LMFDB - Embedded newform 2900.2.c.a.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(349,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.a.349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} -3.00000i q^{13} -2.00000i q^{17} -4.00000 q^{19} -12.0000 q^{21} -6.00000i q^{23} +9.00000i q^{27} +1.00000 q^{29} +9.00000 q^{31} +3.00000i q^{33} +8.00000i q^{37} -9.00000 q^{39} -8.00000 q^{41} -5.00000i q^{43} +7.00000i q^{47} -9.00000 q^{49} -6.00000 q^{51} -5.00000i q^{53} +12.0000i q^{57} +10.0000 q^{59} +10.0000 q^{61} +24.0000i q^{63} -8.00000i q^{67} -18.0000 q^{69} -2.00000 q^{71} +4.00000i q^{77} +1.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -3.00000i q^{87} -12.0000 q^{89} -12.0000 q^{91} -27.0000i q^{93} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9} - 2 q^{11} - 8 q^{19} - 24 q^{21} + 2 q^{29} + 18 q^{31} - 18 q^{39} - 16 q^{41} - 18 q^{49} - 12 q^{51} + 20 q^{59} + 20 q^{61} - 36 q^{69} - 4 q^{71} + 2 q^{79} + 18 q^{81} - 24 q^{89}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 - 2 * q^11 - 8 * q^19 - 24 * q^21 + 2 * q^29 + 18 * q^31 - 18 * q^39 - 16 * q^41 - 18 * q^49 - 12 * q^51 + 20 * q^59 + 20 * q^61 - 36 * q^69 - 4 * q^71 + 2 * q^79 + 18 * q^81 - 24 * q^89 - 24 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−12.0000	−2.61861
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	3.00000i	0.522233i
$34$	0	0
$35$	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
$39$	−9.00000	−1.44115
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	− 5.00000i	− 0.762493i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	− 5.00000i	− 0.762493i	0.924473	−	0.381246i	$-0.124505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 5.00000i	− 0.686803i	−0.939189	−	0.343401i	$-0.888421\pi$
$53$	− 5.00000i	− 0.686803i	0.939189	−	0.343401i	$-0.111579\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000i	1.58944i
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	24.0000i	3.02372i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	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
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	− 27.0000i	− 2.79977i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	18.0000i	1.66410i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	24.0000i	2.16401i
$124$	0	0
$125$	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	$0.884500\pi$
$128$	0	0
$129$	−15.0000	−1.32068
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	3.00000i	0.250873i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	27.0000i	2.22692i
$148$	0	0
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	12.0000i	0.970143i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−15.0000	−1.18958
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	7.00000i	0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$	7.00000i	0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	24.0000	1.83533
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 30.0000i	− 2.25494i
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	− 30.0000i	− 2.21766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	36.0000	2.61861
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	36.0000i	2.50217i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	6.00000i	0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 36.0000i	− 2.44384i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 30.0000i	− 1.99117i	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	− 30.0000i	− 1.99117i	0.0938647	−	0.995585i	$-0.470078\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	23.0000i	1.50678i	0.657574	+	0.753390i	$0.271583\pi$
$233$	23.0000i	1.50678i	−0.657574	+	0.753390i	$0.728417\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.00000i	− 0.194871i
$238$	0	0
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	−31.0000	−1.95670	−0.978351	−	0.206951i	$-0.933646\pi$
$251$	−31.0000	−1.95670	−0.978351	+	0.206951i	$0.933646\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	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
$259$	32.0000	1.98838
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	13.0000i	0.801614i	0.916162	+	0.400807i	$0.131270\pi$
$263$	13.0000i	0.801614i	−0.916162	+	0.400807i	$0.868730\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	36.0000i	2.20316i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	0	0
$273$	36.0000i	2.17882i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	−54.0000	−3.23290
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	32.0000i	1.88890i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.00000i	− 0.522233i
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 15.0000i	− 0.829502i
$328$	0	0
$329$	28.0000	1.54369
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	− 48.0000i	− 2.63038i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−9.00000	−0.487377
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	27.0000	1.44115
$352$	0	0
$353$	22.0000i	1.17094i	0.810693	+	0.585471i	$0.199090\pi$
$353$	22.0000i	1.17094i	−0.810693	+	0.585471i	$0.800910\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.0000i	1.27021i
$358$	0	0
$359$	−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
$361$	−3.00000	−0.157895
$362$	0	0
$363$	30.0000i	1.57459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	0	0
$369$	48.0000	2.49878
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	− 19.0000i	− 0.983783i	−0.870657	−	0.491891i	$-0.836306\pi$
$373$	− 19.0000i	− 0.983783i	0.870657	−	0.491891i	$-0.163694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	30.0000i	1.52499i
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	− 36.0000i	− 1.81596i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13.0000i	− 0.652451i	−0.945292	−	0.326226i	$-0.894223\pi$
$397$	− 13.0000i	− 0.652451i	0.945292	−	0.326226i	$-0.105777\pi$
$398$	0	0
$399$	48.0000	2.40301
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	− 27.0000i	− 1.34497i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	− 40.0000i	− 1.96827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 54.0000i	− 2.64439i
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	− 42.0000i	− 2.04211i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	0	0
$429$	9.00000	0.434524
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	− 28.0000i	− 1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$	− 28.0000i	− 1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	54.0000	2.57143
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	27.0000i	1.27706i
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 12.0000i	− 0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	0	0
$459$	18.0000	0.840168
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	− 10.0000i	− 0.464739i	−0.972628	−	0.232370i	$-0.925352\pi$
$463$	− 10.0000i	− 0.464739i	0.972628	−	0.232370i	$-0.0746479\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.0000i	0.601568i	0.953692	+	0.300784i	$0.0972484\pi$
$467$	13.0000i	0.601568i	−0.953692	+	0.300784i	$0.902752\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	5.00000i	0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	0	0
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	72.0000i	3.27611i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	0	0
$489$	21.0000	0.949653
$490$	0	0
$491$	17.0000	0.767199	0.383600	−	0.923499i	$-0.374684\pi$
$491$	17.0000	0.767199	0.383600	+	0.923499i	$0.374684\pi$
$492$	0	0
$493$	− 2.00000i	− 0.0900755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	23.0000i	1.02552i	0.858532	+	0.512760i	$0.171377\pi$
$503$	23.0000i	1.02552i	−0.858532	+	0.512760i	$0.828623\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 36.0000i	− 1.58944i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.00000i	− 0.307860i
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 18.0000i	− 0.784092i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−60.0000	−2.60378
$532$	0	0
$533$	24.0000i	1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	30.0000i	1.29460i
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	0	0
$543$	− 51.0000i	− 2.18862i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	−60.0000	−2.56074
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 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
$559$	−15.0000	−0.634432
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	− 33.0000i	− 1.39078i	−0.718631	−	0.695392i	$-0.755231\pi$
$563$	− 33.0000i	− 1.39078i	0.718631	−	0.695392i	$-0.244769\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 36.0000i	− 1.51186i
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	5.00000i	0.207079i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.0000i	− 1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$	− 42.0000i	− 1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	−78.0000	−3.20849
$592$	0	0
$593$	− 29.0000i	− 1.19089i	−0.803397	−	0.595444i	$-0.796976\pi$
$593$	− 29.0000i	− 1.19089i	0.803397	−	0.595444i	$-0.203024\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 60.0000i	− 2.45564i
$598$	0	0
$599$	−7.00000	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$599$	−7.00000	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	48.0000i	1.95471i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 47.0000i	− 1.90767i	−0.300329	−	0.953836i	$-0.597097\pi$
$607$	− 47.0000i	− 1.90767i	0.300329	−	0.953836i	$-0.402903\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	21.0000	0.849569
$612$	0	0
$613$	− 21.0000i	− 0.848182i	−0.905620	−	0.424091i	$-0.860594\pi$
$613$	− 21.0000i	− 0.848182i	0.905620	−	0.424091i	$-0.139406\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.0000i	1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$	36.0000i	1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$	0	0
$619$	41.0000	1.64793	0.823965	−	0.566641i	$-0.191757\pi$
$619$	41.0000	1.64793	0.823965	+	0.566641i	$0.191757\pi$
$620$	0	0
$621$	54.0000	2.16695
$622$	0	0
$623$	48.0000i	1.92308i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.0000i	− 0.479234i
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	39.0000i	1.55011i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	27.0000i	1.06978i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22.0000i	− 0.864909i	−0.901656	−	0.432455i	$-0.857648\pi$
$647$	− 22.0000i	− 0.864909i	0.901656	−	0.432455i	$-0.142352\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−108.000	−4.23285
$652$	0	0
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	18.0000i	0.699062i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.00000i	− 0.232321i
$668$	0	0
$669$	42.0000	1.62381
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	25.0000i	0.963679i	0.876259	+	0.481840i	$0.160031\pi$
$673$	25.0000i	0.963679i	−0.876259	+	0.481840i	$0.839969\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−90.0000	−3.44881
$682$	0	0
$683$	8.00000i	0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$	8.00000i	0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.0000i	2.28914i
$688$	0	0
$689$	−15.0000	−0.571454
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	− 24.0000i	− 0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	0	0
$699$	69.0000	2.60982
$700$	0	0
$701$	−35.0000	−1.32193	−0.660966	−	0.750416i	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	0	0
$703$	− 32.0000i	− 1.20690i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.0000i	0.601742i
$708$	0	0
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	− 54.0000i	− 2.02232i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 66.0000i	− 2.46482i
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	− 51.0000i	− 1.89671i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−10.0000	−0.369863
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	0	0
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 36.0000i	− 1.31717i
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	0	0
$753$	93.0000i	3.38911i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.0000i	1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$	34.0000i	1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	− 20.0000i	− 0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 30.0000i	− 1.08324i
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	81.0000	2.91714
$772$	0	0
$773$	36.0000i	1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$	36.0000i	1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 96.0000i	− 3.44398i
$778$	0	0
$779$	32.0000	1.14652
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	9.00000i	0.321634i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	0	0
$789$	39.0000	1.38844
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	− 30.0000i	− 1.06533i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	72.0000	2.54399
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.0000i	1.47847i
$808$	0	0
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	0	0
$813$	15.0000i	0.526073i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.0000i	0.699711i
$818$	0	0
$819$	72.0000	2.51588
$820$	0	0
$821$	−43.0000	−1.50071	−0.750355	−	0.661035i	$-0.770118\pi$
$821$	−43.0000	−1.50071	−0.750355	+	0.661035i	$0.770118\pi$
$822$	0	0
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.0000i	0.452054i	0.974121	+	0.226027i	$0.0725738\pi$
$827$	13.0000i	0.452054i	−0.974121	+	0.226027i	$0.927426\pi$
$828$	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
$831$	30.0000	1.04069
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	81.0000i	2.79977i
$838$	0	0
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	3.00000i	0.103325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	40.0000i	1.37442i
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	− 42.0000i	− 1.43805i	−0.694983	−	0.719026i	$-0.744588\pi$
$853$	− 42.0000i	− 1.43805i	0.694983	−	0.719026i	$-0.255412\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 49.0000i	− 1.67381i	−0.547350	−	0.836904i	$-0.684363\pi$
$857$	− 49.0000i	− 1.67381i	0.547350	−	0.836904i	$-0.315637\pi$
$858$	0	0
$859$	−9.00000	−0.307076	−0.153538	−	0.988143i	$-0.549067\pi$
$859$	−9.00000	−0.307076	−0.153538	+	0.988143i	$0.549067\pi$
$860$	0	0
$861$	96.0000	3.27167
$862$	0	0
$863$	54.0000i	1.83818i	0.394046	+	0.919091i	$0.371075\pi$
$863$	54.0000i	1.83818i	−0.394046	+	0.919091i	$0.628925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 39.0000i	− 1.32451i
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.0000i	0.979260i	0.871930	+	0.489630i	$0.162868\pi$
$877$	29.0000i	0.979260i	−0.871930	+	0.489630i	$0.837132\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	− 24.0000i	− 0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$	− 24.0000i	− 0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.0000i	1.84672i	0.383936	+	0.923360i	$0.374568\pi$
$887$	55.0000i	1.84672i	−0.383936	+	0.923360i	$0.625432\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	− 28.0000i	− 0.936984i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	54.0000i	1.80301i
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	60.0000i	1.99667i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	0	0
$911$	51.0000	1.68971	0.844853	−	0.534999i	$-0.179688\pi$
$911$	51.0000	1.68971	0.844853	+	0.534999i	$0.179688\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	0	0
$921$	−21.0000	−0.691974
$922$	0	0
$923$	6.00000i	0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 36.0000i	− 1.18240i
$928$	0	0
$929$	−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
$931$	36.0000	1.17985
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	3.00000	0.0979013
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−54.0000	−1.75107
$952$	0	0
$953$	3.00000i	0.0971795i	0.998819	+	0.0485898i	$0.0154727\pi$
$953$	3.00000i	0.0971795i	−0.998819	+	0.0485898i	$0.984527\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	0	0
$959$	−16.0000	−0.516667
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	− 48.0000i	− 1.54678i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17.0000i	− 0.546683i	−0.961917	−	0.273342i	$-0.911871\pi$
$967$	− 17.0000i	− 0.546683i	0.961917	−	0.273342i	$-0.0881289\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	− 72.0000i	− 2.30821i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25.0000i	− 0.799821i	−0.916554	−	0.399910i	$-0.869041\pi$
$977$	− 25.0000i	− 0.799821i	0.916554	−	0.399910i	$-0.130959\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	45.0000i	1.43528i	0.696416	+	0.717639i	$0.254777\pi$
$983$	45.0000i	1.43528i	−0.696416	+	0.717639i	$0.745223\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 84.0000i	− 2.67375i
$988$	0	0
$989$	−30.0000	−0.953945
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	− 9.00000i	− 0.285606i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.a.349.1		2
5.2	odd	4		116.2.a.a.1.1	✓	1
5.3	odd	4		2900.2.a.e.1.1		1
5.4	even	2	inner	2900.2.c.a.349.2		2
15.2	even	4		1044.2.a.c.1.1		1
20.7	even	4		464.2.a.g.1.1		1
35.27	even	4		5684.2.a.k.1.1		1
40.27	even	4		1856.2.a.a.1.1		1
40.37	odd	4		1856.2.a.o.1.1		1
60.47	odd	4		4176.2.a.c.1.1		1
145.12	even	4		3364.2.c.b.1681.1		2
145.17	even	4		3364.2.c.b.1681.2		2
145.57	odd	4		3364.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.a.a.1.1	✓	1	5.2	odd	4
464.2.a.g.1.1		1	20.7	even	4
1044.2.a.c.1.1		1	15.2	even	4
1856.2.a.a.1.1		1	40.27	even	4
1856.2.a.o.1.1		1	40.37	odd	4
2900.2.a.e.1.1		1	5.3	odd	4
2900.2.c.a.349.1		2	1.1	even	1	trivial
2900.2.c.a.349.2		2	5.4	even	2	inner
3364.2.a.c.1.1		1	145.57	odd	4
3364.2.c.b.1681.1		2	145.12	even	4
3364.2.c.b.1681.2		2	145.17	even	4
4176.2.a.c.1.1		1	60.47	odd	4
5684.2.a.k.1.1		1	35.27	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} -3.00000i q^{13} -2.00000i q^{17} -4.00000 q^{19} -12.0000 q^{21} -6.00000i q^{23} +9.00000i q^{27} +1.00000 q^{29} +9.00000 q^{31} +3.00000i q^{33} +8.00000i q^{37} -9.00000 q^{39} -8.00000 q^{41} -5.00000i q^{43} +7.00000i q^{47} -9.00000 q^{49} -6.00000 q^{51} -5.00000i q^{53} +12.0000i q^{57} +10.0000 q^{59} +10.0000 q^{61} +24.0000i q^{63} -8.00000i q^{67} -18.0000 q^{69} -2.00000 q^{71} +4.00000i q^{77} +1.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -3.00000i q^{87} -12.0000 q^{89} -12.0000 q^{91} -27.0000i q^{93} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9} - 2 q^{11} - 8 q^{19} - 24 q^{21} + 2 q^{29} + 18 q^{31} - 18 q^{39} - 16 q^{41} - 18 q^{49} - 12 q^{51} + 20 q^{59} + 20 q^{61} - 36 q^{69} - 4 q^{71} + 2 q^{79} + 18 q^{81} - 24 q^{89}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 - 2 * q^11 - 8 * q^19 - 24 * q^21 + 2 * q^29 + 18 * q^31 - 18 * q^39 - 16 * q^41 - 18 * q^49 - 12 * q^51 + 20 * q^59 + 20 * q^61 - 36 * q^69 - 4 * q^71 + 2 * q^79 + 18 * q^81 - 24 * q^89 - 24 * q^91 + 12 * q^99

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