LMFDB - Embedded newform 429.2.b.b.298.6

Newspace parameters

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.42558224671$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
Defining polynomial:	$x^{14} + 23 x^{12} + 201 x^{10} + 835 x^{8} + 1695 x^{6} + 1565 x^{4} + 511 x^{2} + 49$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			298.6
Root			$-0.584778i$ of defining polynomial
Character	$\chi$	$=$	429.298
Dual form			429.2.b.b.298.9

$q$-expansion

$f(q)$	$=$	$q-0.584778i q^{2} +1.00000 q^{3} +1.65803 q^{4} -1.95350i q^{5} -0.584778i q^{6} -1.51078i q^{7} -2.13914i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.584778i q^{2} +1.00000 q^{3} +1.65803 q^{4} -1.95350i q^{5} -0.584778i q^{6} -1.51078i q^{7} -2.13914i q^{8} +1.00000 q^{9} -1.14237 q^{10} -1.00000i q^{11} +1.65803 q^{12} +(-3.49467 + 0.887280i) q^{13} -0.883474 q^{14} -1.95350i q^{15} +2.06515 q^{16} -3.44198 q^{17} -0.584778i q^{18} +5.09587i q^{19} -3.23897i q^{20} -1.51078i q^{21} -0.584778 q^{22} +0.701611 q^{23} -2.13914i q^{24} +1.18383 q^{25} +(0.518862 + 2.04361i) q^{26} +1.00000 q^{27} -2.50493i q^{28} +5.04937 q^{29} -1.14237 q^{30} +7.26957i q^{31} -5.48593i q^{32} -1.00000i q^{33} +2.01280i q^{34} -2.95132 q^{35} +1.65803 q^{36} +2.08160i q^{37} +2.97995 q^{38} +(-3.49467 + 0.887280i) q^{39} -4.17882 q^{40} +1.03658i q^{41} -0.883474 q^{42} +5.48659 q^{43} -1.65803i q^{44} -1.95350i q^{45} -0.410287i q^{46} -3.75225i q^{47} +2.06515 q^{48} +4.71753 q^{49} -0.692276i q^{50} -3.44198 q^{51} +(-5.79429 + 1.47114i) q^{52} -0.502747 q^{53} -0.584778i q^{54} -1.95350 q^{55} -3.23178 q^{56} +5.09587i q^{57} -2.95276i q^{58} +2.65023i q^{59} -3.23897i q^{60} -5.38195 q^{61} +4.25109 q^{62} -1.51078i q^{63} +0.922236 q^{64} +(1.73330 + 6.82685i) q^{65} -0.584778 q^{66} -8.47197i q^{67} -5.70692 q^{68} +0.701611 q^{69} +1.72587i q^{70} -2.31533i q^{71} -2.13914i q^{72} -2.21094i q^{73} +1.21727 q^{74} +1.18383 q^{75} +8.44913i q^{76} -1.51078 q^{77} +(0.518862 + 2.04361i) q^{78} -5.54589 q^{79} -4.03427i q^{80} +1.00000 q^{81} +0.606168 q^{82} +14.5137i q^{83} -2.50493i q^{84} +6.72392i q^{85} -3.20844i q^{86} +5.04937 q^{87} -2.13914 q^{88} +1.80966i q^{89} -1.14237 q^{90} +(1.34049 + 5.27970i) q^{91} +1.16330 q^{92} +7.26957i q^{93} -2.19424 q^{94} +9.95480 q^{95} -5.48593i q^{96} +14.5085i q^{97} -2.75871i q^{98} -1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14q + 14q^{3} - 18q^{4} + 14q^{9} + O(q^{10}) $	$ 14q + 14q^{3} - 18q^{4} + 14q^{9} - 18q^{12} + 16q^{14} + 34q^{16} + 4q^{17} + 6q^{22} - 8q^{23} - 26q^{25} - 6q^{26} + 14q^{27} - 24q^{29} - 8q^{35} - 18q^{36} - 32q^{38} - 20q^{40} + 16q^{42} + 32q^{43} + 34q^{48} - 46q^{49} + 4q^{51} + 4q^{52} + 20q^{53} + 12q^{55} - 32q^{56} - 20q^{61} + 72q^{62} - 58q^{64} + 12q^{65} + 6q^{66} - 20q^{68} - 8q^{69} - 26q^{75} - 12q^{77} - 6q^{78} + 12q^{79} + 14q^{81} + 20q^{82} - 24q^{87} - 30q^{88} + 16q^{91} - 24q^{92} + 64q^{94} - 36q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$.

$n$	$67$	$79$	$287$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	0.584778i		−	0.413501i	−0.978394	−	0.206750i	$-0.933711\pi$
$2$		−	0.584778i		−	0.413501i	0.978394	−	0.206750i	$-0.0662888\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	1.65803			0.829017
$5$		−	1.95350i		−	0.873633i	−0.899551	−	0.436817i	$-0.856106\pi$
$5$		−	1.95350i		−	0.873633i	0.899551	−	0.436817i	$-0.143894\pi$
$6$		−	0.584778i		−	0.238735i
$7$		−	1.51078i		−	0.571023i	−0.958375	−	0.285511i	$-0.907837\pi$
$7$		−	1.51078i		−	0.571023i	0.958375	−	0.285511i	$-0.0921634\pi$
$8$		−	2.13914i		−	0.756300i
$9$	1.00000			0.333333
$10$	−1.14237			−0.361248
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$	1.65803			0.478633
$13$	−3.49467	+	0.887280i	−0.969248	+	0.246087i
$13$	−3.49467	+	0.887280i	−0.969248	+	0.246087i
$14$	−0.883474			−0.236118
$15$		−	1.95350i		−	0.504392i
$16$	2.06515			0.516286
$17$	−3.44198			−0.834803			−0.417401	−	0.908722i	$-0.637059\pi$
$17$	−3.44198			−0.834803			−0.417401	+	0.908722i	$0.637059\pi$
$18$		−	0.584778i		−	0.137834i
$19$			5.09587i			1.16907i	0.811368	+	0.584536i	$0.198724\pi$
$19$			5.09587i			1.16907i	−0.811368	+	0.584536i	$0.801276\pi$
$20$		−	3.23897i		−	0.724257i
$21$		−	1.51078i		−	0.329680i
$22$	−0.584778			−0.124675
$23$	0.701611			0.146296			0.0731480	−	0.997321i	$-0.476695\pi$
$23$	0.701611			0.146296			0.0731480	+	0.997321i	$0.476695\pi$
$24$		−	2.13914i		−	0.436650i
$25$	1.18383			0.236765
$26$	0.518862	+	2.04361i	0.101757	+	0.400785i
$27$	1.00000			0.192450
$28$		−	2.50493i		−	0.473388i
$29$	5.04937			0.937645			0.468822	−	0.883292i	$-0.344678\pi$
$29$	5.04937			0.937645			0.468822	+	0.883292i	$0.344678\pi$
$30$	−1.14237			−0.208567
$31$			7.26957i			1.30565i	0.757507	+	0.652827i	$0.226417\pi$
$31$			7.26957i			1.30565i	−0.757507	+	0.652827i	$0.773583\pi$
$32$		−	5.48593i		−	0.969785i
$33$		−	1.00000i		−	0.174078i
$34$			2.01280i			0.345192i
$35$	−2.95132			−0.498864
$36$	1.65803			0.276339
$37$			2.08160i			0.342213i	0.985253	+	0.171106i	$0.0547342\pi$
$37$			2.08160i			0.342213i	−0.985253	+	0.171106i	$0.945266\pi$
$38$	2.97995			0.483413
$39$	−3.49467	+	0.887280i	−0.559595	+	0.142079i
$40$	−4.17882			−0.660729
$41$			1.03658i			0.161886i	0.996719	+	0.0809430i	$0.0257932\pi$
$41$			1.03658i			0.161886i	−0.996719	+	0.0809430i	$0.974207\pi$
$42$	−0.883474			−0.136323
$43$	5.48659			0.836698			0.418349	−	0.908286i	$-0.362609\pi$
$43$	5.48659			0.836698			0.418349	+	0.908286i	$0.362609\pi$
$44$		−	1.65803i		−	0.249958i
$45$		−	1.95350i		−	0.291211i
$46$		−	0.410287i		−	0.0604935i
$47$		−	3.75225i		−	0.547322i	−0.961826	−	0.273661i	$-0.911765\pi$
$47$		−	3.75225i		−	0.547322i	0.961826	−	0.273661i	$-0.0882347\pi$
$48$	2.06515			0.298078
$49$	4.71753			0.673933
$50$		−	0.692276i		−	0.0979026i
$51$	−3.44198			−0.481974
$52$	−5.79429	+	1.47114i	−0.803523	+	0.204010i
$53$	−0.502747			−0.0690575			−0.0345288	−	0.999404i	$-0.510993\pi$
$53$	−0.502747			−0.0690575			−0.0345288	+	0.999404i	$0.510993\pi$
$54$		−	0.584778i		−	0.0795783i
$55$	−1.95350			−0.263410
$56$	−3.23178			−0.431865
$57$			5.09587i			0.674964i
$58$		−	2.95276i		−	0.387717i
$59$			2.65023i			0.345031i	0.985007	+	0.172515i	$0.0551894\pi$
$59$			2.65023i			0.345031i	−0.985007	+	0.172515i	$0.944811\pi$
$60$		−	3.23897i		−	0.418150i
$61$	−5.38195			−0.689088			−0.344544	−	0.938770i	$-0.611967\pi$
$61$	−5.38195			−0.689088			−0.344544	+	0.938770i	$0.611967\pi$
$62$	4.25109			0.539889
$63$		−	1.51078i		−	0.190341i
$64$	0.922236			0.115279
$65$	1.73330	+	6.82685i	0.214990	+	0.846767i
$66$	−0.584778			−0.0719813
$67$		−	8.47197i		−	1.03502i	−0.855679	−	0.517508i	$-0.826860\pi$
$67$		−	8.47197i		−	1.03502i	0.855679	−	0.517508i	$-0.173140\pi$
$68$	−5.70692			−0.692066
$69$	0.701611			0.0844640
$70$			1.72587i			0.206281i
$71$		−	2.31533i		−	0.274779i	−0.990517	−	0.137390i	$-0.956129\pi$
$71$		−	2.31533i		−	0.274779i	0.990517	−	0.137390i	$-0.0438712\pi$
$72$		−	2.13914i		−	0.252100i
$73$		−	2.21094i		−	0.258771i	−0.991594	−	0.129385i	$-0.958700\pi$
$73$		−	2.21094i		−	0.258771i	0.991594	−	0.129385i	$-0.0413004\pi$
$74$	1.21727			0.141505
$75$	1.18383			0.136696
$76$			8.44913i			0.969181i
$77$	−1.51078			−0.172170
$78$	0.518862	+	2.04361i	0.0587496	+	0.231393i
$79$	−5.54589			−0.623961			−0.311981	−	0.950088i	$-0.600992\pi$
$79$	−5.54589			−0.623961			−0.311981	+	0.950088i	$0.600992\pi$
$80$		−	4.03427i		−	0.451045i
$81$	1.00000			0.111111
$82$	0.606168			0.0669400
$83$			14.5137i			1.59308i	0.604585	+	0.796541i	$0.293339\pi$
$83$			14.5137i			1.59308i	−0.604585	+	0.796541i	$0.706661\pi$
$84$		−	2.50493i		−	0.273310i
$85$			6.72392i			0.729311i
$86$		−	3.20844i		−	0.345975i
$87$	5.04937			0.541350
$88$	−2.13914			−0.228033
$89$			1.80966i			0.191824i	0.995390	+	0.0959119i	$0.0305767\pi$
$89$			1.80966i			0.191824i	−0.995390	+	0.0959119i	$0.969423\pi$
$90$	−1.14237			−0.120416
$91$	1.34049	+	5.27970i	0.140521	+	0.553462i
$92$	1.16330			0.121282
$93$			7.26957i			0.753819i
$94$	−2.19424			−0.226318
$95$	9.95480			1.02134
$96$		−	5.48593i		−	0.559906i
$97$			14.5085i			1.47312i	0.676372	+	0.736560i	$0.263551\pi$
$97$			14.5085i			1.47312i	−0.676372	+	0.736560i	$0.736449\pi$
$98$		−	2.75871i		−	0.278672i
$99$		−	1.00000i		−	0.100504i
$100$	1.96282			0.196282
$101$	−8.63799			−0.859512			−0.429756	−	0.902945i	$-0.641400\pi$
$101$	−8.63799			−0.859512			−0.429756	+	0.902945i	$0.641400\pi$
$102$			2.01280i			0.199296i
$103$	2.90048			0.285792			0.142896	−	0.989738i	$-0.454358\pi$
$103$	2.90048			0.285792			0.142896	+	0.989738i	$0.454358\pi$
$104$	1.89802	+	7.47559i	0.186116	+	0.733042i
$105$	−2.95132			−0.288019
$106$			0.293995i			0.0285554i
$107$	−2.29234			−0.221609			−0.110805	−	0.993842i	$-0.535343\pi$
$107$	−2.29234			−0.221609			−0.110805	+	0.993842i	$0.535343\pi$
$108$	1.65803			0.159544
$109$			12.8818i			1.23386i	0.787019	+	0.616929i	$0.211623\pi$
$109$			12.8818i			1.23386i	−0.787019	+	0.616929i	$0.788377\pi$
$110$			1.14237i			0.108920i
$111$			2.08160i			0.197577i
$112$		−	3.11999i		−	0.294811i
$113$	−12.6273			−1.18788			−0.593940	−	0.804509i	$-0.702428\pi$
$113$	−12.6273			−1.18788			−0.593940	+	0.804509i	$0.702428\pi$
$114$	2.97995			0.279098
$115$		−	1.37060i		−	0.127809i
$116$	8.37203			0.777324
$117$	−3.49467	+	0.887280i	−0.323083	+	0.0820291i
$118$	1.54980			0.142670
$119$			5.20009i			0.476691i
$120$	−4.17882			−0.381472
$121$	−1.00000			−0.0909091
$122$			3.14725i			0.284939i
$123$			1.03658i			0.0934649i
$124$			12.0532i			1.08241i
$125$		−	12.0801i		−	1.08048i
$126$	−0.883474			−0.0787061
$127$	−5.05085			−0.448190			−0.224095	−	0.974567i	$-0.571943\pi$
$127$	−5.05085			−0.448190			−0.224095	+	0.974567i	$0.571943\pi$
$128$		−	11.5112i		−	1.01745i
$129$	5.48659			0.483068
$130$	3.99220	−	1.01360i	0.350139	−	0.0888985i
$131$	15.9772			1.39594			0.697968	−	0.716129i	$-0.254088\pi$
$131$	15.9772			1.39594			0.697968	+	0.716129i	$0.254088\pi$
$132$		−	1.65803i		−	0.144313i
$133$	7.69876			0.667567
$134$	−4.95422			−0.427980
$135$		−	1.95350i		−	0.168131i
$136$			7.36288i			0.631361i
$137$			17.2582i			1.47447i	0.675639	+	0.737233i	$0.263868\pi$
$137$			17.2582i			1.47447i	−0.675639	+	0.737233i	$0.736132\pi$
$138$		−	0.410287i		−	0.0349260i
$139$	−5.70544			−0.483929			−0.241965	−	0.970285i	$-0.577792\pi$
$139$	−5.70544			−0.483929			−0.241965	+	0.970285i	$0.577792\pi$
$140$	−4.89339			−0.413567
$141$		−	3.75225i		−	0.315997i
$142$	−1.35395			−0.113621
$143$	0.887280	+	3.49467i	0.0741981	+	0.292239i
$144$	2.06515			0.172095
$145$		−	9.86396i		−	0.819158i
$146$	−1.29291			−0.107002
$147$	4.71753			0.389095
$148$			3.45136i			0.283700i
$149$		−	0.161015i		−	0.0131908i	−0.999978	−	0.00659542i	$-0.997901\pi$
$149$		−	0.161015i		−	0.0131908i	0.999978	−	0.00659542i	$-0.00209940\pi$
$150$		−	0.692276i		−	0.0565241i
$151$		−	16.3107i		−	1.32734i	−0.748025	−	0.663671i	$-0.768997\pi$
$151$		−	16.3107i		−	1.32734i	0.748025	−	0.663671i	$-0.231003\pi$
$152$	10.9008			0.884170
$153$	−3.44198			−0.278268
$154$			0.883474i			0.0711924i
$155$	14.2011			1.14066
$156$	−5.79429	+	1.47114i	−0.463914	+	0.117786i
$157$	−14.2192			−1.13481			−0.567407	−	0.823437i	$-0.692053\pi$
$157$	−14.2192			−1.13481			−0.567407	+	0.823437i	$0.692053\pi$
$158$			3.24312i			0.258008i
$159$	−0.502747			−0.0398704
$160$	−10.7168			−0.847236
$161$		−	1.05998i		−	0.0835384i
$162$		−	0.584778i		−	0.0459445i
$163$		−	13.0186i		−	1.01970i	−0.860264	−	0.509848i	$-0.829702\pi$
$163$		−	13.0186i		−	1.01970i	0.860264	−	0.509848i	$-0.170298\pi$
$164$			1.71868i			0.134206i
$165$	−1.95350			−0.152080
$166$	8.48728			0.658740
$167$		−	14.0271i		−	1.08545i	−0.839911	−	0.542724i	$-0.817393\pi$
$167$		−	14.0271i		−	1.08545i	0.839911	−	0.542724i	$-0.182607\pi$
$168$	−3.23178			−0.249337
$169$	11.4255	−	6.20151i	0.878882	−	0.477039i
$170$	3.93200			0.301571
$171$			5.09587i			0.389691i
$172$	9.09696			0.693637
$173$	−17.8674			−1.35844			−0.679218	−	0.733936i	$-0.737681\pi$
$173$	−17.8674			−1.35844			−0.679218	+	0.733936i	$0.737681\pi$
$174$		−	2.95276i		−	0.223848i
$175$		−	1.78851i		−	0.135198i
$176$		−	2.06515i		−	0.155666i
$177$			2.65023i			0.199204i
$178$	1.05825			0.0793193
$179$	−17.2723			−1.29099			−0.645495	−	0.763764i	$-0.723349\pi$
$179$	−17.2723			−1.29099			−0.645495	+	0.763764i	$0.723349\pi$
$180$		−	3.23897i		−	0.241419i
$181$	−3.70318			−0.275255			−0.137628	−	0.990484i	$-0.543948\pi$
$181$	−3.70318			−0.275255			−0.137628	+	0.990484i	$0.543948\pi$
$182$	3.08745	−	0.783889i	0.228857	−	0.0581057i
$183$	−5.38195			−0.397845
$184$		−	1.50084i		−	0.110644i
$185$	4.06641			0.298969
$186$	4.25109			0.311705
$187$			3.44198i			0.251703i
$188$		−	6.22136i		−	0.453740i
$189$		−	1.51078i		−	0.109893i
$190$		−	5.82135i		−	0.422325i
$191$	−20.6259			−1.49243			−0.746217	−	0.665702i	$-0.768132\pi$
$191$	−20.6259			−1.49243			−0.746217	+	0.665702i	$0.768132\pi$
$192$	0.922236			0.0665566
$193$		−	18.8475i		−	1.35667i	−0.734753	−	0.678335i	$-0.762702\pi$
$193$		−	18.8475i		−	1.35667i	0.734753	−	0.678335i	$-0.237298\pi$
$194$	8.48429			0.609136
$195$	1.73330	+	6.82685i	0.124124	+	0.488881i
$196$	7.82183			0.558702
$197$		−	7.18370i		−	0.511818i	−0.966701	−	0.255909i	$-0.917625\pi$
$197$		−	7.18370i		−	0.511818i	0.966701	−	0.255909i	$-0.0823747\pi$
$198$	−0.584778			−0.0415584
$199$	18.9591			1.34398			0.671989	−	0.740562i	$-0.265440\pi$
$199$	18.9591			1.34398			0.671989	+	0.740562i	$0.265440\pi$
$200$		−	2.53237i		−	0.179066i
$201$		−	8.47197i		−	0.597566i
$202$			5.05131i			0.355409i
$203$		−	7.62851i		−	0.535417i
$204$	−5.70692			−0.399564
$205$	2.02496			0.141429
$206$		−	1.69614i		−	0.118175i
$207$	0.701611			0.0487653
$208$	−7.21701	+	1.83236i	−0.500409	+	0.127051i
$209$	5.09587			0.352489
$210$			1.72587i			0.119096i
$211$	−19.9916			−1.37628			−0.688139	−	0.725578i	$-0.741572\pi$
$211$	−19.9916			−1.37628			−0.688139	+	0.725578i	$0.741572\pi$
$212$	−0.833571			−0.0572499
$213$		−	2.31533i		−	0.158644i
$214$			1.34051i			0.0916356i
$215$		−	10.7181i		−	0.730967i
$216$		−	2.13914i		−	0.145550i
$217$	10.9828			0.745558
$218$	7.53303			0.510201
$219$		−	2.21094i		−	0.149401i
$220$	−3.23897			−0.218372
$221$	12.0286	−	3.05400i	0.809131	−	0.205434i
$222$	1.21727			0.0816981
$223$		−	27.7784i		−	1.86018i	−0.367330	−	0.930091i	$-0.619728\pi$
$223$		−	27.7784i		−	1.86018i	0.367330	−	0.930091i	$-0.380272\pi$
$224$	−8.28806			−0.553769
$225$	1.18383			0.0789218
$226$			7.38419i			0.491189i
$227$			28.6524i			1.90173i	0.309612	+	0.950863i	$0.399801\pi$
$227$			28.6524i			1.90173i	−0.309612	+	0.950863i	$0.600199\pi$
$228$			8.44913i			0.559557i
$229$		−	4.83459i		−	0.319479i	−0.987159	−	0.159739i	$-0.948935\pi$
$229$		−	4.83459i		−	0.319479i	0.987159	−	0.159739i	$-0.0510654\pi$
$230$	−0.801497			−0.0528491
$231$	−1.51078			−0.0994023
$232$		−	10.8013i		−	0.709141i
$233$	−13.6237			−0.892518			−0.446259	−	0.894904i	$-0.647244\pi$
$233$	−13.6237			−0.892518			−0.446259	+	0.894904i	$0.647244\pi$
$234$	0.518862	+	2.04361i	0.0339191	+	0.133595i
$235$	−7.33004			−0.478159
$236$			4.39417i			0.286036i
$237$	−5.54589			−0.360244
$238$	3.04090			0.197112
$239$		−	5.99903i		−	0.388045i	−0.980997	−	0.194023i	$-0.937846\pi$
$239$		−	5.99903i		−	0.388045i	0.980997	−	0.194023i	$-0.0621535\pi$
$240$		−	4.03427i		−	0.260411i
$241$			12.9290i			0.832831i	0.909174	+	0.416416i	$0.136714\pi$
$241$			12.9290i			0.832831i	−0.909174	+	0.416416i	$0.863286\pi$
$242$			0.584778i			0.0375910i
$243$	1.00000			0.0641500
$244$	−8.92346			−0.571266
$245$		−	9.21571i		−	0.588770i
$246$	0.606168			0.0386478
$247$	−4.52146	−	17.8084i	−0.287694	−	1.13312i
$248$	15.5506			0.987466
$249$			14.5137i			0.919766i
$250$	−7.06420			−0.446779
$251$	23.9724			1.51313			0.756563	−	0.653921i	$-0.226877\pi$
$251$	23.9724			1.51313			0.756563	+	0.653921i	$0.226877\pi$
$252$		−	2.50493i		−	0.157796i
$253$		−	0.701611i		−	0.0441099i
$254$			2.95363i			0.185327i
$255$			6.72392i			0.421068i
$256$	−4.88701			−0.305438
$257$	−16.3441			−1.01952			−0.509758	−	0.860318i	$-0.670265\pi$
$257$	−16.3441			−1.01952			−0.509758	+	0.860318i	$0.670265\pi$
$258$		−	3.20844i		−	0.199749i
$259$	3.14485			0.195411
$260$	2.87388	+	11.3192i	0.178230	+	0.701984i
$261$	5.04937			0.312548
$262$		−	9.34313i		−	0.577220i
$263$	10.1510			0.625936			0.312968	−	0.949764i	$-0.398677\pi$
$263$	10.1510			0.625936			0.312968	+	0.949764i	$0.398677\pi$
$264$	−2.13914			−0.131655
$265$			0.982117i			0.0603310i
$266$		−	4.50207i		−	0.276040i
$267$			1.80966i			0.110750i
$268$		−	14.0468i		−	0.858045i
$269$	16.5294			1.00781			0.503907	−	0.863758i	$-0.331895\pi$
$269$	16.5294			1.00781			0.503907	+	0.863758i	$0.331895\pi$
$270$	−1.14237			−0.0695222
$271$		−	20.8927i		−	1.26914i	−0.772865	−	0.634570i	$-0.781177\pi$
$271$		−	20.8927i		−	1.26914i	0.772865	−	0.634570i	$-0.218823\pi$
$272$	−7.10819			−0.430997
$273$	1.34049	+	5.27970i	0.0811301	+	0.319542i
$274$	10.0922			0.609693
$275$		−	1.18383i		−	0.0713874i
$276$	1.16330			0.0700221
$277$	17.3070			1.03988			0.519938	−	0.854204i	$-0.325955\pi$
$277$	17.3070			1.03988			0.519938	+	0.854204i	$0.325955\pi$
$278$			3.33642i			0.200105i
$279$			7.26957i			0.435218i
$280$			6.31329i			0.377291i
$281$			16.0666i			0.958450i	0.877692	+	0.479225i	$0.159082\pi$
$281$			16.0666i			0.958450i	−0.877692	+	0.479225i	$0.840918\pi$
$282$	−2.19424			−0.130665
$283$	17.2180			1.02350			0.511751	−	0.859134i	$-0.328997\pi$
$283$	17.2180			1.02350			0.511751	+	0.859134i	$0.328997\pi$
$284$		−	3.83890i		−	0.227797i
$285$	9.95480			0.589671
$286$	2.04361	−	0.518862i	0.120841	−	0.0306810i
$287$	1.56604			0.0924406
$288$		−	5.48593i		−	0.323262i
$289$	−5.15277			−0.303104
$290$	−5.76823			−0.338722
$291$			14.5085i			0.850506i
$292$		−	3.66581i		−	0.214525i
$293$		−	5.29670i		−	0.309436i	−0.987959	−	0.154718i	$-0.950553\pi$
$293$		−	5.29670i		−	0.309436i	0.987959	−	0.154718i	$-0.0494470\pi$
$294$		−	2.75871i		−	0.160891i
$295$	5.17723			0.301430
$296$	4.45283			0.258816
$297$		−	1.00000i		−	0.0580259i
$298$	−0.0941580			−0.00545443
$299$	−2.45190	+	0.622525i	−0.141797	+	0.0360016i
$300$	1.96282			0.113324
$301$		−	8.28906i		−	0.477773i
$302$	−9.53812			−0.548857
$303$	−8.63799			−0.496240
$304$			10.5237i			0.603576i
$305$			10.5137i			0.602010i
$306$			2.01280i			0.115064i
$307$		−	26.3020i		−	1.50113i	−0.660794	−	0.750567i	$-0.729780\pi$
$307$		−	26.3020i		−	1.50113i	0.660794	−	0.750567i	$-0.270220\pi$
$308$	−2.50493			−0.142732
$309$	2.90048			0.165002
$310$		−	8.30451i		−	0.471665i
$311$	−32.8590			−1.86326			−0.931632	−	0.363403i	$-0.881615\pi$
$311$	−32.8590			−1.86326			−0.931632	+	0.363403i	$0.881615\pi$
$312$	1.89802	+	7.47559i	0.107454	+	0.423222i
$313$	33.2130			1.87731			0.938655	−	0.344858i	$-0.112073\pi$
$313$	33.2130			1.87731			0.938655	+	0.344858i	$0.112073\pi$
$314$			8.31507i			0.469247i
$315$	−2.95132			−0.166288
$316$	−9.19527			−0.517274
$317$			30.8436i			1.73235i	0.499740	+	0.866175i	$0.333429\pi$
$317$			30.8436i			1.73235i	−0.499740	+	0.866175i	$0.666571\pi$
$318$			0.293995i			0.0164864i
$319$		−	5.04937i		−	0.282711i
$320$		−	1.80159i		−	0.100712i
$321$	−2.29234			−0.127946
$322$	−0.619855			−0.0345432
$323$		−	17.5399i		−	0.975945i
$324$	1.65803			0.0921130
$325$	−4.13709	+	1.05039i	−0.229484	+	0.0582649i
$326$	−7.61300			−0.421645
$327$			12.8818i			0.712368i
$328$	2.21738			0.122434
$329$	−5.66884			−0.312534
$330$			1.14237i			0.0628852i
$331$			5.75414i			0.316276i	0.987417	+	0.158138i	$0.0505491\pi$
$331$			5.75414i			0.316276i	−0.987417	+	0.158138i	$0.949451\pi$
$332$			24.0641i			1.32069i
$333$			2.08160i			0.114071i
$334$	−8.20273			−0.448833
$335$	−16.5500			−0.904224
$336$		−	3.11999i		−	0.170209i
$337$	−10.9245			−0.595094			−0.297547	−	0.954707i	$-0.596169\pi$
$337$	−10.9245			−0.595094			−0.297547	+	0.954707i	$0.596169\pi$
$338$	−3.62651	−	6.68137i	−0.197256	−	0.363419i
$339$	−12.6273			−0.685823
$340$			11.1485i			0.604612i
$341$	7.26957			0.393669
$342$	2.97995			0.161138
$343$		−	17.7027i		−	0.955854i
$344$		−	11.7366i		−	0.632795i
$345$		−	1.37060i		−	0.0737906i
$346$			10.4485i			0.561715i
$347$	16.3474			0.877572			0.438786	−	0.898591i	$-0.355409\pi$
$347$	16.3474			0.877572			0.438786	+	0.898591i	$0.355409\pi$
$348$	8.37203			0.448788
$349$		−	8.62758i		−	0.461824i	−0.972975	−	0.230912i	$-0.925829\pi$
$349$		−	8.62758i		−	0.461824i	0.972975	−	0.230912i	$-0.0741709\pi$
$350$	−1.04588			−0.0559046
$351$	−3.49467	+	0.887280i	−0.186532	+	0.0473595i
$352$	−5.48593			−0.292401
$353$			16.5793i			0.882425i	0.897403	+	0.441212i	$0.145451\pi$
$353$			16.5793i			0.882425i	−0.897403	+	0.441212i	$0.854549\pi$
$354$	1.54980			0.0823708
$355$	−4.52300			−0.240056
$356$			3.00048i			0.159025i
$357$			5.20009i			0.275218i
$358$			10.1005i			0.533826i
$359$			15.5233i			0.819289i	0.912245	+	0.409645i	$0.134347\pi$
$359$			15.5233i			0.819289i	−0.912245	+	0.409645i	$0.865653\pi$
$360$	−4.17882			−0.220243
$361$	−6.96788			−0.366731
$362$			2.16554i			0.113818i
$363$	−1.00000			−0.0524864
$364$	2.22258	+	8.75392i	0.116495	+	0.458830i
$365$	−4.31907			−0.226071
$366$			3.14725i			0.164509i
$367$	23.4682			1.22503			0.612514	−	0.790460i	$-0.290158\pi$
$367$	23.4682			1.22503			0.612514	+	0.790460i	$0.290158\pi$
$368$	1.44893			0.0755306
$369$			1.03658i			0.0539620i
$370$		−	2.37795i		−	0.123624i
$371$			0.759541i			0.0394334i
$372$			12.0532i			0.624929i
$373$	12.3222			0.638018			0.319009	−	0.947752i	$-0.396650\pi$
$373$	12.3222			0.638018			0.319009	+	0.947752i	$0.396650\pi$
$374$	2.01280			0.104079
$375$		−	12.0801i		−	0.623815i
$376$	−8.02659			−0.413940
$377$	−17.6459	+	4.48021i	−0.908810	+	0.230742i
$378$	−0.883474			−0.0454410
$379$		−	22.8695i		−	1.17473i	−0.809322	−	0.587365i	$-0.800165\pi$
$379$		−	22.8695i		−	1.17473i	0.809322	−	0.587365i	$-0.199835\pi$
$380$	16.5054			0.846709
$381$	−5.05085			−0.258763
$382$			12.0616i			0.617123i
$383$		−	35.5259i		−	1.81529i	−0.419737	−	0.907646i	$-0.637878\pi$
$383$		−	35.5259i		−	1.81529i	0.419737	−	0.907646i	$-0.362122\pi$
$384$		−	11.5112i		−	0.587427i
$385$			2.95132i			0.150413i
$386$	−11.0216			−0.560984
$387$	5.48659			0.278899
$388$			24.0557i			1.22124i
$389$	2.67074			0.135412			0.0677060	−	0.997705i	$-0.478432\pi$
$389$	2.67074			0.135412			0.0677060	+	0.997705i	$0.478432\pi$
$390$	3.99220	−	1.01360i	0.202153	−	0.0513256i
$391$	−2.41493			−0.122128
$392$		−	10.0915i		−	0.509696i
$393$	15.9772			0.805944
$394$	−4.20087			−0.211637
$395$			10.8339i			0.545113i
$396$		−	1.65803i		−	0.0833193i
$397$			36.4406i			1.82890i	0.404698	+	0.914451i	$0.367377\pi$
$397$			36.4406i			1.82890i	−0.404698	+	0.914451i	$0.632623\pi$
$398$		−	11.0869i		−	0.555736i
$399$	7.69876			0.385420
$400$	2.44477			0.122239
$401$		−	13.6597i		−	0.682132i	−0.940039	−	0.341066i	$-0.889212\pi$
$401$		−	13.6597i		−	0.682132i	0.940039	−	0.341066i	$-0.110788\pi$
$402$	−4.95422			−0.247094
$403$	−6.45015	−	25.4048i	−0.321305	−	1.26550i
$404$	−14.3221			−0.712550
$405$		−	1.95350i		−	0.0970703i
$406$	−4.46099			−0.221395
$407$	2.08160			0.103181
$408$			7.36288i			0.364517i
$409$			25.8507i			1.27823i	0.769110	+	0.639117i	$0.220700\pi$
$409$			25.8507i			1.27823i	−0.769110	+	0.639117i	$0.779300\pi$
$410$		−	1.18415i		−	0.0584810i
$411$			17.2582i			0.851283i
$412$	4.80909			0.236927
$413$	4.00393			0.197020
$414$		−	0.410287i		−	0.0201645i
$415$	28.3525			1.39177
$416$	4.86756	+	19.1715i	0.238652	+	0.939962i
$417$	−5.70544			−0.279397
$418$		−	2.97995i		−	0.145754i
$419$	22.6067			1.10441			0.552205	−	0.833708i	$-0.313787\pi$
$419$	22.6067			1.10441			0.552205	+	0.833708i	$0.313787\pi$
$420$	−4.89339			−0.238773
$421$			24.6918i			1.20341i	0.798720	+	0.601703i	$0.205511\pi$
$421$			24.6918i			1.20341i	−0.798720	+	0.601703i	$0.794489\pi$
$422$			11.6907i			0.569092i
$423$		−	3.75225i		−	0.182441i
$424$			1.07544i			0.0522282i
$425$	−4.07471			−0.197652
$426$	−1.35395			−0.0655993
$427$			8.13097i			0.393485i
$428$	−3.80079			−0.183718
$429$	0.887280	+	3.49467i	0.0428383	+	0.168724i
$430$	−6.26770			−0.302255
$431$			30.1126i			1.45047i	0.688499	+	0.725237i	$0.258270\pi$
$431$			30.1126i			1.45047i	−0.688499	+	0.725237i	$0.741730\pi$
$432$	2.06515			0.0993594
$433$	−30.9402			−1.48689			−0.743445	−	0.668797i	$-0.766809\pi$
$433$	−30.9402			−1.48689			−0.743445	+	0.668797i	$0.766809\pi$
$434$		−	6.42248i		−	0.308289i
$435$		−	9.86396i		−	0.472941i
$436$			21.3585i			1.02289i
$437$			3.57532i			0.171031i
$438$	−1.29291			−0.0617775
$439$	7.02158			0.335122			0.167561	−	0.985862i	$-0.446411\pi$
$439$	7.02158			0.335122			0.167561	+	0.985862i	$0.446411\pi$
$440$			4.17882i			0.199217i
$441$	4.71753			0.224644
$442$	−1.78591	−	7.03406i	−0.0849472	−	0.334576i
$443$	−22.5411			−1.07096			−0.535480	−	0.844548i	$-0.679869\pi$
$443$	−22.5411			−1.07096			−0.535480	+	0.844548i	$0.679869\pi$
$444$			3.45136i			0.163794i
$445$	3.53518			0.167584
$446$	−16.2442			−0.769187
$447$		−	0.161015i		−	0.00761574i
$448$		−	1.39330i		−	0.0658272i
$449$			9.41670i			0.444401i	0.975001	+	0.222201i	$0.0713240\pi$
$449$			9.41670i			0.444401i	−0.975001	+	0.222201i	$0.928676\pi$
$450$		−	0.692276i		−	0.0326342i
$451$	1.03658			0.0488105
$452$	−20.9365			−0.984772
$453$		−	16.3107i		−	0.766341i
$454$	16.7553			0.786365
$455$	10.3139	−	2.61865i	0.483523	−	0.122764i
$456$	10.9008			0.510476
$457$		−	3.48691i		−	0.163111i	−0.996669	−	0.0815554i	$-0.974011\pi$
$457$		−	3.48691i		−	0.163111i	0.996669	−	0.0815554i	$-0.0259887\pi$
$458$	−2.82717			−0.132105
$459$	−3.44198			−0.160658
$460$		−	2.27250i		−	0.105956i
$461$			15.1983i			0.707855i	0.935273	+	0.353928i	$0.115154\pi$
$461$			15.1983i			0.707855i	−0.935273	+	0.353928i	$0.884846\pi$
$462$			0.883474i			0.0411029i
$463$		−	15.9882i		−	0.743035i	−0.928426	−	0.371518i	$-0.878838\pi$
$463$		−	15.9882i		−	0.743035i	0.928426	−	0.371518i	$-0.121162\pi$
$464$	10.4277			0.484093
$465$	14.2011			0.658562
$466$			7.96684i			0.369057i
$467$	−32.9225			−1.52347			−0.761736	−	0.647888i	$-0.775652\pi$
$467$	−32.9225			−1.52347			−0.761736	+	0.647888i	$0.775652\pi$
$468$	−5.79429	+	1.47114i	−0.267841	+	0.0680035i
$469$	−12.7993			−0.591017
$470$			4.28645i			0.197719i
$471$	−14.2192			−0.655185
$472$	5.66921			0.260947
$473$		−	5.48659i		−	0.252274i
$474$			3.24312i			0.148961i
$475$			6.03262i			0.276796i
$476$			8.62193i			0.395185i
$477$	−0.502747			−0.0230192
$478$	−3.50811			−0.160457
$479$		−	33.6023i		−	1.53533i	−0.640853	−	0.767664i	$-0.721419\pi$
$479$		−	33.6023i		−	1.53533i	0.640853	−	0.767664i	$-0.278581\pi$
$480$	−10.7168			−0.489152
$481$	−1.84696	−	7.27451i	−0.0842142	−	0.331689i
$482$	7.56061			0.344376
$483$		−	1.05998i		−	0.0482309i
$484$	−1.65803			−0.0753652
$485$	28.3425			1.28697
$486$		−	0.584778i		−	0.0265261i
$487$			3.70001i			0.167663i	0.996480	+	0.0838316i	$0.0267158\pi$
$487$			3.70001i			0.167663i	−0.996480	+	0.0838316i	$0.973284\pi$
$488$			11.5127i			0.521158i
$489$		−	13.0186i		−	0.588722i
$490$	−5.38915			−0.243457
$491$	10.1397			0.457599			0.228799	−	0.973474i	$-0.426520\pi$
$491$	10.1397			0.457599			0.228799	+	0.973474i	$0.426520\pi$
$492$			1.71868i			0.0774840i
$493$	−17.3798			−0.782749
$494$	−10.4140	+	2.64405i	−0.468546	+	0.118962i
$495$	−1.95350			−0.0878034
$496$			15.0127i			0.674091i
$497$	−3.49796			−0.156905
$498$	8.48728			0.380324
$499$		−	15.1124i		−	0.676525i	−0.941052	−	0.338263i	$-0.890161\pi$
$499$		−	15.1124i		−	0.676525i	0.941052	−	0.338263i	$-0.109839\pi$
$500$		−	20.0293i		−	0.895736i
$501$		−	14.0271i		−	0.626683i
$502$		−	14.0185i		−	0.625678i
$503$	11.9190			0.531441			0.265721	−	0.964050i	$-0.414390\pi$
$503$	11.9190			0.531441			0.265721	+	0.964050i	$0.414390\pi$
$504$	−3.23178			−0.143955
$505$			16.8743i			0.750898i
$506$	−0.410287			−0.0182395
$507$	11.4255	−	6.20151i	0.507423	−	0.275419i
$508$	−8.37448			−0.371558
$509$		−	35.2059i		−	1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$509$		−	35.2059i		−	1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$510$	3.93200			0.174112
$511$	−3.34025			−0.147764
$512$		−	20.1645i		−	0.891154i
$513$			5.09587i			0.224988i
$514$			9.55767i			0.421571i
$515$		−	5.66609i		−	0.249678i
$516$	9.09696			0.400471
$517$	−3.75225			−0.165024
$518$		−	1.83904i		−	0.0808028i
$519$	−17.8674			−0.784294
$520$	14.6036	−	3.70778i	0.640410	−	0.162597i
$521$	−2.84143			−0.124485			−0.0622426	−	0.998061i	$-0.519825\pi$
$521$	−2.84143			−0.124485			−0.0622426	+	0.998061i	$0.519825\pi$
$522$		−	2.95276i		−	0.129239i
$523$	−17.8564			−0.780804			−0.390402	−	0.920644i	$-0.627664\pi$
$523$	−17.8564			−0.780804			−0.390402	+	0.920644i	$0.627664\pi$
$524$	26.4908			1.15725
$525$		−	1.78851i		−	0.0780568i
$526$		−	5.93607i		−	0.258825i
$527$		−	25.0217i		−	1.08996i
$528$		−	2.06515i		−	0.0898739i
$529$	−22.5077			−0.978597
$530$	0.574321			0.0249469
$531$			2.65023i			0.115010i
$532$	12.7648			0.553424
$533$	−0.919734	−	3.62249i	−0.0398381	−	0.156908i
$534$	1.05825			0.0457950
$535$			4.47810i			0.193605i
$536$	−18.1227			−0.782782
$537$	−17.2723			−0.745354
$538$		−	9.66602i		−	0.416732i
$539$		−	4.71753i		−	0.203198i
$540$		−	3.23897i		−	0.139383i
$541$			3.56883i			0.153436i	0.997053	+	0.0767179i	$0.0244441\pi$
$541$			3.56883i			0.153436i	−0.997053	+	0.0767179i	$0.975556\pi$
$542$	−12.2176			−0.524791
$543$	−3.70318			−0.158919
$544$			18.8825i			0.809579i
$545$	25.1647			1.07794
$546$	3.08745	−	0.783889i	0.132131	−	0.0335473i
$547$	3.50639			0.149923			0.0749613	−	0.997186i	$-0.476117\pi$
$547$	3.50639			0.149923			0.0749613	+	0.997186i	$0.476117\pi$
$548$			28.6146i			1.22236i
$549$	−5.38195			−0.229696
$550$	−0.692276			−0.0295188
$551$			25.7309i			1.09617i
$552$		−	1.50084i		−	0.0638802i
$553$			8.37864i			0.356296i
$554$		−	10.1207i		−	0.429989i
$555$	4.06641			0.172610
$556$	−9.45982			−0.401186
$557$		−	7.38164i		−	0.312770i	−0.987696	−	0.156385i	$-0.950016\pi$
$557$		−	7.38164i		−	0.312770i	0.987696	−	0.156385i	$-0.0499841\pi$
$558$	4.25109			0.179963
$559$	−19.1738	+	4.86815i	−0.810967	+	0.205901i
$560$	−6.09491			−0.257557
$561$			3.44198i			0.145321i
$562$	9.39537			0.396320
$563$	35.5383			1.49776			0.748879	−	0.662706i	$-0.230592\pi$
$563$	35.5383			1.49776			0.748879	+	0.662706i	$0.230592\pi$
$564$		−	6.22136i		−	0.261967i
$565$			24.6675i			1.03777i
$566$		−	10.0687i		−	0.423219i
$567$		−	1.51078i		−	0.0634470i
$568$	−4.95281			−0.207815
$569$	19.6546			0.823966			0.411983	−	0.911192i	$-0.364836\pi$
$569$	19.6546			0.823966			0.411983	+	0.911192i	$0.364836\pi$
$570$		−	5.82135i		−	0.243830i
$571$	18.6081			0.778726			0.389363	−	0.921084i	$-0.372695\pi$
$571$	18.6081			0.778726			0.389363	+	0.921084i	$0.372695\pi$
$572$	1.47114	+	5.79429i	0.0615115	+	0.242271i
$573$	−20.6259			−0.861658
$574$		−	0.915788i		−	0.0382243i
$575$	0.830586			0.0346378
$576$	0.922236			0.0384265
$577$			25.5485i			1.06360i	0.846871	+	0.531799i	$0.178484\pi$
$577$			25.5485i			1.06360i	−0.846871	+	0.531799i	$0.821516\pi$
$578$			3.01323i			0.125334i
$579$		−	18.8475i		−	0.783274i
$580$		−	16.3548i		−	0.679096i
$581$	21.9270			0.909686
$582$	8.48429			0.351685
$583$			0.502747i			0.0208216i
$584$	−4.72950			−0.195708
$585$	1.73330	+	6.82685i	0.0716633	+	0.282256i
$586$	−3.09739			−0.127952
$587$			2.35936i			0.0973813i	0.998814	+	0.0486907i	$0.0155049\pi$
$587$			2.35936i			0.0973813i	−0.998814	+	0.0486907i	$0.984495\pi$
$588$	7.82183			0.322567
$589$	−37.0448			−1.52640
$590$		−	3.02753i		−	0.124642i
$591$		−	7.18370i		−	0.295498i
$592$			4.29881i			0.176680i
$593$		−	11.6707i		−	0.479257i	−0.970865	−	0.239628i	$-0.922974\pi$
$593$		−	11.6707i		−	0.479257i	0.970865	−	0.239628i	$-0.0770256\pi$
$594$	−0.584778			−0.0239938
$595$	10.1584			0.416453
$596$		−	0.266968i		−	0.0109354i
$597$	18.9591			0.775946
$598$	0.364040	+	1.43382i	0.0148867	+	0.0586332i
$599$	9.12181			0.372707			0.186354	−	0.982483i	$-0.440333\pi$
$599$	9.12181			0.372707			0.186354	+	0.982483i	$0.440333\pi$
$600$		−	2.53237i		−	0.103384i
$601$	6.90316			0.281586			0.140793	−	0.990039i	$-0.455035\pi$
$601$	6.90316			0.281586			0.140793	+	0.990039i	$0.455035\pi$
$602$	−4.84726			−0.197560
$603$		−	8.47197i		−	0.345005i
$604$		−	27.0436i		−	1.10039i
$605$			1.95350i			0.0794212i
$606$			5.05131i			0.205196i
$607$	11.5559			0.469038			0.234519	−	0.972112i	$-0.424648\pi$
$607$	11.5559			0.469038			0.234519	+	0.972112i	$0.424648\pi$
$608$	27.9556			1.13375
$609$		−	7.62851i		−	0.309123i
$610$	6.14816			0.248932
$611$	3.32930	+	13.1129i	0.134689	+	0.530491i
$612$	−5.70692			−0.230689
$613$		−	37.7362i		−	1.52415i	−0.647488	−	0.762075i	$-0.724181\pi$
$613$		−	37.7362i		−	1.52415i	0.647488	−	0.762075i	$-0.275819\pi$
$614$	−15.3808			−0.620720
$615$	2.02496			0.0816541
$616$			3.23178i			0.130212i
$617$		−	28.2159i		−	1.13593i	−0.823053	−	0.567964i	$-0.807731\pi$
$617$		−	28.2159i		−	1.13593i	0.823053	−	0.567964i	$-0.192269\pi$
$618$		−	1.69614i		−	0.0682286i
$619$			38.4407i			1.54506i	0.634978	+	0.772530i	$0.281009\pi$
$619$			38.4407i			1.54506i	−0.634978	+	0.772530i	$0.718991\pi$
$620$	23.5460			0.945628
$621$	0.701611			0.0281547
$622$			19.2152i			0.770461i
$623$	2.73401			0.109536
$624$	−7.21701	+	1.83236i	−0.288911	+	0.0733532i
$625$	−17.6794			−0.707177
$626$		−	19.4222i		−	0.776269i
$627$	5.09587			0.203509
$628$	−23.5759			−0.940781
$629$		−	7.16483i		−	0.285680i
$630$			1.72587i			0.0687603i
$631$		−	16.9688i		−	0.675519i	−0.941232	−	0.337759i	$-0.890331\pi$
$631$		−	16.9688i		−	0.675519i	0.941232	−	0.337759i	$-0.109669\pi$
$632$			11.8634i			0.471902i
$633$	−19.9916			−0.794595
$634$	18.0367			0.716328
$635$			9.86685i			0.391554i
$636$	−0.833571			−0.0330532
$637$	−16.4862	+	4.18577i	−0.653208	+	0.165846i
$638$	−2.95276			−0.116901
$639$		−	2.31533i		−	0.0915930i
$640$	−22.4871			−0.888881
$641$	−6.00876			−0.237332			−0.118666	−	0.992934i	$-0.537862\pi$
$641$	−6.00876			−0.237332			−0.118666	+	0.992934i	$0.537862\pi$
$642$			1.34051i			0.0529059i
$643$		−	9.64269i		−	0.380270i	−0.981758	−	0.190135i	$-0.939107\pi$
$643$		−	9.64269i		−	0.380270i	0.981758	−	0.190135i	$-0.0608927\pi$
$644$		−	1.75749i		−	0.0692547i
$645$		−	10.7181i		−	0.422024i
$646$	−10.2569			−0.403554
$647$	34.1912			1.34420			0.672098	−	0.740463i	$-0.265394\pi$
$647$	34.1912			1.34420			0.672098	+	0.740463i	$0.265394\pi$
$648$		−	2.13914i		−	0.0840333i
$649$	2.65023			0.104031
$650$	0.614243	+	2.41928i	0.0240926	+	0.0948919i
$651$	10.9828			0.430448
$652$		−	21.5853i		−	0.845345i
$653$	−2.32307			−0.0909086			−0.0454543	−	0.998966i	$-0.514474\pi$
$653$	−2.32307			−0.0909086			−0.0454543	+	0.998966i	$0.514474\pi$
$654$	7.53303			0.294565
$655$		−	31.2115i		−	1.21954i
$656$			2.14068i			0.0835796i
$657$		−	2.21094i		−	0.0862568i
$658$			3.31502i			0.129233i
$659$	4.07928			0.158906			0.0794532	−	0.996839i	$-0.474683\pi$
$659$	4.07928			0.158906			0.0794532	+	0.996839i	$0.474683\pi$
$660$	−3.23897			−0.126077
$661$			42.2740i			1.64427i	0.569293	+	0.822135i	$0.307217\pi$
$661$			42.2740i			1.64427i	−0.569293	+	0.822135i	$0.692783\pi$
$662$	3.36489			0.130780
$663$	12.0286	−	3.05400i	0.467152	−	0.118608i
$664$	31.0467			1.20485
$665$		−	15.0395i		−	0.583209i
$666$	1.21727			0.0471684
$667$	3.54270			0.137174
$668$		−	23.2574i		−	0.899854i
$669$		−	27.7784i		−	1.07398i
$670$			9.67809i			0.373897i
$671$			5.38195i			0.207768i
$672$	−8.28806			−0.319719
$673$	10.6688			0.411252			0.205626	−	0.978631i	$-0.434077\pi$
$673$	10.6688			0.411252			0.205626	+	0.978631i	$0.434077\pi$
$674$			6.38840i			0.246072i
$675$	1.18383			0.0455655
$676$	18.9438	−	10.2823i	0.728608	−	0.395473i
$677$	24.2174			0.930752			0.465376	−	0.885113i	$-0.345919\pi$
$677$	24.2174			0.930752			0.465376	+	0.885113i	$0.345919\pi$
$678$			7.38419i			0.283588i
$679$	21.9193			0.841185
$680$	14.3834			0.551578
$681$			28.6524i			1.09796i
$682$		−	4.25109i		−	0.162783i
$683$		−	37.1698i		−	1.42226i	−0.703059	−	0.711132i	$-0.748183\pi$
$683$		−	37.1698i		−	1.42226i	0.703059	−	0.711132i	$-0.251817\pi$
$684$			8.44913i			0.323060i
$685$	33.7139			1.28814
$686$	−10.3521			−0.395246
$687$		−	4.83459i		−	0.184451i
$688$	11.3306			0.431976
$689$	1.75693	−	0.446077i	0.0669339	−	0.0169942i
$690$	−0.801497			−0.0305125
$691$		−	15.9632i		−	0.607268i	−0.952789	−	0.303634i	$-0.901800\pi$
$691$		−	15.9632i		−	0.607268i	0.952789	−	0.303634i	$-0.0981999\pi$
$692$	−29.6248			−1.12617
$693$	−1.51078			−0.0573899
$694$		−	9.55959i		−	0.362877i
$695$			11.1456i			0.422777i
$696$		−	10.8013i		−	0.409423i
$697$		−	3.56788i		−	0.135143i
$698$	−5.04522			−0.190965
$699$	−13.6237			−0.515295
$700$		−	2.96540i		−	0.112082i
$701$	2.01736			0.0761947			0.0380974	−	0.999274i	$-0.487870\pi$
$701$	2.01736			0.0761947			0.0380974	+	0.999274i	$0.487870\pi$
$702$	0.518862	+	2.04361i	0.0195832	+	0.0771311i
$703$	−10.6076			−0.400072
$704$		−	0.922236i		−	0.0347581i
$705$	−7.33004			−0.276065
$706$	9.69519			0.364883
$707$			13.0501i			0.490801i
$708$			4.39417i			0.165143i
$709$			29.6428i			1.11326i	0.830761	+	0.556629i	$0.187905\pi$
$709$			29.6428i			1.11326i	−0.830761	+	0.556629i	$0.812095\pi$
$710$			2.64495i			0.0992634i
$711$	−5.54589			−0.207987
$712$	3.87112			0.145076
$713$			5.10041i			0.191012i
$714$	3.04090			0.113803
$715$	6.82685	−	1.73330i	0.255310	−	0.0648219i
$716$	−28.6380			−1.07025
$717$		−	5.99903i		−	0.224038i
$718$	9.07770			0.338777
$719$	−32.3389			−1.20604			−0.603018	−	0.797728i	$-0.706035\pi$
$719$	−32.3389			−1.20604			−0.603018	+	0.797728i	$0.706035\pi$
$720$		−	4.03427i		−	0.150348i
$721$		−	4.38199i		−	0.163194i
$722$			4.07467i			0.151643i
$723$			12.9290i			0.480835i
$724$	−6.13999			−0.228191
$725$	5.97758			0.222002
$726$			0.584778i			0.0217032i
$727$	−37.9325			−1.40684			−0.703419	−	0.710775i	$-0.748344\pi$
$727$	−37.9325			−1.40684			−0.703419	+	0.710775i	$0.748344\pi$
$728$	11.2940	−	2.86749i	0.418584	−	0.106276i
$729$	1.00000			0.0370370
$730$			2.52570i			0.0934803i
$731$	−18.8847			−0.698478
$732$	−8.92346			−0.329821
$733$		−	16.7917i		−	0.620217i	−0.950701	−	0.310109i	$-0.899635\pi$
$733$		−	16.7917i		−	0.620217i	0.950701	−	0.310109i	$-0.100365\pi$
$734$		−	13.7237i		−	0.506550i
$735$		−	9.21571i		−	0.339927i
$736$		−	3.84899i		−	0.141876i
$737$	−8.47197			−0.312069
$738$	0.606168			0.0223133
$739$			25.4629i			0.936668i	0.883551	+	0.468334i	$0.155146\pi$
$739$			25.4629i			0.936668i	−0.883551	+	0.468334i	$0.844854\pi$
$740$	6.74225			0.247850
$741$	−4.52146	−	17.8084i	−0.166100	−	0.654208i
$742$	0.444164			0.0163058
$743$		−	4.81269i		−	0.176560i	−0.996096	−	0.0882802i	$-0.971863\pi$
$743$		−	4.81269i		−	0.176560i	0.996096	−	0.0882802i	$-0.0281371\pi$
$744$	15.5506			0.570114
$745$	−0.314543			−0.0115240
$746$		−	7.20574i		−	0.263821i
$747$			14.5137i			0.531027i
$748$			5.70692i			0.208666i
$749$			3.46324i			0.126544i
$750$	−7.06420			−0.257948
$751$	24.9537			0.910574			0.455287	−	0.890345i	$-0.349537\pi$
$751$	24.9537			0.910574			0.455287	+	0.890345i	$0.349537\pi$
$752$		−	7.74895i		−	0.282575i
$753$	23.9724			0.873603
$754$	2.61993	+	10.3189i	0.0954122	+	0.375794i
$755$	−31.8629			−1.15961
$756$		−	2.50493i		−	0.0911035i
$757$	−33.8373			−1.22984			−0.614919	−	0.788591i	$-0.710811\pi$
$757$	−33.8373			−1.22984			−0.614919	+	0.788591i	$0.710811\pi$
$758$	−13.3736			−0.485752
$759$		−	0.701611i		−	0.0254669i
$760$		−	21.2947i		−	0.772440i
$761$		−	50.5981i		−	1.83418i	−0.398679	−	0.917091i	$-0.630531\pi$
$761$		−	50.5981i		−	1.83418i	0.398679	−	0.917091i	$-0.369469\pi$
$762$			2.95363i			0.106999i
$763$	19.4617			0.704561
$764$	−34.1984			−1.23725
$765$			6.72392i			0.243104i
$766$	−20.7748			−0.750624
$767$	−2.35150	−	9.26169i	−0.0849076	−	0.334420i
$768$	−4.88701			−0.176345
$769$		−	26.1333i		−	0.942389i	−0.882029	−	0.471195i	$-0.843823\pi$
$769$		−	26.1333i		−	0.942389i	0.882029	−	0.471195i	$-0.156177\pi$
$770$	1.72587			0.0621960
$771$	−16.3441			−0.588618
$772$		−	31.2497i		−	1.12470i
$773$			0.778578i			0.0280035i	0.999902	+	0.0140018i	$0.00445704\pi$
$773$			0.778578i			0.0280035i	−0.999902	+	0.0140018i	$0.995543\pi$
$774$		−	3.20844i		−	0.115325i
$775$			8.60591i			0.309133i
$776$	31.0358			1.11412
$777$	3.14485			0.112821
$778$		−	1.56179i		−	0.0559930i
$779$	−5.28226			−0.189257
$780$	2.87388	+	11.3192i	0.102901	+	0.405291i
$781$	−2.31533			−0.0828490
$782$			1.41220i			0.0505002i
$783$	5.04937			0.180450
$784$	9.74239			0.347942
$785$			27.7772i			0.991411i
$786$		−	9.34313i		−	0.333258i
$787$			33.9128i			1.20886i	0.796658	+	0.604431i	$0.206599\pi$
$787$			33.9128i			1.20886i	−0.796658	+	0.604431i	$0.793401\pi$
$788$		−	11.9108i		−	0.424305i
$789$	10.1510			0.361384
$790$	6.33544			0.225405
$791$			19.0772i			0.678306i

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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.584778i q^{2} +1.00000 q^{3} +1.65803 q^{4} -1.95350i q^{5} -0.584778i q^{6} -1.51078i q^{7} -2.13914i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.584778i q^{2} +1.00000 q^{3} +1.65803 q^{4} -1.95350i q^{5} -0.584778i q^{6} -1.51078i q^{7} -2.13914i q^{8} +1.00000 q^{9} -1.14237 q^{10} -1.00000i q^{11} +1.65803 q^{12} +(-3.49467 + 0.887280i) q^{13} -0.883474 q^{14} -1.95350i q^{15} +2.06515 q^{16} -3.44198 q^{17} -0.584778i q^{18} +5.09587i q^{19} -3.23897i q^{20} -1.51078i q^{21} -0.584778 q^{22} +0.701611 q^{23} -2.13914i q^{24} +1.18383 q^{25} +(0.518862 + 2.04361i) q^{26} +1.00000 q^{27} -2.50493i q^{28} +5.04937 q^{29} -1.14237 q^{30} +7.26957i q^{31} -5.48593i q^{32} -1.00000i q^{33} +2.01280i q^{34} -2.95132 q^{35} +1.65803 q^{36} +2.08160i q^{37} +2.97995 q^{38} +(-3.49467 + 0.887280i) q^{39} -4.17882 q^{40} +1.03658i q^{41} -0.883474 q^{42} +5.48659 q^{43} -1.65803i q^{44} -1.95350i q^{45} -0.410287i q^{46} -3.75225i q^{47} +2.06515 q^{48} +4.71753 q^{49} -0.692276i q^{50} -3.44198 q^{51} +(-5.79429 + 1.47114i) q^{52} -0.502747 q^{53} -0.584778i q^{54} -1.95350 q^{55} -3.23178 q^{56} +5.09587i q^{57} -2.95276i q^{58} +2.65023i q^{59} -3.23897i q^{60} -5.38195 q^{61} +4.25109 q^{62} -1.51078i q^{63} +0.922236 q^{64} +(1.73330 + 6.82685i) q^{65} -0.584778 q^{66} -8.47197i q^{67} -5.70692 q^{68} +0.701611 q^{69} +1.72587i q^{70} -2.31533i q^{71} -2.13914i q^{72} -2.21094i q^{73} +1.21727 q^{74} +1.18383 q^{75} +8.44913i q^{76} -1.51078 q^{77} +(0.518862 + 2.04361i) q^{78} -5.54589 q^{79} -4.03427i q^{80} +1.00000 q^{81} +0.606168 q^{82} +14.5137i q^{83} -2.50493i q^{84} +6.72392i q^{85} -3.20844i q^{86} +5.04937 q^{87} -2.13914 q^{88} +1.80966i q^{89} -1.14237 q^{90} +(1.34049 + 5.27970i) q^{91} +1.16330 q^{92} +7.26957i q^{93} -2.19424 q^{94} +9.95480 q^{95} -5.48593i q^{96} +14.5085i q^{97} -2.75871i q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14q + 14q^{3} - 18q^{4} + 14q^{9} + O(q^{10}) \)	\( 14q + 14q^{3} - 18q^{4} + 14q^{9} - 18q^{12} + 16q^{14} + 34q^{16} + 4q^{17} + 6q^{22} - 8q^{23} - 26q^{25} - 6q^{26} + 14q^{27} - 24q^{29} - 8q^{35} - 18q^{36} - 32q^{38} - 20q^{40} + 16q^{42} + 32q^{43} + 34q^{48} - 46q^{49} + 4q^{51} + 4q^{52} + 20q^{53} + 12q^{55} - 32q^{56} - 20q^{61} + 72q^{62} - 58q^{64} + 12q^{65} + 6q^{66} - 20q^{68} - 8q^{69} - 26q^{75} - 12q^{77} - 6q^{78} + 12q^{79} + 14q^{81} + 20q^{82} - 24q^{87} - 30q^{88} + 16q^{91} - 24q^{92} + 64q^{94} - 36q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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