LMFDB - Embedded newform 4800.2.k.q.2401.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4800,2,Mod(2401,4800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4800.2401"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,8,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(57)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$38.3281929702$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 960)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2401.1
Root			$1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	4800.2401
Dual form			4800.2.k.q.2401.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.37780 q^{7} -1.00000 q^{9} +5.58258i q^{11} +4.37780i q^{13} -5.58258 q^{17} +4.00000i q^{19} +4.37780i q^{21} +1.00000i q^{27} -2.55040i q^{29} -5.29150 q^{31} +5.58258 q^{33} -2.55040i q^{37} +4.37780 q^{39} +6.00000 q^{41} -11.1652i q^{43} -6.92820 q^{47} +12.1652 q^{49} +5.58258i q^{51} -7.84190i q^{53} +4.00000 q^{57} -1.58258i q^{59} -10.5830i q^{61} +4.37780 q^{63} +3.16515i q^{67} +6.92820 q^{71} +12.0000 q^{73} -24.4394i q^{77} -5.29150 q^{79} +1.00000 q^{81} +7.16515i q^{83} -2.55040 q^{87} +2.00000 q^{89} -19.1652i q^{91} +5.29150i q^{93} +11.1652 q^{97} -5.58258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{17} + 8 q^{33} + 48 q^{41} + 24 q^{49} + 32 q^{57} + 96 q^{73} + 8 q^{81} + 16 q^{89} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^17 + 8 * q^33 + 48 * q^41 + 24 * q^49 + 32 * q^57 + 96 * q^73 + 8 * q^81 + 16 * q^89 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.37780	−1.65465	−0.827327	−	0.561721i	$-0.810140\pi$
$7$	−4.37780	−1.65465	−0.827327	+	0.561721i	$0.810140\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.58258i	1.68321i	0.540094	+	0.841605i	$0.318389\pi$
$11$	5.58258i	1.68321i	−0.540094	+	0.841605i	$0.681611\pi$
$12$	0	0
$13$	4.37780i	1.21418i	0.794632	+	0.607092i	$0.207664\pi$
$13$	4.37780i	1.21418i	−0.794632	+	0.607092i	$0.792336\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.58258	−1.35397	−0.676987	−	0.735995i	$-0.736715\pi$
$17$	−5.58258	−1.35397	−0.676987	+	0.735995i	$0.736715\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	4.37780i	0.955315i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.55040i	− 0.473598i	−0.971559	−	0.236799i	$-0.923902\pi$
$29$	− 2.55040i	− 0.473598i	0.971559	−	0.236799i	$-0.0760982\pi$
$30$	0	0
$31$	−5.29150	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$32$	0	0
$33$	5.58258	0.971802
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.55040i	− 0.419283i	−0.977778	−	0.209642i	$-0.932770\pi$
$37$	− 2.55040i	− 0.419283i	0.977778	−	0.209642i	$-0.0672297\pi$
$38$	0	0
$39$	4.37780	0.701009
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 11.1652i	− 1.70267i	−0.524623	−	0.851335i	$-0.675794\pi$
$43$	− 11.1652i	− 1.70267i	0.524623	−	0.851335i	$-0.324206\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	12.1652	1.73788
$50$	0	0
$51$	5.58258i	0.781717i
$52$	0	0
$53$	− 7.84190i	− 1.07717i	−0.842572	−	0.538584i	$-0.818959\pi$
$53$	− 7.84190i	− 1.07717i	0.842572	−	0.538584i	$-0.181041\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	− 1.58258i	− 0.206034i	−0.994680	−	0.103017i	$-0.967150\pi$
$59$	− 1.58258i	− 0.206034i	0.994680	−	0.103017i	$-0.0328496\pi$
$60$	0	0
$61$	− 10.5830i	− 1.35501i	−0.735516	−	0.677507i	$-0.763060\pi$
$61$	− 10.5830i	− 1.35501i	0.735516	−	0.677507i	$-0.236940\pi$
$62$	0	0
$63$	4.37780	0.551551
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.16515i	0.386685i	0.981131	+	0.193342i	$0.0619328\pi$
$67$	3.16515i	0.386685i	−0.981131	+	0.193342i	$0.938067\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24.4394i	− 2.78513i
$78$	0	0
$79$	−5.29150	−0.595341	−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150	−0.595341	−0.297670	+	0.954669i	$0.596210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.16515i	0.786478i	0.919436	+	0.393239i	$0.128645\pi$
$83$	7.16515i	0.786478i	−0.919436	+	0.393239i	$0.871355\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.55040	−0.273432
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	− 19.1652i	− 2.00905i
$92$	0	0
$93$	5.29150i	0.548703i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.1652	1.13365	0.566825	−	0.823838i	$-0.308172\pi$
$97$	11.1652	1.13365	0.566825	+	0.823838i	$0.308172\pi$
$98$	0	0
$99$	− 5.58258i	− 0.561070i
$100$	0	0
$101$	− 0.723000i	− 0.0719412i	−0.999353	−	0.0359706i	$-0.988548\pi$
$101$	− 0.723000i	− 0.0719412i	0.999353	−	0.0359706i	$-0.0114523\pi$
$102$	0	0
$103$	−0.723000	−0.0712393	−0.0356197	−	0.999365i	$-0.511340\pi$
$103$	−0.723000	−0.0712393	−0.0356197	+	0.999365i	$0.511340\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	1.82740i	0.175033i	0.996163	+	0.0875166i	$0.0278931\pi$
$109$	1.82740i	0.175033i	−0.996163	+	0.0875166i	$0.972107\pi$
$110$	0	0
$111$	−2.55040	−0.242073
$112$	0	0
$113$	−13.5826	−1.27774	−0.638871	−	0.769314i	$-0.720598\pi$
$113$	−13.5826	−1.27774	−0.638871	+	0.769314i	$0.720598\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.37780i	− 0.404728i
$118$	0	0
$119$	24.4394	2.24036
$120$	0	0
$121$	−20.1652	−1.83320
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.47860	−0.841090	−0.420545	−	0.907272i	$-0.638161\pi$
$127$	−9.47860	−0.841090	−0.420545	+	0.907272i	$0.638161\pi$
$128$	0	0
$129$	−11.1652	−0.983037
$130$	0	0
$131$	6.41742i	0.560693i	0.959899	+	0.280346i	$0.0904494\pi$
$131$	6.41742i	0.560693i	−0.959899	+	0.280346i	$0.909551\pi$
$132$	0	0
$133$	− 17.5112i	− 1.51841i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4174	0.890021	0.445010	−	0.895525i	$-0.353200\pi$
$137$	10.4174	0.890021	0.445010	+	0.895525i	$0.353200\pi$
$138$	0	0
$139$	7.16515i	0.607740i	0.952713	+	0.303870i	$0.0982789\pi$
$139$	7.16515i	0.607740i	−0.952713	+	0.303870i	$0.901721\pi$
$140$	0	0
$141$	6.92820i	0.583460i
$142$	0	0
$143$	−24.4394	−2.04373
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 12.1652i	− 1.00336i
$148$	0	0
$149$	− 14.9608i	− 1.22564i	−0.790224	−	0.612819i	$-0.790036\pi$
$149$	− 14.9608i	− 1.22564i	0.790224	−	0.612819i	$-0.209964\pi$
$150$	0	0
$151$	15.8745	1.29185	0.645925	−	0.763401i	$-0.276472\pi$
$151$	15.8745	1.29185	0.645925	+	0.763401i	$0.276472\pi$
$152$	0	0
$153$	5.58258	0.451324
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.4068i	− 1.30941i	−0.755886	−	0.654703i	$-0.772794\pi$
$157$	− 16.4068i	− 1.30941i	0.755886	−	0.654703i	$-0.227206\pi$
$158$	0	0
$159$	−7.84190	−0.621903
$160$	0	0
$161$	0	0
$162$	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
$167$	−12.0290	−0.930832	−0.465416	−	0.885092i	$-0.654095\pi$
$167$	−12.0290	−0.930832	−0.465416	+	0.885092i	$0.654095\pi$
$168$	0	0
$169$	−6.16515	−0.474242
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 6.01450i	− 0.457274i	−0.973512	−	0.228637i	$-0.926573\pi$
$173$	− 6.01450i	− 0.457274i	0.973512	−	0.228637i	$-0.0734269\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.58258	−0.118954
$178$	0	0
$179$	1.58258i	0.118287i	0.998249	+	0.0591436i	$0.0188370\pi$
$179$	1.58258i	0.118287i	−0.998249	+	0.0591436i	$0.981163\pi$
$180$	0	0
$181$	1.82740i	0.135830i	0.997691	+	0.0679148i	$0.0216346\pi$
$181$	1.82740i	0.135830i	−0.997691	+	0.0679148i	$0.978365\pi$
$182$	0	0
$183$	−10.5830	−0.782318
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 31.1652i	− 2.27902i
$188$	0	0
$189$	− 4.37780i	− 0.318438i
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	−15.1652	−1.09161	−0.545806	−	0.837912i	$-0.683776\pi$
$193$	−15.1652	−1.09161	−0.545806	+	0.837912i	$0.683776\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0797i	1.57311i	0.617518	+	0.786557i	$0.288138\pi$
$197$	22.0797i	1.57311i	−0.617518	+	0.786557i	$0.711862\pi$
$198$	0	0
$199$	−12.2197	−0.866232	−0.433116	−	0.901338i	$-0.642586\pi$
$199$	−12.2197	−0.866232	−0.433116	+	0.901338i	$0.642586\pi$
$200$	0	0
$201$	3.16515	0.223253
$202$	0	0
$203$	11.1652i	0.783640i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−22.3303	−1.54462
$210$	0	0
$211$	− 7.16515i	− 0.493269i	−0.969109	−	0.246635i	$-0.920675\pi$
$211$	− 7.16515i	− 0.493269i	0.969109	−	0.246635i	$-0.0793248\pi$
$212$	0	0
$213$	− 6.92820i	− 0.474713i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	23.1652	1.57255
$218$	0	0
$219$	− 12.0000i	− 0.810885i
$220$	0	0
$221$	− 24.4394i	− 1.64397i
$222$	0	0
$223$	4.37780	0.293159	0.146580	−	0.989199i	$-0.453174\pi$
$223$	4.37780	0.293159	0.146580	+	0.989199i	$0.453174\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 15.1652i	− 1.00655i	−0.864127	−	0.503273i	$-0.832129\pi$
$227$	− 15.1652i	− 1.00655i	0.864127	−	0.503273i	$-0.167871\pi$
$228$	0	0
$229$	12.0290i	0.794899i	0.917624	+	0.397450i	$0.130105\pi$
$229$	12.0290i	0.794899i	−0.917624	+	0.397450i	$0.869895\pi$
$230$	0	0
$231$	−24.4394	−1.60800
$232$	0	0
$233$	10.4174	0.682468	0.341234	−	0.939978i	$-0.389155\pi$
$233$	10.4174	0.682468	0.341234	+	0.939978i	$0.389155\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.29150i	0.343720i
$238$	0	0
$239$	−21.1660	−1.36912	−0.684558	−	0.728959i	$-0.740005\pi$
$239$	−21.1660	−1.36912	−0.684558	+	0.728959i	$0.740005\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−17.5112	−1.11421
$248$	0	0
$249$	7.16515	0.454073
$250$	0	0
$251$	2.41742i	0.152586i	0.997085	+	0.0762932i	$0.0243085\pi$
$251$	2.41742i	0.152586i	−0.997085	+	0.0762932i	$0.975691\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.7477	1.54372	0.771860	−	0.635792i	$-0.219326\pi$
$257$	24.7477	1.54372	0.771860	+	0.635792i	$0.219326\pi$
$258$	0	0
$259$	11.1652i	0.693769i
$260$	0	0
$261$	2.55040i	0.157866i
$262$	0	0
$263$	−1.82740	−0.112682	−0.0563412	−	0.998412i	$-0.517943\pi$
$263$	−1.82740	−0.112682	−0.0563412	+	0.998412i	$0.517943\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	0	0
$269$	6.20520i	0.378338i	0.981945	+	0.189169i	$0.0605794\pi$
$269$	6.20520i	0.378338i	−0.981945	+	0.189169i	$0.939421\pi$
$270$	0	0
$271$	−15.4931	−0.941139	−0.470570	−	0.882363i	$-0.655952\pi$
$271$	−15.4931	−0.941139	−0.470570	+	0.882363i	$0.655952\pi$
$272$	0	0
$273$	−19.1652	−1.15993
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 9.47860i	− 0.569514i	−0.958600	−	0.284757i	$-0.908087\pi$
$277$	− 9.47860i	− 0.569514i	0.958600	−	0.284757i	$-0.0919129\pi$
$278$	0	0
$279$	5.29150	0.316794
$280$	0	0
$281$	32.3303	1.92866	0.964332	−	0.264695i	$-0.0852714\pi$
$281$	32.3303	1.92866	0.964332	+	0.264695i	$0.0852714\pi$
$282$	0	0
$283$	11.1652i	0.663699i	0.943332	+	0.331850i	$0.107673\pi$
$283$	11.1652i	0.663699i	−0.943332	+	0.331850i	$0.892327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.2668	−1.55048
$288$	0	0
$289$	14.1652	0.833244
$290$	0	0
$291$	− 11.1652i	− 0.654513i
$292$	0	0
$293$	0.913701i	0.0533790i	0.999644	+	0.0266895i	$0.00849653\pi$
$293$	0.913701i	0.0533790i	−0.999644	+	0.0266895i	$0.991503\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.58258	−0.323934
$298$	0	0
$299$	0	0
$300$	0	0
$301$	48.8788i	2.81733i
$302$	0	0
$303$	−0.723000	−0.0415353
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0.723000i	0.0411300i
$310$	0	0
$311$	−3.65480	−0.207245	−0.103622	−	0.994617i	$-0.533043\pi$
$311$	−3.65480	−0.207245	−0.103622	+	0.994617i	$0.533043\pi$
$312$	0	0
$313$	0.834849	0.0471884	0.0235942	−	0.999722i	$-0.492489\pi$
$313$	0.834849	0.0471884	0.0235942	+	0.999722i	$0.492489\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 34.1087i	− 1.91574i	−0.287208	−	0.957868i	$-0.592727\pi$
$317$	− 34.1087i	− 1.91574i	0.287208	−	0.957868i	$-0.407273\pi$
$318$	0	0
$319$	14.2378	0.797164
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 22.3303i	− 1.24249i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.82740	0.101056
$328$	0	0
$329$	30.3303	1.67216
$330$	0	0
$331$	10.3303i	0.567805i	0.958853	+	0.283902i	$0.0916292\pi$
$331$	10.3303i	0.567805i	−0.958853	+	0.283902i	$0.908371\pi$
$332$	0	0
$333$	2.55040i	0.139761i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.3303	−1.43430	−0.717151	−	0.696917i	$-0.754554\pi$
$337$	−26.3303	−1.43430	−0.717151	+	0.696917i	$0.754554\pi$
$338$	0	0
$339$	13.5826i	0.737704i
$340$	0	0
$341$	− 29.5402i	− 1.59969i
$342$	0	0
$343$	−22.6120	−1.22093
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 0.834849i	− 0.0448170i	−0.999749	−	0.0224085i	$-0.992867\pi$
$347$	− 0.834849i	− 0.0448170i	0.999749	−	0.0224085i	$-0.00713345\pi$
$348$	0	0
$349$	24.4394i	1.30821i	0.756403	+	0.654106i	$0.226955\pi$
$349$	24.4394i	1.30821i	−0.756403	+	0.654106i	$0.773045\pi$
$350$	0	0
$351$	−4.37780	−0.233670
$352$	0	0
$353$	11.9129	0.634059	0.317029	−	0.948416i	$-0.397315\pi$
$353$	11.9129	0.634059	0.317029	+	0.948416i	$0.397315\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.4394i	− 1.29347i
$358$	0	0
$359$	−6.92820	−0.365657	−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820	−0.365657	−0.182828	+	0.983145i	$0.558525\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	20.1652i	1.05840i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.6874	0.610078	0.305039	−	0.952340i	$-0.401331\pi$
$367$	11.6874	0.610078	0.305039	+	0.952340i	$0.401331\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	34.3303i	1.78234i
$372$	0	0
$373$	− 11.3060i	− 0.585403i	−0.956204	−	0.292701i	$-0.905446\pi$
$373$	− 11.3060i	− 0.585403i	0.956204	−	0.292701i	$-0.0945542\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.1652	0.575035
$378$	0	0
$379$	− 23.1652i	− 1.18991i	−0.803758	−	0.594957i	$-0.797169\pi$
$379$	− 23.1652i	− 1.18991i	0.803758	−	0.594957i	$-0.202831\pi$
$380$	0	0
$381$	9.47860i	0.485604i
$382$	0	0
$383$	29.9216	1.52892	0.764462	−	0.644669i	$-0.223005\pi$
$383$	29.9216	1.52892	0.764462	+	0.644669i	$0.223005\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.1652i	0.567557i
$388$	0	0
$389$	8.03260i	0.407269i	0.979047	+	0.203635i	$0.0652755\pi$
$389$	8.03260i	0.407269i	−0.979047	+	0.203635i	$0.934725\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6.41742	0.323716
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.55040i	0.128001i	0.997950	+	0.0640005i	$0.0203859\pi$
$397$	2.55040i	0.128001i	−0.997950	+	0.0640005i	$0.979614\pi$
$398$	0	0
$399$	−17.5112	−0.876657
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	− 23.1652i	− 1.15394i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.2378	0.705742
$408$	0	0
$409$	−37.1652	−1.83770	−0.918849	−	0.394609i	$-0.870880\pi$
$409$	−37.1652	−1.83770	−0.918849	+	0.394609i	$0.870880\pi$
$410$	0	0
$411$	− 10.4174i	− 0.513854i
$412$	0	0
$413$	6.92820i	0.340915i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.16515	0.350879
$418$	0	0
$419$	− 16.7477i	− 0.818180i	−0.912494	−	0.409090i	$-0.865846\pi$
$419$	− 16.7477i	− 0.818180i	0.912494	−	0.409090i	$-0.134154\pi$
$420$	0	0
$421$	29.5402i	1.43970i	0.694129	+	0.719851i	$0.255790\pi$
$421$	29.5402i	1.43970i	−0.694129	+	0.719851i	$0.744210\pi$
$422$	0	0
$423$	6.92820	0.336861
$424$	0	0
$425$	0	0
$426$	0	0
$427$	46.3303i	2.24208i
$428$	0	0
$429$	24.4394i	1.17995i
$430$	0	0
$431$	38.2958	1.84464	0.922322	−	0.386421i	$-0.126289\pi$
$431$	38.2958	1.84464	0.922322	+	0.386421i	$0.126289\pi$
$432$	0	0
$433$	−1.66970	−0.0802405	−0.0401203	−	0.999195i	$-0.512774\pi$
$433$	−1.66970	−0.0802405	−0.0401203	+	0.999195i	$0.512774\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−29.7309	−1.41898	−0.709490	−	0.704716i	$-0.751074\pi$
$439$	−29.7309	−1.41898	−0.709490	+	0.704716i	$0.751074\pi$
$440$	0	0
$441$	−12.1652	−0.579293
$442$	0	0
$443$	− 31.1652i	− 1.48070i	−0.672221	−	0.740351i	$-0.734660\pi$
$443$	− 31.1652i	− 1.48070i	0.672221	−	0.740351i	$-0.265340\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.9608	−0.707622
$448$	0	0
$449$	28.3303	1.33699	0.668495	−	0.743717i	$-0.266939\pi$
$449$	28.3303	1.33699	0.668495	+	0.743717i	$0.266939\pi$
$450$	0	0
$451$	33.4955i	1.57724i
$452$	0	0
$453$	− 15.8745i	− 0.745849i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.83485	0.226165	0.113082	−	0.993586i	$-0.463928\pi$
$457$	4.83485	0.226165	0.113082	+	0.993586i	$0.463928\pi$
$458$	0	0
$459$	− 5.58258i	− 0.260572i
$460$	0	0
$461$	1.10440i	0.0514371i	0.999669	+	0.0257185i	$0.00818737\pi$
$461$	1.10440i	0.0514371i	−0.999669	+	0.0257185i	$0.991813\pi$
$462$	0	0
$463$	−29.1986	−1.35697	−0.678487	−	0.734612i	$-0.737364\pi$
$463$	−29.1986	−1.35697	−0.678487	+	0.734612i	$0.737364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.834849i	− 0.0386322i	−0.999813	−	0.0193161i	$-0.993851\pi$
$467$	− 0.834849i	− 0.0386322i	0.999813	−	0.0193161i	$-0.00614889\pi$
$468$	0	0
$469$	− 13.8564i	− 0.639829i
$470$	0	0
$471$	−16.4068	−0.755986
$472$	0	0
$473$	62.3303	2.86595
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.84190i	0.359056i
$478$	0	0
$479$	31.7490	1.45065	0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490	1.45065	0.725325	+	0.688407i	$0.241690\pi$
$480$	0	0
$481$	11.1652	0.509087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.7882	−0.760746	−0.380373	−	0.924833i	$-0.624204\pi$
$487$	−16.7882	−0.760746	−0.380373	+	0.924833i	$0.624204\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 12.7477i	− 0.575297i	−0.957736	−	0.287648i	$-0.907127\pi$
$491$	− 12.7477i	− 0.575297i	0.957736	−	0.287648i	$-0.0928735\pi$
$492$	0	0
$493$	14.2378i	0.641239i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−30.3303	−1.36050
$498$	0	0
$499$	− 18.3303i	− 0.820577i	−0.911956	−	0.410289i	$-0.865428\pi$
$499$	− 18.3303i	− 0.820577i	0.911956	−	0.410289i	$-0.134572\pi$
$500$	0	0
$501$	12.0290i	0.537416i
$502$	0	0
$503$	28.0942	1.25266	0.626330	−	0.779558i	$-0.284556\pi$
$503$	28.0942	1.25266	0.626330	+	0.779558i	$0.284556\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.16515i	0.273804i
$508$	0	0
$509$	− 39.4002i	− 1.74638i	−0.487376	−	0.873192i	$-0.662046\pi$
$509$	− 39.4002i	− 1.74638i	0.487376	−	0.873192i	$-0.337954\pi$
$510$	0	0
$511$	−52.5336	−2.32395
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 38.6772i	− 1.70102i
$518$	0	0
$519$	−6.01450	−0.264007
$520$	0	0
$521$	−34.6606	−1.51851	−0.759254	−	0.650794i	$-0.774436\pi$
$521$	−34.6606	−1.51851	−0.759254	+	0.650794i	$0.774436\pi$
$522$	0	0
$523$	18.3303i	0.801528i	0.916181	+	0.400764i	$0.131255\pi$
$523$	18.3303i	0.801528i	−0.916181	+	0.400764i	$0.868745\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.5402	1.28679
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	1.58258i	0.0686779i
$532$	0	0
$533$	26.2668i	1.13774i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.58258	0.0682932
$538$	0	0
$539$	67.9129i	2.92521i
$540$	0	0
$541$	− 36.8498i	− 1.58430i	−0.610328	−	0.792149i	$-0.708962\pi$
$541$	− 36.8498i	− 1.58430i	0.610328	−	0.792149i	$-0.291038\pi$
$542$	0	0
$543$	1.82740	0.0784213
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 25.4955i	− 1.09011i	−0.838401	−	0.545053i	$-0.816509\pi$
$547$	− 25.4955i	− 1.09011i	0.838401	−	0.545053i	$-0.183491\pi$
$548$	0	0
$549$	10.5830i	0.451672i
$550$	0	0
$551$	10.2016	0.434603
$552$	0	0
$553$	23.1652	0.985082
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.18710i	− 0.177413i	−0.996058	−	0.0887066i	$-0.971727\pi$
$557$	− 4.18710i	− 0.177413i	0.996058	−	0.0887066i	$-0.0282733\pi$
$558$	0	0
$559$	48.8788	2.06735
$560$	0	0
$561$	−31.1652	−1.31579
$562$	0	0
$563$	− 29.4955i	− 1.24309i	−0.783380	−	0.621543i	$-0.786506\pi$
$563$	− 29.4955i	− 1.24309i	0.783380	−	0.621543i	$-0.213494\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.37780	−0.183850
$568$	0	0
$569$	−36.3303	−1.52305	−0.761523	−	0.648138i	$-0.775548\pi$
$569$	−36.3303	−1.52305	−0.761523	+	0.648138i	$0.775548\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 13.8564i	− 0.578860i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.1652	−0.631334	−0.315667	−	0.948870i	$-0.602228\pi$
$577$	−15.1652	−0.631334	−0.315667	+	0.948870i	$0.602228\pi$
$578$	0	0
$579$	15.1652i	0.630242i
$580$	0	0
$581$	− 31.3676i	− 1.30135i
$582$	0	0
$583$	43.7780	1.81310
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.66970i	0.234013i	0.993131	+	0.117007i	$0.0373299\pi$
$587$	5.66970i	0.234013i	−0.993131	+	0.117007i	$0.962670\pi$
$588$	0	0
$589$	− 21.1660i	− 0.872130i
$590$	0	0
$591$	22.0797	0.908238
$592$	0	0
$593$	−35.9129	−1.47477	−0.737383	−	0.675475i	$-0.763938\pi$
$593$	−35.9129	−1.47477	−0.737383	+	0.675475i	$0.763938\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.2197i	0.500119i
$598$	0	0
$599$	−14.2378	−0.581741	−0.290871	−	0.956762i	$-0.593945\pi$
$599$	−14.2378	−0.581741	−0.290871	+	0.956762i	$0.593945\pi$
$600$	0	0
$601$	13.1652	0.537018	0.268509	−	0.963277i	$-0.413469\pi$
$601$	13.1652	0.537018	0.268509	+	0.963277i	$0.413469\pi$
$602$	0	0
$603$	− 3.16515i	− 0.128895i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.7454	−1.45086	−0.725431	−	0.688295i	$-0.758359\pi$
$607$	−35.7454	−1.45086	−0.725431	+	0.688295i	$0.758359\pi$
$608$	0	0
$609$	11.1652	0.452435
$610$	0	0
$611$	− 30.3303i	− 1.22703i
$612$	0	0
$613$	32.4720i	1.31153i	0.754964	+	0.655766i	$0.227654\pi$
$613$	32.4720i	1.31153i	−0.754964	+	0.655766i	$0.772346\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.5826	−0.546814	−0.273407	−	0.961898i	$-0.588151\pi$
$617$	−13.5826	−0.546814	−0.273407	+	0.961898i	$0.588151\pi$
$618$	0	0
$619$	− 45.4955i	− 1.82862i	−0.405019	−	0.914308i	$-0.632735\pi$
$619$	− 45.4955i	− 1.82862i	0.405019	−	0.914308i	$-0.367265\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.75560	−0.350786
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22.3303i	0.891786i
$628$	0	0
$629$	14.2378i	0.567699i
$630$	0	0
$631$	−12.2197	−0.486459	−0.243229	−	0.969969i	$-0.578207\pi$
$631$	−12.2197	−0.486459	−0.243229	+	0.969969i	$0.578207\pi$
$632$	0	0
$633$	−7.16515	−0.284789
$634$	0	0
$635$	0	0
$636$	0	0
$637$	53.2566i	2.11010i
$638$	0	0
$639$	−6.92820	−0.274075
$640$	0	0
$641$	−4.33030	−0.171037	−0.0855183	−	0.996337i	$-0.527255\pi$
$641$	−4.33030	−0.171037	−0.0855183	+	0.996337i	$0.527255\pi$
$642$	0	0
$643$	18.3303i	0.722877i	0.932396	+	0.361438i	$0.117714\pi$
$643$	18.3303i	0.722877i	−0.932396	+	0.361438i	$0.882286\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.8926	−0.703431	−0.351716	−	0.936107i	$-0.614402\pi$
$647$	−17.8926	−0.703431	−0.351716	+	0.936107i	$0.614402\pi$
$648$	0	0
$649$	8.83485	0.346798
$650$	0	0
$651$	− 23.1652i	− 0.907914i
$652$	0	0
$653$	− 22.0797i	− 0.864046i	−0.901863	−	0.432023i	$-0.857800\pi$
$653$	− 22.0797i	− 0.864046i	0.901863	−	0.432023i	$-0.142200\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.0000	−0.468165
$658$	0	0
$659$	− 25.5826i	− 0.996556i	−0.867017	−	0.498278i	$-0.833966\pi$
$659$	− 25.5826i	− 0.996556i	0.867017	−	0.498278i	$-0.166034\pi$
$660$	0	0
$661$	− 17.8926i	− 0.695942i	−0.937505	−	0.347971i	$-0.886871\pi$
$661$	− 17.8926i	− 0.695942i	0.937505	−	0.347971i	$-0.113129\pi$
$662$	0	0
$663$	−24.4394	−0.949148
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 4.37780i	− 0.169256i
$670$	0	0
$671$	59.0804	2.28077
$672$	0	0
$673$	−40.0000	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6265i	1.10021i	0.835097	+	0.550103i	$0.185412\pi$
$677$	28.6265i	1.10021i	−0.835097	+	0.550103i	$0.814588\pi$
$678$	0	0
$679$	−48.8788	−1.87580
$680$	0	0
$681$	−15.1652	−0.581130
$682$	0	0
$683$	− 5.49545i	− 0.210278i	−0.994458	−	0.105139i	$-0.966471\pi$
$683$	− 5.49545i	− 0.210278i	0.994458	−	0.105139i	$-0.0335287\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.0290	0.458935
$688$	0	0
$689$	34.3303	1.30788
$690$	0	0
$691$	− 40.6606i	− 1.54680i	−0.633917	−	0.773401i	$-0.718554\pi$
$691$	− 40.6606i	− 1.54680i	0.633917	−	0.773401i	$-0.281446\pi$
$692$	0	0
$693$	24.4394i	0.928376i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−33.4955	−1.26873
$698$	0	0
$699$	− 10.4174i	− 0.394023i
$700$	0	0
$701$	− 45.9470i	− 1.73540i	−0.497093	−	0.867698i	$-0.665599\pi$
$701$	− 45.9470i	− 1.73540i	0.497093	−	0.867698i	$-0.334401\pi$
$702$	0	0
$703$	10.2016	0.384761
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.16515i	0.119038i
$708$	0	0
$709$	− 15.6838i	− 0.589018i	−0.955649	−	0.294509i	$-0.904844\pi$
$709$	− 15.6838i	− 0.589018i	0.955649	−	0.294509i	$-0.0951561\pi$
$710$	0	0
$711$	5.29150	0.198447
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.1660i	0.790459i
$718$	0	0
$719$	30.9862	1.15559	0.577795	−	0.816182i	$-0.303913\pi$
$719$	30.9862	1.15559	0.577795	+	0.816182i	$0.303913\pi$
$720$	0	0
$721$	3.16515	0.117876
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	49.6018	1.83963	0.919815	−	0.392353i	$-0.128339\pi$
$727$	49.6018	1.83963	0.919815	+	0.392353i	$0.128339\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	62.3303i	2.30537i
$732$	0	0
$733$	− 29.1986i	− 1.07848i	−0.842154	−	0.539238i	$-0.818712\pi$
$733$	− 29.1986i	− 1.07848i	0.842154	−	0.539238i	$-0.181288\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.6697	−0.650872
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	17.5112i	0.643290i
$742$	0	0
$743$	−47.4328	−1.74014	−0.870071	−	0.492927i	$-0.835927\pi$
$743$	−47.4328	−1.74014	−0.870071	+	0.492927i	$0.835927\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.16515i	− 0.262159i
$748$	0	0
$749$	17.5112i	0.639846i
$750$	0	0
$751$	−33.0043	−1.20434	−0.602172	−	0.798366i	$-0.705698\pi$
$751$	−33.0043	−1.20434	−0.602172	+	0.798366i	$0.705698\pi$
$752$	0	0
$753$	2.41742	0.0880958
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.2342i	0.662734i	0.943502	+	0.331367i	$0.107510\pi$
$757$	18.2342i	0.662734i	−0.943502	+	0.331367i	$0.892490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.92820	0.250163
$768$	0	0
$769$	−23.4955	−0.847268	−0.423634	−	0.905834i	$-0.639246\pi$
$769$	−23.4955	−0.847268	−0.423634	+	0.905834i	$0.639246\pi$
$770$	0	0
$771$	− 24.7477i	− 0.891268i
$772$	0	0
$773$	30.4539i	1.09535i	0.836691	+	0.547676i	$0.184487\pi$
$773$	30.4539i	1.09535i	−0.836691	+	0.547676i	$0.815513\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.1652	0.400548
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	38.6772i	1.38398i
$782$	0	0
$783$	2.55040	0.0911439
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	0	0
$789$	1.82740i	0.0650572i
$790$	0	0
$791$	59.4618	2.11422
$792$	0	0
$793$	46.3303	1.64524
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.0659i	1.87969i	0.341599	+	0.939846i	$0.389031\pi$
$797$	53.0659i	1.87969i	−0.341599	+	0.939846i	$0.610969\pi$
$798$	0	0
$799$	38.6772	1.36830
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	66.9909i	2.36406i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.20520	0.218433
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	0.834849i	0.0293155i	0.999893	+	0.0146577i	$0.00466587\pi$
$811$	0.834849i	0.0293155i	−0.999893	+	0.0146577i	$0.995334\pi$
$812$	0	0
$813$	15.4931i	0.543367i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	44.6606	1.56248
$818$	0	0
$819$	19.1652i	0.669685i
$820$	0	0
$821$	6.20520i	0.216563i	0.994120	+	0.108282i	$0.0345348\pi$
$821$	6.20520i	0.216563i	−0.994120	+	0.108282i	$0.965465\pi$
$822$	0	0
$823$	−9.47860	−0.330403	−0.165202	−	0.986260i	$-0.552828\pi$
$823$	−9.47860	−0.330403	−0.165202	+	0.986260i	$0.552828\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	− 36.8498i	− 1.27985i	−0.768439	−	0.639924i	$-0.778966\pi$
$829$	− 36.8498i	− 1.27985i	0.768439	−	0.639924i	$-0.221034\pi$
$830$	0	0
$831$	−9.47860	−0.328809
$832$	0	0
$833$	−67.9129	−2.35304
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.29150i	− 0.182901i
$838$	0	0
$839$	55.8070	1.92667	0.963336	−	0.268297i	$-0.0864608\pi$
$839$	55.8070	1.92667	0.963336	+	0.268297i	$0.0864608\pi$
$840$	0	0
$841$	22.4955	0.775705
$842$	0	0
$843$	− 32.3303i	− 1.11351i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	88.2790	3.03330
$848$	0	0
$849$	11.1652	0.383187
$850$	0	0
$851$	0	0
$852$	0	0
$853$	7.65120i	0.261972i	0.991384	+	0.130986i	$0.0418143\pi$
$853$	7.65120i	0.261972i	−0.991384	+	0.130986i	$0.958186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.5826	1.01052	0.505261	−	0.862967i	$-0.331396\pi$
$857$	29.5826	1.01052	0.505261	+	0.862967i	$0.331396\pi$
$858$	0	0
$859$	− 24.8348i	− 0.847354i	−0.905813	−	0.423677i	$-0.860739\pi$
$859$	− 24.8348i	− 0.847354i	0.905813	−	0.423677i	$-0.139261\pi$
$860$	0	0
$861$	26.2668i	0.895171i
$862$	0	0
$863$	50.7062	1.72606	0.863030	−	0.505153i	$-0.168564\pi$
$863$	50.7062	1.72606	0.863030	+	0.505153i	$0.168564\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.1652i	− 0.481074i
$868$	0	0
$869$	− 29.5402i	− 1.00208i
$870$	0	0
$871$	−13.8564	−0.469506
$872$	0	0
$873$	−11.1652	−0.377883
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.8528i	0.602846i	0.953490	+	0.301423i	$0.0974617\pi$
$877$	17.8528i	0.602846i	−0.953490	+	0.301423i	$0.902538\pi$
$878$	0	0
$879$	0.913701	0.0308184
$880$	0	0
$881$	36.3303	1.22400	0.612000	−	0.790858i	$-0.290365\pi$
$881$	36.3303	1.22400	0.612000	+	0.790858i	$0.290365\pi$
$882$	0	0
$883$	34.3303i	1.15531i	0.816282	+	0.577653i	$0.196031\pi$
$883$	34.3303i	1.15531i	−0.816282	+	0.577653i	$0.803969\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.0514	1.57983	0.789916	−	0.613215i	$-0.210124\pi$
$887$	47.0514	1.57983	0.789916	+	0.613215i	$0.210124\pi$
$888$	0	0
$889$	41.4955	1.39171
$890$	0	0
$891$	5.58258i	0.187023i
$892$	0	0
$893$	− 27.7128i	− 0.927374i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.4955i	0.450099i
$900$	0	0
$901$	43.7780i	1.45846i
$902$	0	0
$903$	48.8788	1.62659
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 17.4955i	− 0.580927i	−0.956886	−	0.290464i	$-0.906191\pi$
$907$	− 17.4955i	− 0.580927i	0.956886	−	0.290464i	$-0.0938095\pi$
$908$	0	0
$909$	0.723000i	0.0239804i
$910$	0	0
$911$	7.30960	0.242178	0.121089	−	0.992642i	$-0.461361\pi$
$911$	7.30960	0.242178	0.121089	+	0.992642i	$0.461361\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 28.0942i	− 0.927753i
$918$	0	0
$919$	11.8383	0.390510	0.195255	−	0.980753i	$-0.437447\pi$
$919$	11.8383	0.390510	0.195255	+	0.980753i	$0.437447\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	30.3303i	0.998334i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.723000	0.0237464
$928$	0	0
$929$	−30.6606	−1.00594	−0.502971	−	0.864303i	$-0.667760\pi$
$929$	−30.6606	−1.00594	−0.502971	+	0.864303i	$0.667760\pi$
$930$	0	0
$931$	48.6606i	1.59479i
$932$	0	0
$933$	3.65480i	0.119653i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.3303	0.468151	0.234075	−	0.972218i	$-0.424794\pi$
$937$	14.3303	0.468151	0.234075	+	0.972218i	$0.424794\pi$
$938$	0	0
$939$	− 0.834849i	− 0.0272443i
$940$	0	0
$941$	− 28.4358i	− 0.926981i	−0.886102	−	0.463491i	$-0.846597\pi$
$941$	− 28.4358i	− 0.926981i	0.886102	−	0.463491i	$-0.153403\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	52.5336i	1.70531i
$950$	0	0
$951$	−34.1087	−1.10605
$952$	0	0
$953$	−26.2432	−0.850100	−0.425050	−	0.905170i	$-0.639743\pi$
$953$	−26.2432	−0.850100	−0.425050	+	0.905170i	$0.639743\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 14.2378i	− 0.460243i
$958$	0	0
$959$	−45.6054	−1.47268
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.9898	0.867934	0.433967	−	0.900929i	$-0.357113\pi$
$967$	26.9898	0.867934	0.433967	+	0.900929i	$0.357113\pi$
$968$	0	0
$969$	−22.3303	−0.717353
$970$	0	0
$971$	4.08712i	0.131162i	0.997847	+	0.0655810i	$0.0208901\pi$
$971$	4.08712i	0.131162i	−0.997847	+	0.0655810i	$0.979110\pi$
$972$	0	0
$973$	− 31.3676i	− 1.00560i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.25227	−0.232021	−0.116010	−	0.993248i	$-0.537011\pi$
$977$	−7.25227	−0.232021	−0.116010	+	0.993248i	$0.537011\pi$
$978$	0	0
$979$	11.1652i	0.356840i
$980$	0	0
$981$	− 1.82740i	− 0.0583444i
$982$	0	0
$983$	−2.20880	−0.0704498	−0.0352249	−	0.999379i	$-0.511215\pi$
$983$	−2.20880	−0.0704498	−0.0352249	+	0.999379i	$0.511215\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 30.3303i	− 0.965424i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	11.8383	0.376056	0.188028	−	0.982164i	$-0.439790\pi$
$991$	11.8383	0.376056	0.188028	+	0.982164i	$0.439790\pi$
$992$	0	0
$993$	10.3303	0.327822
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.7520i	0.403860i	0.979400	+	0.201930i	$0.0647214\pi$
$997$	12.7520i	0.403860i	−0.979400	+	0.201930i	$0.935279\pi$
$998$	0	0
$999$	2.55040	0.0806911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.k.q.2401.1		8
4.3	odd	2	inner	4800.2.k.q.2401.8		8
5.2	odd	4		960.2.d.e.289.5	yes	8
5.3	odd	4		960.2.d.f.289.3	yes	8
5.4	even	2		4800.2.k.r.2401.8		8
8.3	odd	2	inner	4800.2.k.q.2401.4		8
8.5	even	2	inner	4800.2.k.q.2401.5		8
15.2	even	4		2880.2.d.j.289.4		8
15.8	even	4		2880.2.d.i.289.6		8
20.3	even	4		960.2.d.e.289.3	✓	8
20.7	even	4		960.2.d.f.289.5	yes	8
20.19	odd	2		4800.2.k.r.2401.1		8
40.3	even	4		960.2.d.f.289.6	yes	8
40.13	odd	4		960.2.d.e.289.6	yes	8
40.19	odd	2		4800.2.k.r.2401.5		8
40.27	even	4		960.2.d.e.289.4	yes	8
40.29	even	2		4800.2.k.r.2401.4		8
40.37	odd	4		960.2.d.f.289.4	yes	8
60.23	odd	4		2880.2.d.j.289.6		8
60.47	odd	4		2880.2.d.i.289.4		8
80.3	even	4		3840.2.f.i.769.1		8
80.13	odd	4		3840.2.f.k.769.5		8
80.27	even	4		3840.2.f.k.769.4		8
80.37	odd	4		3840.2.f.i.769.8		8
80.43	even	4		3840.2.f.k.769.8		8
80.53	odd	4		3840.2.f.i.769.4		8
80.67	even	4		3840.2.f.i.769.5		8
80.77	odd	4		3840.2.f.k.769.1		8
120.53	even	4		2880.2.d.j.289.3		8
120.77	even	4		2880.2.d.i.289.5		8
120.83	odd	4		2880.2.d.i.289.3		8
120.107	odd	4		2880.2.d.j.289.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.e.289.3	✓	8	20.3	even	4
960.2.d.e.289.4	yes	8	40.27	even	4
960.2.d.e.289.5	yes	8	5.2	odd	4
960.2.d.e.289.6	yes	8	40.13	odd	4
960.2.d.f.289.3	yes	8	5.3	odd	4
960.2.d.f.289.4	yes	8	40.37	odd	4
960.2.d.f.289.5	yes	8	20.7	even	4
960.2.d.f.289.6	yes	8	40.3	even	4
2880.2.d.i.289.3		8	120.83	odd	4
2880.2.d.i.289.4		8	60.47	odd	4
2880.2.d.i.289.5		8	120.77	even	4
2880.2.d.i.289.6		8	15.8	even	4
2880.2.d.j.289.3		8	120.53	even	4
2880.2.d.j.289.4		8	15.2	even	4
2880.2.d.j.289.5		8	120.107	odd	4
2880.2.d.j.289.6		8	60.23	odd	4
3840.2.f.i.769.1		8	80.3	even	4
3840.2.f.i.769.4		8	80.53	odd	4
3840.2.f.i.769.5		8	80.67	even	4
3840.2.f.i.769.8		8	80.37	odd	4
3840.2.f.k.769.1		8	80.77	odd	4
3840.2.f.k.769.4		8	80.27	even	4
3840.2.f.k.769.5		8	80.13	odd	4
3840.2.f.k.769.8		8	80.43	even	4
4800.2.k.q.2401.1		8	1.1	even	1	trivial
4800.2.k.q.2401.4		8	8.3	odd	2	inner
4800.2.k.q.2401.5		8	8.5	even	2	inner
4800.2.k.q.2401.8		8	4.3	odd	2	inner
4800.2.k.r.2401.1		8	20.19	odd	2
4800.2.k.r.2401.4		8	40.29	even	2
4800.2.k.r.2401.5		8	40.19	odd	2
4800.2.k.r.2401.8		8	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.37780 q^{7} -1.00000 q^{9} +5.58258i q^{11} +4.37780i q^{13} -5.58258 q^{17} +4.00000i q^{19} +4.37780i q^{21} +1.00000i q^{27} -2.55040i q^{29} -5.29150 q^{31} +5.58258 q^{33} -2.55040i q^{37} +4.37780 q^{39} +6.00000 q^{41} -11.1652i q^{43} -6.92820 q^{47} +12.1652 q^{49} +5.58258i q^{51} -7.84190i q^{53} +4.00000 q^{57} -1.58258i q^{59} -10.5830i q^{61} +4.37780 q^{63} +3.16515i q^{67} +6.92820 q^{71} +12.0000 q^{73} -24.4394i q^{77} -5.29150 q^{79} +1.00000 q^{81} +7.16515i q^{83} -2.55040 q^{87} +2.00000 q^{89} -19.1652i q^{91} +5.29150i q^{93} +11.1652 q^{97} -5.58258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{17} + 8 q^{33} + 48 q^{41} + 24 q^{49} + 32 q^{57} + 96 q^{73} + 8 q^{81} + 16 q^{89} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^17 + 8 * q^33 + 48 * q^41 + 24 * q^49 + 32 * q^57 + 96 * q^73 + 8 * q^81 + 16 * q^89 + 16 * q^97

Introduction

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