3	odd	6
882.3.c.f.685.3		4	84.83	odd	2
882.3.c.f.685.4		4	12.11	even	2
882.3.n.b.19.1		4	84.59	odd	6
882.3.n.b.325.1		4	84.23	even	6
1008.3.cg.l.145.1		4	21.11	odd	6
1008.3.cg.l.577.1		4	21.5	even	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)

\(f(q)\)	\(=\)	\(q+4.18154i q^{3} -3.16693i q^{5} -8.48528 q^{9} +O(q^{10})\)	\(q+4.18154i q^{3} -3.16693i q^{5} -8.48528 q^{9} +13.2426 q^{11} -5.49333i q^{13} +13.2426 q^{15} -13.5592i q^{17} -0.717439i q^{19} +2.27208 q^{23} +14.9706 q^{25} +2.15232i q^{27} +20.4853 q^{29} -24.6180i q^{31} +55.3746i q^{33} +64.9411 q^{37} +22.9706 q^{39} -21.0308i q^{41} -6.48528 q^{43} +26.8723i q^{45} +47.7800i q^{47} +56.6985 q^{51} +22.0294 q^{53} -41.9385i q^{55} +3.00000 q^{57} +83.7539i q^{59} -66.2593i q^{61} -17.3970 q^{65} -92.6396 q^{67} +9.50079i q^{69} +48.4264 q^{71} +130.991i q^{73} +62.6000i q^{75} +76.2132 q^{79} -85.3675 q^{81} -107.981i q^{83} -42.9411 q^{85} +85.6600i q^{87} +167.907i q^{89} +102.941 q^{93} -2.27208 q^{95} +25.5816i q^{97} -112.368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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

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Database

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