LMFDB - Embedded newform 1305.2.c.j.784.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(784,1305)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.4204774638$
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:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			784.8
Root			$-0.604479 + 0.604479i$ of defining polynomial
Character	$\chi$	$=$	1305.784
Dual form			1305.2.c.j.784.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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 $

$2$			1.71457i			1.21238i	0.795319	+	0.606192i	$0.207304\pi$
$2$			1.71457i			1.21238i	−0.795319	+	0.606192i	$0.792696\pi$
$3$		0			0
$3$		0			0
$4$	−0.939748			−0.469874
$5$	−1.51903	−	1.64090i	−0.679330	−	0.733833i
$5$	−1.51903	−	1.64090i	−0.679330	−	0.733833i
$6$		0			0
$7$		−	0.654317i		−	0.247309i	−0.992325	−	0.123654i	$-0.960539\pi$
$7$		−	0.654317i		−	0.247309i	0.992325	−	0.123654i	$-0.0394614\pi$
$8$			1.81788i			0.642716i
$9$		0			0
$10$	2.81344	−	2.60448i	0.889687	−	0.823609i
$11$	−0.163559			−0.0493148			−0.0246574	−	0.999696i	$-0.507849\pi$
$11$	−0.163559			−0.0493148			−0.0246574	+	0.999696i	$0.507849\pi$
$12$		0			0
$13$		−	2.65432i		−	0.736175i	−0.929791	−	0.368088i	$-0.880013\pi$
$13$		−	2.65432i		−	0.736175i	0.929791	−	0.368088i	$-0.119987\pi$
$14$	1.12187			0.299833
$15$		0			0
$16$	−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	$0.844798\pi$
$18$		0			0
$19$	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			0
$22$		−	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	$0.703369\pi$
$24$		0			0
$25$	−0.385107	+	4.98515i	−0.0770214	+	0.997029i
$26$	4.55101			0.892527
$27$		0			0
$28$			0.614893i			0.116204i
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	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			0
$34$	−6.62385			−1.13598
$35$	−1.07367	+	0.993926i	−0.181483	+	0.168004i
$36$		0			0
$37$			10.5904i			1.74106i	0.492118	+	0.870528i	$0.336223\pi$
$37$			10.5904i			1.74106i	−0.492118	+	0.870528i	$0.663777\pi$
$38$			5.96591i			0.967798i
$39$		0			0
$40$	2.98295	−	2.76140i	0.471646	−	0.436616i
$41$	6.17417			0.964244			0.482122	−	0.876104i	$-0.339866\pi$
$41$	6.17417			0.964244			0.482122	+	0.876104i	$0.339866\pi$
$42$		0			0
$43$			10.5547i			1.60958i	0.593559	+	0.804790i	$0.297722\pi$
$43$			10.5547i			1.60958i	−0.593559	+	0.804790i	$0.702278\pi$
$44$	0.153704			0.0231717
$45$		0			0
$46$	−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	$-0.270655\pi$
$48$		0			0
$49$	6.57187			0.938838
$50$	−8.54738	−	0.660292i	−1.20878	−	0.0933795i
$51$		0			0
$52$			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	$0.910330\pi$
$54$		0			0
$55$	0.248450	+	0.268383i	0.0335010	+	0.0361888i
$56$	1.18947			0.158949
$57$		0			0
$58$			1.71457i			0.225134i
$59$	−0.328734			−0.0427975			−0.0213988	−	0.999771i	$-0.506812\pi$
$59$	−0.328734			−0.0427975			−0.0213988	+	0.999771i	$0.506812\pi$
$60$		0			0
$61$	−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			0
$64$	−1.53842			−0.192303
$65$	−4.35547	+	4.03198i	−0.540230	+	0.500106i
$66$		0			0
$67$			3.51985i			0.430019i	0.976612	+	0.215009i	$0.0689782\pi$
$67$			3.51985i			0.430019i	−0.976612	+	0.215009i	$0.931022\pi$
$68$		−	3.63050i		−	0.440263i
$69$		0			0
$70$	−1.70416	−	1.84088i	−0.203686	−	0.220027i
$71$	−11.7457			−1.39396			−0.696981	−	0.717090i	$-0.745474\pi$
$71$	−11.7457			−1.39396			−0.696981	+	0.717090i	$0.745474\pi$
$72$		0			0
$73$			1.12143i			0.131253i	0.997844	+	0.0656267i	$0.0209047\pi$
$73$			1.12143i			0.131253i	−0.997844	+	0.0656267i	$0.979095\pi$
$74$	−18.1580			−2.11083
$75$		0			0
$76$	−3.26989			−0.375082
$77$			0.107019i			0.0121960i
$78$		0			0
$79$	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
$81$		0			0
$82$			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			0
$85$	6.33925	−	5.86842i	0.687588	−	0.636520i
$86$	−18.0968			−1.95143
$87$		0			0
$88$		−	0.297329i		−	0.0316954i
$89$	5.04702			0.534983			0.267491	−	0.963560i	$-0.413805\pi$
$89$	5.04702			0.534983			0.267491	+	0.963560i	$0.413805\pi$
$90$		0			0
$91$	−1.73677			−0.182062
$92$		−	7.23579i		−	0.754383i
$93$		0			0
$94$	17.6663			1.82214
$95$	−5.28551	−	5.70957i	−0.542282	−	0.585790i
$96$		0			0
$97$			8.49076i			0.862106i	0.902327	+	0.431053i	$0.141858\pi$
$97$			8.49076i			0.862106i	−0.902327	+	0.431053i	$0.858142\pi$
$98$			11.2679i			1.13823i
$99$		0			0
$100$	0.361903	−	4.68478i	0.0361903	−	0.468478i
$101$	−1.81126			−0.180227			−0.0901136	−	0.995931i	$-0.528723\pi$
$101$	−1.81126			−0.180227			−0.0901136	+	0.995931i	$0.528723\pi$
$102$		0			0
$103$		−	5.80807i		−	0.572286i	−0.958187	−	0.286143i	$-0.907627\pi$
$103$		−	5.80807i		−	0.572286i	0.958187	−	0.286143i	$-0.0923733\pi$
$104$	4.82522			0.473152
$105$		0			0
$106$	−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			0
$109$	−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
$111$		0			0
$112$			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	$0.969002\pi$
$114$		0			0
$115$	12.6345	−	11.6961i	1.17817	−	1.09067i
$116$	−0.939748			−0.0872534
$117$		0			0
$118$		−	0.563637i		−	0.0518870i
$119$	2.52781			0.231724
$120$		0			0
$121$	−10.9732			−0.997568
$122$		−	9.80827i		−	0.887999i
$123$		0			0
$124$	−4.74830			−0.426410
$125$	8.76512	−	6.94066i	0.783976	−	0.620791i
$126$		0			0
$127$		−	12.9962i		−	1.15323i	−0.817017	−	0.576613i	$-0.804374\pi$
$127$		−	12.9962i		−	1.15323i	0.817017	−	0.576613i	$-0.195626\pi$
$128$		−	12.4995i		−	1.10481i
$129$		0			0
$130$	−6.91311	−	7.46775i	−0.606320	−	0.654965i
$131$	−8.69003			−0.759251			−0.379626	−	0.925140i	$-0.623947\pi$
$131$	−8.69003			−0.759251			−0.379626	+	0.925140i	$0.623947\pi$
$132$		0			0
$133$		−	2.27672i		−	0.197417i
$134$	−6.03504			−0.521348
$135$		0			0
$136$	−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	$-0.211850\pi$
$138$		0			0
$139$	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
$141$		0			0
$142$		−	20.1389i		−	1.69002i
$143$			0.434137i			0.0363043i
$144$		0			0
$145$	−1.51903	−	1.64090i	−0.126148	−	0.136269i
$146$	−1.92277			−0.159129
$147$		0			0
$148$		−	9.95234i		−	0.818077i
$149$	16.1461			1.32274			0.661369	−	0.750060i	$-0.269976\pi$
$149$	16.1461			1.32274			0.661369	+	0.750060i	$0.269976\pi$
$150$		0			0
$151$	−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
$153$		0			0
$154$	−0.183492			−0.0147862
$155$	−7.67526	−	8.29105i	−0.616492	−	0.665953i
$156$		0			0
$157$			10.8270i			0.864085i	0.901853	+	0.432043i	$0.142207\pi$
$157$			10.8270i			0.864085i	−0.901853	+	0.432043i	$0.857793\pi$
$158$			21.2733i			1.69241i
$159$		0			0
$160$	−8.09107	+	7.49013i	−0.639655	+	0.592147i
$161$	5.03806			0.397054
$162$		0			0
$163$			13.5528i			1.06154i	0.847517	+	0.530768i	$0.178096\pi$
$163$			13.5528i			1.06154i	−0.847517	+	0.530768i	$0.821904\pi$
$164$	−5.80216			−0.453073
$165$		0			0
$166$	−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	$-0.253351\pi$
$168$		0			0
$169$	5.95460			0.458046
$170$	10.0618	+	10.8691i	0.771706	+	0.833621i
$171$		0			0
$172$		−	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	$-0.110219\pi$
$174$		0			0
$175$	3.26187	+	0.251982i	0.246574	+	0.0190481i
$176$	0.817200			0.0615987
$177$		0			0
$178$			8.65346i			0.648604i
$179$	−1.86896			−0.139693			−0.0698465	−	0.997558i	$-0.522251\pi$
$179$	−1.86896			−0.139693			−0.0698465	+	0.997558i	$0.522251\pi$
$180$		0			0
$181$	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
$183$		0			0
$184$	−13.9971			−1.03188
$185$	17.3778	−	16.0872i	1.27764	−	1.18275i
$186$		0			0
$187$		−	0.631872i		−	0.0462071i
$188$			9.68282i			0.706192i
$189$		0			0
$190$	9.78946	−	9.06238i	0.710202	−	0.657454i
$191$	−7.58677			−0.548960			−0.274480	−	0.961593i	$-0.588506\pi$
$191$	−7.58677			−0.548960			−0.274480	+	0.961593i	$0.588506\pi$
$192$		0			0
$193$		−	27.2794i		−	1.96362i	−0.189876	−	0.981808i	$-0.560809\pi$
$193$		−	27.2794i		−	1.96362i	0.189876	−	0.981808i	$-0.439191\pi$
$194$	−14.5580			−1.04520
$195$		0			0
$196$	−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			0
$199$	−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
$201$		0			0
$202$		−	3.10553i		−	0.218504i
$203$		−	0.654317i		−	0.0459241i
$204$		0			0
$205$	−9.37874	−	10.1312i	−0.655040	−	0.707594i
$206$	9.95834			0.693831
$207$		0			0
$208$			13.2619i			0.919551i
$209$	−0.569108			−0.0393660
$210$		0			0
$211$	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
$213$		0			0
$214$	5.97644			0.408541
$215$	17.3192	−	16.0329i	1.18116	−	1.09344i
$216$		0			0
$217$		−	3.30610i		−	0.224432i
$218$		−	13.9057i		−	0.941810i
$219$		0			0
$220$	−0.233481	−	0.252213i	−0.0157413	−	0.0170042i
$221$	10.2544			0.689783
$222$		0			0
$223$			19.1383i			1.28160i	0.767708	+	0.640799i	$0.221397\pi$
$223$			19.1383i			1.28160i	−0.767708	+	0.640799i	$0.778603\pi$
$224$	−3.22635			−0.215570
$225$		0			0
$226$	−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	$0.853759\pi$
$228$		0			0
$229$	−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			0
$232$			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	$0.751163\pi$
$234$		0			0
$235$	−16.9072	+	15.6515i	−1.10291	+	1.02099i
$236$	0.308927			0.0201094
$237$		0			0
$238$			4.33410i			0.280938i
$239$	24.1169			1.55999			0.779997	−	0.625784i	$-0.215221\pi$
$239$	24.1169			1.55999			0.779997	+	0.625784i	$0.215221\pi$
$240$		0			0
$241$	−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
$243$		0			0
$244$	5.37587			0.344155
$245$	−9.98285	−	10.7838i	−0.637781	−	0.688951i
$246$		0			0
$247$		−	9.23579i		−	0.587659i
$248$			9.18526i			0.583264i
$249$		0			0
$250$	11.9002	+	15.0284i	0.752637	+	0.950480i
$251$	8.74458			0.551953			0.275977	−	0.961164i	$-0.410999\pi$
$251$	8.74458			0.551953			0.275977	+	0.961164i	$0.410999\pi$
$252$		0			0
$253$		−	1.25936i		−	0.0791750i
$254$	22.2829			1.39815
$255$		0			0
$256$	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	$0.897074\pi$
$258$		0			0
$259$	6.92950			0.430578
$260$	4.09304	−	3.78905i	0.253840	−	0.234987i
$261$		0			0
$262$		−	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			0
$265$	6.64181	−	6.14851i	0.408003	−	0.377700i
$266$	3.90359			0.239345
$267$		0			0
$268$		−	3.30778i		−	0.202055i
$269$	23.0385			1.40468			0.702342	−	0.711839i	$-0.252138\pi$
$269$	23.0385			1.40468			0.702342	+	0.711839i	$0.252138\pi$
$270$		0			0
$271$	−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
$273$		0			0
$274$	24.7842			1.49727
$275$	0.0629876	−	0.815364i	0.00379829	−	0.0491683i
$276$		0			0
$277$			15.7000i			0.943321i	0.881780	+	0.471661i	$0.156345\pi$
$277$			15.7000i			0.943321i	−0.881780	+	0.471661i	$0.843655\pi$
$278$			2.95262i			0.177087i
$279$		0			0
$280$	−1.80683	−	1.95180i	−0.107979	−	0.116642i
$281$	−13.4721			−0.803677			−0.401838	−	0.915711i	$-0.631629\pi$
$281$	−13.4721			−0.803677			−0.401838	+	0.915711i	$0.631629\pi$
$282$		0			0
$283$		−	5.86481i		−	0.348627i	−0.984690	−	0.174313i	$-0.944229\pi$
$283$		−	5.86481i		−	0.348627i	0.984690	−	0.174313i	$-0.0557706\pi$
$284$	11.0380			0.654986
$285$		0			0
$286$	−0.744357			−0.0440148
$287$		−	4.03987i		−	0.238466i
$288$		0			0
$289$	2.07510			0.122065
$290$	2.81344	−	2.60448i	0.165211	−	0.152940i
$291$		0			0
$292$		−	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	$-0.0979022\pi$
$294$		0			0
$295$	0.499356	+	0.539420i	0.0290737	+	0.0314062i
$296$	−19.2521			−1.11901
$297$		0			0
$298$			27.6836i			1.60367i
$299$	20.4375			1.18193
$300$		0			0
$301$	6.90614			0.398063
$302$		−	1.25692i		−	0.0723277i
$303$		0			0
$304$	−17.3850			−0.997101
$305$	8.68967	+	9.38684i	0.497569	+	0.537489i
$306$		0			0
$307$		−	16.0760i		−	0.917504i	−0.888564	−	0.458752i	$-0.848297\pi$
$307$		−	16.0760i		−	0.917504i	0.888564	−	0.458752i	$-0.151703\pi$
$308$		−	0.100571i		−	0.00573057i
$309$		0			0
$310$	14.2156	−	13.1598i	0.807390	−	0.747424i
$311$	−16.9976			−0.963846			−0.481923	−	0.876214i	$-0.660061\pi$
$311$	−16.9976			−0.963846			−0.481923	+	0.876214i	$0.660061\pi$
$312$		0			0
$313$		−	25.2391i		−	1.42660i	−0.700858	−	0.713300i	$-0.747200\pi$
$313$		−	25.2391i		−	1.42660i	0.700858	−	0.713300i	$-0.252800\pi$
$314$	−18.5636			−1.04760
$315$		0			0
$316$	−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	$-0.0274528\pi$
$318$		0			0
$319$	−0.163559			−0.00915753
$320$	2.33690	+	2.52439i	0.130637	+	0.141118i
$321$		0			0
$322$			8.63810i			0.481382i
$323$			13.4424i			0.747955i
$324$		0			0
$325$	13.2322	+	1.02220i	0.733988	+	0.0567012i
$326$	−23.2372			−1.28699
$327$		0			0
$328$			11.2239i			0.619735i
$329$	−6.74185			−0.371690
$330$		0			0
$331$	−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
$333$		0			0
$334$	31.6631			1.73253
$335$	5.77573	−	5.34676i	0.315562	−	0.292125i
$336$		0			0
$337$		−	6.45613i		−	0.351688i	−0.984418	−	0.175844i	$-0.943735\pi$
$337$		−	6.45613i		−	0.351688i	0.984418	−	0.175844i	$-0.0562654\pi$
$338$			10.2096i			0.555328i
$339$		0			0
$340$	−5.95730	+	5.51484i	−0.323080	+	0.299084i
$341$	−0.826420			−0.0447531
$342$		0			0
$343$		−	8.88031i		−	0.479491i
$344$	−19.1872			−1.03450
$345$		0			0
$346$	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			0
$349$	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
$351$		0			0
$352$			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			0
$355$	17.8421	+	19.2736i	0.946960	+	1.02293i
$356$	−4.74292			−0.251374
$357$		0			0
$358$		−	3.20447i		−	0.169361i
$359$	−35.1010			−1.85256			−0.926279	−	0.376838i	$-0.877011\pi$
$359$	−35.1010			−1.85256			−0.926279	+	0.376838i	$0.877011\pi$
$360$		0			0
$361$	−6.89283			−0.362780
$362$			28.9508i			1.52162i
$363$		0			0
$364$	1.63212			0.0855464
$365$	1.84015	−	1.70348i	0.0963181	−	0.0891644i
$366$		0			0
$367$			23.1943i			1.21073i	0.795946	+	0.605367i	$0.206974\pi$
$367$			23.1943i			1.21073i	−0.795946	+	0.605367i	$0.793026\pi$
$368$		−	38.4706i		−	2.00542i
$369$		0			0
$370$	27.5826	+	29.7955i	1.43395	+	1.54900i
$371$	2.64845			0.137501
$372$		0			0
$373$			4.99469i			0.258615i	0.991605	+	0.129308i	$0.0412754\pi$
$373$			4.99469i			0.258615i	−0.991605	+	0.129308i	$0.958725\pi$
$374$	1.08339			0.0560207
$375$		0			0
$376$	18.7307			0.965964
$377$		−	2.65432i		−	0.136704i
$378$		0			0
$379$	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
$381$		0			0
$382$		−	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	$-0.219988\pi$
$384$		0			0
$385$	0.175608	−	0.162565i	0.00894981	−	0.00828509i
$386$	46.7725			2.38066
$387$		0			0
$388$		−	7.97917i		−	0.405081i
$389$	31.0949			1.57657			0.788286	−	0.615308i	$-0.210969\pi$
$389$	31.0949			1.57657			0.788286	+	0.615308i	$0.210969\pi$
$390$		0			0
$391$	−29.7461			−1.50433
$392$			11.9468i			0.603407i
$393$		0			0
$394$	−16.6421			−0.838414
$395$	−18.8472	−	20.3593i	−0.948304	−	1.02439i
$396$		0			0
$397$		−	12.1948i		−	0.612038i	−0.952025	−	0.306019i	$-0.901003\pi$
$397$		−	12.1948i		−	0.612038i	0.952025	−	0.306019i	$-0.0989971\pi$
$398$		−	39.3528i		−	1.97258i
$399$		0			0
$400$	1.92414	−	24.9076i	0.0962068	−	1.24538i
$401$	−4.33039			−0.216249			−0.108125	−	0.994137i	$-0.534485\pi$
$401$	−4.33039			−0.216249			−0.108125	+	0.994137i	$0.534485\pi$
$402$		0			0
$403$		−	13.4116i		−	0.668078i
$404$	1.70213			0.0846841
$405$		0			0
$406$	1.12187			0.0556776
$407$		−	1.73216i		−	0.0858599i
$408$		0			0
$409$	−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
$411$		0			0
$412$			5.45812i			0.268902i
$413$			0.215096i			0.0105842i
$414$		0			0
$415$	12.9517	−	11.9898i	0.635775	−	0.588555i
$416$	−13.0881			−0.641697
$417$		0			0
$418$		−	0.975776i		−	0.0477267i
$419$	14.1945			0.693445			0.346723	−	0.937968i	$-0.387295\pi$
$419$	14.1945			0.693445			0.346723	+	0.937968i	$0.387295\pi$
$420$		0			0
$421$	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
$423$		0			0
$424$	−7.35815			−0.357343
$425$	−19.2590	−	1.48777i	−0.934198	−	0.0721676i
$426$		0			0
$427$			3.74305i			0.181139i
$428$			3.27566i			0.158335i
$429$		0			0
$430$	27.4896	+	29.6950i	1.32566	+	1.43202i
$431$	18.0910			0.871411			0.435706	−	0.900089i	$-0.356499\pi$
$431$	18.0910			0.871411			0.435706	+	0.900089i	$0.356499\pi$
$432$		0			0
$433$			1.71005i			0.0821798i	0.999155	+	0.0410899i	$0.0130830\pi$
$433$			1.71005i			0.0821798i	−0.999155	+	0.0410899i	$0.986917\pi$
$434$	5.66853			0.272098
$435$		0			0
$436$	7.62163			0.365010
$437$			26.7914i			1.28161i
$438$		0			0
$439$	−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
$441$		0			0
$442$			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	$0.851730\pi$
$444$		0			0
$445$	−7.66656	−	8.28165i	−0.363430	−	0.392588i
$446$	−32.8140			−1.55379
$447$		0			0
$448$			1.00661i			0.0475581i
$449$	25.2808			1.19307			0.596537	−	0.802586i	$-0.296543\pi$
$449$	25.2808			1.19307			0.596537	+	0.802586i	$0.296543\pi$
$450$		0			0
$451$	−1.00984			−0.0475515
$452$		−	1.94260i		−	0.0913723i
$453$		0			0
$454$	−22.9102			−1.07523
$455$	2.63820	+	2.84986i	0.123680	+	0.133603i
$456$		0			0
$457$			4.25835i			0.199197i	0.995028	+	0.0995986i	$0.0317559\pi$
$457$			4.25835i			0.199197i	−0.995028	+	0.0995986i	$0.968244\pi$
$458$		−	12.7760i		−	0.596984i
$459$		0			0
$460$	−11.8732	+	10.9914i	−0.553591	+	0.512475i
$461$	−2.48353			−0.115670			−0.0578349	−	0.998326i	$-0.518420\pi$
$461$	−2.48353			−0.115670			−0.0578349	+	0.998326i	$0.518420\pi$
$462$		0			0
$463$		−	27.4876i		−	1.27746i	−0.769433	−	0.638728i	$-0.779461\pi$
$463$		−	27.4876i		−	1.27746i	0.769433	−	0.638728i	$-0.220539\pi$
$464$	−4.99637			−0.231951
$465$		0			0
$466$	−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	$0.953830\pi$
$468$		0			0
$469$	2.30310			0.106347
$470$	−26.8356	−	28.9886i	−1.23783	−	1.33715i
$471$		0			0
$472$		−	0.597598i		−	0.0275067i
$473$		−	1.72632i		−	0.0793761i
$474$		0			0
$475$	−1.33999	+	17.3460i	−0.0614831	+	0.795889i
$476$	−2.37550			−0.108881
$477$		0			0
$478$			41.3501i			1.89131i
$479$	−11.5736			−0.528812			−0.264406	−	0.964412i	$-0.585176\pi$
$479$	−11.5736			−0.528812			−0.264406	+	0.964412i	$0.585176\pi$
$480$		0			0
$481$	28.1104			1.28172
$482$		−	12.3930i		−	0.564485i
$483$		0			0
$484$	10.3121			0.468731
$485$	13.9325	−	12.8977i	0.632642	−	0.585654i
$486$		0			0
$487$		−	31.1947i		−	1.41357i	−0.707431	−	0.706783i	$-0.750146\pi$
$487$		−	31.1947i		−	1.41357i	0.707431	−	0.706783i	$-0.249854\pi$
$488$		−	10.3992i		−	0.470751i
$489$		0			0
$490$	18.4895	−	17.1163i	0.835272	−	0.773235i
$491$	7.93512			0.358107			0.179053	−	0.983839i	$-0.442697\pi$
$491$	7.93512			0.358107			0.179053	+	0.983839i	$0.442697\pi$
$492$		0			0
$493$			3.86328i			0.173993i
$494$	15.8354			0.712469
$495$		0			0
$496$	−25.2454			−1.13355
$497$			7.68543i			0.344739i
$498$		0			0
$499$	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
$501$		0			0
$502$			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	$-0.361971\pi$
$504$		0			0
$505$	2.75136	+	2.97210i	0.122434	+	0.132257i
$506$	2.15925			0.0959905
$507$		0			0
$508$			12.2132i			0.541871i
$509$	−36.9407			−1.63737			−0.818684	−	0.574245i	$-0.805296\pi$
$509$	−36.9407			−1.63737			−0.818684	+	0.574245i	$0.805296\pi$
$510$		0			0
$511$	0.733771			0.0324601
$512$			6.47089i			0.285976i
$513$		0			0
$514$	−17.4676			−0.770462
$515$	−9.53047	+	8.82263i	−0.419963	+	0.388771i
$516$		0			0
$517$			1.68525i			0.0741172i
$518$			11.8811i			0.522026i
$519$		0			0
$520$	−7.32964	−	7.91770i	−0.321426	−	0.347214i
$521$	33.2555			1.45695			0.728476	−	0.685072i	$-0.240229\pi$
$521$	33.2555			1.45695			0.728476	+	0.685072i	$0.240229\pi$
$522$		0			0
$523$			0.341001i			0.0149109i	0.999972	+	0.00745547i	$0.00237317\pi$
$523$			0.341001i			0.0149109i	−0.999972	+	0.00745547i	$0.997627\pi$
$524$	8.16644			0.356752
$525$		0			0
$526$	−20.0252			−0.873139
$527$			19.5201i			0.850310i
$528$		0			0
$529$	−36.2856			−1.57764
$530$	10.5421	+	11.3878i	0.457917	+	0.494656i
$531$		0			0
$532$			2.13954i			0.0927609i
$533$		−	16.3882i		−	0.709852i
$534$		0			0
$535$	−5.71965	+	5.29484i	−0.247282	+	0.228916i
$536$	−6.39866			−0.276380
$537$		0			0
$538$			39.5012i			1.70302i
$539$	−1.07489			−0.0462986
$540$		0			0
$541$	−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
$543$		0			0
$544$	19.0493			0.816732
$545$	12.3198	+	13.3082i	0.527720	+	0.570060i
$546$		0			0
$547$			42.5936i			1.82117i	0.413320	+	0.910586i	$0.364369\pi$
$547$			42.5936i			1.82117i	−0.413320	+	0.910586i	$0.635631\pi$
$548$			13.5841i			0.580284i
$549$		0			0
$550$	1.39800	+	0.107997i	0.0596108	+	0.00460499i
$551$	3.47954			0.148233
$552$		0			0
$553$		−	8.11836i		−	0.345228i
$554$	−26.9187			−1.14367
$555$		0			0
$556$	−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	$-0.121484\pi$
$558$		0			0
$559$	28.0156			1.18493
$560$	5.36445	−	4.96602i	0.226689	−	0.209853i
$561$		0			0
$562$		−	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	$0.957643\pi$
$564$		0			0
$565$	3.39199	−	3.14006i	0.142702	−	0.132103i
$566$	10.0556			0.422669
$567$		0			0
$568$		−	21.3523i		−	0.895921i
$569$	−42.3981			−1.77742			−0.888710	−	0.458469i	$-0.848398\pi$
$569$	−42.3981			−1.77742			−0.888710	+	0.458469i	$0.848398\pi$
$570$		0			0
$571$	−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
$573$		0			0
$574$	6.92663			0.289112
$575$	−38.3842	−	2.96521i	−1.60073	−	0.123658i
$576$		0			0
$577$		−	22.0727i		−	0.918897i	−0.888204	−	0.459448i	$-0.848047\pi$
$577$		−	22.0727i		−	0.918897i	0.888204	−	0.459448i	$-0.151953\pi$
$578$			3.55791i			0.147990i
$579$		0			0
$580$	1.42750	+	1.54203i	0.0592739	+	0.0640294i
$581$	5.16457			0.214262
$582$		0			0
$583$		−	0.662030i		−	0.0274185i
$584$	−2.03862			−0.0843587
$585$		0			0
$586$	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	$0.634519\pi$
$588$		0			0
$589$	17.5812			0.724421
$590$	−0.924873	+	0.856181i	−0.0380764	+	0.0352484i
$591$		0			0
$592$		−	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			0
$595$	−3.83981	−	4.14788i	−0.157417	−	0.170046i
$596$	−15.1732			−0.621520
$597$		0			0
$598$			35.0415i			1.43295i
$599$	7.70544			0.314836			0.157418	−	0.987532i	$-0.449683\pi$
$599$	7.70544			0.314836			0.157418	+	0.987532i	$0.449683\pi$
$600$		0			0
$601$	−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
$603$		0			0
$604$	0.688913			0.0280315
$605$	16.6687	+	18.0060i	0.677678	+	0.732048i
$606$		0			0
$607$		−	13.2516i		−	0.537864i	−0.963159	−	0.268932i	$-0.913329\pi$
$607$		−	13.2516i		−	0.537864i	0.963159	−	0.268932i	$-0.0866707\pi$
$608$		−	17.1571i		−	0.695814i
$609$		0			0
$610$	−16.0944	+	14.8990i	−0.651643	+	0.603244i
$611$	−27.3491			−1.10643
$612$		0			0
$613$		−	29.4609i		−	1.18991i	−0.803758	−	0.594957i	$-0.797169\pi$
$613$		−	29.4609i		−	1.18991i	0.803758	−	0.594957i	$-0.202831\pi$
$614$	27.5634			1.11237
$615$		0			0
$616$	−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	$0.963974\pi$
$618$		0			0
$619$	−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
$621$		0			0
$622$		−	29.1436i		−	1.16855i
$623$		−	3.30235i		−	0.132306i
$624$		0			0
$625$	−24.7034	−	3.83963i	−0.988135	−	0.153585i
$626$	43.2743			1.72959
$627$		0			0
$628$		−	10.1746i		−	0.406011i
$629$	−40.9138			−1.63134
$630$		0			0
$631$	−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
$633$		0			0
$634$	5.25911			0.208866
$635$	−21.3255	+	19.7416i	−0.846276	+	0.783421i
$636$		0			0
$637$		−	17.4438i		−	0.691149i
$638$		−	0.280433i		−	0.0111024i
$639$		0			0
$640$	−20.5104	+	18.9870i	−0.810744	+	0.750529i
$641$	44.0804			1.74107			0.870535	−	0.492107i	$-0.163773\pi$
$641$	44.0804			1.74107			0.870535	+	0.492107i	$0.163773\pi$
$642$		0			0
$643$		−	3.56303i		−	0.140512i	−0.997529	−	0.0702561i	$-0.977618\pi$
$643$		−	3.56303i		−	0.140512i	0.997529	−	0.0702561i	$-0.0223816\pi$
$644$	−4.73450			−0.186566
$645$		0			0
$646$	−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	$0.704265\pi$
$648$		0			0
$649$	0.0537673			0.00211055
$650$	−1.75263	+	22.6875i	−0.0687436	+	0.889875i
$651$		0			0
$652$		−	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	$-0.0734592\pi$
$654$		0			0
$655$	13.2004	+	14.2595i	0.515782	+	0.557164i
$656$	−30.8484			−1.20443
$657$		0			0
$658$		−	11.5594i		−	0.450631i
$659$	−28.1795			−1.09772			−0.548858	−	0.835915i	$-0.684937\pi$
$659$	−28.1795			−1.09772			−0.548858	+	0.835915i	$0.684937\pi$
$660$		0			0
$661$	−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
$663$		0			0
$664$	−14.3486			−0.556834
$665$	−3.73587	+	3.45840i	−0.144871	+	0.134111i
$666$		0			0
$667$			7.69972i			0.298134i
$668$			17.3544i			0.671461i
$669$		0			0
$670$	9.16739	+	9.90289i	0.354167	+	0.382582i
$671$	0.935645			0.0361202
$672$		0			0
$673$		−	44.7915i		−	1.72659i	−0.504703	−	0.863293i	$-0.668398\pi$
$673$		−	44.7915i		−	1.72659i	0.504703	−	0.863293i	$-0.331602\pi$
$674$	11.0695			0.426380
$675$		0			0
$676$	−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	$-0.105119\pi$
$678$		0			0
$679$	5.55565			0.213206
$680$	10.6681	+	11.5240i	0.409102	+	0.441924i
$681$		0			0
$682$		−	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	$-0.184447\pi$
$684$		0			0
$685$	−23.7193	+	21.9576i	−0.906267	+	0.838958i
$686$	15.2259			0.581328
$687$		0			0
$688$		−	52.7353i		−	2.01051i
$689$	10.7438			0.409305
$690$		0			0
$691$	−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			0
$694$	−25.6182			−0.972452
$695$	−2.61589	−	2.82576i	−0.0992262	−	0.107187i
$696$		0			0
$697$			23.8525i			0.903479i
$698$			20.7557i			0.785615i
$699$		0			0
$700$	−3.06533	−	0.236800i	−0.115859	−	0.00895018i
$701$	−49.6636			−1.87577			−0.937885	−	0.346946i	$-0.887219\pi$
$701$	−49.6636			−1.87577			−0.937885	+	0.346946i	$0.887219\pi$
$702$		0			0
$703$			36.8498i			1.38982i
$704$	0.251622			0.00948336
$705$		0			0
$706$	31.1944			1.17402
$707$			1.18514i			0.0445717i
$708$		0			0
$709$	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
$711$		0			0
$712$			9.17485i			0.343842i
$713$			38.9047i			1.45699i
$714$		0			0
$715$	0.712375	−	0.659466i	0.0266413	−	0.0246626i
$716$	1.75636			0.0656381
$717$		0			0
$718$		−	60.1830i		−	2.24601i
$719$	−34.3195			−1.27990			−0.639951	−	0.768415i	$-0.721046\pi$
$719$	−34.3195			−1.27990			−0.639951	+	0.768415i	$0.721046\pi$
$720$		0			0
$721$	−3.80032			−0.141531
$722$		−	11.8182i		−	0.439829i
$723$		0			0
$724$	−15.8678			−0.589723
$725$	−0.385107	+	4.98515i	−0.0143025	+	0.185144i
$726$		0			0
$727$			19.3206i			0.716560i	0.933614	+	0.358280i	$0.116637\pi$
$727$			19.3206i			0.716560i	−0.933614	+	0.358280i	$0.883363\pi$
$728$		−	3.15722i		−	0.117014i
$729$		0			0
$730$	2.92074	+	3.15507i	0.108101	+	0.116774i
$731$	−40.7758			−1.50815
$732$		0			0
$733$		−	3.10588i		−	0.114718i	−0.998354	−	0.0573591i	$-0.981732\pi$
$733$		−	3.10588i		−	0.114718i	0.998354	−	0.0573591i	$-0.0182680\pi$
$734$	−39.7683			−1.46787
$735$		0			0
$736$	37.9663			1.39946
$737$		−	0.575703i		−	0.0212063i
$738$		0			0
$739$	−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
$741$		0			0
$742$			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	$-0.0493091\pi$
$744$		0			0
$745$	−24.5264	−	26.4941i	−0.898576	−	0.970669i
$746$	−8.56373			−0.313541
$747$		0			0
$748$			0.593801i			0.0217115i
$749$	−2.28074			−0.0833364
$750$		0			0
$751$	−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
$753$		0			0
$754$	4.55101			0.165738
$755$	1.11357	+	1.20292i	0.0405271	+	0.0437786i
$756$		0			0
$757$		−	41.6526i		−	1.51389i	−0.653479	−	0.756945i	$-0.726691\pi$
$757$		−	41.6526i		−	1.51389i	0.653479	−	0.756945i	$-0.273309\pi$
$758$			49.3347i			1.79192i
$759$		0			0
$760$	10.3793	−	9.60841i	0.376496	−	0.348534i
$761$	28.3957			1.02934			0.514671	−	0.857388i	$-0.327914\pi$
$761$	28.3957			1.02934			0.514671	+	0.857388i	$0.327914\pi$
$762$		0			0
$763$			5.30670i			0.192115i
$764$	7.12965			0.257942
$765$		0			0
$766$	42.7753			1.54554
$767$			0.872565i			0.0315065i
$768$		0			0
$769$	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
$771$		0			0
$772$			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	$-0.452873\pi$
$774$		0			0
$775$	−1.94585	+	25.1887i	−0.0698969	+	0.904804i
$776$	−15.4351			−0.554089
$777$		0			0
$778$			53.3143i			1.91141i
$779$	21.4833			0.769717
$780$		0			0
$781$	1.92112			0.0687429
$782$		−	51.0018i		−	1.82382i
$783$		0			0
$784$	−32.8355			−1.17270
$785$	17.7660	−	16.4464i	0.634094	−	0.586999i
$786$		0			0
$787$			32.7472i			1.16731i	0.812002	+	0.583655i	$0.198378\pi$
$787$			32.7472i			1.16731i	−0.812002	+	0.583655i	$0.801622\pi$
$788$		−	9.12143i		−	0.324938i
$789$		0			0
$790$	34.9074	−	32.3148i	1.24195	−	1.14971i
$791$	1.35257			0.0480920
$792$		0

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more