LMFDB - Embedded newform 4050.2.c.p.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(649,4050)] 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("4050.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$4050 = 2 \cdot 3^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,6,0,0,-8,0,2,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3394128186$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.p.649.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} +4.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -5.00000 q^{19} -3.00000i q^{22} -6.00000i q^{23} +4.00000 q^{26} +4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} -4.00000i q^{37} +5.00000i q^{38} -3.00000 q^{41} -11.0000i q^{43} -3.00000 q^{44} -6.00000 q^{46} -9.00000 q^{49} -4.00000i q^{52} -6.00000i q^{53} +4.00000 q^{56} +6.00000i q^{58} +3.00000 q^{59} -10.0000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +5.00000i q^{67} -3.00000i q^{68} +6.00000 q^{71} +7.00000i q^{73} -4.00000 q^{74} +5.00000 q^{76} -12.0000i q^{77} -14.0000 q^{79} +3.00000i q^{82} -12.0000i q^{83} -11.0000 q^{86} +3.00000i q^{88} -6.00000 q^{89} +16.0000 q^{91} +6.00000i q^{92} +11.0000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{4} + 6 q^{11} - 8 q^{14} + 2 q^{16} - 10 q^{19} + 8 q^{26} - 12 q^{29} + 4 q^{31} + 6 q^{34} - 6 q^{41} - 6 q^{44} - 12 q^{46} - 18 q^{49} + 8 q^{56} + 6 q^{59} - 20 q^{61} - 2 q^{64} + 12 q^{71}+ \cdots + 32 q^{91}+O(q^{100})$ 2 * q - 2 * q^4 + 6 * q^11 - 8 * q^14 + 2 * q^16 - 10 * q^19 + 8 * q^26 - 12 * q^29 + 4 * q^31 + 6 * q^34 - 6 * q^41 - 6 * q^44 - 12 * q^46 - 18 * q^49 + 8 * q^56 + 6 * q^59 - 20 * q^61 - 2 * q^64 + 12 * q^71 - 8 * q^74 + 10 * q^76 - 28 * q^79 - 22 * q^86 - 12 * q^89 + 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$ .

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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

)

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	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	$-0.272814\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	− 3.00000i	− 0.639602i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	4.00000i	0.755929i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	5.00000i	0.811107i
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	− 11.0000i	− 1.67748i	−0.544529	−	0.838742i	$-0.683292\pi$
$43$	− 11.0000i	− 1.67748i	0.544529	−	0.838742i	$-0.316708\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	6.00000i	0.787839i
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 2.00000i	− 0.254000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	5.00000	0.573539
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	0	0
$82$	3.00000i	0.331295i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−11.0000	−1.18616
$87$	0	0
$88$	3.00000i	0.319801i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	6.00000i	0.625543i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	− 3.00000i	− 0.276172i
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	10.0000i	0.905357i
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	20.0000i	1.73422i
$134$	5.00000	0.431934
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	0	0
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	7.00000	0.579324
$147$	0	0
$148$	4.00000i	0.328798i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 5.00000i	− 0.405554i
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	8.00000i	0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$	8.00000i	0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$	14.0000i	1.11378i
$159$	0	0
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	11.0000i	0.838742i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000i	0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 16.0000i	− 1.18600i
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	9.00000i	0.658145i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	13.0000i	0.935760i	0.883792	+	0.467880i	$0.154982\pi$
$193$	13.0000i	0.935760i	−0.883792	+	0.467880i	$0.845018\pi$
$194$	11.0000	0.789754
$195$	0	0
$196$	9.00000	0.642857
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	12.0000i	0.844317i
$203$	24.0000i	1.68447i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	4.00000i	0.277350i
$209$	−15.0000	−1.03757
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−9.00000	−0.615227
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	− 4.00000i	− 0.270914i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 21.0000i	− 1.37576i	−0.725826	−	0.687878i	$-0.758542\pi$
$233$	− 21.0000i	− 1.37576i	0.725826	−	0.687878i	$-0.241458\pi$
$234$	0	0
$235$	0	0
$236$	−3.00000	−0.195283
$237$	0	0
$238$	− 12.0000i	− 0.777844i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 20.0000i	− 1.27257i
$248$	2.00000i	0.127000i
$249$	0	0
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000i	0.561405i	0.959795	+	0.280702i	$0.0905674\pi$
$257$	9.00000i	0.561405i	−0.959795	+	0.280702i	$0.909433\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	20.0000	1.22628
$267$	0	0
$268$	− 5.00000i	− 0.305424i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	− 1.00000i	− 0.0599760i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	12.0000i	0.708338i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 7.00000i	− 0.409644i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−44.0000	−2.53612
$302$	10.0000i	0.575435i
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	14.0000	0.787562
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	24.0000i	1.33747i
$323$	− 15.0000i	− 0.834622i
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	− 3.00000i	− 0.165647i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	− 31.0000i	− 1.68868i	−0.535810	−	0.844339i	$-0.679994\pi$
$337$	− 31.0000i	− 1.68868i	0.535810	−	0.844339i	$-0.320006\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	8.00000i	0.431959i
$344$	11.0000	0.593080
$345$	0	0
$346$	18.0000	0.967686
$347$	21.0000i	1.12734i	0.826000	+	0.563670i	$0.190611\pi$
$347$	21.0000i	1.12734i	−0.826000	+	0.563670i	$0.809389\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	− 3.00000i	− 0.159901i
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	16.0000i	0.840941i
$363$	0	0
$364$	−16.0000	−0.838628
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	0	0
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	13.0000	0.661683
$387$	0	0
$388$	− 11.0000i	− 0.558440i
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	20.0000i	1.00251i
$399$	0	0
$400$	0	0
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	12.0000	0.597022
$405$	0	0
$406$	24.0000	1.19110
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	31.0000	1.53285	0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000	1.53285	0.766426	+	0.642333i	$0.222033\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	15.0000i	0.733674i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	40.0000i	1.93574i
$428$	9.00000i	0.435031i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	13.0000i	0.624740i	0.949960	+	0.312370i	$0.101123\pi$
$433$	13.0000i	0.624740i	−0.949960	+	0.312370i	$0.898877\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−4.00000	−0.191565
$437$	30.0000i	1.43509i
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	12.0000i	0.570782i
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	0	0
$445$	0	0
$446$	22.0000	1.04173
$447$	0	0
$448$	4.00000i	0.188982i
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	18.0000i	0.846649i
$453$	0	0
$454$	−3.00000	−0.140797
$455$	0	0
$456$	0	0
$457$	− 1.00000i	− 0.0467780i	−0.999726	−	0.0233890i	$-0.992554\pi$
$457$	− 1.00000i	− 0.0467780i	0.999726	−	0.0233890i	$-0.00744563\pi$
$458$	20.0000i	0.934539i
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−21.0000	−0.972806
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	3.00000i	0.138086i
$473$	− 33.0000i	− 1.51734i
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	0	0
$478$	6.00000i	0.274434i
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 17.0000i	− 0.774329i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 10.0000i	− 0.452679i
$489$	0	0
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	−20.0000	−0.899843
$495$	0	0
$496$	2.00000	0.0898027
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	31.0000	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$500$	0	0
$501$	0	0
$502$	21.0000i	0.937276i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	−18.0000	−0.800198
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	9.00000	0.396973
$515$	0	0
$516$	0	0
$517$	0	0
$518$	16.0000i	0.703000i
$519$	0	0
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	6.00000i	0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 20.0000i	− 0.867110i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	6.00000i	0.258678i
$539$	−27.0000	−1.16297
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	16.0000i	0.687259i
$543$	0	0
$544$	3.00000	0.128624
$545$	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	$-0.00680549\pi$
$548$	9.00000i	0.384461i
$549$	0	0
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	56.0000i	2.38136i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−1.00000	−0.0424094
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	44.0000	1.86100
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	3.00000i	0.126435i	0.998000	+	0.0632175i	$0.0201362\pi$
$563$	3.00000i	0.126435i	−0.998000	+	0.0632175i	$0.979864\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	6.00000i	0.251754i
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	− 12.0000i	− 0.501745i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 7.00000i	− 0.291414i	−0.989328	−	0.145707i	$-0.953454\pi$
$577$	− 7.00000i	− 0.291414i	0.989328	−	0.145707i	$-0.0465456\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	−7.00000	−0.289662
$585$	0	0
$586$	18.0000	0.743573
$587$	39.0000i	1.60970i	0.593477	+	0.804851i	$0.297755\pi$
$587$	39.0000i	1.60970i	−0.593477	+	0.804851i	$0.702245\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	− 4.00000i	− 0.164399i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 24.0000i	− 0.981433i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	44.0000i	1.79331i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	5.00000i	0.202777i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	12.0000	0.483494
$617$	− 39.0000i	− 1.57008i	−0.619445	−	0.785040i	$-0.712642\pi$
$617$	− 39.0000i	− 1.57008i	0.619445	−	0.785040i	$-0.287358\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	0	0
$622$	6.00000i	0.240578i
$623$	24.0000i	0.961540i
$624$	0	0
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	− 8.00000i	− 0.319235i
$629$	12.0000	0.478471
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	− 14.0000i	− 0.556890i
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	0	0
$637$	− 36.0000i	− 1.42637i
$638$	18.0000i	0.712627i
$639$	0	0
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	−15.0000	−0.590167
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	− 16.0000i	− 0.626608i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	18.0000i	0.696441i
$669$	0	0
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−31.0000	−1.19408
$675$	0	0
$676$	3.00000	0.115385
$677$	− 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
$679$	44.0000	1.68857
$680$	0	0
$681$	0	0
$682$	− 6.00000i	− 0.229752i
$683$	21.0000i	0.803543i	0.915740	+	0.401771i	$0.131605\pi$
$683$	21.0000i	0.803543i	−0.915740	+	0.401771i	$0.868395\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	− 11.0000i	− 0.419371i
$689$	24.0000	0.914327
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	21.0000	0.797149
$695$	0	0
$696$	0	0
$697$	− 9.00000i	− 0.340899i
$698$	− 16.0000i	− 0.605609i
$699$	0	0
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	20.0000i	0.754314i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−9.00000	−0.338719
$707$	48.0000i	1.80523i
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 6.00000i	− 0.223297i
$723$	0	0
$724$	16.0000	0.594635
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	16.0000i	0.592999i
$729$	0	0
$730$	0	0
$731$	33.0000	1.22055
$732$	0	0
$733$	− 32.0000i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$733$	− 32.0000i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−6.00000	−0.221163
$737$	15.0000i	0.552532i
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	− 9.00000i	− 0.329073i
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	29.0000i	1.05333i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 16.0000i	− 0.579239i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	12.0000i	0.433295i
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	− 13.0000i	− 0.467880i
$773$	30.0000i	1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$	30.0000i	1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$	0	0
$775$	0	0
$776$	−11.0000	−0.394877
$777$	0	0
$778$	36.0000i	1.29066i
$779$	15.0000	0.537431
$780$	0	0
$781$	18.0000	0.644091
$782$	− 18.0000i	− 0.643679i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	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
$789$	0	0
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	8.00000	0.283909
$795$	0	0
$796$	20.0000	0.708881
$797$	− 48.0000i	− 1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$	− 48.0000i	− 1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 33.0000i	− 1.16527i
$803$	21.0000i	0.741074i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	− 24.0000i	− 0.842235i
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	55.0000i	1.92421i
$818$	− 31.0000i	− 1.08389i
$819$	0	0
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	0	0
$832$	− 4.00000i	− 0.138675i
$833$	− 27.0000i	− 0.935495i
$834$	0	0
$835$	0	0
$836$	15.0000	0.518786
$837$	0	0
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 2.00000i	− 0.0689246i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	8.00000i	0.274883i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	9.00000	0.307614
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	0	0
$866$	13.0000	0.441758
$867$	0	0
$868$	8.00000i	0.271538i
$869$	−42.0000	−1.42475
$870$	0	0
$871$	−20.0000	−0.677674
$872$	4.00000i	0.135457i
$873$	0	0
$874$	30.0000	1.01477
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	− 10.0000i	− 0.337484i
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	− 41.0000i	− 1.37976i	−0.723924	−	0.689880i	$-0.757663\pi$
$883$	− 41.0000i	− 1.37976i	0.723924	−	0.689880i	$-0.242337\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−3.00000	−0.100787
$887$	− 42.0000i	− 1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$	− 42.0000i	− 1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	− 22.0000i	− 0.736614i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	− 15.0000i	− 0.500556i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	18.0000	0.599667
$902$	9.00000i	0.299667i
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	0	0
$907$	5.00000i	0.166022i	0.996549	+	0.0830111i	$0.0264537\pi$
$907$	5.00000i	0.166022i	−0.996549	+	0.0830111i	$0.973546\pi$
$908$	3.00000i	0.0995585i
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	20.0000	0.660819
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	0	0
$928$	6.00000i	0.196960i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	45.0000	1.47482
$932$	21.0000i	0.687878i
$933$	0	0
$934$	21.0000	0.687141
$935$	0	0
$936$	0	0
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	− 20.0000i	− 0.653023i
$939$	0	0
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	3.00000	0.0976417
$945$	0	0
$946$	−33.0000	−1.07292
$947$	27.0000i	0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$	27.0000i	0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	− 51.0000i	− 1.65205i	−0.563632	−	0.826026i	$-0.690596\pi$
$953$	− 51.0000i	− 1.65205i	0.563632	−	0.826026i	$-0.309404\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 16.0000i	− 0.515861i
$963$	0	0
$964$	−17.0000	−0.547533
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 9.00000i	− 0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$977$	− 9.00000i	− 0.287936i	0.989582	−	0.143968i	$-0.0459862\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	33.0000i	1.05307i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	20.0000i	0.636285i
$989$	−66.0000	−2.09868
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	− 31.0000i	− 0.981288i
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.p.649.1		2
3.2	odd	2		4050.2.c.d.649.2		2
5.2	odd	4		4050.2.a.bi.1.1		1
5.3	odd	4		810.2.a.a.1.1		1
5.4	even	2	inner	4050.2.c.p.649.2		2
9.2	odd	6		1350.2.j.c.199.2		4
9.4	even	3		450.2.j.a.349.2		4
9.5	odd	6		1350.2.j.c.1099.1		4
9.7	even	3		450.2.j.a.49.1		4
15.2	even	4		4050.2.a.q.1.1		1
15.8	even	4		810.2.a.e.1.1		1
15.14	odd	2		4050.2.c.d.649.1		2
20.3	even	4		6480.2.a.z.1.1		1
45.2	even	12		1350.2.e.g.901.1		2
45.4	even	6		450.2.j.a.349.1		4
45.7	odd	12		450.2.e.d.301.1		2
45.13	odd	12		90.2.e.b.61.1	yes	2
45.14	odd	6		1350.2.j.c.1099.2		4
45.22	odd	12		450.2.e.d.151.1		2
45.23	even	12		270.2.e.a.181.1		2
45.29	odd	6		1350.2.j.c.199.1		4
45.32	even	12		1350.2.e.g.451.1		2
45.34	even	6		450.2.j.a.49.2		4
45.38	even	12		270.2.e.a.91.1		2
45.43	odd	12		90.2.e.b.31.1	✓	2
60.23	odd	4		6480.2.a.l.1.1		1
180.23	odd	12		2160.2.q.d.721.1		2
180.43	even	12		720.2.q.c.481.1		2
180.83	odd	12		2160.2.q.d.1441.1		2
180.103	even	12		720.2.q.c.241.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.b.31.1	✓	2	45.43	odd	12
90.2.e.b.61.1	yes	2	45.13	odd	12
270.2.e.a.91.1		2	45.38	even	12
270.2.e.a.181.1		2	45.23	even	12
450.2.e.d.151.1		2	45.22	odd	12
450.2.e.d.301.1		2	45.7	odd	12
450.2.j.a.49.1		4	9.7	even	3
450.2.j.a.49.2		4	45.34	even	6
450.2.j.a.349.1		4	45.4	even	6
450.2.j.a.349.2		4	9.4	even	3
720.2.q.c.241.1		2	180.103	even	12
720.2.q.c.481.1		2	180.43	even	12
810.2.a.a.1.1		1	5.3	odd	4
810.2.a.e.1.1		1	15.8	even	4
1350.2.e.g.451.1		2	45.32	even	12
1350.2.e.g.901.1		2	45.2	even	12
1350.2.j.c.199.1		4	45.29	odd	6
1350.2.j.c.199.2		4	9.2	odd	6
1350.2.j.c.1099.1		4	9.5	odd	6
1350.2.j.c.1099.2		4	45.14	odd	6
2160.2.q.d.721.1		2	180.23	odd	12
2160.2.q.d.1441.1		2	180.83	odd	12
4050.2.a.q.1.1		1	15.2	even	4
4050.2.a.bi.1.1		1	5.2	odd	4
4050.2.c.d.649.1		2	15.14	odd	2
4050.2.c.d.649.2		2	3.2	odd	2
4050.2.c.p.649.1		2	1.1	even	1	trivial
4050.2.c.p.649.2		2	5.4	even	2	inner
6480.2.a.l.1.1		1	60.23	odd	4
6480.2.a.z.1.1		1	20.3	even	4

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

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4050.2.c.p.649.1

Level	$4050$
Weight	$2$
Character	4050.649
Analytic conductor	$32.339$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$