LMFDB - Embedded newform 4950.2.c.r.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(199,4950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4950.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.5259490005$
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 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.r.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{11} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +1.00000i q^{22} +6.00000i q^{23} -4.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -8.00000i q^{43} -1.00000 q^{44} -6.00000 q^{46} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{52} +2.00000 q^{56} +6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{71} -2.00000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +2.00000i q^{77} -14.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -1.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -6.00000i q^{92} -6.00000 q^{94} +14.0000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 8 q^{19} - 8 q^{26} + 12 q^{29} + 16 q^{31} - 12 q^{34} - 12 q^{41} - 2 q^{44} - 12 q^{46} + 6 q^{49} + 4 q^{56} + 16 q^{61} - 2 q^{64} - 12 q^{71} + 20 q^{74} - 8 q^{76} - 28 q^{79} + 16 q^{86} - 12 q^{89} - 16 q^{91} - 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 8 * q^19 - 8 * q^26 + 12 * q^29 + 16 * q^31 - 12 * q^34 - 12 * q^41 - 2 * q^44 - 12 * q^46 + 6 * q^49 + 4 * q^56 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 20 * q^74 - 8 * q^76 - 28 * q^79 + 16 * q^86 - 12 * q^89 - 16 * q^91 - 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$.

$n$	$551$	$2377$	$4501$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	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
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	0.213201i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	6.00000i	0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	2.00000i	0.227921i
$78$	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
$82$	− 6.00000i	− 0.662589i
$83$	− 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
$86$	8.00000	0.862662
$87$	0	0
$88$	− 1.00000i	− 0.106600i
$89$	−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
$91$	−8.00000	−0.838628
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	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	$0.936859\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	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
$112$	2.00000i	0.188982i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.00000i	0.724286i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	14.0000i	1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$	14.0000i	1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$	− 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
$133$	8.00000i	0.693688i
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	10.0000i	0.821995i
$149$	−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
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	−2.00000	−0.161165
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 14.0000i	− 1.11378i
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	− 8.00000i	− 0.592999i
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	6.00000i	0.438763i
$188$	− 6.00000i	− 0.437595i
$189$	0	0
$190$	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
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	4.00000i	0.277350i
$209$	4.00000	0.276686
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	4.00000i	0.270914i
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	− 12.0000i	− 0.777844i
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	− 16.0000i	− 0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$	− 16.0000i	− 0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 8.00000i	− 0.475551i	−0.971320	−	0.237775i	$-0.923582\pi$
$283$	− 8.00000i	− 0.475551i	0.971320	−	0.237775i	$-0.0764182\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−4.00000	−0.236525
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	16.0000	0.922225
$302$	− 10.0000i	− 0.575435i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	− 2.00000i	− 0.113961i
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	14.0000	0.787562
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	− 12.0000i	− 0.668734i
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000i	0.331295i
$329$	−12.0000	−0.661581
$330$	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
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 3.00000i	− 0.163178i
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	20.0000i	1.07990i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 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
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	0	0
$352$	1.00000i	0.0533002i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	24.0000i	1.26844i
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 22.0000i	− 1.15629i
$363$	0	0
$364$	8.00000	0.419314
$365$	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
$371$	0	0
$372$	0	0
$373$	− 20.0000i	− 1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$	− 20.0000i	− 1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	6.00000	0.309426
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	4.00000i	0.200502i
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	32.0000i	1.59403i
$404$	6.00000	0.298511
$405$	0	0
$406$	−12.0000	−0.595550
$407$	− 10.0000i	− 0.495682i
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	4.00000i	0.195646i
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.0000i	0.774294i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−4.00000	−0.191565
$437$	24.0000i	1.14808i
$438$	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
$442$	− 24.0000i	− 1.14156i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	22.0000i	1.02799i
$459$	0	0
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	− 12.0000i	− 0.548867i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	− 8.00000i	− 0.362143i
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	8.00000	0.359211
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	− 14.0000i	− 0.621150i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−30.0000	−1.32324
$515$	0	0
$516$	0	0
$517$	6.00000i	0.263880i
$518$	20.0000i	0.878750i
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	48.0000i	2.09091i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	− 24.0000i	− 1.03471i
$539$	3.00000	0.129219
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	2.00000i	0.0859074i
$543$	0	0
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	− 28.0000i	− 1.19068i
$554$	16.0000	0.679775
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 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
$566$	8.00000	0.336265
$567$	0	0
$568$	6.00000i	0.251754i
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	− 24.0000i	− 0.981433i
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	16.0000i	0.652111i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	2.00000	0.0805823
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	60.0000	2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	14.0000i	0.556890i
$633$	0	0
$634$	12.0000	0.476581
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 36.0000i	− 1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$	− 36.0000i	− 1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	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
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	3.00000	0.115385
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	0	0
$682$	8.00000i	0.306336i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	36.0000	1.36654
$695$	0	0
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	4.00000i	0.151402i
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	− 40.0000i	− 1.50863i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	48.0000i	1.79761i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	− 12.0000i	− 0.447836i
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	22.0000	0.817624
$725$	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	$-0.173782\pi$
$728$	8.00000i	0.296500i
$729$	0	0
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−6.00000	−0.221163
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	20.0000	0.732252
$747$	0	0
$748$	− 6.00000i	− 0.219382i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	6.00000i	0.218797i
$753$	0	0
$754$	−24.0000	−0.874028
$755$	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	$-0.212006\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	18.0000	0.651217
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	14.0000i	0.503871i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−6.00000	−0.214697
$782$	− 36.0000i	− 1.28736i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	32.0000i	1.13635i
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−4.00000	−0.141776
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	0	0
$802$	− 30.0000i	− 1.05934i
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	0	0
$806$	−32.0000	−1.12715
$807$	0	0
$808$	6.00000i	0.211079i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	10.0000	0.350500
$815$	0	0
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	34.0000i	1.18878i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	− 4.00000i	− 0.138675i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	24.0000i	0.829066i
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 10.0000i	− 0.344623i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	− 8.00000i	− 0.273915i	−0.990577	−	0.136957i	$-0.956268\pi$
$853$	− 8.00000i	− 0.273915i	0.990577	−	0.136957i	$-0.0437323\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	12.0000	0.410152
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.0000i	1.42970i	0.699280	+	0.714848i	$0.253504\pi$
$863$	42.0000i	1.42970i	−0.699280	+	0.714848i	$0.746496\pi$
$864$	0	0
$865$	0	0
$866$	26.0000	0.883516
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	−14.0000	−0.474917
$870$	0	0
$871$	16.0000	0.542139
$872$	− 4.00000i	− 0.135457i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	− 52.0000i	− 1.75592i	−0.478738	−	0.877958i	$-0.658906\pi$
$877$	− 52.0000i	− 1.75592i	0.478738	−	0.877958i	$-0.341094\pi$
$878$	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
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	0	0
$889$	−28.0000	−0.939090
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	6.00000i	0.200223i
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	− 6.00000i	− 0.199778i
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	10.0000	0.330771
$915$	0	0
$916$	−22.0000	−0.726900
$917$	24.0000i	0.792550i
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	0	0
$922$	− 42.0000i	− 1.38320i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	6.00000i	0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	18.0000i	0.589610i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	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	$-0.117003\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	− 36.0000i	− 1.17232i
$944$	0	0
$945$	0	0
$946$	8.00000	0.260102
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	− 42.0000i	− 1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	− 42.0000i	− 1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	33.0000	1.06452
$962$	40.0000i	1.28965i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	8.00000	0.256074
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	0	0
$982$	− 12.0000i	− 0.382935i
$983$	30.0000i	0.956851i	0.878128	+	0.478426i	$0.158792\pi$
$983$	30.0000i	0.956851i	−0.878128	+	0.478426i	$0.841208\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	48.0000	1.52631
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	44.0000i	1.39349i	0.717317	+	0.696747i	$0.245370\pi$
$997$	44.0000i	1.39349i	−0.717317	+	0.696747i	$0.754630\pi$
$998$	4.00000i	0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.r.199.2		2
3.2	odd	2		1650.2.c.d.199.1		2
5.2	odd	4		4950.2.a.g.1.1		1
5.3	odd	4		198.2.a.e.1.1		1
5.4	even	2	inner	4950.2.c.r.199.1		2
15.2	even	4		1650.2.a.m.1.1		1
15.8	even	4		66.2.a.a.1.1	✓	1
15.14	odd	2		1650.2.c.d.199.2		2
20.3	even	4		1584.2.a.h.1.1		1
35.13	even	4		9702.2.a.bu.1.1		1
40.3	even	4		6336.2.a.bf.1.1		1
40.13	odd	4		6336.2.a.bj.1.1		1
45.13	odd	12		1782.2.e.f.1189.1		2
45.23	even	12		1782.2.e.s.1189.1		2
45.38	even	12		1782.2.e.s.595.1		2
45.43	odd	12		1782.2.e.f.595.1		2
55.43	even	4		2178.2.a.b.1.1		1
60.23	odd	4		528.2.a.d.1.1		1
105.83	odd	4		3234.2.a.d.1.1		1
120.53	even	4		2112.2.a.i.1.1		1
120.83	odd	4		2112.2.a.v.1.1		1
165.8	odd	20		726.2.e.b.493.1		4
165.38	even	20		726.2.e.k.487.1		4
165.53	even	20		726.2.e.k.565.1		4
165.68	odd	20		726.2.e.b.565.1		4
165.83	odd	20		726.2.e.b.487.1		4
165.98	odd	4		726.2.a.i.1.1		1
165.113	even	20		726.2.e.k.493.1		4
165.128	odd	20		726.2.e.b.511.1		4
165.158	even	20		726.2.e.k.511.1		4
660.263	even	4		5808.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.a.1.1	✓	1	15.8	even	4
198.2.a.e.1.1		1	5.3	odd	4
528.2.a.d.1.1		1	60.23	odd	4
726.2.a.i.1.1		1	165.98	odd	4
726.2.e.b.487.1		4	165.83	odd	20
726.2.e.b.493.1		4	165.8	odd	20
726.2.e.b.511.1		4	165.128	odd	20
726.2.e.b.565.1		4	165.68	odd	20
726.2.e.k.487.1		4	165.38	even	20
726.2.e.k.493.1		4	165.113	even	20
726.2.e.k.511.1		4	165.158	even	20
726.2.e.k.565.1		4	165.53	even	20
1584.2.a.h.1.1		1	20.3	even	4
1650.2.a.m.1.1		1	15.2	even	4
1650.2.c.d.199.1		2	3.2	odd	2
1650.2.c.d.199.2		2	15.14	odd	2
1782.2.e.f.595.1		2	45.43	odd	12
1782.2.e.f.1189.1		2	45.13	odd	12
1782.2.e.s.595.1		2	45.38	even	12
1782.2.e.s.1189.1		2	45.23	even	12
2112.2.a.i.1.1		1	120.53	even	4
2112.2.a.v.1.1		1	120.83	odd	4
2178.2.a.b.1.1		1	55.43	even	4
3234.2.a.d.1.1		1	105.83	odd	4
4950.2.a.g.1.1		1	5.2	odd	4
4950.2.c.r.199.1		2	5.4	even	2	inner
4950.2.c.r.199.2		2	1.1	even	1	trivial
5808.2.a.l.1.1		1	660.263	even	4
6336.2.a.bf.1.1		1	40.3	even	4
6336.2.a.bj.1.1		1	40.13	odd	4
9702.2.a.bu.1.1		1	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{11} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +1.00000i q^{22} +6.00000i q^{23} -4.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} -10.0000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -8.00000i q^{43} -1.00000 q^{44} -6.00000 q^{46} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{52} +2.00000 q^{56} +6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{71} -2.00000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +2.00000i q^{77} -14.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -1.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -6.00000i q^{92} -6.00000 q^{94} +14.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 8 q^{19} - 8 q^{26} + 12 q^{29} + 16 q^{31} - 12 q^{34} - 12 q^{41} - 2 q^{44} - 12 q^{46} + 6 q^{49} + 4 q^{56} + 16 q^{61} - 2 q^{64} - 12 q^{71} + 20 q^{74} - 8 q^{76} - 28 q^{79} + 16 q^{86} - 12 q^{89} - 16 q^{91} - 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 8 * q^19 - 8 * q^26 + 12 * q^29 + 16 * q^31 - 12 * q^34 - 12 * q^41 - 2 * q^44 - 12 * q^46 + 6 * q^49 + 4 * q^56 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 20 * q^74 - 8 * q^76 - 28 * q^79 + 16 * q^86 - 12 * q^89 - 16 * q^91 - 12 * q^94

Introduction

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Modular forms

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