LMFDB - Embedded newform 370.2.b.c.149.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(149,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("370.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.b (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:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.3
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	370.149
Dual form			370.2.b.c.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.44949 q^{6} -0.449490i q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.44949 q^{6} -0.449490i q^{7} -1.00000i q^{8} -3.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} -4.89898 q^{11} +2.44949i q^{12} -4.00000i q^{13} +0.449490 q^{14} +(2.44949 + 4.89898i) q^{15} +1.00000 q^{16} +4.89898i q^{17} -3.00000i q^{18} -8.44949 q^{19} +(2.00000 - 1.00000i) q^{20} -1.10102 q^{21} -4.89898i q^{22} -0.898979i q^{23} -2.44949 q^{24} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +0.449490i q^{28} +(-4.89898 + 2.44949i) q^{30} -6.44949 q^{31} +1.00000i q^{32} +12.0000i q^{33} -4.89898 q^{34} +(0.449490 + 0.898979i) q^{35} +3.00000 q^{36} +1.00000i q^{37} -8.44949i q^{38} -9.79796 q^{39} +(1.00000 + 2.00000i) q^{40} +2.00000 q^{41} -1.10102i q^{42} -4.00000i q^{43} +4.89898 q^{44} +(6.00000 - 3.00000i) q^{45} +0.898979 q^{46} -0.449490i q^{47} -2.44949i q^{48} +6.79796 q^{49} +(4.00000 + 3.00000i) q^{50} +12.0000 q^{51} +4.00000i q^{52} -7.79796i q^{53} +(9.79796 - 4.89898i) q^{55} -0.449490 q^{56} +20.6969i q^{57} +8.44949 q^{59} +(-2.44949 - 4.89898i) q^{60} +12.0000 q^{61} -6.44949i q^{62} +1.34847i q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -12.0000 q^{66} -10.4495i q^{67} -4.89898i q^{68} -2.20204 q^{69} +(-0.898979 + 0.449490i) q^{70} -4.89898 q^{71} +3.00000i q^{72} -4.00000i q^{73} -1.00000 q^{74} +(-9.79796 - 7.34847i) q^{75} +8.44949 q^{76} +2.20204i q^{77} -9.79796i q^{78} -1.55051 q^{79} +(-2.00000 + 1.00000i) q^{80} -9.00000 q^{81} +2.00000i q^{82} +14.4495i q^{83} +1.10102 q^{84} +(-4.89898 - 9.79796i) q^{85} +4.00000 q^{86} +4.89898i q^{88} +3.79796 q^{89} +(3.00000 + 6.00000i) q^{90} -1.79796 q^{91} +0.898979i q^{92} +15.7980i q^{93} +0.449490 q^{94} +(16.8990 - 8.44949i) q^{95} +2.44949 q^{96} -2.00000i q^{97} +6.79796i q^{98} +14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9	$ 4 q - 4 q^{4} - 8 q^{5} - 12 q^{9} - 4 q^{10} - 8 q^{14} + 4 q^{16} - 24 q^{19} + 8 q^{20} - 24 q^{21} + 12 q^{25} + 16 q^{26} - 16 q^{31} - 8 q^{35} + 12 q^{36} + 4 q^{40} + 8 q^{41} + 24 q^{45} - 16 q^{46} - 12 q^{49} + 16 q^{50} + 48 q^{51} + 8 q^{56} + 24 q^{59} + 48 q^{61} - 4 q^{64} + 16 q^{65} - 48 q^{66} - 48 q^{69} + 16 q^{70} - 4 q^{74} + 24 q^{76} - 16 q^{79} - 8 q^{80} - 36 q^{81} + 24 q^{84} + 16 q^{86} - 24 q^{89} + 12 q^{90} + 32 q^{91} - 8 q^{94} + 48 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9 - 4 * q^10 - 8 * q^14 + 4 * q^16 - 24 * q^19 + 8 * q^20 - 24 * q^21 + 12 * q^25 + 16 * q^26 - 16 * q^31 - 8 * q^35 + 12 * q^36 + 4 * q^40 + 8 * q^41 + 24 * q^45 - 16 * q^46 - 12 * q^49 + 16 * q^50 + 48 * q^51 + 8 * q^56 + 24 * q^59 + 48 * q^61 - 4 * q^64 + 16 * q^65 - 48 * q^66 - 48 * q^69 + 16 * q^70 - 4 * q^74 + 24 * q^76 - 16 * q^79 - 8 * q^80 - 36 * q^81 + 24 * q^84 + 16 * q^86 - 24 * q^89 + 12 * q^90 + 32 * q^91 - 8 * q^94 + 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$.

$n$	$261$	$297$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	2.44949i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$		−	2.44949i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$	2.44949			1.00000
$7$		−	0.449490i		−	0.169891i	−0.996386	−	0.0849456i	$-0.972928\pi$
$7$		−	0.449490i		−	0.169891i	0.996386	−	0.0849456i	$-0.0270716\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−3.00000			−1.00000
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$	−4.89898			−1.47710			−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898			−1.47710			−0.738549	+	0.674200i	$0.764489\pi$
$12$			2.44949i			0.707107i
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0.449490			0.120131
$15$	2.44949	+	4.89898i	0.632456	+	1.26491i
$16$	1.00000			0.250000
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		−	3.00000i		−	0.707107i
$19$	−8.44949			−1.93845			−0.969223	−	0.246185i	$-0.920823\pi$
$19$	−8.44949			−1.93845			−0.969223	+	0.246185i	$0.920823\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$	−1.10102			−0.240262
$22$		−	4.89898i		−	1.04447i
$23$		−	0.898979i		−	0.187450i	−0.995598	−	0.0937251i	$-0.970123\pi$
$23$		−	0.898979i		−	0.187450i	0.995598	−	0.0937251i	$-0.0298775\pi$
$24$	−2.44949			−0.500000
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	4.00000			0.784465
$27$		0			0
$28$			0.449490i			0.0849456i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−4.89898	+	2.44949i	−0.894427	+	0.447214i
$31$	−6.44949			−1.15836			−0.579181	−	0.815199i	$-0.696628\pi$
$31$	−6.44949			−1.15836			−0.579181	+	0.815199i	$0.696628\pi$
$32$			1.00000i			0.176777i
$33$			12.0000i			2.08893i
$34$	−4.89898			−0.840168
$35$	0.449490	+	0.898979i	0.0759776	+	0.151955i
$36$	3.00000			0.500000
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		−	8.44949i		−	1.37069i
$39$	−9.79796			−1.56893
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		−	1.10102i		−	0.169891i
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.89898			0.738549
$45$	6.00000	−	3.00000i	0.894427	−	0.447214i
$46$	0.898979			0.132547
$47$		−	0.449490i		−	0.0655648i	−0.999463	−	0.0327824i	$-0.989563\pi$
$47$		−	0.449490i		−	0.0655648i	0.999463	−	0.0327824i	$-0.0104368\pi$
$48$		−	2.44949i		−	0.353553i
$49$	6.79796			0.971137
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$	12.0000			1.68034
$52$			4.00000i			0.554700i
$53$		−	7.79796i		−	1.07113i	−0.844493	−	0.535566i	$-0.820098\pi$
$53$		−	7.79796i		−	1.07113i	0.844493	−	0.535566i	$-0.179902\pi$
$54$		0			0
$55$	9.79796	−	4.89898i	1.32116	−	0.660578i
$56$	−0.449490			−0.0600656
$57$			20.6969i			2.74138i
$58$		0			0
$59$	8.44949			1.10003			0.550015	−	0.835155i	$-0.314622\pi$
$59$	8.44949			1.10003			0.550015	+	0.835155i	$0.314622\pi$
$60$	−2.44949	−	4.89898i	−0.316228	−	0.632456i
$61$	12.0000			1.53644			0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000			1.53644			0.768221	+	0.640184i	$0.221142\pi$
$62$		−	6.44949i		−	0.819086i
$63$			1.34847i			0.169891i
$64$	−1.00000			−0.125000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$	−12.0000			−1.47710
$67$		−	10.4495i		−	1.27661i	−0.769784	−	0.638304i	$-0.779636\pi$
$67$		−	10.4495i		−	1.27661i	0.769784	−	0.638304i	$-0.220364\pi$
$68$		−	4.89898i		−	0.594089i
$69$	−2.20204			−0.265095
$70$	−0.898979	+	0.449490i	−0.107449	+	0.0537243i
$71$	−4.89898			−0.581402			−0.290701	−	0.956814i	$-0.593888\pi$
$71$	−4.89898			−0.581402			−0.290701	+	0.956814i	$0.593888\pi$
$72$			3.00000i			0.353553i
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−1.00000			−0.116248
$75$	−9.79796	−	7.34847i	−1.13137	−	0.848528i
$76$	8.44949			0.969223
$77$			2.20204i			0.250946i
$78$		−	9.79796i		−	1.10940i
$79$	−1.55051			−0.174446			−0.0872230	−	0.996189i	$-0.527799\pi$
$79$	−1.55051			−0.174446			−0.0872230	+	0.996189i	$0.527799\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$	−9.00000			−1.00000
$82$			2.00000i			0.220863i
$83$			14.4495i			1.58604i	0.609197	+	0.793019i	$0.291492\pi$
$83$			14.4495i			1.58604i	−0.609197	+	0.793019i	$0.708508\pi$
$84$	1.10102			0.120131
$85$	−4.89898	−	9.79796i	−0.531369	−	1.06274i
$86$	4.00000			0.431331
$87$		0			0
$88$			4.89898i			0.522233i
$89$	3.79796			0.402583			0.201291	−	0.979531i	$-0.435486\pi$
$89$	3.79796			0.402583			0.201291	+	0.979531i	$0.435486\pi$
$90$	3.00000	+	6.00000i	0.316228	+	0.632456i
$91$	−1.79796			−0.188477
$92$			0.898979i			0.0937251i
$93$			15.7980i			1.63817i
$94$	0.449490			0.0463613
$95$	16.8990	−	8.44949i	1.73380	−	0.866899i
$96$	2.44949			0.250000
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$			6.79796i			0.686698i
$99$	14.6969			1.47710
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−18.6969			−1.86041			−0.930207	−	0.367034i	$-0.880373\pi$
$101$	−18.6969			−1.86041			−0.930207	+	0.367034i	$0.880373\pi$
$102$			12.0000i			1.18818i
$103$			9.79796i			0.965422i	0.875780	+	0.482711i	$0.160348\pi$
$103$			9.79796i			0.965422i	−0.875780	+	0.482711i	$0.839652\pi$
$104$	−4.00000			−0.392232
$105$	2.20204	−	1.10102i	0.214897	−	0.107449i
$106$	7.79796			0.757405
$107$		−	10.4495i		−	1.01019i	−0.863064	−	0.505095i	$-0.831457\pi$
$107$		−	10.4495i		−	1.01019i	0.863064	−	0.505095i	$-0.168543\pi$
$108$		0			0
$109$	−13.7980			−1.32160			−0.660802	−	0.750560i	$-0.729784\pi$
$109$	−13.7980			−1.32160			−0.660802	+	0.750560i	$0.729784\pi$
$110$	4.89898	+	9.79796i	0.467099	+	0.934199i
$111$	2.44949			0.232495
$112$		−	0.449490i		−	0.0424728i
$113$			12.8990i			1.21343i	0.794918	+	0.606717i	$0.207514\pi$
$113$			12.8990i			1.21343i	−0.794918	+	0.606717i	$0.792486\pi$
$114$	−20.6969			−1.93845
$115$	0.898979	+	1.79796i	0.0838303	+	0.167661i
$116$		0			0
$117$			12.0000i			1.10940i
$118$			8.44949i			0.777839i
$119$	2.20204			0.201861
$120$	4.89898	−	2.44949i	0.447214	−	0.223607i
$121$	13.0000			1.18182
$122$			12.0000i			1.08643i
$123$		−	4.89898i		−	0.441726i
$124$	6.44949			0.579181
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$	−1.34847			−0.120131
$127$		−	17.3485i		−	1.53943i	−0.638389	−	0.769714i	$-0.720399\pi$
$127$		−	17.3485i		−	1.53943i	0.638389	−	0.769714i	$-0.279601\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−9.79796			−0.862662
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	14.2474			1.24481			0.622403	−	0.782697i	$-0.286157\pi$
$131$	14.2474			1.24481			0.622403	+	0.782697i	$0.286157\pi$
$132$		−	12.0000i		−	1.04447i
$133$			3.79796i			0.329325i
$134$	10.4495			0.902698
$135$		0			0
$136$	4.89898			0.420084
$137$		−	19.5959i		−	1.67419i	−0.547056	−	0.837096i	$-0.684251\pi$
$137$		−	19.5959i		−	1.67419i	0.547056	−	0.837096i	$-0.315749\pi$
$138$		−	2.20204i		−	0.187450i
$139$	−13.7980			−1.17033			−0.585164	−	0.810915i	$-0.698970\pi$
$139$	−13.7980			−1.17033			−0.585164	+	0.810915i	$0.698970\pi$
$140$	−0.449490	−	0.898979i	−0.0379888	−	0.0759776i
$141$	−1.10102			−0.0927227
$142$		−	4.89898i		−	0.411113i
$143$			19.5959i			1.63869i
$144$	−3.00000			−0.250000
$145$		0			0
$146$	4.00000			0.331042
$147$		−	16.6515i		−	1.37340i
$148$		−	1.00000i		−	0.0821995i
$149$	−10.6969			−0.876327			−0.438164	−	0.898895i	$-0.644371\pi$
$149$	−10.6969			−0.876327			−0.438164	+	0.898895i	$0.644371\pi$
$150$	7.34847	−	9.79796i	0.600000	−	0.800000i
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$			8.44949i			0.685344i
$153$		−	14.6969i		−	1.18818i
$154$	−2.20204			−0.177446
$155$	12.8990	−	6.44949i	1.03607	−	0.518035i
$156$	9.79796			0.784465
$157$			11.7980i			0.941580i	0.882245	+	0.470790i	$0.156031\pi$
$157$			11.7980i			0.941580i	−0.882245	+	0.470790i	$0.843969\pi$
$158$		−	1.55051i		−	0.123352i
$159$	−19.1010			−1.51481
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$	−0.404082			−0.0318461
$162$		−	9.00000i		−	0.707107i
$163$			2.20204i			0.172477i	0.996275	+	0.0862386i	$0.0274847\pi$
$163$			2.20204i			0.172477i	−0.996275	+	0.0862386i	$0.972515\pi$
$164$	−2.00000			−0.156174
$165$	−12.0000	−	24.0000i	−0.934199	−	1.86840i
$166$	−14.4495			−1.12150
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$			1.10102i			0.0849456i
$169$	−3.00000			−0.230769
$170$	9.79796	−	4.89898i	0.751469	−	0.375735i
$171$	25.3485			1.93845
$172$			4.00000i			0.304997i
$173$		−	0.202041i		−	0.0153609i	−0.999971	−	0.00768045i	$-0.997555\pi$
$173$		−	0.202041i		−	0.0153609i	0.999971	−	0.00768045i	$-0.00244479\pi$
$174$		0			0
$175$	−1.79796	−	1.34847i	−0.135913	−	0.101935i
$176$	−4.89898			−0.369274
$177$		−	20.6969i		−	1.55568i
$178$			3.79796i			0.284669i
$179$	−5.34847			−0.399763			−0.199882	−	0.979820i	$-0.564056\pi$
$179$	−5.34847			−0.399763			−0.199882	+	0.979820i	$0.564056\pi$
$180$	−6.00000	+	3.00000i	−0.447214	+	0.223607i
$181$	−18.6969			−1.38973			−0.694866	−	0.719139i	$-0.744536\pi$
$181$	−18.6969			−1.38973			−0.694866	+	0.719139i	$0.744536\pi$
$182$		−	1.79796i		−	0.133274i
$183$		−	29.3939i		−	2.17286i
$184$	−0.898979			−0.0662736
$185$	−1.00000	−	2.00000i	−0.0735215	−	0.147043i
$186$	−15.7980			−1.15836
$187$		−	24.0000i		−	1.75505i
$188$			0.449490i			0.0327824i
$189$		0			0
$190$	8.44949	+	16.8990i	0.612990	+	1.22598i
$191$	−3.34847			−0.242287			−0.121143	−	0.992635i	$-0.538656\pi$
$191$	−3.34847			−0.242287			−0.121143	+	0.992635i	$0.538656\pi$
$192$			2.44949i			0.176777i
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	2.00000			0.143592
$195$	19.5959	−	9.79796i	1.40329	−	0.701646i
$196$	−6.79796			−0.485568
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$			14.6969i			1.04447i
$199$	−1.55051			−0.109913			−0.0549564	−	0.998489i	$-0.517502\pi$
$199$	−1.55051			−0.109913			−0.0549564	+	0.998489i	$0.517502\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$	−25.5959			−1.80540
$202$		−	18.6969i		−	1.31551i
$203$		0			0
$204$	−12.0000			−0.840168
$205$	−4.00000	+	2.00000i	−0.279372	+	0.139686i
$206$	−9.79796			−0.682656
$207$			2.69694i			0.187450i
$208$		−	4.00000i		−	0.277350i
$209$	41.3939			2.86327
$210$	1.10102	+	2.20204i	0.0759776	+	0.151955i
$211$	22.6969			1.56252			0.781261	−	0.624205i	$-0.214577\pi$
$211$	22.6969			1.56252			0.781261	+	0.624205i	$0.214577\pi$
$212$			7.79796i			0.535566i
$213$			12.0000i			0.822226i
$214$	10.4495			0.714312
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			2.89898i			0.196796i
$218$		−	13.7980i		−	0.934516i
$219$	−9.79796			−0.662085
$220$	−9.79796	+	4.89898i	−0.660578	+	0.330289i
$221$	19.5959			1.31816
$222$			2.44949i			0.164399i
$223$			4.44949i			0.297960i	0.988840	+	0.148980i	$0.0475990\pi$
$223$			4.44949i			0.297960i	−0.988840	+	0.148980i	$0.952401\pi$
$224$	0.449490			0.0300328
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$	−12.8990			−0.858027
$227$		−	15.1010i		−	1.00229i	−0.865363	−	0.501145i	$-0.832912\pi$
$227$		−	15.1010i		−	1.00229i	0.865363	−	0.501145i	$-0.167088\pi$
$228$		−	20.6969i		−	1.37069i
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	−1.79796	+	0.898979i	−0.118554	+	0.0592770i
$231$	5.39388			0.354891
$232$		0			0
$233$			9.79796i			0.641886i	0.947099	+	0.320943i	$0.104000\pi$
$233$			9.79796i			0.641886i	−0.947099	+	0.320943i	$0.896000\pi$
$234$	−12.0000			−0.784465
$235$	0.449490	+	0.898979i	0.0293215	+	0.0586430i
$236$	−8.44949			−0.550015
$237$			3.79796i			0.246704i
$238$			2.20204i			0.142737i
$239$	−26.0454			−1.68474			−0.842369	−	0.538902i	$-0.818839\pi$
$239$	−26.0454			−1.68474			−0.842369	+	0.538902i	$0.818839\pi$
$240$	2.44949	+	4.89898i	0.158114	+	0.316228i
$241$	−11.7980			−0.759973			−0.379987	−	0.924992i	$-0.624071\pi$
$241$	−11.7980			−0.759973			−0.379987	+	0.924992i	$0.624071\pi$
$242$			13.0000i			0.835672i
$243$			22.0454i			1.41421i
$244$	−12.0000			−0.768221
$245$	−13.5959	+	6.79796i	−0.868611	+	0.434306i
$246$	4.89898			0.312348
$247$			33.7980i			2.15051i
$248$			6.44949i			0.409543i
$249$	35.3939			2.24300
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	6.65153			0.419841			0.209920	−	0.977718i	$-0.432679\pi$
$251$	6.65153			0.419841			0.209920	+	0.977718i	$0.432679\pi$
$252$		−	1.34847i		−	0.0849456i
$253$			4.40408i			0.276882i
$254$	17.3485			1.08854
$255$	−24.0000	+	12.0000i	−1.50294	+	0.751469i
$256$	1.00000			0.0625000
$257$			4.89898i			0.305590i	0.988258	+	0.152795i	$0.0488274\pi$
$257$			4.89898i			0.305590i	−0.988258	+	0.152795i	$0.951173\pi$
$258$		−	9.79796i		−	0.609994i
$259$	0.449490			0.0279299
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$		0			0
$262$			14.2474i			0.880210i
$263$		−	26.2474i		−	1.61849i	−0.587473	−	0.809244i	$-0.699877\pi$
$263$		−	26.2474i		−	1.61849i	0.587473	−	0.809244i	$-0.300123\pi$
$264$	12.0000			0.738549
$265$	7.79796	+	15.5959i	0.479025	+	0.958050i
$266$	−3.79796			−0.232868
$267$		−	9.30306i		−	0.569338i
$268$			10.4495i			0.638304i
$269$	−10.6969			−0.652204			−0.326102	−	0.945335i	$-0.605735\pi$
$269$	−10.6969			−0.652204			−0.326102	+	0.945335i	$0.605735\pi$
$270$		0			0
$271$	16.4949			1.00199			0.500997	−	0.865449i	$-0.332967\pi$
$271$	16.4949			1.00199			0.500997	+	0.865449i	$0.332967\pi$
$272$			4.89898i			0.297044i
$273$			4.40408i			0.266547i
$274$	19.5959			1.18383
$275$	−14.6969	+	19.5959i	−0.886259	+	1.18168i
$276$	2.20204			0.132547
$277$		−	19.5959i		−	1.17740i	−0.808350	−	0.588702i	$-0.799639\pi$
$277$		−	19.5959i		−	1.17740i	0.808350	−	0.588702i	$-0.200361\pi$
$278$		−	13.7980i		−	0.827547i
$279$	19.3485			1.15836
$280$	0.898979	−	0.449490i	0.0537243	−	0.0268622i
$281$	8.20204			0.489293			0.244646	−	0.969612i	$-0.421328\pi$
$281$	8.20204			0.489293			0.244646	+	0.969612i	$0.421328\pi$
$282$		−	1.10102i		−	0.0655648i
$283$			23.5959i			1.40263i	0.712851	+	0.701316i	$0.247404\pi$
$283$			23.5959i			1.40263i	−0.712851	+	0.701316i	$0.752596\pi$
$284$	4.89898			0.290701
$285$	−20.6969	−	41.3939i	−1.22598	−	2.45196i
$286$	−19.5959			−1.15873
$287$		−	0.898979i		−	0.0530651i
$288$		−	3.00000i		−	0.176777i
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−4.89898			−0.287183
$292$			4.00000i			0.234082i
$293$			13.5959i			0.794282i	0.917758	+	0.397141i	$0.129998\pi$
$293$			13.5959i			0.794282i	−0.917758	+	0.397141i	$0.870002\pi$
$294$	16.6515			0.971137
$295$	−16.8990	+	8.44949i	−0.983897	+	0.491948i
$296$	1.00000			0.0581238
$297$		0			0
$298$		−	10.6969i		−	0.619657i
$299$	−3.59592			−0.207957
$300$	9.79796	+	7.34847i	0.565685	+	0.424264i
$301$	−1.79796			−0.103633
$302$		−	8.00000i		−	0.460348i
$303$			45.7980i			2.63102i
$304$	−8.44949			−0.484611
$305$	−24.0000	+	12.0000i	−1.37424	+	0.687118i
$306$	14.6969			0.840168
$307$		−	21.1464i		−	1.20689i	−0.797404	−	0.603445i	$-0.793794\pi$
$307$		−	21.1464i		−	1.20689i	0.797404	−	0.603445i	$-0.206206\pi$
$308$		−	2.20204i		−	0.125473i
$309$	24.0000			1.36531
$310$	6.44949	+	12.8990i	0.366306	+	0.732613i
$311$	7.34847			0.416693			0.208347	−	0.978055i	$-0.433192\pi$
$311$	7.34847			0.416693			0.208347	+	0.978055i	$0.433192\pi$
$312$			9.79796i			0.554700i
$313$		−	21.5959i		−	1.22067i	−0.792142	−	0.610337i	$-0.791034\pi$
$313$		−	21.5959i		−	1.22067i	0.792142	−	0.610337i	$-0.208966\pi$
$314$	−11.7980			−0.665797
$315$	−1.34847	−	2.69694i	−0.0759776	−	0.151955i
$316$	1.55051			0.0872230
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		−	19.1010i		−	1.07113i
$319$		0			0
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$	−25.5959			−1.42862
$322$		−	0.404082i		−	0.0225186i
$323$		−	41.3939i		−	2.30322i
$324$	9.00000			0.500000
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	−2.20204			−0.121960
$327$			33.7980i			1.86903i
$328$		−	2.00000i		−	0.110432i
$329$	−0.202041			−0.0111389
$330$	24.0000	−	12.0000i	1.32116	−	0.660578i
$331$	−10.2474			−0.563251			−0.281625	−	0.959524i	$-0.590874\pi$
$331$	−10.2474			−0.563251			−0.281625	+	0.959524i	$0.590874\pi$
$332$		−	14.4495i		−	0.793019i
$333$		−	3.00000i		−	0.164399i
$334$	−8.00000			−0.437741
$335$	10.4495	+	20.8990i	0.570917	+	1.14183i
$336$	−1.10102			−0.0600656
$337$			35.5959i			1.93903i	0.245026	+	0.969517i	$0.421204\pi$
$337$			35.5959i			1.93903i	−0.245026	+	0.969517i	$0.578796\pi$
$338$		−	3.00000i		−	0.163178i
$339$	31.5959			1.71605
$340$	4.89898	+	9.79796i	0.265684	+	0.531369i
$341$	31.5959			1.71101
$342$			25.3485i			1.37069i
$343$		−	6.20204i		−	0.334879i
$344$	−4.00000			−0.215666
$345$	4.40408	−	2.20204i	0.237108	−	0.118554i
$346$	0.202041			0.0108618
$347$			21.7980i			1.17018i	0.810970	+	0.585088i	$0.198940\pi$
$347$			21.7980i			1.17018i	−0.810970	+	0.585088i	$0.801060\pi$
$348$		0			0
$349$	−16.8990			−0.904582			−0.452291	−	0.891871i	$-0.649393\pi$
$349$	−16.8990			−0.904582			−0.452291	+	0.891871i	$0.649393\pi$
$350$	1.34847	−	1.79796i	0.0720787	−	0.0961049i
$351$		0			0
$352$		−	4.89898i		−	0.261116i
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	20.6969			1.10003
$355$	9.79796	−	4.89898i	0.520022	−	0.260011i
$356$	−3.79796			−0.201291
$357$		−	5.39388i		−	0.285474i
$358$		−	5.34847i		−	0.282675i
$359$	3.10102			0.163666			0.0818328	−	0.996646i	$-0.473923\pi$
$359$	3.10102			0.163666			0.0818328	+	0.996646i	$0.473923\pi$
$360$	−3.00000	−	6.00000i	−0.158114	−	0.316228i
$361$	52.3939			2.75757
$362$		−	18.6969i		−	0.982689i
$363$		−	31.8434i		−	1.67134i
$364$	1.79796			0.0942387
$365$	4.00000	+	8.00000i	0.209370	+	0.418739i
$366$	29.3939			1.53644
$367$		−	17.3485i		−	0.905583i	−0.891617	−	0.452791i	$-0.850428\pi$
$367$		−	17.3485i		−	0.905583i	0.891617	−	0.452791i	$-0.149572\pi$
$368$		−	0.898979i		−	0.0468625i
$369$	−6.00000			−0.312348
$370$	2.00000	−	1.00000i	0.103975	−	0.0519875i
$371$	−3.50510			−0.181976
$372$		−	15.7980i		−	0.819086i
$373$		−	7.79796i		−	0.403763i	−0.979410	−	0.201882i	$-0.935294\pi$
$373$		−	7.79796i		−	0.403763i	0.979410	−	0.201882i	$-0.0647056\pi$
$374$	24.0000			1.24101
$375$	26.9444	+	4.89898i	1.39140	+	0.252982i
$376$	−0.449490			−0.0231807
$377$		0			0
$378$		0			0
$379$	−7.59592			−0.390176			−0.195088	−	0.980786i	$-0.562499\pi$
$379$	−7.59592			−0.390176			−0.195088	+	0.980786i	$0.562499\pi$
$380$	−16.8990	+	8.44949i	−0.866899	+	0.433450i
$381$	−42.4949			−2.17708
$382$		−	3.34847i		−	0.171323i
$383$			5.30306i			0.270974i	0.990779	+	0.135487i	$0.0432599\pi$
$383$			5.30306i			0.270974i	−0.990779	+	0.135487i	$0.956740\pi$
$384$	−2.44949			−0.125000
$385$	−2.20204	−	4.40408i	−0.112226	−	0.224453i
$386$	14.0000			0.712581
$387$			12.0000i			0.609994i
$388$			2.00000i			0.101535i
$389$	6.20204			0.314456			0.157228	−	0.987562i	$-0.449744\pi$
$389$	6.20204			0.314456			0.157228	+	0.987562i	$0.449744\pi$
$390$	9.79796	+	19.5959i	0.496139	+	0.992278i
$391$	4.40408			0.222724
$392$		−	6.79796i		−	0.343349i
$393$		−	34.8990i		−	1.76042i
$394$	2.00000			0.100759
$395$	3.10102	−	1.55051i	0.156029	−	0.0780146i
$396$	−14.6969			−0.738549
$397$		−	35.7980i		−	1.79665i	−0.439334	−	0.898324i	$-0.644785\pi$
$397$		−	35.7980i		−	1.79665i	0.439334	−	0.898324i	$-0.355215\pi$
$398$		−	1.55051i		−	0.0777201i
$399$	9.30306			0.465736
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−4.20204			−0.209840			−0.104920	−	0.994481i	$-0.533459\pi$
$401$	−4.20204			−0.209840			−0.104920	+	0.994481i	$0.533459\pi$
$402$		−	25.5959i		−	1.27661i
$403$			25.7980i			1.28509i
$404$	18.6969			0.930207
$405$	18.0000	−	9.00000i	0.894427	−	0.447214i
$406$		0			0
$407$		−	4.89898i		−	0.242833i
$408$		−	12.0000i		−	0.594089i
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	−2.00000	−	4.00000i	−0.0987730	−	0.197546i
$411$	−48.0000			−2.36767
$412$		−	9.79796i		−	0.482711i
$413$		−	3.79796i		−	0.186885i
$414$	−2.69694			−0.132547
$415$	−14.4495	−	28.8990i	−0.709298	−	1.41860i
$416$	4.00000			0.196116
$417$			33.7980i			1.65509i
$418$			41.3939i			2.02464i
$419$	13.7980			0.674074			0.337037	−	0.941491i	$-0.390575\pi$
$419$	13.7980			0.674074			0.337037	+	0.941491i	$0.390575\pi$
$420$	−2.20204	+	1.10102i	−0.107449	+	0.0537243i
$421$	19.5959			0.955047			0.477523	−	0.878619i	$-0.341535\pi$
$421$	19.5959			0.955047			0.477523	+	0.878619i	$0.341535\pi$
$422$			22.6969i			1.10487i
$423$			1.34847i			0.0655648i
$424$	−7.79796			−0.378702
$425$	19.5959	+	14.6969i	0.950542	+	0.712906i
$426$	−12.0000			−0.581402
$427$		−	5.39388i		−	0.261028i
$428$			10.4495i			0.505095i
$429$	48.0000			2.31746
$430$	−8.00000	+	4.00000i	−0.385794	+	0.192897i
$431$	−40.2474			−1.93865			−0.969326	−	0.245780i	$-0.920956\pi$
$431$	−40.2474			−1.93865			−0.969326	+	0.245780i	$0.920956\pi$
$432$		0			0
$433$			37.3939i			1.79704i	0.438938	+	0.898518i	$0.355355\pi$
$433$			37.3939i			1.79704i	−0.438938	+	0.898518i	$0.644645\pi$
$434$	−2.89898			−0.139155
$435$		0			0
$436$	13.7980			0.660802
$437$			7.59592i			0.363362i
$438$		−	9.79796i		−	0.468165i
$439$	−35.3485			−1.68709			−0.843545	−	0.537058i	$-0.819536\pi$
$439$	−35.3485			−1.68709			−0.843545	+	0.537058i	$0.819536\pi$
$440$	−4.89898	−	9.79796i	−0.233550	−	0.467099i
$441$	−20.3939			−0.971137
$442$			19.5959i			0.932083i
$443$		−	39.3485i		−	1.86950i	−0.355303	−	0.934751i	$-0.615622\pi$
$443$		−	39.3485i		−	1.86950i	0.355303	−	0.934751i	$-0.384378\pi$
$444$	−2.44949			−0.116248
$445$	−7.59592	+	3.79796i	−0.360081	+	0.180041i
$446$	−4.44949			−0.210689
$447$			26.2020i			1.23931i
$448$			0.449490i			0.0212364i
$449$	−3.79796			−0.179237			−0.0896184	−	0.995976i	$-0.528565\pi$
$449$	−3.79796			−0.179237			−0.0896184	+	0.995976i	$0.528565\pi$
$450$	−12.0000	−	9.00000i	−0.565685	−	0.424264i
$451$	−9.79796			−0.461368
$452$		−	12.8990i		−	0.606717i
$453$			19.5959i			0.920697i
$454$	15.1010			0.708726
$455$	3.59592	−	1.79796i	0.168579	−	0.0842896i
$456$	20.6969			0.969223
$457$		−	2.00000i		−	0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$		−	2.00000i		−	0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$			10.0000i			0.467269i
$459$		0			0
$460$	−0.898979	−	1.79796i	−0.0419151	−	0.0838303i
$461$	−14.2020			−0.661455			−0.330727	−	0.943726i	$-0.607294\pi$
$461$	−14.2020			−0.661455			−0.330727	+	0.943726i	$0.607294\pi$
$462$			5.39388i			0.250946i
$463$		−	28.4949i		−	1.32427i	−0.749384	−	0.662135i	$-0.769650\pi$
$463$		−	28.4949i		−	1.32427i	0.749384	−	0.662135i	$-0.230350\pi$
$464$		0			0
$465$	−15.7980	−	31.5959i	−0.732613	−	1.46523i
$466$	−9.79796			−0.453882
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		−	12.0000i		−	0.554700i
$469$	−4.69694			−0.216884
$470$	−0.898979	+	0.449490i	−0.0414668	+	0.0207334i
$471$	28.8990			1.33159
$472$		−	8.44949i		−	0.388919i
$473$			19.5959i			0.901021i
$474$	−3.79796			−0.174446
$475$	−25.3485	+	33.7980i	−1.16307	+	1.55076i
$476$	−2.20204			−0.100930
$477$			23.3939i			1.07113i
$478$		−	26.0454i		−	1.19129i
$479$	29.1464			1.33173			0.665867	−	0.746070i	$-0.268062\pi$
$479$	29.1464			1.33173			0.665867	+	0.746070i	$0.268062\pi$
$480$	−4.89898	+	2.44949i	−0.223607	+	0.111803i
$481$	4.00000			0.182384
$482$		−	11.7980i		−	0.537382i
$483$			0.989795i			0.0450372i
$484$	−13.0000			−0.590909
$485$	2.00000	+	4.00000i	0.0908153	+	0.181631i
$486$	−22.0454			−1.00000
$487$			29.3939i			1.33196i	0.745968	+	0.665982i	$0.231987\pi$
$487$			29.3939i			1.33196i	−0.745968	+	0.665982i	$0.768013\pi$
$488$		−	12.0000i		−	0.543214i
$489$	5.39388			0.243920
$490$	−6.79796	−	13.5959i	−0.307100	−	0.614201i
$491$	−38.6969			−1.74637			−0.873184	−	0.487390i	$-0.837949\pi$
$491$	−38.6969			−1.74637			−0.873184	+	0.487390i	$0.837949\pi$
$492$			4.89898i			0.220863i
$493$		0			0
$494$	−33.7980			−1.52064
$495$	−29.3939	+	14.6969i	−1.32116	+	0.660578i
$496$	−6.44949			−0.289591
$497$			2.20204i			0.0987750i
$498$			35.3939i			1.58604i
$499$	25.3485			1.13475			0.567377	−	0.823458i	$-0.307958\pi$
$499$	25.3485			1.13475			0.567377	+	0.823458i	$0.307958\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$	19.5959			0.875481
$502$			6.65153i			0.296872i
$503$		−	17.7980i		−	0.793572i	−0.917911	−	0.396786i	$-0.870126\pi$
$503$		−	17.7980i		−	0.793572i	0.917911	−	0.396786i	$-0.129874\pi$
$504$	1.34847			0.0600656
$505$	37.3939	−	18.6969i	1.66401	−	0.832003i
$506$	−4.40408			−0.195785
$507$			7.34847i			0.326357i
$508$			17.3485i			0.769714i
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$	−12.0000	−	24.0000i	−0.531369	−	1.06274i
$511$	−1.79796			−0.0795370
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−4.89898			−0.216085
$515$	−9.79796	−	19.5959i	−0.431750	−	0.863499i
$516$	9.79796			0.431331
$517$			2.20204i			0.0968457i
$518$			0.449490i			0.0197494i
$519$	−0.494897			−0.0217236
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$	−1.79796			−0.0787700			−0.0393850	−	0.999224i	$-0.512540\pi$
$521$	−1.79796			−0.0787700			−0.0393850	+	0.999224i	$0.512540\pi$
$522$		0			0
$523$		−	0.898979i		−	0.0393096i	−0.999807	−	0.0196548i	$-0.993743\pi$
$523$		−	0.898979i		−	0.0393096i	0.999807	−	0.0196548i	$-0.00625672\pi$
$524$	−14.2474			−0.622403
$525$	−3.30306	+	4.40408i	−0.144157	+	0.192210i
$526$	26.2474			1.14444
$527$		−	31.5959i		−	1.37634i
$528$			12.0000i			0.522233i
$529$	22.1918			0.964862
$530$	−15.5959	+	7.79796i	−0.677443	+	0.338722i
$531$	−25.3485			−1.10003
$532$		−	3.79796i		−	0.164662i
$533$		−	8.00000i		−	0.346518i
$534$	9.30306			0.402583
$535$	10.4495	+	20.8990i	0.451771	+	0.903542i
$536$	−10.4495			−0.451349
$537$			13.1010i			0.565351i
$538$		−	10.6969i		−	0.461178i
$539$	−33.3031			−1.43446
$540$		0			0
$541$	39.5959			1.70236			0.851181	−	0.524873i	$-0.175887\pi$
$541$	39.5959			1.70236			0.851181	+	0.524873i	$0.175887\pi$
$542$			16.4949i			0.708517i
$543$			45.7980i			1.96538i
$544$	−4.89898			−0.210042
$545$	27.5959	−	13.7980i	1.18208	−	0.591040i
$546$	−4.40408			−0.188477
$547$			18.6969i			0.799423i	0.916641	+	0.399712i	$0.130890\pi$
$547$			18.6969i			0.799423i	−0.916641	+	0.399712i	$0.869110\pi$
$548$			19.5959i			0.837096i
$549$	−36.0000			−1.53644
$550$	−19.5959	−	14.6969i	−0.835573	−	0.626680i
$551$		0			0
$552$			2.20204i			0.0937251i
$553$			0.696938i			0.0296368i
$554$	19.5959			0.832551
$555$	−4.89898	+	2.44949i	−0.207950	+	0.103975i
$556$	13.7980			0.585164
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$			19.3485i			0.819086i
$559$	−16.0000			−0.676728
$560$	0.449490	+	0.898979i	0.0189944	+	0.0379888i
$561$	−58.7878			−2.48202
$562$			8.20204i			0.345982i
$563$			5.30306i			0.223497i	0.993736	+	0.111749i	$0.0356452\pi$
$563$			5.30306i			0.223497i	−0.993736	+	0.111749i	$0.964355\pi$
$564$	1.10102			0.0463613
$565$	−12.8990	−	25.7980i	−0.542664	−	1.08533i
$566$	−23.5959			−0.991810
$567$			4.04541i			0.169891i
$568$			4.89898i			0.205557i
$569$	17.5959			0.737659			0.368830	−	0.929497i	$-0.379759\pi$
$569$	17.5959			0.737659			0.368830	+	0.929497i	$0.379759\pi$
$570$	41.3939	−	20.6969i	1.73380	−	0.866899i
$571$	33.3939			1.39749			0.698745	−	0.715371i	$-0.253742\pi$
$571$	33.3939			1.39749			0.698745	+	0.715371i	$0.253742\pi$
$572$		−	19.5959i		−	0.819346i
$573$			8.20204i			0.342645i
$574$	0.898979			0.0375227
$575$	−3.59592	−	2.69694i	−0.149960	−	0.112470i
$576$	3.00000			0.125000
$577$		−	1.30306i		−	0.0542472i	−0.999632	−	0.0271236i	$-0.991365\pi$
$577$		−	1.30306i		−	0.0542472i	0.999632	−	0.0271236i	$-0.00863476\pi$
$578$		−	7.00000i		−	0.291162i
$579$	−34.2929			−1.42516
$580$		0			0
$581$	6.49490			0.269454
$582$		−	4.89898i		−	0.203069i
$583$			38.2020i			1.58217i
$584$	−4.00000			−0.165521
$585$	−12.0000	−	24.0000i	−0.496139	−	0.992278i
$586$	−13.5959			−0.561642
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$			16.6515i			0.686698i
$589$	54.4949			2.24542
$590$	−8.44949	−	16.8990i	−0.347860	−	0.695720i
$591$	−4.89898			−0.201517
$592$			1.00000i			0.0410997i
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$	−4.40408	+	2.20204i	−0.180550	+	0.0902749i
$596$	10.6969			0.438164
$597$			3.79796i			0.155440i
$598$		−	3.59592i		−	0.147048i
$599$	27.5959			1.12754			0.563769	−	0.825932i	$-0.309351\pi$
$599$	27.5959			1.12754			0.563769	+	0.825932i	$0.309351\pi$
$600$	−7.34847	+	9.79796i	−0.300000	+	0.400000i
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		−	1.79796i		−	0.0732793i
$603$			31.3485i			1.27661i
$604$	8.00000			0.325515
$605$	−26.0000	+	13.0000i	−1.05705	+	0.528525i
$606$	−45.7980			−1.86041
$607$		−	2.69694i		−	0.109465i	−0.998501	−	0.0547327i	$-0.982569\pi$
$607$		−	2.69694i		−	0.109465i	0.998501	−	0.0547327i	$-0.0174307\pi$
$608$		−	8.44949i		−	0.342672i
$609$		0			0
$610$	−12.0000	−	24.0000i	−0.485866	−	0.971732i
$611$	−1.79796			−0.0727376
$612$			14.6969i			0.594089i
$613$		−	20.2020i		−	0.815953i	−0.912992	−	0.407976i	$-0.866235\pi$
$613$		−	20.2020i		−	0.815953i	0.912992	−	0.407976i	$-0.133765\pi$
$614$	21.1464			0.853400
$615$	4.89898	+	9.79796i	0.197546	+	0.395092i
$616$	2.20204			0.0887228
$617$		−	19.5959i		−	0.788902i	−0.918917	−	0.394451i	$-0.870935\pi$
$617$		−	19.5959i		−	0.788902i	0.918917	−	0.394451i	$-0.129065\pi$
$618$			24.0000i			0.965422i
$619$	3.10102			0.124641			0.0623203	−	0.998056i	$-0.480150\pi$
$619$	3.10102			0.124641			0.0623203	+	0.998056i	$0.480150\pi$
$620$	−12.8990	+	6.44949i	−0.518035	+	0.259018i
$621$		0			0
$622$			7.34847i			0.294647i
$623$		−	1.70714i		−	0.0683953i
$624$	−9.79796			−0.392232
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	21.5959			0.863146
$627$		−	101.394i		−	4.04928i
$628$		−	11.7980i		−	0.470790i
$629$	−4.89898			−0.195335
$630$	2.69694	−	1.34847i	0.107449	−	0.0537243i
$631$	4.24745			0.169088			0.0845441	−	0.996420i	$-0.473057\pi$
$631$	4.24745			0.169088			0.0845441	+	0.996420i	$0.473057\pi$
$632$			1.55051i			0.0616760i
$633$		−	55.5959i		−	2.20974i
$634$	−18.0000			−0.714871
$635$	17.3485	+	34.6969i	0.688453	+	1.37691i
$636$	19.1010			0.757405
$637$		−	27.1918i		−	1.07738i
$638$		0			0
$639$	14.6969			0.581402
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−1.79796			−0.0710151			−0.0355076	−	0.999369i	$-0.511305\pi$
$641$	−1.79796			−0.0710151			−0.0355076	+	0.999369i	$0.511305\pi$
$642$		−	25.5959i		−	1.01019i
$643$			32.8990i			1.29741i	0.761040	+	0.648705i	$0.224689\pi$
$643$			32.8990i			1.29741i	−0.761040	+	0.648705i	$0.775311\pi$
$644$	0.404082			0.0159231
$645$	19.5959	−	9.79796i	0.771589	−	0.385794i
$646$	41.3939			1.62862
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$			9.00000i			0.353553i
$649$	−41.3939			−1.62485
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$	7.10102			0.278311
$652$		−	2.20204i		−	0.0862386i
$653$		−	34.0000i		−	1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$		−	34.0000i		−	1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	−33.7980			−1.32160
$655$	−28.4949	+	14.2474i	−1.11339	+	0.556694i
$656$	2.00000			0.0780869
$657$			12.0000i			0.468165i
$658$		−	0.202041i		−	0.00787638i
$659$	7.59592			0.295895			0.147947	−	0.988995i	$-0.452733\pi$
$659$	7.59592			0.295895			0.147947	+	0.988995i	$0.452733\pi$
$660$	12.0000	+	24.0000i	0.467099	+	0.934199i
$661$	47.1918			1.83555			0.917775	−	0.397101i	$-0.129984\pi$
$661$	47.1918			1.83555			0.917775	+	0.397101i	$0.129984\pi$
$662$		−	10.2474i		−	0.398278i
$663$		−	48.0000i		−	1.86417i
$664$	14.4495			0.560749
$665$	−3.79796	−	7.59592i	−0.147279	−	0.294557i
$666$	3.00000			0.116248
$667$		0			0
$668$		−	8.00000i		−	0.309529i
$669$	10.8990			0.421379
$670$	−20.8990	+	10.4495i	−0.807398	+	0.403699i
$671$	−58.7878			−2.26948
$672$		−	1.10102i		−	0.0424728i
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$	−35.5959			−1.37110
$675$		0			0
$676$	3.00000			0.115385
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$			31.5959i			1.21343i
$679$	−0.898979			−0.0344997
$680$	−9.79796	+	4.89898i	−0.375735	+	0.187867i
$681$	−36.9898			−1.41745
$682$			31.5959i			1.20987i
$683$			23.5959i			0.902873i	0.892303	+	0.451436i	$0.149088\pi$
$683$			23.5959i			0.902873i	−0.892303	+	0.451436i	$0.850912\pi$
$684$	−25.3485			−0.969223
$685$	19.5959	+	39.1918i	0.748722	+	1.49744i
$686$	6.20204			0.236795
$687$		−	24.4949i		−	0.934539i
$688$		−	4.00000i		−	0.152499i
$689$	−31.1918			−1.18831
$690$	2.20204	+	4.40408i	0.0838303	+	0.167661i
$691$	18.2020			0.692438			0.346219	−	0.938154i	$-0.387465\pi$
$691$	18.2020			0.692438			0.346219	+	0.938154i	$0.387465\pi$
$692$			0.202041i			0.00768045i
$693$		−	6.60612i		−	0.250946i
$694$	−21.7980			−0.827439
$695$	27.5959	−	13.7980i	1.04677	−	0.523386i
$696$		0			0
$697$			9.79796i			0.371124i
$698$		−	16.8990i		−	0.639636i
$699$	24.0000			0.907763
$700$	1.79796	+	1.34847i	0.0679565	+	0.0509673i
$701$	5.79796			0.218986			0.109493	−	0.993988i	$-0.465077\pi$
$701$	5.79796			0.218986			0.109493	+	0.993988i	$0.465077\pi$
$702$		0			0
$703$		−	8.44949i		−	0.318679i
$704$	4.89898			0.184637
$705$	2.20204	−	1.10102i	0.0829337	−	0.0414668i
$706$	−6.00000			−0.225813
$707$			8.40408i			0.316068i
$708$			20.6969i			0.777839i
$709$	13.7980			0.518193			0.259097	−	0.965851i	$-0.416575\pi$
$709$	13.7980			0.518193			0.259097	+	0.965851i	$0.416575\pi$
$710$	4.89898	+	9.79796i	0.183855	+	0.367711i
$711$	4.65153			0.174446
$712$		−	3.79796i		−	0.142335i
$713$			5.79796i			0.217135i
$714$	5.39388			0.201861
$715$	−19.5959	−	39.1918i	−0.732846	−	1.46569i
$716$	5.34847			0.199882
$717$			63.7980i			2.38258i
$718$			3.10102i			0.115729i
$719$	−12.4041			−0.462594			−0.231297	−	0.972883i	$-0.574297\pi$
$719$	−12.4041			−0.462594			−0.231297	+	0.972883i	$0.574297\pi$
$720$	6.00000	−	3.00000i	0.223607	−	0.111803i
$721$	4.40408			0.164017
$722$			52.3939i			1.94990i
$723$			28.8990i			1.07476i
$724$	18.6969			0.694866
$725$		0			0
$726$	31.8434			1.18182
$727$		−	8.89898i		−	0.330045i	−0.986290	−	0.165022i	$-0.947230\pi$
$727$		−	8.89898i		−	0.330045i	0.986290	−	0.165022i	$-0.0527697\pi$
$728$			1.79796i			0.0666368i
$729$	27.0000			1.00000
$730$	−8.00000	+	4.00000i	−0.296093	+	0.148047i
$731$	19.5959			0.724781
$732$			29.3939i			1.08643i
$733$		−	21.5959i		−	0.797663i	−0.917024	−	0.398832i	$-0.869416\pi$
$733$		−	21.5959i		−	0.797663i	0.917024	−	0.398832i	$-0.130584\pi$
$734$	17.3485			0.640344
$735$	16.6515	+	33.3031i	0.614201	+	1.22840i
$736$	0.898979			0.0331368
$737$			51.1918i			1.88568i
$738$		−	6.00000i		−	0.220863i
$739$	26.2020			0.963858			0.481929	−	0.876210i	$-0.339936\pi$
$739$	26.2020			0.963858			0.481929	+	0.876210i	$0.339936\pi$
$740$	1.00000	+	2.00000i	0.0367607	+	0.0735215i
$741$	82.7878			3.04128
$742$		−	3.50510i		−	0.128676i
$743$			38.2474i			1.40316i	0.712589	+	0.701581i	$0.247522\pi$
$743$			38.2474i			1.40316i	−0.712589	+	0.701581i	$0.752478\pi$
$744$	15.7980			0.579181
$745$	21.3939	−	10.6969i	0.783811	−	0.391906i
$746$	7.79796			0.285504
$747$		−	43.3485i		−	1.58604i
$748$			24.0000i			0.877527i
$749$	−4.69694			−0.171622
$750$	−4.89898	+	26.9444i	−0.178885	+	0.983870i
$751$	1.30306			0.0475494			0.0237747	−	0.999717i	$-0.492432\pi$
$751$	1.30306			0.0475494			0.0237747	+	0.999717i	$0.492432\pi$
$752$		−	0.449490i		−	0.0163912i
$753$		−	16.2929i		−	0.593745i
$754$		0			0
$755$	16.0000	−	8.00000i	0.582300	−	0.291150i
$756$		0			0
$757$			5.59592i			0.203387i	0.994816	+	0.101694i	$0.0324261\pi$
$757$			5.59592i			0.203387i	−0.994816	+	0.101694i	$0.967574\pi$
$758$		−	7.59592i		−	0.275896i
$759$	10.7878			0.391571
$760$	−8.44949	−	16.8990i	−0.306495	−	0.612990i
$761$	−53.1918			−1.92820			−0.964101	−	0.265535i	$-0.914451\pi$
$761$	−53.1918			−1.92820			−0.964101	+	0.265535i	$0.914451\pi$
$762$		−	42.4949i		−	1.53943i
$763$			6.20204i			0.224529i
$764$	3.34847			0.121143
$765$	14.6969	+	29.3939i	0.531369	+	1.06274i
$766$	−5.30306			−0.191607
$767$		−	33.7980i		−	1.22037i
$768$		−	2.44949i		−	0.0883883i
$769$	16.2020			0.584261			0.292130	−	0.956379i	$-0.405636\pi$
$769$	16.2020			0.584261			0.292130	+	0.956379i	$0.405636\pi$
$770$	4.40408	−	2.20204i	0.158712	−	0.0793561i
$771$	12.0000			0.432169
$772$			14.0000i			0.503871i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−12.0000			−0.431331
$775$	−19.3485	+	25.7980i	−0.695018	+	0.926690i
$776$	−2.00000			−0.0717958
$777$		−	1.10102i		−	0.0394989i
$778$			6.20204i			0.222354i
$779$	−16.8990			−0.605469
$780$	−19.5959	+	9.79796i	−0.701646	+	0.350823i
$781$	24.0000			0.858788
$782$			4.40408i			0.157490i
$783$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.44949 q^{6} -0.449490i q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.44949 q^{6} -0.449490i q^{7} -1.00000i q^{8} -3.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} -4.89898 q^{11} +2.44949i q^{12} -4.00000i q^{13} +0.449490 q^{14} +(2.44949 + 4.89898i) q^{15} +1.00000 q^{16} +4.89898i q^{17} -3.00000i q^{18} -8.44949 q^{19} +(2.00000 - 1.00000i) q^{20} -1.10102 q^{21} -4.89898i q^{22} -0.898979i q^{23} -2.44949 q^{24} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +0.449490i q^{28} +(-4.89898 + 2.44949i) q^{30} -6.44949 q^{31} +1.00000i q^{32} +12.0000i q^{33} -4.89898 q^{34} +(0.449490 + 0.898979i) q^{35} +3.00000 q^{36} +1.00000i q^{37} -8.44949i q^{38} -9.79796 q^{39} +(1.00000 + 2.00000i) q^{40} +2.00000 q^{41} -1.10102i q^{42} -4.00000i q^{43} +4.89898 q^{44} +(6.00000 - 3.00000i) q^{45} +0.898979 q^{46} -0.449490i q^{47} -2.44949i q^{48} +6.79796 q^{49} +(4.00000 + 3.00000i) q^{50} +12.0000 q^{51} +4.00000i q^{52} -7.79796i q^{53} +(9.79796 - 4.89898i) q^{55} -0.449490 q^{56} +20.6969i q^{57} +8.44949 q^{59} +(-2.44949 - 4.89898i) q^{60} +12.0000 q^{61} -6.44949i q^{62} +1.34847i q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -12.0000 q^{66} -10.4495i q^{67} -4.89898i q^{68} -2.20204 q^{69} +(-0.898979 + 0.449490i) q^{70} -4.89898 q^{71} +3.00000i q^{72} -4.00000i q^{73} -1.00000 q^{74} +(-9.79796 - 7.34847i) q^{75} +8.44949 q^{76} +2.20204i q^{77} -9.79796i q^{78} -1.55051 q^{79} +(-2.00000 + 1.00000i) q^{80} -9.00000 q^{81} +2.00000i q^{82} +14.4495i q^{83} +1.10102 q^{84} +(-4.89898 - 9.79796i) q^{85} +4.00000 q^{86} +4.89898i q^{88} +3.79796 q^{89} +(3.00000 + 6.00000i) q^{90} -1.79796 q^{91} +0.898979i q^{92} +15.7980i q^{93} +0.449490 q^{94} +(16.8990 - 8.44949i) q^{95} +2.44949 q^{96} -2.00000i q^{97} +6.79796i q^{98} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9	\( 4 q - 4 q^{4} - 8 q^{5} - 12 q^{9} - 4 q^{10} - 8 q^{14} + 4 q^{16} - 24 q^{19} + 8 q^{20} - 24 q^{21} + 12 q^{25} + 16 q^{26} - 16 q^{31} - 8 q^{35} + 12 q^{36} + 4 q^{40} + 8 q^{41} + 24 q^{45} - 16 q^{46} - 12 q^{49} + 16 q^{50} + 48 q^{51} + 8 q^{56} + 24 q^{59} + 48 q^{61} - 4 q^{64} + 16 q^{65} - 48 q^{66} - 48 q^{69} + 16 q^{70} - 4 q^{74} + 24 q^{76} - 16 q^{79} - 8 q^{80} - 36 q^{81} + 24 q^{84} + 16 q^{86} - 24 q^{89} + 12 q^{90} + 32 q^{91} - 8 q^{94} + 48 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9 - 4 * q^10 - 8 * q^14 + 4 * q^16 - 24 * q^19 + 8 * q^20 - 24 * q^21 + 12 * q^25 + 16 * q^26 - 16 * q^31 - 8 * q^35 + 12 * q^36 + 4 * q^40 + 8 * q^41 + 24 * q^45 - 16 * q^46 - 12 * q^49 + 16 * q^50 + 48 * q^51 + 8 * q^56 + 24 * q^59 + 48 * q^61 - 4 * q^64 + 16 * q^65 - 48 * q^66 - 48 * q^69 + 16 * q^70 - 4 * q^74 + 24 * q^76 - 16 * q^79 - 8 * q^80 - 36 * q^81 + 24 * q^84 + 16 * q^86 - 24 * q^89 + 12 * q^90 + 32 * q^91 - 8 * q^94 + 48 * q^95

Introduction

L-functions

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