LMFDB - Embedded newform 2300.2.c.d.1749.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,2,Mod(1749,2300)] 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("2300.1749"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,4,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.d.1749.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.00000i q^{7} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{13} -2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} -9.00000 q^{29} +5.00000 q^{31} +6.00000i q^{33} +2.00000i q^{37} +1.00000 q^{39} -9.00000 q^{41} +4.00000i q^{43} -3.00000i q^{47} -9.00000 q^{49} +6.00000i q^{53} +2.00000i q^{57} +2.00000 q^{61} -8.00000i q^{63} -10.0000i q^{67} -1.00000 q^{69} -3.00000 q^{71} +7.00000i q^{73} +24.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +4.00000 q^{91} -5.00000i q^{93} +8.00000i q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 12 q^{11} - 4 q^{19} - 8 q^{21} - 18 q^{29} + 10 q^{31} + 2 q^{39} - 18 q^{41} - 18 q^{49} + 4 q^{61} - 2 q^{69} - 6 q^{71} + 20 q^{79} + 2 q^{81} + 8 q^{91} - 24 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 12 * q^11 - 4 * q^19 - 8 * q^21 - 18 * q^29 + 10 * q^31 + 2 * q^39 - 18 * q^41 - 18 * q^49 + 4 * q^61 - 2 * q^69 - 6 * q^71 + 20 * q^79 + 2 * q^81 + 8 * q^91 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	− 8.00000i	− 1.00791i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	24.0000i	2.73505i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.00000i	0.964901i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	− 5.00000i	− 0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	9.00000i	0.811503i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.0000i	0.976092i	0.872818	+	0.488046i	$0.162290\pi$
$127$	11.0000i	0.976092i	−0.872818	+	0.488046i	$0.837710\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−23.0000	−1.95083	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−23.0000	−1.95083	−0.975417	+	0.220366i	$0.929275\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	19.0000i	1.36765i	0.729646	+	0.683825i	$0.239685\pi$
$193$	19.0000i	1.36765i	−0.729646	+	0.683825i	$0.760315\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.0000i	1.49619i	0.663593	+	0.748094i	$0.269031\pi$
$197$	21.0000i	1.49619i	−0.663593	+	0.748094i	$0.730969\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	36.0000i	2.52670i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00000i	0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$	3.00000i	0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	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
$273$	− 4.00000i	− 0.242091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 31.0000i	− 1.86261i	−0.364241	−	0.931305i	$-0.618672\pi$
$277$	− 31.0000i	− 1.86261i	0.364241	−	0.931305i	$-0.381328\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.0000i	2.12501i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	30.0000i	1.74078i
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	− 32.0000i	− 1.80875i	−0.426742	−	0.904373i	$-0.640339\pi$
$313$	− 32.0000i	− 1.80875i	0.426742	−	0.904373i	$-0.359661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	54.0000	3.02342
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.0000i	1.10600i
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 36.0000i	− 1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$	− 36.0000i	− 1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	3.00000i	0.159674i	0.996808	+	0.0798369i	$0.0254400\pi$
$353$	3.00000i	0.159674i	−0.996808	+	0.0798369i	$0.974560\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.00000i	− 0.463524i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	15.0000i	0.756650i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000i	1.15434i	0.816625	+	0.577168i	$0.195842\pi$
$397$	23.0000i	1.15434i	−0.816625	+	0.577168i	$0.804158\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	5.00000i	0.249068i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.0000i	1.12631i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	54.0000	2.54276
$452$	0	0
$453$	− 5.00000i	− 0.234920i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000i	0.538274i
$498$	0	0
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	6.00000i	0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$	6.00000i	0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	10.0000i	0.441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	18.0000i	0.791639i
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.00000i	− 0.389833i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0000i	0.647298i
$538$	0	0
$539$	54.0000	2.32594
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00000i	− 0.0427569i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	− 1.00000i	− 0.0427569i	0.999771	−	0.0213785i	$-0.00680549\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	− 40.0000i	− 1.70097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 43.0000i	− 1.79011i	−0.445952	−	0.895057i	$-0.647135\pi$
$577$	− 43.0000i	− 1.79011i	0.445952	−	0.895057i	$-0.352865\pi$
$578$	0	0
$579$	19.0000	0.789613
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	− 36.0000i	− 1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.0000i	1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$	33.0000i	1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	21.0000	0.863825
$592$	0	0
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.0000i	0.572982i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	− 20.0000i	− 0.814463i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.00000i	− 0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$	− 4.00000i	− 0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.0000i	− 0.479234i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	9.00000i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$653$	9.00000i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	− 3.00000i	− 0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$683$	− 3.00000i	− 0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.00000i	0.305219i
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	48.0000i	1.82337i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	− 5.00000i	− 0.187251i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.00000i	− 0.112037i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	22.0000i	0.818189i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	40.0000i	1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$	40.0000i	1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.0000i	2.21013i
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.0000i	0.878114i
$748$	0	0
$749$	−72.0000	−2.63082
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	80.0000i	2.89619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	45.0000i	1.60817i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	−48.0000	−1.70668
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 42.0000i	− 1.48215i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 27.0000i	− 0.950445i
$808$	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
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	31.0000i	1.08059i	0.841475	+	0.540296i	$0.181688\pi$
$823$	31.0000i	1.08059i	−0.841475	+	0.540296i	$0.818312\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−31.0000	−1.07538
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 25.0000i	− 0.864126i
$838$	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
$841$	52.0000	1.79310
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 100.000i	− 3.43604i
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	− 38.0000i	− 1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$	− 38.0000i	− 1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.0000i	1.33221i	0.745856	+	0.666107i	$0.232041\pi$
$857$	39.0000i	1.33221i	−0.745856	+	0.666107i	$0.767959\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	0	0
$863$	45.0000i	1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$863$	45.0000i	1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 17.0000i	− 0.577350i
$868$	0	0
$869$	−60.0000	−2.03536
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	16.0000i	0.541518i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.00000i	− 0.100730i	−0.998731	−	0.0503651i	$-0.983962\pi$
$887$	− 3.00000i	− 0.100730i	0.998731	−	0.0503651i	$-0.0160385\pi$
$888$	0	0
$889$	44.0000	1.47571
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.00000i	− 0.0333890i
$898$	0	0
$899$	−45.0000	−1.50083
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 10.0000i	− 0.332045i	−0.986122	−	0.166022i	$-0.946908\pi$
$907$	− 10.0000i	− 0.332045i	0.986122	−	0.166022i	$-0.0530924\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	− 72.0000i	− 2.38285i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60.0000i	1.98137i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	− 3.00000i	− 0.0987462i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	− 21.0000i	− 0.687509i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	−32.0000	−1.04428
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	9.00000i	0.293080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.0000i	1.07236i	0.844105	+	0.536178i	$0.180132\pi$
$947$	33.0000i	1.07236i	−0.844105	+	0.536178i	$0.819868\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 54.0000i	− 1.74557i
$958$	0	0
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	− 36.0000i	− 1.16008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.0000i	0.739630i	0.929105	+	0.369815i	$0.120579\pi$
$967$	23.0000i	0.739630i	−0.929105	+	0.369815i	$0.879421\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	92.0000i	2.94938i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−40.0000	−1.27710
$982$	0	0
$983$	− 54.0000i	− 1.72233i	−0.508323	−	0.861166i	$-0.669735\pi$
$983$	− 54.0000i	− 1.72233i	0.508323	−	0.861166i	$-0.330265\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 35.0000i	− 1.11069i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.d.1749.1		2
5.2	odd	4		2300.2.a.d.1.1		1
5.3	odd	4		460.2.a.c.1.1	✓	1
5.4	even	2	inner	2300.2.c.d.1749.2		2
15.8	even	4		4140.2.a.f.1.1		1
20.3	even	4		1840.2.a.c.1.1		1
20.7	even	4		9200.2.a.y.1.1		1
40.3	even	4		7360.2.a.v.1.1		1
40.13	odd	4		7360.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.c.1.1	✓	1	5.3	odd	4
1840.2.a.c.1.1		1	20.3	even	4
2300.2.a.d.1.1		1	5.2	odd	4
2300.2.c.d.1749.1		2	1.1	even	1	trivial
2300.2.c.d.1749.2		2	5.4	even	2	inner
4140.2.a.f.1.1		1	15.8	even	4
7360.2.a.i.1.1		1	40.13	odd	4
7360.2.a.v.1.1		1	40.3	even	4
9200.2.a.y.1.1		1	20.7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.00000i q^{7} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{13} -2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} -9.00000 q^{29} +5.00000 q^{31} +6.00000i q^{33} +2.00000i q^{37} +1.00000 q^{39} -9.00000 q^{41} +4.00000i q^{43} -3.00000i q^{47} -9.00000 q^{49} +6.00000i q^{53} +2.00000i q^{57} +2.00000 q^{61} -8.00000i q^{63} -10.0000i q^{67} -1.00000 q^{69} -3.00000 q^{71} +7.00000i q^{73} +24.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +4.00000 q^{91} -5.00000i q^{93} +8.00000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 12 q^{11} - 4 q^{19} - 8 q^{21} - 18 q^{29} + 10 q^{31} + 2 q^{39} - 18 q^{41} - 18 q^{49} + 4 q^{61} - 2 q^{69} - 6 q^{71} + 20 q^{79} + 2 q^{81} + 8 q^{91} - 24 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 12 * q^11 - 4 * q^19 - 8 * q^21 - 18 * q^29 + 10 * q^31 + 2 * q^39 - 18 * q^41 - 18 * q^49 + 4 * q^61 - 2 * q^69 - 6 * q^71 + 20 * q^79 + 2 * q^81 + 8 * q^91 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more