LMFDB - Embedded newform 1792.2.b.f.897.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (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:	$14.3091920422$
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 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.f.897.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -4.00000i q^{13} -2.00000 q^{17} -6.00000i q^{19} -2.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -4.00000i q^{27} +2.00000i q^{29} -4.00000 q^{31} +8.00000 q^{33} -10.0000i q^{37} -8.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{47} +1.00000 q^{49} +4.00000i q^{51} +2.00000i q^{53} -12.0000 q^{57} -10.0000i q^{59} -8.00000i q^{61} -1.00000 q^{63} -8.00000i q^{67} +16.0000i q^{69} +6.00000 q^{73} -10.0000i q^{75} +4.00000i q^{77} -16.0000 q^{79} -11.0000 q^{81} +2.00000i q^{83} +4.00000 q^{87} -18.0000 q^{89} -4.00000i q^{91} +8.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{17} - 16 q^{23} + 10 q^{25} - 8 q^{31} + 16 q^{33} - 16 q^{39} + 20 q^{41} + 8 q^{47} + 2 q^{49} - 24 q^{57} - 2 q^{63} + 12 q^{73} - 32 q^{79} - 22 q^{81} + 8 q^{87} - 36 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 - 4 * q^17 - 16 * q^23 + 10 * q^25 - 8 * q^31 + 16 * q^33 - 16 * q^39 + 20 * q^41 + 8 * q^47 + 2 * q^49 - 24 * q^57 - 2 * q^63 + 12 * q^73 - 32 * q^79 - 22 * q^81 + 8 * q^87 - 36 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	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
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	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
$33$	8.00000	1.39262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	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
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000i	0.560112i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	− 10.0000i	− 1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$	− 10.0000i	− 1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	16.0000i	1.92617i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	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
$75$	− 10.0000i	− 1.15470i
$76$	0	0
$77$	4.00000i	0.455842i
$78$	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
$81$	−11.0000	−1.22222
$82$	0	0
$83$	2.00000i	0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$	2.00000i	0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	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
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	−20.0000	−1.89832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 20.0000i	− 1.80334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	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
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	10.0000i	0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$	10.0000i	0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	16.0000	1.33799
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0000i	0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$	12.0000i	0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	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
$175$	5.00000	0.377964
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	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
$183$	−16.0000	−1.18275
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	− 12.0000i	− 0.810885i
$220$	0	0
$221$	8.00000i	0.538138i
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	− 22.0000i	− 1.46019i	−0.683345	−	0.730096i	$-0.739475\pi$
$227$	− 22.0000i	− 1.46019i	0.683345	−	0.730096i	$-0.260525\pi$
$228$	0	0
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.0000i	2.07862i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	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
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	10.0000i	0.631194i	0.948893	+	0.315597i	$0.102205\pi$
$251$	10.0000i	0.631194i	−0.948893	+	0.315597i	$0.897795\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	− 10.0000i	− 0.621370i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	36.0000i	2.20316i
$268$	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	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	20.0000i	1.20605i
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	4.00000i	0.234484i
$292$	0	0
$293$	− 8.00000i	− 0.467365i	−0.972313	−	0.233682i	$-0.924922\pi$
$293$	− 8.00000i	− 0.467365i	0.972313	−	0.233682i	$-0.0750776\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	32.0000i	1.85061i
$300$	0	0
$301$	− 4.00000i	− 0.230556i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.0000i	0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$	10.0000i	0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	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
$317$	− 34.0000i	− 1.90963i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	− 34.0000i	− 1.90963i	0.297200	−	0.954815i	$-0.403947\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	32.0000	1.78607
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	− 20.0000i	− 1.10940i
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	12.0000i	0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$	12.0000i	0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	− 12.0000i	− 0.651751i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	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
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	2.00000i	0.103835i
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	32.0000i	1.63941i
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	0	0
$396$	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
$399$	−12.0000	−0.600751
$400$	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
$403$	16.0000i	0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	− 28.0000i	− 1.38114i
$412$	0	0
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	6.00000i	0.293119i	0.989202	+	0.146560i	$0.0468200\pi$
$419$	6.00000i	0.293119i	−0.989202	+	0.146560i	$0.953180\pi$
$420$	0	0
$421$	18.0000i	0.877266i	0.898666	+	0.438633i	$0.144537\pi$
$421$	18.0000i	0.877266i	−0.898666	+	0.438633i	$0.855463\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−10.0000	−0.485071
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	− 32.0000i	− 1.54497i
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.0000i	2.29615i
$438$	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
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	40.0000i	1.88353i
$452$	0	0
$453$	− 32.0000i	− 1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	8.00000i	0.373408i
$460$	0	0
$461$	28.0000i	1.30409i	0.758180	+	0.652045i	$0.226089\pi$
$461$	28.0000i	1.30409i	−0.758180	+	0.652045i	$0.773911\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	0	0
$469$	− 8.00000i	− 0.369406i
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	− 30.0000i	− 1.37649i
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	− 8.00000i	− 0.361035i	−0.983572	−	0.180517i	$-0.942223\pi$
$491$	− 8.00000i	− 0.361035i	0.983572	−	0.180517i	$-0.0577772\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 8.00000i	− 0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$	− 8.00000i	− 0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$	0	0
$501$	40.0000i	1.78707i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.00000i	0.266469i
$508$	0	0
$509$	20.0000i	0.886484i	0.896402	+	0.443242i	$0.146172\pi$
$509$	20.0000i	0.886484i	−0.896402	+	0.443242i	$0.853828\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−24.0000	−1.05348
$520$	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
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	0	0
$525$	− 10.0000i	− 0.436436i
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	10.0000i	0.433963i
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000i	0.172292i
$540$	0	0
$541$	30.0000i	1.28980i	0.764267	+	0.644900i	$0.223101\pi$
$541$	30.0000i	1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$	0	0
$543$	24.0000	1.02994
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	8.00000i	0.341432i
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	46.0000i	1.93867i	0.245745	+	0.969334i	$0.420967\pi$
$563$	46.0000i	1.93867i	−0.245745	+	0.969334i	$0.579033\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.0000	−0.461957
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−40.0000	−1.66812
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	− 28.0000i	− 1.16364i
$580$	0	0
$581$	2.00000i	0.0829740i
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	0	0
$589$	24.0000i	0.988903i
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	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
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	0	0
$605$	0	0
$606$	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
$609$	4.00000	0.162088
$610$	0	0
$611$	− 16.0000i	− 0.647291i
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	22.0000i	0.884255i	0.896952	+	0.442127i	$0.145776\pi$
$619$	22.0000i	0.884255i	−0.896952	+	0.442127i	$0.854224\pi$
$620$	0	0
$621$	32.0000i	1.28412i
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 48.0000i	− 1.91694i
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	− 34.0000i	− 1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$	− 34.0000i	− 1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	8.00000i	0.313545i
$652$	0	0
$653$	− 38.0000i	− 1.48705i	−0.668705	−	0.743527i	$-0.733151\pi$
$653$	− 38.0000i	− 1.48705i	0.668705	−	0.743527i	$-0.266849\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	0	0
$661$	− 24.0000i	− 0.933492i	−0.884391	−	0.466746i	$-0.845426\pi$
$661$	− 24.0000i	− 0.933492i	0.884391	−	0.466746i	$-0.154574\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	− 48.0000i	− 1.85579i
$670$	0	0
$671$	32.0000	1.23535
$672$	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
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−44.0000	−1.68608
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	40.0000	1.52610
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	− 22.0000i	− 0.836919i	−0.908235	−	0.418460i	$-0.862570\pi$
$691$	− 22.0000i	− 0.836919i	0.908235	−	0.418460i	$-0.137430\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	− 6.00000i	− 0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$	− 6.00000i	− 0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$	0	0
$703$	−60.0000	−2.26294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 26.0000i	− 0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$	− 26.0000i	− 0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 48.0000i	− 1.79259i
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	− 28.0000i	− 1.04133i
$724$	0	0
$725$	10.0000i	0.371391i
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	8.00000i	0.295487i	0.989026	+	0.147743i	$0.0472010\pi$
$733$	8.00000i	0.295487i	−0.989026	+	0.147743i	$0.952799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	4.00000i	0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$	4.00000i	0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	− 2.00000i	− 0.0731762i
$748$	0	0
$749$	16.0000i	0.584627i
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	−64.0000	−2.32305
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	10.0000i	0.362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	− 4.00000i	− 0.144056i
$772$	0	0
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	− 60.0000i	− 2.14972i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 46.0000i	− 1.63972i	−0.572562	−	0.819861i	$-0.694050\pi$
$787$	− 46.0000i	− 1.63972i	0.572562	−	0.819861i	$-0.305950\pi$
$788$	0	0
$789$	− 32.0000i	− 1.13923i
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.0000i	0.708436i	0.935163	+	0.354218i	$0.115253\pi$
$797$	20.0000i	0.708436i	−0.935163	+	0.354218i	$0.884747\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	− 34.0000i	− 1.19390i	−0.802278	−	0.596951i	$-0.796379\pi$
$811$	− 34.0000i	− 1.19390i	0.802278	−	0.596951i	$-0.203621\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	4.00000i	0.139771i
$820$	0	0
$821$	2.00000i	0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$	2.00000i	0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$	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
$825$	40.0000	1.39262
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	24.0000i	0.833554i	0.909009	+	0.416777i	$0.136840\pi$
$829$	24.0000i	0.833554i	−0.909009	+	0.416777i	$0.863160\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	80.0000i	2.74236i
$852$	0	0
$853$	36.0000i	1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$	36.0000i	1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$	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
$859$	26.0000i	0.887109i	0.896248	+	0.443554i	$0.146283\pi$
$859$	26.0000i	0.887109i	−0.896248	+	0.443554i	$0.853717\pi$
$860$	0	0
$861$	− 20.0000i	− 0.681598i
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.0000i	0.883006i
$868$	0	0
$869$	− 64.0000i	− 2.17105i
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	2.00000	0.0676897
$874$	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
$879$	−16.0000	−0.539667
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	− 44.0000i	− 1.47406i
$892$	0	0
$893$	− 24.0000i	− 0.803129i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	64.0000	2.13690
$898$	0	0
$899$	− 8.00000i	− 0.266815i
$900$	0	0
$901$	− 4.00000i	− 0.133259i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 50.0000i	− 1.64399i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	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
$931$	− 6.00000i	− 0.196642i
$932$	0	0
$933$	− 48.0000i	− 1.57145i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	− 52.0000i	− 1.69696i
$940$	0	0
$941$	− 40.0000i	− 1.30396i	−0.758235	−	0.651981i	$-0.773938\pi$
$941$	− 40.0000i	− 1.30396i	0.758235	−	0.651981i	$-0.226062\pi$
$942$	0	0
$943$	−80.0000	−2.60516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	− 24.0000i	− 0.779073i
$950$	0	0
$951$	−68.0000	−2.20505
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.0000i	0.517207i
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 16.0000i	− 0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	− 42.0000i	− 1.34784i	−0.738802	−	0.673922i	$-0.764608\pi$
$971$	− 42.0000i	− 1.34784i	0.738802	−	0.673922i	$-0.235392\pi$
$972$	0	0
$973$	10.0000i	0.320585i
$974$	0	0
$975$	−40.0000	−1.28103
$976$	0	0
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	0	0
$979$	− 72.0000i	− 2.30113i
$980$	0	0
$981$	− 10.0000i	− 0.319275i
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.00000i	− 0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$	− 8.00000i	− 0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.f.897.1		2
4.3	odd	2		1792.2.b.b.897.2		2
8.3	odd	2		1792.2.b.b.897.1		2
8.5	even	2	inner	1792.2.b.f.897.2		2
16.3	odd	4		448.2.a.b.1.1		1
16.5	even	4		224.2.a.a.1.1	✓	1
16.11	odd	4		224.2.a.b.1.1	yes	1
16.13	even	4		448.2.a.f.1.1		1
48.5	odd	4		2016.2.a.e.1.1		1
48.11	even	4		2016.2.a.g.1.1		1
48.29	odd	4		4032.2.a.p.1.1		1
48.35	even	4		4032.2.a.z.1.1		1
80.59	odd	4		5600.2.a.c.1.1		1
80.69	even	4		5600.2.a.t.1.1		1
112.5	odd	12		1568.2.i.c.1537.1		2
112.11	odd	12		1568.2.i.b.961.1		2
112.13	odd	4		3136.2.a.f.1.1		1
112.27	even	4		1568.2.a.b.1.1		1
112.37	even	12		1568.2.i.k.1537.1		2
112.53	even	12		1568.2.i.k.961.1		2
112.59	even	12		1568.2.i.j.961.1		2
112.69	odd	4		1568.2.a.h.1.1		1
112.75	even	12		1568.2.i.j.1537.1		2
112.83	even	4		3136.2.a.y.1.1		1
112.101	odd	12		1568.2.i.c.961.1		2
112.107	odd	12		1568.2.i.b.1537.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.a.1.1	✓	1	16.5	even	4
224.2.a.b.1.1	yes	1	16.11	odd	4
448.2.a.b.1.1		1	16.3	odd	4
448.2.a.f.1.1		1	16.13	even	4
1568.2.a.b.1.1		1	112.27	even	4
1568.2.a.h.1.1		1	112.69	odd	4
1568.2.i.b.961.1		2	112.11	odd	12
1568.2.i.b.1537.1		2	112.107	odd	12
1568.2.i.c.961.1		2	112.101	odd	12
1568.2.i.c.1537.1		2	112.5	odd	12
1568.2.i.j.961.1		2	112.59	even	12
1568.2.i.j.1537.1		2	112.75	even	12
1568.2.i.k.961.1		2	112.53	even	12
1568.2.i.k.1537.1		2	112.37	even	12
1792.2.b.b.897.1		2	8.3	odd	2
1792.2.b.b.897.2		2	4.3	odd	2
1792.2.b.f.897.1		2	1.1	even	1	trivial
1792.2.b.f.897.2		2	8.5	even	2	inner
2016.2.a.e.1.1		1	48.5	odd	4
2016.2.a.g.1.1		1	48.11	even	4
3136.2.a.f.1.1		1	112.13	odd	4
3136.2.a.y.1.1		1	112.83	even	4
4032.2.a.p.1.1		1	48.29	odd	4
4032.2.a.z.1.1		1	48.35	even	4
5600.2.a.c.1.1		1	80.59	odd	4
5600.2.a.t.1.1		1	80.69	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -4.00000i q^{13} -2.00000 q^{17} -6.00000i q^{19} -2.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -4.00000i q^{27} +2.00000i q^{29} -4.00000 q^{31} +8.00000 q^{33} -10.0000i q^{37} -8.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{47} +1.00000 q^{49} +4.00000i q^{51} +2.00000i q^{53} -12.0000 q^{57} -10.0000i q^{59} -8.00000i q^{61} -1.00000 q^{63} -8.00000i q^{67} +16.0000i q^{69} +6.00000 q^{73} -10.0000i q^{75} +4.00000i q^{77} -16.0000 q^{79} -11.0000 q^{81} +2.00000i q^{83} +4.00000 q^{87} -18.0000 q^{89} -4.00000i q^{91} +8.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} - 4 q^{17} - 16 q^{23} + 10 q^{25} - 8 q^{31} + 16 q^{33} - 16 q^{39} + 20 q^{41} + 8 q^{47} + 2 q^{49} - 24 q^{57} - 2 q^{63} + 12 q^{73} - 32 q^{79} - 22 q^{81} + 8 q^{87} - 36 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 - 4 * q^17 - 16 * q^23 + 10 * q^25 - 8 * q^31 + 16 * q^33 - 16 * q^39 + 20 * q^41 + 8 * q^47 + 2 * q^49 - 24 * q^57 - 2 * q^63 + 12 * q^73 - 32 * q^79 - 22 * q^81 + 8 * q^87 - 36 * q^89 - 4 * q^97

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