LMFDB - Embedded newform 5850.2.e.bj.5149.2

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:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bj.5149.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.60555i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.60555i q^{7} +1.00000i q^{8} +1.00000i q^{13} +2.60555 q^{14} +1.00000 q^{16} -4.60555i q^{17} -6.60555 q^{19} +4.60555i q^{23} +1.00000 q^{26} -2.60555i q^{28} +4.60555 q^{29} +2.00000 q^{31} -1.00000i q^{32} -4.60555 q^{34} -11.2111i q^{37} +6.60555i q^{38} +3.21110 q^{41} +5.21110i q^{43} +4.60555 q^{46} +9.21110i q^{47} +0.211103 q^{49} -1.00000i q^{52} -2.60555 q^{56} -4.60555i q^{58} +9.21110 q^{59} -7.21110 q^{61} -2.00000i q^{62} -1.00000 q^{64} +7.21110i q^{67} +4.60555i q^{68} -12.0000 q^{71} +6.60555i q^{73} -11.2111 q^{74} +6.60555 q^{76} +1.21110 q^{79} -3.21110i q^{82} +5.21110 q^{86} -3.21110 q^{89} -2.60555 q^{91} -4.60555i q^{92} +9.21110 q^{94} -6.60555i q^{97} -0.211103i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} - 12 q^{19} + 4 q^{26} + 4 q^{29} + 8 q^{31} - 4 q^{34} - 16 q^{41} + 4 q^{46} - 28 q^{49} + 4 q^{56} + 8 q^{59} - 4 q^{64} - 48 q^{71} - 16 q^{74} + 12 q^{76} - 24 q^{79} - 8 q^{86} + 16 q^{89} + 4 q^{91} + 8 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 - 12 * q^19 + 4 * q^26 + 4 * q^29 + 8 * q^31 - 4 * q^34 - 16 * q^41 + 4 * q^46 - 28 * q^49 + 4 * q^56 + 8 * q^59 - 4 * q^64 - 48 * q^71 - 16 * q^74 + 12 * q^76 - 24 * q^79 - 8 * q^86 + 16 * q^89 + 4 * q^91 + 8 * 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
$7$	2.60555i	0.984806i	0.870367	+	0.492403i	$0.163881\pi$
$7$	2.60555i	0.984806i	−0.870367	+	0.492403i	$0.836119\pi$
$8$	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
$14$	2.60555	0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.60555i	− 1.11701i	−0.829501	−	0.558505i	$-0.811375\pi$
$17$	− 4.60555i	− 1.11701i	0.829501	−	0.558505i	$-0.188625\pi$
$18$	0	0
$19$	−6.60555	−1.51542	−0.757709	−	0.652593i	$-0.773681\pi$
$19$	−6.60555	−1.51542	−0.757709	+	0.652593i	$0.773681\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.60555i	0.960324i	0.877180	+	0.480162i	$0.159422\pi$
$23$	4.60555i	0.960324i	−0.877180	+	0.480162i	$0.840578\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 2.60555i	− 0.492403i
$29$	4.60555	0.855229	0.427615	−	0.903961i	$-0.359354\pi$
$29$	4.60555	0.855229	0.427615	+	0.903961i	$0.359354\pi$
$30$	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
$34$	−4.60555	−0.789846
$35$	0	0
$36$	0	0
$37$	− 11.2111i	− 1.84309i	−0.388267	−	0.921547i	$-0.626926\pi$
$37$	− 11.2111i	− 1.84309i	0.388267	−	0.921547i	$-0.373074\pi$
$38$	6.60555i	1.07156i
$39$	0	0
$40$	0	0
$41$	3.21110	0.501490	0.250745	−	0.968053i	$-0.419324\pi$
$41$	3.21110	0.501490	0.250745	+	0.968053i	$0.419324\pi$
$42$	0	0
$43$	5.21110i	0.794686i	0.917670	+	0.397343i	$0.130068\pi$
$43$	5.21110i	0.794686i	−0.917670	+	0.397343i	$0.869932\pi$
$44$	0	0
$45$	0	0
$46$	4.60555	0.679051
$47$	9.21110i	1.34358i	0.740743	+	0.671789i	$0.234474\pi$
$47$	9.21110i	1.34358i	−0.740743	+	0.671789i	$0.765526\pi$
$48$	0	0
$49$	0.211103	0.0301575
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−2.60555	−0.348181
$57$	0	0
$58$	− 4.60555i	− 0.604739i
$59$	9.21110	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$59$	9.21110	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	− 2.00000i	− 0.254000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	7.21110i	0.880976i	0.897758	+	0.440488i	$0.145195\pi$
$67$	7.21110i	0.880976i	−0.897758	+	0.440488i	$0.854805\pi$
$68$	4.60555i	0.558505i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.60555i	0.773121i	0.922264	+	0.386561i	$0.126337\pi$
$73$	6.60555i	0.773121i	−0.922264	+	0.386561i	$0.873663\pi$
$74$	−11.2111	−1.30326
$75$	0	0
$76$	6.60555	0.757709
$77$	0	0
$78$	0	0
$79$	1.21110	0.136260	0.0681298	−	0.997676i	$-0.478297\pi$
$79$	1.21110	0.136260	0.0681298	+	0.997676i	$0.478297\pi$
$80$	0	0
$81$	0	0
$82$	− 3.21110i	− 0.354607i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	5.21110	0.561928
$87$	0	0
$88$	0	0
$89$	−3.21110	−0.340376	−0.170188	−	0.985412i	$-0.554438\pi$
$89$	−3.21110	−0.340376	−0.170188	+	0.985412i	$0.554438\pi$
$90$	0	0
$91$	−2.60555	−0.273136
$92$	− 4.60555i	− 0.480162i
$93$	0	0
$94$	9.21110	0.950053
$95$	0	0
$96$	0	0
$97$	− 6.60555i	− 0.670692i	−0.942095	−	0.335346i	$-0.891147\pi$
$97$	− 6.60555i	− 0.670692i	0.942095	−	0.335346i	$-0.108853\pi$
$98$	− 0.211103i	− 0.0213246i
$99$	0	0
$100$	0	0
$101$	−13.8167	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$101$	−13.8167	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−1.00000	−0.0980581
$105$	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
$109$	−18.6056	−1.78209	−0.891044	−	0.453916i	$-0.850026\pi$
$109$	−18.6056	−1.78209	−0.891044	+	0.453916i	$0.850026\pi$
$110$	0	0
$111$	0	0
$112$	2.60555i	0.246201i
$113$	7.39445i	0.695611i	0.937567	+	0.347806i	$0.113073\pi$
$113$	7.39445i	0.695611i	−0.937567	+	0.347806i	$0.886927\pi$
$114$	0	0
$115$	0	0
$116$	−4.60555	−0.427615
$117$	0	0
$118$	− 9.21110i	− 0.847951i
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	7.21110i	0.652863i
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	1.21110i	0.107468i	0.998555	+	0.0537340i	$0.0171123\pi$
$127$	1.21110i	0.107468i	−0.998555	+	0.0537340i	$0.982888\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−16.6056	−1.45083	−0.725417	−	0.688310i	$-0.758353\pi$
$131$	−16.6056	−1.45083	−0.725417	+	0.688310i	$0.758353\pi$
$132$	0	0
$133$	− 17.2111i	− 1.49239i
$134$	7.21110	0.622944
$135$	0	0
$136$	4.60555	0.394923
$137$	− 21.6333i	− 1.84826i	−0.382080	−	0.924129i	$-0.624792\pi$
$137$	− 21.6333i	− 1.84826i	0.382080	−	0.924129i	$-0.375208\pi$
$138$	0	0
$139$	−14.4222	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$139$	−14.4222	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	6.60555	0.546679
$147$	0	0
$148$	11.2111i	0.921547i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.42221	0.685389	0.342695	−	0.939447i	$-0.388660\pi$
$151$	8.42221	0.685389	0.342695	+	0.939447i	$0.388660\pi$
$152$	− 6.60555i	− 0.535781i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.2111i	− 0.894743i	−0.894348	−	0.447372i	$-0.852360\pi$
$157$	− 11.2111i	− 0.894743i	0.894348	−	0.447372i	$-0.147640\pi$
$158$	− 1.21110i	− 0.0963501i
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	−3.21110	−0.250745
$165$	0	0
$166$	0	0
$167$	18.4222i	1.42555i	0.701391	+	0.712777i	$0.252563\pi$
$167$	18.4222i	1.42555i	−0.701391	+	0.712777i	$0.747437\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 5.21110i	− 0.397343i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	3.21110i	0.240682i
$179$	−11.0278	−0.824253	−0.412127	−	0.911127i	$-0.635214\pi$
$179$	−11.0278	−0.824253	−0.412127	+	0.911127i	$0.635214\pi$
$180$	0	0
$181$	−19.2111	−1.42795	−0.713975	−	0.700171i	$-0.753107\pi$
$181$	−19.2111	−1.42795	−0.713975	+	0.700171i	$0.753107\pi$
$182$	2.60555i	0.193136i
$183$	0	0
$184$	−4.60555	−0.339526
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 9.21110i	− 0.671789i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 14.6056i	− 1.05133i	−0.850691	−	0.525665i	$-0.823816\pi$
$193$	− 14.6056i	− 1.05133i	0.850691	−	0.525665i	$-0.176184\pi$
$194$	−6.60555	−0.474251
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	24.4222i	1.74001i	0.493043	+	0.870005i	$0.335885\pi$
$197$	24.4222i	1.74001i	−0.493043	+	0.870005i	$0.664115\pi$
$198$	0	0
$199$	−5.21110	−0.369405	−0.184703	−	0.982794i	$-0.559132\pi$
$199$	−5.21110	−0.369405	−0.184703	+	0.982794i	$0.559132\pi$
$200$	0	0
$201$	0	0
$202$	13.8167i	0.972136i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−25.2111	−1.73560	−0.867802	−	0.496910i	$-0.834468\pi$
$211$	−25.2111	−1.73560	−0.867802	+	0.496910i	$0.834468\pi$
$212$	0	0
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	5.21110i	0.353753i
$218$	18.6056i	1.26013i
$219$	0	0
$220$	0	0
$221$	4.60555	0.309803
$222$	0	0
$223$	− 17.3944i	− 1.16482i	−0.812896	−	0.582409i	$-0.802110\pi$
$223$	− 17.3944i	− 1.16482i	0.812896	−	0.582409i	$-0.197890\pi$
$224$	2.60555	0.174091
$225$	0	0
$226$	7.39445	0.491871
$227$	− 6.42221i	− 0.426257i	−0.977024	−	0.213128i	$-0.931635\pi$
$227$	− 6.42221i	− 0.426257i	0.977024	−	0.213128i	$-0.0683653\pi$
$228$	0	0
$229$	−0.183346	−0.0121159	−0.00605793	−	0.999982i	$-0.501928\pi$
$229$	−0.183346	−0.0121159	−0.00605793	+	0.999982i	$0.501928\pi$
$230$	0	0
$231$	0	0
$232$	4.60555i	0.302369i
$233$	− 1.81665i	− 0.119013i	−0.998228	−	0.0595065i	$-0.981047\pi$
$233$	− 1.81665i	− 0.119013i	0.998228	−	0.0595065i	$-0.0189527\pi$
$234$	0	0
$235$	0	0
$236$	−9.21110	−0.599592
$237$	0	0
$238$	− 12.0000i	− 0.777844i
$239$	−18.4222	−1.19163	−0.595817	−	0.803120i	$-0.703172\pi$
$239$	−18.4222	−1.19163	−0.595817	+	0.803120i	$0.703172\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	7.21110	0.461644
$245$	0	0
$246$	0	0
$247$	− 6.60555i	− 0.420301i
$248$	2.00000i	0.127000i
$249$	0	0
$250$	0	0
$251$	4.60555	0.290700	0.145350	−	0.989380i	$-0.453569\pi$
$251$	4.60555	0.290700	0.145350	+	0.989380i	$0.453569\pi$
$252$	0	0
$253$	0	0
$254$	1.21110	0.0759913
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.0278i	0.687893i	0.938989	+	0.343946i	$0.111764\pi$
$257$	11.0278i	0.687893i	−0.938989	+	0.343946i	$0.888236\pi$
$258$	0	0
$259$	29.2111	1.81509
$260$	0	0
$261$	0	0
$262$	16.6056i	1.02589i
$263$	− 11.0278i	− 0.680001i	−0.940425	−	0.340000i	$-0.889573\pi$
$263$	− 11.0278i	− 0.680001i	0.940425	−	0.340000i	$-0.110427\pi$
$264$	0	0
$265$	0	0
$266$	−17.2111	−1.05528
$267$	0	0
$268$	− 7.21110i	− 0.440488i
$269$	−10.1833	−0.620890	−0.310445	−	0.950591i	$-0.600478\pi$
$269$	−10.1833	−0.620890	−0.310445	+	0.950591i	$0.600478\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 4.60555i	− 0.279253i
$273$	0	0
$274$	−21.6333	−1.30692
$275$	0	0
$276$	0	0
$277$	31.2111i	1.87529i	0.347590	+	0.937647i	$0.387000\pi$
$277$	31.2111i	1.87529i	−0.347590	+	0.937647i	$0.613000\pi$
$278$	14.4222i	0.864986i
$279$	0	0
$280$	0	0
$281$	−3.21110	−0.191558	−0.0957792	−	0.995403i	$-0.530534\pi$
$281$	−3.21110	−0.191558	−0.0957792	+	0.995403i	$0.530534\pi$
$282$	0	0
$283$	− 1.21110i	− 0.0719926i	−0.999352	−	0.0359963i	$-0.988540\pi$
$283$	− 1.21110i	− 0.0719926i	0.999352	−	0.0359963i	$-0.0114604\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	8.36669i	0.493870i
$288$	0	0
$289$	−4.21110	−0.247712
$290$	0	0
$291$	0	0
$292$	− 6.60555i	− 0.386561i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	11.2111	0.651632
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−4.60555	−0.266346
$300$	0	0
$301$	−13.5778	−0.782611
$302$	− 8.42221i	− 0.484643i
$303$	0	0
$304$	−6.60555	−0.378854
$305$	0	0
$306$	0	0
$307$	− 4.78890i	− 0.273317i	−0.990618	−	0.136658i	$-0.956364\pi$
$307$	− 4.78890i	− 0.273317i	0.990618	−	0.136658i	$-0.0436363\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	11.2111i	0.633689i	0.948478	+	0.316844i	$0.102623\pi$
$313$	11.2111i	0.633689i	−0.948478	+	0.316844i	$0.897377\pi$
$314$	−11.2111	−0.632679
$315$	0	0
$316$	−1.21110	−0.0681298
$317$	12.4222i	0.697701i	0.937178	+	0.348850i	$0.113428\pi$
$317$	12.4222i	0.697701i	−0.937178	+	0.348850i	$0.886572\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	30.4222i	1.69274i
$324$	0	0
$325$	0	0
$326$	2.00000	0.110770
$327$	0	0
$328$	3.21110i	0.177303i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−5.39445	−0.296506	−0.148253	−	0.988949i	$-0.547365\pi$
$331$	−5.39445	−0.296506	−0.148253	+	0.988949i	$0.547365\pi$
$332$	0	0
$333$	0	0
$334$	18.4222	1.00802
$335$	0	0
$336$	0	0
$337$	− 26.8444i	− 1.46231i	−0.682212	−	0.731154i	$-0.738982\pi$
$337$	− 26.8444i	− 1.46231i	0.682212	−	0.731154i	$-0.261018\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.7889i	1.01451i
$344$	−5.21110	−0.280964
$345$	0	0
$346$	12.0000	0.645124
$347$	− 14.7889i	− 0.793910i	−0.917838	−	0.396955i	$-0.870067\pi$
$347$	− 14.7889i	− 0.793910i	0.917838	−	0.396955i	$-0.129933\pi$
$348$	0	0
$349$	11.8167	0.632531	0.316265	−	0.948671i	$-0.397571\pi$
$349$	11.8167	0.632531	0.316265	+	0.948671i	$0.397571\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.6333i	1.15142i	0.817652	+	0.575712i	$0.195275\pi$
$353$	21.6333i	1.15142i	−0.817652	+	0.575712i	$0.804725\pi$
$354$	0	0
$355$	0	0
$356$	3.21110	0.170188
$357$	0	0
$358$	11.0278i	0.582835i
$359$	30.4222	1.60562	0.802811	−	0.596233i	$-0.203337\pi$
$359$	30.4222	1.60562	0.802811	+	0.596233i	$0.203337\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	19.2111i	1.00971i
$363$	0	0
$364$	2.60555	0.136568
$365$	0	0
$366$	0	0
$367$	6.78890i	0.354378i	0.984177	+	0.177189i	$0.0567003\pi$
$367$	6.78890i	0.354378i	−0.984177	+	0.177189i	$0.943300\pi$
$368$	4.60555i	0.240081i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 0.788897i	− 0.0408476i	−0.999791	−	0.0204238i	$-0.993498\pi$
$373$	− 0.788897i	− 0.0408476i	0.999791	−	0.0204238i	$-0.00650154\pi$
$374$	0	0
$375$	0	0
$376$	−9.21110	−0.475026
$377$	4.60555i	0.237198i
$378$	0	0
$379$	26.6056	1.36664	0.683318	−	0.730121i	$-0.260536\pi$
$379$	26.6056	1.36664	0.683318	+	0.730121i	$0.260536\pi$
$380$	0	0
$381$	0	0
$382$	12.0000i	0.613973i
$383$	18.4222i	0.941331i	0.882312	+	0.470665i	$0.155986\pi$
$383$	18.4222i	0.941331i	−0.882312	+	0.470665i	$0.844014\pi$
$384$	0	0
$385$	0	0
$386$	−14.6056	−0.743403
$387$	0	0
$388$	6.60555i	0.335346i
$389$	1.81665	0.0921080	0.0460540	−	0.998939i	$-0.485335\pi$
$389$	1.81665	0.0921080	0.0460540	+	0.998939i	$0.485335\pi$
$390$	0	0
$391$	21.2111	1.07269
$392$	0.211103i	0.0106623i
$393$	0	0
$394$	24.4222	1.23037
$395$	0	0
$396$	0	0
$397$	− 35.2111i	− 1.76719i	−0.468248	−	0.883597i	$-0.655114\pi$
$397$	− 35.2111i	− 1.76719i	0.468248	−	0.883597i	$-0.344886\pi$
$398$	5.21110i	0.261209i
$399$	0	0
$400$	0	0
$401$	33.6333	1.67957	0.839784	−	0.542921i	$-0.182682\pi$
$401$	33.6333	1.67957	0.839784	+	0.542921i	$0.182682\pi$
$402$	0	0
$403$	2.00000i	0.0996271i
$404$	13.8167	0.687404
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	13.6333	0.674124	0.337062	−	0.941483i	$-0.390567\pi$
$409$	13.6333	0.674124	0.337062	+	0.941483i	$0.390567\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	29.4500	1.43872	0.719362	−	0.694635i	$-0.244434\pi$
$419$	29.4500	1.43872	0.719362	+	0.694635i	$0.244434\pi$
$420$	0	0
$421$	−9.02776	−0.439986	−0.219993	−	0.975501i	$-0.570603\pi$
$421$	−9.02776	−0.439986	−0.219993	+	0.975501i	$0.570603\pi$
$422$	25.2111i	1.22726i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 18.7889i	− 0.909258i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	− 22.8444i	− 1.09783i	−0.835877	−	0.548916i	$-0.815041\pi$
$433$	− 22.8444i	− 1.09783i	0.835877	−	0.548916i	$-0.184959\pi$
$434$	5.21110	0.250141
$435$	0	0
$436$	18.6056	0.891044
$437$	− 30.4222i	− 1.45529i
$438$	0	0
$439$	−32.8444	−1.56758	−0.783789	−	0.621027i	$-0.786716\pi$
$439$	−32.8444	−1.56758	−0.783789	+	0.621027i	$0.786716\pi$
$440$	0	0
$441$	0	0
$442$	− 4.60555i	− 0.219064i
$443$	− 39.6333i	− 1.88304i	−0.336964	−	0.941518i	$-0.609400\pi$
$443$	− 39.6333i	− 1.88304i	0.336964	−	0.941518i	$-0.390600\pi$
$444$	0	0
$445$	0	0
$446$	−17.3944	−0.823651
$447$	0	0
$448$	− 2.60555i	− 0.123101i
$449$	9.63331	0.454624	0.227312	−	0.973822i	$-0.427006\pi$
$449$	9.63331	0.454624	0.227312	+	0.973822i	$0.427006\pi$
$450$	0	0
$451$	0	0
$452$	− 7.39445i	− 0.347806i
$453$	0	0
$454$	−6.42221	−0.301409
$455$	0	0
$456$	0	0
$457$	17.3944i	0.813678i	0.913500	+	0.406839i	$0.133369\pi$
$457$	17.3944i	0.813678i	−0.913500	+	0.406839i	$0.866631\pi$
$458$	0.183346i	0.00856720i
$459$	0	0
$460$	0	0
$461$	−3.21110	−0.149556	−0.0747780	−	0.997200i	$-0.523825\pi$
$461$	−3.21110	−0.149556	−0.0747780	+	0.997200i	$0.523825\pi$
$462$	0	0
$463$	12.1833i	0.566208i	0.959089	+	0.283104i	$0.0913642\pi$
$463$	12.1833i	0.566208i	−0.959089	+	0.283104i	$0.908636\pi$
$464$	4.60555	0.213807
$465$	0	0
$466$	−1.81665	−0.0841549
$467$	5.57779i	0.258110i	0.991637	+	0.129055i	$0.0411943\pi$
$467$	5.57779i	0.258110i	−0.991637	+	0.129055i	$0.958806\pi$
$468$	0	0
$469$	−18.7889	−0.867591
$470$	0	0
$471$	0	0
$472$	9.21110i	0.423975i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	0	0
$478$	18.4222i	0.842612i
$479$	−18.4222	−0.841732	−0.420866	−	0.907123i	$-0.638274\pi$
$479$	−18.4222	−0.841732	−0.420866	+	0.907123i	$0.638274\pi$
$480$	0	0
$481$	11.2111	0.511182
$482$	10.0000i	0.455488i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	− 2.97224i	− 0.134685i	−0.997730	−	0.0673426i	$-0.978548\pi$
$487$	− 2.97224i	− 0.134685i	0.997730	−	0.0673426i	$-0.0214521\pi$
$488$	− 7.21110i	− 0.326431i
$489$	0	0
$490$	0	0
$491$	32.2389	1.45492	0.727460	−	0.686150i	$-0.240701\pi$
$491$	32.2389	1.45492	0.727460	+	0.686150i	$0.240701\pi$
$492$	0	0
$493$	− 21.2111i	− 0.955300i
$494$	−6.60555	−0.297198
$495$	0	0
$496$	2.00000	0.0898027
$497$	− 31.2666i	− 1.40250i
$498$	0	0
$499$	−27.8167	−1.24524	−0.622622	−	0.782523i	$-0.713933\pi$
$499$	−27.8167	−1.24524	−0.622622	+	0.782523i	$0.713933\pi$
$500$	0	0
$501$	0	0
$502$	− 4.60555i	− 0.205556i
$503$	20.2389i	0.902406i	0.892421	+	0.451203i	$0.149005\pi$
$503$	20.2389i	0.902406i	−0.892421	+	0.451203i	$0.850995\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 1.21110i	− 0.0537340i
$509$	−9.63331	−0.426989	−0.213494	−	0.976944i	$-0.568485\pi$
$509$	−9.63331	−0.426989	−0.213494	+	0.976944i	$0.568485\pi$
$510$	0	0
$511$	−17.2111	−0.761374
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	11.0278	0.486413
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 29.2111i	− 1.28346i
$519$	0	0
$520$	0	0
$521$	−21.2111	−0.929275	−0.464638	−	0.885501i	$-0.653815\pi$
$521$	−21.2111	−0.929275	−0.464638	+	0.885501i	$0.653815\pi$
$522$	0	0
$523$	38.4222i	1.68009i	0.542520	+	0.840043i	$0.317470\pi$
$523$	38.4222i	1.68009i	−0.542520	+	0.840043i	$0.682530\pi$
$524$	16.6056	0.725417
$525$	0	0
$526$	−11.0278	−0.480833
$527$	− 9.21110i	− 0.401242i
$528$	0	0
$529$	1.78890	0.0777781
$530$	0	0
$531$	0	0
$532$	17.2111i	0.746196i
$533$	3.21110i	0.139088i
$534$	0	0
$535$	0	0
$536$	−7.21110	−0.311472
$537$	0	0
$538$	10.1833i	0.439035i
$539$	0	0
$540$	0	0
$541$	−9.02776	−0.388134	−0.194067	−	0.980988i	$-0.562168\pi$
$541$	−9.02776	−0.388134	−0.194067	+	0.980988i	$0.562168\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	0	0
$544$	−4.60555	−0.197461
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	21.6333i	0.924129i
$549$	0	0
$550$	0	0
$551$	−30.4222	−1.29603
$552$	0	0
$553$	3.15559i	0.134189i
$554$	31.2111	1.32603
$555$	0	0
$556$	14.4222	0.611638
$557$	18.8444i	0.798463i	0.916850	+	0.399232i	$0.130723\pi$
$557$	18.8444i	0.798463i	−0.916850	+	0.399232i	$0.869277\pi$
$558$	0	0
$559$	−5.21110	−0.220406
$560$	0	0
$561$	0	0
$562$	3.21110i	0.135452i
$563$	− 15.6333i	− 0.658865i	−0.944179	−	0.329433i	$-0.893143\pi$
$563$	− 15.6333i	− 0.658865i	0.944179	−	0.329433i	$-0.106857\pi$
$564$	0	0
$565$	0	0
$566$	−1.21110	−0.0509064
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	26.7889	1.12305	0.561525	−	0.827460i	$-0.310215\pi$
$569$	26.7889	1.12305	0.561525	+	0.827460i	$0.310215\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	8.36669	0.349219
$575$	0	0
$576$	0	0
$577$	− 27.8167i	− 1.15802i	−0.815320	−	0.579011i	$-0.803439\pi$
$577$	− 27.8167i	− 1.15802i	0.815320	−	0.579011i	$-0.196561\pi$
$578$	4.21110i	0.175159i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−6.60555	−0.273340
$585$	0	0
$586$	30.0000	1.23929
$587$	6.42221i	0.265073i	0.991178	+	0.132536i	$0.0423121\pi$
$587$	6.42221i	0.265073i	−0.991178	+	0.132536i	$0.957688\pi$
$588$	0	0
$589$	−13.2111	−0.544354
$590$	0	0
$591$	0	0
$592$	− 11.2111i	− 0.460773i
$593$	20.7889i	0.853698i	0.904323	+	0.426849i	$0.140376\pi$
$593$	20.7889i	0.853698i	−0.904323	+	0.426849i	$0.859624\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	4.60555i	0.188335i
$599$	36.8444	1.50542	0.752711	−	0.658351i	$-0.228746\pi$
$599$	36.8444	1.50542	0.752711	+	0.658351i	$0.228746\pi$
$600$	0	0
$601$	17.6333	0.719278	0.359639	−	0.933092i	$-0.382900\pi$
$601$	17.6333	0.719278	0.359639	+	0.933092i	$0.382900\pi$
$602$	13.5778i	0.553390i
$603$	0	0
$604$	−8.42221	−0.342695
$605$	0	0
$606$	0	0
$607$	− 41.2111i	− 1.67271i	−0.548190	−	0.836354i	$-0.684683\pi$
$607$	− 41.2111i	− 1.67271i	0.548190	−	0.836354i	$-0.315317\pi$
$608$	6.60555i	0.267890i
$609$	0	0
$610$	0	0
$611$	−9.21110	−0.372641
$612$	0	0
$613$	5.63331i	0.227527i	0.993508	+	0.113764i	$0.0362906\pi$
$613$	5.63331i	0.227527i	−0.993508	+	0.113764i	$0.963709\pi$
$614$	−4.78890	−0.193264
$615$	0	0
$616$	0	0
$617$	18.8444i	0.758647i	0.925264	+	0.379324i	$0.123843\pi$
$617$	18.8444i	0.758647i	−0.925264	+	0.379324i	$0.876157\pi$
$618$	0	0
$619$	2.60555	0.104726	0.0523630	−	0.998628i	$-0.483325\pi$
$619$	2.60555	0.104726	0.0523630	+	0.998628i	$0.483325\pi$
$620$	0	0
$621$	0	0
$622$	12.0000i	0.481156i
$623$	− 8.36669i	− 0.335204i
$624$	0	0
$625$	0	0
$626$	11.2111	0.448086
$627$	0	0
$628$	11.2111i	0.447372i
$629$	−51.6333	−2.05875
$630$	0	0
$631$	23.2111	0.924019	0.462010	−	0.886875i	$-0.347129\pi$
$631$	23.2111	0.924019	0.462010	+	0.886875i	$0.347129\pi$
$632$	1.21110i	0.0481751i
$633$	0	0
$634$	12.4222	0.493349
$635$	0	0
$636$	0	0
$637$	0.211103i	0.00836419i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	48.0555i	1.89512i	0.319571	+	0.947562i	$0.396461\pi$
$643$	48.0555i	1.89512i	−0.319571	+	0.947562i	$0.603539\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	30.4222	1.19695
$647$	35.0278i	1.37708i	0.725197	+	0.688542i	$0.241749\pi$
$647$	35.0278i	1.37708i	−0.725197	+	0.688542i	$0.758251\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 2.00000i	− 0.0783260i
$653$	− 2.78890i	− 0.109138i	−0.998510	−	0.0545690i	$-0.982622\pi$
$653$	− 2.78890i	− 0.109138i	0.998510	−	0.0545690i	$-0.0173785\pi$
$654$	0	0
$655$	0	0
$656$	3.21110	0.125372
$657$	0	0
$658$	24.0000i	0.935617i
$659$	−13.8167	−0.538220	−0.269110	−	0.963109i	$-0.586730\pi$
$659$	−13.8167	−0.538220	−0.269110	+	0.963109i	$0.586730\pi$
$660$	0	0
$661$	−21.0278	−0.817885	−0.408942	−	0.912560i	$-0.634102\pi$
$661$	−21.0278	−0.817885	−0.408942	+	0.912560i	$0.634102\pi$
$662$	5.39445i	0.209661i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.2111i	0.821297i
$668$	− 18.4222i	− 0.712777i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 19.2111i	− 0.740534i	−0.928925	−	0.370267i	$-0.879266\pi$
$673$	− 19.2111i	− 0.740534i	0.928925	−	0.370267i	$-0.120734\pi$
$674$	−26.8444	−1.03401
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 9.21110i	− 0.354011i	−0.984210	−	0.177006i	$-0.943359\pi$
$677$	− 9.21110i	− 0.354011i	0.984210	−	0.177006i	$-0.0566411\pi$
$678$	0	0
$679$	17.2111	0.660501
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.57779i	0.213428i	0.994290	+	0.106714i	$0.0340330\pi$
$683$	5.57779i	0.213428i	−0.994290	+	0.106714i	$0.965967\pi$
$684$	0	0
$685$	0	0
$686$	18.7889	0.717363
$687$	0	0
$688$	5.21110i	0.198671i
$689$	0	0
$690$	0	0
$691$	−23.8167	−0.906028	−0.453014	−	0.891503i	$-0.649651\pi$
$691$	−23.8167	−0.906028	−0.453014	+	0.891503i	$0.649651\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	−14.7889	−0.561379
$695$	0	0
$696$	0	0
$697$	− 14.7889i	− 0.560169i
$698$	− 11.8167i	− 0.447267i
$699$	0	0
$700$	0	0
$701$	16.6056	0.627183	0.313592	−	0.949558i	$-0.398468\pi$
$701$	16.6056	0.627183	0.313592	+	0.949558i	$0.398468\pi$
$702$	0	0
$703$	74.0555i	2.79306i
$704$	0	0
$705$	0	0
$706$	21.6333	0.814180
$707$	− 36.0000i	− 1.35392i
$708$	0	0
$709$	−31.4500	−1.18113	−0.590564	−	0.806991i	$-0.701095\pi$
$709$	−31.4500	−1.18113	−0.590564	+	0.806991i	$0.701095\pi$
$710$	0	0
$711$	0	0
$712$	− 3.21110i	− 0.120341i
$713$	9.21110i	0.344959i
$714$	0	0
$715$	0	0
$716$	11.0278	0.412127
$717$	0	0
$718$	− 30.4222i	− 1.13535i
$719$	−21.2111	−0.791041	−0.395520	−	0.918457i	$-0.629436\pi$
$719$	−21.2111	−0.791041	−0.395520	+	0.918457i	$0.629436\pi$
$720$	0	0
$721$	10.4222	0.388143
$722$	− 24.6333i	− 0.916757i
$723$	0	0
$724$	19.2111	0.713975
$725$	0	0
$726$	0	0
$727$	43.6333i	1.61827i	0.587623	+	0.809135i	$0.300064\pi$
$727$	43.6333i	1.61827i	−0.587623	+	0.809135i	$0.699936\pi$
$728$	− 2.60555i	− 0.0965682i
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	41.6333i	1.53776i	0.639392	+	0.768881i	$0.279186\pi$
$733$	41.6333i	1.53776i	−0.639392	+	0.768881i	$0.720814\pi$
$734$	6.78890	0.250583
$735$	0	0
$736$	4.60555	0.169763
$737$	0	0
$738$	0	0
$739$	26.6056	0.978701	0.489351	−	0.872087i	$-0.337234\pi$
$739$	26.6056	0.978701	0.489351	+	0.872087i	$0.337234\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 46.0555i	− 1.68961i	−0.535072	−	0.844806i	$-0.679716\pi$
$743$	− 46.0555i	− 1.68961i	0.535072	−	0.844806i	$-0.320284\pi$
$744$	0	0
$745$	0	0
$746$	−0.788897	−0.0288836
$747$	0	0
$748$	0	0
$749$	−31.2666	−1.14246
$750$	0	0
$751$	−35.2666	−1.28690	−0.643449	−	0.765489i	$-0.722497\pi$
$751$	−35.2666	−1.28690	−0.643449	+	0.765489i	$0.722497\pi$
$752$	9.21110i	0.335894i
$753$	0	0
$754$	4.60555	0.167724
$755$	0	0
$756$	0	0
$757$	34.8444i	1.26644i	0.773971	+	0.633221i	$0.218268\pi$
$757$	34.8444i	1.26644i	−0.773971	+	0.633221i	$0.781732\pi$
$758$	− 26.6056i	− 0.966357i
$759$	0	0
$760$	0	0
$761$	−9.63331	−0.349207	−0.174604	−	0.984639i	$-0.555864\pi$
$761$	−9.63331	−0.349207	−0.174604	+	0.984639i	$0.555864\pi$
$762$	0	0
$763$	− 48.4777i	− 1.75501i
$764$	12.0000	0.434145
$765$	0	0
$766$	18.4222	0.665621
$767$	9.21110i	0.332594i
$768$	0	0
$769$	34.8444	1.25652	0.628261	−	0.778003i	$-0.283767\pi$
$769$	34.8444	1.25652	0.628261	+	0.778003i	$0.283767\pi$
$770$	0	0
$771$	0	0
$772$	14.6056i	0.525665i
$773$	− 12.4222i	− 0.446796i	−0.974727	−	0.223398i	$-0.928285\pi$
$773$	− 12.4222i	− 0.446796i	0.974727	−	0.223398i	$-0.0717149\pi$
$774$	0	0
$775$	0	0
$776$	6.60555	0.237125
$777$	0	0
$778$	− 1.81665i	− 0.0651302i
$779$	−21.2111	−0.759967
$780$	0	0
$781$	0	0
$782$	− 21.2111i	− 0.758507i
$783$	0	0
$784$	0.211103	0.00753938
$785$	0	0
$786$	0	0
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	− 24.4222i	− 0.870005i
$789$	0	0
$790$	0	0
$791$	−19.2666	−0.685042
$792$	0	0
$793$	− 7.21110i	− 0.256074i
$794$	−35.2111	−1.24960
$795$	0	0
$796$	5.21110	0.184703
$797$	− 42.4222i	− 1.50267i	−0.659920	−	0.751336i	$-0.729410\pi$
$797$	− 42.4222i	− 1.50267i	0.659920	−	0.751336i	$-0.270590\pi$
$798$	0	0
$799$	42.4222	1.50079
$800$	0	0
$801$	0	0
$802$	− 33.6333i	− 1.18763i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	2.00000	0.0704470
$807$	0	0
$808$	− 13.8167i	− 0.486068i
$809$	−6.42221	−0.225793	−0.112896	−	0.993607i	$-0.536013\pi$
$809$	−6.42221	−0.225793	−0.112896	+	0.993607i	$0.536013\pi$
$810$	0	0
$811$	49.0278	1.72160	0.860799	−	0.508946i	$-0.169965\pi$
$811$	49.0278	1.72160	0.860799	+	0.508946i	$0.169965\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 34.4222i	− 1.20428i
$818$	− 13.6333i	− 0.476677i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	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
$832$	− 1.00000i	− 0.0346688i
$833$	− 0.972244i	− 0.0336862i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 29.4500i	− 1.01733i
$839$	42.4222	1.46458	0.732289	−	0.680994i	$-0.238452\pi$
$839$	42.4222	1.46458	0.732289	+	0.680994i	$0.238452\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	9.02776i	0.311117i
$843$	0	0
$844$	25.2111	0.867802
$845$	0	0
$846$	0	0
$847$	− 28.6611i	− 0.984806i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	51.6333	1.76997
$852$	0	0
$853$	24.0555i	0.823645i	0.911264	+	0.411823i	$0.135108\pi$
$853$	24.0555i	0.823645i	−0.911264	+	0.411823i	$0.864892\pi$
$854$	−18.7889	−0.642943
$855$	0	0
$856$	−12.0000	−0.410152
$857$	29.4500i	1.00599i	0.864289	+	0.502996i	$0.167769\pi$
$857$	29.4500i	1.00599i	−0.864289	+	0.502996i	$0.832231\pi$
$858$	0	0
$859$	−4.36669	−0.148990	−0.0744948	−	0.997221i	$-0.523734\pi$
$859$	−4.36669	−0.148990	−0.0744948	+	0.997221i	$0.523734\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 42.4222i	− 1.44407i	−0.691857	−	0.722034i	$-0.743207\pi$
$863$	− 42.4222i	− 1.44407i	0.691857	−	0.722034i	$-0.256793\pi$
$864$	0	0
$865$	0	0
$866$	−22.8444	−0.776285
$867$	0	0
$868$	− 5.21110i	− 0.176876i
$869$	0	0
$870$	0	0
$871$	−7.21110	−0.244339
$872$	− 18.6056i	− 0.630063i
$873$	0	0
$874$	−30.4222	−1.02905
$875$	0	0
$876$	0	0
$877$	− 26.0000i	− 0.877958i	−0.898497	−	0.438979i	$-0.855340\pi$
$877$	− 26.0000i	− 0.877958i	0.898497	−	0.438979i	$-0.144660\pi$
$878$	32.8444i	1.10845i
$879$	0	0
$880$	0	0
$881$	26.7889	0.902541	0.451270	−	0.892387i	$-0.350971\pi$
$881$	26.7889	0.902541	0.451270	+	0.892387i	$0.350971\pi$
$882$	0	0
$883$	14.4222i	0.485346i	0.970108	+	0.242673i	$0.0780242\pi$
$883$	14.4222i	0.485346i	−0.970108	+	0.242673i	$0.921976\pi$
$884$	−4.60555	−0.154901
$885$	0	0
$886$	−39.6333	−1.33151
$887$	− 4.60555i	− 0.154639i	−0.997006	−	0.0773196i	$-0.975364\pi$
$887$	− 4.60555i	− 0.154639i	0.997006	−	0.0773196i	$-0.0246362\pi$
$888$	0	0
$889$	−3.15559	−0.105835
$890$	0	0
$891$	0	0
$892$	17.3944i	0.582409i
$893$	− 60.8444i	− 2.03608i
$894$	0	0
$895$	0	0
$896$	−2.60555	−0.0870454
$897$	0	0
$898$	− 9.63331i	− 0.321468i
$899$	9.21110	0.307207
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−7.39445	−0.245936
$905$	0	0
$906$	0	0
$907$	12.3667i	0.410629i	0.978696	+	0.205315i	$0.0658218\pi$
$907$	12.3667i	0.410629i	−0.978696	+	0.205315i	$0.934178\pi$
$908$	6.42221i	0.213128i
$909$	0	0
$910$	0	0
$911$	15.6333	0.517955	0.258977	−	0.965883i	$-0.416615\pi$
$911$	15.6333	0.517955	0.258977	+	0.965883i	$0.416615\pi$
$912$	0	0
$913$	0	0
$914$	17.3944	0.575357
$915$	0	0
$916$	0.183346	0.00605793
$917$	− 43.2666i	− 1.42879i
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	0	0
$922$	3.21110i	0.105752i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	12.1833	0.400370
$927$	0	0
$928$	− 4.60555i	− 0.151185i
$929$	40.0555	1.31418	0.657089	−	0.753813i	$-0.271787\pi$
$929$	40.0555	1.31418	0.657089	+	0.753813i	$0.271787\pi$
$930$	0	0
$931$	−1.39445	−0.0457012
$932$	1.81665i	0.0595065i
$933$	0	0
$934$	5.57779	0.182511
$935$	0	0
$936$	0	0
$937$	13.6333i	0.445381i	0.974889	+	0.222690i	$0.0714839\pi$
$937$	13.6333i	0.445381i	−0.974889	+	0.222690i	$0.928516\pi$
$938$	18.7889i	0.613479i
$939$	0	0
$940$	0	0
$941$	21.6333	0.705226	0.352613	−	0.935769i	$-0.385293\pi$
$941$	21.6333	0.705226	0.352613	+	0.935769i	$0.385293\pi$
$942$	0	0
$943$	14.7889i	0.481593i
$944$	9.21110	0.299796
$945$	0	0
$946$	0	0
$947$	− 42.4222i	− 1.37854i	−0.724506	−	0.689268i	$-0.757932\pi$
$947$	− 42.4222i	− 1.37854i	0.724506	−	0.689268i	$-0.242068\pi$
$948$	0	0
$949$	−6.60555	−0.214425
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	16.6056i	0.537907i	0.963153	+	0.268953i	$0.0866777\pi$
$953$	16.6056i	0.537907i	−0.963153	+	0.268953i	$0.913322\pi$
$954$	0	0
$955$	0	0
$956$	18.4222	0.595817
$957$	0	0
$958$	18.4222i	0.595194i
$959$	56.3667	1.82018
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 11.2111i	− 0.361460i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	26.6056i	0.855577i	0.903879	+	0.427788i	$0.140707\pi$
$967$	26.6056i	0.855577i	−0.903879	+	0.427788i	$0.859293\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−19.3944	−0.622397	−0.311199	−	0.950345i	$-0.600730\pi$
$971$	−19.3944	−0.622397	−0.311199	+	0.950345i	$0.600730\pi$
$972$	0	0
$973$	− 37.5778i	− 1.20469i
$974$	−2.97224	−0.0952368
$975$	0	0
$976$	−7.21110	−0.230822
$977$	− 57.6333i	− 1.84385i	−0.387365	−	0.921926i	$-0.626615\pi$
$977$	− 57.6333i	− 1.84385i	0.387365	−	0.921926i	$-0.373385\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 32.2389i	− 1.02878i
$983$	27.6333i	0.881366i	0.897663	+	0.440683i	$0.145264\pi$
$983$	27.6333i	0.881366i	−0.897663	+	0.440683i	$0.854736\pi$
$984$	0	0
$985$	0	0
$986$	−21.2111	−0.675499
$987$	0	0
$988$	6.60555i	0.210151i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−31.6333	−1.00487	−0.502433	−	0.864616i	$-0.667561\pi$
$991$	−31.6333	−1.00487	−0.502433	+	0.864616i	$0.667561\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	0	0
$994$	−31.2666	−0.991717
$995$	0	0
$996$	0	0
$997$	− 4.78890i	− 0.151666i	−0.997121	−	0.0758330i	$-0.975838\pi$
$997$	− 4.78890i	− 0.151666i	0.997121	−	0.0758330i	$-0.0241616\pi$
$998$	27.8167i	0.880521i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bj.5149.2		4
3.2	odd	2		5850.2.e.bl.5149.4		4
5.2	odd	4		1170.2.a.q.1.1	yes	2
5.3	odd	4		5850.2.a.ce.1.2		2
5.4	even	2	inner	5850.2.e.bj.5149.3		4
15.2	even	4		1170.2.a.p.1.1	✓	2
15.8	even	4		5850.2.a.ck.1.2		2
15.14	odd	2		5850.2.e.bl.5149.1		4
20.7	even	4		9360.2.a.cn.1.2		2
60.47	odd	4		9360.2.a.cf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.p.1.1	✓	2	15.2	even	4
1170.2.a.q.1.1	yes	2	5.2	odd	4
5850.2.a.ce.1.2		2	5.3	odd	4
5850.2.a.ck.1.2		2	15.8	even	4
5850.2.e.bj.5149.2		4	1.1	even	1	trivial
5850.2.e.bj.5149.3		4	5.4	even	2	inner
5850.2.e.bl.5149.1		4	15.14	odd	2
5850.2.e.bl.5149.4		4	3.2	odd	2
9360.2.a.cf.1.2		2	60.47	odd	4
9360.2.a.cn.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.60555i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.60555i q^{7} +1.00000i q^{8} +1.00000i q^{13} +2.60555 q^{14} +1.00000 q^{16} -4.60555i q^{17} -6.60555 q^{19} +4.60555i q^{23} +1.00000 q^{26} -2.60555i q^{28} +4.60555 q^{29} +2.00000 q^{31} -1.00000i q^{32} -4.60555 q^{34} -11.2111i q^{37} +6.60555i q^{38} +3.21110 q^{41} +5.21110i q^{43} +4.60555 q^{46} +9.21110i q^{47} +0.211103 q^{49} -1.00000i q^{52} -2.60555 q^{56} -4.60555i q^{58} +9.21110 q^{59} -7.21110 q^{61} -2.00000i q^{62} -1.00000 q^{64} +7.21110i q^{67} +4.60555i q^{68} -12.0000 q^{71} +6.60555i q^{73} -11.2111 q^{74} +6.60555 q^{76} +1.21110 q^{79} -3.21110i q^{82} +5.21110 q^{86} -3.21110 q^{89} -2.60555 q^{91} -4.60555i q^{92} +9.21110 q^{94} -6.60555i q^{97} -0.211103i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} - 12 q^{19} + 4 q^{26} + 4 q^{29} + 8 q^{31} - 4 q^{34} - 16 q^{41} + 4 q^{46} - 28 q^{49} + 4 q^{56} + 8 q^{59} - 4 q^{64} - 48 q^{71} - 16 q^{74} + 12 q^{76} - 24 q^{79} - 8 q^{86} + 16 q^{89} + 4 q^{91} + 8 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 - 12 * q^19 + 4 * q^26 + 4 * q^29 + 8 * q^31 - 4 * q^34 - 16 * q^41 + 4 * q^46 - 28 * q^49 + 4 * q^56 + 8 * q^59 - 4 * q^64 - 48 * q^71 - 16 * q^74 + 12 * q^76 - 24 * q^79 - 8 * q^86 + 16 * q^89 + 4 * q^91 + 8 * q^94

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