LMFDB - Embedded newform 3136.2.b.a.1569.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1569,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, 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("3136.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.b (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:	$25.0410860739$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 448)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3136.1569
Dual form			3136.2.b.a.1569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{9} +2.00000i q^{11} +4.00000i q^{13} -8.00000 q^{15} -2.00000 q^{17} -6.00000i q^{19} -11.0000 q^{25} +4.00000i q^{27} +8.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} -8.00000i q^{37} -8.00000 q^{39} -10.0000 q^{41} +2.00000i q^{43} -4.00000i q^{45} -8.00000 q^{47} -4.00000i q^{51} -8.00000 q^{55} +12.0000 q^{57} +10.0000i q^{59} -4.00000i q^{61} -16.0000 q^{65} +2.00000i q^{67} +8.00000 q^{71} +6.00000 q^{73} -22.0000i q^{75} +8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} -8.00000i q^{85} -16.0000 q^{87} -10.0000 q^{89} +16.0000i q^{93} +24.0000 q^{95} -2.00000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 16 q^{15} - 4 q^{17} - 22 q^{25} + 16 q^{31} - 8 q^{33} - 16 q^{39} - 20 q^{41} - 16 q^{47} - 16 q^{55} + 24 q^{57} - 32 q^{65} + 16 q^{71} + 12 q^{73} + 16 q^{79} - 22 q^{81} - 32 q^{87} - 20 q^{89} + 48 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^9 - 16 * q^15 - 4 * q^17 - 22 * q^25 + 16 * q^31 - 8 * q^33 - 16 * q^39 - 20 * q^41 - 16 * q^47 - 16 * q^55 + 24 * q^57 - 32 * q^65 + 16 * q^71 + 12 * q^73 + 16 * q^79 - 22 * q^81 - 32 * q^87 - 20 * q^89 + 48 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	4.00000i	1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$	4.00000i	1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	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
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−8.00000	−2.06559
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−11.0000	−2.20000
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	8.00000i	1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$	8.00000i	1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$	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
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	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
$49$	0	0
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	10.0000i	1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$	10.0000i	1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$	0	0
$61$	− 4.00000i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0000	−1.98456
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 22.0000i	− 2.54034i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	− 8.00000i	− 0.867722i
$86$	0	0
$87$	−16.0000	−1.71538
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16.0000i	1.65912i
$94$	0	0
$95$	24.0000	2.46235
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	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
$109$	8.00000i	0.766261i	0.923694	+	0.383131i	$0.125154\pi$
$109$	8.00000i	0.766261i	−0.923694	+	0.383131i	$0.874846\pi$
$110$	0	0
$111$	16.0000	1.51865
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 20.0000i	− 1.80334i
$124$	0	0
$125$	− 24.0000i	− 2.14663i
$126$	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
$129$	−4.00000	−0.352180
$130$	0	0
$131$	− 6.00000i	− 0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$	− 6.00000i	− 0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.0000	−1.37706
$136$	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
$139$	2.00000i	0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$	2.00000i	0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$	0	0
$141$	− 16.0000i	− 1.34744i
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−32.0000	−2.65746
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.0000i	1.31077i	0.755295	+	0.655386i	$0.227494\pi$
$149$	16.0000i	1.31077i	−0.755295	+	0.655386i	$0.772506\pi$
$150$	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
$153$	2.00000	0.161690
$154$	0	0
$155$	32.0000i	2.57030i
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 22.0000i	− 1.72317i	−0.507611	−	0.861586i	$-0.669471\pi$
$163$	− 22.0000i	− 1.72317i	0.507611	−	0.861586i	$-0.330529\pi$
$164$	0	0
$165$	− 16.0000i	− 1.24560i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	6.00000i	0.458831i
$172$	0	0
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	− 14.0000i	− 1.04641i	−0.852207	−	0.523205i	$-0.824736\pi$
$179$	− 14.0000i	− 1.04641i	0.852207	−	0.523205i	$-0.175264\pi$
$180$	0	0
$181$	12.0000i	0.891953i	0.895045	+	0.445976i	$0.147144\pi$
$181$	12.0000i	0.891953i	−0.895045	+	0.445976i	$0.852856\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	32.0000	2.35269
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	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
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	− 32.0000i	− 2.29157i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 40.0000i	− 2.79372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	2.00000i	0.137686i	0.997628	+	0.0688428i	$0.0219307\pi$
$211$	2.00000i	0.137686i	−0.997628	+	0.0688428i	$0.978069\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.0000i	0.810885i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	0	0
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	− 32.0000i	− 2.08745i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	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
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	18.0000i	1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$	18.0000i	1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	16.0000	1.00196
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 8.00000i	− 0.495188i
$262$	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
$267$	− 20.0000i	− 1.22398i
$268$	0	0
$269$	− 12.0000i	− 0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$	− 12.0000i	− 0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$	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
$273$	0	0
$274$	0	0
$275$	− 22.0000i	− 1.32665i
$276$	0	0
$277$	16.0000i	0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$	16.0000i	0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	48.0000i	2.84327i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	0	0
$293$	4.00000i	0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$	4.00000i	0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	0	0
$295$	−40.0000	−2.32889
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	32.0000i	1.82042i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	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
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	− 44.0000i	− 2.44068i
$326$	0	0
$327$	−16.0000	−0.884802
$328$	0	0
$329$	0	0
$330$	0	0
$331$	18.0000i	0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$	18.0000i	0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	4.00000i	0.217250i
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	20.0000i	1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$	20.0000i	1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	32.0000i	1.69838i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	14.0000i	0.734809i
$364$	0	0
$365$	24.0000i	1.25622i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	48.0000	2.47871
$376$	0	0
$377$	−32.0000	−1.64808
$378$	0	0
$379$	34.0000i	1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$	34.0000i	1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	0	0
$381$	32.0000i	1.63941i
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.00000i	− 0.101666i
$388$	0	0
$389$	24.0000i	1.21685i	0.793612	+	0.608424i	$0.208198\pi$
$389$	24.0000i	1.21685i	−0.793612	+	0.608424i	$0.791802\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	32.0000i	1.61009i
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	32.0000i	1.59403i
$404$	0	0
$405$	− 44.0000i	− 2.18638i
$406$	0	0
$407$	16.0000	0.793091
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	− 12.0000i	− 0.591916i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	− 14.0000i	− 0.683945i	−0.939710	−	0.341972i	$-0.888905\pi$
$419$	− 14.0000i	− 0.683945i	0.939710	−	0.341972i	$-0.111095\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 16.0000i	− 0.772487i
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	− 64.0000i	− 3.06857i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	0	0
$445$	− 40.0000i	− 1.89618i
$446$	0	0
$447$	−32.0000	−1.51355
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	− 20.0000i	− 0.941763i
$452$	0	0
$453$	− 32.0000i	− 1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	− 8.00000i	− 0.373408i
$460$	0	0
$461$	20.0000i	0.931493i	0.884918	+	0.465746i	$0.154214\pi$
$461$	20.0000i	0.931493i	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	−64.0000	−2.96793
$466$	0	0
$467$	26.0000i	1.20314i	0.798821	+	0.601568i	$0.205457\pi$
$467$	26.0000i	1.20314i	−0.798821	+	0.601568i	$0.794543\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−8.00000	−0.368621
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	66.0000i	3.02829i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.00000i	− 0.363261i
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	44.0000	1.98975
$490$	0	0
$491$	− 6.00000i	− 0.270776i	−0.990793	−	0.135388i	$-0.956772\pi$
$491$	− 6.00000i	− 0.270776i	0.990793	−	0.135388i	$-0.0432281\pi$
$492$	0	0
$493$	− 16.0000i	− 0.720604i
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	34.0000i	1.52205i	0.648723	+	0.761025i	$0.275303\pi$
$499$	34.0000i	1.52205i	−0.648723	+	0.761025i	$0.724697\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	0	0
$507$	− 6.00000i	− 0.266469i
$508$	0	0
$509$	− 12.0000i	− 0.531891i	−0.963988	−	0.265945i	$-0.914316\pi$
$509$	− 12.0000i	− 0.531891i	0.963988	−	0.265945i	$-0.0856841\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	24.0000	1.05963
$514$	0	0
$515$	64.0000i	2.82018i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 10.0000i	− 0.433963i
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	28.0000	1.20829
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.0000i	0.687894i	0.938989	+	0.343947i	$0.111764\pi$
$541$	16.0000i	0.687894i	−0.938989	+	0.343947i	$0.888236\pi$
$542$	0	0
$543$	−24.0000	−1.02994
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	− 22.0000i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$	− 22.0000i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$	0	0
$549$	4.00000i	0.170716i
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	0	0
$554$	0	0
$555$	64.0000i	2.71665i
$556$	0	0
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	18.0000i	0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$	18.0000i	0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$	0	0
$565$	8.00000i	0.336563i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	18.0000i	0.753277i	0.926360	+	0.376638i	$0.122920\pi$
$571$	18.0000i	0.753277i	−0.926360	+	0.376638i	$0.877080\pi$
$572$	0	0
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	20.0000i	0.831172i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	16.0000	0.661519
$586$	0	0
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	0	0
$589$	− 48.0000i	− 1.97781i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	32.0000i	1.30967i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	0	0
$605$	28.0000i	1.13836i
$606$	0	0
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 32.0000i	− 1.29458i
$612$	0	0
$613$	8.00000i	0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$	8.00000i	0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	0	0
$615$	80.0000	3.22591
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	10.0000i	0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$	10.0000i	0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	24.0000i	0.958468i
$628$	0	0
$629$	16.0000i	0.637962i
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	64.0000i	2.53976i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	0	0
$645$	− 16.0000i	− 0.629999i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	24.0000	0.937758
$656$	0	0
$657$	−6.00000	−0.234082
$658$	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
$661$	36.0000i	1.40024i	0.714026	+	0.700119i	$0.246870\pi$
$661$	36.0000i	1.40024i	−0.714026	+	0.700119i	$0.753130\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	− 44.0000i	− 1.69356i
$676$	0	0
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	26.0000i	0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$	26.0000i	0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$	0	0
$685$	− 24.0000i	− 0.916993i
$686$	0	0
$687$	8.00000	0.305219
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.0000i	0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$	10.0000i	0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	36.0000i	1.36165i
$700$	0	0
$701$	− 24.0000i	− 0.906467i	−0.891392	−	0.453234i	$-0.850270\pi$
$701$	− 24.0000i	− 0.906467i	0.891392	−	0.453234i	$-0.149730\pi$
$702$	0	0
$703$	−48.0000	−1.81035
$704$	0	0
$705$	64.0000	2.41038
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.00000i	0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$	8.00000i	0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 32.0000i	− 1.19673i
$716$	0	0
$717$	− 32.0000i	− 1.19506i
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 4.00000i	− 0.148762i
$724$	0	0
$725$	− 88.0000i	− 3.26824i
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 4.00000i	− 0.147945i
$732$	0	0
$733$	28.0000i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$733$	28.0000i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	− 22.0000i	− 0.809283i	−0.914475	−	0.404642i	$-0.867396\pi$
$739$	− 22.0000i	− 0.809283i	0.914475	−	0.404642i	$-0.132604\pi$
$740$	0	0
$741$	48.0000i	1.76332i
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−64.0000	−2.34478
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	−36.0000	−1.31191
$754$	0	0
$755$	− 64.0000i	− 2.32920i
$756$	0	0
$757$	40.0000i	1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$	40.0000i	1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000i	0.289241i
$766$	0	0
$767$	−40.0000	−1.44432
$768$	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
$771$	− 36.0000i	− 1.29651i
$772$	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
$775$	−88.0000	−3.16105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	60.0000i	2.14972i
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	−32.0000	−1.14359
$784$	0	0
$785$	−16.0000	−0.571064
$786$	0	0
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28.0000i	− 0.991811i	−0.868377	−	0.495905i	$-0.834836\pi$
$797$	− 28.0000i	− 0.991811i	0.868377	−	0.495905i	$-0.165164\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	12.0000i	0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	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
$811$	− 6.00000i	− 0.210688i	−0.994436	−	0.105344i	$-0.966406\pi$
$811$	− 6.00000i	− 0.210688i	0.994436	−	0.105344i	$-0.0335944\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	88.0000	3.08251
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.0000i	− 1.67521i	−0.546275	−	0.837606i	$-0.683955\pi$
$821$	− 48.0000i	− 1.67521i	0.546275	−	0.837606i	$-0.316045\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	44.0000	1.53188
$826$	0	0
$827$	42.0000i	1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$	42.0000i	1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$	0	0
$829$	12.0000i	0.416777i	0.978046	+	0.208389i	$0.0668219\pi$
$829$	12.0000i	0.416777i	−0.978046	+	0.208389i	$0.933178\pi$
$830$	0	0
$831$	−32.0000	−1.11007
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	32.0000i	1.10608i
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	4.00000i	0.137767i
$844$	0	0
$845$	− 12.0000i	− 0.412813i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	44.0000	1.51008
$850$	0	0
$851$	0	0
$852$	0	0
$853$	12.0000i	0.410872i	0.978671	+	0.205436i	$0.0658613\pi$
$853$	12.0000i	0.410872i	−0.978671	+	0.205436i	$0.934139\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	34.0000i	1.16007i	0.814593	+	0.580033i	$0.196960\pi$
$859$	34.0000i	1.16007i	−0.814593	+	0.580033i	$0.803040\pi$
$860$	0	0
$861$	0	0
$862$	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
$865$	48.0000	1.63205
$866$	0	0
$867$	− 26.0000i	− 0.883006i
$868$	0	0
$869$	16.0000i	0.542763i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	0	0
$879$	−8.00000	−0.269833
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	− 30.0000i	− 1.00958i	−0.863242	−	0.504790i	$-0.831570\pi$
$883$	− 30.0000i	− 1.00958i	0.863242	−	0.504790i	$-0.168430\pi$
$884$	0	0
$885$	− 80.0000i	− 2.68917i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 22.0000i	− 0.737028i
$892$	0	0
$893$	48.0000i	1.60626i
$894$	0	0
$895$	56.0000	1.87187
$896$	0	0
$897$	0	0
$898$	0	0
$899$	64.0000i	2.13452i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48.0000	−1.59557
$906$	0	0
$907$	− 22.0000i	− 0.730498i	−0.930910	−	0.365249i	$-0.880984\pi$
$907$	− 22.0000i	− 0.730498i	0.930910	−	0.365249i	$-0.119016\pi$
$908$	0	0
$909$	− 4.00000i	− 0.132672i
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	32.0000i	1.05789i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	44.0000	1.44985
$922$	0	0
$923$	32.0000i	1.05329i
$924$	0	0
$925$	88.0000i	2.89342i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	12.0000i	0.391605i
$940$	0	0
$941$	− 52.0000i	− 1.69515i	−0.530674	−	0.847576i	$-0.678061\pi$
$941$	− 52.0000i	− 1.69515i	0.530674	−	0.847576i	$-0.321939\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.0000i	− 1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$947$	− 38.0000i	− 1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$948$	0	0
$949$	24.0000i	0.779073i
$950$	0	0
$951$	−64.0000	−2.07534
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	− 32.0000i	− 1.03550i
$956$	0	0
$957$	− 32.0000i	− 1.03441i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	40.0000i	1.28765i
$966$	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
$969$	−24.0000	−0.770991
$970$	0	0
$971$	− 6.00000i	− 0.192549i	−0.995355	−	0.0962746i	$-0.969307\pi$
$971$	− 6.00000i	− 0.192549i	0.995355	−	0.0962746i	$-0.0306927\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	88.0000	2.81826
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	− 20.0000i	− 0.639203i
$980$	0	0
$981$	− 8.00000i	− 0.255420i
$982$	0	0
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	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
$993$	−36.0000	−1.14243
$994$	0	0
$995$	64.0000i	2.02894i
$996$	0	0
$997$	20.0000i	0.633406i	0.948525	+	0.316703i	$0.102576\pi$
$997$	20.0000i	0.633406i	−0.948525	+	0.316703i	$0.897424\pi$
$998$	0	0
$999$	32.0000	1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.b.a.1569.2		2
4.3	odd	2		3136.2.b.c.1569.1		2
7.6	odd	2		448.2.b.b.225.1	yes	2
8.3	odd	2		3136.2.b.c.1569.2		2
8.5	even	2	inner	3136.2.b.a.1569.1		2
21.20	even	2		4032.2.c.f.2017.2		2
28.27	even	2		448.2.b.a.225.2	yes	2
56.13	odd	2		448.2.b.b.225.2	yes	2
56.27	even	2		448.2.b.a.225.1	✓	2
84.83	odd	2		4032.2.c.b.2017.2		2
112.13	odd	4		1792.2.a.e.1.1		1
112.27	even	4		1792.2.a.h.1.1		1
112.69	odd	4		1792.2.a.d.1.1		1
112.83	even	4		1792.2.a.a.1.1		1
168.83	odd	2		4032.2.c.b.2017.1		2
168.125	even	2		4032.2.c.f.2017.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.b.a.225.1	✓	2	56.27	even	2
448.2.b.a.225.2	yes	2	28.27	even	2
448.2.b.b.225.1	yes	2	7.6	odd	2
448.2.b.b.225.2	yes	2	56.13	odd	2
1792.2.a.a.1.1		1	112.83	even	4
1792.2.a.d.1.1		1	112.69	odd	4
1792.2.a.e.1.1		1	112.13	odd	4
1792.2.a.h.1.1		1	112.27	even	4
3136.2.b.a.1569.1		2	8.5	even	2	inner
3136.2.b.a.1569.2		2	1.1	even	1	trivial
3136.2.b.c.1569.1		2	4.3	odd	2
3136.2.b.c.1569.2		2	8.3	odd	2
4032.2.c.b.2017.1		2	168.83	odd	2
4032.2.c.b.2017.2		2	84.83	odd	2
4032.2.c.f.2017.1		2	168.125	even	2
4032.2.c.f.2017.2		2	21.20	even	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{9} +2.00000i q^{11} +4.00000i q^{13} -8.00000 q^{15} -2.00000 q^{17} -6.00000i q^{19} -11.0000 q^{25} +4.00000i q^{27} +8.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} -8.00000i q^{37} -8.00000 q^{39} -10.0000 q^{41} +2.00000i q^{43} -4.00000i q^{45} -8.00000 q^{47} -4.00000i q^{51} -8.00000 q^{55} +12.0000 q^{57} +10.0000i q^{59} -4.00000i q^{61} -16.0000 q^{65} +2.00000i q^{67} +8.00000 q^{71} +6.00000 q^{73} -22.0000i q^{75} +8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} -8.00000i q^{85} -16.0000 q^{87} -10.0000 q^{89} +16.0000i q^{93} +24.0000 q^{95} -2.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 16 q^{15} - 4 q^{17} - 22 q^{25} + 16 q^{31} - 8 q^{33} - 16 q^{39} - 20 q^{41} - 16 q^{47} - 16 q^{55} + 24 q^{57} - 32 q^{65} + 16 q^{71} + 12 q^{73} + 16 q^{79} - 22 q^{81} - 32 q^{87} - 20 q^{89} + 48 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^9 - 16 * q^15 - 4 * q^17 - 22 * q^25 + 16 * q^31 - 8 * q^33 - 16 * q^39 - 20 * q^41 - 16 * q^47 - 16 * q^55 + 24 * q^57 - 32 * q^65 + 16 * q^71 + 12 * q^73 + 16 * q^79 - 22 * q^81 - 32 * q^87 - 20 * q^89 + 48 * q^95 - 4 * q^97

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