LMFDB - Embedded newform 420.2.q.d.121.2

Newspace parameters

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.q (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	420.121
Dual form			420.2.q.d.361.2

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.62132 + 2.09077i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.62132 + 2.09077i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.12132 + 3.67423i) q^{11} +3.24264 q^{13} -1.00000 q^{15} +(-2.12132 - 3.67423i) q^{17} +(3.50000 - 6.06218i) q^{19} +(1.00000 - 2.44949i) q^{21} +(2.12132 - 3.67423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -1.75736 q^{29} +(4.74264 + 8.21449i) q^{31} +(2.12132 - 3.67423i) q^{33} +(2.62132 - 0.358719i) q^{35} +(-1.62132 + 2.80821i) q^{37} +(-1.62132 - 2.80821i) q^{39} -4.24264 q^{41} +3.24264 q^{43} +(0.500000 + 0.866025i) q^{45} +(3.00000 - 5.19615i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(-2.12132 + 3.67423i) q^{51} +(-4.24264 - 7.34847i) q^{53} +4.24264 q^{55} -7.00000 q^{57} +(5.12132 + 8.87039i) q^{59} +(-2.24264 + 3.88437i) q^{61} +(-2.62132 + 0.358719i) q^{63} +(1.62132 - 2.80821i) q^{65} +(2.62132 + 4.54026i) q^{67} -4.24264 q^{69} -12.7279 q^{71} +(-4.62132 - 8.00436i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-4.24264 + 10.3923i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-0.500000 - 0.866025i) q^{81} -10.2426 q^{83} -4.24264 q^{85} +(0.878680 + 1.52192i) q^{87} +(5.12132 - 8.87039i) q^{89} +(5.25736 + 6.77962i) q^{91} +(4.74264 - 8.21449i) q^{93} +(-3.50000 - 6.06218i) q^{95} -0.485281 q^{97} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 - 2 * q^9	$ 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{15} + 14 q^{19} + 4 q^{21} - 2 q^{25} + 4 q^{27} - 24 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 28 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} - 2 q^{65} + 2 q^{67} - 10 q^{73} - 2 q^{75} - 22 q^{79} - 2 q^{81} - 24 q^{83} + 12 q^{87} + 12 q^{89} + 38 q^{91} + 2 q^{93} - 14 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 - 2 * q^9 - 4 * q^13 - 4 * q^15 + 14 * q^19 + 4 * q^21 - 2 * q^25 + 4 * q^27 - 24 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^43 + 2 * q^45 + 12 * q^47 + 10 * q^49 - 28 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^63 - 2 * q^65 + 2 * q^67 - 10 * q^73 - 2 * q^75 - 22 * q^79 - 2 * q^81 - 24 * q^83 + 12 * q^87 + 12 * q^89 + 38 * q^91 + 2 * q^93 - 14 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	1.62132	+	2.09077i	0.612801	+	0.790237i
$7$	1.62132	+	2.09077i	0.612801	+	0.790237i
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	2.12132	+	3.67423i	0.639602	+	1.10782i	0.985520	+	0.169559i	$0.0542342\pi$
$11$	2.12132	+	3.67423i	0.639602	+	1.10782i	−0.345918	+	0.938265i	$0.612432\pi$
$12$		0			0
$13$	3.24264			0.899347			0.449673	−	0.893193i	$-0.351540\pi$
$13$	3.24264			0.899347			0.449673	+	0.893193i	$0.351540\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−2.12132	−	3.67423i	−0.514496	−	0.891133i	−0.999859	−	0.0168199i	$-0.994646\pi$
$17$	−2.12132	−	3.67423i	−0.514496	−	0.891133i	0.485363	−	0.874313i	$-0.338688\pi$
$18$		0			0
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$	1.00000	−	2.44949i	0.218218	−	0.534522i
$22$		0			0
$23$	2.12132	−	3.67423i	0.442326	−	0.766131i	−0.555536	−	0.831493i	$-0.687487\pi$
$23$	2.12132	−	3.67423i	0.442326	−	0.766131i	0.997862	+	0.0653618i	$0.0208201\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−1.75736			−0.326333			−0.163167	−	0.986599i	$-0.552171\pi$
$29$	−1.75736			−0.326333			−0.163167	+	0.986599i	$0.552171\pi$
$30$		0			0
$31$	4.74264	+	8.21449i	0.851803	+	1.47537i	0.879579	+	0.475753i	$0.157824\pi$
$31$	4.74264	+	8.21449i	0.851803	+	1.47537i	−0.0277757	+	0.999614i	$0.508842\pi$
$32$		0			0
$33$	2.12132	−	3.67423i	0.369274	−	0.639602i
$34$		0			0
$35$	2.62132	−	0.358719i	0.443084	−	0.0606347i
$36$		0			0
$37$	−1.62132	+	2.80821i	−0.266543	+	0.461667i	−0.967967	−	0.251078i	$-0.919215\pi$
$37$	−1.62132	+	2.80821i	−0.266543	+	0.461667i	0.701423	+	0.712745i	$0.252548\pi$
$38$		0			0
$39$	−1.62132	−	2.80821i	−0.259619	−	0.449673i
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$	3.24264			0.494498			0.247249	−	0.968952i	$-0.420473\pi$
$43$	3.24264			0.494498			0.247249	+	0.968952i	$0.420473\pi$
$44$		0			0
$45$	0.500000	+	0.866025i	0.0745356	+	0.129099i
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	−1.74264	+	6.77962i	−0.248949	+	0.968517i
$50$		0			0
$51$	−2.12132	+	3.67423i	−0.297044	+	0.514496i
$52$		0			0
$53$	−4.24264	−	7.34847i	−0.582772	−	1.00939i	−0.995149	−	0.0983769i	$-0.968635\pi$
$53$	−4.24264	−	7.34847i	−0.582772	−	1.00939i	0.412378	−	0.911013i	$-0.364698\pi$
$54$		0			0
$55$	4.24264			0.572078
$56$		0			0
$57$	−7.00000			−0.927173
$58$		0			0
$59$	5.12132	+	8.87039i	0.666739	+	1.15483i	0.978811	+	0.204767i	$0.0656438\pi$
$59$	5.12132	+	8.87039i	0.666739	+	1.15483i	−0.312072	+	0.950059i	$0.601023\pi$
$60$		0			0
$61$	−2.24264	+	3.88437i	−0.287141	+	0.497342i	−0.973126	−	0.230273i	$-0.926038\pi$
$61$	−2.24264	+	3.88437i	−0.287141	+	0.497342i	0.685985	+	0.727615i	$0.259371\pi$
$62$		0			0
$63$	−2.62132	+	0.358719i	−0.330255	+	0.0451944i
$64$		0			0
$65$	1.62132	−	2.80821i	0.201100	−	0.348315i
$66$		0			0
$67$	2.62132	+	4.54026i	0.320245	+	0.554681i	0.980539	−	0.196327i	$-0.0629013\pi$
$67$	2.62132	+	4.54026i	0.320245	+	0.554681i	−0.660293	+	0.751008i	$0.729568\pi$
$68$		0			0
$69$	−4.24264			−0.510754
$70$		0			0
$71$	−12.7279			−1.51053			−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279			−1.51053			−0.755263	+	0.655422i	$0.772491\pi$
$72$		0			0
$73$	−4.62132	−	8.00436i	−0.540885	−	0.936840i	−0.998854	−	0.0478714i	$-0.984756\pi$
$73$	−4.62132	−	8.00436i	−0.540885	−	0.936840i	0.457969	−	0.888968i	$-0.348577\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.0577350	+	0.100000i
$76$		0			0
$77$	−4.24264	+	10.3923i	−0.483494	+	1.18431i
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	$0.379047\pi$
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	−0.989705	+	0.143120i	$0.954286\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−10.2426			−1.12428			−0.562138	−	0.827043i	$-0.690021\pi$
$83$	−10.2426			−1.12428			−0.562138	+	0.827043i	$0.690021\pi$
$84$		0			0
$85$	−4.24264			−0.460179
$86$		0			0
$87$	0.878680	+	1.52192i	0.0942043	+	0.163167i
$88$		0			0
$89$	5.12132	−	8.87039i	0.542859	−	0.940259i	−0.455879	−	0.890042i	$-0.650675\pi$
$89$	5.12132	−	8.87039i	0.542859	−	0.940259i	0.998738	−	0.0502176i	$-0.0159915\pi$
$90$		0			0
$91$	5.25736	+	6.77962i	0.551121	+	0.710697i
$92$		0			0
$93$	4.74264	−	8.21449i	0.491789	−	0.851803i
$94$		0			0
$95$	−3.50000	−	6.06218i	−0.359092	−	0.621966i
$96$		0			0
$97$	−0.485281			−0.0492729			−0.0246364	−	0.999696i	$-0.507843\pi$
$97$	−0.485281			−0.0492729			−0.0246364	+	0.999696i	$0.507843\pi$
$98$		0			0
$99$	−4.24264			−0.426401
$100$		0			0
$101$	3.87868	+	6.71807i	0.385943	+	0.668473i	0.991900	−	0.127025i	$-0.0405428\pi$
$101$	3.87868	+	6.71807i	0.385943	+	0.668473i	−0.605956	+	0.795498i	$0.707209\pi$
$102$		0			0
$103$	8.62132	−	14.9326i	0.849484	−	1.47135i	−0.0321856	−	0.999482i	$-0.510247\pi$
$103$	8.62132	−	14.9326i	0.849484	−	1.47135i	0.881670	−	0.471867i	$-0.156420\pi$
$104$		0			0
$105$	−1.62132	−	2.09077i	−0.158225	−	0.204038i
$106$		0			0
$107$	−6.36396	+	11.0227i	−0.615227	+	1.06561i	0.375117	+	0.926977i	$0.377602\pi$
$107$	−6.36396	+	11.0227i	−0.615227	+	1.06561i	−0.990345	+	0.138628i	$0.955731\pi$
$108$		0			0
$109$	4.74264	+	8.21449i	0.454263	+	0.786806i	0.998645	−	0.0520310i	$-0.0165695\pi$
$109$	4.74264	+	8.21449i	0.454263	+	0.786806i	−0.544383	+	0.838837i	$0.683236\pi$
$110$		0			0
$111$	3.24264			0.307778
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	−2.12132	−	3.67423i	−0.197814	−	0.342624i
$116$		0			0
$117$	−1.62132	+	2.80821i	−0.149891	+	0.259619i
$118$		0			0
$119$	4.24264	−	10.3923i	0.388922	−	0.952661i
$120$		0			0
$121$	−3.50000	+	6.06218i	−0.318182	+	0.551107i
$122$		0			0
$123$	2.12132	+	3.67423i	0.191273	+	0.331295i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−2.75736			−0.244676			−0.122338	−	0.992488i	$-0.539039\pi$
$127$	−2.75736			−0.244676			−0.122338	+	0.992488i	$0.539039\pi$
$128$		0			0
$129$	−1.62132	−	2.80821i	−0.142749	−	0.247249i
$130$		0			0
$131$	−7.24264	+	12.5446i	−0.632792	+	1.09603i	0.354186	+	0.935175i	$0.384758\pi$
$131$	−7.24264	+	12.5446i	−0.632792	+	1.09603i	−0.986978	+	0.160854i	$0.948575\pi$
$132$		0			0
$133$	18.3492	−	2.51104i	1.59108	−	0.217734i
$134$		0			0
$135$	0.500000	−	0.866025i	0.0430331	−	0.0745356i
$136$		0			0
$137$	−2.12132	−	3.67423i	−0.181237	−	0.313911i	0.761065	−	0.648675i	$-0.224677\pi$
$137$	−2.12132	−	3.67423i	−0.181237	−	0.313911i	−0.942302	+	0.334764i	$0.891343\pi$
$138$		0			0
$139$	−15.4853			−1.31344			−0.656722	−	0.754133i	$-0.728058\pi$
$139$	−15.4853			−1.31344			−0.656722	+	0.754133i	$0.728058\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	6.87868	+	11.9142i	0.575224	+	0.996317i
$144$		0			0
$145$	−0.878680	+	1.52192i	−0.0729704	+	0.126388i
$146$		0			0
$147$	6.74264	−	1.88064i	0.556124	−	0.155112i
$148$		0			0
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	−0.508413	−	0.861113i	$-0.669768\pi$
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	0.999953	+	0.00974235i	$0.00310113\pi$
$150$		0			0
$151$	−11.2426	−	19.4728i	−0.914913	−	1.58468i	−0.807028	−	0.590513i	$-0.798926\pi$
$151$	−11.2426	−	19.4728i	−0.914913	−	1.58468i	−0.107885	−	0.994163i	$-0.534408\pi$
$152$		0			0
$153$	4.24264			0.342997
$154$		0			0
$155$	9.48528			0.761876
$156$		0			0
$157$	3.24264	+	5.61642i	0.258791	+	0.448239i	0.965918	−	0.258847i	$-0.0833426\pi$
$157$	3.24264	+	5.61642i	0.258791	+	0.448239i	−0.707127	+	0.707086i	$0.750009\pi$
$158$		0			0
$159$	−4.24264	+	7.34847i	−0.336463	+	0.582772i
$160$		0			0
$161$	11.1213	−	1.52192i	0.876483	−	0.119944i
$162$		0			0
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$	−2.12132	−	3.67423i	−0.165145	−	0.286039i
$166$		0			0
$167$	18.7279			1.44921			0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279			1.44921			0.724605	+	0.689164i	$0.242022\pi$
$168$		0			0
$169$	−2.48528			−0.191175
$170$		0			0
$171$	3.50000	+	6.06218i	0.267652	+	0.463586i
$172$		0			0
$173$	10.2426	−	17.7408i	0.778734	−	1.34881i	−0.153938	−	0.988080i	$-0.549196\pi$
$173$	10.2426	−	17.7408i	0.778734	−	1.34881i	0.932672	−	0.360726i	$-0.117471\pi$
$174$		0			0
$175$	1.00000	−	2.44949i	0.0755929	−	0.185164i
$176$		0			0
$177$	5.12132	−	8.87039i	0.384942	−	0.666739i
$178$		0			0
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	0.956088	−	0.293079i	$-0.0946798\pi$
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	−0.731858	+	0.681457i	$0.761346\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$		0			0
$183$	4.48528			0.331562
$184$		0			0
$185$	1.62132	+	2.80821i	0.119202	+	0.206464i
$186$		0			0
$187$	9.00000	−	15.5885i	0.658145	−	1.13994i
$188$		0			0
$189$	1.62132	+	2.09077i	0.117934	+	0.152081i
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	−3.37868	−	5.85204i	−0.243203	−	0.421239i	0.718422	−	0.695607i	$-0.244865\pi$
$193$	−3.37868	−	5.85204i	−0.243203	−	0.421239i	−0.961625	+	0.274368i	$0.911531\pi$
$194$		0			0
$195$	−3.24264			−0.232210
$196$		0			0
$197$	−16.2426			−1.15724			−0.578620	−	0.815597i	$-0.696409\pi$
$197$	−16.2426			−1.15724			−0.578620	+	0.815597i	$0.696409\pi$
$198$		0			0
$199$	−5.24264	−	9.08052i	−0.371641	−	0.643701i	0.618177	−	0.786039i	$-0.287871\pi$
$199$	−5.24264	−	9.08052i	−0.371641	−	0.643701i	−0.989818	+	0.142338i	$0.954538\pi$
$200$		0			0
$201$	2.62132	−	4.54026i	0.184894	−	0.320245i
$202$		0			0
$203$	−2.84924	−	3.67423i	−0.199978	−	0.257881i
$204$		0			0
$205$	−2.12132	+	3.67423i	−0.148159	+	0.256620i
$206$		0			0
$207$	2.12132	+	3.67423i	0.147442	+	0.255377i
$208$		0			0
$209$	29.6985			2.05429
$210$		0			0
$211$	22.4853			1.54795			0.773975	−	0.633216i	$-0.218265\pi$
$211$	22.4853			1.54795			0.773975	+	0.633216i	$0.218265\pi$
$212$		0			0
$213$	6.36396	+	11.0227i	0.436051	+	0.755263i
$214$		0			0
$215$	1.62132	−	2.80821i	0.110573	−	0.191518i
$216$		0			0
$217$	−9.48528	+	23.2341i	−0.643903	+	1.57723i
$218$		0			0
$219$	−4.62132	+	8.00436i	−0.312280	+	0.540885i
$220$		0			0
$221$	−6.87868	−	11.9142i	−0.462710	−	0.801437i
$222$		0			0
$223$	−7.51472			−0.503223			−0.251611	−	0.967828i	$-0.580960\pi$
$223$	−7.51472			−0.503223			−0.251611	+	0.967828i	$0.580960\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	7.60660	+	13.1750i	0.504868	+	0.874457i	0.999984	+	0.00563010i	$0.00179213\pi$
$227$	7.60660	+	13.1750i	0.504868	+	0.874457i	−0.495116	+	0.868827i	$0.664875\pi$
$228$		0			0
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$	11.1213	−	1.52192i	0.731729	−	0.100135i
$232$		0			0
$233$	−7.24264	+	12.5446i	−0.474481	+	0.821825i	−0.999573	−	0.0292201i	$-0.990698\pi$
$233$	−7.24264	+	12.5446i	−0.474481	+	0.821825i	0.525092	+	0.851046i	$0.324031\pi$
$234$		0			0
$235$	−3.00000	−	5.19615i	−0.195698	−	0.338960i
$236$		0			0
$237$	11.0000			0.714527
$238$		0			0
$239$	10.9706			0.709627			0.354813	−	0.934937i	$-0.384544\pi$
$239$	10.9706			0.709627			0.354813	+	0.934937i	$0.384544\pi$
$240$		0			0
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	0.923224	−	0.384262i	$-0.125544\pi$
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	−0.794393	+	0.607404i	$0.792211\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	5.00000	+	4.89898i	0.319438	+	0.312984i
$246$		0			0
$247$	11.3492	−	19.6575i	0.722135	−	1.25077i
$248$		0			0
$249$	5.12132	+	8.87039i	0.324550	+	0.562138i
$250$		0			0
$251$	−18.7279			−1.18210			−0.591048	−	0.806636i	$-0.701286\pi$
$251$	−18.7279			−1.18210			−0.591048	+	0.806636i	$0.701286\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$	2.12132	+	3.67423i	0.132842	+	0.230089i
$256$		0			0
$257$	−5.12132	+	8.87039i	−0.319459	+	0.553320i	−0.980375	−	0.197140i	$-0.936835\pi$
$257$	−5.12132	+	8.87039i	−0.319459	+	0.553320i	0.660916	+	0.750460i	$0.270168\pi$
$258$		0			0
$259$	−8.50000	+	1.16320i	−0.528164	+	0.0722776i
$260$		0			0
$261$	0.878680	−	1.52192i	0.0543889	−	0.0942043i
$262$		0			0
$263$	4.24264	+	7.34847i	0.261612	+	0.453126i	0.966671	−	0.256023i	$-0.0824124\pi$
$263$	4.24264	+	7.34847i	0.261612	+	0.453126i	−0.705058	+	0.709150i	$0.749079\pi$
$264$		0			0
$265$	−8.48528			−0.521247
$266$		0			0
$267$	−10.2426			−0.626839
$268$		0			0
$269$	1.24264	+	2.15232i	0.0757651	+	0.131229i	0.901419	−	0.432948i	$-0.142527\pi$
$269$	1.24264	+	2.15232i	0.0757651	+	0.131229i	−0.825654	+	0.564177i	$0.809193\pi$
$270$		0			0
$271$	11.7279	−	20.3134i	0.712421	−	1.23395i	−0.251525	−	0.967851i	$-0.580932\pi$
$271$	11.7279	−	20.3134i	0.712421	−	1.23395i	0.963946	−	0.266098i	$-0.0857344\pi$
$272$		0			0
$273$	3.24264	−	7.94282i	0.196254	−	0.480721i
$274$		0			0
$275$	2.12132	−	3.67423i	0.127920	−	0.221565i
$276$		0			0
$277$	−8.86396	−	15.3528i	−0.532584	−	0.922462i	−0.999276	−	0.0380425i	$-0.987888\pi$
$277$	−8.86396	−	15.3528i	−0.532584	−	0.922462i	0.466692	−	0.884420i	$-0.345446\pi$
$278$		0			0
$279$	−9.48528			−0.567869
$280$		0			0
$281$	28.9706			1.72824			0.864119	−	0.503287i	$-0.167876\pi$
$281$	28.9706			1.72824			0.864119	+	0.503287i	$0.167876\pi$
$282$		0			0
$283$	−11.8640	−	20.5490i	−0.705239	−	1.22151i	−0.966605	−	0.256270i	$-0.917506\pi$
$283$	−11.8640	−	20.5490i	−0.705239	−	1.22151i	0.261366	−	0.965240i	$-0.415827\pi$
$284$		0			0
$285$	−3.50000	+	6.06218i	−0.207322	+	0.359092i
$286$		0			0
$287$	−6.87868	−	8.87039i	−0.406036	−	0.523602i
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.0294118	+	0.0509427i
$290$		0			0
$291$	0.242641	+	0.420266i	0.0142238	+	0.0246364i
$292$		0			0
$293$	−4.97056			−0.290383			−0.145192	−	0.989404i	$-0.546380\pi$
$293$	−4.97056			−0.290383			−0.145192	+	0.989404i	$0.546380\pi$
$294$		0			0
$295$	10.2426			0.596350
$296$		0			0
$297$	2.12132	+	3.67423i	0.123091	+	0.213201i
$298$		0			0
$299$	6.87868	−	11.9142i	0.397804	−	0.689017i
$300$		0			0
$301$	5.25736	+	6.77962i	0.303029	+	0.390771i
$302$		0			0
$303$	3.87868	−	6.71807i	0.222824	−	0.385943i
$304$		0			0
$305$	2.24264	+	3.88437i	0.128413	+	0.222418i
$306$		0			0
$307$	3.24264			0.185067			0.0925336	−	0.995710i	$-0.470503\pi$
$307$	3.24264			0.185067			0.0925336	+	0.995710i	$0.470503\pi$
$308$		0			0
$309$	−17.2426			−0.980900
$310$		0			0
$311$	−10.6066	−	18.3712i	−0.601445	−	1.04173i	−0.992602	−	0.121410i	$-0.961258\pi$
$311$	−10.6066	−	18.3712i	−0.601445	−	1.04173i	0.391157	−	0.920324i	$-0.372075\pi$
$312$		0			0
$313$	−11.8640	+	20.5490i	−0.670591	+	1.16150i	0.307146	+	0.951662i	$0.400626\pi$
$313$	−11.8640	+	20.5490i	−0.670591	+	1.16150i	−0.977737	+	0.209835i	$0.932707\pi$
$314$		0			0
$315$	−1.00000	+	2.44949i	−0.0563436	+	0.138013i
$316$		0			0
$317$	12.3640	−	21.4150i	0.694429	−	1.20279i	−0.275943	−	0.961174i	$-0.588990\pi$
$317$	12.3640	−	21.4150i	0.694429	−	1.20279i	0.970373	−	0.241613i	$-0.0776764\pi$
$318$		0			0
$319$	−3.72792	−	6.45695i	−0.208724	−	0.361520i
$320$		0			0
$321$	12.7279			0.710403
$322$		0			0
$323$	−29.6985			−1.65247
$324$		0			0
$325$	−1.62132	−	2.80821i	−0.0899347	−	0.155771i
$326$		0			0
$327$	4.74264	−	8.21449i	0.262269	−	0.454263i
$328$		0			0
$329$	15.7279	−	2.15232i	0.867108	−	0.118661i
$330$		0			0
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	−0.999298	−	0.0374662i	$-0.988071\pi$
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	0.532096	+	0.846684i	$0.321405\pi$
$332$		0			0
$333$	−1.62132	−	2.80821i	−0.0888478	−	0.153889i
$334$		0			0
$335$	5.24264			0.286436
$336$		0			0
$337$	−13.7279			−0.747808			−0.373904	−	0.927467i	$-0.621981\pi$
$337$	−13.7279			−0.747808			−0.373904	+	0.927467i	$0.621981\pi$
$338$		0			0
$339$	9.00000	+	15.5885i	0.488813	+	0.846649i
$340$		0			0
$341$	−20.1213	+	34.8511i	−1.08963	+	1.88730i
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$		0			0
$345$	−2.12132	+	3.67423i	−0.114208	+	0.197814i
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$	3.24264			0.173079
$352$		0			0
$353$	5.12132	+	8.87039i	0.272580	+	0.472123i	0.969522	−	0.245005i	$-0.0787896\pi$
$353$	5.12132	+	8.87039i	0.272580	+	0.472123i	−0.696941	+	0.717128i	$0.745456\pi$
$354$		0			0
$355$	−6.36396	+	11.0227i	−0.337764	+	0.585024i
$356$		0			0
$357$	−11.1213	+	1.52192i	−0.588603	+	0.0805484i
$358$		0			0
$359$	−0.878680	+	1.52192i	−0.0463749	+	0.0803237i	−0.888281	−	0.459300i	$-0.848100\pi$
$359$	−0.878680	+	1.52192i	−0.0463749	+	0.0803237i	0.841906	+	0.539624i	$0.181434\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$	−9.24264			−0.483782
$366$		0			0
$367$	−17.8640	−	30.9413i	−0.932491	−	1.61512i	−0.779048	−	0.626965i	$-0.784297\pi$
$367$	−17.8640	−	30.9413i	−0.932491	−	1.61512i	−0.153443	−	0.988157i	$-0.549036\pi$
$368$		0			0
$369$	2.12132	−	3.67423i	0.110432	−	0.191273i
$370$		0			0
$371$	8.48528	−	20.7846i	0.440534	−	1.07908i
$372$		0			0
$373$	−8.86396	+	15.3528i	−0.458959	+	0.794939i	−0.998906	−	0.0467591i	$-0.985111\pi$
$373$	−8.86396	+	15.3528i	−0.458959	+	0.794939i	0.539948	+	0.841699i	$0.318444\pi$
$374$		0			0
$375$	0.500000	+	0.866025i	0.0258199	+	0.0447214i
$376$		0			0
$377$	−5.69848			−0.293487
$378$		0			0
$379$	−20.4558			−1.05075			−0.525373	−	0.850872i	$-0.676074\pi$
$379$	−20.4558			−1.05075			−0.525373	+	0.850872i	$0.676074\pi$
$380$		0			0
$381$	1.37868	+	2.38794i	0.0706319	+	0.122338i
$382$		0			0
$383$	−12.7279	+	22.0454i	−0.650366	+	1.12647i	0.332668	+	0.943044i	$0.392051\pi$
$383$	−12.7279	+	22.0454i	−0.650366	+	1.12647i	−0.983034	+	0.183424i	$0.941282\pi$
$384$		0			0
$385$	6.87868	+	8.87039i	0.350570	+	0.452077i
$386$		0			0
$387$	−1.62132	+	2.80821i	−0.0824163	+	0.142749i
$388$		0			0
$389$	10.6066	+	18.3712i	0.537776	+	0.931455i	0.999023	+	0.0441839i	$0.0140687\pi$
$389$	10.6066	+	18.3712i	0.537776	+	0.931455i	−0.461247	+	0.887272i	$0.652598\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$		0			0
$393$	14.4853			0.730686
$394$		0			0
$395$	5.50000	+	9.52628i	0.276735	+	0.479319i
$396$		0			0
$397$	4.37868	−	7.58410i	0.219760	−	0.380635i	−0.734975	−	0.678094i	$-0.762806\pi$
$397$	4.37868	−	7.58410i	0.219760	−	0.380635i	0.954734	+	0.297460i	$0.0961394\pi$
$398$		0			0
$399$	−11.3492	−	14.6354i	−0.568173	−	0.732686i
$400$		0			0
$401$	1.75736	−	3.04384i	0.0877583	−	0.152002i	−0.818805	−	0.574072i	$-0.805363\pi$
$401$	1.75736	−	3.04384i	0.0877583	−	0.152002i	0.906563	+	0.422070i	$0.138696\pi$
$402$		0			0
$403$	15.3787	+	26.6367i	0.766067	+	1.32687i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	−13.7574			−0.681927
$408$		0			0
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	0.575670	−	0.817682i	$-0.304741\pi$
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	−0.995968	+	0.0897044i	$0.971408\pi$
$410$		0			0
$411$	−2.12132	+	3.67423i	−0.104637	+	0.181237i
$412$		0			0
$413$	−10.2426	+	25.0892i	−0.504007	+	1.23456i
$414$		0			0
$415$	−5.12132	+	8.87039i	−0.251396	+	0.435430i
$416$		0			0
$417$	7.74264	+	13.4106i	0.379159	+	0.656722i
$418$		0			0
$419$	−14.4853			−0.707652			−0.353826	−	0.935311i	$-0.615120\pi$
$419$	−14.4853			−0.707652			−0.353826	+	0.935311i	$0.615120\pi$
$420$		0			0
$421$	31.4853			1.53450			0.767249	−	0.641349i	$-0.221625\pi$
$421$	31.4853			1.53450			0.767249	+	0.641349i	$0.221625\pi$
$422$		0			0
$423$	3.00000	+	5.19615i	0.145865	+	0.252646i
$424$		0			0
$425$	−2.12132	+	3.67423i	−0.102899	+	0.178227i
$426$		0			0
$427$	−11.7574	+	1.60896i	−0.568978	+	0.0778629i
$428$		0			0
$429$	6.87868	−	11.9142i	0.332106	−	0.575224i
$430$		0			0
$431$	−9.72792	−	16.8493i	−0.468578	−	0.811600i	0.530777	−	0.847511i	$-0.321900\pi$
$431$	−9.72792	−	16.8493i	−0.468578	−	0.811600i	−0.999355	+	0.0359112i	$0.988567\pi$
$432$		0			0
$433$	33.2426			1.59754			0.798770	−	0.601637i	$-0.205485\pi$
$433$	33.2426			1.59754			0.798770	+	0.601637i	$0.205485\pi$
$434$		0			0
$435$	1.75736			0.0842589
$436$		0			0
$437$	−14.8492	−	25.7196i	−0.710336	−	1.23034i
$438$		0			0
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	−0.721686	−	0.692220i	$-0.756633\pi$
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	0.960323	+	0.278889i	$0.0899661\pi$
$440$		0			0
$441$	−5.00000	−	4.89898i	−0.238095	−	0.233285i
$442$		0			0
$443$	−10.2426	+	17.7408i	−0.486643	+	0.842890i	−0.999882	−	0.0153558i	$-0.995112\pi$
$443$	−10.2426	+	17.7408i	−0.486643	+	0.842890i	0.513240	+	0.858245i	$0.328445\pi$
$444$		0			0
$445$	−5.12132	−	8.87039i	−0.242774	−	0.420497i
$446$		0			0
$447$	−12.0000			−0.567581
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−9.00000	−	15.5885i	−0.423793	−	0.734032i
$452$		0			0
$453$	−11.2426	+	19.4728i	−0.528225	+	0.914913i
$454$		0			0
$455$	8.50000	−	1.16320i	0.398486	−	0.0545316i
$456$		0			0
$457$	−16.1066	+	27.8975i	−0.753435	+	1.30499i	0.192714	+	0.981255i	$0.438271\pi$
$457$	−16.1066	+	27.8975i	−0.753435	+	1.30499i	−0.946149	+	0.323733i	$0.895062\pi$
$458$		0			0
$459$	−2.12132	−	3.67423i	−0.0990148	−	0.171499i
$460$		0			0
$461$	18.7279			0.872246			0.436123	−	0.899887i	$-0.356351\pi$
$461$	18.7279			0.872246			0.436123	+	0.899887i	$0.356351\pi$
$462$		0			0
$463$	10.2721			0.477384			0.238692	−	0.971095i	$-0.423281\pi$
$463$	10.2721			0.477384			0.238692	+	0.971095i	$0.423281\pi$
$464$		0			0
$465$	−4.74264	−	8.21449i	−0.219935	−	0.380938i
$466$		0			0
$467$	5.48528	−	9.50079i	0.253829	−	0.439644i	−0.710748	−	0.703447i	$-0.751643\pi$
$467$	5.48528	−	9.50079i	0.253829	−	0.439644i	0.964577	+	0.263803i	$0.0849768\pi$
$468$		0			0
$469$	−5.24264	+	12.8418i	−0.242083	+	0.592979i
$470$		0			0
$471$	3.24264	−	5.61642i	0.149413	−	0.258791i
$472$		0			0
$473$	6.87868	+	11.9142i	0.316282	+	0.547817i
$474$		0			0
$475$	−7.00000			−0.321182
$476$		0			0
$477$	8.48528			0.388514
$478$		0			0
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	0.695773	−	0.718262i	$-0.255062\pi$
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	−0.969920	+	0.243426i	$0.921729\pi$
$480$		0			0
$481$	−5.25736	+	9.10601i	−0.239715	+	0.415198i
$482$		0			0
$483$	−6.87868	−	8.87039i	−0.312991	−	0.403617i
$484$		0			0
$485$	−0.242641	+	0.420266i	−0.0110177	+	0.0190833i
$486$		0			0
$487$	18.8640	+	32.6733i	0.854808	+	1.48057i	0.876823	+	0.480813i	$0.159658\pi$
$487$	18.8640	+	32.6733i	0.854808	+	1.48057i	−0.0220157	+	0.999758i	$0.507008\pi$
$488$		0			0
$489$	8.00000			0.361773
$490$		0			0
$491$	15.5147			0.700169			0.350085	−	0.936718i	$-0.386153\pi$
$491$	15.5147			0.700169			0.350085	+	0.936718i	$0.386153\pi$
$492$		0			0
$493$	3.72792	+	6.45695i	0.167897	+	0.290806i
$494$		0			0
$495$	−2.12132	+	3.67423i	−0.0953463	+	0.165145i
$496$		0			0
$497$	−20.6360	−	26.6112i	−0.925653	−	1.19367i
$498$		0			0
$499$	−9.74264	+	16.8747i	−0.436140	+	0.755417i	−0.997388	−	0.0722305i	$-0.976988\pi$
$499$	−9.74264	+	16.8747i	−0.436140	+	0.755417i	0.561247	+	0.827648i	$0.310322\pi$
$500$		0			0
$501$	−9.36396	−	16.2189i	−0.418351	−	0.724605i
$502$		0			0
$503$	−26.4853			−1.18092			−0.590460	−	0.807067i	$-0.701054\pi$
$503$	−26.4853			−1.18092			−0.590460	+	0.807067i	$0.701054\pi$
$504$		0			0
$505$	7.75736			0.345198
$506$		0			0
$507$	1.24264	+	2.15232i	0.0551876	+	0.0955877i
$508$		0			0
$509$	9.72792	−	16.8493i	0.431183	−	0.746830i	−0.565793	−	0.824547i	$-0.691430\pi$
$509$	9.72792	−	16.8493i	0.431183	−	0.746830i	0.996975	+	0.0777173i	$0.0247632\pi$
$510$		0			0
$511$	9.24264	−	22.6398i	0.408870	−	1.00152i
$512$		0			0
$513$	3.50000	−	6.06218i	0.154529	−	0.267652i
$514$		0			0
$515$	−8.62132	−	14.9326i	−0.379901	−	0.658007i
$516$		0			0
$517$	25.4558			1.11955
$518$		0			0
$519$	−20.4853			−0.899204
$520$		0			0
$521$	6.72792	+	11.6531i	0.294756	+	0.510532i	0.974928	−	0.222520i	$-0.0714284\pi$
$521$	6.72792	+	11.6531i	0.294756	+	0.510532i	−0.680172	+	0.733052i	$0.738095\pi$
$522$		0			0
$523$	2.62132	−	4.54026i	0.114622	−	0.198532i	−0.803006	−	0.595970i	$-0.796768\pi$
$523$	2.62132	−	4.54026i	0.114622	−	0.198532i	0.917629	+	0.397439i	$0.130101\pi$
$524$		0			0
$525$	−2.62132	+	0.358719i	−0.114404	+	0.0156558i
$526$		0			0
$527$	20.1213	−	34.8511i	0.876498	−	1.51814i
$528$		0			0
$529$	2.50000	+	4.33013i	0.108696	+	0.188266i
$530$		0			0
$531$	−10.2426			−0.444493
$532$		0			0
$533$	−13.7574			−0.595897
$534$		0			0
$535$	6.36396	+	11.0227i	0.275138	+	0.476553i
$536$		0			0
$537$	3.00000	−	5.19615i	0.129460	−	0.224231i
$538$		0			0
$539$	−28.6066	+	7.97887i	−1.23217	+	0.343674i
$540$		0			0
$541$	20.4706	−	35.4561i	0.880098	−	1.52437i	0.0288675	−	0.999583i	$-0.490810\pi$
$541$	20.4706	−	35.4561i	0.880098	−	1.52437i	0.851231	−	0.524792i	$-0.175857\pi$
$542$		0			0
$543$	6.50000	+	11.2583i	0.278942	+	0.483141i
$544$		0			0
$545$	9.48528			0.406305
$546$		0			0
$547$	33.4558			1.43047			0.715234	−	0.698885i	$-0.246320\pi$
$547$	33.4558			1.43047			0.715234	+	0.698885i	$0.246320\pi$
$548$		0			0
$549$	−2.24264	−	3.88437i	−0.0957136	−	0.165781i
$550$		0			0
$551$	−6.15076	+	10.6534i	−0.262031	+	0.453851i
$552$		0			0
$553$	−28.8345	+	3.94591i	−1.22617	+	0.167797i
$554$		0			0
$555$	1.62132	−	2.80821i	0.0688212	−	0.119202i
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	10.5147			0.444725
$560$		0			0
$561$	−18.0000			−0.759961
$562$		0			0
$563$	3.00000	+	5.19615i	0.126435	+	0.218992i	0.922293	−	0.386492i	$-0.126313\pi$
$563$	3.00000	+	5.19615i	0.126435	+	0.218992i	−0.795858	+	0.605483i	$0.792980\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$	1.00000	−	2.44949i	0.0419961	−	0.102869i
$568$		0			0
$569$	20.8492	−	36.1119i	0.874046	−	1.51389i	0.0162699	−	0.999868i	$-0.494821\pi$
$569$	20.8492	−	36.1119i	0.874046	−	1.51389i	0.857776	−	0.514024i	$-0.171846\pi$
$570$		0			0
$571$	14.4706	+	25.0637i	0.605574	+	1.04889i	0.991960	+	0.126548i	$0.0403898\pi$
$571$	14.4706	+	25.0637i	0.605574	+	1.04889i	−0.386386	+	0.922337i	$0.626277\pi$
$572$		0			0
$573$	6.00000			0.250654
$574$		0			0
$575$	−4.24264			−0.176930
$576$		0			0
$577$	−6.37868	−	11.0482i	−0.265548	−	0.459942i	0.702159	−	0.712020i	$-0.252220\pi$
$577$	−6.37868	−	11.0482i	−0.265548	−	0.459942i	−0.967707	+	0.252078i	$0.918886\pi$
$578$		0			0
$579$	−3.37868	+	5.85204i	−0.140413	+	0.243203i
$580$		0			0
$581$	−16.6066	−	21.4150i	−0.688958	−	0.888444i
$582$		0			0
$583$	18.0000	−	31.1769i	0.745484	−	1.29122i
$584$		0			0
$585$	1.62132	+	2.80821i	0.0670333	+	0.116105i
$586$		0			0
$587$	45.2132			1.86615			0.933074	−	0.359684i	$-0.117115\pi$
$587$	45.2132			1.86615			0.933074	+	0.359684i	$0.117115\pi$
$588$		0			0
$589$	66.3970			2.73584
$590$		0			0
$591$	8.12132	+	14.0665i	0.334066	+	0.578620i
$592$		0			0
$593$	−1.60660	+	2.78272i	−0.0659752	+	0.114272i	−0.897126	−	0.441774i	$-0.854349\pi$
$593$	−1.60660	+	2.78272i	−0.0659752	+	0.114272i	0.831151	+	0.556047i	$0.187682\pi$
$594$		0			0
$595$	−6.87868	−	8.87039i	−0.281998	−	0.363650i
$596$		0			0
$597$	−5.24264	+	9.08052i	−0.214567	+	0.371641i
$598$		0			0
$599$	16.2426	+	28.1331i	0.663656	+	1.14949i	0.979648	+	0.200724i	$0.0643296\pi$
$599$	16.2426	+	28.1331i	0.663656	+	1.14949i	−0.315991	+	0.948762i	$0.602337\pi$
$600$		0			0
$601$	−3.48528			−0.142168			−0.0710838	−	0.997470i	$-0.522646\pi$
$601$	−3.48528			−0.142168			−0.0710838	+	0.997470i	$0.522646\pi$
$602$		0			0
$603$	−5.24264			−0.213497
$604$		0			0
$605$	3.50000	+	6.06218i	0.142295	+	0.246463i
$606$		0			0
$607$	14.6213	−	25.3249i	0.593461	−	1.02790i	−0.400301	−	0.916384i	$-0.631094\pi$
$607$	14.6213	−	25.3249i	0.593461	−	1.02790i	0.993762	−	0.111521i	$-0.0355722\pi$
$608$		0			0
$609$	−1.75736	+	4.30463i	−0.0712118	+	0.174433i
$610$		0			0
$611$	9.72792	−	16.8493i	0.393550	−	0.681648i
$612$		0			0
$613$	2.72792	+	4.72490i	0.110180	+	0.190837i	0.915843	−	0.401537i	$-0.131524\pi$
$613$	2.72792	+	4.72490i	0.110180	+	0.190837i	−0.805663	+	0.592374i	$0.798191\pi$
$614$		0			0
$615$	4.24264			0.171080
$616$		0			0
$617$	−26.4853			−1.06626			−0.533129	−	0.846034i	$-0.678984\pi$
$617$	−26.4853			−1.06626			−0.533129	+	0.846034i	$0.678984\pi$
$618$		0			0
$619$	5.98528	+	10.3668i	0.240569	+	0.416677i	0.960876	−	0.276977i	$-0.0893327\pi$
$619$	5.98528	+	10.3668i	0.240569	+	0.416677i	−0.720308	+	0.693655i	$0.755999\pi$
$620$		0			0
$621$	2.12132	−	3.67423i	0.0851257	−	0.147442i
$622$		0			0
$623$	26.8492	−	3.67423i	1.07569	−	0.147205i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−14.8492	−	25.7196i	−0.593022	−	1.02714i
$628$		0			0
$629$	13.7574			0.548542
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$	−11.2426	−	19.4728i	−0.446855	−	0.773975i
$634$		0			0
$635$	−1.37868	+	2.38794i	−0.0547112	+	0.0947626i
$636$		0			0
$637$	−5.65076	+	21.9839i	−0.223891	+	0.871032i
$638$		0			0
$639$	6.36396	−	11.0227i	0.251754	−	0.436051i
$640$		0			0
$641$	−17.1213	−	29.6550i	−0.676251	−	1.17130i	−0.976101	−	0.217316i	$-0.930270\pi$
$641$	−17.1213	−	29.6550i	−0.676251	−	1.17130i	0.299850	−	0.953986i	$-0.403063\pi$
$642$		0			0
$643$	−19.7279			−0.777993			−0.388997	−	0.921239i	$-0.627178\pi$
$643$	−19.7279			−0.777993			−0.388997	+	0.921239i	$0.627178\pi$
$644$		0			0
$645$	−3.24264			−0.127679
$646$		0			0
$647$	−5.12132	−	8.87039i	−0.201340	−	0.348731i	0.747621	−	0.664126i	$-0.231196\pi$
$647$	−5.12132	−	8.87039i	−0.201340	−	0.348731i	−0.948960	+	0.315395i	$0.897863\pi$
$648$		0			0
$649$	−21.7279	+	37.6339i	−0.852896	+	1.47726i
$650$		0			0
$651$	24.8640	−	3.40256i	0.974495	−	0.133357i
$652$		0			0
$653$	−5.12132	+	8.87039i	−0.200413	+	0.347125i	−0.948661	−	0.316293i	$-0.897562\pi$
$653$	−5.12132	+	8.87039i	−0.200413	+	0.347125i	0.748249	+	0.663418i	$0.230895\pi$
$654$		0			0
$655$	7.24264	+	12.5446i	0.282993	+	0.490159i
$656$		0			0
$657$	9.24264			0.360590
$658$		0			0
$659$	40.9706			1.59599			0.797993	−	0.602666i	$-0.205895\pi$
$659$	40.9706			1.59599			0.797993	+	0.602666i	$0.205895\pi$
$660$		0			0
$661$	1.01472	+	1.75754i	0.0394680	+	0.0683605i	0.885085	−	0.465430i	$-0.154100\pi$
$661$	1.01472	+	1.75754i	0.0394680	+	0.0683605i	−0.845617	+	0.533791i	$0.820767\pi$
$662$		0			0
$663$	−6.87868	+	11.9142i	−0.267146	+	0.462710i
$664$		0			0
$665$	7.00000	−	17.1464i	0.271448	−	0.664910i
$666$		0			0
$667$	−3.72792	+	6.45695i	−0.144346	+	0.250014i
$668$		0			0
$669$	3.75736	+	6.50794i	0.145268	+	0.251611i
$670$		0			0
$671$	−19.0294			−0.734623
$672$		0			0
$673$	29.7279			1.14593			0.572964	−	0.819581i	$-0.305794\pi$
$673$	29.7279			1.14593			0.572964	+	0.819581i	$0.305794\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.0192450	−	0.0333333i
$676$		0			0
$677$	−6.36396	+	11.0227i	−0.244587	+	0.423637i	−0.962015	−	0.272995i	$-0.911986\pi$
$677$	−6.36396	+	11.0227i	−0.244587	+	0.423637i	0.717428	+	0.696632i	$0.245319\pi$
$678$		0			0
$679$	−0.786797	−	1.01461i	−0.0301945	−	0.0389372i
$680$		0			0
$681$	7.60660	−	13.1750i	0.291486	−	0.504868i
$682$		0			0
$683$	−16.6066	−	28.7635i	−0.635434	−	1.10060i	−0.986423	−	0.164224i	$-0.947488\pi$
$683$	−16.6066	−	28.7635i	−0.635434	−	1.10060i	0.350989	−	0.936380i	$-0.385845\pi$
$684$		0			0
$685$	−4.24264			−0.162103
$686$		0			0
$687$	−7.00000			−0.267067
$688$		0			0
$689$	−13.7574	−	23.8284i	−0.524114	−	0.907791i
$690$		0			0
$691$	−13.4706	+	23.3317i	−0.512444	+	0.887580i	0.487452	+	0.873150i	$0.337927\pi$
$691$	−13.4706	+	23.3317i	−0.512444	+	0.887580i	−0.999896	+	0.0144296i	$0.995407\pi$
$692$		0			0
$693$	−6.87868	−	8.87039i	−0.261299	−	0.336958i
$694$		0			0
$695$	−7.74264	+	13.4106i	−0.293695	+	0.508695i
$696$		0			0
$697$	9.00000	+	15.5885i	0.340899	+	0.590455i
$698$		0			0
$699$	14.4853			0.547884
$700$		0			0
$701$	8.78680			0.331873			0.165936	−	0.986136i	$-0.446935\pi$
$701$	8.78680			0.331873			0.165936	+	0.986136i	$0.446935\pi$
$702$		0			0
$703$	11.3492	+	19.6575i	0.428045	+	0.741395i
$704$		0			0
$705$	−3.00000	+	5.19615i	−0.112987	+	0.195698i
$706$		0			0
$707$	−7.75736	+	19.0016i	−0.291746	+	0.714628i
$708$		0			0
$709$	18.2426	−	31.5972i	0.685117	−	1.18666i	−0.288283	−	0.957545i	$-0.593085\pi$
$709$	18.2426	−	31.5972i	0.685117	−	1.18666i	0.973400	−	0.229112i	$-0.0735822\pi$
$710$		0			0
$711$	−5.50000	−	9.52628i	−0.206266	−	0.357263i
$712$		0			0
$713$	40.2426			1.50710
$714$		0			0
$715$	13.7574			0.514496
$716$		0			0
$717$	−5.48528	−	9.50079i	−0.204852	−	0.354813i
$718$		0			0
$719$	−13.2426	+	22.9369i	−0.493867	+	0.855403i	−0.999975	−	0.00706717i	$-0.997750\pi$
$719$	−13.2426	+	22.9369i	−0.493867	+	0.855403i	0.506108	+	0.862470i	$0.331084\pi$
$720$		0			0
$721$	45.1985	−	6.18527i	1.68328	−	0.230352i
$722$		0			0
$723$	2.00000	−	3.46410i	0.0743808	−	0.128831i
$724$		0			0
$725$	0.878680	+	1.52192i	0.0326333	+	0.0565226i
$726$		0			0
$727$	0.757359			0.0280889			0.0140445	−	0.999901i	$-0.495529\pi$
$727$	0.757359			0.0280889			0.0140445	+	0.999901i	$0.495529\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−6.87868	−	11.9142i	−0.254417	−	0.440663i
$732$		0			0
$733$	−0.893398	+	1.54741i	−0.0329984	+	0.0571549i	−0.882053	−	0.471150i	$-0.843839\pi$
$733$	−0.893398	+	1.54741i	−0.0329984	+	0.0571549i	0.849055	+	0.528305i	$0.177172\pi$
$734$		0			0
$735$	1.74264	−	6.77962i	0.0642783	−	0.250070i
$736$		0			0
$737$	−11.1213	+	19.2627i	−0.409659	+	0.709550i
$738$		0			0
$739$	7.74264	+	13.4106i	0.284818	+	0.493319i	0.972565	−	0.232632i	$-0.0747336\pi$
$739$	7.74264	+	13.4106i	0.284818	+	0.493319i	−0.687747	+	0.725950i	$0.741400\pi$
$740$		0			0
$741$	−22.6985			−0.833850
$742$		0			0
$743$	15.2132			0.558118			0.279059	−	0.960274i	$-0.409977\pi$
$743$	15.2132			0.558118			0.279059	+	0.960274i	$0.409977\pi$
$744$		0			0
$745$	−6.00000	−	10.3923i	−0.219823	−	0.380745i
$746$		0			0
$747$	5.12132	−	8.87039i	0.187379	−	0.324550i
$748$		0			0
$749$	−33.3640	+	4.56575i	−1.21909	+	0.166829i
$750$		0			0
$751$	−22.4706	+	38.9202i	−0.819962	+	1.42022i	0.0857467	+	0.996317i	$0.472672\pi$
$751$	−22.4706	+	38.9202i	−0.819962	+	1.42022i	−0.905709	+	0.423900i	$0.860661\pi$
$752$		0			0
$753$	9.36396	+	16.2189i	0.341242	+	0.591048i
$754$		0			0
$755$	−22.4853			−0.818323
$756$		0			0
$757$	9.02944			0.328180			0.164090	−	0.986445i	$-0.447531\pi$
$757$	9.02944			0.328180			0.164090	+	0.986445i	$0.447531\pi$
$758$		0			0
$759$	−9.00000	−	15.5885i	−0.326679	−	0.565825i
$760$		0			0
$761$	−23.1213	+	40.0473i	−0.838147	+	1.45171i	0.0532948	+	0.998579i	$0.483028\pi$
$761$	−23.1213	+	40.0473i	−0.838147	+	1.45171i	−0.891442	+	0.453135i	$0.850306\pi$
$762$		0			0
$763$	−9.48528	+	23.2341i	−0.343390	+	0.841131i
$764$		0			0
$765$	2.12132	−	3.67423i	0.0766965	−	0.132842i
$766$		0			0
$767$	16.6066	+	28.7635i	0.599630	+	1.03859i
$768$		0			0
$769$	5.00000			0.180305			0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000			0.180305			0.0901523	+	0.995928i	$0.471265\pi$
$770$		0			0
$771$	10.2426			0.368880
$772$		0			0
$773$	−5.84924	−	10.1312i	−0.210383	−	0.364393i	0.741452	−	0.671006i	$-0.234138\pi$
$773$	−5.84924	−	10.1312i	−0.210383	−	0.364393i	−0.951834	+	0.306613i	$0.900804\pi$
$774$		0			0
$775$	4.74264	−	8.21449i	0.170361	−	0.295073i
$776$		0			0
$777$	5.25736	+	6.77962i	0.188607	+	0.243217i
$778$		0			0
$779$	−14.8492	+	25.7196i	−0.532029	+	0.921502i
$780$		0			0
$781$	−27.0000	−	46.7654i	−0.966136	−	1.67340i
$782$		0			0
$783$	−1.75736			−0.0628029
$784$		0			0
$785$	6.48528			0.231470
$786$		0			0
$787$	−14.2426	−	24.6690i	−0.507695	−	0.879354i	−0.999960	−	0.00890869i	$-0.997164\pi$
$787$	−14.2426	−	24.6690i	−0.507695	−	0.879354i	0.492265	−	0.870445i	$-0.336169\pi$
$788$		0			0
$789$	4.24264	−	7.34847i	0.151042	−	0.261612i
$790$		0			0
$791$	−29.1838	−	37.6339i	−1.03766	−	1.33811i
$792$		0			0
$793$	−7.27208	+	12.5956i	−0.258239	+	0.447283i
$794$		0			0
$795$	4.24264	+	7.34847i

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.62132 + 2.09077i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.62132 + 2.09077i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.12132 + 3.67423i) q^{11} +3.24264 q^{13} -1.00000 q^{15} +(-2.12132 - 3.67423i) q^{17} +(3.50000 - 6.06218i) q^{19} +(1.00000 - 2.44949i) q^{21} +(2.12132 - 3.67423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -1.75736 q^{29} +(4.74264 + 8.21449i) q^{31} +(2.12132 - 3.67423i) q^{33} +(2.62132 - 0.358719i) q^{35} +(-1.62132 + 2.80821i) q^{37} +(-1.62132 - 2.80821i) q^{39} -4.24264 q^{41} +3.24264 q^{43} +(0.500000 + 0.866025i) q^{45} +(3.00000 - 5.19615i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(-2.12132 + 3.67423i) q^{51} +(-4.24264 - 7.34847i) q^{53} +4.24264 q^{55} -7.00000 q^{57} +(5.12132 + 8.87039i) q^{59} +(-2.24264 + 3.88437i) q^{61} +(-2.62132 + 0.358719i) q^{63} +(1.62132 - 2.80821i) q^{65} +(2.62132 + 4.54026i) q^{67} -4.24264 q^{69} -12.7279 q^{71} +(-4.62132 - 8.00436i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-4.24264 + 10.3923i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-0.500000 - 0.866025i) q^{81} -10.2426 q^{83} -4.24264 q^{85} +(0.878680 + 1.52192i) q^{87} +(5.12132 - 8.87039i) q^{89} +(5.25736 + 6.77962i) q^{91} +(4.74264 - 8.21449i) q^{93} +(-3.50000 - 6.06218i) q^{95} -0.485281 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 - 2 * q^9	\( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{15} + 14 q^{19} + 4 q^{21} - 2 q^{25} + 4 q^{27} - 24 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 28 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} - 2 q^{65} + 2 q^{67} - 10 q^{73} - 2 q^{75} - 22 q^{79} - 2 q^{81} - 24 q^{83} + 12 q^{87} + 12 q^{89} + 38 q^{91} + 2 q^{93} - 14 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 - 2 * q^9 - 4 * q^13 - 4 * q^15 + 14 * q^19 + 4 * q^21 - 2 * q^25 + 4 * q^27 - 24 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^43 + 2 * q^45 + 12 * q^47 + 10 * q^49 - 28 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^63 - 2 * q^65 + 2 * q^67 - 10 * q^73 - 2 * q^75 - 22 * q^79 - 2 * q^81 - 24 * q^83 + 12 * q^87 + 12 * q^89 + 38 * q^91 + 2 * q^93 - 14 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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