LMFDB - Embedded newform 435.2.c.e.349.6

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			349.6
Root			$0.561843 - 0.561843i$ of defining polynomial
Character	$\chi$	$=$	435.349
Dual form			435.2.c.e.349.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.712495i q^{2} +1.00000i q^{3} +1.49235 q^{4} +(2.01848 + 0.962154i) q^{5} -0.712495 q^{6} -2.77986i q^{7} +2.48828i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.712495i q^{2} +1.00000i q^{3} +1.49235 q^{4} +(2.01848 + 0.962154i) q^{5} -0.712495 q^{6} -2.77986i q^{7} +2.48828i q^{8} -1.00000 q^{9} +(-0.685530 + 1.43816i) q^{10} +4.26814 q^{11} +1.49235i q^{12} -0.779856i q^{13} +1.98063 q^{14} +(-0.962154 + 2.01848i) q^{15} +1.21181 q^{16} -1.90354i q^{17} -0.712495i q^{18} -6.72036 q^{19} +(3.01228 + 1.43587i) q^{20} +2.77986 q^{21} +3.04103i q^{22} -2.17168i q^{23} -2.48828 q^{24} +(3.14852 + 3.88418i) q^{25} +0.555643 q^{26} -1.00000i q^{27} -4.14852i q^{28} -1.00000 q^{29} +(-1.43816 - 0.685530i) q^{30} -8.82061 q^{31} +5.83998i q^{32} +4.26814i q^{33} +1.35626 q^{34} +(2.67465 - 5.61108i) q^{35} -1.49235 q^{36} +1.48402i q^{37} -4.78822i q^{38} +0.779856 q^{39} +(-2.39411 + 5.02255i) q^{40} -7.71389 q^{41} +1.98063i q^{42} +8.19624i q^{43} +6.36956 q^{44} +(-2.01848 - 0.962154i) q^{45} +1.54731 q^{46} -5.19381i q^{47} +1.21181i q^{48} -0.727598 q^{49} +(-2.76746 + 2.24330i) q^{50} +1.90354 q^{51} -1.16382i q^{52} +11.7853i q^{53} +0.712495 q^{54} +(8.61515 + 4.10661i) q^{55} +6.91707 q^{56} -6.72036i q^{57} -0.712495i q^{58} +4.46028 q^{59} +(-1.43587 + 3.01228i) q^{60} -5.24905 q^{61} -6.28464i q^{62} +2.77986i q^{63} -1.73733 q^{64} +(0.750341 - 1.57412i) q^{65} -3.04103 q^{66} -8.49375i q^{67} -2.84075i q^{68} +2.17168 q^{69} +(3.99787 + 1.90567i) q^{70} +0.663102 q^{71} -2.48828i q^{72} -16.5345i q^{73} -1.05736 q^{74} +(-3.88418 + 3.14852i) q^{75} -10.0291 q^{76} -11.8648i q^{77} +0.555643i q^{78} -9.54554 q^{79} +(2.44602 + 1.16595i) q^{80} +1.00000 q^{81} -5.49611i q^{82} +0.0123998i q^{83} +4.14852 q^{84} +(1.83150 - 3.84226i) q^{85} -5.83978 q^{86} -1.00000i q^{87} +10.6203i q^{88} +5.46783 q^{89} +(0.685530 - 1.43816i) q^{90} -2.16789 q^{91} -3.24091i q^{92} -8.82061i q^{93} +3.70056 q^{94} +(-13.5649 - 6.46602i) q^{95} -5.83998 q^{96} -0.952006i q^{97} -0.518410i q^{98} -4.26814 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			0.712495i			0.503810i	0.967752	+	0.251905i	$0.0810571\pi$
$2$			0.712495i			0.503810i	−0.967752	+	0.251905i	$0.918943\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.49235			0.746175
$5$	2.01848	+	0.962154i	0.902692	+	0.430288i
$5$	2.01848	+	0.962154i	0.902692	+	0.430288i
$6$	−0.712495			−0.290875
$7$		−	2.77986i		−	1.05069i	−0.850890	−	0.525343i	$-0.823937\pi$
$7$		−	2.77986i		−	1.05069i	0.850890	−	0.525343i	$-0.176063\pi$
$8$			2.48828i			0.879741i
$9$	−1.00000			−0.333333
$10$	−0.685530	+	1.43816i	−0.216784	+	0.454785i
$11$	4.26814			1.28689			0.643446	−	0.765491i	$-0.277504\pi$
$11$	4.26814			1.28689			0.643446	+	0.765491i	$0.277504\pi$
$12$			1.49235i			0.430805i
$13$		−	0.779856i		−	0.216293i	−0.994135	−	0.108147i	$-0.965508\pi$
$13$		−	0.779856i		−	0.216293i	0.994135	−	0.108147i	$-0.0344916\pi$
$14$	1.98063			0.529347
$15$	−0.962154	+	2.01848i	−0.248427	+	0.521169i
$16$	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	$-0.0741469\pi$
$18$		−	0.712495i		−	0.167937i
$19$	−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
$21$	2.77986			0.606614
$22$			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	$-0.0727000\pi$
$24$	−2.48828			−0.507919
$25$	3.14852	+	3.88418i	0.629704	+	0.776835i
$26$	0.555643			0.108971
$27$		−	1.00000i		−	0.192450i
$28$		−	4.14852i		−	0.783997i
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$	−1.43816	−	0.685530i	−0.262570	−	0.125160i
$31$	−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
$33$			4.26814i			0.742988i
$34$	1.35626			0.232597
$35$	2.67465	−	5.61108i	0.452098	−	0.948446i
$36$	−1.49235			−0.248725
$37$			1.48402i			0.243971i	0.992532	+	0.121986i	$0.0389262\pi$
$37$			1.48402i			0.243971i	−0.992532	+	0.121986i	$0.961074\pi$
$38$		−	4.78822i		−	0.776752i
$39$	0.779856			0.124877
$40$	−2.39411	+	5.02255i	−0.378542	+	0.794135i
$41$	−7.71389			−1.20471			−0.602354	−	0.798229i	$-0.705770\pi$
$41$	−7.71389			−1.20471			−0.602354	+	0.798229i	$0.705770\pi$
$42$			1.98063i			0.305618i
$43$			8.19624i			1.24991i	0.780659	+	0.624957i	$0.214884\pi$
$43$			8.19624i			1.24991i	−0.780659	+	0.624957i	$0.785116\pi$
$44$	6.36956			0.960247
$45$	−2.01848	−	0.962154i	−0.300897	−	0.143429i
$46$	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	$-0.123662\pi$
$48$			1.21181i			0.174910i
$49$	−0.727598			−0.103943
$50$	−2.76746	+	2.24330i	−0.391377	+	0.317251i
$51$	1.90354			0.266549
$52$		−	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	$0.699782\pi$
$54$	0.712495			0.0969583
$55$	8.61515	+	4.10661i	1.16167	+	0.553735i
$56$	6.91707			0.924332
$57$		−	6.72036i		−	0.890133i
$58$		−	0.712495i		−	0.0935552i
$59$	4.46028			0.580679			0.290339	−	0.956924i	$-0.406232\pi$
$59$	4.46028			0.580679			0.290339	+	0.956924i	$0.406232\pi$
$60$	−1.43587	+	3.01228i	−0.185370	+	0.388884i
$61$	−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
$63$			2.77986i			0.350229i
$64$	−1.73733			−0.217166
$65$	0.750341	−	1.57412i	0.0930684	−	0.195246i
$66$	−3.04103			−0.374325
$67$		−	8.49375i		−	1.03768i	−0.854872	−	0.518838i	$-0.826365\pi$
$67$		−	8.49375i		−	1.03768i	0.854872	−	0.518838i	$-0.173635\pi$
$68$		−	2.84075i		−	0.344492i
$69$	2.17168			0.261440
$70$	3.99787	+	1.90567i	0.477837	+	0.227772i
$71$	0.663102			0.0786957			0.0393479	−	0.999226i	$-0.487472\pi$
$71$	0.663102			0.0786957			0.0393479	+	0.999226i	$0.487472\pi$
$72$		−	2.48828i		−	0.293247i
$73$		−	16.5345i		−	1.93522i	−0.252455	−	0.967609i	$-0.581238\pi$
$73$		−	16.5345i		−	1.93522i	0.252455	−	0.967609i	$-0.418762\pi$
$74$	−1.05736			−0.122915
$75$	−3.88418	+	3.14852i	−0.448506	+	0.363560i
$76$	−10.0291			−1.15042
$77$		−	11.8648i		−	1.35212i
$78$			0.555643i			0.0629142i
$79$	−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
$81$	1.00000			0.111111
$82$		−	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	$0.999783\pi$
$84$	4.14852			0.452641
$85$	1.83150	−	3.84226i	0.198654	−	0.416752i
$86$	−5.83978			−0.629720
$87$		−	1.00000i		−	0.107211i
$88$			10.6203i			1.13213i
$89$	5.46783			0.579588			0.289794	−	0.957089i	$-0.406413\pi$
$89$	5.46783			0.579588			0.289794	+	0.957089i	$0.406413\pi$
$90$	0.685530	−	1.43816i	0.0722612	−	0.151595i
$91$	−2.16789			−0.227256
$92$		−	3.24091i		−	0.337888i
$93$		−	8.82061i		−	0.914655i
$94$	3.70056			0.381684
$95$	−13.5649	−	6.46602i	−1.39173	−	0.663399i
$96$	−5.83998			−0.596040
$97$		−	0.952006i		−	0.0966615i	−0.998831	−	0.0483308i	$-0.984610\pi$
$97$		−	0.952006i		−	0.0966615i	0.998831	−	0.0483308i	$-0.0153902\pi$
$98$		−	0.518410i		−	0.0523673i
$99$	−4.26814			−0.428964
$100$	4.69870	+	5.79655i	0.469870	+	0.579655i
$101$	8.31781			0.827653			0.413826	−	0.910356i	$-0.364192\pi$
$101$	8.31781			0.827653			0.413826	+	0.910356i	$0.364192\pi$
$102$			1.35626i			0.134290i
$103$		−	12.4091i		−	1.22271i	−0.791357	−	0.611355i	$-0.790625\pi$
$103$		−	12.4091i		−	1.22271i	0.791357	−	0.611355i	$-0.209375\pi$
$104$	1.94050			0.190282
$105$	5.61108	+	2.67465i	0.547586	+	0.261019i
$106$	−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	$-0.322611\pi$
$108$		−	1.49235i		−	0.143602i
$109$	−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
$111$	−1.48402			−0.140857
$112$		−	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	$0.738232\pi$
$114$	4.78822			0.448458
$115$	2.08949	−	4.38349i	0.194846	−	0.408763i
$116$	−1.49235			−0.138561
$117$			0.779856i			0.0720977i
$118$			3.17792i			0.292552i
$119$	−5.29157			−0.485078
$120$	−5.02255	−	2.39411i	−0.458494	−	0.218551i
$121$	7.21701			0.656091
$122$		−	3.73992i		−	0.338596i
$123$		−	7.71389i		−	0.695538i
$124$	−13.1634			−1.18211
$125$	2.61805	+	10.8695i	0.234165	+	0.972197i
$126$	−1.98063			−0.176449
$127$		−	16.9536i		−	1.50438i	−0.658943	−	0.752192i	$-0.728996\pi$
$127$		−	16.9536i		−	1.50438i	0.658943	−	0.752192i	$-0.271004\pi$
$128$			10.4421i			0.922961i
$129$	−8.19624			−0.721639
$130$	1.12155	+	0.534614i	0.0983669	+	0.0468888i
$131$	11.9324			1.04254			0.521268	−	0.853393i	$-0.325459\pi$
$131$	11.9324			1.04254			0.521268	+	0.853393i	$0.325459\pi$
$132$			6.36956i			0.554399i
$133$			18.6816i			1.61990i
$134$	6.05175			0.522792
$135$	0.962154	−	2.01848i	0.0828090	−	0.173723i
$136$	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	$-0.00326054\pi$
$138$			1.54731i			0.131716i
$139$	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
$141$	5.19381			0.437398
$142$			0.472457i			0.0396477i
$143$		−	3.32853i		−	0.278346i
$144$	−1.21181			−0.100984
$145$	−2.01848	−	0.962154i	−0.167626	−	0.0799025i
$146$	11.7808			0.974982
$147$		−	0.727598i		−	0.0600113i
$148$			2.21468i			0.182045i
$149$	4.13046			0.338380			0.169190	−	0.985583i	$-0.445885\pi$
$149$	4.13046			0.338380			0.169190	+	0.985583i	$0.445885\pi$
$150$	−2.24330	−	2.76746i	−0.183165	−	0.225962i
$151$	−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
$153$			1.90354i			0.153892i
$154$	8.45362			0.681212
$155$	−17.8042	−	8.48678i	−1.43007	−	0.681675i
$156$	1.16382			0.0931800
$157$			6.84237i			0.546081i	0.962003	+	0.273040i	$0.0880292\pi$
$157$			6.84237i			0.546081i	−0.962003	+	0.273040i	$0.911971\pi$
$158$		−	6.80115i		−	0.541070i
$159$	−11.7853			−0.934637
$160$	−5.61895	+	11.7879i	−0.444217	+	0.931913i
$161$	−6.03696			−0.475779
$162$			0.712495i			0.0559789i
$163$			14.0502i			1.10050i	0.835001	+	0.550248i	$0.185467\pi$
$163$			14.0502i			1.10050i	−0.835001	+	0.550248i	$0.814533\pi$
$164$	−11.5118			−0.898923
$165$	−4.10661	+	8.61515i	−0.319699	+	0.670689i
$166$	−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	$0.756947\pi$
$168$			6.91707i			0.533663i
$169$	12.3918			0.953217
$170$	2.73759	+	1.30494i	0.209964	+	0.100084i
$171$	6.72036			0.513919
$172$			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	$0.965748\pi$
$174$	0.712495			0.0540141
$175$	10.7974	−	8.75243i	0.816210	−	0.661622i
$176$	5.17218			0.389868
$177$			4.46028i			0.335255i
$178$			3.89580i			0.292002i
$179$	−20.2829			−1.51601			−0.758006	−	0.652247i	$-0.773826\pi$
$179$	−20.2829			−1.51601			−0.758006	+	0.652247i	$0.773826\pi$
$180$	−3.01228	−	1.43587i	−0.224522	−	0.107024i
$181$	−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
$183$		−	5.24905i		−	0.388021i
$184$	5.40376			0.398370
$185$	−1.42785	+	2.99546i	−0.104978	+	0.220231i
$186$	6.28464			0.460812
$187$		−	8.12458i		−	0.594128i
$188$		−	7.75099i		−	0.565299i
$189$	−2.77986			−0.202205
$190$	4.60701	−	9.66493i	0.334227	−	0.701168i
$191$	0.896850			0.0648938			0.0324469	−	0.999473i	$-0.489670\pi$
$191$	0.896850			0.0648938			0.0324469	+	0.999473i	$0.489670\pi$
$192$		−	1.73733i		−	0.125381i
$193$			10.4506i			0.752251i	0.926569	+	0.376126i	$0.122744\pi$
$193$			10.4506i			0.752251i	−0.926569	+	0.376126i	$0.877256\pi$
$194$	0.678299			0.0486991
$195$	1.57412	+	0.750341i	0.112725	+	0.0537331i
$196$	−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	$0.889173\pi$
$198$		−	3.04103i		−	0.216116i
$199$	−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
$201$	8.49375			0.599103
$202$			5.92640i			0.416980i
$203$			2.77986i			0.195108i
$204$	2.84075			0.198892
$205$	−15.5703	−	7.42195i	−1.08748	−	0.518372i
$206$	8.84145			0.616013
$207$			2.17168i			0.150942i
$208$		−	0.945039i		−	0.0655267i
$209$	−28.6834			−1.98407
$210$	−1.90567	+	3.99787i	−0.131504	+	0.275879i
$211$	−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
$213$			0.663102i			0.0454350i
$214$	12.5099			0.855162
$215$	−7.88604	+	16.5439i	−0.537824	+	1.12829i
$216$	2.48828			0.169306
$217$			24.5200i			1.66453i
$218$		−	0.719428i		−	0.0487258i
$219$	16.5345			1.11730
$220$	12.8568	+	6.12850i	0.866807	+	0.413183i
$221$	−1.48449			−0.0998575
$222$		−	1.05736i		−	0.0709651i
$223$		−	16.5537i		−	1.10852i	−0.832344	−	0.554260i	$-0.813001\pi$
$223$		−	16.5537i		−	1.10852i	0.832344	−	0.554260i	$-0.186999\pi$
$224$	16.2343			1.08470
$225$	−3.14852	−	3.88418i	−0.209901	−	0.258945i
$226$	−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	$-0.260619\pi$
$228$		−	10.0291i		−	0.664195i
$229$	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
$231$	11.8648			0.780647
$232$		−	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	$0.941615\pi$
$234$	−0.555643			−0.0363235
$235$	4.99725	−	10.4836i	0.325984	−	0.683875i
$236$	6.65630			0.433288
$237$		−	9.54554i		−	0.620049i
$238$		−	3.77022i		−	0.244387i
$239$	−6.90640			−0.446738			−0.223369	−	0.974734i	$-0.571705\pi$
$239$	−6.90640			−0.446738			−0.223369	+	0.974734i	$0.571705\pi$
$240$	−1.16595	+	2.44602i	−0.0752618	+	0.157890i
$241$	−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
$243$			1.00000i			0.0641500i
$244$	−7.83342			−0.501483
$245$	−1.46864	−	0.700061i	−0.0938281	−	0.0447253i
$246$	5.49611			0.350419
$247$			5.24091i			0.333471i
$248$		−	21.9482i		−	1.39371i
$249$	−0.0123998			−0.000785805
$250$	−7.74446	+	1.86535i	−0.489803	+	0.117975i
$251$	28.3687			1.79062			0.895309	−	0.445446i	$-0.146955\pi$
$251$	28.3687			1.79062			0.895309	+	0.445446i	$0.146955\pi$
$252$			4.14852i			0.261332i
$253$		−	9.26903i		−	0.582739i
$254$	12.0793			0.757924
$255$	3.84226	+	1.83150i	0.240612	+	0.114693i
$256$	−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	$0.912418\pi$
$258$		−	5.83978i		−	0.363569i
$259$	4.12536			0.256337
$260$	1.11977	−	2.34914i	0.0694453	−	0.145688i
$261$	1.00000			0.0618984
$262$			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	$0.866447\pi$
$264$	−10.6203			−0.653636
$265$	−11.3393	+	23.7884i	−0.696567	+	1.46131i
$266$	−13.3106			−0.816123
$267$			5.46783i			0.334625i
$268$		−	12.6757i		−	0.774289i
$269$	10.3447			0.630725			0.315363	−	0.948971i	$-0.397874\pi$
$269$	10.3447			0.630725			0.315363	+	0.948971i	$0.397874\pi$
$270$	1.43816	+	0.685530i	0.0875234	+	0.0417200i
$271$	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
$273$		−	2.16789i		−	0.131206i
$274$	0.170845			0.0103212
$275$	13.4383	+	16.5782i	0.810361	+	0.999703i
$276$	3.24091			0.195080
$277$			1.87503i			0.112659i	0.998412	+	0.0563297i	$0.0179398\pi$
$277$			1.87503i			0.112659i	−0.998412	+	0.0563297i	$0.982060\pi$
$278$			12.1306i			0.727542i
$279$	8.82061			0.528076
$280$	13.9620	+	6.65528i	0.834387	+	0.397729i
$281$	−21.2930			−1.27024			−0.635118	−	0.772415i	$-0.719049\pi$
$281$	−21.2930			−1.27024			−0.635118	+	0.772415i	$0.719049\pi$
$282$			3.70056i			0.220365i
$283$			15.8729i			0.943544i	0.881721	+	0.471772i	$0.156385\pi$
$283$			15.8729i			0.943544i	−0.881721	+	0.471772i	$0.843615\pi$
$284$	0.989581			0.0587208
$285$	6.46602	−	13.5649i	0.383014	−	0.803516i
$286$	2.37156			0.140233
$287$			21.4435i			1.26577i
$288$		−	5.83998i		−	0.344124i
$289$	13.3765			0.786855
$290$	0.685530	−	1.43816i	0.0402557	−	0.0844515i
$291$	0.952006			0.0558076
$292$		−	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	$0.580822\pi$
$294$	0.518410			0.0302343
$295$	9.00298	+	4.29147i	0.524174	+	0.249859i
$296$	−3.69266			−0.214631
$297$		−	4.26814i		−	0.247663i
$298$			2.94293i			0.170479i
$299$	−1.69360			−0.0979433
$300$	−5.79655	+	4.69870i	−0.334664	+	0.271279i
$301$	22.7844			1.31327
$302$		−	2.29124i		−	0.131846i
$303$			8.31781i			0.477845i
$304$	−8.14381			−0.467080
$305$	−10.5951	−	5.05039i	−0.606673	−	0.289184i
$306$	−1.35626			−0.0775325
$307$		−	17.2605i		−	0.985109i	−0.870282	−	0.492555i	$-0.836063\pi$
$307$		−	17.2605i		−	0.985109i	0.870282	−	0.492555i	$-0.163937\pi$
$308$		−	17.7065i		−	1.00892i
$309$	12.4091			0.705932
$310$	6.04679	−	12.6854i	0.343435	−	0.720483i
$311$	−32.5116			−1.84356			−0.921781	−	0.387711i	$-0.873266\pi$
$311$	−32.5116			−1.84356			−0.921781	+	0.387711i	$0.873266\pi$
$312$			1.94050i			0.109859i
$313$			6.26304i			0.354008i	0.984210	+	0.177004i	$0.0566405\pi$
$313$			6.26304i			0.354008i	−0.984210	+	0.177004i	$0.943359\pi$
$314$	−4.87516			−0.275121
$315$	−2.67465	+	5.61108i	−0.150699	+	0.316149i
$316$	−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	$0.822407\pi$
$318$		−	8.39698i		−	0.470879i
$319$	−4.26814			−0.238970
$320$	−3.50676	−	1.67158i	−0.196034	−	0.0934440i
$321$	17.5579			0.979989
$322$		−	4.30130i		−	0.239702i
$323$			12.7925i			0.711793i
$324$	1.49235			0.0829084
$325$	3.02910	−	2.45539i	0.168024	−	0.136201i
$326$	−10.0107			−0.554441
$327$		−	1.00973i		−	0.0558382i
$328$		−	19.1943i		−	1.05983i
$329$	−14.4380			−0.795995
$330$	−6.13825	−	2.92594i	−0.337900	−	0.161067i
$331$	−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
$333$		−	1.48402i		−	0.0813238i
$334$	−12.7341			−0.696776
$335$	8.17229	−	17.1445i	0.446500	−	0.936702i
$336$	3.36866			0.183776
$337$		−	32.6414i		−	1.77809i	−0.457818	−	0.889046i	$-0.651369\pi$
$337$		−	32.6414i		−	1.77809i	0.457818	−	0.889046i	$-0.348631\pi$
$338$			8.82911i			0.480240i
$339$	−15.5787			−0.846119
$340$	2.73324	−	5.73400i	0.148231	−	0.310970i
$341$	−37.6476			−2.03873
$342$			4.78822i			0.258917i
$343$		−	17.4364i		−	0.941476i
$344$	−20.3946			−1.09960
$345$	4.38349	+	2.08949i	0.235999	+	0.112494i
$346$	−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	$0.884731\pi$
$348$		−	1.49235i		−	0.0799984i
$349$	−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
$351$	−0.779856			−0.0416256
$352$			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	$-0.0480848\pi$
$354$	−3.17792			−0.168905
$355$	1.33846	+	0.638006i	0.0710380	+	0.0338619i
$356$	8.15991			0.432475
$357$		−	5.29157i		−	0.280060i
$358$		−	14.4514i		−	0.763782i
$359$	−25.2693			−1.33366			−0.666832	−	0.745208i	$-0.732350\pi$
$359$	−25.2693			−1.33366			−0.666832	+	0.745208i	$0.732350\pi$
$360$	2.39411	−	5.02255i	0.126181	−	0.264712i
$361$	26.1632			1.37701
$362$		−	6.15441i		−	0.323469i
$363$			7.21701i			0.378795i
$364$	−3.23525			−0.169573
$365$	15.9087	−	33.3746i	0.832701	−	1.74690i
$366$	3.73992			0.195489
$367$			26.2788i			1.37174i	0.727722	+	0.685872i	$0.240579\pi$
$367$			26.2788i			1.37174i	−0.727722	+	0.685872i	$0.759421\pi$
$368$		−	2.63167i		−	0.137185i
$369$	7.71389			0.401569
$370$	−2.13425	−	1.01734i	−0.110954	−	0.0528890i
$371$	32.7615			1.70089
$372$		−	13.1634i		−	0.682493i
$373$		−	1.43033i		−	0.0740596i	−0.999314	−	0.0370298i	$-0.988210\pi$
$373$		−	1.43033i		−	0.0740596i	0.999314	−	0.0370298i	$-0.0117897\pi$
$374$	5.78872			0.299328
$375$	−10.8695	+	2.61805i	−0.561298	+	0.135195i
$376$	12.9237			0.666487
$377$			0.779856i			0.0401646i
$378$		−	1.98063i		−	0.101873i
$379$	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
$381$	16.9536			0.868557
$382$			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	$-0.305666\pi$
$384$	−10.4421			−0.532872
$385$	11.4158	−	23.9489i	0.581802	−	1.22055i
$386$	−7.44601			−0.378992
$387$		−	8.19624i		−	0.416638i
$388$		−	1.42073i		−	0.0721265i
$389$	−1.63922			−0.0831117			−0.0415558	−	0.999136i	$-0.513231\pi$
$389$	−1.63922			−0.0831117			−0.0415558	+	0.999136i	$0.513231\pi$
$390$	−0.534614	+	1.12155i	−0.0270713	+	0.0567921i
$391$	−4.13389			−0.209060
$392$		−	1.81047i		−	0.0914425i
$393$			11.9324i			0.601908i
$394$	−6.82385			−0.343781
$395$	−19.2675	−	9.18428i	−0.969452	−	0.462111i
$396$	−6.36956			−0.320082
$397$		−	2.04098i		−	0.102434i	−0.998688	−	0.0512170i	$-0.983690\pi$
$397$		−	2.04098i		−	0.102434i	0.998688	−	0.0512170i	$-0.0163100\pi$
$398$		−	4.00257i		−	0.200631i
$399$	−18.6816			−0.935251
$400$	3.81542	+	4.70689i	0.190771	+	0.235345i
$401$	17.4108			0.869455			0.434727	−	0.900562i	$-0.356845\pi$
$401$	17.4108			0.869455			0.434727	+	0.900562i	$0.356845\pi$
$402$			6.05175i			0.301834i
$403$			6.87880i			0.342658i
$404$	12.4131			0.617574
$405$	2.01848	+	0.962154i	0.100299	+	0.0478098i
$406$	−1.98063			−0.0982972
$407$			6.33400i			0.313965i
$408$			4.73655i			0.234494i
$409$	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
$411$	0.239785			0.0118277
$412$		−	18.5188i		−	0.912356i
$413$		−	12.3989i		−	0.610111i
$414$	−1.54731			−0.0760462
$415$	−0.0119305	+	0.0250287i	−0.000585646	+	0.00122861i
$416$	4.55434			0.223295
$417$			17.0255i			0.833740i
$418$		−	20.4368i		−	0.999596i
$419$	−4.32941			−0.211506			−0.105753	−	0.994392i	$-0.533725\pi$
$419$	−4.32941			−0.211506			−0.105753	+	0.994392i	$0.533725\pi$
$420$	8.37370	+	3.99151i	0.408595	+	0.194766i
$421$	−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
$423$			5.19381i			0.252532i
$424$	−29.3252			−1.42416
$425$	7.39369	−	5.99334i	0.358647	−	0.290720i
$426$	−0.472457			−0.0228906
$427$			14.5916i			0.706137i
$428$		−	26.2026i		−	1.26655i
$429$	3.32853			0.160703
$430$	−11.7875	−	5.61877i	−0.568443	−	0.270961i
$431$	34.4885			1.66125			0.830626	−	0.556831i	$-0.187983\pi$
$431$	34.4885			1.66125			0.830626	+	0.556831i	$0.187983\pi$
$432$		−	1.21181i		−	0.0583034i
$433$			3.02258i			0.145256i	0.997359	+	0.0726279i	$0.0231386\pi$
$433$			3.02258i			0.145256i	−0.997359	+	0.0726279i	$0.976861\pi$
$434$	−17.4704			−0.838606
$435$	0.962154	−	2.01848i	0.0461317	−	0.0967787i
$436$	−1.50687			−0.0721661
$437$			14.5945i			0.698148i
$438$			11.7808i			0.562906i
$439$	−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
$441$	0.727598			0.0346475
$442$		−	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	$-0.251192\pi$
$444$	−2.21468			−0.105104
$445$	11.0367	+	5.26089i	0.523189	+	0.249390i
$446$	11.7945			0.558484
$447$			4.13046i			0.195364i
$448$			4.82952i			0.228174i
$449$	−12.0780			−0.569994			−0.284997	−	0.958528i	$-0.591993\pi$
$449$	−12.0780			−0.569994			−0.284997	+	0.958528i	$0.591993\pi$
$450$	2.76746	−	2.24330i	0.130459	−	0.105750i
$451$	−32.9240			−1.55033
$452$			23.2489i			1.09354i
$453$		−	3.21580i		−	0.151092i
$454$	15.6793			0.735864
$455$	−4.37583	−	2.08584i	−0.205142	−	0.0977857i
$456$	16.7221			0.783086
$457$		−	3.03357i		−	0.141905i	−0.997480	−	0.0709523i	$-0.977396\pi$
$457$		−	3.03357i		−	0.141905i	0.997480	−	0.0709523i	$-0.0226038\pi$
$458$			17.5022i			0.817826i
$459$	−1.90354			−0.0888497
$460$	3.11825	−	6.54171i	0.145389	−	0.305009i
$461$	−4.44508			−0.207028			−0.103514	−	0.994628i	$-0.533009\pi$
$461$	−4.44508			−0.207028			−0.103514	+	0.994628i	$0.533009\pi$
$462$			8.45362i			0.393298i
$463$		−	9.04875i		−	0.420531i	−0.977644	−	0.210266i	$-0.932567\pi$
$463$		−	9.04875i		−	0.420531i	0.977644	−	0.210266i	$-0.0674329\pi$
$464$	−1.21181			−0.0562570
$465$	8.48678	−	17.8042i	0.393565	−	0.825651i
$466$	−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	$-0.0601093\pi$
$468$			1.16382i			0.0537975i
$469$	−23.6114			−1.09027
$470$	7.46951	+	3.56051i	0.344543	+	0.164234i
$471$	−6.84237			−0.315280
$472$			11.0984i			0.510847i
$473$			34.9827i			1.60851i
$474$	6.80115			0.312387
$475$	−21.1592	−	26.1030i	−0.970850	−	1.19769i
$476$	−7.89688			−0.361953
$477$		−	11.7853i		−	0.539613i
$478$		−	4.92078i		−	0.225071i
$479$	−6.62947			−0.302908			−0.151454	−	0.988464i	$-0.548396\pi$
$479$	−6.62947			−0.302908			−0.151454	+	0.988464i	$0.548396\pi$
$480$	−11.7879	−	5.61895i	−0.538040	−	0.256469i
$481$	1.15732			0.0527693
$482$		−	17.9541i		−	0.817787i
$483$		−	6.03696i		−	0.274691i
$484$	10.7703			0.489559
$485$	0.915976	−	1.92160i	0.0415923	−	0.0872556i
$486$	−0.712495			−0.0323194
$487$			29.2047i			1.32339i	0.749773	+	0.661695i	$0.230163\pi$
$487$			29.2047i			1.32339i	−0.749773	+	0.661695i	$0.769837\pi$
$488$		−	13.0611i		−	0.591249i
$489$	−14.0502			−0.635372
$490$	0.498790	−	1.04640i	0.0225330	−	0.0472715i
$491$	−4.32464			−0.195168			−0.0975841	−	0.995227i	$-0.531111\pi$
$491$	−4.32464			−0.195168			−0.0975841	+	0.995227i	$0.531111\pi$
$492$		−	11.5118i		−	0.518994i
$493$			1.90354i			0.0857312i
$494$	−3.73412			−0.168006
$495$	−8.61515	−	4.10661i	−0.387222	−	0.184578i
$496$	−10.6889			−0.479947
$497$		−	1.84333i		−	0.0826846i
$498$		−	0.00883480i		−	0.000395897i
$499$	−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
$501$	−17.8725			−0.798483
$502$			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	$-0.102915\pi$
$504$	−6.91707			−0.308111
$505$	16.7893	+	8.00301i	0.747115	+	0.356129i
$506$	6.60414			0.293590
$507$			12.3918i			0.550340i
$508$		−	25.3007i		−	1.12254i
$509$	19.4974			0.864206			0.432103	−	0.901824i	$-0.357772\pi$
$509$	19.4974			0.864206			0.432103	+	0.901824i	$0.357772\pi$
$510$	−1.30494	+	2.73759i	−0.0577835	+	0.121223i
$511$	−45.9635			−2.03331
$512$			13.1076i			0.579280i
$513$			6.72036i			0.296711i
$514$	−6.20654			−0.273759
$515$	11.9395	−	25.0476i	0.526117	−	1.10373i
$516$	−12.2317			−0.538469
$517$		−	22.1679i		−	0.974943i
$518$			2.93930i			0.129145i
$519$	−2.82518			−0.124012
$520$	3.91686	+	1.86706i	0.171766	+	0.0818760i
$521$	24.5229			1.07437			0.537185	−	0.843465i	$-0.319488\pi$
$521$	24.5229			1.07437			0.537185	+	0.843465i	$0.319488\pi$
$522$			0.712495i			0.0311851i
$523$		−	10.8566i		−	0.474725i	−0.971421	−	0.237362i	$-0.923717\pi$
$523$		−	10.8566i		−	0.474725i	0.971421	−	0.237362i	$-0.0762829\pi$
$524$	17.8073			0.777914
$525$	8.75243	+	10.7974i	0.381987	+	0.471239i
$526$	−9.41406			−0.410472
$527$			16.7904i			0.731401i
$528$			5.17218i			0.225090i
$529$	18.2838			0.794948
$530$	−16.9491	−	8.07919i	−0.736224	−	0.350938i
$531$	−4.46028			−0.193560
$532$			27.8795i			1.20873i
$533$			6.01572i			0.260570i
$534$	−3.89580			−0.168588
$535$	16.8934	−	35.4403i	0.730367	−	1.53222i
$536$	21.1348			0.912886
$537$		−	20.2829i		−	0.875270i
$538$			7.37052i			0.317766i
$539$	−3.10549			−0.133763
$540$	1.43587	−	3.01228i	0.0617901	−	0.129628i
$541$	−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
$543$		−	8.63783i		−	0.370685i
$544$	11.1166			0.476622
$545$	−2.03812	−	0.971516i	−0.0873035	−	0.0416152i
$546$	1.54461			0.0661031
$547$			32.8229i			1.40340i	0.712471	+	0.701702i	$0.247576\pi$
$547$			32.8229i			1.40340i	−0.712471	+	0.701702i	$0.752424\pi$
$548$		−	0.357843i		−	0.0152863i
$549$	5.24905			0.224024
$550$	−11.8119	+	9.57474i	−0.503660	+	0.408268i
$551$	6.72036			0.286297
$552$			5.40376i			0.229999i
$553$			26.5352i			1.12839i
$554$	−1.33595			−0.0567590
$555$	−2.99546	−	1.42785i	−0.127150	−	0.0606091i
$556$	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	$-0.188308\pi$
$558$			6.28464i			0.266050i
$559$	6.39189			0.270348
$560$	3.24117	−	6.79958i	0.136965	−	0.287335i
$561$	8.12458			0.343020
$562$		−	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	$0.795152\pi$
$564$	7.75099			0.326375
$565$	−14.9891	+	31.4453i	−0.630597	+	1.32291i
$566$	−11.3093			−0.475367
$567$		−	2.77986i		−	0.116743i
$568$			1.64999i			0.0692318i
$569$	14.7555			0.618582			0.309291	−	0.950967i	$-0.399908\pi$
$569$	14.7555			0.618582			0.309291	+	0.950967i	$0.399908\pi$
$570$	9.66493	+	4.60701i	0.404819	+	0.192966i
$571$	−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
$573$			0.896850i			0.0374664i
$574$	−15.2784			−0.637708
$575$	8.43519	−	6.83758i	0.351772	−	0.285147i
$576$	1.73733			0.0723887
$577$			37.3300i			1.55407i	0.629457	+	0.777035i	$0.283277\pi$
$577$			37.3300i			1.55407i	−0.629457	+	0.777035i	$0.716723\pi$
$578$			9.53071i			0.396425i
$579$	−10.4506			−0.434312
$580$	−3.01228	−	1.43587i	−0.125078	−	0.0596213i
$581$	0.0344697			0.00143004
$582$			0.678299i			0.0281164i
$583$			50.3014i			2.08327i
$584$	41.1425			1.70249
$585$	−0.750341	+	1.57412i	−0.0310228	+	0.0650820i
$586$	−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	$0.928119\pi$
$588$		−	1.08583i		−	0.0447789i
$589$	59.2776			2.44249
$590$	−3.05765	+	6.41458i	−0.125882	+	0.264084i
$591$	−9.57740			−0.393962
$592$			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	$0.906798\pi$
$594$	3.04103			0.124775
$595$	−10.6809	−	5.09131i	−0.437876	−	0.208723i
$596$	6.16409			0.252491
$597$		−	5.61768i		−	0.229916i
$598$		−	1.20668i		−	0.0493448i
$599$	5.52446			0.225724			0.112862	−	0.993611i	$-0.463998\pi$
$599$	5.52446			0.225724			0.112862	+	0.993611i	$0.463998\pi$
$600$	−7.83441	−	9.66493i	−0.319838	−	0.394569i
$601$	−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
$603$			8.49375i			0.345892i
$604$	−4.79910			−0.195273
$605$	14.5674	+	6.94387i	0.592248	+	0.282308i
$606$	−5.92640			−0.240743
$607$		−	30.4979i		−	1.23787i	−0.785442	−	0.618935i	$-0.787564\pi$
$607$		−	30.4979i		−	1.23787i	0.785442	−	0.618935i	$-0.212436\pi$
$608$		−	39.2467i		−	1.59166i
$609$	−2.77986			−0.112645
$610$	3.59838	−	7.54895i	0.145694	−	0.305648i
$611$	−4.05042			−0.163863
$612$			2.84075i			0.114831i
$613$			29.9211i			1.20850i	0.796794	+	0.604250i	$0.206527\pi$
$613$			29.9211i			1.20850i	−0.796794	+	0.604250i	$0.793473\pi$
$614$	12.2980			0.496308
$615$	7.42195	−	15.5703i	0.299282	−	0.627857i
$616$	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	$0.993316\pi$
$618$			8.84145i			0.355655i
$619$	−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
$621$	−2.17168			−0.0871465
$622$		−	23.1643i		−	0.928805i
$623$		−	15.1998i		−	0.608966i
$624$	0.945039			0.0378318
$625$	−5.17364	+	24.4588i	−0.206946	+	0.978352i
$626$	−4.46238			−0.178353
$627$		−	28.6834i		−	1.14551i
$628$			10.2112i			0.407472i
$629$	2.82489			0.112636
$630$	−3.99787	−	1.90567i	−0.159279	−	0.0759239i
$631$	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
$633$		−	0.605216i		−	0.0240552i
$634$	−13.4322			−0.533460
$635$	16.3119	−	34.2204i	0.647319	−	1.35800i
$636$	−17.5878			−0.697403
$637$			0.567422i			0.0224821i
$638$		−	3.04103i		−	0.120395i
$639$	−0.663102			−0.0262319
$640$	−10.0469	+	21.0772i	−0.397139	+	0.833149i
$641$	5.93422			0.234388			0.117194	−	0.993109i	$-0.462610\pi$
$641$	5.93422			0.234388			0.117194	+	0.993109i	$0.462610\pi$
$642$			12.5099i			0.493728i
$643$			0.951317i			0.0375163i	0.999824	+	0.0187581i	$0.00597125\pi$
$643$			0.951317i			0.0375163i	−0.999824	+	0.0187581i	$0.994029\pi$
$644$	−9.00926			−0.355015
$645$	−16.5439	−	7.88604i	−0.651417	−	0.310513i
$646$	−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	$0.757295\pi$
$648$			2.48828i			0.0977490i
$649$	19.0371			0.747271
$650$	1.74945	+	2.15822i	0.0686192	+	0.0846522i
$651$	−24.5200			−0.961015
$652$			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	$-0.356377\pi$
$654$	0.719428			0.0281319
$655$	24.0852	+	11.4808i	0.941088	+	0.448591i
$656$	−9.34779			−0.364970
$657$			16.5345i			0.645072i
$658$		−	10.2870i		−	0.401030i
$659$	19.3162			0.752451			0.376226	−	0.926528i	$-0.377222\pi$
$659$	19.3162			0.752451			0.376226	+	0.926528i	$0.377222\pi$
$660$	−6.12850	+	12.8568i	−0.238551	+	0.500451i
$661$	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
$663$		−	1.48449i		−	0.0576528i
$664$	−0.0308542			−0.00119738
$665$	−17.9746	+	37.7085i	−0.697025	+	1.46227i
$666$	1.05736			0.0409717
$667$			2.17168i			0.0840878i
$668$			26.6720i			1.03197i
$669$	16.5537			0.640004
$670$	12.2153	+	5.82272i	0.471920	+	0.224951i
$671$	−22.4037			−0.864883
$672$			16.2343i			0.626251i
$673$			12.7421i			0.491170i	0.969375	+	0.245585i	$0.0789801\pi$
$673$			12.7421i			0.491170i	−0.969375	+	0.245585i	$0.921020\pi$
$674$	23.2569			0.895821
$675$	3.88418	−	3.14852i	0.149502	−	0.121187i
$676$	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	$0.942082\pi$
$678$		−	11.0997i		−	0.426283i
$679$	−2.64644			−0.101561
$680$	9.56063	+	4.55729i	0.366634	+	0.174764i
$681$	22.0061			0.843276
$682$		−	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	$0.957714\pi$
$684$	10.0291			0.383473
$685$	0.230710	−	0.484001i	0.00881497	−	0.0184927i
$686$	12.4233			0.474325
$687$			24.5647i			0.937202i
$688$			9.93231i			0.378666i
$689$	9.19085			0.350144
$690$	−1.48875	+	3.12322i	−0.0566758	+	0.118899i
$691$	−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
$693$			11.8648i			0.450707i
$694$	−9.40384			−0.356965
$695$	34.3655	+	16.3811i	1.30356	+	0.621371i
$696$	2.48828			0.0943181
$697$			14.6837i			0.556186i
$698$		−	11.1443i		−	0.421818i
$699$	−5.56824			−0.210610
$700$	16.1136	−	13.0617i	0.609036	−	0.493686i
$701$	2.88580			0.108995			0.0544975	−	0.998514i	$-0.482644\pi$
$701$	2.88580			0.108995			0.0544975	+	0.998514i	$0.482644\pi$
$702$		−	0.555643i		−	0.0209714i
$703$		−	9.97314i		−	0.376144i
$704$	−7.41516			−0.279469
$705$	10.4836	+	4.99725i	0.394835	+	0.188207i
$706$	4.02906			0.151636
$707$		−	23.1223i		−	0.869604i
$708$			6.65630i			0.250159i
$709$	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
$711$	9.54554			0.357986
$712$			13.6055i			0.509887i
$713$			19.1555i			0.717381i
$714$	3.77022			0.141097
$715$	3.20256	−	6.71857i	0.119769	−	0.251260i
$716$	−30.2692			−1.13121
$717$		−	6.90640i		−	0.257924i
$718$		−	18.0043i		−	0.671913i
$719$	12.9089			0.481420			0.240710	−	0.970597i	$-0.422620\pi$
$719$	12.9089			0.481420			0.240710	+	0.970597i	$0.422620\pi$
$720$	−2.44602	−	1.16595i	−0.0911577	−	0.0434524i
$721$	−34.4956			−1.28468
$722$			18.6412i			0.693752i
$723$		−	25.1989i		−	0.937158i
$724$	−12.8907			−0.479078
$725$	−3.14852	−	3.88418i	−0.116933	−	0.144255i
$726$	−5.14208			−0.190841
$727$			38.3405i			1.42197i	0.703207	+	0.710985i	$0.251751\pi$
$727$			38.3405i			1.42197i	−0.703207	+	0.710985i	$0.748249\pi$
$728$		−	5.39431i		−	0.199927i
$729$	−1.00000			−0.0370370
$730$	23.7792	+	11.3349i	0.880108	+	0.419523i
$731$	15.6019			0.577057
$732$		−	7.83342i		−	0.289531i
$733$		−	34.8966i		−	1.28894i	−0.764631	−	0.644468i	$-0.777079\pi$
$733$		−	34.8966i		−	1.28894i	0.764631	−	0.644468i	$-0.222921\pi$
$734$	−18.7235			−0.691098
$735$	0.700061	−	1.46864i	0.0258222	−	0.0541717i
$736$	12.6826			0.467485
$737$		−	36.2525i		−	1.33538i
$738$			5.49611i			0.202315i
$739$	−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
$741$	−5.24091			−0.192530
$742$			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	$-0.300311\pi$
$744$	21.9482			0.804659
$745$	8.33725	+	3.97414i	0.305453	+	0.145601i
$746$	1.01910			0.0373120
$747$		−	0.0123998i		−	0.000453685i
$748$		−	12.1247i		−	0.443324i
$749$	−48.8085			−1.78343
$750$	−1.86535	−	7.74446i	−0.0681128	−	0.282788i
$751$	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
$753$			28.3687i			1.03381i
$754$	−0.555643			−0.0202353
$755$	−6.49103	−	3.09410i	−0.236233	−	0.112606i
$756$	−4.14852			−0.150880
$757$		−	30.1910i		−	1.09731i	−0.836048	−	0.548656i	$-0.815140\pi$
$757$		−	30.1910i		−	1.09731i	0.836048	−	0.548656i	$-0.184860\pi$
$758$			25.1038i			0.911811i
$759$	9.26903			0.336445
$760$	16.0893	−	33.7533i	0.583619	−	1.22436i
$761$	8.27038			0.299801			0.149900	−	0.988701i	$-0.452105\pi$
$761$	8.27038			0.299801			0.149900	+	0.988701i	$0.452105\pi$
$762$			12.0793i			0.437588i
$763$			2.80690i			0.101617i
$764$	1.33841			0.0484221
$765$	−1.83150	+	3.84226i	−0.0662180	+	0.138917i
$766$	22.8498			0.825595
$767$		−	3.47837i		−	0.125597i
$768$		−	10.9146i		−	0.393847i
$769$	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
$771$	−8.71100			−0.313719
$772$			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	$-0.161071\pi$
$774$	5.83978			0.209907
$775$	−27.7719	−	34.2608i	−0.997595	−	1.23068i
$776$	2.36886			0.0850371
$777$			4.12536i			0.147996i
$778$		−	1.16794i		−	0.0418725i
$779$	51.8401			1.85737
$780$	2.34914	+	1.11977i	0.0841128	+	0.0400943i
$781$	2.83021			0.101273

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)

\(f(q)\)	\(=\)	\(q+0.712495i q^{2} +1.00000i q^{3} +1.49235 q^{4} +(2.01848 + 0.962154i) q^{5} -0.712495 q^{6} -2.77986i q^{7} +2.48828i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.712495i q^{2} +1.00000i q^{3} +1.49235 q^{4} +(2.01848 + 0.962154i) q^{5} -0.712495 q^{6} -2.77986i q^{7} +2.48828i q^{8} -1.00000 q^{9} +(-0.685530 + 1.43816i) q^{10} +4.26814 q^{11} +1.49235i q^{12} -0.779856i q^{13} +1.98063 q^{14} +(-0.962154 + 2.01848i) q^{15} +1.21181 q^{16} -1.90354i q^{17} -0.712495i q^{18} -6.72036 q^{19} +(3.01228 + 1.43587i) q^{20} +2.77986 q^{21} +3.04103i q^{22} -2.17168i q^{23} -2.48828 q^{24} +(3.14852 + 3.88418i) q^{25} +0.555643 q^{26} -1.00000i q^{27} -4.14852i q^{28} -1.00000 q^{29} +(-1.43816 - 0.685530i) q^{30} -8.82061 q^{31} +5.83998i q^{32} +4.26814i q^{33} +1.35626 q^{34} +(2.67465 - 5.61108i) q^{35} -1.49235 q^{36} +1.48402i q^{37} -4.78822i q^{38} +0.779856 q^{39} +(-2.39411 + 5.02255i) q^{40} -7.71389 q^{41} +1.98063i q^{42} +8.19624i q^{43} +6.36956 q^{44} +(-2.01848 - 0.962154i) q^{45} +1.54731 q^{46} -5.19381i q^{47} +1.21181i q^{48} -0.727598 q^{49} +(-2.76746 + 2.24330i) q^{50} +1.90354 q^{51} -1.16382i q^{52} +11.7853i q^{53} +0.712495 q^{54} +(8.61515 + 4.10661i) q^{55} +6.91707 q^{56} -6.72036i q^{57} -0.712495i q^{58} +4.46028 q^{59} +(-1.43587 + 3.01228i) q^{60} -5.24905 q^{61} -6.28464i q^{62} +2.77986i q^{63} -1.73733 q^{64} +(0.750341 - 1.57412i) q^{65} -3.04103 q^{66} -8.49375i q^{67} -2.84075i q^{68} +2.17168 q^{69} +(3.99787 + 1.90567i) q^{70} +0.663102 q^{71} -2.48828i q^{72} -16.5345i q^{73} -1.05736 q^{74} +(-3.88418 + 3.14852i) q^{75} -10.0291 q^{76} -11.8648i q^{77} +0.555643i q^{78} -9.54554 q^{79} +(2.44602 + 1.16595i) q^{80} +1.00000 q^{81} -5.49611i q^{82} +0.0123998i q^{83} +4.14852 q^{84} +(1.83150 - 3.84226i) q^{85} -5.83978 q^{86} -1.00000i q^{87} +10.6203i q^{88} +5.46783 q^{89} +(0.685530 - 1.43816i) q^{90} -2.16789 q^{91} -3.24091i q^{92} -8.82061i q^{93} +3.70056 q^{94} +(-13.5649 - 6.46602i) q^{95} -5.83998 q^{96} -0.952006i q^{97} -0.518410i q^{98} -4.26814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9	\( 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35} + 10 q^{36} - 4 q^{39} - 14 q^{40} - 28 q^{41} - 40 q^{44} - 12 q^{46} + 14 q^{49} - 12 q^{50} - 6 q^{54} + 2 q^{55} - 4 q^{56} - 8 q^{59} - 2 q^{60} - 24 q^{61} + 18 q^{64} + 6 q^{65} + 28 q^{66} - 16 q^{69} + 20 q^{70} + 60 q^{71} - 4 q^{74} + 12 q^{75} - 88 q^{76} + 36 q^{79} - 30 q^{80} + 10 q^{81} + 12 q^{84} - 14 q^{85} + 60 q^{86} + 44 q^{89} - 4 q^{90} - 24 q^{91} + 100 q^{94} - 36 q^{95} + 2 q^{96} - 24 q^{99}+O(q^{100}) \) 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

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