LMFDB - Embedded newform 768.2.d.e.385.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(385,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$768 = 2^{8} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	768.d (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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.2.d.e.385.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} -6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -2.00000i q^{45} -7.00000 q^{49} -2.00000i q^{51} +2.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} -2.00000i q^{61} +4.00000 q^{65} -4.00000i q^{67} -8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +4.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} -8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{9} + 4 q^{15} + 4 q^{17} + 16 q^{23} + 2 q^{25} + 16 q^{31} - 8 q^{33} - 4 q^{39} + 12 q^{41} - 14 q^{49} + 16 q^{55} - 8 q^{57} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots + 4 q^{97}+O(q^{100})$ 2 * q - 2 * q^9 + 4 * q^15 + 4 * q^17 + 16 * q^23 + 2 * q^25 + 16 * q^31 - 8 * q^33 - 4 * q^39 + 12 * q^41 - 14 * q^49 + 16 * q^55 - 8 * q^57 + 8 * q^65 - 16 * q^71 - 20 * q^73 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 12 * q^89 + 16 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$ .

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	18.0000i	1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$	18.0000i	1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	16.0000i	1.49201i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	− 14.0000i	− 1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	− 14.0000i	− 1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	16.0000i	1.28515i
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 6.00000i	− 0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$	− 6.00000i	− 0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 14.0000i	− 0.894427i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	− 10.0000i	− 0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$	− 10.0000i	− 0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	− 16.0000i	− 0.925304i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	16.0000i	0.910208i
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	− 18.0000i	− 0.977626i
$340$	0	0
$341$	− 32.0000i	− 1.73290i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	30.0000i	1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$	30.0000i	1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	− 20.0000i	− 1.04685i
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	8.00000i	0.409852i
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	2.00000i	0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$	2.00000i	0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	12.0000i	0.586238i	0.956076	+	0.293119i	$0.0946933\pi$
$419$	12.0000i	0.586238i	−0.956076	+	0.293119i	$0.905307\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000i	0.386244i
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	− 26.0000i	− 1.21094i	−0.795868	−	0.605470i	$-0.792985\pi$
$461$	− 26.0000i	− 1.21094i	0.795868	−	0.605470i	$-0.207015\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.00000i	0.181631i
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	12.0000i	0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$	12.0000i	0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.0000i	0.537194i	0.963253	+	0.268597i	$0.0865599\pi$
$499$	12.0000i	0.537194i	−0.963253	+	0.268597i	$0.913440\pi$
$500$	0	0
$501$	24.0000i	1.07224i
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	28.0000i	1.20605i
$540$	0	0
$541$	− 18.0000i	− 0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$	− 18.0000i	− 0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	0	0
$549$	2.00000i	0.0853579i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 12.0000i	− 0.509372i
$556$	0	0
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	28.0000i	1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$	28.0000i	1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$	0	0
$565$	36.0000i	1.51453i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	− 36.0000i	− 1.50655i	−0.657704	−	0.753277i	$-0.728472\pi$
$571$	− 36.0000i	− 1.50655i	0.657704	−	0.753277i	$-0.271528\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	0	0
$589$	− 32.0000i	− 1.31854i
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	44.0000i	1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$	44.0000i	1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$	0	0
$621$	8.00000i	0.321029i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	16.0000i	0.638978i
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	14.0000i	0.554700i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	− 8.00000i	− 0.315000i
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	0	0
$661$	10.0000i	0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$	10.0000i	0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.0000i	1.85857i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	2.00000i	0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$	2.00000i	0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	12.0000i	0.458496i
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	− 4.00000i	− 0.152167i	−0.997101	−	0.0760836i	$-0.975758\pi$
$691$	− 4.00000i	− 0.152167i	0.997101	−	0.0760836i	$-0.0242416\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	10.0000i	0.378235i
$700$	0	0
$701$	6.00000i	0.226617i	0.993560	+	0.113308i	$0.0361448\pi$
$701$	6.00000i	0.226617i	−0.993560	+	0.113308i	$0.963855\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	64.0000	2.39682
$714$	0	0
$715$	− 16.0000i	− 0.598366i
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 18.0000i	− 0.669427i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	−14.0000	−0.516398
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	− 4.00000i	− 0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$	− 4.00000i	− 0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	28.0000	1.02584
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	− 2.00000i	− 0.0720282i
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	32.0000i	1.14505i
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	− 8.00000i	− 0.284808i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	4.00000i	0.141865i
$796$	0	0
$797$	22.0000i	0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$	22.0000i	0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	40.0000i	1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 30.0000i	− 1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$	− 30.0000i	− 1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	− 50.0000i	− 1.73657i	−0.496064	−	0.868286i	$-0.665222\pi$
$829$	− 50.0000i	− 1.73657i	0.496064	−	0.868286i	$-0.334778\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	− 48.0000i	− 1.66111i
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	26.0000i	0.895488i
$844$	0	0
$845$	18.0000i	0.619219i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	28.0000	0.960958
$850$	0	0
$851$	− 48.0000i	− 1.64542i
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	12.0000i	0.409435i	0.978821	+	0.204717i	$0.0656275\pi$
$859$	12.0000i	0.409435i	−0.978821	+	0.204717i	$0.934372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	32.0000i	1.08553i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	48.0000i	1.60089i
$900$	0	0
$901$	4.00000i	0.133259i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	− 18.0000i	− 0.597022i
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	28.0000i	0.917663i
$932$	0	0
$933$	− 24.0000i	− 0.785725i
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	− 6.00000i	− 0.195803i
$940$	0	0
$941$	6.00000i	0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$	6.00000i	0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	20.0000i	0.649227i
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	4.00000i	0.128765i
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	− 36.0000i	− 1.15529i	−0.816286	−	0.577647i	$-0.803971\pi$
$971$	− 36.0000i	− 1.15529i	0.816286	−	0.577647i	$-0.196029\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	0	0
$999$	6.00000	0.189832

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.d.e.385.1		2
3.2	odd	2		2304.2.d.i.1153.1		2
4.3	odd	2		768.2.d.d.385.2		2
8.3	odd	2		768.2.d.d.385.1		2
8.5	even	2	inner	768.2.d.e.385.2		2
12.11	even	2		2304.2.d.k.1153.1		2
16.3	odd	4		192.2.a.b.1.1		1
16.5	even	4		24.2.a.a.1.1	✓	1
16.11	odd	4		48.2.a.a.1.1		1
16.13	even	4		192.2.a.d.1.1		1
24.5	odd	2		2304.2.d.i.1153.2		2
24.11	even	2		2304.2.d.k.1153.2		2
48.5	odd	4		72.2.a.a.1.1		1
48.11	even	4		144.2.a.b.1.1		1
48.29	odd	4		576.2.a.d.1.1		1
48.35	even	4		576.2.a.b.1.1		1
80.3	even	4		4800.2.f.bg.3649.1		2
80.13	odd	4		4800.2.f.d.3649.2		2
80.19	odd	4		4800.2.a.cc.1.1		1
80.27	even	4		1200.2.f.b.49.1		2
80.29	even	4		4800.2.a.q.1.1		1
80.37	odd	4		600.2.f.e.49.2		2
80.43	even	4		1200.2.f.b.49.2		2
80.53	odd	4		600.2.f.e.49.1		2
80.59	odd	4		1200.2.a.d.1.1		1
80.67	even	4		4800.2.f.bg.3649.2		2
80.69	even	4		600.2.a.h.1.1		1
80.77	odd	4		4800.2.f.d.3649.1		2
112.5	odd	12		1176.2.q.a.361.1		2
112.11	odd	12		2352.2.q.l.961.1		2
112.13	odd	4		9408.2.a.h.1.1		1
112.27	even	4		2352.2.a.i.1.1		1
112.37	even	12		1176.2.q.i.361.1		2
112.53	even	12		1176.2.q.i.961.1		2
112.59	even	12		2352.2.q.r.961.1		2
112.69	odd	4		1176.2.a.i.1.1		1
112.75	even	12		2352.2.q.r.1537.1		2
112.83	even	4		9408.2.a.cc.1.1		1
112.101	odd	12		1176.2.q.a.961.1		2
112.107	odd	12		2352.2.q.l.1537.1		2
144.5	odd	12		648.2.i.b.217.1		2
144.11	even	12		1296.2.i.e.433.1		2
144.43	odd	12		1296.2.i.m.433.1		2
144.59	even	12		1296.2.i.e.865.1		2
144.85	even	12		648.2.i.g.217.1		2
144.101	odd	12		648.2.i.b.433.1		2
144.133	even	12		648.2.i.g.433.1		2
144.139	odd	12		1296.2.i.m.865.1		2
176.21	odd	4		2904.2.a.c.1.1		1
176.43	even	4		5808.2.a.s.1.1		1
208.5	odd	4		4056.2.c.e.337.2		2
208.21	odd	4		4056.2.c.e.337.1		2
208.155	odd	4		8112.2.a.be.1.1		1
208.181	even	4		4056.2.a.i.1.1		1
240.53	even	4		1800.2.f.c.649.1		2
240.59	even	4		3600.2.a.v.1.1		1
240.107	odd	4		3600.2.f.r.2449.2		2
240.149	odd	4		1800.2.a.m.1.1		1
240.197	even	4		1800.2.f.c.649.2		2
240.203	odd	4		3600.2.f.r.2449.1		2
272.101	even	4		6936.2.a.p.1.1		1
304.37	odd	4		8664.2.a.j.1.1		1
336.5	even	12		3528.2.s.y.361.1		2
336.53	odd	12		3528.2.s.j.3313.1		2
336.101	even	12		3528.2.s.y.3313.1		2
336.149	odd	12		3528.2.s.j.361.1		2
336.251	odd	4		7056.2.a.q.1.1		1
336.293	even	4		3528.2.a.d.1.1		1
528.197	even	4		8712.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.a.a.1.1	✓	1	16.5	even	4
48.2.a.a.1.1		1	16.11	odd	4
72.2.a.a.1.1		1	48.5	odd	4
144.2.a.b.1.1		1	48.11	even	4
192.2.a.b.1.1		1	16.3	odd	4
192.2.a.d.1.1		1	16.13	even	4
576.2.a.b.1.1		1	48.35	even	4
576.2.a.d.1.1		1	48.29	odd	4
600.2.a.h.1.1		1	80.69	even	4
600.2.f.e.49.1		2	80.53	odd	4
600.2.f.e.49.2		2	80.37	odd	4
648.2.i.b.217.1		2	144.5	odd	12
648.2.i.b.433.1		2	144.101	odd	12
648.2.i.g.217.1		2	144.85	even	12
648.2.i.g.433.1		2	144.133	even	12
768.2.d.d.385.1		2	8.3	odd	2
768.2.d.d.385.2		2	4.3	odd	2
768.2.d.e.385.1		2	1.1	even	1	trivial
768.2.d.e.385.2		2	8.5	even	2	inner
1176.2.a.i.1.1		1	112.69	odd	4
1176.2.q.a.361.1		2	112.5	odd	12
1176.2.q.a.961.1		2	112.101	odd	12
1176.2.q.i.361.1		2	112.37	even	12
1176.2.q.i.961.1		2	112.53	even	12
1200.2.a.d.1.1		1	80.59	odd	4
1200.2.f.b.49.1		2	80.27	even	4
1200.2.f.b.49.2		2	80.43	even	4
1296.2.i.e.433.1		2	144.11	even	12
1296.2.i.e.865.1		2	144.59	even	12
1296.2.i.m.433.1		2	144.43	odd	12
1296.2.i.m.865.1		2	144.139	odd	12
1800.2.a.m.1.1		1	240.149	odd	4
1800.2.f.c.649.1		2	240.53	even	4
1800.2.f.c.649.2		2	240.197	even	4
2304.2.d.i.1153.1		2	3.2	odd	2
2304.2.d.i.1153.2		2	24.5	odd	2
2304.2.d.k.1153.1		2	12.11	even	2
2304.2.d.k.1153.2		2	24.11	even	2
2352.2.a.i.1.1		1	112.27	even	4
2352.2.q.l.961.1		2	112.11	odd	12
2352.2.q.l.1537.1		2	112.107	odd	12
2352.2.q.r.961.1		2	112.59	even	12
2352.2.q.r.1537.1		2	112.75	even	12
2904.2.a.c.1.1		1	176.21	odd	4
3528.2.a.d.1.1		1	336.293	even	4
3528.2.s.j.361.1		2	336.149	odd	12
3528.2.s.j.3313.1		2	336.53	odd	12
3528.2.s.y.361.1		2	336.5	even	12
3528.2.s.y.3313.1		2	336.101	even	12
3600.2.a.v.1.1		1	240.59	even	4
3600.2.f.r.2449.1		2	240.203	odd	4
3600.2.f.r.2449.2		2	240.107	odd	4
4056.2.a.i.1.1		1	208.181	even	4
4056.2.c.e.337.1		2	208.21	odd	4
4056.2.c.e.337.2		2	208.5	odd	4
4800.2.a.q.1.1		1	80.29	even	4
4800.2.a.cc.1.1		1	80.19	odd	4
4800.2.f.d.3649.1		2	80.77	odd	4
4800.2.f.d.3649.2		2	80.13	odd	4
4800.2.f.bg.3649.1		2	80.3	even	4
4800.2.f.bg.3649.2		2	80.67	even	4
5808.2.a.s.1.1		1	176.43	even	4
6936.2.a.p.1.1		1	272.101	even	4
7056.2.a.q.1.1		1	336.251	odd	4
8112.2.a.be.1.1		1	208.155	odd	4
8664.2.a.j.1.1		1	304.37	odd	4
8712.2.a.u.1.1		1	528.197	even	4
9408.2.a.h.1.1		1	112.13	odd	4
9408.2.a.cc.1.1		1	112.83	even	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	768.2.d.e.385.1

Level	$768$
Weight	$2$
Character	768.385
Analytic conductor	$6.133$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$