LMFDB - Embedded newform 5850.2.e.s.5149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
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 1950)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.s.5149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} +1.00000i q^{8} +1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -5.00000 q^{19} +1.00000 q^{26} -4.00000i q^{28} +3.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +7.00000i q^{37} +5.00000i q^{38} -3.00000 q^{41} +2.00000i q^{43} +9.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -9.00000i q^{53} -4.00000 q^{56} -3.00000i q^{58} +6.00000 q^{59} +8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -5.00000i q^{67} +3.00000 q^{71} -4.00000i q^{73} +7.00000 q^{74} +5.00000 q^{76} -11.0000 q^{79} +3.00000i q^{82} +6.00000i q^{83} +2.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} +9.00000 q^{94} -8.00000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{14} + 2 q^{16} - 10 q^{19} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 6 q^{41} - 18 q^{49} - 8 q^{56} + 12 q^{59} + 16 q^{61} - 2 q^{64} + 6 q^{71} + 14 q^{74} + 10 q^{76} - 22 q^{79} + 4 q^{86} + 12 q^{89} - 8 q^{91} + 18 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^14 + 2 * q^16 - 10 * q^19 + 2 * q^26 + 6 * q^29 - 8 * q^31 - 6 * q^41 - 18 * q^49 - 8 * q^56 + 12 * q^59 + 16 * q^61 - 2 * q^64 + 6 * q^71 + 14 * q^74 + 10 * q^76 - 22 * q^79 + 4 * q^86 + 12 * q^89 - 8 * q^91 + 18 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	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
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	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
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	− 3.00000i	− 0.393919i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	3.00000i	0.331295i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	9.00000	0.928279
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	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
$116$	−3.00000	−0.278543
$117$	0	0
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	− 20.0000i	− 1.73422i
$134$	−5.00000	−0.431934
$135$	0	0
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 3.00000i	− 0.251754i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 7.00000i	− 0.575396i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 5.00000i	− 0.405554i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	11.0000i	0.875113i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	− 21.0000i	− 1.62503i	−0.582941	−	0.812514i	$-0.698098\pi$
$167$	− 21.0000i	− 1.62503i	0.582941	−	0.812514i	$-0.301902\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	− 21.0000i	− 1.59660i	−0.602260	−	0.798300i	$-0.705733\pi$
$173$	− 21.0000i	− 1.59660i	0.602260	−	0.798300i	$-0.294267\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 9.00000i	− 0.656392i
$189$	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
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	18.0000i	1.26648i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	9.00000i	0.618123i
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	0	0
$217$	− 16.0000i	− 1.08615i
$218$	11.0000i	0.745014i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	− 5.00000i	− 0.318142i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	0	0
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	−28.0000	−1.73984
$260$	0	0
$261$	0	0
$262$	3.00000i	0.185341i
$263$	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
$266$	−20.0000	−1.22628
$267$	0	0
$268$	5.00000i	0.305424i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	32.0000i	1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$	32.0000i	1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	− 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
$296$	−7.00000	−0.406867
$297$	0	0
$298$	12.0000i	0.695141i
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	16.0000i	0.920697i
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	0	0
$309$	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
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	11.0000	0.618798
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	− 3.00000i	− 0.165647i
$329$	−36.0000	−1.98474
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 6.00000i	− 0.329293i
$333$	0	0
$334$	−21.0000	−1.14907
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−21.0000	−1.12897
$347$	− 9.00000i	− 0.483145i	−0.970383	−	0.241573i	$-0.922337\pi$
$347$	− 9.00000i	− 0.483145i	0.970383	−	0.241573i	$-0.0776632\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	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
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	16.0000i	0.840941i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	19.0000i	0.991792i	0.868382	+	0.495896i	$0.165160\pi$
$367$	19.0000i	0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	36.0000	1.86903
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	0	0
$376$	−9.00000	−0.464140
$377$	3.00000i	0.154508i
$378$	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
$383$	− 15.0000i	− 0.766464i	−0.923652	−	0.383232i	$-0.874811\pi$
$383$	− 15.0000i	− 0.766464i	0.923652	−	0.383232i	$-0.125189\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	8.00000i	0.406138i
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	31.0000i	1.55585i	0.628360	+	0.777923i	$0.283727\pi$
$397$	31.0000i	1.55585i	−0.628360	+	0.777923i	$0.716273\pi$
$398$	− 7.00000i	− 0.350878i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	18.0000	0.895533
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	0	0
$412$	16.0000i	0.788263i
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	32.0000i	1.54859i
$428$	− 9.00000i	− 0.435031i
$429$	0	0
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	11.0000	0.526804
$437$	0	0
$438$	0	0
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.00000i	0.142534i	0.997457	+	0.0712672i	$0.0227043\pi$
$443$	3.00000i	0.142534i	−0.997457	+	0.0712672i	$0.977296\pi$
$444$	0	0
$445$	0	0
$446$	14.0000	0.662919
$447$	0	0
$448$	− 4.00000i	− 0.188982i
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	− 7.00000i	− 0.327089i
$459$	0	0
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−24.0000	−1.11178
$467$	− 33.0000i	− 1.52706i	−0.645774	−	0.763529i	$-0.723465\pi$
$467$	− 33.0000i	− 1.52706i	0.645774	−	0.763529i	$-0.276535\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	6.00000i	0.276172i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	24.0000i	1.09773i
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	− 8.00000i	− 0.364390i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	−5.00000	−0.224961
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.0000i	0.538274i
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	0	0
$502$	9.00000i	0.401690i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 7.00000i	− 0.310575i
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	0	0
$518$	28.0000i	1.23025i
$519$	0	0
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	20.0000i	0.867110i
$533$	− 3.00000i	− 0.129944i
$534$	0	0
$535$	0	0
$536$	5.00000	0.215967
$537$	0	0
$538$	21.0000i	0.905374i
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	28.0000i	1.20270i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	− 3.00000i	− 0.128154i
$549$	0	0
$550$	0	0
$551$	−15.0000	−0.639021
$552$	0	0
$553$	− 44.0000i	− 1.87107i
$554$	−14.0000	−0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	− 15.0000i	− 0.632737i
$563$	21.0000i	0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$	21.0000i	0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$	0	0
$565$	0	0
$566$	32.0000	1.34506
$567$	0	0
$568$	3.00000i	0.125877i
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	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
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	0	0
$584$	4.00000	0.165521
$585$	0	0
$586$	−24.0000	−0.991431
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	7.00000i	0.287698i
$593$	− 3.00000i	− 0.123195i	−0.998101	−	0.0615976i	$-0.980380\pi$
$593$	− 3.00000i	− 0.123195i	0.998101	−	0.0615976i	$-0.0196196\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	− 23.0000i	− 0.933541i	−0.884378	−	0.466771i	$-0.845417\pi$
$607$	− 23.0000i	− 0.933541i	0.884378	−	0.466771i	$-0.154583\pi$
$608$	5.00000i	0.202777i
$609$	0	0
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	0	0
$617$	33.0000i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$617$	33.0000i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	24.0000i	0.961540i
$624$	0	0
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 11.0000i	− 0.437557i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	23.0000i	0.907031i	0.891248	+	0.453516i	$0.149830\pi$
$643$	23.0000i	0.907031i	−0.891248	+	0.453516i	$0.850170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	36.0000i	1.40343i
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	0	0
$668$	21.0000i	0.812514i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000i	1.11787i	0.829212	+	0.558934i	$0.188789\pi$
$673$	29.0000i	1.11787i	−0.829212	+	0.558934i	$0.811211\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	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
$681$	0	0
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	2.00000i	0.0762493i
$689$	9.00000	0.342873
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	21.0000i	0.798300i
$693$	0	0
$694$	−9.00000	−0.341635
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.00000i	0.0757011i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	− 35.0000i	− 1.32005i
$704$	0	0
$705$	0	0
$706$	3.00000	0.112906
$707$	− 72.0000i	− 2.70784i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	3.00000i	0.111959i
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	− 6.00000i	− 0.223297i
$723$	0	0
$724$	16.0000	0.594635
$725$	0	0
$726$	0	0
$727$	52.0000i	1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$	52.0000i	1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$	− 4.00000i	− 0.148250i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.00000i	0.184679i	0.995728	+	0.0923396i	$0.0294345\pi$
$733$	5.00000i	0.184679i	−0.995728	+	0.0923396i	$0.970565\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	0	0
$742$	− 36.0000i	− 1.32160i
$743$	39.0000i	1.43077i	0.698730	+	0.715386i	$0.253749\pi$
$743$	39.0000i	1.43077i	−0.698730	+	0.715386i	$0.746251\pi$
$744$	0	0
$745$	0	0
$746$	2.00000	0.0732252
$747$	0	0
$748$	0	0
$749$	−36.0000	−1.31541
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	9.00000i	0.328196i
$753$	0	0
$754$	3.00000	0.109254
$755$	0	0
$756$	0	0
$757$	− 32.0000i	− 1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$	− 32.0000i	− 1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	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
$763$	− 44.0000i	− 1.59291i
$764$	18.0000	0.651217
$765$	0	0
$766$	−15.0000	−0.541972
$767$	6.00000i	0.216647i
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0000i	− 0.719816i
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	8.00000	0.287183
$777$	0	0
$778$	− 27.0000i	− 0.967997i
$779$	15.0000	0.537431
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	− 24.0000i	− 0.854965i
$789$	0	0
$790$	0	0
$791$	72.0000	2.56003
$792$	0	0
$793$	8.00000i	0.284088i
$794$	31.0000	1.10015
$795$	0	0
$796$	−7.00000	−0.248108
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 30.0000i	− 1.05934i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 18.0000i	− 0.633238i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.0000i	− 0.349856i
$818$	− 4.00000i	− 0.139857i
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	23.0000i	0.801730i	0.916137	+	0.400865i	$0.131290\pi$
$823$	23.0000i	0.801730i	−0.916137	+	0.400865i	$0.868710\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	24.0000	0.835067
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	27.0000i	0.932700i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	22.0000i	0.758170i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	− 9.00000i	− 0.309061i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	−9.00000	−0.307614
$857$	− 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
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	0	0
$861$	0	0
$862$	− 33.0000i	− 1.12398i
$863$	27.0000i	0.919091i	0.888154	+	0.459545i	$0.151988\pi$
$863$	27.0000i	0.919091i	−0.888154	+	0.459545i	$0.848012\pi$
$864$	0	0
$865$	0	0
$866$	11.0000	0.373795
$867$	0	0
$868$	16.0000i	0.543075i
$869$	0	0
$870$	0	0
$871$	5.00000	0.169419
$872$	− 11.0000i	− 0.372507i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 35.0000i	− 1.18187i	−0.806721	−	0.590933i	$-0.798760\pi$
$877$	− 35.0000i	− 1.18187i	0.806721	−	0.590933i	$-0.201240\pi$
$878$	29.0000i	0.978703i
$879$	0	0
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	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
$892$	− 14.0000i	− 0.468755i
$893$	− 45.0000i	− 1.50587i
$894$	0	0
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	− 21.0000i	− 0.700779i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	− 6.00000i	− 0.199117i
$909$	0	0
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	0	0
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−7.00000	−0.231287
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	0	0
$922$	24.0000i	0.790398i
$923$	3.00000i	0.0987462i
$924$	0	0
$925$	0	0
$926$	26.0000	0.854413
$927$	0	0
$928$	− 3.00000i	− 0.0984798i
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	45.0000	1.47482
$932$	24.0000i	0.786146i
$933$	0	0
$934$	−33.0000	−1.07979
$935$	0	0
$936$	0	0
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	− 20.0000i	− 0.653023i
$939$	0	0
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	21.0000i	0.678479i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	7.00000i	0.225689i
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	39.0000	1.25157	0.625785	−	0.779996i	$-0.284779\pi$
$971$	39.0000	1.25157	0.625785	+	0.779996i	$0.284779\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	34.0000	1.08943
$975$	0	0
$976$	8.00000	0.256074
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	5.00000i	0.159071i
$989$	0	0
$990$	0	0
$991$	41.0000	1.30241	0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000	1.30241	0.651204	+	0.758903i	$0.274264\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	11.0000i	0.348199i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.s.5149.1		2
3.2	odd	2		1950.2.e.c.1249.2		2
5.2	odd	4		5850.2.a.be.1.1		1
5.3	odd	4		5850.2.a.y.1.1		1
5.4	even	2	inner	5850.2.e.s.5149.2		2
15.2	even	4		1950.2.a.f.1.1	✓	1
15.8	even	4		1950.2.a.t.1.1	yes	1
15.14	odd	2		1950.2.e.c.1249.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.f.1.1	✓	1	15.2	even	4
1950.2.a.t.1.1	yes	1	15.8	even	4
1950.2.e.c.1249.1		2	15.14	odd	2
1950.2.e.c.1249.2		2	3.2	odd	2
5850.2.a.y.1.1		1	5.3	odd	4
5850.2.a.be.1.1		1	5.2	odd	4
5850.2.e.s.5149.1		2	1.1	even	1	trivial
5850.2.e.s.5149.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} +1.00000i q^{8} +1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -5.00000 q^{19} +1.00000 q^{26} -4.00000i q^{28} +3.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +7.00000i q^{37} +5.00000i q^{38} -3.00000 q^{41} +2.00000i q^{43} +9.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -9.00000i q^{53} -4.00000 q^{56} -3.00000i q^{58} +6.00000 q^{59} +8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -5.00000i q^{67} +3.00000 q^{71} -4.00000i q^{73} +7.00000 q^{74} +5.00000 q^{76} -11.0000 q^{79} +3.00000i q^{82} +6.00000i q^{83} +2.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} +9.00000 q^{94} -8.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{14} + 2 q^{16} - 10 q^{19} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 6 q^{41} - 18 q^{49} - 8 q^{56} + 12 q^{59} + 16 q^{61} - 2 q^{64} + 6 q^{71} + 14 q^{74} + 10 q^{76} - 22 q^{79} + 4 q^{86} + 12 q^{89} - 8 q^{91} + 18 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^14 + 2 * q^16 - 10 * q^19 + 2 * q^26 + 6 * q^29 - 8 * q^31 - 6 * q^41 - 18 * q^49 - 8 * q^56 + 12 * q^59 + 16 * q^61 - 2 * q^64 + 6 * q^71 + 14 * q^74 + 10 * q^76 - 22 * q^79 + 4 * q^86 + 12 * q^89 - 8 * q^91 + 18 * q^94

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