LMFDB - Embedded newform 768.2.d.g.385.2

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 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +6.00000i q^{13} +6.00000 q^{17} +2.00000i q^{21} +4.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} +4.00000i q^{29} -10.0000 q^{31} +4.00000 q^{33} -2.00000i q^{37} -6.00000 q^{39} +2.00000 q^{41} +8.00000i q^{43} +12.0000 q^{47} -3.00000 q^{49} +6.00000i q^{51} +12.0000i q^{53} -4.00000i q^{59} +2.00000i q^{61} -2.00000 q^{63} -4.00000i q^{67} +4.00000i q^{69} -4.00000 q^{71} +10.0000 q^{73} +5.00000i q^{75} -8.00000i q^{77} +6.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -4.00000 q^{87} -2.00000 q^{89} +12.0000i q^{91} -10.0000i q^{93} -6.00000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^7 - 2 * q^9	$ 2 q + 4 q^{7} - 2 q^{9} + 12 q^{17} + 8 q^{23} + 10 q^{25} - 20 q^{31} + 8 q^{33} - 12 q^{39} + 4 q^{41} + 24 q^{47} - 6 q^{49} - 4 q^{63} - 8 q^{71} + 20 q^{73} + 12 q^{79} + 2 q^{81} - 8 q^{87} - 4 q^{89} - 12 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^9 + 12 * q^17 + 8 * q^23 + 10 * q^25 - 20 * q^31 + 8 * q^33 - 12 * q^39 + 4 * q^41 + 24 * q^47 - 6 * q^49 - 4 * q^63 - 8 * q^71 + 20 * q^73 + 12 * q^79 + 2 * q^81 - 8 * q^87 - 4 * q^89 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	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
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.00000i	0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$	4.00000i	0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	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	$0.0408676\pi$
$61$	2.00000i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	0	0
$66$	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
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	5.00000i	0.577350i
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	12.0000i	1.25794i
$92$	0	0
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	− 4.00000i	− 0.398015i	−0.979998	−	0.199007i	$-0.936228\pi$
$101$	− 4.00000i	− 0.398015i	0.979998	−	0.199007i	$-0.0637718\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	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
$111$	2.00000	0.189832
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	− 16.0000i	− 1.31077i	−0.755295	−	0.655386i	$-0.772506\pi$
$149$	− 16.0000i	− 1.31077i	0.755295	−	0.655386i	$-0.227494\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	10.0000	0.755929
$176$	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
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	− 4.00000i	− 0.274075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	36.0000i	2.42162i
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	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
$237$	6.00000i	0.389742i
$238$	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
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	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
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	− 4.00000i	− 0.248548i
$260$	0	0
$261$	− 4.00000i	− 0.247594i
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	0	0
$269$	− 28.0000i	− 1.70719i	−0.520937	−	0.853595i	$-0.674417\pi$
$269$	− 28.0000i	− 1.70719i	0.520937	−	0.853595i	$-0.325583\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	− 20.0000i	− 1.20605i
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	− 4.00000i	− 0.233682i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	− 4.00000i	− 0.233682i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	24.0000i	1.38796i
$300$	0	0
$301$	16.0000i	0.922225i
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	0	0
$306$	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
$309$	− 10.0000i	− 0.568880i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	30.0000i	1.66410i
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	− 14.0000i	− 0.760376i
$340$	0	0
$341$	40.0000i	2.16612i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.0000i	1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$	36.0000i	1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.0000i	0.635107i
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	0	0
$381$	− 14.0000i	− 0.717242i
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 60.0000i	− 2.98881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	4.00000i	0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$	4.00000i	0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	30.0000	1.45521
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	24.0000i	1.15873i
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.0000	0.756774
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	− 8.00000i	− 0.376705i
$452$	0	0
$453$	10.0000i	0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	− 8.00000i	− 0.369406i
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	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
$493$	24.0000i	1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	36.0000i	1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$	36.0000i	1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	16.0000i	0.714827i
$502$	0	0
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	28.0000i	1.24108i	0.784176	+	0.620539i	$0.213086\pi$
$509$	28.0000i	1.24108i	−0.784176	+	0.620539i	$0.786914\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	40.0000i	1.74908i	0.484955	+	0.874539i	$0.338836\pi$
$523$	40.0000i	1.74908i	−0.484955	+	0.874539i	$0.661164\pi$
$524$	0	0
$525$	10.0000i	0.436436i
$526$	0	0
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	12.0000i	0.516877i
$540$	0	0
$541$	38.0000i	1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$	38.0000i	1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.0000i	− 1.69485i	−0.530912	−	0.847427i	$-0.678150\pi$
$557$	− 40.0000i	− 1.69485i	0.530912	−	0.847427i	$-0.321850\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	20.0000	0.834058
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	− 24.0000i	− 0.995688i
$582$	0	0
$583$	48.0000	1.98796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.0000i	− 0.736691i
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	72.0000i	2.91281i
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	− 20.0000i	− 0.783862i
$652$	0	0
$653$	− 8.00000i	− 0.313064i	−0.987673	−	0.156532i	$-0.949969\pi$
$653$	− 8.00000i	− 0.313064i	0.987673	−	0.156532i	$-0.0500315\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	− 20.0000i	− 0.779089i	−0.921008	−	0.389545i	$-0.872632\pi$
$659$	− 20.0000i	− 0.779089i	0.921008	−	0.389545i	$-0.127368\pi$
$660$	0	0
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	−36.0000	−1.39812
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	2.00000i	0.0773245i
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	− 24.0000i	− 0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$	− 24.0000i	− 0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−72.0000	−2.74298
$690$	0	0
$691$	− 16.0000i	− 0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$	− 16.0000i	− 0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	12.0000i	0.453234i	0.973984	+	0.226617i	$0.0727665\pi$
$701$	12.0000i	0.453234i	−0.973984	+	0.226617i	$0.927233\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.00000i	− 0.300871i
$708$	0	0
$709$	− 6.00000i	− 0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$	− 6.00000i	− 0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−40.0000	−1.49801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	20.0000i	0.742781i
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000i	1.77534i
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	28.0000i	1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$	28.0000i	1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	− 24.0000i	− 0.876941i
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	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
$759$	16.0000	0.580763
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	− 14.0000i	− 0.504198i
$772$	0	0
$773$	36.0000i	1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$	36.0000i	1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$	0	0
$775$	−50.0000	−1.79605
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	0	0
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	0	0
$789$	− 8.00000i	− 0.284808i
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.0000i	− 0.850124i	−0.905164	−	0.425062i	$-0.860252\pi$
$797$	− 24.0000i	− 0.850124i	0.905164	−	0.425062i	$-0.139748\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.0000	0.985647
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	32.0000i	1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$	32.0000i	1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$	0	0
$813$	2.00000i	0.0701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 12.0000i	− 0.419314i
$820$	0	0
$821$	4.00000i	0.139601i	0.997561	+	0.0698005i	$0.0222363\pi$
$821$	4.00000i	0.139601i	−0.997561	+	0.0698005i	$0.977764\pi$
$822$	0	0
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	− 34.0000i	− 1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$	− 34.0000i	− 1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	14.0000i	0.482186i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	− 8.00000i	− 0.274236i
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	16.0000i	0.545913i	0.962026	+	0.272956i	$0.0880015\pi$
$859$	16.0000i	0.545913i	−0.962026	+	0.272956i	$0.911998\pi$
$860$	0	0
$861$	4.00000i	0.136320i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	− 24.0000i	− 0.814144i
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	4.00000	0.134917
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	48.0000i	1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$	48.0000i	1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$	0	0
$885$	0	0
$886$	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
$889$	−28.0000	−0.939090
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	− 40.0000i	− 1.33407i
$900$	0	0
$901$	72.0000i	2.39867i
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	4.00000i	0.132672i
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 24.0000i	− 0.792550i
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	26.0000i	0.848478i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	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
$949$	60.0000i	1.94768i
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.0000i	0.517207i
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.00000i	0.128366i	0.997938	+	0.0641831i	$0.0204442\pi$
$971$	4.00000i	0.128366i	−0.997938	+	0.0641831i	$0.979556\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	−30.0000	−0.960769
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	8.00000i	0.255681i
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.d.g.385.2		2
3.2	odd	2		2304.2.d.m.1153.2		2
4.3	odd	2		768.2.d.b.385.1		2
8.3	odd	2		768.2.d.b.385.2		2
8.5	even	2	inner	768.2.d.g.385.1		2
12.11	even	2		2304.2.d.d.1153.1		2
16.3	odd	4		384.2.a.g.1.1	yes	1
16.5	even	4		384.2.a.f.1.1	yes	1
16.11	odd	4		384.2.a.c.1.1	yes	1
16.13	even	4		384.2.a.b.1.1	✓	1
24.5	odd	2		2304.2.d.m.1153.1		2
24.11	even	2		2304.2.d.d.1153.2		2
48.5	odd	4		1152.2.a.i.1.1		1
48.11	even	4		1152.2.a.l.1.1		1
48.29	odd	4		1152.2.a.j.1.1		1
48.35	even	4		1152.2.a.k.1.1		1
80.19	odd	4		9600.2.a.h.1.1		1
80.29	even	4		9600.2.a.bw.1.1		1
80.59	odd	4		9600.2.a.bh.1.1		1
80.69	even	4		9600.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.a.b.1.1	✓	1	16.13	even	4
384.2.a.c.1.1	yes	1	16.11	odd	4
384.2.a.f.1.1	yes	1	16.5	even	4
384.2.a.g.1.1	yes	1	16.3	odd	4
768.2.d.b.385.1		2	4.3	odd	2
768.2.d.b.385.2		2	8.3	odd	2
768.2.d.g.385.1		2	8.5	even	2	inner
768.2.d.g.385.2		2	1.1	even	1	trivial
1152.2.a.i.1.1		1	48.5	odd	4
1152.2.a.j.1.1		1	48.29	odd	4
1152.2.a.k.1.1		1	48.35	even	4
1152.2.a.l.1.1		1	48.11	even	4
2304.2.d.d.1153.1		2	12.11	even	2
2304.2.d.d.1153.2		2	24.11	even	2
2304.2.d.m.1153.1		2	24.5	odd	2
2304.2.d.m.1153.2		2	3.2	odd	2
9600.2.a.h.1.1		1	80.19	odd	4
9600.2.a.w.1.1		1	80.69	even	4
9600.2.a.bh.1.1		1	80.59	odd	4
9600.2.a.bw.1.1		1	80.29	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +6.00000i q^{13} +6.00000 q^{17} +2.00000i q^{21} +4.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} +4.00000i q^{29} -10.0000 q^{31} +4.00000 q^{33} -2.00000i q^{37} -6.00000 q^{39} +2.00000 q^{41} +8.00000i q^{43} +12.0000 q^{47} -3.00000 q^{49} +6.00000i q^{51} +12.0000i q^{53} -4.00000i q^{59} +2.00000i q^{61} -2.00000 q^{63} -4.00000i q^{67} +4.00000i q^{69} -4.00000 q^{71} +10.0000 q^{73} +5.00000i q^{75} -8.00000i q^{77} +6.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -4.00000 q^{87} -2.00000 q^{89} +12.0000i q^{91} -10.0000i q^{93} -6.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^7 - 2 * q^9	\( 2 q + 4 q^{7} - 2 q^{9} + 12 q^{17} + 8 q^{23} + 10 q^{25} - 20 q^{31} + 8 q^{33} - 12 q^{39} + 4 q^{41} + 24 q^{47} - 6 q^{49} - 4 q^{63} - 8 q^{71} + 20 q^{73} + 12 q^{79} + 2 q^{81} - 8 q^{87} - 4 q^{89} - 12 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^9 + 12 * q^17 + 8 * q^23 + 10 * q^25 - 20 * q^31 + 8 * q^33 - 12 * q^39 + 4 * q^41 + 24 * q^47 - 6 * q^49 - 4 * q^63 - 8 * q^71 + 20 * q^73 + 12 * q^79 + 2 * q^81 - 8 * q^87 - 4 * q^89 - 12 * q^97

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