LMFDB - Embedded newform 1305.2.c.j.784.6

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			784.6
Root			$0.561843 - 0.561843i$ of defining polynomial
Character	$\chi$	$=$	1305.784
Dual form			1305.2.c.j.784.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.712495i q^{2} +1.49235 q^{4} +(-2.01848 + 0.962154i) q^{5} +2.77986i q^{7} +2.48828i q^{8} +O(q^{10})$	\(q+0.712495i q^{2} +1.49235 q^{4} +(-2.01848 + 0.962154i) q^{5} +2.77986i q^{7} +2.48828i q^{8} +(-0.685530 - 1.43816i) q^{10} -4.26814 q^{11} +0.779856i q^{13} -1.98063 q^{14} +1.21181 q^{16} -1.90354i q^{17} -6.72036 q^{19} +(-3.01228 + 1.43587i) q^{20} -3.04103i q^{22} -2.17168i q^{23} +(3.14852 - 3.88418i) q^{25} -0.555643 q^{26} +4.14852i q^{28} +1.00000 q^{29} -8.82061 q^{31} +5.83998i q^{32} +1.35626 q^{34} +(-2.67465 - 5.61108i) q^{35} -1.48402i q^{37} -4.78822i q^{38} +(-2.39411 - 5.02255i) q^{40} +7.71389 q^{41} -8.19624i q^{43} -6.36956 q^{44} +1.54731 q^{46} -5.19381i q^{47} -0.727598 q^{49} +(2.76746 + 2.24330i) q^{50} +1.16382i q^{52} +11.7853i q^{53} +(8.61515 - 4.10661i) q^{55} -6.91707 q^{56} +0.712495i q^{58} -4.46028 q^{59} -5.24905 q^{61} -6.28464i q^{62} -1.73733 q^{64} +(-0.750341 - 1.57412i) q^{65} +8.49375i q^{67} -2.84075i q^{68} +(3.99787 - 1.90567i) q^{70} -0.663102 q^{71} +16.5345i q^{73} +1.05736 q^{74} -10.0291 q^{76} -11.8648i q^{77} -9.54554 q^{79} +(-2.44602 + 1.16595i) q^{80} +5.49611i q^{82} +0.0123998i q^{83} +(1.83150 + 3.84226i) q^{85} +5.83978 q^{86} -10.6203i q^{88} -5.46783 q^{89} -2.16789 q^{91} -3.24091i q^{92} +3.70056 q^{94} +(13.5649 - 6.46602i) q^{95} +0.952006i q^{97} -0.518410i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{4}+O(q^{10}) $ 10 * q - 10 * q^4	$ 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}+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

Display 100 coefficients 10 coefficients

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 $

$2$			0.712495i			0.503810i	0.967752	+	0.251905i	$0.0810571\pi$
$2$			0.712495i			0.503810i	−0.967752	+	0.251905i	$0.918943\pi$
$3$		0			0
$3$		0			0
$4$	1.49235			0.746175
$5$	−2.01848	+	0.962154i	−0.902692	+	0.430288i
$5$	−2.01848	+	0.962154i	−0.902692	+	0.430288i
$6$		0			0
$7$			2.77986i			1.05069i	0.850890	+	0.525343i	$0.176063\pi$
$7$			2.77986i			1.05069i	−0.850890	+	0.525343i	$0.823937\pi$
$8$			2.48828i			0.879741i
$9$		0			0
$10$	−0.685530	−	1.43816i	−0.216784	−	0.454785i
$11$	−4.26814			−1.28689			−0.643446	−	0.765491i	$-0.722496\pi$
$11$	−4.26814			−1.28689			−0.643446	+	0.765491i	$0.722496\pi$
$12$		0			0
$13$			0.779856i			0.216293i	0.994135	+	0.108147i	$0.0344916\pi$
$13$			0.779856i			0.216293i	−0.994135	+	0.108147i	$0.965508\pi$
$14$	−1.98063			−0.529347
$15$		0			0
$16$	1.21181			0.302953
$17$		−	1.90354i		−	0.461677i	−0.972992	−	0.230838i	$-0.925853\pi$
$17$		−	1.90354i		−	0.461677i	0.972992	−	0.230838i	$-0.0741469\pi$
$18$		0			0
$19$	−6.72036			−1.54176			−0.770878	−	0.636983i	$-0.780182\pi$
$19$	−6.72036			−1.54176			−0.770878	+	0.636983i	$0.780182\pi$
$20$	−3.01228	+	1.43587i	−0.673566	+	0.321071i
$21$		0			0
$22$		−	3.04103i		−	0.648349i
$23$		−	2.17168i		−	0.452827i	−0.974031	−	0.226413i	$-0.927300\pi$
$23$		−	2.17168i		−	0.452827i	0.974031	−	0.226413i	$-0.0727000\pi$
$24$		0			0
$25$	3.14852	−	3.88418i	0.629704	−	0.776835i
$26$	−0.555643			−0.108971
$27$		0			0
$28$			4.14852i			0.783997i
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	−8.82061			−1.58423			−0.792114	−	0.610373i	$-0.791020\pi$
$31$	−8.82061			−1.58423			−0.792114	+	0.610373i	$0.791020\pi$
$32$			5.83998i			1.03237i
$33$		0			0
$34$	1.35626			0.232597
$35$	−2.67465	−	5.61108i	−0.452098	−	0.948446i
$36$		0			0
$37$		−	1.48402i		−	0.243971i	−0.992532	−	0.121986i	$-0.961074\pi$
$37$		−	1.48402i		−	0.243971i	0.992532	−	0.121986i	$-0.0389262\pi$
$38$		−	4.78822i		−	0.776752i
$39$		0			0
$40$	−2.39411	−	5.02255i	−0.378542	−	0.794135i
$41$	7.71389			1.20471			0.602354	−	0.798229i	$-0.294230\pi$
$41$	7.71389			1.20471			0.602354	+	0.798229i	$0.294230\pi$
$42$		0			0
$43$		−	8.19624i		−	1.24991i	−0.780659	−	0.624957i	$-0.785116\pi$
$43$		−	8.19624i		−	1.24991i	0.780659	−	0.624957i	$-0.214884\pi$
$44$	−6.36956			−0.960247
$45$		0			0
$46$	1.54731			0.228139
$47$		−	5.19381i		−	0.757595i	−0.925480	−	0.378798i	$-0.876338\pi$
$47$		−	5.19381i		−	0.757595i	0.925480	−	0.378798i	$-0.123662\pi$
$48$		0			0
$49$	−0.727598			−0.103943
$50$	2.76746	+	2.24330i	0.391377	+	0.317251i
$51$		0			0
$52$			1.16382i			0.161393i
$53$			11.7853i			1.61884i	0.587231	+	0.809419i	$0.300218\pi$
$53$			11.7853i			1.61884i	−0.587231	+	0.809419i	$0.699782\pi$
$54$		0			0
$55$	8.61515	−	4.10661i	1.16167	−	0.553735i
$56$	−6.91707			−0.924332
$57$		0			0
$58$			0.712495i			0.0935552i
$59$	−4.46028			−0.580679			−0.290339	−	0.956924i	$-0.593768\pi$
$59$	−4.46028			−0.580679			−0.290339	+	0.956924i	$0.593768\pi$
$60$		0			0
$61$	−5.24905			−0.672071			−0.336036	−	0.941849i	$-0.609086\pi$
$61$	−5.24905			−0.672071			−0.336036	+	0.941849i	$0.609086\pi$
$62$		−	6.28464i		−	0.798150i
$63$		0			0
$64$	−1.73733			−0.217166
$65$	−0.750341	−	1.57412i	−0.0930684	−	0.195246i
$66$		0			0
$67$			8.49375i			1.03768i	0.854872	+	0.518838i	$0.173635\pi$
$67$			8.49375i			1.03768i	−0.854872	+	0.518838i	$0.826365\pi$
$68$		−	2.84075i		−	0.344492i
$69$		0			0
$70$	3.99787	−	1.90567i	0.477837	−	0.227772i
$71$	−0.663102			−0.0786957			−0.0393479	−	0.999226i	$-0.512528\pi$
$71$	−0.663102			−0.0786957			−0.0393479	+	0.999226i	$0.512528\pi$
$72$		0			0
$73$			16.5345i			1.93522i	0.252455	+	0.967609i	$0.418762\pi$
$73$			16.5345i			1.93522i	−0.252455	+	0.967609i	$0.581238\pi$
$74$	1.05736			0.122915
$75$		0			0
$76$	−10.0291			−1.15042
$77$		−	11.8648i		−	1.35212i
$78$		0			0
$79$	−9.54554			−1.07396			−0.536978	−	0.843596i	$-0.680434\pi$
$79$	−9.54554			−1.07396			−0.536978	+	0.843596i	$0.680434\pi$
$80$	−2.44602	+	1.16595i	−0.273473	+	0.130357i
$81$		0			0
$82$			5.49611i			0.606944i
$83$			0.0123998i			0.00136106i	1.00000		0.000680528i	$0.000216619\pi$
$83$			0.0123998i			0.00136106i	−1.00000		0.000680528i	$0.999783\pi$
$84$		0			0
$85$	1.83150	+	3.84226i	0.198654	+	0.416752i
$86$	5.83978			0.629720
$87$		0			0
$88$		−	10.6203i		−	1.13213i
$89$	−5.46783			−0.579588			−0.289794	−	0.957089i	$-0.593587\pi$
$89$	−5.46783			−0.579588			−0.289794	+	0.957089i	$0.593587\pi$
$90$		0			0
$91$	−2.16789			−0.227256
$92$		−	3.24091i		−	0.337888i
$93$		0			0
$94$	3.70056			0.381684
$95$	13.5649	−	6.46602i	1.39173	−	0.663399i
$96$		0			0
$97$			0.952006i			0.0966615i	0.998831	+	0.0483308i	$0.0153902\pi$
$97$			0.952006i			0.0966615i	−0.998831	+	0.0483308i	$0.984610\pi$
$98$		−	0.518410i		−	0.0523673i
$99$		0			0
$100$	4.69870	−	5.79655i	0.469870	−	0.579655i
$101$	−8.31781			−0.827653			−0.413826	−	0.910356i	$-0.635808\pi$
$101$	−8.31781			−0.827653			−0.413826	+	0.910356i	$0.635808\pi$
$102$		0			0
$103$			12.4091i			1.22271i	0.791357	+	0.611355i	$0.209375\pi$
$103$			12.4091i			1.22271i	−0.791357	+	0.611355i	$0.790625\pi$
$104$	−1.94050			−0.190282
$105$		0			0
$106$	−8.39698			−0.815587
$107$		−	17.5579i		−	1.69739i	−0.528883	−	0.848695i	$-0.677389\pi$
$107$		−	17.5579i		−	1.69739i	0.528883	−	0.848695i	$-0.322611\pi$
$108$		0			0
$109$	−1.00973			−0.0967146			−0.0483573	−	0.998830i	$-0.515399\pi$
$109$	−1.00973			−0.0967146			−0.0483573	+	0.998830i	$0.515399\pi$
$110$	2.92594	+	6.13825i	0.278977	+	0.585259i
$111$		0			0
$112$			3.36866i			0.318309i
$113$			15.5787i			1.46552i	0.680487	+	0.732761i	$0.261768\pi$
$113$			15.5787i			1.46552i	−0.680487	+	0.732761i	$0.738232\pi$
$114$		0			0
$115$	2.08949	+	4.38349i	0.194846	+	0.408763i
$116$	1.49235			0.138561
$117$		0			0
$118$		−	3.17792i		−	0.292552i
$119$	5.29157			0.485078
$120$		0			0
$121$	7.21701			0.656091
$122$		−	3.73992i		−	0.338596i
$123$		0			0
$124$	−13.1634			−1.18211
$125$	−2.61805	+	10.8695i	−0.234165	+	0.972197i
$126$		0			0
$127$			16.9536i			1.50438i	0.658943	+	0.752192i	$0.271004\pi$
$127$			16.9536i			1.50438i	−0.658943	+	0.752192i	$0.728996\pi$
$128$			10.4421i			0.922961i
$129$		0			0
$130$	1.12155	−	0.534614i	0.0983669	−	0.0468888i
$131$	−11.9324			−1.04254			−0.521268	−	0.853393i	$-0.674541\pi$
$131$	−11.9324			−1.04254			−0.521268	+	0.853393i	$0.674541\pi$
$132$		0			0
$133$		−	18.6816i		−	1.61990i
$134$	−6.05175			−0.522792
$135$		0			0
$136$	4.73655			0.406156
$137$		−	0.239785i		−	0.0204862i	−0.999948	−	0.0102431i	$-0.996739\pi$
$137$		−	0.239785i		−	0.0204862i	0.999948	−	0.0102431i	$-0.00326054\pi$
$138$		0			0
$139$	17.0255			1.44408			0.722040	−	0.691851i	$-0.243205\pi$
$139$	17.0255			1.44408			0.722040	+	0.691851i	$0.243205\pi$
$140$	−3.99151	−	8.37370i	−0.337345	−	0.707707i
$141$		0			0
$142$		−	0.472457i		−	0.0396477i
$143$		−	3.32853i		−	0.278346i
$144$		0			0
$145$	−2.01848	+	0.962154i	−0.167626	+	0.0799025i
$146$	−11.7808			−0.974982
$147$		0			0
$148$		−	2.21468i		−	0.182045i
$149$	−4.13046			−0.338380			−0.169190	−	0.985583i	$-0.554115\pi$
$149$	−4.13046			−0.338380			−0.169190	+	0.985583i	$0.554115\pi$
$150$		0			0
$151$	−3.21580			−0.261698			−0.130849	−	0.991402i	$-0.541770\pi$
$151$	−3.21580			−0.261698			−0.130849	+	0.991402i	$0.541770\pi$
$152$		−	16.7221i		−	1.35635i
$153$		0			0
$154$	8.45362			0.681212
$155$	17.8042	−	8.48678i	1.43007	−	0.681675i
$156$		0			0
$157$		−	6.84237i		−	0.546081i	−0.962003	−	0.273040i	$-0.911971\pi$
$157$		−	6.84237i		−	0.546081i	0.962003	−	0.273040i	$-0.0880292\pi$
$158$		−	6.80115i		−	0.541070i
$159$		0			0
$160$	−5.61895	−	11.7879i	−0.444217	−	0.931913i
$161$	6.03696			0.475779
$162$		0			0
$163$		−	14.0502i		−	1.10050i	−0.835001	−	0.550248i	$-0.814533\pi$
$163$		−	14.0502i		−	1.10050i	0.835001	−	0.550248i	$-0.185467\pi$
$164$	11.5118			0.898923
$165$		0			0
$166$	−0.00883480			−0.000685713
$167$			17.8725i			1.38301i	0.722370	+	0.691507i	$0.243053\pi$
$167$			17.8725i			1.38301i	−0.722370	+	0.691507i	$0.756947\pi$
$168$		0			0
$169$	12.3918			0.953217
$170$	−2.73759	+	1.30494i	−0.209964	+	0.100084i
$171$		0			0
$172$		−	12.2317i		−	0.932656i
$173$			2.82518i			0.214795i	0.994216	+	0.107397i	$0.0342517\pi$
$173$			2.82518i			0.214795i	−0.994216	+	0.107397i	$0.965748\pi$
$174$		0			0
$175$	10.7974	+	8.75243i	0.816210	+	0.661622i
$176$	−5.17218			−0.389868
$177$		0			0
$178$		−	3.89580i		−	0.292002i
$179$	20.2829			1.51601			0.758006	−	0.652247i	$-0.226174\pi$
$179$	20.2829			1.51601			0.758006	+	0.652247i	$0.226174\pi$
$180$		0			0
$181$	−8.63783			−0.642045			−0.321023	−	0.947072i	$-0.604027\pi$
$181$	−8.63783			−0.642045			−0.321023	+	0.947072i	$0.604027\pi$
$182$		−	1.54461i		−	0.114494i
$183$		0			0
$184$	5.40376			0.398370
$185$	1.42785	+	2.99546i	0.104978	+	0.220231i
$186$		0			0
$187$			8.12458i			0.594128i
$188$		−	7.75099i		−	0.565299i
$189$		0			0
$190$	4.60701	+	9.66493i	0.334227	+	0.701168i
$191$	−0.896850			−0.0648938			−0.0324469	−	0.999473i	$-0.510330\pi$
$191$	−0.896850			−0.0648938			−0.0324469	+	0.999473i	$0.510330\pi$
$192$		0			0
$193$		−	10.4506i		−	0.752251i	−0.926569	−	0.376126i	$-0.877256\pi$
$193$		−	10.4506i		−	0.752251i	0.926569	−	0.376126i	$-0.122744\pi$
$194$	−0.678299			−0.0486991
$195$		0			0
$196$	−1.08583			−0.0775594
$197$			9.57740i			0.682361i	0.939998	+	0.341181i	$0.110827\pi$
$197$			9.57740i			0.682361i	−0.939998	+	0.341181i	$0.889173\pi$
$198$		0			0
$199$	−5.61768			−0.398227			−0.199113	−	0.979976i	$-0.563806\pi$
$199$	−5.61768			−0.398227			−0.199113	+	0.979976i	$0.563806\pi$
$200$	9.66493	+	7.83441i	0.683414	+	0.553976i
$201$		0			0
$202$		−	5.92640i		−	0.416980i
$203$			2.77986i			0.195108i
$204$		0			0
$205$	−15.5703	+	7.42195i	−1.08748	+	0.518372i
$206$	−8.84145			−0.616013
$207$		0			0
$208$			0.945039i			0.0655267i
$209$	28.6834			1.98407
$210$		0			0
$211$	−0.605216			−0.0416648			−0.0208324	−	0.999783i	$-0.506632\pi$
$211$	−0.605216			−0.0416648			−0.0208324	+	0.999783i	$0.506632\pi$
$212$			17.5878i			1.20794i
$213$		0			0
$214$	12.5099			0.855162
$215$	7.88604	+	16.5439i	0.537824	+	1.12829i
$216$		0			0
$217$		−	24.5200i		−	1.66453i
$218$		−	0.719428i		−	0.0487258i
$219$		0			0
$220$	12.8568	−	6.12850i	0.866807	−	0.413183i
$221$	1.48449			0.0998575
$222$		0			0
$223$			16.5537i			1.10852i	0.832344	+	0.554260i	$0.186999\pi$
$223$			16.5537i			1.10852i	−0.832344	+	0.554260i	$0.813001\pi$
$224$	−16.2343			−1.08470
$225$		0			0
$226$	−11.0997			−0.738344
$227$		−	22.0061i		−	1.46060i	−0.683128	−	0.730299i	$-0.739381\pi$
$227$		−	22.0061i		−	1.46060i	0.683128	−	0.730299i	$-0.260619\pi$
$228$		0			0
$229$	24.5647			1.62328			0.811641	−	0.584157i	$-0.198575\pi$
$229$	24.5647			1.62328			0.811641	+	0.584157i	$0.198575\pi$
$230$	−3.12322	+	1.48875i	−0.205939	+	0.0981654i
$231$		0			0
$232$			2.48828i			0.163364i
$233$			5.56824i			0.364787i	0.983226	+	0.182394i	$0.0583845\pi$
$233$			5.56824i			0.364787i	−0.983226	+	0.182394i	$0.941615\pi$
$234$		0			0
$235$	4.99725	+	10.4836i	0.325984	+	0.683875i
$236$	−6.65630			−0.433288
$237$		0			0
$238$			3.77022i			0.244387i
$239$	6.90640			0.446738			0.223369	−	0.974734i	$-0.428295\pi$
$239$	6.90640			0.446738			0.223369	+	0.974734i	$0.428295\pi$
$240$		0			0
$241$	−25.1989			−1.62320			−0.811602	−	0.584210i	$-0.801404\pi$
$241$	−25.1989			−1.62320			−0.811602	+	0.584210i	$0.801404\pi$
$242$			5.14208i			0.330545i
$243$		0			0
$244$	−7.83342			−0.501483
$245$	1.46864	−	0.700061i	0.0938281	−	0.0447253i
$246$		0			0
$247$		−	5.24091i		−	0.333471i
$248$		−	21.9482i		−	1.39371i
$249$		0			0
$250$	−7.74446	−	1.86535i	−0.489803	−	0.117975i
$251$	−28.3687			−1.79062			−0.895309	−	0.445446i	$-0.853045\pi$
$251$	−28.3687			−1.79062			−0.895309	+	0.445446i	$0.853045\pi$
$252$		0			0
$253$			9.26903i			0.582739i
$254$	−12.0793			−0.757924
$255$		0			0
$256$	−10.9146			−0.682163
$257$			8.71100i			0.543377i	0.962385	+	0.271688i	$0.0875820\pi$
$257$			8.71100i			0.543377i	−0.962385	+	0.271688i	$0.912418\pi$
$258$		0			0
$259$	4.12536			0.256337
$260$	−1.11977	−	2.34914i	−0.0694453	−	0.145688i
$261$		0			0
$262$		−	8.50175i		−	0.525240i
$263$			13.2128i			0.814736i	0.913264	+	0.407368i	$0.133553\pi$
$263$			13.2128i			0.814736i	−0.913264	+	0.407368i	$0.866447\pi$
$264$		0			0
$265$	−11.3393	−	23.7884i	−0.696567	−	1.46131i
$266$	13.3106			0.816123
$267$		0			0
$268$			12.6757i			0.774289i
$269$	−10.3447			−0.630725			−0.315363	−	0.948971i	$-0.602126\pi$
$269$	−10.3447			−0.630725			−0.315363	+	0.948971i	$0.602126\pi$
$270$		0			0
$271$	9.76022			0.592891			0.296445	−	0.955050i	$-0.404199\pi$
$271$	9.76022			0.592891			0.296445	+	0.955050i	$0.404199\pi$
$272$		−	2.30674i		−	0.139866i
$273$		0			0
$274$	0.170845			0.0103212
$275$	−13.4383	+	16.5782i	−0.810361	+	0.999703i
$276$		0			0
$277$		−	1.87503i		−	0.112659i	−0.998412	−	0.0563297i	$-0.982060\pi$
$277$		−	1.87503i		−	0.112659i	0.998412	−	0.0563297i	$-0.0179398\pi$
$278$			12.1306i			0.727542i
$279$		0			0
$280$	13.9620	−	6.65528i	0.834387	−	0.397729i
$281$	21.2930			1.27024			0.635118	−	0.772415i	$-0.280951\pi$
$281$	21.2930			1.27024			0.635118	+	0.772415i	$0.280951\pi$
$282$		0			0
$283$		−	15.8729i		−	0.943544i	−0.881721	−	0.471772i	$-0.843615\pi$
$283$		−	15.8729i		−	0.943544i	0.881721	−	0.471772i	$-0.156385\pi$
$284$	−0.989581			−0.0587208
$285$		0			0
$286$	2.37156			0.140233
$287$			21.4435i			1.26577i
$288$		0			0
$289$	13.3765			0.786855
$290$	−0.685530	−	1.43816i	−0.0402557	−	0.0844515i
$291$		0			0
$292$			24.6753i			1.44401i
$293$			33.1369i			1.93588i	0.251189	+	0.967938i	$0.419178\pi$
$293$			33.1369i			1.93588i	−0.251189	+	0.967938i	$0.580822\pi$
$294$		0			0
$295$	9.00298	−	4.29147i	0.524174	−	0.249859i
$296$	3.69266			0.214631
$297$		0			0
$298$		−	2.94293i		−	0.170479i
$299$	1.69360			0.0979433
$300$		0			0
$301$	22.7844			1.31327
$302$		−	2.29124i		−	0.131846i
$303$		0			0
$304$	−8.14381			−0.467080
$305$	10.5951	−	5.05039i	0.606673	−	0.289184i
$306$		0			0
$307$			17.2605i			0.985109i	0.870282	+	0.492555i	$0.163937\pi$
$307$			17.2605i			0.985109i	−0.870282	+	0.492555i	$0.836063\pi$
$308$		−	17.7065i		−	1.00892i
$309$		0			0
$310$	6.04679	+	12.6854i	0.343435	+	0.720483i
$311$	32.5116			1.84356			0.921781	−	0.387711i	$-0.126734\pi$
$311$	32.5116			1.84356			0.921781	+	0.387711i	$0.126734\pi$
$312$		0			0
$313$		−	6.26304i		−	0.354008i	−0.984210	−	0.177004i	$-0.943359\pi$
$313$		−	6.26304i		−	0.354008i	0.984210	−	0.177004i	$-0.0566405\pi$
$314$	4.87516			0.275121
$315$		0			0
$316$	−14.2453			−0.801360
$317$			18.8523i			1.05885i	0.848356	+	0.529426i	$0.177593\pi$
$317$			18.8523i			1.05885i	−0.848356	+	0.529426i	$0.822407\pi$
$318$		0			0
$319$	−4.26814			−0.238970
$320$	3.50676	−	1.67158i	0.196034	−	0.0934440i
$321$		0			0
$322$			4.30130i			0.239702i
$323$			12.7925i			0.711793i
$324$		0			0
$325$	3.02910	+	2.45539i	0.168024	+	0.136201i
$326$	10.0107			0.554441
$327$		0			0
$328$			19.1943i			1.05983i
$329$	14.4380			0.795995
$330$		0			0
$331$	−0.596245			−0.0327726			−0.0163863	−	0.999866i	$-0.505216\pi$
$331$	−0.596245			−0.0327726			−0.0163863	+	0.999866i	$0.505216\pi$
$332$			0.0185049i			0.00101559i
$333$		0			0
$334$	−12.7341			−0.696776
$335$	−8.17229	−	17.1445i	−0.446500	−	0.936702i
$336$		0			0
$337$			32.6414i			1.77809i	0.457818	+	0.889046i	$0.348631\pi$
$337$			32.6414i			1.77809i	−0.457818	+	0.889046i	$0.651369\pi$
$338$			8.82911i			0.480240i
$339$		0			0
$340$	2.73324	+	5.73400i	0.148231	+	0.310970i
$341$	37.6476			2.03873
$342$		0			0
$343$			17.4364i			0.941476i
$344$	20.3946			1.09960
$345$		0			0
$346$	−2.01293			−0.108216
$347$			13.1985i			0.708531i	0.935145	+	0.354265i	$0.115269\pi$
$347$			13.1985i			0.708531i	−0.935145	+	0.354265i	$0.884731\pi$
$348$		0			0
$349$	−15.6412			−0.837255			−0.418628	−	0.908158i	$-0.637489\pi$
$349$	−15.6412			−0.837255			−0.418628	+	0.908158i	$0.637489\pi$
$350$	−6.23606	+	7.69313i	−0.333332	+	0.411215i
$351$		0			0
$352$		−	24.9258i		−	1.32855i
$353$		−	5.65486i		−	0.300978i	−0.988612	−	0.150489i	$-0.951915\pi$
$353$		−	5.65486i		−	0.300978i	0.988612	−	0.150489i	$-0.0480848\pi$
$354$		0			0
$355$	1.33846	−	0.638006i	0.0710380	−	0.0338619i
$356$	−8.15991			−0.432475
$357$		0			0
$358$			14.4514i			0.763782i
$359$	25.2693			1.33366			0.666832	−	0.745208i	$-0.267650\pi$
$359$	25.2693			1.33366			0.666832	+	0.745208i	$0.267650\pi$
$360$		0			0
$361$	26.1632			1.37701
$362$		−	6.15441i		−	0.323469i
$363$		0			0
$364$	−3.23525			−0.169573
$365$	−15.9087	−	33.3746i	−0.832701	−	1.74690i
$366$		0			0
$367$		−	26.2788i		−	1.37174i	−0.727722	−	0.685872i	$-0.759421\pi$
$367$		−	26.2788i		−	1.37174i	0.727722	−	0.685872i	$-0.240579\pi$
$368$		−	2.63167i		−	0.137185i
$369$		0			0
$370$	−2.13425	+	1.01734i	−0.110954	+	0.0528890i
$371$	−32.7615			−1.70089
$372$		0			0
$373$			1.43033i			0.0740596i	0.999314	+	0.0370298i	$0.0117897\pi$
$373$			1.43033i			0.0740596i	−0.999314	+	0.0370298i	$0.988210\pi$
$374$	−5.78872			−0.299328
$375$		0			0
$376$	12.9237			0.666487
$377$			0.779856i			0.0401646i
$378$		0			0
$379$	35.2336			1.80983			0.904915	−	0.425591i	$-0.139934\pi$
$379$	35.2336			1.80983			0.904915	+	0.425591i	$0.139934\pi$
$380$	20.2436	−	9.64957i	1.03847	−	0.495012i
$381$		0			0
$382$		−	0.639001i		−	0.0326941i
$383$		−	32.0701i		−	1.63870i	−0.573291	−	0.819352i	$-0.694334\pi$
$383$		−	32.0701i		−	1.63870i	0.573291	−	0.819352i	$-0.305666\pi$
$384$		0			0
$385$	11.4158	+	23.9489i	0.581802	+	1.22055i
$386$	7.44601			0.378992
$387$		0			0
$388$			1.42073i			0.0721265i
$389$	1.63922			0.0831117			0.0415558	−	0.999136i	$-0.486769\pi$
$389$	1.63922			0.0831117			0.0415558	+	0.999136i	$0.486769\pi$
$390$		0			0
$391$	−4.13389			−0.209060
$392$		−	1.81047i		−	0.0914425i
$393$		0			0
$394$	−6.82385			−0.343781
$395$	19.2675	−	9.18428i	0.969452	−	0.462111i
$396$		0			0
$397$			2.04098i			0.102434i	0.998688	+	0.0512170i	$0.0163100\pi$
$397$			2.04098i			0.102434i	−0.998688	+	0.0512170i	$0.983690\pi$
$398$		−	4.00257i		−	0.200631i
$399$		0			0
$400$	3.81542	−	4.70689i	0.190771	−	0.235345i
$401$	−17.4108			−0.869455			−0.434727	−	0.900562i	$-0.643155\pi$
$401$	−17.4108			−0.869455			−0.434727	+	0.900562i	$0.643155\pi$
$402$		0			0
$403$		−	6.87880i		−	0.342658i
$404$	−12.4131			−0.617574
$405$		0			0
$406$	−1.98063			−0.0982972
$407$			6.33400i			0.313965i
$408$		0			0
$409$	37.3010			1.84441			0.922207	−	0.386696i	$-0.126384\pi$
$409$	37.3010			1.84441			0.922207	+	0.386696i	$0.126384\pi$
$410$	−5.28810	−	11.0938i	−0.261161	−	0.547883i
$411$		0			0
$412$			18.5188i			0.912356i
$413$		−	12.3989i		−	0.610111i
$414$		0			0
$415$	−0.0119305	−	0.0250287i	−0.000585646	−	0.00122861i
$416$	−4.55434			−0.223295
$417$		0			0
$418$			20.4368i			0.999596i
$419$	4.32941			0.211506			0.105753	−	0.994392i	$-0.466275\pi$
$419$	4.32941			0.211506			0.105753	+	0.994392i	$0.466275\pi$
$420$		0			0
$421$	−34.2505			−1.66927			−0.834634	−	0.550806i	$-0.814321\pi$
$421$	−34.2505			−1.66927			−0.834634	+	0.550806i	$0.814321\pi$
$422$		−	0.431214i		−	0.0209911i
$423$		0			0
$424$	−29.3252			−1.42416
$425$	−7.39369	−	5.99334i	−0.358647	−	0.290720i
$426$		0			0
$427$		−	14.5916i		−	0.706137i
$428$		−	26.2026i		−	1.26655i
$429$		0			0
$430$	−11.7875	+	5.61877i	−0.568443	+	0.270961i
$431$	−34.4885			−1.66125			−0.830626	−	0.556831i	$-0.812017\pi$
$431$	−34.4885			−1.66125			−0.830626	+	0.556831i	$0.812017\pi$
$432$		0			0
$433$		−	3.02258i		−	0.145256i	−0.997359	−	0.0726279i	$-0.976861\pi$
$433$		−	3.02258i		−	0.145256i	0.997359	−	0.0726279i	$-0.0231386\pi$
$434$	17.4704			0.838606
$435$		0			0
$436$	−1.50687			−0.0721661
$437$			14.5945i			0.698148i
$438$		0			0
$439$	−8.37392			−0.399665			−0.199833	−	0.979830i	$-0.564040\pi$
$439$	−8.37392			−0.399665			−0.199833	+	0.979830i	$0.564040\pi$
$440$	10.2184	+	21.4369i	0.487143	+	1.02197i
$441$		0			0
$442$			1.05769i			0.0503092i
$443$		−	29.8770i		−	1.41950i	−0.704454	−	0.709749i	$-0.748808\pi$
$443$		−	29.8770i		−	1.41950i	0.704454	−	0.709749i	$-0.251192\pi$
$444$		0			0
$445$	11.0367	−	5.26089i	0.523189	−	0.249390i
$446$	−11.7945			−0.558484
$447$		0			0
$448$		−	4.82952i		−	0.228174i
$449$	12.0780			0.569994			0.284997	−	0.958528i	$-0.408007\pi$
$449$	12.0780			0.569994			0.284997	+	0.958528i	$0.408007\pi$
$450$		0			0
$451$	−32.9240			−1.55033
$452$			23.2489i			1.09354i
$453$		0			0
$454$	15.6793			0.735864
$455$	4.37583	−	2.08584i	0.205142	−	0.0977857i
$456$		0			0
$457$			3.03357i			0.141905i	0.997480	+	0.0709523i	$0.0226038\pi$
$457$			3.03357i			0.141905i	−0.997480	+	0.0709523i	$0.977396\pi$
$458$			17.5022i			0.817826i
$459$		0			0
$460$	3.11825	+	6.54171i	0.145389	+	0.305009i
$461$	4.44508			0.207028			0.103514	−	0.994628i	$-0.466991\pi$
$461$	4.44508			0.207028			0.103514	+	0.994628i	$0.466991\pi$
$462$		0			0
$463$			9.04875i			0.420531i	0.977644	+	0.210266i	$0.0674329\pi$
$463$			9.04875i			0.420531i	−0.977644	+	0.210266i	$0.932567\pi$
$464$	1.21181			0.0562570
$465$		0			0
$466$	−3.96734			−0.183784
$467$		−	8.11327i		−	0.375437i	−0.982223	−	0.187719i	$-0.939891\pi$
$467$		−	8.11327i		−	0.375437i	0.982223	−	0.187719i	$-0.0601093\pi$
$468$		0			0
$469$	−23.6114			−1.09027
$470$	−7.46951	+	3.56051i	−0.344543	+	0.164234i
$471$		0			0
$472$		−	11.0984i		−	0.510847i
$473$			34.9827i			1.60851i
$474$		0			0
$475$	−21.1592	+	26.1030i	−0.970850	+	1.19769i
$476$	7.89688			0.361953
$477$		0			0
$478$			4.92078i			0.225071i
$479$	6.62947			0.302908			0.151454	−	0.988464i	$-0.451604\pi$
$479$	6.62947			0.302908			0.151454	+	0.988464i	$0.451604\pi$
$480$		0			0
$481$	1.15732			0.0527693
$482$		−	17.9541i		−	0.817787i
$483$		0			0
$484$	10.7703			0.489559
$485$	−0.915976	−	1.92160i	−0.0415923	−	0.0872556i
$486$		0			0
$487$		−	29.2047i		−	1.32339i	−0.749773	−	0.661695i	$-0.769837\pi$
$487$		−	29.2047i		−	1.32339i	0.749773	−	0.661695i	$-0.230163\pi$
$488$		−	13.0611i		−	0.591249i
$489$		0			0
$490$	0.498790	+	1.04640i	0.0225330	+	0.0472715i
$491$	4.32464			0.195168			0.0975841	−	0.995227i	$-0.468889\pi$
$491$	4.32464			0.195168			0.0975841	+	0.995227i	$0.468889\pi$
$492$		0			0
$493$		−	1.90354i		−	0.0857312i
$494$	3.73412			0.168006
$495$		0			0
$496$	−10.6889			−0.479947
$497$		−	1.84333i		−	0.0826846i
$498$		0			0
$499$	−4.29688			−0.192355			−0.0961774	−	0.995364i	$-0.530662\pi$
$499$	−4.29688			−0.192355			−0.0961774	+	0.995364i	$0.530662\pi$
$500$	−3.90705	+	16.2211i	−0.174728	+	0.725429i
$501$		0			0
$502$		−	20.2126i		−	0.902131i
$503$		−	14.2512i		−	0.635429i	−0.948186	−	0.317715i	$-0.897085\pi$
$503$		−	14.2512i		−	0.635429i	0.948186	−	0.317715i	$-0.102915\pi$
$504$		0			0
$505$	16.7893	−	8.00301i	0.747115	−	0.356129i
$506$	−6.60414			−0.293590
$507$		0			0
$508$			25.3007i			1.12254i
$509$	−19.4974			−0.864206			−0.432103	−	0.901824i	$-0.642228\pi$
$509$	−19.4974			−0.864206			−0.432103	+	0.901824i	$0.642228\pi$
$510$		0			0
$511$	−45.9635			−2.03331
$512$			13.1076i			0.579280i
$513$		0			0
$514$	−6.20654			−0.273759
$515$	−11.9395	−	25.0476i	−0.526117	−	1.10373i
$516$		0			0
$517$			22.1679i			0.974943i
$518$			2.93930i			0.129145i
$519$		0			0
$520$	3.91686	−	1.86706i	0.171766	−	0.0818760i
$521$	−24.5229			−1.07437			−0.537185	−	0.843465i	$-0.680512\pi$
$521$	−24.5229			−1.07437			−0.537185	+	0.843465i	$0.680512\pi$
$522$		0			0
$523$			10.8566i			0.474725i	0.971421	+	0.237362i	$0.0762829\pi$
$523$			10.8566i			0.474725i	−0.971421	+	0.237362i	$0.923717\pi$
$524$	−17.8073			−0.777914
$525$		0			0
$526$	−9.41406			−0.410472
$527$			16.7904i			0.731401i
$528$		0			0
$529$	18.2838			0.794948
$530$	16.9491	−	8.07919i	0.736224	−	0.350938i
$531$		0			0
$532$		−	27.8795i		−	1.20873i
$533$			6.01572i			0.260570i
$534$		0			0
$535$	16.8934	+	35.4403i	0.730367	+	1.53222i
$536$	−21.1348			−0.912886
$537$		0			0
$538$		−	7.37052i		−	0.317766i
$539$	3.10549			0.133763
$540$		0			0
$541$	−22.7032			−0.976085			−0.488043	−	0.872820i	$-0.662289\pi$
$541$	−22.7032			−0.976085			−0.488043	+	0.872820i	$0.662289\pi$
$542$			6.95410i			0.298704i
$543$		0			0
$544$	11.1166			0.476622
$545$	2.03812	−	0.971516i	0.0873035	−	0.0416152i
$546$		0			0
$547$		−	32.8229i		−	1.40340i	−0.712471	−	0.701702i	$-0.752424\pi$
$547$		−	32.8229i		−	1.40340i	0.712471	−	0.701702i	$-0.247576\pi$
$548$		−	0.357843i		−	0.0152863i
$549$		0			0
$550$	−11.8119	−	9.57474i	−0.503660	−	0.408268i
$551$	−6.72036			−0.286297
$552$		0			0
$553$		−	26.5352i		−	1.12839i
$554$	1.33595			0.0567590
$555$		0			0
$556$	25.4080			1.07754
$557$		−	26.3234i		−	1.11536i	−0.830057	−	0.557679i	$-0.811692\pi$
$557$		−	26.3234i		−	1.11536i	0.830057	−	0.557679i	$-0.188308\pi$
$558$		0			0
$559$	6.39189			0.270348
$560$	−3.24117	−	6.79958i	−0.136965	−	0.287335i
$561$		0			0
$562$			15.1712i			0.639958i
$563$			28.4750i			1.20008i	0.799971	+	0.600039i	$0.204848\pi$
$563$			28.4750i			1.20008i	−0.799971	+	0.600039i	$0.795152\pi$
$564$		0			0
$565$	−14.9891	−	31.4453i	−0.630597	−	1.32291i
$566$	11.3093			0.475367
$567$		0			0
$568$		−	1.64999i		−	0.0692318i
$569$	−14.7555			−0.618582			−0.309291	−	0.950967i	$-0.600092\pi$
$569$	−14.7555			−0.618582			−0.309291	+	0.950967i	$0.600092\pi$
$570$		0			0
$571$	−0.204397			−0.00855376			−0.00427688	−	0.999991i	$-0.501361\pi$
$571$	−0.204397			−0.00855376			−0.00427688	+	0.999991i	$0.501361\pi$
$572$		−	4.96734i		−	0.207695i
$573$		0			0
$574$	−15.2784			−0.637708
$575$	−8.43519	−	6.83758i	−0.351772	−	0.285147i
$576$		0			0
$577$		−	37.3300i		−	1.55407i	−0.629457	−	0.777035i	$-0.716723\pi$
$577$		−	37.3300i		−	1.55407i	0.629457	−	0.777035i	$-0.283277\pi$
$578$			9.53071i			0.396425i
$579$		0			0
$580$	−3.01228	+	1.43587i	−0.125078	+	0.0596213i
$581$	−0.0344697			−0.00143004
$582$		0			0
$583$		−	50.3014i		−	2.08327i
$584$	−41.1425			−1.70249
$585$		0			0
$586$	−23.6098			−0.975314
$587$			10.8497i			0.447813i	0.974611	+	0.223907i	$0.0718811\pi$
$587$			10.8497i			0.447813i	−0.974611	+	0.223907i	$0.928119\pi$
$588$		0			0
$589$	59.2776			2.44249
$590$	3.05765	+	6.41458i	0.125882	+	0.264084i
$591$		0			0
$592$		−	1.79835i		−	0.0739119i
$593$			14.0575i			0.577272i	0.957439	+	0.288636i	$0.0932018\pi$
$593$			14.0575i			0.577272i	−0.957439	+	0.288636i	$0.906798\pi$
$594$		0			0
$595$	−10.6809	+	5.09131i	−0.437876	+	0.208723i
$596$	−6.16409			−0.252491
$597$		0			0
$598$			1.20668i			0.0493448i
$599$	−5.52446			−0.225724			−0.112862	−	0.993611i	$-0.536002\pi$
$599$	−5.52446			−0.225724			−0.112862	+	0.993611i	$0.536002\pi$
$600$		0			0
$601$	−6.85394			−0.279578			−0.139789	−	0.990181i	$-0.544642\pi$
$601$	−6.85394			−0.279578			−0.139789	+	0.990181i	$0.544642\pi$
$602$			16.2337i			0.661638i
$603$		0			0
$604$	−4.79910			−0.195273
$605$	−14.5674	+	6.94387i	−0.592248	+	0.282308i
$606$		0			0
$607$			30.4979i			1.23787i	0.785442	+	0.618935i	$0.212436\pi$
$607$			30.4979i			1.23787i	−0.785442	+	0.618935i	$0.787564\pi$
$608$		−	39.2467i		−	1.59166i
$609$		0			0
$610$	3.59838	+	7.54895i	0.145694	+	0.305648i
$611$	4.05042			0.163863
$612$		0			0
$613$		−	29.9211i		−	1.20850i	−0.796794	−	0.604250i	$-0.793473\pi$
$613$		−	29.9211i		−	1.20850i	0.796794	−	0.604250i	$-0.206527\pi$
$614$	−12.2980			−0.496308
$615$		0			0
$616$	29.5230			1.18952
$617$			1.04305i			0.0419917i	0.999780	+	0.0209959i	$0.00668368\pi$
$617$			1.04305i			0.0419917i	−0.999780	+	0.0209959i	$0.993316\pi$
$618$		0			0
$619$	−22.1661			−0.890930			−0.445465	−	0.895299i	$-0.646962\pi$
$619$	−22.1661			−0.890930			−0.445465	+	0.895299i	$0.646962\pi$
$620$	26.5701	−	12.6653i	1.06708	−	0.508649i
$621$		0			0
$622$			23.1643i			0.928805i
$623$		−	15.1998i		−	0.608966i
$624$		0			0
$625$	−5.17364	−	24.4588i	−0.206946	−	0.978352i
$626$	4.46238			0.178353
$627$		0			0
$628$		−	10.2112i		−	0.407472i
$629$	−2.82489			−0.112636
$630$		0			0
$631$	20.1089			0.800524			0.400262	−	0.916401i	$-0.368919\pi$
$631$	20.1089			0.800524			0.400262	+	0.916401i	$0.368919\pi$
$632$		−	23.7520i		−	0.944804i
$633$		0			0
$634$	−13.4322			−0.533460
$635$	−16.3119	−	34.2204i	−0.647319	−	1.35800i
$636$		0			0
$637$		−	0.567422i		−	0.0224821i
$638$		−	3.04103i		−	0.120395i
$639$		0			0
$640$	−10.0469	−	21.0772i	−0.397139	−	0.833149i
$641$	−5.93422			−0.234388			−0.117194	−	0.993109i	$-0.537390\pi$
$641$	−5.93422			−0.234388			−0.117194	+	0.993109i	$0.537390\pi$
$642$		0			0
$643$		−	0.951317i		−	0.0375163i	−0.999824	−	0.0187581i	$-0.994029\pi$
$643$		−	0.951317i		−	0.0375163i	0.999824	−	0.0187581i	$-0.00597125\pi$
$644$	9.00926			0.355015
$645$		0			0
$646$	−9.11458			−0.358608
$647$			35.1384i			1.38143i	0.723126	+	0.690716i	$0.242705\pi$
$647$			35.1384i			1.38143i	−0.723126	+	0.690716i	$0.757295\pi$
$648$		0			0
$649$	19.0371			0.747271
$650$	−1.74945	+	2.15822i	−0.0686192	+	0.0846522i
$651$		0			0
$652$		−	20.9678i		−	0.821163i
$653$		−	45.9930i		−	1.79984i	−0.436050	−	0.899922i	$-0.643623\pi$
$653$		−	45.9930i		−	1.79984i	0.436050	−	0.899922i	$-0.356377\pi$
$654$		0			0
$655$	24.0852	−	11.4808i	0.941088	−	0.448591i
$656$	9.34779			0.364970
$657$		0			0
$658$			10.2870i			0.401030i
$659$	−19.3162			−0.752451			−0.376226	−	0.926528i	$-0.622778\pi$
$659$	−19.3162			−0.752451			−0.376226	+	0.926528i	$0.622778\pi$
$660$		0			0
$661$	44.3224			1.72394			0.861970	−	0.506959i	$-0.169230\pi$
$661$	44.3224			1.72394			0.861970	+	0.506959i	$0.169230\pi$
$662$		−	0.424821i		−	0.0165111i
$663$		0			0
$664$	−0.0308542			−0.00119738
$665$	17.9746	+	37.7085i	0.697025	+	1.46227i
$666$		0			0
$667$		−	2.17168i		−	0.0840878i
$668$			26.6720i			1.03197i
$669$		0			0
$670$	12.2153	−	5.82272i	0.471920	−	0.224951i
$671$	22.4037			0.864883
$672$		0			0
$673$		−	12.7421i		−	0.491170i	−0.969375	−	0.245585i	$-0.921020\pi$
$673$		−	12.7421i		−	0.491170i	0.969375	−	0.245585i	$-0.0789801\pi$
$674$	−23.2569			−0.895821
$675$		0			0
$676$	18.4930			0.711267
$677$			9.41654i			0.361907i	0.983492	+	0.180954i	$0.0579184\pi$
$677$			9.41654i			0.361907i	−0.983492	+	0.180954i	$0.942082\pi$
$678$		0			0
$679$	−2.64644			−0.101561
$680$	−9.56063	+	4.55729i	−0.366634	+	0.174764i
$681$		0			0
$682$			26.8237i			1.02713i
$683$			6.92316i			0.264907i	0.991189	+	0.132454i	$0.0422856\pi$
$683$			6.92316i			0.264907i	−0.991189	+	0.132454i	$0.957714\pi$
$684$		0			0
$685$	0.230710	+	0.484001i	0.00881497	+	0.0184927i
$686$	−12.4233			−0.474325
$687$		0			0
$688$		−	9.93231i		−	0.378666i
$689$	−9.19085			−0.350144
$690$		0			0
$691$	−48.7272			−1.85367			−0.926835	−	0.375469i	$-0.877482\pi$
$691$	−48.7272			−1.85367			−0.926835	+	0.375469i	$0.877482\pi$
$692$			4.21616i			0.160274i
$693$		0			0
$694$	−9.40384			−0.356965
$695$	−34.3655	+	16.3811i	−1.30356	+	0.621371i
$696$		0			0
$697$		−	14.6837i		−	0.556186i
$698$		−	11.1443i		−	0.421818i
$699$		0			0
$700$	16.1136	+	13.0617i	0.609036	+	0.493686i
$701$	−2.88580			−0.108995			−0.0544975	−	0.998514i	$-0.517356\pi$
$701$	−2.88580			−0.108995			−0.0544975	+	0.998514i	$0.517356\pi$
$702$		0			0
$703$			9.97314i			0.376144i
$704$	7.41516			0.279469
$705$		0			0
$706$	4.02906			0.151636
$707$		−	23.1223i		−	0.869604i
$708$		0			0
$709$	1.73056			0.0649924			0.0324962	−	0.999472i	$-0.489654\pi$
$709$	1.73056			0.0649924			0.0324962	+	0.999472i	$0.489654\pi$
$710$	0.454576	+	0.953645i	0.0170599	+	0.0357896i
$711$		0			0
$712$		−	13.6055i		−	0.509887i
$713$			19.1555i			0.717381i
$714$		0			0
$715$	3.20256	+	6.71857i	0.119769	+	0.251260i
$716$	30.2692			1.13121
$717$		0			0
$718$			18.0043i			0.671913i
$719$	−12.9089			−0.481420			−0.240710	−	0.970597i	$-0.577380\pi$
$719$	−12.9089			−0.481420			−0.240710	+	0.970597i	$0.577380\pi$
$720$		0			0
$721$	−34.4956			−1.28468
$722$			18.6412i			0.693752i
$723$		0			0
$724$	−12.8907			−0.479078
$725$	3.14852	−	3.88418i	0.116933	−	0.144255i
$726$		0			0
$727$		−	38.3405i		−	1.42197i	−0.703207	−	0.710985i	$-0.748249\pi$
$727$		−	38.3405i		−	1.42197i	0.703207	−	0.710985i	$-0.251751\pi$
$728$		−	5.39431i		−	0.199927i
$729$		0			0
$730$	23.7792	−	11.3349i	0.880108	−	0.419523i
$731$	−15.6019			−0.577057
$732$		0			0
$733$			34.8966i			1.28894i	0.764631	+	0.644468i	$0.222921\pi$
$733$			34.8966i			1.28894i	−0.764631	+	0.644468i	$0.777079\pi$
$734$	18.7235			0.691098
$735$		0			0
$736$	12.6826			0.467485
$737$		−	36.2525i		−	1.33538i
$738$		0			0
$739$	−14.2969			−0.525921			−0.262960	−	0.964807i	$-0.584699\pi$
$739$	−14.2969			−0.525921			−0.262960	+	0.964807i	$0.584699\pi$
$740$	2.13086	+	4.47028i	0.0783320	+	0.164331i
$741$		0			0
$742$		−	23.3424i		−	0.856927i
$743$		−	44.1357i		−	1.61918i	−0.586995	−	0.809591i	$-0.699689\pi$
$743$		−	44.1357i		−	1.61918i	0.586995	−	0.809591i	$-0.300311\pi$
$744$		0			0
$745$	8.33725	−	3.97414i	0.305453	−	0.145601i
$746$	−1.01910			−0.0373120
$747$		0			0
$748$			12.1247i			0.443324i
$749$	48.8085			1.78343
$750$		0			0
$751$	9.56375			0.348986			0.174493	−	0.984658i	$-0.444171\pi$
$751$	9.56375			0.348986			0.174493	+	0.984658i	$0.444171\pi$
$752$		−	6.29393i		−	0.229516i
$753$		0			0
$754$	−0.555643			−0.0202353
$755$	6.49103	−	3.09410i	0.236233	−	0.112606i
$756$		0			0
$757$			30.1910i			1.09731i	0.836048	+	0.548656i	$0.184860\pi$
$757$			30.1910i			1.09731i	−0.836048	+	0.548656i	$0.815140\pi$
$758$			25.1038i			0.911811i
$759$		0			0
$760$	16.0893	+	33.7533i	0.583619	+	1.22436i
$761$	−8.27038			−0.299801			−0.149900	−	0.988701i	$-0.547895\pi$
$761$	−8.27038			−0.299801			−0.149900	+	0.988701i	$0.547895\pi$
$762$		0			0
$763$		−	2.80690i		−	0.101617i
$764$	−1.33841			−0.0484221
$765$		0			0
$766$	22.8498			0.825595
$767$		−	3.47837i		−	0.125597i
$768$		0			0
$769$	22.4592			0.809901			0.404950	−	0.914339i	$-0.367289\pi$
$769$	22.4592			0.809901			0.404950	+	0.914339i	$0.367289\pi$
$770$	−17.0635	+	8.13368i	−0.614924	+	0.293118i
$771$		0			0
$772$		−	15.5960i		−	0.561311i
$773$		−	26.9520i		−	0.969398i	−0.874681	−	0.484699i	$-0.838929\pi$
$773$		−	26.9520i		−	0.969398i	0.874681	−	0.484699i	$-0.161071\pi$
$774$		0			0
$775$	−27.7719	+	34.2608i	−0.997595	+	1.23068i
$776$	−2.36886			−0.0850371
$777$		0			0
$778$			1.16794i			0.0418725i
$779$	−51.8401			−1.85737
$780$		0			0
$781$	2.83021			0.101273
$782$		−	2.94537i		−	0.105326i
$783$		0			0
$784$	−0.881713			−0.0314897
$785$	6.58341	+	13.8112i	0.234972	+	0.492942i
$786$		0			0
$787$			1.16423i			0.0415004i	0.999785	+	0.0207502i	$0.00660546\pi$
$787$			1.16423i			0.0415004i	−0.999785	+	0.0207502i	$0.993395\pi$
$788$			14.2928i			0.509161i
$789$		0			0
$790$	6.54375	+	13.7280i	0.232816	+	0.488420i

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)

\(f(q)\)	\(=\)	\(q+0.712495i q^{2} +1.49235 q^{4} +(-2.01848 + 0.962154i) q^{5} +2.77986i q^{7} +2.48828i q^{8} +O(q^{10})\)	\(q+0.712495i q^{2} +1.49235 q^{4} +(-2.01848 + 0.962154i) q^{5} +2.77986i q^{7} +2.48828i q^{8} +(-0.685530 - 1.43816i) q^{10} -4.26814 q^{11} +0.779856i q^{13} -1.98063 q^{14} +1.21181 q^{16} -1.90354i q^{17} -6.72036 q^{19} +(-3.01228 + 1.43587i) q^{20} -3.04103i q^{22} -2.17168i q^{23} +(3.14852 - 3.88418i) q^{25} -0.555643 q^{26} +4.14852i q^{28} +1.00000 q^{29} -8.82061 q^{31} +5.83998i q^{32} +1.35626 q^{34} +(-2.67465 - 5.61108i) q^{35} -1.48402i q^{37} -4.78822i q^{38} +(-2.39411 - 5.02255i) q^{40} +7.71389 q^{41} -8.19624i q^{43} -6.36956 q^{44} +1.54731 q^{46} -5.19381i q^{47} -0.727598 q^{49} +(2.76746 + 2.24330i) q^{50} +1.16382i q^{52} +11.7853i q^{53} +(8.61515 - 4.10661i) q^{55} -6.91707 q^{56} +0.712495i q^{58} -4.46028 q^{59} -5.24905 q^{61} -6.28464i q^{62} -1.73733 q^{64} +(-0.750341 - 1.57412i) q^{65} +8.49375i q^{67} -2.84075i q^{68} +(3.99787 - 1.90567i) q^{70} -0.663102 q^{71} +16.5345i q^{73} +1.05736 q^{74} -10.0291 q^{76} -11.8648i q^{77} -9.54554 q^{79} +(-2.44602 + 1.16595i) q^{80} +5.49611i q^{82} +0.0123998i q^{83} +(1.83150 + 3.84226i) q^{85} +5.83978 q^{86} -10.6203i q^{88} -5.46783 q^{89} -2.16789 q^{91} -3.24091i q^{92} +3.70056 q^{94} +(13.5649 - 6.46602i) q^{95} +0.952006i q^{97} -0.518410i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{4}+O(q^{10}) \) 10 * q - 10 * q^4	\( 10 q - 10 q^{4} + 4 q^{10} - 24 q^{11} + 12 q^{14} + 2 q^{16} + 4 q^{19} - 8 q^{20} + 2 q^{25} + 10 q^{29} + 4 q^{31} - 8 q^{34} - 2 q^{35} - 14 q^{40} + 28 q^{41} + 40 q^{44} - 12 q^{46} + 14 q^{49} + 12 q^{50} + 2 q^{55} + 4 q^{56} + 8 q^{59} - 24 q^{61} + 18 q^{64} - 6 q^{65} + 20 q^{70} - 60 q^{71} + 4 q^{74} - 88 q^{76} + 36 q^{79} + 30 q^{80} - 14 q^{85} - 60 q^{86} - 44 q^{89} - 24 q^{91} + 100 q^{94} + 36 q^{95}+O(q^{100}) \) 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

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