LMFDB - Embedded newform 6720.2.g.c.3361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6720,2,Mod(3361,6720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6720.3361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6720.g (of order $2$, degree $1$, 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:	$53.6594701583$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3361.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	6720.3361
Dual form			6720.2.g.c.3361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} -6.00000 q^{17} +1.00000i q^{21} +8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} -1.00000i q^{35} -2.00000 q^{39} +10.0000 q^{41} +6.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} +6.00000i q^{51} +12.0000i q^{53} -2.00000 q^{55} -12.0000i q^{59} +2.00000i q^{61} +1.00000 q^{63} +2.00000 q^{65} -6.00000i q^{67} -8.00000i q^{69} +10.0000 q^{73} +1.00000i q^{75} -2.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -16.0000i q^{83} -6.00000i q^{85} -4.00000 q^{87} +6.00000 q^{89} +2.00000i q^{91} +4.00000i q^{93} +18.0000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{7} - 2 q^{9} + 2 q^{15} - 12 q^{17} + 16 q^{23} - 2 q^{25} - 8 q^{31} + 4 q^{33} - 4 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} - 4 q^{55} + 2 q^{63} + 4 q^{65} + 20 q^{73} - 32 q^{79} + 2 q^{81} - 8 q^{87} + 12 q^{89} + 36 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 2 * q^9 + 2 * q^15 - 12 * q^17 + 16 * q^23 - 2 * q^25 - 8 * q^31 + 4 * q^33 - 4 * q^39 + 20 * q^41 - 16 * q^47 + 2 * q^49 - 4 * q^55 + 2 * q^63 + 4 * q^65 + 20 * q^73 - 32 * q^79 + 2 * q^81 - 8 * q^87 + 12 * q^89 + 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6720\mathbb{Z}\right)^\times$.

$n$	$1471$	$1921$	$3781$	$4481$	$5377$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	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
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	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
$29$	− 4.00000i	− 0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	− 4.00000i	− 0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	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
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	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
$63$	1.00000	0.125988
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	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
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	−4.00000	−0.428845
$88$	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
$91$	2.00000i	0.209657i
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	12.0000i	1.14939i	0.818367	+	0.574696i	$0.194880\pi$
$109$	12.0000i	1.14939i	−0.818367	+	0.574696i	$0.805120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 10.0000i	− 0.901670i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	8.00000i	0.698963i	0.936943	+	0.349482i	$0.113642\pi$
$131$	8.00000i	0.698963i	−0.936943	+	0.349482i	$0.886358\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	22.0000i	1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$	22.0000i	1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	2.00000i	0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$	2.00000i	0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	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
$195$	− 2.00000i	− 0.143223i
$196$	0	0
$197$	− 20.0000i	− 1.42494i	−0.701702	−	0.712470i	$-0.747576\pi$
$197$	− 20.0000i	− 1.42494i	0.701702	−	0.712470i	$-0.252424\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	10.0000i	0.698430i
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 26.0000i	− 1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$	− 26.0000i	− 1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	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
$231$	−2.00000	−0.131590
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	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
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.00000i	0.247594i
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	6.00000i	0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$	6.00000i	0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 18.0000i	− 1.05518i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	− 16.0000i	− 0.925304i
$300$	0	0
$301$	− 6.00000i	− 0.345834i
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	0	0
$309$	− 4.00000i	− 0.227552i
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	1.00000i	0.0563436i
$316$	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
$319$	8.00000	0.447914
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	− 22.0000i	− 1.20923i	−0.796518	−	0.604615i	$-0.793327\pi$
$331$	− 22.0000i	− 1.20923i	0.796518	−	0.604615i	$-0.206673\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	2.00000i	0.108625i
$340$	0	0
$341$	− 8.00000i	− 0.433224i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	− 2.00000i	− 0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$	− 2.00000i	− 0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$	0	0
$349$	− 10.0000i	− 0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	− 10.0000i	− 0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	10.0000i	0.523424i
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	− 14.0000i	− 0.719132i	−0.933120	−	0.359566i	$-0.882925\pi$
$379$	− 14.0000i	− 0.719132i	0.933120	−	0.359566i	$-0.117075\pi$
$380$	0	0
$381$	− 16.0000i	− 0.819705i
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	36.0000i	1.82527i	0.408773	+	0.912636i	$0.365957\pi$
$389$	36.0000i	1.82527i	−0.408773	+	0.912636i	$0.634043\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	12.0000i	0.590481i
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	− 8.00000i	− 0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$421$	− 8.00000i	− 0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	0	0
$429$	− 4.00000i	− 0.193122i
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	− 4.00000i	− 0.191785i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 14.0000i	− 0.665160i	−0.943075	−	0.332580i	$-0.892081\pi$
$443$	− 14.0000i	− 0.665160i	0.943075	−	0.332580i	$-0.107919\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	−16.0000	−0.756774
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	20.0000i	0.941763i
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	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
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	− 34.0000i	− 1.58354i	−0.610821	−	0.791769i	$-0.709160\pi$
$461$	− 34.0000i	− 1.58354i	0.610821	−	0.791769i	$-0.290840\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	6.00000i	0.277054i
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−12.0000	−0.551761
$474$	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
$481$	0	0
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	18.0000i	0.817338i
$486$	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
$489$	22.0000	0.994874
$490$	0	0
$491$	− 34.0000i	− 1.53440i	−0.641409	−	0.767199i	$-0.721650\pi$
$491$	− 34.0000i	− 1.53440i	0.641409	−	0.767199i	$-0.278350\pi$
$492$	0	0
$493$	24.0000i	1.08091i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 2.00000i	− 0.0895323i	−0.998997	−	0.0447661i	$-0.985746\pi$
$499$	− 2.00000i	− 0.0895323i	0.998997	−	0.0447661i	$-0.0142543\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.00000i	0.176261i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	2.00000i	0.0861461i
$540$	0	0
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	−12.0000	−0.506640
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	− 2.00000i	− 0.0841406i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	34.0000i	1.42286i	0.702759	+	0.711428i	$0.251951\pi$
$571$	34.0000i	1.42286i	−0.702759	+	0.711428i	$0.748049\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	2.00000i	0.0831172i
$580$	0	0
$581$	16.0000i	0.663792i
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	6.00000i	0.245976i
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	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
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	6.00000i	0.244339i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	40.0000i	1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$	40.0000i	1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$	0	0
$615$	10.0000	0.403239
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	8.00000i	0.321029i
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	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
$633$	−26.0000	−1.03341
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	6.00000i	0.236250i
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	− 4.00000i	− 0.156772i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	10.0000i	0.389545i	0.980848	+	0.194772i	$0.0623968\pi$
$659$	10.0000i	0.389545i	−0.980848	+	0.194772i	$0.937603\pi$
$660$	0	0
$661$	38.0000i	1.47803i	0.673690	+	0.739014i	$0.264708\pi$
$661$	38.0000i	1.47803i	−0.673690	+	0.739014i	$0.735292\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 32.0000i	− 1.23904i
$668$	0	0
$669$	− 4.00000i	− 0.154649i
$670$	0	0
$671$	−4.00000	−0.154418
$672$	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
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	−18.0000	−0.690777
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	− 38.0000i	− 1.45403i	−0.686622	−	0.727015i	$-0.740907\pi$
$683$	− 38.0000i	− 1.45403i	0.686622	−	0.727015i	$-0.259093\pi$
$684$	0	0
$685$	18.0000i	0.687745i
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	32.0000i	1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$	32.0000i	1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$	0	0
$693$	2.00000i	0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	− 26.0000i	− 0.983410i
$700$	0	0
$701$	− 28.0000i	− 1.05755i	−0.848763	−	0.528773i	$-0.822652\pi$
$701$	− 28.0000i	− 1.05755i	0.848763	−	0.528773i	$-0.177348\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	− 12.0000i	− 0.450669i	−0.974281	−	0.225335i	$-0.927652\pi$
$709$	− 12.0000i	− 0.450669i	0.974281	−	0.225335i	$-0.0723476\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	4.00000i	0.149592i
$716$	0	0
$717$	8.00000i	0.298765i
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	4.00000i	0.148556i
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 36.0000i	− 1.33151i
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	− 14.0000i	− 0.514998i	−0.966279	−	0.257499i	$-0.917102\pi$
$739$	− 14.0000i	− 0.514998i	0.966279	−	0.257499i	$-0.0828985\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	6.00000i	0.219235i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 16.0000i	− 0.582300i
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 12.0000i	− 0.434429i
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	− 14.0000i	− 0.504198i
$772$	0	0
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	− 24.0000i	− 0.854423i
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	20.0000i	0.705785i
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	−22.0000	−0.770626
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 2.00000i	− 0.0698857i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	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
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	− 22.0000i	− 0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$	− 22.0000i	− 0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$	0	0
$829$	− 46.0000i	− 1.59765i	−0.601566	−	0.798823i	$-0.705456\pi$
$829$	− 46.0000i	− 1.59765i	0.601566	−	0.798823i	$-0.294544\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	24.0000i	0.818869i	0.912339	+	0.409435i	$0.134274\pi$
$859$	24.0000i	0.818869i	−0.912339	+	0.409435i	$0.865726\pi$
$860$	0	0
$861$	10.0000i	0.340799i
$862$	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
$865$	18.0000	0.612018
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	− 32.0000i	− 1.08553i
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−18.0000	−0.609208
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	40.0000i	1.35070i	0.737496	+	0.675352i	$0.236008\pi$
$877$	40.0000i	1.35070i	−0.737496	+	0.675352i	$0.763992\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	− 6.00000i	− 0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$	− 6.00000i	− 0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$	0	0
$885$	− 12.0000i	− 0.403376i
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	16.0000i	0.533630i
$900$	0	0
$901$	− 72.0000i	− 2.39867i
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	0	0
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	0	0
$915$	2.00000i	0.0661180i
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.0000i	0.654771i
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	0	0
$943$	80.0000	2.60516
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	− 20.0000i	− 0.649227i
$950$	0	0
$951$	28.0000	0.907962
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	− 2.00000i	− 0.0643823i
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	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
$979$	12.0000i	0.383522i
$980$	0	0
$981$	− 12.0000i	− 0.383131i
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	−22.0000	−0.698149
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	30.0000i	0.950110i	0.879956	+	0.475055i	$0.157572\pi$
$997$	30.0000i	0.950110i	−0.879956	+	0.475055i	$0.842428\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.g.c.3361.1	✓	2
4.3	odd	2		6720.2.g.e.3361.2	yes	2
8.3	odd	2		6720.2.g.e.3361.1	yes	2
8.5	even	2	inner	6720.2.g.c.3361.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6720.2.g.c.3361.1	✓	2	1.1	even	1	trivial
6720.2.g.c.3361.2	yes	2	8.5	even	2	inner
6720.2.g.e.3361.1	yes	2	8.3	odd	2
6720.2.g.e.3361.2	yes	2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6720.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} -6.00000 q^{17} +1.00000i q^{21} +8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} -1.00000i q^{35} -2.00000 q^{39} +10.0000 q^{41} +6.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} +6.00000i q^{51} +12.0000i q^{53} -2.00000 q^{55} -12.0000i q^{59} +2.00000i q^{61} +1.00000 q^{63} +2.00000 q^{65} -6.00000i q^{67} -8.00000i q^{69} +10.0000 q^{73} +1.00000i q^{75} -2.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -16.0000i q^{83} -6.00000i q^{85} -4.00000 q^{87} +6.00000 q^{89} +2.00000i q^{91} +4.00000i q^{93} +18.0000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{7} - 2 q^{9} + 2 q^{15} - 12 q^{17} + 16 q^{23} - 2 q^{25} - 8 q^{31} + 4 q^{33} - 4 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} - 4 q^{55} + 2 q^{63} + 4 q^{65} + 20 q^{73} - 32 q^{79} + 2 q^{81} - 8 q^{87} + 12 q^{89} + 36 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 2 * q^9 + 2 * q^15 - 12 * q^17 + 16 * q^23 - 2 * q^25 - 8 * q^31 + 4 * q^33 - 4 * q^39 + 20 * q^41 - 16 * q^47 + 2 * q^49 - 4 * q^55 + 2 * q^63 + 4 * q^65 + 20 * q^73 - 32 * q^79 + 2 * q^81 - 8 * q^87 + 12 * q^89 + 36 * q^97

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