LMFDB - Embedded newform 435.2.c.b.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(349,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.c (of order $2$, degree $1$, 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:	$3.47349248793$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	435.349
Dual form			435.2.c.b.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +(2.00000 - 1.00000i) q^{10} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +(-2.00000 + 1.00000i) q^{15} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +(1.00000 + 2.00000i) q^{20} -2.00000 q^{21} +2.00000i q^{23} +3.00000 q^{24} +(-3.00000 + 4.00000i) q^{25} +4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -1.00000 q^{29} +(1.00000 + 2.00000i) q^{30} +4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +(-4.00000 + 2.00000i) q^{35} -1.00000 q^{36} -2.00000i q^{37} -4.00000 q^{39} +(6.00000 - 3.00000i) q^{40} +10.0000 q^{41} +2.00000i q^{42} +(-1.00000 - 2.00000i) q^{45} +2.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(4.00000 + 3.00000i) q^{50} +2.00000 q^{51} +4.00000i q^{52} -12.0000i q^{53} -1.00000 q^{54} +6.00000 q^{56} +1.00000i q^{58} -4.00000 q^{59} +(-2.00000 + 1.00000i) q^{60} +2.00000 q^{61} -4.00000i q^{62} -2.00000i q^{63} -7.00000 q^{64} +(-8.00000 + 4.00000i) q^{65} -2.00000i q^{67} -2.00000i q^{68} -2.00000 q^{69} +(2.00000 + 4.00000i) q^{70} -8.00000 q^{71} +3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} +(-4.00000 - 3.00000i) q^{75} +4.00000i q^{78} -8.00000 q^{79} +(-1.00000 - 2.00000i) q^{80} +1.00000 q^{81} -10.0000i q^{82} -6.00000i q^{83} -2.00000 q^{84} +(4.00000 - 2.00000i) q^{85} -1.00000i q^{87} -10.0000 q^{89} +(-2.00000 + 1.00000i) q^{90} -8.00000 q^{91} +2.00000i q^{92} +4.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 4 q^{14} - 4 q^{15} - 2 q^{16} + 2 q^{20} - 4 q^{21} + 6 q^{24} - 6 q^{25} + 8 q^{26} - 2 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{34} - 8 q^{35} - 2 q^{36} - 8 q^{39} + 12 q^{40} + 20 q^{41} - 2 q^{45} + 4 q^{46} + 6 q^{49} + 8 q^{50} + 4 q^{51} - 2 q^{54} + 12 q^{56} - 8 q^{59} - 4 q^{60} + 4 q^{61} - 14 q^{64} - 16 q^{65} - 4 q^{69} + 4 q^{70} - 16 q^{71} - 4 q^{74} - 8 q^{75} - 16 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{84} + 8 q^{85} - 20 q^{89} - 4 q^{90} - 16 q^{91} - 24 q^{94} + 10 q^{96}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 + 4 * q^14 - 4 * q^15 - 2 * q^16 + 2 * q^20 - 4 * q^21 + 6 * q^24 - 6 * q^25 + 8 * q^26 - 2 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^34 - 8 * q^35 - 2 * q^36 - 8 * q^39 + 12 * q^40 + 20 * q^41 - 2 * q^45 + 4 * q^46 + 6 * q^49 + 8 * q^50 + 4 * q^51 - 2 * q^54 + 12 * q^56 - 8 * q^59 - 4 * q^60 + 4 * q^61 - 14 * q^64 - 16 * q^65 - 4 * q^69 + 4 * q^70 - 16 * q^71 - 4 * q^74 - 8 * q^75 - 16 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^84 + 8 * q^85 - 20 * q^89 - 4 * q^90 - 16 * q^91 - 24 * q^94 + 10 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$.

$n$	$31$	$146$	$262$
$\chi(n)$	$1$	$1$	$-1$

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	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$	1.00000			0.408248
$7$			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
$9$	−1.00000			−0.333333
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$			1.00000i			0.288675i
$13$			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
$15$	−2.00000	+	1.00000i	−0.516398	+	0.258199i
$16$	−1.00000			−0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$			1.00000i			0.235702i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	1.00000	+	2.00000i	0.223607	+	0.447214i
$21$	−2.00000			−0.436436
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	3.00000			0.612372
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	4.00000			0.784465
$27$		−	1.00000i		−	0.192450i
$28$			2.00000i			0.377964i
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$	1.00000	+	2.00000i	0.182574	+	0.365148i
$31$	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
$36$	−1.00000			−0.166667
$37$		−	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
$39$	−4.00000			−0.640513
$40$	6.00000	−	3.00000i	0.948683	−	0.474342i
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$			2.00000i			0.308607i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1.00000	−	2.00000i	−0.149071	−	0.298142i
$46$	2.00000			0.294884
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$	2.00000			0.280056
$52$			4.00000i			0.554700i
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	−1.00000			−0.136083
$55$		0			0
$56$	6.00000			0.801784
$57$		0			0
$58$			1.00000i			0.131306i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$	−2.00000	+	1.00000i	−0.258199	+	0.129099i
$61$	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
$63$		−	2.00000i		−	0.251976i
$64$	−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
$69$	−2.00000			−0.240772
$70$	2.00000	+	4.00000i	0.239046	+	0.478091i
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$			3.00000i			0.353553i
$73$			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
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$		0			0
$78$			4.00000i			0.452911i
$79$	−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
$81$	1.00000			0.111111
$82$		−	10.0000i		−	1.10432i
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−2.00000			−0.218218
$85$	4.00000	−	2.00000i	0.433861	−	0.216930i
$86$		0			0
$87$		−	1.00000i		−	0.107211i
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$	−2.00000	+	1.00000i	−0.210819	+	0.105409i
$91$	−8.00000			−0.838628
$92$			2.00000i			0.208514i
$93$			4.00000i			0.414781i
$94$	−12.0000			−1.23771
$95$		0			0
$96$	5.00000			0.510310
$97$		−	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	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	2.00000i		−	0.198030i
$103$		−	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
$105$	−2.00000	−	4.00000i	−0.195180	−	0.390360i
$106$	−12.0000			−1.16554
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	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
$111$	2.00000			0.189832
$112$		−	2.00000i		−	0.188982i
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	−4.00000	+	2.00000i	−0.373002	+	0.186501i
$116$	−1.00000			−0.0928477
$117$		−	4.00000i		−	0.369800i
$118$			4.00000i			0.368230i
$119$	4.00000			0.366679
$120$	3.00000	+	6.00000i	0.273861	+	0.547723i
$121$	−11.0000			−1.00000
$122$		−	2.00000i		−	0.181071i
$123$			10.0000i			0.901670i
$124$	4.00000			0.359211
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$	−2.00000			−0.178174
$127$		−	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	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$	−2.00000			−0.172774
$135$	2.00000	−	1.00000i	0.172133	−	0.0860663i
$136$	−6.00000			−0.514496
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$			2.00000i			0.170251i
$139$	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
$141$	12.0000			1.01058
$142$			8.00000i			0.671345i
$143$		0			0
$144$	1.00000			0.0833333
$145$	−1.00000	−	2.00000i	−0.0830455	−	0.166091i
$146$	14.0000			1.15865
$147$			3.00000i			0.247436i
$148$		−	2.00000i		−	0.164399i
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$	−3.00000	+	4.00000i	−0.244949	+	0.326599i
$151$	−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
$153$			2.00000i			0.161690i
$154$		0			0
$155$	4.00000	+	8.00000i	0.321288	+	0.642575i
$156$	−4.00000			−0.320256
$157$		−	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
$159$	12.0000			0.951662
$160$	10.0000	−	5.00000i	0.790569	−	0.395285i
$161$	−4.00000			−0.315244
$162$		−	1.00000i		−	0.0785674i
$163$		−	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	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$			6.00000i			0.462910i
$169$	−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			$\pi$
$174$	−1.00000			−0.0758098
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$		0			0
$177$		−	4.00000i		−	0.300658i
$178$			10.0000i			0.749532i
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−	2.00000i	−0.0745356	−	0.149071i
$181$	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
$183$			2.00000i			0.147844i
$184$	6.00000			0.442326
$185$	4.00000	−	2.00000i	0.294086	−	0.147043i
$186$	4.00000			0.293294
$187$		0			0
$188$		−	12.0000i		−	0.875190i
$189$	2.00000			0.145479
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		−	7.00000i		−	0.505181i
$193$			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
$195$	−4.00000	−	8.00000i	−0.286446	−	0.572892i
$196$	3.00000			0.214286
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		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
$201$	2.00000			0.141069
$202$		−	2.00000i		−	0.140720i
$203$		−	2.00000i		−	0.140372i
$204$	2.00000			0.140028
$205$	10.0000	+	20.0000i	0.698430	+	1.39686i
$206$	−2.00000			−0.139347
$207$		−	2.00000i		−	0.139010i
$208$		−	4.00000i		−	0.277350i
$209$		0			0
$210$	−4.00000	+	2.00000i	−0.276026	+	0.138013i
$211$	−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
$213$		−	8.00000i		−	0.548151i
$214$	−6.00000			−0.410152
$215$		0			0
$216$	−3.00000			−0.204124
$217$			8.00000i			0.543075i
$218$		−	14.0000i		−	0.948200i
$219$	−14.0000			−0.946032
$220$		0			0
$221$	8.00000			0.538138
$222$		−	2.00000i		−	0.134231i
$223$		−	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
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$	14.0000			0.931266
$227$			6.00000i			0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$			6.00000i			0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$		0			0
$229$	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	$0.227383\pi$
$233$			20.0000i			1.31024i	−0.755523	+	0.655122i	$0.772617\pi$
$234$	−4.00000			−0.261488
$235$	24.0000	−	12.0000i	1.56559	−	0.782794i
$236$	−4.00000			−0.260378
$237$		−	8.00000i		−	0.519656i
$238$		−	4.00000i		−	0.259281i
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$	2.00000	−	1.00000i	0.129099	−	0.0645497i
$241$	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
$243$			1.00000i			0.0641500i
$244$	2.00000			0.128037
$245$	3.00000	+	6.00000i	0.191663	+	0.383326i
$246$	10.0000			0.637577
$247$		0			0
$248$		−	12.0000i		−	0.762001i
$249$	6.00000			0.380235
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		−	2.00000i		−	0.125988i
$253$		0			0
$254$	−12.0000			−0.752947
$255$	2.00000	+	4.00000i	0.125245	+	0.250490i
$256$	−17.0000			−1.06250
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$	−8.00000	+	4.00000i	−0.496139	+	0.248069i
$261$	1.00000			0.0618984
$262$			12.0000i			0.741362i
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$	24.0000	−	12.0000i	1.47431	−	0.737154i
$266$		0			0
$267$		−	10.0000i		−	0.611990i
$268$		−	2.00000i		−	0.122169i
$269$	−26.0000			−1.58525			−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000			−1.58525			−0.792624	+	0.609711i	$0.791286\pi$
$270$	−1.00000	−	2.00000i	−0.0608581	−	0.121716i
$271$	−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
$273$		−	8.00000i		−	0.484182i
$274$	14.0000			0.845771
$275$		0			0
$276$	−2.00000			−0.120386
$277$			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
$279$	−4.00000			−0.239474
$280$	6.00000	+	12.0000i	0.358569	+	0.717137i
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		−	12.0000i		−	0.714590i
$283$		−	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
$288$			5.00000i			0.294628i
$289$	13.0000			0.764706
$290$	−2.00000	+	1.00000i	−0.117444	+	0.0587220i
$291$	10.0000			0.586210
$292$			14.0000i			0.819288i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	3.00000			0.174964
$295$	−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
$300$	−4.00000	−	3.00000i	−0.230940	−	0.173205i
$301$		0			0
$302$			24.0000i			1.38104i
$303$			2.00000i			0.114897i
$304$		0			0
$305$	2.00000	+	4.00000i	0.114520	+	0.229039i
$306$	2.00000			0.114332
$307$			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
$309$	2.00000			0.113776
$310$	8.00000	−	4.00000i	0.454369	−	0.227185i
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$			12.0000i			0.679366i
$313$			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
$315$	4.00000	−	2.00000i	0.225374	−	0.112687i
$316$	−8.00000			−0.450035
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		−	12.0000i		−	0.672927i
$319$		0			0
$320$	−7.00000	−	14.0000i	−0.391312	−	0.782624i
$321$	6.00000			0.334887
$322$			4.00000i			0.222911i
$323$		0			0
$324$	1.00000			0.0555556
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	−4.00000			−0.221540
$327$			14.0000i			0.774202i
$328$		−	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
$333$			2.00000i			0.109599i
$334$	−2.00000			−0.109435
$335$	4.00000	−	2.00000i	0.218543	−	0.109272i
$336$	2.00000			0.109109
$337$		−	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
$339$	−14.0000			−0.760376
$340$	4.00000	−	2.00000i	0.216930	−	0.108465i
$341$		0			0
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$	−2.00000	−	4.00000i	−0.107676	−	0.215353i
$346$		0			0
$347$			6.00000i			0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$			6.00000i			0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		−	1.00000i		−	0.0536056i
$349$	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
$351$	4.00000			0.213504
$352$		0			0
$353$			28.0000i			1.49029i	0.666903	+	0.745145i	$0.267620\pi$
$353$			28.0000i			1.49029i	−0.666903	+	0.745145i	$0.732380\pi$
$354$	−4.00000			−0.212598
$355$	−8.00000	−	16.0000i	−0.424596	−	0.849192i
$356$	−10.0000			−0.529999
$357$			4.00000i			0.211702i
$358$		−	20.0000i		−	1.05703i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−6.00000	+	3.00000i	−0.316228	+	0.158114i
$361$	−19.0000			−1.00000
$362$		−	2.00000i		−	0.105118i
$363$		−	11.0000i		−	0.577350i
$364$	−8.00000			−0.419314
$365$	−28.0000	+	14.0000i	−1.46559	+	0.732793i
$366$	2.00000			0.104542
$367$			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
$369$	−10.0000			−0.520579
$370$	−2.00000	−	4.00000i	−0.103975	−	0.207950i
$371$	24.0000			1.24602
$372$			4.00000i			0.207390i
$373$			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
$375$	2.00000	−	11.0000i	0.103280	−	0.568038i
$376$	−36.0000			−1.85656
$377$		−	4.00000i		−	0.206010i
$378$		−	2.00000i		−	0.102869i
$379$	−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
$381$	12.0000			0.614779
$382$		−	12.0000i		−	0.613973i
$383$			18.0000i			0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$			18.0000i			0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$	3.00000			0.153093
$385$		0			0
$386$	6.00000			0.305392
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−8.00000	+	4.00000i	−0.405096	+	0.202548i
$391$	4.00000			0.202289
$392$		−	9.00000i		−	0.454569i
$393$		−	12.0000i		−	0.605320i
$394$	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	$-0.822758\pi$
$401$	−34.0000			−1.69788			−0.848939	+	0.528490i	$0.822758\pi$
$402$		−	2.00000i		−	0.0997509i
$403$			16.0000i			0.797017i
$404$	2.00000			0.0995037
$405$	1.00000	+	2.00000i	0.0496904	+	0.0993808i
$406$	−2.00000			−0.0992583
$407$		0			0
$408$		−	6.00000i		−	0.297044i
$409$	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
$411$	−14.0000			−0.690569
$412$		−	2.00000i		−	0.0985329i
$413$		−	8.00000i		−	0.393654i
$414$	−2.00000			−0.0982946
$415$	12.0000	−	6.00000i	0.589057	−	0.294528i
$416$	20.0000			0.980581
$417$			4.00000i			0.195881i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$	−2.00000	−	4.00000i	−0.0975900	−	0.195180i
$421$	−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
$423$			12.0000i			0.583460i
$424$	−36.0000			−1.74831
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$	−8.00000			−0.387601
$427$			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	$0.5\pi$
$432$			1.00000i			0.0481125i
$433$		−	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
$435$	2.00000	−	1.00000i	0.0958927	−	0.0479463i
$436$	14.0000			0.670478
$437$		0			0
$438$			14.0000i			0.668946i
$439$	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
$441$	−3.00000			−0.142857
$442$		−	8.00000i		−	0.380521i
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	2.00000			0.0949158
$445$	−10.0000	−	20.0000i	−0.474045	−	0.948091i
$446$	−18.0000			−0.852325
$447$			6.00000i			0.283790i
$448$		−	14.0000i		−	0.661438i
$449$	34.0000			1.60456			0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000			1.60456			0.802280	+	0.596948i	$0.203620\pi$
$450$	−4.00000	−	3.00000i	−0.188562	−	0.141421i
$451$		0			0
$452$			14.0000i			0.658505i
$453$		−	24.0000i		−	1.12762i
$454$	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
$459$	−2.00000			−0.0933520
$460$	−4.00000	+	2.00000i	−0.186501	+	0.0932505i
$461$	26.0000			1.21094			0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000			1.21094			0.605470	+	0.795868i	$0.292985\pi$
$462$		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
$465$	−8.00000	+	4.00000i	−0.370991	+	0.185496i
$466$	20.0000			0.926482
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		−	4.00000i		−	0.184900i
$469$	4.00000			0.184703
$470$	−12.0000	−	24.0000i	−0.553519	−	1.10704i
$471$	22.0000			1.01371
$472$			12.0000i			0.552345i
$473$		0			0
$474$	−8.00000			−0.367452
$475$		0			0
$476$	4.00000			0.183340
$477$			12.0000i			0.549442i
$478$			24.0000i			1.09773i
$479$	−36.0000			−1.64488			−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000			−1.64488			−0.822441	+	0.568850i	$0.807388\pi$
$480$	5.00000	+	10.0000i	0.228218	+	0.456435i
$481$	8.00000			0.364769
$482$		−	30.0000i		−	1.36646i
$483$		−	4.00000i		−	0.182006i
$484$	−11.0000			−0.500000
$485$	20.0000	−	10.0000i	0.908153	−	0.454077i
$486$	1.00000			0.0453609
$487$		−	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
$489$	4.00000			0.180886
$490$	6.00000	−	3.00000i	0.271052	−	0.135526i
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$			10.0000i			0.450835i
$493$			2.00000i			0.0900755i
$494$		0			0
$495$		0			0
$496$	−4.00000			−0.179605
$497$		−	16.0000i		−	0.717698i
$498$		−	6.00000i		−	0.268866i
$499$	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
$501$	2.00000			0.0893534
$502$		0			0
$503$			4.00000i			0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$			4.00000i			0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	−6.00000			−0.267261
$505$	2.00000	+	4.00000i	0.0889988	+	0.177998i
$506$		0			0
$507$		−	3.00000i		−	0.133235i
$508$		−	12.0000i		−	0.532414i
$509$	−26.0000			−1.15243			−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000			−1.15243			−0.576215	+	0.817298i	$0.695471\pi$
$510$	4.00000	−	2.00000i	0.177123	−	0.0885615i
$511$	−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	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		−	1.00000i		−	0.0437688i
$523$			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
$525$	6.00000	−	8.00000i	0.261861	−	0.349149i
$526$	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
$531$	4.00000			0.173585
$532$		0			0
$533$			40.0000i			1.73259i
$534$	−10.0000			−0.432742
$535$	12.0000	−	6.00000i	0.518805	−	0.259403i
$536$	−6.00000			−0.259161
$537$			20.0000i			0.863064i
$538$			26.0000i			1.12094i
$539$		0			0
$540$	2.00000	−	1.00000i	0.0860663	−	0.0430331i
$541$	−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
$543$			2.00000i			0.0858282i
$544$	−10.0000			−0.428746
$545$	14.0000	+	28.0000i	0.599694	+	1.19939i
$546$	−8.00000			−0.342368
$547$			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
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$			6.00000i			0.255377i
$553$		−	16.0000i		−	0.680389i
$554$	8.00000			0.339887
$555$	2.00000	+	4.00000i	0.0848953	+	0.169791i
$556$	4.00000			0.169638
$557$			20.0000i			0.847427i	0.905796	+	0.423714i	$0.139274\pi$
$557$			20.0000i			0.847427i	−0.905796	+	0.423714i	$0.860726\pi$
$558$			4.00000i			0.169334i
$559$		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	$-0.799116\pi$
$563$		−	28.0000i		−	1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	12.0000			0.505291
$565$	−28.0000	+	14.0000i	−1.17797	+	0.588984i
$566$	−10.0000			−0.420331
$567$			2.00000i			0.0839921i
$568$			24.0000i			1.00702i
$569$	−14.0000			−0.586911			−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000			−0.586911			−0.293455	+	0.955973i	$0.594805\pi$
$570$		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
$573$			12.0000i			0.501307i
$574$	20.0000			0.834784
$575$	−8.00000	−	6.00000i	−0.333623	−	0.250217i
$576$	7.00000			0.291667
$577$		−	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
$579$	−6.00000			−0.249351
$580$	−1.00000	−	2.00000i	−0.0415227	−	0.0830455i
$581$	12.0000			0.497844
$582$		−	10.0000i		−	0.414513i
$583$		0			0
$584$	42.0000			1.73797
$585$	8.00000	−	4.00000i	0.330759	−	0.165380i
$586$	6.00000			0.247858
$587$		−	42.0000i		−	1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$		−	42.0000i		−	1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$			3.00000i			0.123718i
$589$		0			0
$590$	−8.00000	+	4.00000i	−0.329355	+	0.164677i
$591$	−8.00000			−0.329076
$592$			2.00000i			0.0821995i
$593$			36.0000i			1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$			36.0000i			1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$		0			0
$595$	4.00000	+	8.00000i	0.163984	+	0.327968i
$596$	6.00000			0.245770
$597$			16.0000i			0.654836i
$598$			8.00000i			0.327144i
$599$	36.0000			1.47092			0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000			1.47092			0.735460	+	0.677568i	$0.236966\pi$
$600$	−9.00000	+	12.0000i	−0.367423	+	0.489898i
$601$	−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
$603$			2.00000i			0.0814463i
$604$	−24.0000			−0.976546
$605$	−11.0000	−	22.0000i	−0.447214	−	0.894427i
$606$	2.00000			0.0812444
$607$			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
$609$	2.00000			0.0810441
$610$	4.00000	−	2.00000i	0.161955	−	0.0809776i
$611$	48.0000			1.94187
$612$			2.00000i			0.0808452i
$613$			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
$615$	−20.0000	+	10.0000i	−0.806478	+	0.403239i
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		−	2.00000i		−	0.0804518i
$619$	−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
$621$	2.00000			0.0802572
$622$		−	8.00000i		−	0.320771i
$623$		−	20.0000i		−	0.801283i
$624$	4.00000			0.160128
$625$	−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
$630$	−2.00000	−	4.00000i	−0.0796819	−	0.159364i
$631$	−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
$633$		−	12.0000i		−	0.476957i
$634$	6.00000			0.238290
$635$	24.0000	−	12.0000i	0.952411	−	0.476205i
$636$	12.0000			0.475831
$637$			12.0000i			0.475457i
$638$		0			0
$639$	8.00000			0.316475
$640$	6.00000	−	3.00000i	0.237171	−	0.118585i
$641$	50.0000			1.97488			0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000			1.97488			0.987441	+	0.157991i	$0.0505015\pi$
$642$		−	6.00000i		−	0.236801i
$643$		−	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	$0.115120\pi$
$647$			18.0000i			0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$		−	3.00000i		−	0.117851i
$649$		0			0
$650$	−12.0000	+	16.0000i	−0.470679	+	0.627572i
$651$	−8.00000			−0.313545
$652$		−	4.00000i		−	0.156652i
$653$		−	6.00000i		−	0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$		−	6.00000i		−	0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	14.0000			0.547443
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$	−10.0000			−0.390434
$657$		−	14.0000i		−	0.546192i
$658$		−	24.0000i		−	0.935617i
$659$	−28.0000			−1.09073			−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000			−1.09073			−0.545363	+	0.838200i	$0.683608\pi$
$660$		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
$663$			8.00000i			0.310694i
$664$	−18.0000			−0.698535
$665$		0			0
$666$	2.00000			0.0774984
$667$		−	2.00000i		−	0.0774403i
$668$		−	2.00000i		−	0.0773823i
$669$	18.0000			0.695920
$670$	−2.00000	−	4.00000i	−0.0772667	−	0.154533i
$671$		0			0
$672$			10.0000i			0.385758i
$673$		−	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
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$	−3.00000			−0.115385
$677$		−	14.0000i		−	0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$		−	14.0000i		−	0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$			14.0000i			0.537667i
$679$	20.0000			0.767530
$680$	−6.00000	−	12.0000i	−0.230089	−	0.460179i
$681$	−6.00000			−0.229920
$682$		0			0
$683$		−	18.0000i		−	0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$		−	18.0000i		−	0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$		0			0
$685$	−28.0000	+	14.0000i	−1.06983	+	0.534913i
$686$	20.0000			0.763604
$687$			22.0000i			0.839352i
$688$		0			0
$689$	48.0000			1.82865
$690$	−4.00000	+	2.00000i	−0.152277	+	0.0761387i
$691$	−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
$696$	−3.00000			−0.113715
$697$		−	20.0000i		−	0.757554i
$698$		−	2.00000i		−	0.0757011i
$699$	−20.0000			−0.756469
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		−	4.00000i		−	0.150970i
$703$		0			0
$704$		0			0
$705$	12.0000	+	24.0000i	0.451946	+	0.903892i
$706$	28.0000			1.05379
$707$			4.00000i			0.150435i
$708$		−	4.00000i		−	0.150329i
$709$	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
$711$	8.00000			0.300023
$712$			30.0000i			1.12430i
$713$			8.00000i			0.299602i
$714$	4.00000			0.149696
$715$		0			0
$716$	20.0000			0.747435
$717$		−	24.0000i		−	0.896296i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	1.00000	+	2.00000i	0.0372678	+	0.0745356i
$721$	4.00000			0.148968
$722$			19.0000i			0.707107i
$723$			30.0000i			1.11571i
$724$	2.00000			0.0743294
$725$	3.00000	−	4.00000i	0.111417	−	0.148556i
$726$	−11.0000			−0.408248
$727$			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
$729$	−1.00000			−0.0370370
$730$	14.0000	+	28.0000i	0.518163	+	1.03633i
$731$		0			0
$732$			2.00000i			0.0739221i
$733$		−	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
$735$	−6.00000	+	3.00000i	−0.221313	+	0.110657i
$736$	10.0000			0.368605
$737$		0			0
$738$			10.0000i			0.368105i
$739$	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	$0.119569\pi$
$743$			20.0000i			0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	12.0000			0.439941
$745$	6.00000	+	12.0000i	0.219823	+	0.439646i
$746$	4.00000			0.146450
$747$			6.00000i			0.219529i
$748$		0			0
$749$	12.0000			0.438470
$750$	−11.0000	−	2.00000i	−0.401663	−	0.0730297i
$751$	−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
$756$	2.00000			0.0727393
$757$		−	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	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		−	12.0000i		−	0.434714i
$763$			28.0000i			1.01367i
$764$	12.0000			0.434145
$765$	−4.00000	+	2.00000i	−0.144620	+	0.0723102i
$766$	18.0000			0.650366
$767$		−	16.0000i		−	0.577727i
$768$		−	17.0000i		−	0.613435i
$769$	−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
$771$	8.00000			0.288113
$772$			6.00000i			0.215945i
$773$		−	30.0000i		−	1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$		−	30.0000i		−	1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$		0			0
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	−30.0000			−1.07694
$777$			4.00000i			0.143499i
$778$			2.00000i			0.0717035i
$779$		0			0
$780$	−4.00000	−	8.00000i	−0.143223	−	0.286446i
$781$		0			0
$782$		−	4.00000i		−	0.143040i
$783$			1.00000i			0.0357371i
$784$	−3.00000			−0.107143
$785$	44.0000	−	22.0000i	1.57043	−	0.785214i
$786$	−12.0000

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:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +(2.00000 - 1.00000i) q^{10} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +(-2.00000 + 1.00000i) q^{15} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +(1.00000 + 2.00000i) q^{20} -2.00000 q^{21} +2.00000i q^{23} +3.00000 q^{24} +(-3.00000 + 4.00000i) q^{25} +4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -1.00000 q^{29} +(1.00000 + 2.00000i) q^{30} +4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +(-4.00000 + 2.00000i) q^{35} -1.00000 q^{36} -2.00000i q^{37} -4.00000 q^{39} +(6.00000 - 3.00000i) q^{40} +10.0000 q^{41} +2.00000i q^{42} +(-1.00000 - 2.00000i) q^{45} +2.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(4.00000 + 3.00000i) q^{50} +2.00000 q^{51} +4.00000i q^{52} -12.0000i q^{53} -1.00000 q^{54} +6.00000 q^{56} +1.00000i q^{58} -4.00000 q^{59} +(-2.00000 + 1.00000i) q^{60} +2.00000 q^{61} -4.00000i q^{62} -2.00000i q^{63} -7.00000 q^{64} +(-8.00000 + 4.00000i) q^{65} -2.00000i q^{67} -2.00000i q^{68} -2.00000 q^{69} +(2.00000 + 4.00000i) q^{70} -8.00000 q^{71} +3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} +(-4.00000 - 3.00000i) q^{75} +4.00000i q^{78} -8.00000 q^{79} +(-1.00000 - 2.00000i) q^{80} +1.00000 q^{81} -10.0000i q^{82} -6.00000i q^{83} -2.00000 q^{84} +(4.00000 - 2.00000i) q^{85} -1.00000i q^{87} -10.0000 q^{89} +(-2.00000 + 1.00000i) q^{90} -8.00000 q^{91} +2.00000i q^{92} +4.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 4 q^{14} - 4 q^{15} - 2 q^{16} + 2 q^{20} - 4 q^{21} + 6 q^{24} - 6 q^{25} + 8 q^{26} - 2 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{34} - 8 q^{35} - 2 q^{36} - 8 q^{39} + 12 q^{40} + 20 q^{41} - 2 q^{45} + 4 q^{46} + 6 q^{49} + 8 q^{50} + 4 q^{51} - 2 q^{54} + 12 q^{56} - 8 q^{59} - 4 q^{60} + 4 q^{61} - 14 q^{64} - 16 q^{65} - 4 q^{69} + 4 q^{70} - 16 q^{71} - 4 q^{74} - 8 q^{75} - 16 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{84} + 8 q^{85} - 20 q^{89} - 4 q^{90} - 16 q^{91} - 24 q^{94} + 10 q^{96}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 + 4 * q^14 - 4 * q^15 - 2 * q^16 + 2 * q^20 - 4 * q^21 + 6 * q^24 - 6 * q^25 + 8 * q^26 - 2 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^34 - 8 * q^35 - 2 * q^36 - 8 * q^39 + 12 * q^40 + 20 * q^41 - 2 * q^45 + 4 * q^46 + 6 * q^49 + 8 * q^50 + 4 * q^51 - 2 * q^54 + 12 * q^56 - 8 * q^59 - 4 * q^60 + 4 * q^61 - 14 * q^64 - 16 * q^65 - 4 * q^69 + 4 * q^70 - 16 * q^71 - 4 * q^74 - 8 * q^75 - 16 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^84 + 8 * q^85 - 20 * q^89 - 4 * q^90 - 16 * q^91 - 24 * q^94 + 10 * q^96

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