LMFDB - Embedded newform 5850.2.e.be.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:	$2$
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			5149.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.be.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +5.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} -7.00000 q^{29} -9.00000 q^{31} +1.00000i q^{32} -5.00000 q^{34} +8.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} -5.00000 q^{44} -9.00000i q^{47} +6.00000 q^{49} +1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} -7.00000i q^{58} +1.00000 q^{59} -7.00000 q^{61} -9.00000i q^{62} -1.00000 q^{64} +15.0000i q^{67} -5.00000i q^{68} +8.00000 q^{71} +4.00000i q^{73} -8.00000 q^{74} +5.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} +9.00000i q^{83} -8.00000 q^{86} -5.00000i q^{88} +16.0000 q^{89} +1.00000 q^{91} +9.00000 q^{94} -2.00000i q^{97} +6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{26} - 14 q^{29} - 18 q^{31} - 10 q^{34} + 4 q^{41} - 10 q^{44} + 12 q^{49} + 2 q^{56} + 2 q^{59} - 14 q^{61} - 2 q^{64} + 16 q^{71} - 16 q^{74} + 8 q^{79} - 16 q^{86} + 32 q^{89} + 2 q^{91} + 18 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^26 - 14 * q^29 - 18 * q^31 - 10 * q^34 + 4 * q^41 - 10 * q^44 + 12 * q^49 + 2 * q^56 + 2 * q^59 - 14 * q^61 - 2 * q^64 + 16 * q^71 - 16 * q^74 + 8 * q^79 - 16 * q^86 + 32 * q^89 + 2 * 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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	5.00000i	1.06600i
$23$	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
$28$	− 1.00000i	− 0.188982i
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	0	0
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 7.00000i	− 0.919145i
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	− 9.00000i	− 1.14300i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	15.0000i	1.83254i	0.400559	+	0.916271i	$0.368816\pi$
$67$	15.0000i	1.83254i	−0.400559	+	0.916271i	$0.631184\pi$
$68$	− 5.00000i	− 0.606339i
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	0	0
$77$	5.00000i	0.569803i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	2.00000i	0.220863i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	− 5.00000i	− 0.533002i
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	9.00000	0.928279
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	11.0000	1.06841
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	7.00000	0.649934
$117$	0	0
$118$	1.00000i	0.0920575i
$119$	−5.00000	−0.458349
$120$	0	0
$121$	14.0000	1.27273
$122$	− 7.00000i	− 0.633750i
$123$	0	0
$124$	9.00000	0.808224
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	0	0
$134$	−15.0000	−1.29580
$135$	0	0
$136$	5.00000	0.428746
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−21.0000	−1.70896	−0.854478	−	0.519488i	$-0.826123\pi$
$151$	−21.0000	−1.70896	−0.854478	+	0.519488i	$0.826123\pi$
$152$	0	0
$153$	0	0
$154$	−5.00000	−0.402911
$155$	0	0
$156$	0	0
$157$	5.00000i	0.399043i	0.979893	+	0.199522i	$0.0639388\pi$
$157$	5.00000i	0.399043i	−0.979893	+	0.199522i	$0.936061\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−9.00000	−0.698535
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 8.00000i	− 0.609994i
$173$	11.0000i	0.836315i	0.908375	+	0.418157i	$0.137324\pi$
$173$	11.0000i	0.836315i	−0.908375	+	0.418157i	$0.862676\pi$
$174$	0	0
$175$	0	0
$176$	5.00000	0.376889
$177$	0	0
$178$	16.0000i	1.19925i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	25.0000i	1.82818i
$188$	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
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	7.00000i	0.492518i
$203$	− 7.00000i	− 0.491304i
$204$	0	0
$205$	0	0
$206$	−6.00000	−0.418040
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	11.0000i	0.755483i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	− 9.00000i	− 0.610960i
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	29.0000i	1.92480i	0.271640	+	0.962399i	$0.412434\pi$
$227$	29.0000i	1.92480i	−0.271640	+	0.962399i	$0.587566\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	7.00000i	0.459573i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−1.00000	−0.0650945
$237$	0	0
$238$	− 5.00000i	− 0.324102i
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	14.0000i	0.899954i
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	0	0
$248$	9.00000i	0.571501i
$249$	0	0
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.0000i	1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$	27.0000i	1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	2.00000i	0.123560i
$263$	− 26.0000i	− 1.60323i	−0.597841	−	0.801614i	$-0.703975\pi$
$263$	− 26.0000i	− 1.60323i	0.597841	−	0.801614i	$-0.296025\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 15.0000i	− 0.916271i
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	5.00000i	0.303170i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	2.00000i	0.119952i
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	5.00000	0.295656
$287$	2.00000i	0.118056i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	− 22.0000i	− 1.27443i
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 21.0000i	− 1.20841i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	− 5.00000i	− 0.284901i
$309$	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
$313$	21.0000i	1.18699i	0.804838	+	0.593495i	$0.202252\pi$
$313$	21.0000i	1.18699i	−0.804838	+	0.593495i	$0.797748\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−4.00000	−0.225018
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−35.0000	−1.95962
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	− 2.00000i	− 0.110432i
$329$	9.00000	0.496186
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 9.00000i	− 0.493939i
$333$	0	0
$334$	4.00000	0.218870
$335$	0	0
$336$	0	0
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−45.0000	−2.43689
$342$	0	0
$343$	13.0000i	0.701934i
$344$	8.00000	0.431331
$345$	0	0
$346$	−11.0000	−0.591364
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	0	0
$351$	0	0
$352$	5.00000i	0.266501i
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	0	0
$355$	0	0
$356$	−16.0000	−0.847998
$357$	0	0
$358$	2.00000i	0.105703i
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 11.0000i	− 0.578147i
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	0	0
$367$	36.0000i	1.87918i	0.342296	+	0.939592i	$0.388796\pi$
$367$	36.0000i	1.87918i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.0000	0.571092
$372$	0	0
$373$	− 17.0000i	− 0.880227i	−0.897942	−	0.440113i	$-0.854938\pi$
$373$	− 17.0000i	− 0.880227i	0.897942	−	0.440113i	$-0.145062\pi$
$374$	−25.0000	−1.29272
$375$	0	0
$376$	−9.00000	−0.464140
$377$	7.00000i	0.360518i
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	2.00000i	0.101535i
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	0	0
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	− 18.0000i	− 0.902258i
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	9.00000i	0.448322i
$404$	−7.00000	−0.348263
$405$	0	0
$406$	7.00000	0.347404
$407$	40.0000i	1.98273i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	− 6.00000i	− 0.295599i
$413$	1.00000i	0.0492068i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	16.0000i	0.778868i
$423$	0	0
$424$	−11.0000	−0.534207
$425$	0	0
$426$	0	0
$427$	− 7.00000i	− 0.338754i
$428$	− 6.00000i	− 0.290021i
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	5.00000i	0.237826i
$443$	32.0000i	1.52037i	0.649709	+	0.760183i	$0.274891\pi$
$443$	32.0000i	1.52037i	−0.649709	+	0.760183i	$0.725109\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	2.00000i	0.0940721i
$453$	0	0
$454$	−29.0000	−1.36104
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	2.00000i	0.0934539i
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 11.0000i	− 0.511213i	−0.966781	−	0.255607i	$-0.917725\pi$
$463$	− 11.0000i	− 0.511213i	0.966781	−	0.255607i	$-0.0822752\pi$
$464$	−7.00000	−0.324967
$465$	0	0
$466$	6.00000	0.277945
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	0	0
$472$	− 1.00000i	− 0.0460287i
$473$	40.0000i	1.83920i
$474$	0	0
$475$	0	0
$476$	5.00000	0.229175
$477$	0	0
$478$	1.00000i	0.0457389i
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	−14.0000	−0.636364
$485$	0	0
$486$	0	0
$487$	1.00000i	0.0453143i	0.999743	+	0.0226572i	$0.00721262\pi$
$487$	1.00000i	0.0453143i	−0.999743	+	0.0226572i	$0.992787\pi$
$488$	7.00000i	0.316875i
$489$	0	0
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	− 35.0000i	− 1.57632i
$494$	0	0
$495$	0	0
$496$	−9.00000	−0.404112
$497$	8.00000i	0.358849i
$498$	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
$502$	6.00000i	0.267793i
$503$	2.00000i	0.0891756i	0.999005	+	0.0445878i	$0.0141974\pi$
$503$	2.00000i	0.0891756i	−0.999005	+	0.0445878i	$0.985803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−27.0000	−1.19092
$515$	0	0
$516$	0	0
$517$	− 45.0000i	− 1.97910i
$518$	− 8.00000i	− 0.351500i
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	− 18.0000i	− 0.787085i	−0.919306	−	0.393543i	$-0.871249\pi$
$523$	− 18.0000i	− 0.787085i	0.919306	−	0.393543i	$-0.128751\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	26.0000	1.13365
$527$	− 45.0000i	− 1.96023i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00000i	− 0.0866296i
$534$	0	0
$535$	0	0
$536$	15.0000	0.647901
$537$	0	0
$538$	9.00000i	0.388018i
$539$	30.0000	1.29219
$540$	0	0
$541$	28.0000	1.20381	0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000	1.20381	0.601907	+	0.798566i	$0.294408\pi$
$542$	7.00000i	0.300676i
$543$	0	0
$544$	−5.00000	−0.214373
$545$	0	0
$546$	0	0
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4.00000i	0.170097i
$554$	26.0000	1.10463
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	− 30.0000i	− 1.26547i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	22.0000	0.924729
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	5.00000i	0.209061i
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	0	0
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	− 55.0000i	− 2.27787i
$584$	4.00000	0.165521
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	21.0000	0.854478
$605$	0	0
$606$	0	0
$607$	− 2.00000i	− 0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$	− 2.00000i	− 0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	5.00000	0.201456
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	16.0000i	0.641026i
$624$	0	0
$625$	0	0
$626$	−21.0000	−0.839329
$627$	0	0
$628$	− 5.00000i	− 0.199522i
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	− 35.0000i	− 1.38566i
$639$	0	0
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 48.0000i	− 1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$	− 48.0000i	− 1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	27.0000i	1.05659i	0.849060	+	0.528296i	$0.177169\pi$
$653$	27.0000i	1.05659i	−0.849060	+	0.528296i	$0.822831\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	0	0
$658$	9.00000i	0.350857i
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	0	0
$668$	4.00000i	0.154765i
$669$	0	0
$670$	0	0
$671$	−35.0000	−1.35116
$672$	0	0
$673$	11.0000i	0.424019i	0.977268	+	0.212009i	$0.0680008\pi$
$673$	11.0000i	0.424019i	−0.977268	+	0.212009i	$0.931999\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	− 45.0000i	− 1.72314i
$683$	− 19.0000i	− 0.727015i	−0.931591	−	0.363507i	$-0.881579\pi$
$683$	− 19.0000i	− 0.727015i	0.931591	−	0.363507i	$-0.118421\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	0	0
$688$	8.00000i	0.304997i
$689$	−11.0000	−0.419067
$690$	0	0
$691$	7.00000	0.266293	0.133146	−	0.991096i	$-0.457492\pi$
$691$	7.00000	0.266293	0.133146	+	0.991096i	$0.457492\pi$
$692$	− 11.0000i	− 0.418157i
$693$	0	0
$694$	6.00000	0.227757
$695$	0	0
$696$	0	0
$697$	10.0000i	0.378777i
$698$	− 32.0000i	− 1.21122i
$699$	0	0
$700$	0	0
$701$	25.0000	0.944237	0.472118	−	0.881535i	$-0.343489\pi$
$701$	25.0000	0.944237	0.472118	+	0.881535i	$0.343489\pi$
$702$	0	0
$703$	0	0
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	7.00000i	0.263262i
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	0	0
$712$	− 16.0000i	− 0.599625i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	27.0000i	1.00763i
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	11.0000	0.408812
$725$	0	0
$726$	0	0
$727$	38.0000i	1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$	38.0000i	1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	−40.0000	−1.47945
$732$	0	0
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	0	0
$737$	75.0000i	2.76266i
$738$	0	0
$739$	−3.00000	−0.110357	−0.0551784	−	0.998477i	$-0.517573\pi$
$739$	−3.00000	−0.110357	−0.0551784	+	0.998477i	$0.517573\pi$
$740$	0	0
$741$	0	0
$742$	11.0000i	0.403823i
$743$	− 39.0000i	− 1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$	− 39.0000i	− 1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$	0	0
$745$	0	0
$746$	17.0000	0.622414
$747$	0	0
$748$	− 25.0000i	− 0.914091i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	− 9.00000i	− 0.328196i
$753$	0	0
$754$	−7.00000	−0.254925
$755$	0	0
$756$	0	0
$757$	− 33.0000i	− 1.19941i	−0.800223	−	0.599703i	$-0.795286\pi$
$757$	− 33.0000i	− 1.19941i	0.800223	−	0.599703i	$-0.204714\pi$
$758$	15.0000i	0.544825i
$759$	0	0
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	− 6.00000i	− 0.217215i
$764$	18.0000	0.651217
$765$	0	0
$766$	0	0
$767$	− 1.00000i	− 0.0361079i
$768$	0	0
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	2.00000i	0.0717035i
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	5.00000i	0.178231i	0.996021	+	0.0891154i	$0.0284040\pi$
$787$	5.00000i	0.178231i	−0.996021	+	0.0891154i	$0.971596\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	7.00000i	0.248577i
$794$	−34.0000	−1.20661
$795$	0	0
$796$	18.0000	0.637993
$797$	− 13.0000i	− 0.460484i	−0.973133	−	0.230242i	$-0.926048\pi$
$797$	− 13.0000i	− 0.460484i	0.973133	−	0.230242i	$-0.0739517\pi$
$798$	0	0
$799$	45.0000	1.59199
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20.0000i	0.705785i
$804$	0	0
$805$	0	0
$806$	−9.00000	−0.317011
$807$	0	0
$808$	− 7.00000i	− 0.246259i
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	−29.0000	−1.01833	−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000	−1.01833	−0.509164	+	0.860670i	$0.670045\pi$
$812$	7.00000i	0.245652i
$813$	0	0
$814$	−40.0000	−1.40200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	14.0000i	0.489499i
$819$	0	0
$820$	0	0
$821$	16.0000	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$821$	16.0000	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−1.00000	−0.0347945
$827$	− 21.0000i	− 0.730242i	−0.930960	−	0.365121i	$-0.881028\pi$
$827$	− 21.0000i	− 0.730242i	0.930960	−	0.365121i	$-0.118972\pi$
$828$	0	0
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	30.0000i	1.03944i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 2.00000i	− 0.0690889i
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	28.0000i	0.964944i
$843$	0	0
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	− 11.0000i	− 0.377742i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 8.00000i	− 0.273915i	−0.990577	−	0.136957i	$-0.956268\pi$
$853$	− 8.00000i	− 0.273915i	0.990577	−	0.136957i	$-0.0437323\pi$
$854$	7.00000	0.239535
$855$	0	0
$856$	6.00000	0.205076
$857$	− 26.0000i	− 0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$	− 26.0000i	− 0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	− 27.0000i	− 0.919091i	−0.888154	−	0.459545i	$-0.848012\pi$
$863$	− 27.0000i	− 0.919091i	0.888154	−	0.459545i	$-0.151988\pi$
$864$	0	0
$865$	0	0
$866$	−34.0000	−1.15537
$867$	0	0
$868$	9.00000i	0.305480i
$869$	20.0000	0.678454
$870$	0	0
$871$	15.0000	0.508256
$872$	6.00000i	0.203186i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 40.0000i	− 1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$	− 40.0000i	− 1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$	− 4.00000i	− 0.134993i
$879$	0	0
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	0	0
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	−32.0000	−1.07506
$887$	4.00000i	0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$	4.00000i	0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	36.0000i	1.20134i
$899$	63.0000	2.10117
$900$	0	0
$901$	55.0000	1.83232
$902$	10.0000i	0.332964i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	0	0
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	− 29.0000i	− 0.962399i
$909$	0	0
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	45.0000i	1.48928i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	2.00000i	0.0660458i
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	11.0000	0.361482
$927$	0	0
$928$	− 7.00000i	− 0.229786i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	6.00000i	0.196537i
$933$	0	0
$934$	32.0000	1.04707
$935$	0	0
$936$	0	0
$937$	− 51.0000i	− 1.66610i	−0.553200	−	0.833049i	$-0.686593\pi$
$937$	− 51.0000i	− 1.66610i	0.553200	−	0.833049i	$-0.313407\pi$
$938$	− 15.0000i	− 0.489767i
$939$	0	0
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	0	0
$944$	1.00000	0.0325472
$945$	0	0
$946$	−40.0000	−1.30051
$947$	− 41.0000i	− 1.33232i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	− 41.0000i	− 1.33232i	0.745808	−	0.666160i	$-0.232063\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	5.00000i	0.162051i
$953$	− 51.0000i	− 1.65205i	−0.563632	−	0.826026i	$-0.690596\pi$
$953$	− 51.0000i	− 1.65205i	0.563632	−	0.826026i	$-0.309404\pi$
$954$	0	0
$955$	0	0
$956$	−1.00000	−0.0323423
$957$	0	0
$958$	29.0000i	0.936947i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	50.0000	1.61290
$962$	8.00000i	0.257930i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	17.0000i	0.546683i	0.961917	+	0.273342i	$0.0881289\pi$
$967$	17.0000i	0.546683i	−0.961917	+	0.273342i	$0.911871\pi$
$968$	− 14.0000i	− 0.449977i
$969$	0	0
$970$	0	0
$971$	34.0000	1.09111	0.545556	−	0.838074i	$-0.316319\pi$
$971$	34.0000	1.09111	0.545556	+	0.838074i	$0.316319\pi$
$972$	0	0
$973$	2.00000i	0.0641171i
$974$	−1.00000	−0.0320421
$975$	0	0
$976$	−7.00000	−0.224065
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	80.0000	2.55681
$980$	0	0
$981$	0	0
$982$	− 30.0000i	− 0.957338i
$983$	31.0000i	0.988746i	0.869250	+	0.494373i	$0.164602\pi$
$983$	31.0000i	0.988746i	−0.869250	+	0.494373i	$0.835398\pi$
$984$	0	0
$985$	0	0
$986$	35.0000	1.11463
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	− 9.00000i	− 0.285750i
$993$	0	0
$994$	−8.00000	−0.253745
$995$	0	0
$996$	0	0
$997$	11.0000i	0.348373i	0.984713	+	0.174187i	$0.0557296\pi$
$997$	11.0000i	0.348373i	−0.984713	+	0.174187i	$0.944270\pi$
$998$	− 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.be.5149.2		2
3.2	odd	2		1950.2.e.a.1249.1		2
5.2	odd	4		5850.2.a.k.1.1		1
5.3	odd	4		5850.2.a.bu.1.1		1
5.4	even	2	inner	5850.2.e.be.5149.1		2
15.2	even	4		1950.2.a.p.1.1	yes	1
15.8	even	4		1950.2.a.l.1.1	✓	1
15.14	odd	2		1950.2.e.a.1249.2		2

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

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +5.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} -7.00000 q^{29} -9.00000 q^{31} +1.00000i q^{32} -5.00000 q^{34} +8.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} -5.00000 q^{44} -9.00000i q^{47} +6.00000 q^{49} +1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} -7.00000i q^{58} +1.00000 q^{59} -7.00000 q^{61} -9.00000i q^{62} -1.00000 q^{64} +15.0000i q^{67} -5.00000i q^{68} +8.00000 q^{71} +4.00000i q^{73} -8.00000 q^{74} +5.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} +9.00000i q^{83} -8.00000 q^{86} -5.00000i q^{88} +16.0000 q^{89} +1.00000 q^{91} +9.00000 q^{94} -2.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{26} - 14 q^{29} - 18 q^{31} - 10 q^{34} + 4 q^{41} - 10 q^{44} + 12 q^{49} + 2 q^{56} + 2 q^{59} - 14 q^{61} - 2 q^{64} + 16 q^{71} - 16 q^{74} + 8 q^{79} - 16 q^{86} + 32 q^{89} + 2 q^{91} + 18 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^26 - 14 * q^29 - 18 * q^31 - 10 * q^34 + 4 * q^41 - 10 * q^44 + 12 * q^49 + 2 * q^56 + 2 * q^59 - 14 * q^61 - 2 * q^64 + 16 * q^71 - 16 * q^74 + 8 * q^79 - 16 * q^86 + 32 * q^89 + 2 * q^91 + 18 * q^94

Introduction

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Modular forms

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