LMFDB - Embedded newform 1305.2.c.b.784.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(784,1305)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.784"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,-2,0,0,0,0,4,0] 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:	$10.4204774638$
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 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			784.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1305.784
Dual form			1305.2.c.b.784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +(2.00000 - 1.00000i) q^{10} +4.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +(-1.00000 - 2.00000i) q^{20} -2.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +2.00000i q^{28} +1.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +(4.00000 - 2.00000i) q^{35} -2.00000i q^{37} +(6.00000 - 3.00000i) q^{40} -10.0000 q^{41} +2.00000 q^{46} +12.0000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +12.0000i q^{53} -6.00000 q^{56} +1.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -2.00000i q^{67} +2.00000i q^{68} +(2.00000 + 4.00000i) q^{70} +8.00000 q^{71} +14.0000i q^{73} +2.00000 q^{74} -8.00000 q^{79} +(1.00000 + 2.00000i) q^{80} -10.0000i q^{82} +6.00000i q^{83} +(4.00000 - 2.00000i) q^{85} +10.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} -12.0000 q^{94} -10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} - 4 q^{14} - 2 q^{16} - 2 q^{20} - 6 q^{25} - 8 q^{26} + 2 q^{29} + 8 q^{31} - 4 q^{34} + 8 q^{35} + 12 q^{40} - 20 q^{41} + 4 q^{46} + 6 q^{49} - 8 q^{50} - 12 q^{56}+ \cdots - 24 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 - 4 * q^14 - 2 * q^16 - 2 * q^20 - 6 * q^25 - 8 * q^26 + 2 * q^29 + 8 * q^31 - 4 * q^34 + 8 * q^35 + 12 * q^40 - 20 * q^41 + 4 * q^46 + 6 * q^49 - 8 * q^50 - 12 * q^56 + 8 * q^59 + 4 * q^61 - 14 * q^64 + 16 * q^65 + 4 * q^70 + 16 * q^71 + 4 * q^74 - 16 * q^79 + 2 * q^80 + 8 * q^85 + 20 * q^89 - 16 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$1$

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	0.935414	+	0.353553i	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			3.00000i			1.06066i
$9$		0			0
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000			−0.534522
$15$		0			0
$16$	−1.00000			−0.250000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.00000	−	2.00000i	−0.223607	−	0.447214i
$21$		0			0
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$			2.00000i			0.377964i
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$			5.00000i			0.883883i
$33$		0			0
$34$	−2.00000			−0.342997
$35$	4.00000	−	2.00000i	0.676123	−	0.338062i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	6.00000	−	3.00000i	0.948683	−	0.474342i
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	2.00000			0.294884
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$		0			0
$56$	−6.00000			−0.801784
$57$		0			0
$58$			1.00000i			0.131306i
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$			4.00000i			0.508001i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	8.00000	−	4.00000i	0.992278	−	0.496139i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$			2.00000i			0.242536i
$69$		0			0
$70$	2.00000	+	4.00000i	0.239046	+	0.478091i
$71$	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
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	2.00000			0.232495
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	+	2.00000i	0.111803	+	0.223607i
$81$		0			0
$82$		−	10.0000i		−	1.10432i
$83$			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
$85$	4.00000	−	2.00000i	0.433861	−	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		−	2.00000i		−	0.208514i
$93$		0			0
$94$	−12.0000			−1.23771
$95$		0			0
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$			3.00000i			0.303046i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$		−	2.00000i		−	0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$		−	2.00000i		−	0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	−12.0000			−1.17670
$105$		0			0
$106$	−12.0000			−1.16554
$107$			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
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$	−4.00000	+	2.00000i	−0.373002	+	0.186501i
$116$	1.00000			0.0928477
$117$		0			0
$118$			4.00000i			0.368230i
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−11.0000			−1.00000
$122$			2.00000i			0.181071i
$123$		0			0
$124$	4.00000			0.359211
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$		−	12.0000i		−	1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$		−	12.0000i		−	1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$			3.00000i			0.265165i
$129$		0			0
$130$	4.00000	+	8.00000i	0.350823	+	0.701646i
$131$	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
$133$		0			0
$134$	2.00000			0.172774
$135$		0			0
$136$	−6.00000			−0.514496
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	−	2.00000i	0.338062	−	0.169031i
$141$		0			0
$142$			8.00000i			0.671345i
$143$		0			0
$144$		0			0
$145$	−1.00000	−	2.00000i	−0.0830455	−	0.166091i
$146$	−14.0000			−1.15865
$147$		0			0
$148$		−	2.00000i		−	0.164399i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	−24.0000			−1.95309			−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000			−1.95309			−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000	−	8.00000i	−0.321288	−	0.642575i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	10.0000	−	5.00000i	0.790569	−	0.395285i
$161$	4.00000			0.315244
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−10.0000			−0.780869
$165$		0			0
$166$	−6.00000			−0.465690
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	2.00000	+	4.00000i	0.153393	+	0.306786i
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$		0			0
$177$		0			0
$178$			10.0000i			0.749532i
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		−	8.00000i		−	0.592999i
$183$		0			0
$184$	6.00000			0.442326
$185$	−4.00000	+	2.00000i	−0.294086	+	0.147043i
$186$		0			0
$187$		0			0
$188$			12.0000i			0.875190i
$189$		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
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	3.00000			0.214286
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$	−12.0000	−	9.00000i	−0.848528	−	0.636396i
$201$		0			0
$202$		−	2.00000i		−	0.140720i
$203$			2.00000i			0.140372i
$204$		0			0
$205$	10.0000	+	20.0000i	0.698430	+	1.39686i
$206$	2.00000			0.139347
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$			12.0000i			0.824163i
$213$		0			0
$214$	−6.00000			−0.410152
$215$		0			0
$216$		0			0
$217$			8.00000i			0.543075i
$218$			14.0000i			0.948200i
$219$		0			0
$220$		0			0
$221$	−8.00000			−0.538138
$222$		0			0
$223$		−	18.0000i		−	1.20537i	−0.797980	−	0.602685i	$-0.794098\pi$
$223$		−	18.0000i		−	1.20537i	0.797980	−	0.602685i	$-0.205902\pi$
$224$	−10.0000			−0.668153
$225$		0			0
$226$	14.0000			0.931266
$227$		−	6.00000i		−	0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$		−	6.00000i		−	0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$	−2.00000	−	4.00000i	−0.131876	−	0.263752i
$231$		0			0
$232$			3.00000i			0.196960i
$233$		−	20.0000i		−	1.31024i	−0.755523	−	0.655122i	$-0.772617\pi$
$233$		−	20.0000i		−	1.31024i	0.755523	−	0.655122i	$-0.227383\pi$
$234$		0			0
$235$	24.0000	−	12.0000i	1.56559	−	0.782794i
$236$	4.00000			0.260378
$237$		0			0
$238$		−	4.00000i		−	0.259281i
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	30.0000			1.93247			0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000			1.93247			0.966235	+	0.257663i	$0.0829523\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	2.00000			0.128037
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		0			0
$248$			12.0000i			0.762001i
$249$		0			0
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	12.0000			0.752947
$255$		0			0
$256$	−17.0000			−1.06250
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$	8.00000	−	4.00000i	0.496139	−	0.248069i
$261$		0			0
$262$			12.0000i			0.741362i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$	24.0000	−	12.0000i	1.47431	−	0.737154i
$266$		0			0
$267$		0			0
$268$		−	2.00000i		−	0.122169i
$269$	26.0000			1.58525			0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000			1.58525			0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		−	2.00000i		−	0.121268i
$273$		0			0
$274$	14.0000			0.845771
$275$		0			0
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$			4.00000i			0.239904i
$279$		0			0
$280$	6.00000	+	12.0000i	0.358569	+	0.717137i
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	10.0000i		−	0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$		−	10.0000i		−	0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	8.00000			0.474713
$285$		0			0
$286$		0			0
$287$		−	20.0000i		−	1.18056i
$288$		0			0
$289$	13.0000			0.764706
$290$	2.00000	−	1.00000i	0.117444	−	0.0587220i
$291$		0			0
$292$			14.0000i			0.819288i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	−4.00000	−	8.00000i	−0.232889	−	0.465778i
$296$	6.00000			0.348743
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$	8.00000			0.462652
$300$		0			0
$301$		0			0
$302$		−	24.0000i		−	1.38104i
$303$		0			0
$304$		0			0
$305$	−2.00000	−	4.00000i	−0.114520	−	0.229039i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$		0			0
$310$	8.00000	−	4.00000i	0.454369	−	0.227185i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$			28.0000i			1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$			28.0000i			1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	22.0000			1.24153
$315$		0			0
$316$	−8.00000			−0.450035
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$		0			0
$320$	7.00000	+	14.0000i	0.391312	+	0.782624i
$321$		0			0
$322$			4.00000i			0.222911i
$323$		0			0
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	4.00000			0.221540
$327$		0			0
$328$		−	30.0000i		−	1.65647i
$329$	−24.0000			−1.32316
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			6.00000i			0.329293i
$333$		0			0
$334$	−2.00000			−0.109435
$335$	−4.00000	+	2.00000i	−0.218543	+	0.109272i
$336$		0			0
$337$		−	2.00000i		−	0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$		−	2.00000i		−	0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$	4.00000	−	2.00000i	0.216930	−	0.108465i
$341$		0			0
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	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
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$	6.00000	−	8.00000i	0.320713	−	0.427618i
$351$		0			0
$352$		0			0
$353$		−	28.0000i		−	1.49029i	−0.666903	−	0.745145i	$-0.732380\pi$
$353$		−	28.0000i		−	1.49029i	0.666903	−	0.745145i	$-0.267620\pi$
$354$		0			0
$355$	−8.00000	−	16.0000i	−0.424596	−	0.849192i
$356$	10.0000			0.529999
$357$		0			0
$358$		−	20.0000i		−	1.05703i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$			2.00000i			0.105118i
$363$		0			0
$364$	−8.00000			−0.419314
$365$	28.0000	−	14.0000i	1.46559	−	0.732793i
$366$		0			0
$367$			24.0000i			1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$			24.0000i			1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$			2.00000i			0.104257i
$369$		0			0
$370$	−2.00000	−	4.00000i	−0.103975	−	0.207950i
$371$	−24.0000			−1.24602
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$		0			0
$375$		0			0
$376$	−36.0000			−1.85656
$377$			4.00000i			0.206010i
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		−	12.0000i		−	0.613973i
$383$		−	18.0000i		−	0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$		−	18.0000i		−	0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$		0			0
$385$		0			0
$386$	−6.00000			−0.305392
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	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
$391$	4.00000			0.202289
$392$			9.00000i			0.454569i
$393$		0			0
$394$	8.00000			0.403034
$395$	8.00000	+	16.0000i	0.402524	+	0.805047i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$			16.0000i			0.802008i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	−2.00000			−0.0992583
$407$		0			0
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$	−20.0000	+	10.0000i	−0.987730	+	0.493865i
$411$		0			0
$412$		−	2.00000i		−	0.0985329i
$413$			8.00000i			0.393654i
$414$		0			0
$415$	12.0000	−	6.00000i	0.589057	−	0.294528i
$416$	−20.0000			−0.980581
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$	−36.0000			−1.74831
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$			4.00000i			0.193574i
$428$			6.00000i			0.290021i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		−	14.0000i		−	0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$		−	14.0000i		−	0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	−8.00000			−0.384012
$435$		0			0
$436$	14.0000			0.670478
$437$		0			0
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$		0			0
$442$		−	8.00000i		−	0.380521i
$443$			24.0000i			1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$			24.0000i			1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$		0			0
$445$	−10.0000	−	20.0000i	−0.474045	−	0.948091i
$446$	18.0000			0.852325
$447$		0			0
$448$		−	14.0000i		−	0.661438i
$449$	−34.0000			−1.60456			−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000			−1.60456			−0.802280	+	0.596948i	$0.796380\pi$
$450$		0			0
$451$		0			0
$452$		−	14.0000i		−	0.658505i
$453$		0			0
$454$	6.00000			0.281594
$455$	8.00000	+	16.0000i	0.375046	+	0.750092i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$			22.0000i			1.02799i
$459$		0			0
$460$	−4.00000	+	2.00000i	−0.186501	+	0.0932505i
$461$	−26.0000			−1.21094			−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000			−1.21094			−0.605470	+	0.795868i	$0.707015\pi$
$462$		0			0
$463$		−	30.0000i		−	1.39422i	−0.716965	−	0.697109i	$-0.754469\pi$
$463$		−	30.0000i		−	1.39422i	0.716965	−	0.697109i	$-0.245531\pi$
$464$	−1.00000			−0.0464238
$465$		0			0
$466$	20.0000			0.926482
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$	12.0000	+	24.0000i	0.553519	+	1.10704i
$471$		0			0
$472$			12.0000i			0.552345i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−4.00000			−0.183340
$477$		0			0
$478$			24.0000i			1.09773i
$479$	36.0000			1.64488			0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000			1.64488			0.822441	+	0.568850i	$0.192612\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$			30.0000i			1.36646i
$483$		0			0
$484$	−11.0000			−0.500000
$485$	−20.0000	+	10.0000i	−0.908153	+	0.454077i
$486$		0			0
$487$		−	14.0000i		−	0.634401i	−0.948359	−	0.317200i	$-0.897257\pi$
$487$		−	14.0000i		−	0.634401i	0.948359	−	0.317200i	$-0.102743\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	6.00000	−	3.00000i	0.271052	−	0.135526i
$491$	4.00000			0.180517			0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000			0.180517			0.0902587	+	0.995918i	$0.471231\pi$
$492$		0			0
$493$			2.00000i			0.0900755i
$494$		0			0
$495$		0			0
$496$	−4.00000			−0.179605
$497$			16.0000i			0.717698i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	11.0000	+	2.00000i	0.491935	+	0.0894427i
$501$		0			0
$502$		0			0
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	2.00000	+	4.00000i	0.0889988	+	0.177998i
$506$		0			0
$507$		0			0
$508$		−	12.0000i		−	0.532414i
$509$	26.0000			1.15243			0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000			1.15243			0.576215	+	0.817298i	$0.304529\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$		−	11.0000i		−	0.486136i
$513$		0			0
$514$	−8.00000			−0.352865
$515$	−4.00000	+	2.00000i	−0.176261	+	0.0881305i
$516$		0			0
$517$		0			0
$518$			4.00000i			0.175750i
$519$		0			0
$520$	12.0000	+	24.0000i	0.526235	+	1.05247i
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$	16.0000			0.697633
$527$			8.00000i			0.348485i
$528$		0			0
$529$	19.0000			0.826087
$530$	12.0000	+	24.0000i	0.521247	+	1.04249i
$531$		0			0
$532$		0			0
$533$		−	40.0000i		−	1.73259i
$534$		0			0
$535$	12.0000	−	6.00000i	0.518805	−	0.259403i
$536$	6.00000			0.259161
$537$		0			0
$538$			26.0000i			1.12094i
$539$		0			0
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		−	24.0000i		−	1.03089i
$543$		0			0
$544$	−10.0000			−0.428746
$545$	−14.0000	−	28.0000i	−0.599694	−	1.19939i
$546$		0			0
$547$			46.0000i			1.96682i	0.181402	+	0.983409i	$0.441936\pi$
$547$			46.0000i			1.96682i	−0.181402	+	0.983409i	$0.558064\pi$
$548$		−	14.0000i		−	0.598050i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$	−8.00000			−0.339887
$555$		0			0
$556$	4.00000			0.169638
$557$		−	20.0000i		−	0.847427i	−0.905796	−	0.423714i	$-0.860726\pi$
$557$		−	20.0000i		−	0.847427i	0.905796	−	0.423714i	$-0.139274\pi$
$558$		0			0
$559$		0			0
$560$	−4.00000	+	2.00000i	−0.169031	+	0.0845154i
$561$		0			0
$562$			6.00000i			0.253095i
$563$			28.0000i			1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$			28.0000i			1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$		0			0
$565$	−28.0000	+	14.0000i	−1.17797	+	0.588984i
$566$	10.0000			0.420331
$567$		0			0
$568$			24.0000i			1.00702i
$569$	14.0000			0.586911			0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000			0.586911			0.293455	+	0.955973i	$0.405195\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$	20.0000			0.834784
$575$	8.00000	+	6.00000i	0.333623	+	0.250217i
$576$		0			0
$577$		−	6.00000i		−	0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$		−	6.00000i		−	0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$	−1.00000	−	2.00000i	−0.0415227	−	0.0830455i
$581$	−12.0000			−0.497844
$582$		0			0
$583$		0			0
$584$	−42.0000			−1.73797
$585$		0			0
$586$	6.00000			0.247858
$587$			42.0000i			1.73353i	0.498721	+	0.866763i	$0.333803\pi$
$587$			42.0000i			1.73353i	−0.498721	+	0.866763i	$0.666197\pi$
$588$		0			0
$589$		0			0
$590$	8.00000	−	4.00000i	0.329355	−	0.164677i
$591$		0			0
$592$			2.00000i			0.0821995i
$593$		−	36.0000i		−	1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	4.00000	+	8.00000i	0.163984	+	0.327968i
$596$	−6.00000			−0.245770
$597$		0			0
$598$			8.00000i			0.327144i
$599$	−36.0000			−1.47092			−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000			−1.47092			−0.735460	+	0.677568i	$0.763034\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$		0			0
$604$	−24.0000			−0.976546
$605$	11.0000	+	22.0000i	0.447214	+	0.894427i
$606$		0			0
$607$			28.0000i			1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$			28.0000i			1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$		0			0
$609$		0			0
$610$	4.00000	−	2.00000i	0.161955	−	0.0809776i
$611$	−48.0000			−1.94187
$612$		0			0
$613$			40.0000i			1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$			40.0000i			1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$	−8.00000			−0.322854
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	−36.0000			−1.44696			−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$	−4.00000	−	8.00000i	−0.160644	−	0.321288i
$621$		0			0
$622$		−	8.00000i		−	0.320771i
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−28.0000			−1.11911
$627$		0			0
$628$		−	22.0000i		−	0.877896i
$629$	4.00000			0.159490
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		−	24.0000i		−	0.954669i
$633$		0			0
$634$	6.00000			0.238290
$635$	−24.0000	+	12.0000i	−0.952411	+	0.476205i
$636$		0			0
$637$			12.0000i			0.475457i
$638$		0			0
$639$		0			0
$640$	6.00000	−	3.00000i	0.237171	−	0.118585i
$641$	−50.0000			−1.97488			−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000			−1.97488			−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		−	10.0000i		−	0.394362i	−0.980367	−	0.197181i	$-0.936821\pi$
$643$		−	10.0000i		−	0.394362i	0.980367	−	0.197181i	$-0.0631786\pi$
$644$	4.00000			0.157622
$645$		0			0
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$		0			0
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$	10.0000			0.390434
$657$		0			0
$658$		−	24.0000i		−	0.935617i
$659$	28.0000			1.09073			0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000			1.09073			0.545363	+	0.838200i	$0.316392\pi$
$660$		0			0
$661$	26.0000			1.01128			0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000			1.01128			0.505641	+	0.862744i	$0.331256\pi$
$662$			8.00000i			0.310929i
$663$		0			0
$664$	−18.0000			−0.698535
$665$		0			0
$666$		0			0
$667$		−	2.00000i		−	0.0774403i
$668$			2.00000i			0.0773823i
$669$		0			0
$670$	−2.00000	−	4.00000i	−0.0772667	−	0.154533i
$671$		0			0
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$	2.00000			0.0770371
$675$		0			0
$676$	−3.00000			−0.115385
$677$			14.0000i			0.538064i	0.963131	+	0.269032i	$0.0867037\pi$
$677$			14.0000i			0.538064i	−0.963131	+	0.269032i	$0.913296\pi$
$678$		0			0
$679$	20.0000			0.767530
$680$	6.00000	+	12.0000i	0.230089	+	0.460179i
$681$		0			0
$682$		0			0
$683$			18.0000i			0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$			18.0000i			0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$		0			0
$685$	−28.0000	+	14.0000i	−1.06983	+	0.534913i
$686$	−20.0000			−0.763604
$687$		0			0
$688$		0			0
$689$	−48.0000			−1.82865
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$		0			0
$693$		0			0
$694$	6.00000			0.227757
$695$	−4.00000	−	8.00000i	−0.151729	−	0.303457i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$			2.00000i			0.0757011i
$699$		0			0
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	28.0000			1.05379
$707$		−	4.00000i		−	0.150435i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$	16.0000	−	8.00000i	0.600469	−	0.300235i
$711$		0			0
$712$			30.0000i			1.12430i
$713$		−	8.00000i		−	0.299602i
$714$		0			0
$715$		0			0
$716$	−20.0000			−0.747435
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$		−	19.0000i		−	0.707107i
$723$		0			0
$724$	2.00000			0.0743294
$725$	−3.00000	+	4.00000i	−0.111417	+	0.148556i
$726$		0			0
$727$			40.0000i			1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$			40.0000i			1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$		−	24.0000i		−	0.889499i
$729$		0			0
$730$	14.0000	+	28.0000i	0.518163	+	1.03633i
$731$		0			0
$732$		0			0
$733$		−	50.0000i		−	1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$		−	50.0000i		−	1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	−24.0000			−0.885856
$735$		0			0
$736$	10.0000			0.368605
$737$		0			0
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$	−4.00000	+	2.00000i	−0.147043	+	0.0735215i
$741$		0			0
$742$		−	24.0000i		−	0.881068i
$743$		−	20.0000i		−	0.733729i	−0.930274	−	0.366864i	$-0.880431\pi$
$743$		−	20.0000i		−	0.733729i	0.930274	−	0.366864i	$-0.119569\pi$
$744$		0			0
$745$	6.00000	+	12.0000i	0.219823	+	0.439646i
$746$	−4.00000			−0.146450
$747$		0			0
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		−	12.0000i		−	0.437595i
$753$		0			0
$754$	−4.00000			−0.145671
$755$	24.0000	+	48.0000i	0.873449	+	1.74690i
$756$		0			0
$757$		−	42.0000i		−	1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$		−	42.0000i		−	1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$			28.0000i			1.01367i
$764$	−12.0000			−0.434145
$765$		0			0
$766$	18.0000			0.650366
$767$			16.0000i			0.577727i
$768$		0			0
$769$	−38.0000			−1.37032			−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000			−1.37032			−0.685158	+	0.728395i	$0.740267\pi$
$770$		0			0
$771$		0			0
$772$			6.00000i			0.215945i
$773$			30.0000i			1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$			30.0000i			1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$		0			0
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	30.0000			1.07694
$777$		0			0
$778$			2.00000i			0.0717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$			4.00000i			0.143040i
$783$		0			0
$784$	−3.00000			−0.107143
$785$	−44.0000	+	22.0000i	−1.57043	+	0.785214i
$786$		0			0
$787$			34.0000i			1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$			34.0000i			1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$		−	8.00000i		−	0.284988i
$789$		0			0
$790$	−16.0000	+	8.00000i	−0.569254	+	0.284627i
$791$	28.0000			0.995565
$792$		0			0
$793$			8.00000i			0.284088i
$794$	20.0000			0.709773
$795$		0			0
$796$	16.0000			0.567105
$797$		−	22.0000i		−	0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$		−	22.0000i		−	0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$	−20.0000	−	15.0000i	−0.707107	−	0.530330i
$801$		0			0
$802$			34.0000i			1.20058i
$803$		0			0
$804$		0			0
$805$	−4.00000	−	8.00000i	−0.140981	−	0.281963i
$806$	−16.0000			−0.563576
$807$		0			0
$808$		−	6.00000i		−	0.211079i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$			2.00000i			0.0701862i
$813$		0			0
$814$		0			0
$815$	−8.00000	+	4.00000i	−0.280228	+	0.140114i
$816$		0			0
$817$		0			0
$818$			22.0000i			0.769212i
$819$		0			0
$820$	10.0000	+	20.0000i	0.349215	+	0.698430i
$821$	42.0000			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$822$		0			0
$823$		−	40.0000i		−	1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$		−	40.0000i		−	1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	6.00000			0.209020
$825$		0			0
$826$	−8.00000			−0.278356
$827$			8.00000i			0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$			8.00000i			0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$	6.00000	+	12.0000i	0.208263	+	0.416526i
$831$		0			0
$832$		−	28.0000i		−	0.970725i
$833$			6.00000i			0.207888i
$834$		0			0
$835$	4.00000	−	2.00000i	0.138426	−	0.0692129i
$836$		0			0
$837$		0			0
$838$			12.0000i			0.414533i
$839$	20.0000			0.690477			0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000			0.690477			0.345238	+	0.938515i	$0.387798\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		−	22.0000i		−	0.758170i
$843$		0			0
$844$	−12.0000			−0.413057
$845$	3.00000	+	6.00000i	0.103203	+	0.206406i
$846$		0			0
$847$		−	22.0000i		−	0.755929i
$848$		−	12.0000i		−	0.412082i
$849$		0			0
$850$	6.00000	−	8.00000i	0.205798	−	0.274398i
$851$	−4.00000			−0.137118
$852$		0			0
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−4.00000			−0.136877
$855$		0			0
$856$	−18.0000			−0.615227
$857$			32.0000i			1.09310i	0.837427	+	0.546550i	$0.184059\pi$
$857$			32.0000i			1.09310i	−0.837427	+	0.546550i	$0.815941\pi$
$858$		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
$862$		0			0
$863$		−	50.0000i		−	1.70202i	−0.525150	−	0.851010i	$-0.675991\pi$
$863$		−	50.0000i		−	1.70202i	0.525150	−	0.851010i	$-0.324009\pi$
$864$		0			0
$865$		0			0
$866$	14.0000			0.475739
$867$		0			0
$868$			8.00000i			0.271538i
$869$		0			0
$870$		0			0
$871$	8.00000			0.271070
$872$			42.0000i			1.42230i
$873$		0			0
$874$		0			0
$875$	−4.00000	+	22.0000i	−0.135225	+	0.743736i
$876$		0			0
$877$		−	8.00000i		−	0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$		−	8.00000i		−	0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$			16.0000i			0.539974i
$879$		0			0
$880$		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
$883$		−	38.0000i		−	1.27880i	−0.768874	−	0.639401i	$-0.779182\pi$
$883$		−	38.0000i		−	1.27880i	0.768874	−	0.639401i	$-0.220818\pi$
$884$	−8.00000			−0.269069
$885$		0			0
$886$	−24.0000			−0.806296
$887$			44.0000i			1.47738i	0.674048	+	0.738688i	$0.264554\pi$
$887$			44.0000i			1.47738i	−0.674048	+	0.738688i	$0.735446\pi$
$888$		0			0
$889$	24.0000			0.804934
$890$	20.0000	−	10.0000i	0.670402	−	0.335201i
$891$		0			0
$892$		−	18.0000i		−	0.602685i
$893$		0			0
$894$		0			0
$895$	20.0000	+	40.0000i	0.668526	+	1.33705i
$896$	−6.00000			−0.200446
$897$		0			0
$898$		−	34.0000i		−	1.13459i
$899$	4.00000			0.133407
$900$		0			0
$901$	−24.0000			−0.799556
$902$		0			0
$903$		0			0
$904$	42.0000			1.39690
$905$	−2.00000	−	4.00000i	−0.0664822	−	0.132964i
$906$		0			0
$907$		−	8.00000i		−	0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$		−	8.00000i		−	0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$		−	6.00000i		−	0.199117i
$909$		0			0
$910$	−16.0000	+	8.00000i	−0.530395	+	0.265197i
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		0			0
$914$	−8.00000			−0.264616
$915$		0			0
$916$	22.0000			0.726900
$917$			24.0000i			0.792550i
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$	−6.00000	−	12.0000i	−0.197814	−	0.395628i
$921$		0			0
$922$		−	26.0000i		−	0.856264i
$923$			32.0000i			1.05329i
$924$		0			0
$925$	8.00000	+	6.00000i	0.263038	+	0.197279i
$926$	30.0000			0.985861
$927$		0			0
$928$			5.00000i			0.164133i
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$		0			0
$932$		−	20.0000i		−	0.655122i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			16.0000i			0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$			16.0000i			0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$			4.00000i			0.130605i
$939$		0			0
$940$	24.0000	−	12.0000i	0.782794	−	0.391397i
$941$	42.0000			1.36916			0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000			1.36916			0.684580	+	0.728937i	$0.259985\pi$
$942$		0			0
$943$			20.0000i			0.651290i
$944$	−4.00000			−0.130189
$945$		0			0
$946$		0			0
$947$			48.0000i			1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$			48.0000i			1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$		0			0
$949$	−56.0000			−1.81784
$950$		0			0
$951$		0			0
$952$		−	12.0000i		−	0.388922i
$953$		−	12.0000i		−	0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$		−	12.0000i		−	0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$		0			0
$955$	12.0000	+	24.0000i	0.388311	+	0.776622i
$956$	24.0000			0.776215
$957$		0			0
$958$			36.0000i			1.16311i
$959$	28.0000			0.904167
$960$		0			0
$961$	−15.0000			−0.483871
$962$			8.00000i			0.257930i
$963$		0			0
$964$	30.0000			0.966235
$965$	12.0000	−	6.00000i	0.386294	−	0.193147i
$966$		0			0
$967$			48.0000i			1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$			48.0000i			1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$		−	33.0000i		−	1.06066i
$969$		0			0
$970$	−10.0000	−	20.0000i	−0.321081	−	0.642161i
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$	14.0000			0.448589
$975$		0			0
$976$	−2.00000			−0.0640184
$977$		−	32.0000i		−	1.02377i	−0.859054	−	0.511885i	$-0.828947\pi$
$977$		−	32.0000i		−	1.02377i	0.859054	−	0.511885i	$-0.171053\pi$
$978$		0			0
$979$		0			0
$980$	−3.00000	−	6.00000i	−0.0958315	−	0.191663i
$981$		0			0
$982$			4.00000i			0.127645i
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	−16.0000	+	8.00000i	−0.509802	+	0.254901i
$986$	−2.00000			−0.0636930
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$			20.0000i			0.635001i
$993$		0			0
$994$	−16.0000			−0.507489
$995$	−16.0000	−	32.0000i	−0.507234	−	1.01447i
$996$		0			0
$997$		−	26.0000i		−	0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$		−	26.0000i		−	0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$			4.00000i			0.126618i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.c.b.784.2		2
3.2	odd	2		435.2.c.b.349.1	✓	2
5.2	odd	4		6525.2.a.c.1.1		1
5.3	odd	4		6525.2.a.k.1.1		1
5.4	even	2	inner	1305.2.c.b.784.1		2
15.2	even	4		2175.2.a.i.1.1		1
15.8	even	4		2175.2.a.c.1.1		1
15.14	odd	2		435.2.c.b.349.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.b.349.1	✓	2	3.2	odd	2
435.2.c.b.349.2	yes	2	15.14	odd	2
1305.2.c.b.784.1		2	5.4	even	2	inner
1305.2.c.b.784.2		2	1.1	even	1	trivial
2175.2.a.c.1.1		1	15.8	even	4
2175.2.a.i.1.1		1	15.2	even	4
6525.2.a.c.1.1		1	5.2	odd	4
6525.2.a.k.1.1		1	5.3	odd	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +(2.00000 - 1.00000i) q^{10} +4.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +(-1.00000 - 2.00000i) q^{20} -2.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +2.00000i q^{28} +1.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +(4.00000 - 2.00000i) q^{35} -2.00000i q^{37} +(6.00000 - 3.00000i) q^{40} -10.0000 q^{41} +2.00000 q^{46} +12.0000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +12.0000i q^{53} -6.00000 q^{56} +1.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -2.00000i q^{67} +2.00000i q^{68} +(2.00000 + 4.00000i) q^{70} +8.00000 q^{71} +14.0000i q^{73} +2.00000 q^{74} -8.00000 q^{79} +(1.00000 + 2.00000i) q^{80} -10.0000i q^{82} +6.00000i q^{83} +(4.00000 - 2.00000i) q^{85} +10.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} -12.0000 q^{94} -10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} - 4 q^{14} - 2 q^{16} - 2 q^{20} - 6 q^{25} - 8 q^{26} + 2 q^{29} + 8 q^{31} - 4 q^{34} + 8 q^{35} + 12 q^{40} - 20 q^{41} + 4 q^{46} + 6 q^{49} - 8 q^{50} - 12 q^{56}+ \cdots - 24 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 - 4 * q^14 - 2 * q^16 - 2 * q^20 - 6 * q^25 - 8 * q^26 + 2 * q^29 + 8 * q^31 - 4 * q^34 + 8 * q^35 + 12 * q^40 - 20 * q^41 + 4 * q^46 + 6 * q^49 - 8 * q^50 - 12 * q^56 + 8 * q^59 + 4 * q^61 - 14 * q^64 + 16 * q^65 + 4 * q^70 + 16 * q^71 + 4 * q^74 - 16 * q^79 + 2 * q^80 + 8 * q^85 + 20 * q^89 - 16 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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