LMFDB - Embedded newform 435.2.c.e.349.8

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:	$10$
Coefficient field:	10.0.3899266318336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 $ x^10 - 2x^9 + 2x^8 + 6x^7 + 19x^6 - 12x^5 + 4x^4 + 2x^3 + 9x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.8
Root			$-0.604479 + 0.604479i$ of defining polynomial
Character	$\chi$	$=$	435.349
Dual form			435.2.c.e.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.71457i q^{2} +1.00000i q^{3} -0.939748 q^{4} +(1.51903 - 1.64090i) q^{5} -1.71457 q^{6} +0.654317i q^{7} +1.81788i q^{8} -1.00000 q^{9} +(2.81344 + 2.60448i) q^{10} +0.163559 q^{11} -0.939748i q^{12} +2.65432i q^{13} -1.12187 q^{14} +(1.64090 + 1.51903i) q^{15} -4.99637 q^{16} +3.86328i q^{17} -1.71457i q^{18} +3.47954 q^{19} +(-1.42750 + 1.54203i) q^{20} -0.654317 q^{21} +0.280433i q^{22} +7.69972i q^{23} -1.81788 q^{24} +(-0.385107 - 4.98515i) q^{25} -4.55101 q^{26} -1.00000i q^{27} -0.614893i q^{28} -1.00000 q^{29} +(-2.60448 + 2.81344i) q^{30} +5.05274 q^{31} -4.93087i q^{32} +0.163559i q^{33} -6.62385 q^{34} +(1.07367 + 0.993926i) q^{35} +0.939748 q^{36} -10.5904i q^{37} +5.96591i q^{38} -2.65432 q^{39} +(2.98295 + 2.76140i) q^{40} -6.17417 q^{41} -1.12187i q^{42} -10.5547i q^{43} -0.153704 q^{44} +(-1.51903 + 1.64090i) q^{45} -13.2017 q^{46} -10.3036i q^{47} -4.99637i q^{48} +6.57187 q^{49} +(8.54738 - 0.660292i) q^{50} -3.86328 q^{51} -2.49439i q^{52} +4.04766i q^{53} +1.71457 q^{54} +(0.248450 - 0.268383i) q^{55} -1.18947 q^{56} +3.47954i q^{57} -1.71457i q^{58} +0.328734 q^{59} +(-1.54203 - 1.42750i) q^{60} -5.72054 q^{61} +8.66328i q^{62} -0.654317i q^{63} -1.53842 q^{64} +(4.35547 + 4.03198i) q^{65} -0.280433 q^{66} -3.51985i q^{67} -3.63050i q^{68} -7.69972 q^{69} +(-1.70416 + 1.84088i) q^{70} +11.7457 q^{71} -1.81788i q^{72} -1.12143i q^{73} +18.1580 q^{74} +(4.98515 - 0.385107i) q^{75} -3.26989 q^{76} +0.107019i q^{77} -4.55101i q^{78} +12.4074 q^{79} +(-7.58963 + 8.19854i) q^{80} +1.00000 q^{81} -10.5860i q^{82} +7.89306i q^{83} +0.614893 q^{84} +(6.33925 + 5.86842i) q^{85} +18.0968 q^{86} -1.00000i q^{87} +0.297329i q^{88} -5.04702 q^{89} +(-2.81344 - 2.60448i) q^{90} -1.73677 q^{91} -7.23579i q^{92} +5.05274i q^{93} +17.6663 q^{94} +(5.28551 - 5.70957i) q^{95} +4.93087 q^{96} -8.49076i q^{97} +11.2679i q^{98} -0.163559 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) $ 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9 + 4 * q^10 + 24 * q^11 - 12 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^19 + 8 * q^20 + 16 * q^21 - 18 * q^24 + 2 * q^25 - 10 * q^29 - 18 * q^30 + 4 * q^31 - 8 * q^34 + 2 * q^35 + 10 * q^36 - 4 * q^39 - 14 * q^40 - 28 * q^41 - 40 * q^44 - 12 * q^46 + 14 * q^49 - 12 * q^50 - 6 * q^54 + 2 * q^55 - 4 * q^56 - 8 * q^59 - 2 * q^60 - 24 * q^61 + 18 * q^64 + 6 * q^65 + 28 * q^66 - 16 * q^69 + 20 * q^70 + 60 * q^71 - 4 * q^74 + 12 * q^75 - 88 * q^76 + 36 * q^79 - 30 * q^80 + 10 * q^81 + 12 * q^84 - 14 * q^85 + 60 * q^86 + 44 * q^89 - 4 * q^90 - 24 * q^91 + 100 * q^94 - 36 * q^95 + 2 * q^96 - 24 * q^99

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 $

$2$			1.71457i			1.21238i	0.795319	+	0.606192i	$0.207304\pi$
$2$			1.71457i			1.21238i	−0.795319	+	0.606192i	$0.792696\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−0.939748			−0.469874
$5$	1.51903	−	1.64090i	0.679330	−	0.733833i
$5$	1.51903	−	1.64090i	0.679330	−	0.733833i
$6$	−1.71457			−0.699970
$7$			0.654317i			0.247309i	0.992325	+	0.123654i	$0.0394614\pi$
$7$			0.654317i			0.247309i	−0.992325	+	0.123654i	$0.960539\pi$
$8$			1.81788i			0.642716i
$9$	−1.00000			−0.333333
$10$	2.81344	+	2.60448i	0.889687	+	0.823609i
$11$	0.163559			0.0493148			0.0246574	−	0.999696i	$-0.492151\pi$
$11$	0.163559			0.0493148			0.0246574	+	0.999696i	$0.492151\pi$
$12$		−	0.939748i		−	0.271282i
$13$			2.65432i			0.736175i	0.929791	+	0.368088i	$0.119987\pi$
$13$			2.65432i			0.736175i	−0.929791	+	0.368088i	$0.880013\pi$
$14$	−1.12187			−0.299833
$15$	1.64090	+	1.51903i	0.423679	+	0.392211i
$16$	−4.99637			−1.24909
$17$			3.86328i			0.936982i	0.883468	+	0.468491i	$0.155202\pi$
$17$			3.86328i			0.936982i	−0.883468	+	0.468491i	$0.844798\pi$
$18$		−	1.71457i		−	0.404128i
$19$	3.47954			0.798260			0.399130	−	0.916894i	$-0.369312\pi$
$19$	3.47954			0.798260			0.399130	+	0.916894i	$0.369312\pi$
$20$	−1.42750	+	1.54203i	−0.319199	+	0.344809i
$21$	−0.654317			−0.142784
$22$			0.280433i			0.0597884i
$23$			7.69972i			1.60550i	0.596314	+	0.802751i	$0.296631\pi$
$23$			7.69972i			1.60550i	−0.596314	+	0.802751i	$0.703369\pi$
$24$	−1.81788			−0.371072
$25$	−0.385107	−	4.98515i	−0.0770214	−	0.997029i
$26$	−4.55101			−0.892527
$27$		−	1.00000i		−	0.192450i
$28$		−	0.614893i		−	0.116204i
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$	−2.60448	+	2.81344i	−0.475511	+	0.513661i
$31$	5.05274			0.907499			0.453750	−	0.891129i	$-0.350086\pi$
$31$	5.05274			0.907499			0.453750	+	0.891129i	$0.350086\pi$
$32$		−	4.93087i		−	0.871663i
$33$			0.163559i			0.0284719i
$34$	−6.62385			−1.13598
$35$	1.07367	+	0.993926i	0.181483	+	0.168004i
$36$	0.939748			0.156625
$37$		−	10.5904i		−	1.74106i	−0.492118	−	0.870528i	$-0.663777\pi$
$37$		−	10.5904i		−	1.74106i	0.492118	−	0.870528i	$-0.336223\pi$
$38$			5.96591i			0.967798i
$39$	−2.65432			−0.425031
$40$	2.98295	+	2.76140i	0.471646	+	0.436616i
$41$	−6.17417			−0.964244			−0.482122	−	0.876104i	$-0.660134\pi$
$41$	−6.17417			−0.964244			−0.482122	+	0.876104i	$0.660134\pi$
$42$		−	1.12187i		−	0.173109i
$43$		−	10.5547i		−	1.60958i	−0.593559	−	0.804790i	$-0.702278\pi$
$43$		−	10.5547i		−	1.60958i	0.593559	−	0.804790i	$-0.297722\pi$
$44$	−0.153704			−0.0231717
$45$	−1.51903	+	1.64090i	−0.226443	+	0.244611i
$46$	−13.2017			−1.94648
$47$		−	10.3036i		−	1.50294i	−0.659767	−	0.751470i	$-0.729345\pi$
$47$		−	10.3036i		−	1.50294i	0.659767	−	0.751470i	$-0.270655\pi$
$48$		−	4.99637i		−	0.721164i
$49$	6.57187			0.938838
$50$	8.54738	−	0.660292i	1.20878	−	0.0933795i
$51$	−3.86328			−0.540967
$52$		−	2.49439i		−	0.345909i
$53$			4.04766i			0.555989i	0.960583	+	0.277995i	$0.0896697\pi$
$53$			4.04766i			0.555989i	−0.960583	+	0.277995i	$0.910330\pi$
$54$	1.71457			0.233323
$55$	0.248450	−	0.268383i	0.0335010	−	0.0361888i
$56$	−1.18947			−0.158949
$57$			3.47954i			0.460876i
$58$		−	1.71457i		−	0.225134i
$59$	0.328734			0.0427975			0.0213988	−	0.999771i	$-0.493188\pi$
$59$	0.328734			0.0427975			0.0213988	+	0.999771i	$0.493188\pi$
$60$	−1.54203	−	1.42750i	−0.199076	−	0.184290i
$61$	−5.72054			−0.732441			−0.366220	−	0.930528i	$-0.619348\pi$
$61$	−5.72054			−0.732441			−0.366220	+	0.930528i	$0.619348\pi$
$62$			8.66328i			1.10024i
$63$		−	0.654317i		−	0.0824362i
$64$	−1.53842			−0.192303
$65$	4.35547	+	4.03198i	0.540230	+	0.500106i
$66$	−0.280433			−0.0345189
$67$		−	3.51985i		−	0.430019i	−0.976612	−	0.215009i	$-0.931022\pi$
$67$		−	3.51985i		−	0.430019i	0.976612	−	0.215009i	$-0.0689782\pi$
$68$		−	3.63050i		−	0.440263i
$69$	−7.69972			−0.926937
$70$	−1.70416	+	1.84088i	−0.203686	+	0.220027i
$71$	11.7457			1.39396			0.696981	−	0.717090i	$-0.254526\pi$
$71$	11.7457			1.39396			0.696981	+	0.717090i	$0.254526\pi$
$72$		−	1.81788i		−	0.214239i
$73$		−	1.12143i		−	0.131253i	−0.997844	−	0.0656267i	$-0.979095\pi$
$73$		−	1.12143i		−	0.131253i	0.997844	−	0.0656267i	$-0.0209047\pi$
$74$	18.1580			2.11083
$75$	4.98515	−	0.385107i	0.575635	−	0.0444683i
$76$	−3.26989			−0.375082
$77$			0.107019i			0.0121960i
$78$		−	4.55101i		−	0.515300i
$79$	12.4074			1.39594			0.697970	−	0.716127i	$-0.254087\pi$
$79$	12.4074			1.39594			0.697970	+	0.716127i	$0.254087\pi$
$80$	−7.58963	+	8.19854i	−0.848546	+	0.916625i
$81$	1.00000			0.111111
$82$		−	10.5860i		−	1.16903i
$83$			7.89306i			0.866376i	0.901304	+	0.433188i	$0.142611\pi$
$83$			7.89306i			0.866376i	−0.901304	+	0.433188i	$0.857389\pi$
$84$	0.614893			0.0670903
$85$	6.33925	+	5.86842i	0.687588	+	0.636520i
$86$	18.0968			1.95143
$87$		−	1.00000i		−	0.107211i
$88$			0.297329i			0.0316954i
$89$	−5.04702			−0.534983			−0.267491	−	0.963560i	$-0.586195\pi$
$89$	−5.04702			−0.534983			−0.267491	+	0.963560i	$0.586195\pi$
$90$	−2.81344	−	2.60448i	−0.296562	−	0.274536i
$91$	−1.73677			−0.182062
$92$		−	7.23579i		−	0.754383i
$93$			5.05274i			0.523945i
$94$	17.6663			1.82214
$95$	5.28551	−	5.70957i	0.542282	−	0.585790i
$96$	4.93087			0.503255
$97$		−	8.49076i		−	0.862106i	−0.902327	−	0.431053i	$-0.858142\pi$
$97$		−	8.49076i		−	0.862106i	0.902327	−	0.431053i	$-0.141858\pi$
$98$			11.2679i			1.13823i
$99$	−0.163559			−0.0164383
$100$	0.361903	+	4.68478i	0.0361903	+	0.468478i
$101$	1.81126			0.180227			0.0901136	−	0.995931i	$-0.471277\pi$
$101$	1.81126			0.180227			0.0901136	+	0.995931i	$0.471277\pi$
$102$		−	6.62385i		−	0.655859i
$103$			5.80807i			0.572286i	0.958187	+	0.286143i	$0.0923733\pi$
$103$			5.80807i			0.572286i	−0.958187	+	0.286143i	$0.907627\pi$
$104$	−4.82522			−0.473152
$105$	−0.993926	+	1.07367i	−0.0969973	+	0.104779i
$106$	−6.94000			−0.674072
$107$		−	3.48568i		−	0.336973i	−0.985704	−	0.168487i	$-0.946112\pi$
$107$		−	3.48568i		−	0.336973i	0.985704	−	0.168487i	$-0.0538880\pi$
$108$			0.939748i			0.0904273i
$109$	−8.11029			−0.776825			−0.388412	−	0.921486i	$-0.626976\pi$
$109$	−8.11029			−0.776825			−0.388412	+	0.921486i	$0.626976\pi$
$110$	0.460162	+	0.425985i	0.0438747	+	0.0406161i
$111$	10.5904			1.00520
$112$		−	3.26921i		−	0.308911i
$113$			2.06715i			0.194461i	0.995262	+	0.0972307i	$0.0309985\pi$
$113$			2.06715i			0.194461i	−0.995262	+	0.0972307i	$0.969002\pi$
$114$	−5.96591			−0.558758
$115$	12.6345	+	11.6961i	1.17817	+	1.09067i
$116$	0.939748			0.0872534
$117$		−	2.65432i		−	0.245392i
$118$			0.563637i			0.0518870i
$119$	−2.52781			−0.231724
$120$	−2.76140	+	2.98295i	−0.252081	+	0.272305i
$121$	−10.9732			−0.997568
$122$		−	9.80827i		−	0.887999i
$123$		−	6.17417i		−	0.556706i
$124$	−4.74830			−0.426410
$125$	−8.76512	−	6.94066i	−0.783976	−	0.620791i
$126$	1.12187			0.0999443
$127$			12.9962i			1.15323i	0.817017	+	0.576613i	$0.195626\pi$
$127$			12.9962i			1.15323i	−0.817017	+	0.576613i	$0.804374\pi$
$128$		−	12.4995i		−	1.10481i
$129$	10.5547			0.929292
$130$	−6.91311	+	7.46775i	−0.606320	+	0.654965i
$131$	8.69003			0.759251			0.379626	−	0.925140i	$-0.376053\pi$
$131$	8.69003			0.759251			0.379626	+	0.925140i	$0.376053\pi$
$132$		−	0.153704i		−	0.0133782i
$133$			2.27672i			0.197417i
$134$	6.03504			0.521348
$135$	−1.64090	−	1.51903i	−0.141226	−	0.130737i
$136$	−7.02295			−0.602213
$137$		−	14.4550i		−	1.23498i	−0.786580	−	0.617489i	$-0.788150\pi$
$137$		−	14.4550i		−	1.23498i	0.786580	−	0.617489i	$-0.211850\pi$
$138$		−	13.2017i		−	1.12380i
$139$	1.72208			0.146065			0.0730324	−	0.997330i	$-0.476732\pi$
$139$	1.72208			0.146065			0.0730324	+	0.997330i	$0.476732\pi$
$140$	−1.00898	−	0.934040i	−0.0852742	−	0.0789408i
$141$	10.3036			0.867723
$142$			20.1389i			1.69002i
$143$			0.434137i			0.0363043i
$144$	4.99637			0.416364
$145$	−1.51903	+	1.64090i	−0.126148	+	0.136269i
$146$	1.92277			0.159129
$147$			6.57187i			0.542039i
$148$			9.95234i			0.818077i
$149$	−16.1461			−1.32274			−0.661369	−	0.750060i	$-0.730024\pi$
$149$	−16.1461			−1.32274			−0.661369	+	0.750060i	$0.730024\pi$
$150$	0.660292	+	8.54738i	0.0539127	+	0.697891i
$151$	−0.733083			−0.0596574			−0.0298287	−	0.999555i	$-0.509496\pi$
$151$	−0.733083			−0.0596574			−0.0298287	+	0.999555i	$0.509496\pi$
$152$			6.32536i			0.513055i
$153$		−	3.86328i		−	0.312327i
$154$	−0.183492			−0.0147862
$155$	7.67526	−	8.29105i	0.616492	−	0.665953i
$156$	2.49439			0.199711
$157$		−	10.8270i		−	0.864085i	−0.901853	−	0.432043i	$-0.857793\pi$
$157$		−	10.8270i		−	0.864085i	0.901853	−	0.432043i	$-0.142207\pi$
$158$			21.2733i			1.69241i
$159$	−4.04766			−0.321000
$160$	−8.09107	−	7.49013i	−0.639655	−	0.592147i
$161$	−5.03806			−0.397054
$162$			1.71457i			0.134709i
$163$		−	13.5528i		−	1.06154i	−0.847517	−	0.530768i	$-0.821904\pi$
$163$		−	13.5528i		−	1.06154i	0.847517	−	0.530768i	$-0.178096\pi$
$164$	5.80216			0.453073
$165$	0.268383	+	0.248450i	0.0208936	+	0.0193418i
$166$	−13.5332			−1.05038
$167$		−	18.4671i		−	1.42902i	−0.699623	−	0.714512i	$-0.746649\pi$
$167$		−	18.4671i		−	1.42902i	0.699623	−	0.714512i	$-0.253351\pi$
$168$		−	1.18947i		−	0.0917694i
$169$	5.95460			0.458046
$170$	−10.0618	+	10.8691i	−0.771706	+	0.833621i
$171$	−3.47954			−0.266087
$172$			9.91878i			0.756300i
$173$		−	8.92785i		−	0.678772i	−0.940647	−	0.339386i	$-0.889781\pi$
$173$		−	8.92785i		−	0.678772i	0.940647	−	0.339386i	$-0.110219\pi$
$174$	1.71457			0.129981
$175$	3.26187	−	0.251982i	0.246574	−	0.0190481i
$176$	−0.817200			−0.0615987
$177$			0.328734i			0.0247092i
$178$		−	8.65346i		−	0.648604i
$179$	1.86896			0.139693			0.0698465	−	0.997558i	$-0.477749\pi$
$179$	1.86896			0.139693			0.0698465	+	0.997558i	$0.477749\pi$
$180$	1.42750	−	1.54203i	0.106400	−	0.114936i
$181$	16.8852			1.25507			0.627533	−	0.778590i	$-0.284065\pi$
$181$	16.8852			1.25507			0.627533	+	0.778590i	$0.284065\pi$
$182$		−	2.97780i		−	0.220730i
$183$		−	5.72054i		−	0.422875i
$184$	−13.9971			−1.03188
$185$	−17.3778	−	16.0872i	−1.27764	−	1.18275i
$186$	−8.66328			−0.635222
$187$			0.631872i			0.0462071i
$188$			9.68282i			0.706192i
$189$	0.654317			0.0475946
$190$	9.78946	+	9.06238i	0.710202	+	0.657454i
$191$	7.58677			0.548960			0.274480	−	0.961593i	$-0.411494\pi$
$191$	7.58677			0.548960			0.274480	+	0.961593i	$0.411494\pi$
$192$		−	1.53842i		−	0.111026i
$193$			27.2794i			1.96362i	0.189876	+	0.981808i	$0.439191\pi$
$193$			27.2794i			1.96362i	−0.189876	+	0.981808i	$0.560809\pi$
$194$	14.5580			1.04520
$195$	−4.03198	+	4.35547i	−0.288736	+	0.311902i
$196$	−6.17590			−0.441136
$197$			9.70626i			0.691542i	0.938319	+	0.345771i	$0.112383\pi$
$197$			9.70626i			0.691542i	−0.938319	+	0.345771i	$0.887617\pi$
$198$		−	0.280433i		−	0.0199295i
$199$	−22.9520			−1.62703			−0.813513	−	0.581547i	$-0.802448\pi$
$199$	−22.9520			−1.62703			−0.813513	+	0.581547i	$0.802448\pi$
$200$	9.06238	−	0.700077i	0.640807	−	0.0495029i
$201$	3.51985			0.248271
$202$			3.10553i			0.218504i
$203$		−	0.654317i		−	0.0459241i
$204$	3.63050			0.254186
$205$	−9.37874	+	10.1312i	−0.655040	+	0.707594i
$206$	−9.95834			−0.693831
$207$		−	7.69972i		−	0.535167i
$208$		−	13.2619i		−	0.919551i
$209$	0.569108			0.0393660
$210$	−1.84088	−	1.70416i	−0.127033	−	0.117598i
$211$	18.0020			1.23931			0.619655	−	0.784874i	$-0.287273\pi$
$211$	18.0020			1.23931			0.619655	+	0.784874i	$0.287273\pi$
$212$		−	3.80378i		−	0.261245i
$213$			11.7457i			0.804804i
$214$	5.97644			0.408541
$215$	−17.3192	−	16.0329i	−1.18116	−	1.09344i
$216$	1.81788			0.123691
$217$			3.30610i			0.224432i
$218$		−	13.9057i		−	0.941810i
$219$	1.12143			0.0757792
$220$	−0.233481	+	0.252213i	−0.0157413	+	0.0170042i
$221$	−10.2544			−0.689783
$222$			18.1580i			1.21869i
$223$		−	19.1383i		−	1.28160i	−0.767708	−	0.640799i	$-0.778603\pi$
$223$		−	19.1383i		−	1.28160i	0.767708	−	0.640799i	$-0.221397\pi$
$224$	3.22635			0.215570
$225$	0.385107	+	4.98515i	0.0256738	+	0.332343i
$226$	−3.54428			−0.235762
$227$			13.3621i			0.886872i	0.896306	+	0.443436i	$0.146241\pi$
$227$			13.3621i			0.886872i	−0.896306	+	0.443436i	$0.853759\pi$
$228$		−	3.26989i		−	0.216554i
$229$	−7.45145			−0.492405			−0.246203	−	0.969218i	$-0.579183\pi$
$229$	−7.45145			−0.492405			−0.246203	+	0.969218i	$0.579183\pi$
$230$	−20.0538	+	21.6627i	−1.32231	+	1.42839i
$231$	−0.107019			−0.00704135
$232$		−	1.81788i		−	0.119349i
$233$			21.5080i			1.40904i	0.709687	+	0.704518i	$0.248837\pi$
$233$			21.5080i			1.40904i	−0.709687	+	0.704518i	$0.751163\pi$
$234$	4.55101			0.297509
$235$	−16.9072	−	15.6515i	−1.10291	−	1.02099i
$236$	−0.308927			−0.0201094
$237$			12.4074i			0.805946i
$238$		−	4.33410i		−	0.280938i
$239$	−24.1169			−1.55999			−0.779997	−	0.625784i	$-0.784779\pi$
$239$	−24.1169			−1.55999			−0.779997	+	0.625784i	$0.784779\pi$
$240$	−8.19854	−	7.58963i	−0.529214	−	0.489908i
$241$	−7.22805			−0.465600			−0.232800	−	0.972525i	$-0.574789\pi$
$241$	−7.22805			−0.465600			−0.232800	+	0.972525i	$0.574789\pi$
$242$		−	18.8144i		−	1.20944i
$243$			1.00000i			0.0641500i
$244$	5.37587			0.344155
$245$	9.98285	−	10.7838i	0.637781	−	0.688951i
$246$	10.5860			0.674942
$247$			9.23579i			0.587659i
$248$			9.18526i			0.583264i
$249$	−7.89306			−0.500203
$250$	11.9002	−	15.0284i	0.752637	−	0.950480i
$251$	−8.74458			−0.551953			−0.275977	−	0.961164i	$-0.589001\pi$
$251$	−8.74458			−0.551953			−0.275977	+	0.961164i	$0.589001\pi$
$252$			0.614893i			0.0387346i
$253$			1.25936i			0.0791750i
$254$	−22.2829			−1.39815
$255$	−5.86842	+	6.33925i	−0.367495	+	0.396979i
$256$	18.3544			1.14715
$257$			10.1877i			0.635494i	0.948176	+	0.317747i	$0.102926\pi$
$257$			10.1877i			0.635494i	−0.948176	+	0.317747i	$0.897074\pi$
$258$			18.0968i			1.12666i
$259$	6.92950			0.430578
$260$	−4.09304	−	3.78905i	−0.253840	−	0.234987i
$261$	1.00000			0.0618984
$262$			14.8997i			0.920504i
$263$			11.6794i			0.720184i	0.932917	+	0.360092i	$0.117255\pi$
$263$			11.6794i			0.720184i	−0.932917	+	0.360092i	$0.882745\pi$
$264$	−0.297329			−0.0182994
$265$	6.64181	+	6.14851i	0.408003	+	0.377700i
$266$	−3.90359			−0.239345
$267$		−	5.04702i		−	0.308872i
$268$			3.30778i			0.202055i
$269$	−23.0385			−1.40468			−0.702342	−	0.711839i	$-0.747862\pi$
$269$	−23.0385			−1.40468			−0.702342	+	0.711839i	$0.747862\pi$
$270$	2.60448	−	2.81344i	0.158504	−	0.171220i
$271$	−4.45505			−0.270625			−0.135312	−	0.990803i	$-0.543204\pi$
$271$	−4.45505			−0.270625			−0.135312	+	0.990803i	$0.543204\pi$
$272$		−	19.3024i		−	1.17038i
$273$		−	1.73677i		−	0.105114i
$274$	24.7842			1.49727
$275$	−0.0629876	−	0.815364i	−0.00379829	−	0.0491683i
$276$	7.23579			0.435543
$277$		−	15.7000i		−	0.943321i	−0.881780	−	0.471661i	$-0.843655\pi$
$277$		−	15.7000i		−	0.943321i	0.881780	−	0.471661i	$-0.156345\pi$
$278$			2.95262i			0.177087i
$279$	−5.05274			−0.302500
$280$	−1.80683	+	1.95180i	−0.107979	+	0.116642i
$281$	13.4721			0.803677			0.401838	−	0.915711i	$-0.368371\pi$
$281$	13.4721			0.803677			0.401838	+	0.915711i	$0.368371\pi$
$282$			17.6663i			1.05201i
$283$			5.86481i			0.348627i	0.984690	+	0.174313i	$0.0557706\pi$
$283$			5.86481i			0.348627i	−0.984690	+	0.174313i	$0.944229\pi$
$284$	−11.0380			−0.654986
$285$	5.70957	+	5.28551i	0.338206	+	0.313087i
$286$	−0.744357			−0.0440148
$287$		−	4.03987i		−	0.238466i
$288$			4.93087i			0.290554i
$289$	2.07510			0.122065
$290$	−2.81344	−	2.60448i	−0.165211	−	0.152940i
$291$	8.49076			0.497737
$292$			1.05386i			0.0616726i
$293$		−	10.3642i		−	0.605485i	−0.953072	−	0.302742i	$-0.902098\pi$
$293$		−	10.3642i		−	0.605485i	0.953072	−	0.302742i	$-0.0979022\pi$
$294$	−11.2679			−0.657159
$295$	0.499356	−	0.539420i	0.0290737	−	0.0314062i
$296$	19.2521			1.11901
$297$		−	0.163559i		−	0.00949064i
$298$		−	27.6836i		−	1.60367i
$299$	−20.4375			−1.18193
$300$	−4.68478	+	0.361903i	−0.270476	+	0.0208945i
$301$	6.90614			0.398063
$302$		−	1.25692i		−	0.0723277i
$303$			1.81126i			0.104054i
$304$	−17.3850			−0.997101
$305$	−8.68967	+	9.38684i	−0.497569	+	0.537489i
$306$	6.62385			0.378660
$307$			16.0760i			0.917504i	0.888564	+	0.458752i	$0.151703\pi$
$307$			16.0760i			0.917504i	−0.888564	+	0.458752i	$0.848297\pi$
$308$		−	0.100571i		−	0.00573057i
$309$	−5.80807			−0.330410
$310$	14.2156	+	13.1598i	0.807390	+	0.747424i
$311$	16.9976			0.963846			0.481923	−	0.876214i	$-0.339939\pi$
$311$	16.9976			0.963846			0.481923	+	0.876214i	$0.339939\pi$
$312$		−	4.82522i		−	0.273174i
$313$			25.2391i			1.42660i	0.700858	+	0.713300i	$0.252800\pi$
$313$			25.2391i			1.42660i	−0.700858	+	0.713300i	$0.747200\pi$
$314$	18.5636			1.04760
$315$	−1.07367	−	0.993926i	−0.0604944	−	0.0560014i
$316$	−11.6598			−0.655916
$317$		−	3.06731i		−	0.172277i	−0.996283	−	0.0861386i	$-0.972547\pi$
$317$		−	3.06731i		−	0.172277i	0.996283	−	0.0861386i	$-0.0274528\pi$
$318$		−	6.94000i		−	0.389176i
$319$	−0.163559			−0.00915753
$320$	−2.33690	+	2.52439i	−0.130637	+	0.141118i
$321$	3.48568			0.194552
$322$		−	8.63810i		−	0.481382i
$323$			13.4424i			0.747955i
$324$	−0.939748			−0.0522082
$325$	13.2322	−	1.02220i	0.733988	−	0.0567012i
$326$	23.2372			1.28699
$327$		−	8.11029i		−	0.448500i
$328$		−	11.2239i		−	0.619735i
$329$	6.74185			0.371690
$330$	−0.425985	+	0.460162i	−0.0234497	+	0.0253311i
$331$	−19.9971			−1.09914			−0.549571	−	0.835447i	$-0.685209\pi$
$331$	−19.9971			−1.09914			−0.549571	+	0.835447i	$0.685209\pi$
$332$		−	7.41749i		−	0.407088i
$333$			10.5904i			0.580352i
$334$	31.6631			1.73253
$335$	−5.77573	−	5.34676i	−0.315562	−	0.292125i
$336$	3.26921			0.178350
$337$			6.45613i			0.351688i	0.984418	+	0.175844i	$0.0562654\pi$
$337$			6.45613i			0.351688i	−0.984418	+	0.175844i	$0.943735\pi$
$338$			10.2096i			0.555328i
$339$	−2.06715			−0.112272
$340$	−5.95730	−	5.51484i	−0.323080	−	0.299084i
$341$	0.826420			0.0447531
$342$		−	5.96591i		−	0.322599i
$343$			8.88031i			0.479491i
$344$	19.1872			1.03450
$345$	−11.6961	+	12.6345i	−0.629696	+	0.680217i
$346$	15.3074			0.822932
$347$			14.9415i			0.802099i	0.916056	+	0.401050i	$0.131355\pi$
$347$			14.9415i			0.802099i	−0.916056	+	0.401050i	$0.868645\pi$
$348$			0.939748i			0.0503758i
$349$	12.1055			0.647992			0.323996	−	0.946058i	$-0.394974\pi$
$349$	12.1055			0.647992			0.323996	+	0.946058i	$0.394974\pi$
$350$	0.432041	+	5.59270i	0.0230935	+	0.298942i
$351$	2.65432			0.141677
$352$		−	0.806487i		−	0.0429859i
$353$		−	18.1937i		−	0.968355i	−0.874970	−	0.484178i	$-0.839119\pi$
$353$		−	18.1937i		−	0.968355i	0.874970	−	0.484178i	$-0.160881\pi$
$354$	−0.563637			−0.0299570
$355$	17.8421	−	19.2736i	0.946960	−	1.02293i
$356$	4.74292			0.251374
$357$		−	2.52781i		−	0.133786i
$358$			3.20447i			0.169361i
$359$	35.1010			1.85256			0.926279	−	0.376838i	$-0.122989\pi$
$359$	35.1010			1.85256			0.926279	+	0.376838i	$0.122989\pi$
$360$	−2.98295	−	2.76140i	−0.157215	−	0.145539i
$361$	−6.89283			−0.362780
$362$			28.9508i			1.52162i
$363$		−	10.9732i		−	0.575946i
$364$	1.63212			0.0855464
$365$	−1.84015	−	1.70348i	−0.0963181	−	0.0891644i
$366$	9.80827			0.512686
$367$		−	23.1943i		−	1.21073i	−0.795946	−	0.605367i	$-0.793026\pi$
$367$		−	23.1943i		−	1.21073i	0.795946	−	0.605367i	$-0.206974\pi$
$368$		−	38.4706i		−	2.00542i
$369$	6.17417			0.321415
$370$	27.5826	−	29.7955i	1.43395	−	1.54900i
$371$	−2.64845			−0.137501
$372$		−	4.74830i		−	0.246188i
$373$		−	4.99469i		−	0.258615i	−0.991605	−	0.129308i	$-0.958725\pi$
$373$		−	4.99469i		−	0.258615i	0.991605	−	0.129308i	$-0.0412754\pi$
$374$	−1.08339			−0.0560207
$375$	6.94066	−	8.76512i	0.358414	−	0.452629i
$376$	18.7307			0.965964
$377$		−	2.65432i		−	0.136704i
$378$			1.12187i			0.0577029i
$379$	28.7738			1.47801			0.739006	−	0.673699i	$-0.235296\pi$
$379$	28.7738			1.47801			0.739006	+	0.673699i	$0.235296\pi$
$380$	−4.96705	+	5.36556i	−0.254804	+	0.275247i
$381$	−12.9962			−0.665816
$382$			13.0080i			0.665550i
$383$		−	24.9482i		−	1.27479i	−0.770537	−	0.637396i	$-0.780012\pi$
$383$		−	24.9482i		−	1.27479i	0.770537	−	0.637396i	$-0.219988\pi$
$384$	12.4995			0.637861
$385$	0.175608	+	0.162565i	0.00894981	+	0.00828509i
$386$	−46.7725			−2.38066
$387$			10.5547i			0.536527i
$388$			7.97917i			0.405081i
$389$	−31.0949			−1.57657			−0.788286	−	0.615308i	$-0.789031\pi$
$389$	−31.0949			−1.57657			−0.788286	+	0.615308i	$0.789031\pi$
$390$	−7.46775	−	6.91311i	−0.378144	−	0.350059i
$391$	−29.7461			−1.50433
$392$			11.9468i			0.603407i
$393$			8.69003i			0.438354i
$394$	−16.6421			−0.838414
$395$	18.8472	−	20.3593i	0.948304	−	1.02439i
$396$	0.153704			0.00772391
$397$			12.1948i			0.612038i	0.952025	+	0.306019i	$0.0989971\pi$
$397$			12.1948i			0.612038i	−0.952025	+	0.306019i	$0.901003\pi$
$398$		−	39.3528i		−	1.97258i
$399$	−2.27672			−0.113979
$400$	1.92414	+	24.9076i	0.0962068	+	1.24538i
$401$	4.33039			0.216249			0.108125	−	0.994137i	$-0.465515\pi$
$401$	4.33039			0.216249			0.108125	+	0.994137i	$0.465515\pi$
$402$			6.03504i			0.301000i
$403$			13.4116i			0.668078i
$404$	−1.70213			−0.0846841
$405$	1.51903	−	1.64090i	0.0754811	−	0.0815370i
$406$	1.12187			0.0556776
$407$		−	1.73216i		−	0.0858599i
$408$		−	7.02295i		−	0.347688i
$409$	−5.12178			−0.253256			−0.126628	−	0.991950i	$-0.540415\pi$
$409$	−5.12178			−0.253256			−0.126628	+	0.991950i	$0.540415\pi$
$410$	−17.3706	−	16.0805i	−0.857875	−	0.794159i
$411$	14.4550			0.713015
$412$		−	5.45812i		−	0.268902i
$413$			0.215096i			0.0105842i
$414$	13.2017			0.648828
$415$	12.9517	+	11.9898i	0.635775	+	0.588555i
$416$	13.0881			0.641697
$417$			1.72208i			0.0843306i
$418$			0.975776i			0.0477267i
$419$	−14.1945			−0.693445			−0.346723	−	0.937968i	$-0.612705\pi$
$419$	−14.1945			−0.693445			−0.346723	+	0.937968i	$0.612705\pi$
$420$	0.934040	−	1.00898i	0.0455765	−	0.0492331i
$421$	4.83364			0.235577			0.117789	−	0.993039i	$-0.462419\pi$
$421$	4.83364			0.235577			0.117789	+	0.993039i	$0.462419\pi$
$422$			30.8657i			1.50252i
$423$			10.3036i			0.500980i
$424$	−7.35815			−0.357343
$425$	19.2590	−	1.48777i	0.934198	−	0.0721676i
$426$	−20.1389			−0.975731
$427$		−	3.74305i		−	0.181139i
$428$			3.27566i			0.158335i
$429$	−0.434137			−0.0209603
$430$	27.4896	−	29.6950i	1.32566	−	1.43202i
$431$	−18.0910			−0.871411			−0.435706	−	0.900089i	$-0.643501\pi$
$431$	−18.0910			−0.871411			−0.435706	+	0.900089i	$0.643501\pi$
$432$			4.99637i			0.240388i
$433$		−	1.71005i		−	0.0821798i	−0.999155	−	0.0410899i	$-0.986917\pi$
$433$		−	1.71005i		−	0.0821798i	0.999155	−	0.0410899i	$-0.0130830\pi$
$434$	−5.66853			−0.272098
$435$	−1.64090	−	1.51903i	−0.0786751	−	0.0728318i
$436$	7.62163			0.365010
$437$			26.7914i			1.28161i
$438$			1.92277i			0.0918734i
$439$	−24.3533			−1.16232			−0.581160	−	0.813789i	$-0.697401\pi$
$439$	−24.3533			−1.16232			−0.581160	+	0.813789i	$0.697401\pi$
$440$	0.487888	+	0.451652i	0.0232591	+	0.0215316i
$441$	−6.57187			−0.312946
$442$		−	17.5818i		−	0.836281i
$443$			18.9066i			0.898280i	0.893461	+	0.449140i	$0.148270\pi$
$443$			18.9066i			0.898280i	−0.893461	+	0.449140i	$0.851730\pi$
$444$	−9.95234			−0.472317
$445$	−7.66656	+	8.28165i	−0.363430	+	0.392588i
$446$	32.8140			1.55379
$447$		−	16.1461i		−	0.763684i
$448$		−	1.00661i		−	0.0475581i
$449$	−25.2808			−1.19307			−0.596537	−	0.802586i	$-0.703457\pi$
$449$	−25.2808			−1.19307			−0.596537	+	0.802586i	$0.703457\pi$
$450$	−8.54738	+	0.660292i	−0.402927	+	0.0311265i
$451$	−1.00984			−0.0475515
$452$		−	1.94260i		−	0.0913723i
$453$		−	0.733083i		−	0.0344432i
$454$	−22.9102			−1.07523
$455$	−2.63820	+	2.84986i	−0.123680	+	0.133603i
$456$	−6.32536			−0.296212
$457$		−	4.25835i		−	0.199197i	−0.995028	−	0.0995986i	$-0.968244\pi$
$457$		−	4.25835i		−	0.199197i	0.995028	−	0.0995986i	$-0.0317559\pi$
$458$		−	12.7760i		−	0.596984i
$459$	3.86328			0.180322
$460$	−11.8732	−	10.9914i	−0.553591	−	0.512475i
$461$	2.48353			0.115670			0.0578349	−	0.998326i	$-0.481580\pi$
$461$	2.48353			0.115670			0.0578349	+	0.998326i	$0.481580\pi$
$462$		−	0.183492i		−	0.00853682i
$463$			27.4876i			1.27746i	0.769433	+	0.638728i	$0.220539\pi$
$463$			27.4876i			1.27746i	−0.769433	+	0.638728i	$0.779461\pi$
$464$	4.99637			0.231951
$465$	8.29105	+	7.67526i	0.384488	+	0.355932i
$466$	−36.8769			−1.70829
$467$			6.24698i			0.289076i	0.989499	+	0.144538i	$0.0461695\pi$
$467$			6.24698i			0.289076i	−0.989499	+	0.144538i	$0.953830\pi$
$468$			2.49439i			0.115303i
$469$	2.30310			0.106347
$470$	26.8356	−	28.9886i	1.23783	−	1.33715i
$471$	10.8270			0.498880
$472$			0.597598i			0.0275067i
$473$		−	1.72632i		−	0.0793761i
$474$	−21.2733			−0.977116
$475$	−1.33999	−	17.3460i	−0.0614831	−	0.795889i
$476$	2.37550			0.108881
$477$		−	4.04766i		−	0.185330i
$478$		−	41.3501i		−	1.89131i
$479$	11.5736			0.528812			0.264406	−	0.964412i	$-0.414824\pi$
$479$	11.5736			0.528812			0.264406	+	0.964412i	$0.414824\pi$
$480$	7.49013	−	8.09107i	0.341876	−	0.369305i
$481$	28.1104			1.28172
$482$		−	12.3930i		−	0.564485i
$483$		−	5.03806i		−	0.229240i
$484$	10.3121			0.468731
$485$	−13.9325	−	12.8977i	−0.632642	−	0.585654i
$486$	−1.71457			−0.0777744
$487$			31.1947i			1.41357i	0.707431	+	0.706783i	$0.249854\pi$
$487$			31.1947i			1.41357i	−0.707431	+	0.706783i	$0.750146\pi$
$488$		−	10.3992i		−	0.470751i
$489$	13.5528			0.612878
$490$	18.4895	+	17.1163i	0.835272	+	0.773235i
$491$	−7.93512			−0.358107			−0.179053	−	0.983839i	$-0.557303\pi$
$491$	−7.93512			−0.358107			−0.179053	+	0.983839i	$0.557303\pi$
$492$			5.80216i			0.261582i
$493$		−	3.86328i		−	0.173993i
$494$	−15.8354			−0.712469
$495$	−0.248450	+	0.268383i	−0.0111670	+	0.0120629i
$496$	−25.2454			−1.13355
$497$			7.68543i			0.344739i
$498$		−	13.5332i		−	0.606437i
$499$	32.8984			1.47274			0.736368	−	0.676581i	$-0.236539\pi$
$499$	32.8984			1.47274			0.736368	+	0.676581i	$0.236539\pi$
$500$	8.23700	+	6.52247i	0.368370	+	0.291694i
$501$	18.4671			0.825047
$502$		−	14.9932i		−	0.669179i
$503$		−	40.7038i		−	1.81489i	−0.420168	−	0.907446i	$-0.638029\pi$
$503$		−	40.7038i		−	1.81489i	0.420168	−	0.907446i	$-0.361971\pi$
$504$	1.18947			0.0529831
$505$	2.75136	−	2.97210i	0.122434	−	0.132257i
$506$	−2.15925			−0.0959905
$507$			5.95460i			0.264453i
$508$		−	12.2132i		−	0.541871i
$509$	36.9407			1.63737			0.818684	−	0.574245i	$-0.194704\pi$
$509$	36.9407			1.63737			0.818684	+	0.574245i	$0.194704\pi$
$510$	−10.8691	−	10.0618i	−0.481291	−	0.445545i
$511$	0.733771			0.0324601
$512$			6.47089i			0.285976i
$513$		−	3.47954i		−	0.153625i
$514$	−17.4676			−0.770462
$515$	9.53047	+	8.82263i	0.419963	+	0.388771i
$516$	−9.91878			−0.436650
$517$		−	1.68525i		−	0.0741172i
$518$			11.8811i			0.522026i
$519$	8.92785			0.391889
$520$	−7.32964	+	7.91770i	−0.321426	+	0.347214i
$521$	−33.2555			−1.45695			−0.728476	−	0.685072i	$-0.759771\pi$
$521$	−33.2555			−1.45695			−0.728476	+	0.685072i	$0.759771\pi$
$522$			1.71457i			0.0750447i
$523$		−	0.341001i		−	0.0149109i	−0.999972	−	0.00745547i	$-0.997627\pi$
$523$		−	0.341001i		−	0.0149109i	0.999972	−	0.00745547i	$-0.00237317\pi$
$524$	−8.16644			−0.356752
$525$	0.251982	+	3.26187i	0.0109974	+	0.142360i
$526$	−20.0252			−0.873139
$527$			19.5201i			0.850310i
$528$		−	0.817200i		−	0.0355640i
$529$	−36.2856			−1.57764
$530$	−10.5421	+	11.3878i	−0.457917	+	0.494656i
$531$	−0.328734			−0.0142658
$532$		−	2.13954i		−	0.0927609i
$533$		−	16.3882i		−	0.709852i
$534$	8.65346			0.374472
$535$	−5.71965	−	5.29484i	−0.247282	−	0.228916i
$536$	6.39866			0.276380
$537$			1.86896i			0.0806518i
$538$		−	39.5012i		−	1.70302i
$539$	1.07489			0.0462986
$540$	1.54203	+	1.42750i	0.0663585	+	0.0614300i
$541$	−18.4196			−0.791919			−0.395960	−	0.918268i	$-0.629588\pi$
$541$	−18.4196			−0.791919			−0.395960	+	0.918268i	$0.629588\pi$
$542$		−	7.63849i		−	0.328101i
$543$			16.8852i			0.724613i
$544$	19.0493			0.816732
$545$	−12.3198	+	13.3082i	−0.527720	+	0.570060i
$546$	2.97780			0.127438
$547$		−	42.5936i		−	1.82117i	−0.413320	−	0.910586i	$-0.635631\pi$
$547$		−	42.5936i		−	1.82117i	0.413320	−	0.910586i	$-0.364369\pi$
$548$			13.5841i			0.580284i
$549$	5.72054			0.244147
$550$	1.39800	−	0.107997i	0.0596108	−	0.00460499i
$551$	−3.47954			−0.148233
$552$		−	13.9971i		−	0.595757i
$553$			8.11836i			0.345228i
$554$	26.9187			1.14367
$555$	16.0872	−	17.3778i	0.682862	−	0.737648i
$556$	−1.61832			−0.0686321
$557$		−	17.5805i		−	0.744908i	−0.928051	−	0.372454i	$-0.878516\pi$
$557$		−	17.5805i		−	0.744908i	0.928051	−	0.372454i	$-0.121484\pi$
$558$		−	8.66328i		−	0.366746i
$559$	28.0156			1.18493
$560$	−5.36445	−	4.96602i	−0.226689	−	0.209853i
$561$	−0.631872			−0.0266777
$562$			23.0988i			0.974365i
$563$			6.29621i			0.265354i	0.991159	+	0.132677i	$0.0423572\pi$
$563$			6.29621i			0.265354i	−0.991159	+	0.132677i	$0.957643\pi$
$564$	−9.68282			−0.407720
$565$	3.39199	+	3.14006i	0.142702	+	0.132103i
$566$	−10.0556			−0.422669
$567$			0.654317i			0.0274787i
$568$			21.3523i			0.895921i
$569$	42.3981			1.77742			0.888710	−	0.458469i	$-0.151602\pi$
$569$	42.3981			1.77742			0.888710	+	0.458469i	$0.151602\pi$
$570$	−9.06238	+	9.78946i	−0.379581	+	0.410035i
$571$	−18.4882			−0.773708			−0.386854	−	0.922141i	$-0.626438\pi$
$571$	−18.4882			−0.773708			−0.386854	+	0.922141i	$0.626438\pi$
$572$		−	0.407979i		−	0.0170585i
$573$			7.58677i			0.316942i
$574$	6.92663			0.289112
$575$	38.3842	−	2.96521i	1.60073	−	0.123658i
$576$	1.53842			0.0641009
$577$			22.0727i			0.918897i	0.888204	+	0.459448i	$0.151953\pi$
$577$			22.0727i			0.918897i	−0.888204	+	0.459448i	$0.848047\pi$
$578$			3.55791i			0.147990i
$579$	−27.2794			−1.13369
$580$	1.42750	−	1.54203i	0.0592739	−	0.0640294i
$581$	−5.16457			−0.214262
$582$			14.5580i			0.603448i
$583$			0.662030i			0.0274185i
$584$	2.03862			0.0843587
$585$	−4.35547	−	4.03198i	−0.180077	−	0.166702i
$586$	17.7702			0.734080
$587$			44.1932i			1.82405i	0.410136	+	0.912024i	$0.365481\pi$
$587$			44.1932i			1.82405i	−0.410136	+	0.912024i	$0.634519\pi$
$588$		−	6.17590i		−	0.254690i
$589$	17.5812			0.724421
$590$	0.924873	+	0.856181i	0.0380764	+	0.0352484i
$591$	−9.70626			−0.399262
$592$			52.9137i			2.17474i
$593$			17.4159i			0.715184i	0.933878	+	0.357592i	$0.116402\pi$
$593$			17.4159i			0.715184i	−0.933878	+	0.357592i	$0.883598\pi$
$594$	0.280433			0.0115063
$595$	−3.83981	+	4.14788i	−0.157417	+	0.170046i
$596$	15.1732			0.621520
$597$		−	22.9520i		−	0.939364i
$598$		−	35.0415i		−	1.43295i
$599$	−7.70544			−0.314836			−0.157418	−	0.987532i	$-0.550317\pi$
$599$	−7.70544			−0.314836			−0.157418	+	0.987532i	$0.550317\pi$
$600$	0.700077	+	9.06238i	0.0285805	+	0.369970i
$601$	−31.5500			−1.28695			−0.643476	−	0.765466i	$-0.722508\pi$
$601$	−31.5500			−1.28695			−0.643476	+	0.765466i	$0.722508\pi$
$602$			11.8410i			0.482605i
$603$			3.51985i			0.143340i
$604$	0.688913			0.0280315
$605$	−16.6687	+	18.0060i	−0.677678	+	0.732048i
$606$	−3.10553			−0.126154
$607$			13.2516i			0.537864i	0.963159	+	0.268932i	$0.0866707\pi$
$607$			13.2516i			0.537864i	−0.963159	+	0.268932i	$0.913329\pi$
$608$		−	17.1571i		−	0.695814i
$609$	0.654317			0.0265143
$610$	−16.0944	−	14.8990i	−0.651643	−	0.603244i
$611$	27.3491			1.10643
$612$			3.63050i			0.146754i
$613$			29.4609i			1.18991i	0.803758	+	0.594957i	$0.202831\pi$
$613$			29.4609i			1.18991i	−0.803758	+	0.594957i	$0.797169\pi$
$614$	−27.5634			−1.11237
$615$	−10.1312	−	9.37874i	−0.408529	−	0.378187i
$616$	−0.194548			−0.00783855
$617$			5.61068i			0.225877i	0.993602	+	0.112939i	$0.0360264\pi$
$617$			5.61068i			0.225877i	−0.993602	+	0.112939i	$0.963974\pi$
$618$		−	9.95834i		−	0.400583i
$619$	−40.5248			−1.62883			−0.814415	−	0.580282i	$-0.802942\pi$
$619$	−40.5248			−1.62883			−0.814415	+	0.580282i	$0.802942\pi$
$620$	−7.21281	+	7.79149i	−0.289673	+	0.312914i
$621$	7.69972			0.308979
$622$			29.1436i			1.16855i
$623$		−	3.30235i		−	0.132306i
$624$	13.2619			0.530903
$625$	−24.7034	+	3.83963i	−0.988135	+	0.153585i
$626$	−43.2743			−1.72959
$627$			0.569108i			0.0227280i
$628$			10.1746i			0.406011i
$629$	40.9138			1.63134
$630$	1.70416	−	1.84088i	0.0678952	−	0.0733424i
$631$	−20.2197			−0.804935			−0.402467	−	0.915434i	$-0.631847\pi$
$631$	−20.2197			−0.804935			−0.402467	+	0.915434i	$0.631847\pi$
$632$			22.5551i			0.897193i
$633$			18.0020i			0.715516i
$634$	5.25911			0.208866
$635$	21.3255	+	19.7416i	0.846276	+	0.783421i
$636$	3.80378			0.150830
$637$			17.4438i			0.691149i
$638$		−	0.280433i		−	0.0111024i
$639$	−11.7457			−0.464654
$640$	−20.5104	−	18.9870i	−0.810744	−	0.750529i
$641$	−44.0804			−1.74107			−0.870535	−	0.492107i	$-0.836227\pi$
$641$	−44.0804			−1.74107			−0.870535	+	0.492107i	$0.836227\pi$
$642$			5.97644i			0.235871i
$643$			3.56303i			0.140512i	0.997529	+	0.0702561i	$0.0223816\pi$
$643$			3.56303i			0.140512i	−0.997529	+	0.0702561i	$0.977618\pi$
$644$	4.73450			0.186566
$645$	16.0329	−	17.3192i	0.631296	−	0.681945i
$646$	−23.0479			−0.906809
$647$			40.7523i			1.60214i	0.598573	+	0.801069i	$0.295735\pi$
$647$			40.7523i			1.60214i	−0.598573	+	0.801069i	$0.704265\pi$
$648$			1.81788i			0.0714129i
$649$	0.0537673			0.00211055
$650$	1.75263	+	22.6875i	0.0687436	+	0.889875i
$651$	−3.30610			−0.129576
$652$			12.7362i			0.498788i
$653$		−	11.6902i		−	0.457472i	−0.973489	−	0.228736i	$-0.926541\pi$
$653$		−	11.6902i		−	0.457472i	0.973489	−	0.228736i	$-0.0734592\pi$
$654$	13.9057			0.543754
$655$	13.2004	−	14.2595i	0.515782	−	0.557164i
$656$	30.8484			1.20443
$657$			1.12143i			0.0437511i
$658$			11.5594i			0.450631i
$659$	28.1795			1.09772			0.548858	−	0.835915i	$-0.315063\pi$
$659$	28.1795			1.09772			0.548858	+	0.835915i	$0.315063\pi$
$660$	−0.252213	−	0.233481i	−0.00981737	−	0.00908822i
$661$	−20.6736			−0.804108			−0.402054	−	0.915616i	$-0.631704\pi$
$661$	−20.6736			−0.804108			−0.402054	+	0.915616i	$0.631704\pi$
$662$		−	34.2865i		−	1.33258i
$663$		−	10.2544i		−	0.398246i
$664$	−14.3486			−0.556834
$665$	3.73587	+	3.45840i	0.144871	+	0.134111i
$666$	−18.1580			−0.703610
$667$		−	7.69972i		−	0.298134i
$668$			17.3544i			0.671461i
$669$	19.1383			0.739931
$670$	9.16739	−	9.90289i	0.354167	−	0.382582i
$671$	−0.935645			−0.0361202
$672$			3.22635i			0.124459i
$673$			44.7915i			1.72659i	0.504703	+	0.863293i	$0.331602\pi$
$673$			44.7915i			1.72659i	−0.504703	+	0.863293i	$0.668398\pi$
$674$	−11.0695			−0.426380
$675$	−4.98515	+	0.385107i	−0.191878	+	0.0148228i
$676$	−5.59582			−0.215224
$677$		−	16.8746i		−	0.648542i	−0.945964	−	0.324271i	$-0.894881\pi$
$677$		−	16.8746i		−	0.648542i	0.945964	−	0.324271i	$-0.105119\pi$
$678$		−	3.54428i		−	0.136117i
$679$	5.55565			0.213206
$680$	−10.6681	+	11.5240i	−0.409102	+	0.441924i
$681$	−13.3621			−0.512036
$682$			1.41695i			0.0542580i
$683$		−	28.6207i		−	1.09514i	−0.836759	−	0.547571i	$-0.815553\pi$
$683$		−	28.6207i		−	1.09514i	0.836759	−	0.547571i	$-0.184447\pi$
$684$	3.26989			0.125027
$685$	−23.7193	−	21.9576i	−0.906267	−	0.838958i
$686$	−15.2259			−0.581328
$687$		−	7.45145i		−	0.284290i
$688$			52.7353i			2.01051i
$689$	−10.7438			−0.409305
$690$	−21.6627	−	20.0538i	−0.824684	−	0.763433i
$691$	−28.5928			−1.08772			−0.543861	−	0.839175i	$-0.683038\pi$
$691$	−28.5928			−1.08772			−0.543861	+	0.839175i	$0.683038\pi$
$692$			8.38993i			0.318937i
$693$		−	0.107019i		−	0.00406532i
$694$	−25.6182			−0.972452
$695$	2.61589	−	2.82576i	0.0992262	−	0.107187i
$696$	1.81788			0.0689064
$697$		−	23.8525i		−	0.903479i
$698$			20.7557i			0.785615i
$699$	−21.5080			−0.813507
$700$	−3.06533	+	0.236800i	−0.115859	+	0.00895018i
$701$	49.6636			1.87577			0.937885	−	0.346946i	$-0.112781\pi$
$701$	49.6636			1.87577			0.937885	+	0.346946i	$0.112781\pi$
$702$			4.55101i			0.171767i
$703$		−	36.8498i		−	1.38982i
$704$	−0.251622			−0.00948336
$705$	15.6515	−	16.9072i	0.589470	−	0.636763i
$706$	31.1944			1.17402
$707$			1.18514i			0.0445717i
$708$		−	0.308927i		−	0.0116102i
$709$	19.4755			0.731419			0.365710	−	0.930729i	$-0.380826\pi$
$709$	19.4755			0.731419			0.365710	+	0.930729i	$0.380826\pi$
$710$	33.0459	+	30.5915i	1.24019	+	1.14808i
$711$	−12.4074			−0.465313
$712$		−	9.17485i		−	0.343842i
$713$			38.9047i			1.45699i
$714$	4.33410			0.162200
$715$	0.712375	+	0.659466i	0.0266413	+	0.0246626i
$716$	−1.75636			−0.0656381
$717$		−	24.1169i		−	0.900663i
$718$			60.1830i			2.24601i
$719$	34.3195			1.27990			0.639951	−	0.768415i	$-0.278954\pi$
$719$	34.3195			1.27990			0.639951	+	0.768415i	$0.278954\pi$
$720$	7.58963	−	8.19854i	0.282849	−	0.305542i
$721$	−3.80032			−0.141531
$722$		−	11.8182i		−	0.439829i
$723$		−	7.22805i		−	0.268814i
$724$	−15.8678			−0.589723
$725$	0.385107	+	4.98515i	0.0143025	+	0.185144i
$726$	18.8144			0.698268
$727$		−	19.3206i		−	0.716560i	−0.933614	−	0.358280i	$-0.883363\pi$
$727$		−	19.3206i		−	0.716560i	0.933614	−	0.358280i	$-0.116637\pi$
$728$		−	3.15722i		−	0.117014i
$729$	−1.00000			−0.0370370
$730$	2.92074	−	3.15507i	0.108101	−	0.116774i
$731$	40.7758			1.50815
$732$			5.37587i			0.198698i
$733$			3.10588i			0.114718i	0.998354	+	0.0573591i	$0.0182680\pi$
$733$			3.10588i			0.114718i	−0.998354	+	0.0573591i	$0.981732\pi$
$734$	39.7683			1.46787
$735$	10.7838	+	9.98285i	0.397766	+	0.368223i
$736$	37.9663			1.39946
$737$		−	0.575703i		−	0.0212063i
$738$			10.5860i			0.389678i
$739$	−38.7106			−1.42399			−0.711997	−	0.702183i	$-0.752209\pi$
$739$	−38.7106			−1.42399			−0.711997	+	0.702183i	$0.752209\pi$
$740$	16.3308	+	15.1179i	0.600332	+	0.555744i
$741$	−9.23579			−0.339285
$742$		−	4.54096i		−	0.166704i
$743$		−	8.41130i		−	0.308581i	−0.988026	−	0.154290i	$-0.950691\pi$
$743$		−	8.41130i		−	0.308581i	0.988026	−	0.154290i	$-0.0493091\pi$
$744$	−9.18526			−0.336748
$745$	−24.5264	+	26.4941i	−0.898576	+	0.970669i
$746$	8.56373			0.313541
$747$		−	7.89306i		−	0.288792i
$748$		−	0.593801i		−	0.0217115i
$749$	2.28074			0.0833364
$750$	15.0284	+	11.9002i	0.548760	+	0.434535i
$751$	−30.5930			−1.11635			−0.558177	−	0.829722i	$-0.688499\pi$
$751$	−30.5930			−1.11635			−0.558177	+	0.829722i	$0.688499\pi$
$752$			51.4808i			1.87731i
$753$		−	8.74458i		−	0.318670i
$754$	4.55101			0.165738
$755$	−1.11357	+	1.20292i	−0.0405271	+	0.0437786i
$756$	−0.614893			−0.0223634
$757$			41.6526i			1.51389i	0.653479	+	0.756945i	$0.273309\pi$
$757$			41.6526i			1.51389i	−0.653479	+	0.756945i	$0.726691\pi$
$758$			49.3347i			1.79192i
$759$	−1.25936			−0.0457117
$760$	10.3793	+	9.60841i	0.376496	+	0.348534i
$761$	−28.3957			−1.02934			−0.514671	−	0.857388i	$-0.672086\pi$
$761$	−28.3957			−1.02934			−0.514671	+	0.857388i	$0.672086\pi$
$762$		−	22.2829i		−	0.807224i
$763$		−	5.30670i		−	0.192115i
$764$	−7.12965			−0.257942
$765$	−6.33925	−	5.86842i	−0.229196	−	0.212173i
$766$	42.7753			1.54554
$767$			0.872565i			0.0315065i
$768$			18.3544i			0.662306i
$769$	48.6782			1.75538			0.877690	−	0.479229i	$-0.159084\pi$
$769$	48.6782			1.75538			0.877690	+	0.479229i	$0.159084\pi$
$770$	−0.278729	+	0.301092i	−0.0100447	+	0.0108506i
$771$	−10.1877			−0.366902
$772$		−	25.6358i		−	0.922652i
$773$		−	54.9974i		−	1.97812i	−0.147513	−	0.989060i	$-0.547127\pi$
$773$		−	54.9974i		−	1.97812i	0.147513	−	0.989060i	$-0.452873\pi$
$774$	−18.0968			−0.650476
$775$	−1.94585	−	25.1887i	−0.0698969	−	0.904804i
$776$	15.4351			0.554089
$777$			6.92950i			0.248595i
$778$		−	53.3143i		−	1.91141i
$779$	−21.4833			−0.769717
$780$	3.78905	−	4.09304i	0.135670	−	0.146554i
$781$	1.92112			0.0687429
$782$		−	51.0018i		−	1.82382i
$783$			1.00000i

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)

\(f(q)\)	\(=\)	\(q+1.71457i q^{2} +1.00000i q^{3} -0.939748 q^{4} +(1.51903 - 1.64090i) q^{5} -1.71457 q^{6} +0.654317i q^{7} +1.81788i q^{8} -1.00000 q^{9} +(2.81344 + 2.60448i) q^{10} +0.163559 q^{11} -0.939748i q^{12} +2.65432i q^{13} -1.12187 q^{14} +(1.64090 + 1.51903i) q^{15} -4.99637 q^{16} +3.86328i q^{17} -1.71457i q^{18} +3.47954 q^{19} +(-1.42750 + 1.54203i) q^{20} -0.654317 q^{21} +0.280433i q^{22} +7.69972i q^{23} -1.81788 q^{24} +(-0.385107 - 4.98515i) q^{25} -4.55101 q^{26} -1.00000i q^{27} -0.614893i q^{28} -1.00000 q^{29} +(-2.60448 + 2.81344i) q^{30} +5.05274 q^{31} -4.93087i q^{32} +0.163559i q^{33} -6.62385 q^{34} +(1.07367 + 0.993926i) q^{35} +0.939748 q^{36} -10.5904i q^{37} +5.96591i q^{38} -2.65432 q^{39} +(2.98295 + 2.76140i) q^{40} -6.17417 q^{41} -1.12187i q^{42} -10.5547i q^{43} -0.153704 q^{44} +(-1.51903 + 1.64090i) q^{45} -13.2017 q^{46} -10.3036i q^{47} -4.99637i q^{48} +6.57187 q^{49} +(8.54738 - 0.660292i) q^{50} -3.86328 q^{51} -2.49439i q^{52} +4.04766i q^{53} +1.71457 q^{54} +(0.248450 - 0.268383i) q^{55} -1.18947 q^{56} +3.47954i q^{57} -1.71457i q^{58} +0.328734 q^{59} +(-1.54203 - 1.42750i) q^{60} -5.72054 q^{61} +8.66328i q^{62} -0.654317i q^{63} -1.53842 q^{64} +(4.35547 + 4.03198i) q^{65} -0.280433 q^{66} -3.51985i q^{67} -3.63050i q^{68} -7.69972 q^{69} +(-1.70416 + 1.84088i) q^{70} +11.7457 q^{71} -1.81788i q^{72} -1.12143i q^{73} +18.1580 q^{74} +(4.98515 - 0.385107i) q^{75} -3.26989 q^{76} +0.107019i q^{77} -4.55101i q^{78} +12.4074 q^{79} +(-7.58963 + 8.19854i) q^{80} +1.00000 q^{81} -10.5860i q^{82} +7.89306i q^{83} +0.614893 q^{84} +(6.33925 + 5.86842i) q^{85} +18.0968 q^{86} -1.00000i q^{87} +0.297329i q^{88} -5.04702 q^{89} +(-2.81344 - 2.60448i) q^{90} -1.73677 q^{91} -7.23579i q^{92} +5.05274i q^{93} +17.6663 q^{94} +(5.28551 - 5.70957i) q^{95} +4.93087 q^{96} -8.49076i q^{97} +11.2679i q^{98} -0.163559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9 + 4 * q^10 + 24 * q^11 - 12 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^19 + 8 * q^20 + 16 * q^21 - 18 * q^24 + 2 * q^25 - 10 * q^29 - 18 * q^30 + 4 * q^31 - 8 * q^34 + 2 * q^35 + 10 * q^36 - 4 * q^39 - 14 * q^40 - 28 * q^41 - 40 * q^44 - 12 * q^46 + 14 * q^49 - 12 * q^50 - 6 * q^54 + 2 * q^55 - 4 * q^56 - 8 * q^59 - 2 * q^60 - 24 * q^61 + 18 * q^64 + 6 * q^65 + 28 * q^66 - 16 * q^69 + 20 * q^70 + 60 * q^71 - 4 * q^74 + 12 * q^75 - 88 * q^76 + 36 * q^79 - 30 * q^80 + 10 * q^81 + 12 * q^84 - 14 * q^85 + 60 * q^86 + 44 * q^89 - 4 * q^90 - 24 * q^91 + 100 * q^94 - 36 * q^95 + 2 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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