LMFDB - Embedded newform 336.2.h.b.239.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	336.h (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:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.3
Root			$0.500000 + 1.56488i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.2.h.b.239.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.06488 - 1.36603i) q^{3} +2.12976i q^{5} -1.00000i q^{7} +(-0.732051 + 2.90931i) q^{9} +O(q^{10})$	\(q+(-1.06488 - 1.36603i) q^{3} +2.12976i q^{5} -1.00000i q^{7} +(-0.732051 + 2.90931i) q^{9} +5.81863 q^{11} +4.19615 q^{13} +(2.90931 - 2.26795i) q^{15} -5.81863i q^{17} -2.73205i q^{19} +(-1.36603 + 1.06488i) q^{21} -4.25953 q^{23} +0.464102 q^{25} +(4.75374 - 2.09808i) q^{27} +5.81863i q^{29} +2.53590i q^{31} +(-6.19615 - 7.94839i) q^{33} +2.12976 q^{35} +11.4641 q^{37} +(-4.46841 - 5.73205i) q^{39} +1.55910i q^{41} +2.00000i q^{43} +(-6.19615 - 1.55910i) q^{45} -1.00000 q^{49} +(-7.94839 + 6.19615i) q^{51} -1.55910i q^{53} +12.3923i q^{55} +(-3.73205 + 2.90931i) q^{57} -9.50749 q^{59} -1.26795 q^{61} +(2.90931 + 0.732051i) q^{63} +8.93682i q^{65} +3.46410i q^{67} +(4.53590 + 5.81863i) q^{69} -1.55910 q^{71} -11.4641 q^{73} +(-0.494214 - 0.633975i) q^{75} -5.81863i q^{77} +12.0000i q^{79} +(-7.92820 - 4.25953i) q^{81} -9.50749 q^{83} +12.3923 q^{85} +(7.94839 - 6.19615i) q^{87} -13.1963i q^{89} -4.19615i q^{91} +(3.46410 - 2.70043i) q^{93} +5.81863 q^{95} -4.92820 q^{97} +(-4.25953 + 16.9282i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 - 8 * q^13 - 4 * q^21 - 24 * q^25 - 8 * q^33 + 64 * q^37 - 8 * q^45 - 8 * q^49 - 16 * q^57 - 24 * q^61 + 64 * q^69 - 64 * q^73 - 8 * q^81 + 16 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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			0
$2$		0			0
$3$	−1.06488	−	1.36603i	−0.614810	−	0.788675i
$3$	−1.06488	−	1.36603i	−0.614810	−	0.788675i
$4$		0			0
$5$			2.12976i			0.952460i	0.879321	+	0.476230i	$0.157997\pi$
$5$			2.12976i			0.952460i	−0.879321	+	0.476230i	$0.842003\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.732051	+	2.90931i	−0.244017	+	0.969771i
$10$		0			0
$11$	5.81863			1.75438			0.877191	−	0.480142i	$-0.159415\pi$
$11$	5.81863			1.75438			0.877191	+	0.480142i	$0.159415\pi$
$12$		0			0
$13$	4.19615			1.16380			0.581902	−	0.813259i	$-0.302309\pi$
$13$	4.19615			1.16380			0.581902	+	0.813259i	$0.302309\pi$
$14$		0			0
$15$	2.90931	−	2.26795i	0.751181	−	0.585582i
$16$		0			0
$17$		−	5.81863i		−	1.41122i	−0.708598	−	0.705612i	$-0.750672\pi$
$17$		−	5.81863i		−	1.41122i	0.708598	−	0.705612i	$-0.249328\pi$
$18$		0			0
$19$		−	2.73205i		−	0.626775i	−0.949625	−	0.313388i	$-0.898536\pi$
$19$		−	2.73205i		−	0.626775i	0.949625	−	0.313388i	$-0.101464\pi$
$20$		0			0
$21$	−1.36603	+	1.06488i	−0.298091	+	0.232376i
$22$		0			0
$23$	−4.25953			−0.888173			−0.444087	−	0.895984i	$-0.646472\pi$
$23$	−4.25953			−0.888173			−0.444087	+	0.895984i	$0.646472\pi$
$24$		0			0
$25$	0.464102			0.0928203
$26$		0			0
$27$	4.75374	−	2.09808i	0.914858	−	0.403775i
$28$		0			0
$29$			5.81863i			1.08049i	0.841507	+	0.540246i	$0.181669\pi$
$29$			5.81863i			1.08049i	−0.841507	+	0.540246i	$0.818331\pi$
$30$		0			0
$31$			2.53590i			0.455461i	0.973724	+	0.227730i	$0.0731305\pi$
$31$			2.53590i			0.455461i	−0.973724	+	0.227730i	$0.926870\pi$
$32$		0			0
$33$	−6.19615	−	7.94839i	−1.07861	−	1.38364i
$34$		0			0
$35$	2.12976			0.359996
$36$		0			0
$37$	11.4641			1.88469			0.942343	−	0.334648i	$-0.108617\pi$
$37$	11.4641			1.88469			0.942343	+	0.334648i	$0.108617\pi$
$38$		0			0
$39$	−4.46841	−	5.73205i	−0.715518	−	0.917863i
$40$		0			0
$41$			1.55910i			0.243490i	0.992561	+	0.121745i	$0.0388490\pi$
$41$			1.55910i			0.243490i	−0.992561	+	0.121745i	$0.961151\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	−6.19615	−	1.55910i	−0.923668	−	0.232416i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.94839	+	6.19615i	−1.11300	+	0.867635i
$52$		0			0
$53$		−	1.55910i		−	0.214158i	−0.994250	−	0.107079i	$-0.965850\pi$
$53$		−	1.55910i		−	0.214158i	0.994250	−	0.107079i	$-0.0341498\pi$
$54$		0			0
$55$			12.3923i			1.67098i
$56$		0			0
$57$	−3.73205	+	2.90931i	−0.494322	+	0.385348i
$58$		0			0
$59$	−9.50749			−1.23777			−0.618885	−	0.785482i	$-0.712415\pi$
$59$	−9.50749			−1.23777			−0.618885	+	0.785482i	$0.712415\pi$
$60$		0			0
$61$	−1.26795			−0.162344			−0.0811721	−	0.996700i	$-0.525866\pi$
$61$	−1.26795			−0.162344			−0.0811721	+	0.996700i	$0.525866\pi$
$62$		0			0
$63$	2.90931	+	0.732051i	0.366539	+	0.0922297i
$64$		0			0
$65$			8.93682i			1.10848i
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$	4.53590	+	5.81863i	0.546058	+	0.700480i
$70$		0			0
$71$	−1.55910			−0.185031			−0.0925153	−	0.995711i	$-0.529491\pi$
$71$	−1.55910			−0.185031			−0.0925153	+	0.995711i	$0.529491\pi$
$72$		0			0
$73$	−11.4641			−1.34177			−0.670886	−	0.741561i	$-0.734086\pi$
$73$	−11.4641			−1.34177			−0.670886	+	0.741561i	$0.734086\pi$
$74$		0			0
$75$	−0.494214	−	0.633975i	−0.0570669	−	0.0732051i
$76$		0			0
$77$		−	5.81863i		−	0.663094i
$78$		0			0
$79$			12.0000i			1.35011i	0.737769	+	0.675053i	$0.235879\pi$
$79$			12.0000i			1.35011i	−0.737769	+	0.675053i	$0.764121\pi$
$80$		0			0
$81$	−7.92820	−	4.25953i	−0.880911	−	0.473281i
$82$		0			0
$83$	−9.50749			−1.04358			−0.521791	−	0.853073i	$-0.674736\pi$
$83$	−9.50749			−1.04358			−0.521791	+	0.853073i	$0.674736\pi$
$84$		0			0
$85$	12.3923			1.34413
$86$		0			0
$87$	7.94839	−	6.19615i	0.852157	−	0.664297i
$88$		0			0
$89$		−	13.1963i		−	1.39881i	−0.714726	−	0.699405i	$-0.753448\pi$
$89$		−	13.1963i		−	1.39881i	0.714726	−	0.699405i	$-0.246552\pi$
$90$		0			0
$91$		−	4.19615i		−	0.439876i
$92$		0			0
$93$	3.46410	−	2.70043i	0.359211	−	0.280022i
$94$		0			0
$95$	5.81863			0.596978
$96$		0			0
$97$	−4.92820			−0.500383			−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820			−0.500383			−0.250192	+	0.968196i	$0.580494\pi$
$98$		0			0
$99$	−4.25953	+	16.9282i	−0.428099	+	1.70135i
$100$		0			0
$101$		−	9.50749i		−	0.946030i	−0.881054	−	0.473015i	$-0.843166\pi$
$101$		−	9.50749i		−	0.946030i	0.881054	−	0.473015i	$-0.156834\pi$
$102$		0			0
$103$			1.07180i			0.105607i	0.998605	+	0.0528036i	$0.0168157\pi$
$103$			1.07180i			0.105607i	−0.998605	+	0.0528036i	$0.983184\pi$
$104$		0			0
$105$	−2.26795	−	2.90931i	−0.221329	−	0.283920i
$106$		0			0
$107$	1.55910			0.150724			0.0753618	−	0.997156i	$-0.475989\pi$
$107$	1.55910			0.150724			0.0753618	+	0.997156i	$0.475989\pi$
$108$		0			0
$109$	−14.3923			−1.37853			−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923			−1.37853			−0.689266	+	0.724508i	$0.742067\pi$
$110$		0			0
$111$	−12.2079	−	15.6603i	−1.15872	−	1.48641i
$112$		0			0
$113$		−	14.3377i		−	1.34878i	−0.738377	−	0.674388i	$-0.764408\pi$
$113$		−	14.3377i		−	1.34878i	0.738377	−	0.674388i	$-0.235592\pi$
$114$		0			0
$115$		−	9.07180i		−	0.845949i
$116$		0			0
$117$	−3.07180	+	12.2079i	−0.283988	+	1.12862i
$118$		0			0
$119$	−5.81863			−0.533393
$120$		0			0
$121$	22.8564			2.07786
$122$		0			0
$123$	2.12976	−	1.66025i	0.192034	−	0.149700i
$124$		0			0
$125$			11.6373i			1.04087i
$126$		0			0
$127$		−	14.3923i		−	1.27711i	−0.769576	−	0.638555i	$-0.779532\pi$
$127$		−	14.3923i		−	1.27711i	0.769576	−	0.638555i	$-0.220468\pi$
$128$		0			0
$129$	2.73205	−	2.12976i	0.240544	−	0.187515i
$130$		0			0
$131$	18.0265			1.57499			0.787493	−	0.616323i	$-0.211378\pi$
$131$	18.0265			1.57499			0.787493	+	0.616323i	$0.211378\pi$
$132$		0			0
$133$	−2.73205			−0.236899
$134$		0			0
$135$	4.46841	+	10.1244i	0.384579	+	0.871366i
$136$		0			0
$137$			8.51906i			0.727832i	0.931432	+	0.363916i	$0.118561\pi$
$137$			8.51906i			0.727832i	−0.931432	+	0.363916i	$0.881439\pi$
$138$		0			0
$139$			11.1244i			0.943556i	0.881717	+	0.471778i	$0.156388\pi$
$139$			11.1244i			0.943556i	−0.881717	+	0.471778i	$0.843612\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	24.4158			2.04176
$144$		0			0
$145$	−12.3923			−1.02912
$146$		0			0
$147$	1.06488	+	1.36603i	0.0878300	+	0.112668i
$148$		0			0
$149$			21.7154i			1.77900i	0.456940	+	0.889498i	$0.348946\pi$
$149$			21.7154i			1.77900i	−0.456940	+	0.889498i	$0.651054\pi$
$150$		0			0
$151$		−	0.535898i		−	0.0436108i	−0.999762	−	0.0218054i	$-0.993059\pi$
$151$		−	0.535898i		−	0.0436108i	0.999762	−	0.0218054i	$-0.00694142\pi$
$152$		0			0
$153$	16.9282	+	4.25953i	1.36856	+	0.344363i
$154$		0			0
$155$	−5.40087			−0.433808
$156$		0			0
$157$	−1.26795			−0.101193			−0.0505967	−	0.998719i	$-0.516112\pi$
$157$	−1.26795			−0.101193			−0.0505967	+	0.998719i	$0.516112\pi$
$158$		0			0
$159$	−2.12976	+	1.66025i	−0.168901	+	0.131667i
$160$		0			0
$161$			4.25953i			0.335698i
$162$		0			0
$163$		−	20.9282i		−	1.63922i	−0.572919	−	0.819612i	$-0.694189\pi$
$163$		−	20.9282i		−	1.63922i	0.572919	−	0.819612i	$-0.305811\pi$
$164$		0			0
$165$	16.9282	−	13.1963i	1.31786	−	1.02733i
$166$		0			0
$167$	−1.14134			−0.0883194			−0.0441597	−	0.999024i	$-0.514061\pi$
$167$	−1.14134			−0.0883194			−0.0441597	+	0.999024i	$0.514061\pi$
$168$		0			0
$169$	4.60770			0.354438
$170$		0			0
$171$	7.94839	+	2.00000i	0.607829	+	0.152944i
$172$		0			0
$173$		−	6.38929i		−	0.485769i	−0.970055	−	0.242885i	$-0.921906\pi$
$173$		−	6.38929i		−	0.485769i	0.970055	−	0.242885i	$-0.0780936\pi$
$174$		0			0
$175$		−	0.464102i		−	0.0350828i
$176$		0			0
$177$	10.1244	+	12.9875i	0.760993	+	0.976198i
$178$		0			0
$179$	−21.7154			−1.62309			−0.811543	−	0.584293i	$-0.801372\pi$
$179$	−21.7154			−1.62309			−0.811543	+	0.584293i	$0.801372\pi$
$180$		0			0
$181$	−10.7321			−0.797707			−0.398854	−	0.917015i	$-0.630592\pi$
$181$	−10.7321			−0.797707			−0.398854	+	0.917015i	$0.630592\pi$
$182$		0			0
$183$	1.35022	+	1.73205i	0.0998109	+	0.128037i
$184$		0			0
$185$			24.4158i			1.79509i
$186$		0			0
$187$		−	33.8564i		−	2.47583i
$188$		0			0
$189$	−2.09808	−	4.75374i	−0.152613	−	0.345784i
$190$		0			0
$191$	−10.0782			−0.729230			−0.364615	−	0.931158i	$-0.618799\pi$
$191$	−10.0782			−0.729230			−0.364615	+	0.931158i	$0.618799\pi$
$192$		0			0
$193$	−2.53590			−0.182538			−0.0912690	−	0.995826i	$-0.529092\pi$
$193$	−2.53590			−0.182538			−0.0912690	+	0.995826i	$0.529092\pi$
$194$		0			0
$195$	12.2079	−	9.51666i	0.874227	−	0.681502i
$196$		0			0
$197$			10.0782i			0.718039i	0.933330	+	0.359019i	$0.116889\pi$
$197$			10.0782i			0.718039i	−0.933330	+	0.359019i	$0.883111\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$	4.73205	−	3.68886i	0.333773	−	0.260192i
$202$		0			0
$203$	5.81863			0.408387
$204$		0			0
$205$	−3.32051			−0.231914
$206$		0			0
$207$	3.11819	−	12.3923i	0.216729	−	0.861325i
$208$		0			0
$209$		−	15.8968i		−	1.09960i
$210$		0			0
$211$			0.535898i			0.0368928i	0.999830	+	0.0184464i	$0.00587200\pi$
$211$			0.535898i			0.0368928i	−0.999830	+	0.0184464i	$0.994128\pi$
$212$		0			0
$213$	1.66025	+	2.12976i	0.113759	+	0.145929i
$214$		0			0
$215$	−4.25953			−0.290498
$216$		0			0
$217$	2.53590			0.172148
$218$		0			0
$219$	12.2079	+	15.6603i	0.824935	+	1.05822i
$220$		0			0
$221$		−	24.4158i		−	1.64239i
$222$		0			0
$223$			27.3205i			1.82952i	0.404002	+	0.914758i	$0.367619\pi$
$223$			27.3205i			1.82952i	−0.404002	+	0.914758i	$0.632381\pi$
$224$		0			0
$225$	−0.339746	+	1.35022i	−0.0226497	+	0.0900145i
$226$		0			0
$227$	16.8852			1.12071			0.560355	−	0.828252i	$-0.310665\pi$
$227$	16.8852			1.12071			0.560355	+	0.828252i	$0.310665\pi$
$228$		0			0
$229$	6.73205			0.444866			0.222433	−	0.974948i	$-0.428600\pi$
$229$	6.73205			0.444866			0.222433	+	0.974948i	$0.428600\pi$
$230$		0			0
$231$	−7.94839	+	6.19615i	−0.522966	+	0.407677i
$232$		0			0
$233$		−	8.51906i		−	0.558102i	−0.960276	−	0.279051i	$-0.909980\pi$
$233$		−	8.51906i		−	0.558102i	0.960276	−	0.279051i	$-0.0900199\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	16.3923	−	12.7786i	1.06479	−	0.830059i
$238$		0			0
$239$	1.14134			0.0738270			0.0369135	−	0.999318i	$-0.488247\pi$
$239$	1.14134			0.0738270			0.0369135	+	0.999318i	$0.488247\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	2.62398	+	15.3660i	0.168328	+	0.985731i
$244$		0			0
$245$		−	2.12976i		−	0.136066i
$246$		0			0
$247$		−	11.4641i		−	0.729443i
$248$		0			0
$249$	10.1244	+	12.9875i	0.641605	+	0.823047i
$250$		0			0
$251$	13.7670			0.868966			0.434483	−	0.900680i	$-0.356931\pi$
$251$	13.7670			0.868966			0.434483	+	0.900680i	$0.356931\pi$
$252$		0			0
$253$	−24.7846			−1.55820
$254$		0			0
$255$	−13.1963	−	16.9282i	−0.826387	−	1.06009i
$256$		0			0
$257$			14.3377i			0.894360i	0.894444	+	0.447180i	$0.147572\pi$
$257$			14.3377i			0.894360i	−0.894444	+	0.447180i	$0.852428\pi$
$258$		0			0
$259$		−	11.4641i		−	0.712345i
$260$		0			0
$261$	−16.9282	−	4.25953i	−1.04783	−	0.263658i
$262$		0			0
$263$	6.95996			0.429170			0.214585	−	0.976705i	$-0.431160\pi$
$263$	6.95996			0.429170			0.214585	+	0.976705i	$0.431160\pi$
$264$		0			0
$265$	3.32051			0.203977
$266$		0			0
$267$	−18.0265	+	14.0526i	−1.10321	+	0.860003i
$268$		0			0
$269$		−	14.9084i		−	0.908978i	−0.890752	−	0.454489i	$-0.849822\pi$
$269$		−	14.9084i		−	0.908978i	0.890752	−	0.454489i	$-0.150178\pi$
$270$		0			0
$271$			15.3205i			0.930655i	0.885139	+	0.465327i	$0.154063\pi$
$271$			15.3205i			0.930655i	−0.885139	+	0.465327i	$0.845937\pi$
$272$		0			0
$273$	−5.73205	+	4.46841i	−0.346919	+	0.270440i
$274$		0			0
$275$	2.70043			0.162842
$276$		0			0
$277$	−16.9282			−1.01712			−0.508559	−	0.861027i	$-0.669821\pi$
$277$	−16.9282			−1.01712			−0.508559	+	0.861027i	$0.669821\pi$
$278$		0			0
$279$	−7.37772	−	1.85641i	−0.441693	−	0.111140i
$280$		0			0
$281$			17.0381i			1.01641i	0.861236	+	0.508204i	$0.169691\pi$
$281$			17.0381i			1.01641i	−0.861236	+	0.508204i	$0.830309\pi$
$282$		0			0
$283$			29.6603i			1.76312i	0.472073	+	0.881560i	$0.343506\pi$
$283$			29.6603i			1.76312i	−0.472073	+	0.881560i	$0.656494\pi$
$284$		0			0
$285$	−6.19615	−	7.94839i	−0.367028	−	0.470822i
$286$		0			0
$287$	1.55910			0.0920305
$288$		0			0
$289$	−16.8564			−0.991553
$290$		0			0
$291$	5.24796	+	6.73205i	0.307641	+	0.394640i
$292$		0			0
$293$		−	5.24796i		−	0.306589i	−0.988181	−	0.153294i	$-0.951012\pi$
$293$		−	5.24796i		−	0.306589i	0.988181	−	0.153294i	$-0.0489883\pi$
$294$		0			0
$295$		−	20.2487i		−	1.17893i
$296$		0			0
$297$	27.6603	−	12.2079i	1.60501	−	0.708375i
$298$		0			0
$299$	−17.8736			−1.03366
$300$		0			0
$301$	2.00000			0.115278
$302$		0			0
$303$	−12.9875	+	10.1244i	−0.746111	+	0.581629i
$304$		0			0
$305$		−	2.70043i		−	0.154626i
$306$		0			0
$307$		−	8.87564i		−	0.506560i	−0.967393	−	0.253280i	$-0.918491\pi$
$307$		−	8.87564i		−	0.506560i	0.967393	−	0.253280i	$-0.0815094\pi$
$308$		0			0
$309$	1.46410	−	1.14134i	0.0832898	−	0.0649284i
$310$		0			0
$311$	−27.5340			−1.56131			−0.780656	−	0.624961i	$-0.785115\pi$
$311$	−27.5340			−1.56131			−0.780656	+	0.624961i	$0.785115\pi$
$312$		0			0
$313$	8.53590			0.482478			0.241239	−	0.970466i	$-0.422446\pi$
$313$	8.53590			0.482478			0.241239	+	0.970466i	$0.422446\pi$
$314$		0			0
$315$	−1.55910	+	6.19615i	−0.0878451	+	0.349114i
$316$		0			0
$317$		−	24.8336i		−	1.39479i	−0.716685	−	0.697397i	$-0.754341\pi$
$317$		−	24.8336i		−	1.39479i	0.716685	−	0.697397i	$-0.245659\pi$
$318$		0			0
$319$			33.8564i			1.89559i
$320$		0			0
$321$	−1.66025	−	2.12976i	−0.0926663	−	0.118872i
$322$		0			0
$323$	−15.8968			−0.884521
$324$		0			0
$325$	1.94744			0.108025
$326$		0			0
$327$	15.3261	+	19.6603i	0.847536	+	1.08721i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	10.7846i		−	0.592776i	−0.955068	−	0.296388i	$-0.904218\pi$
$331$		−	10.7846i		−	0.592776i	0.955068	−	0.296388i	$-0.0957821\pi$
$332$		0			0
$333$	−8.39230	+	33.3527i	−0.459895	+	1.82771i
$334$		0			0
$335$	−7.37772			−0.403088
$336$		0			0
$337$	20.3923			1.11084			0.555420	−	0.831570i	$-0.312558\pi$
$337$	20.3923			1.11084			0.555420	+	0.831570i	$0.312558\pi$
$338$		0			0
$339$	−19.5856	+	15.2679i	−1.06375	+	0.829241i
$340$		0			0
$341$			14.7554i			0.799052i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−12.3923	+	9.66040i	−0.667179	+	0.520098i
$346$		0			0
$347$	2.70043			0.144967			0.0724834	−	0.997370i	$-0.476908\pi$
$347$	2.70043			0.144967			0.0724834	+	0.997370i	$0.476908\pi$
$348$		0			0
$349$	−35.1244			−1.88016			−0.940082	−	0.340949i	$-0.889252\pi$
$349$	−35.1244			−1.88016			−0.940082	+	0.340949i	$0.889252\pi$
$350$		0			0
$351$	19.9474	−	8.80385i	1.06472	−	0.469915i
$352$		0			0
$353$		−	5.81863i		−	0.309694i	−0.987938	−	0.154847i	$-0.950512\pi$
$353$		−	5.81863i		−	0.309694i	0.987938	−	0.154847i	$-0.0494885\pi$
$354$		0			0
$355$		−	3.32051i		−	0.176234i
$356$		0			0
$357$	6.19615	+	7.94839i	0.327935	+	0.420673i
$358$		0			0
$359$	4.25953			0.224809			0.112405	−	0.993663i	$-0.464145\pi$
$359$	4.25953			0.224809			0.112405	+	0.993663i	$0.464145\pi$
$360$		0			0
$361$	11.5359			0.607153
$362$		0			0
$363$	−24.3394	−	31.2224i	−1.27749	−	1.63875i
$364$		0			0
$365$		−	24.4158i		−	1.27798i
$366$		0			0
$367$			5.46410i			0.285224i	0.989779	+	0.142612i	$0.0455501\pi$
$367$			5.46410i			0.285224i	−0.989779	+	0.142612i	$0.954450\pi$
$368$		0			0
$369$	−4.53590	−	1.14134i	−0.236129	−	0.0594157i
$370$		0			0
$371$	−1.55910			−0.0809442
$372$		0			0
$373$	10.7846			0.558406			0.279203	−	0.960232i	$-0.409930\pi$
$373$	10.7846			0.558406			0.279203	+	0.960232i	$0.409930\pi$
$374$		0			0
$375$	15.8968	−	12.3923i	0.820906	−	0.639936i
$376$		0			0
$377$			24.4158i			1.25748i
$378$		0			0
$379$			14.7846i			0.759434i	0.925103	+	0.379717i	$0.123979\pi$
$379$			14.7846i			0.759434i	−0.925103	+	0.379717i	$0.876021\pi$
$380$		0			0
$381$	−19.6603	+	15.3261i	−1.00723	+	0.785181i
$382$		0			0
$383$	−37.1944			−1.90055			−0.950273	−	0.311417i	$-0.899196\pi$
$383$	−37.1944			−1.90055			−0.950273	+	0.311417i	$0.899196\pi$
$384$		0			0
$385$	12.3923			0.631570
$386$		0			0
$387$	−5.81863	−	1.46410i	−0.295777	−	0.0744245i
$388$		0			0
$389$			8.93682i			0.453115i	0.973998	+	0.226557i	$0.0727471\pi$
$389$			8.93682i			0.453115i	−0.973998	+	0.226557i	$0.927253\pi$
$390$		0			0
$391$			24.7846i			1.25341i
$392$		0			0
$393$	−19.1962	−	24.6247i	−0.968318	−	1.24215i
$394$		0			0
$395$	−25.5572			−1.28592
$396$		0			0
$397$	2.73205			0.137118			0.0685588	−	0.997647i	$-0.478160\pi$
$397$	2.73205			0.137118			0.0685588	+	0.997647i	$0.478160\pi$
$398$		0			0
$399$	2.90931	+	3.73205i	0.145648	+	0.186836i
$400$		0			0
$401$		−	20.5741i		−	1.02742i	−0.857964	−	0.513710i	$-0.828271\pi$
$401$		−	20.5741i		−	1.02742i	0.857964	−	0.513710i	$-0.171729\pi$
$402$		0			0
$403$			10.6410i			0.530067i
$404$		0			0
$405$	9.07180	−	16.8852i	0.450781	−	0.839033i
$406$		0			0
$407$	66.7053			3.30646
$408$		0			0
$409$	17.3205			0.856444			0.428222	−	0.903674i	$-0.359140\pi$
$409$	17.3205			0.856444			0.428222	+	0.903674i	$0.359140\pi$
$410$		0			0
$411$	11.6373	−	9.07180i	0.574023	−	0.447479i
$412$		0			0
$413$			9.50749i			0.467833i
$414$		0			0
$415$		−	20.2487i		−	0.993970i
$416$		0			0
$417$	15.1962	−	11.8461i	0.744159	−	0.580108i
$418$		0			0
$419$	38.1829			1.86535			0.932677	−	0.360712i	$-0.117466\pi$
$419$	38.1829			1.86535			0.932677	+	0.360712i	$0.117466\pi$
$420$		0			0
$421$	3.85641			0.187950			0.0939749	−	0.995575i	$-0.470043\pi$
$421$	3.85641			0.187950			0.0939749	+	0.995575i	$0.470043\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.70043i		−	0.130990i
$426$		0			0
$427$			1.26795i			0.0613604i
$428$		0			0
$429$	−26.0000	−	33.3527i	−1.25529	−	1.61028i
$430$		0			0
$431$	−15.8968			−0.765721			−0.382861	−	0.923806i	$-0.625061\pi$
$431$	−15.8968			−0.765721			−0.382861	+	0.923806i	$0.625061\pi$
$432$		0			0
$433$	−22.7846			−1.09496			−0.547479	−	0.836819i	$-0.684412\pi$
$433$	−22.7846			−1.09496			−0.547479	+	0.836819i	$0.684412\pi$
$434$		0			0
$435$	13.1963	+	16.9282i	0.632716	+	0.811645i
$436$		0			0
$437$			11.6373i			0.556685i
$438$		0			0
$439$		−	1.85641i		−	0.0886014i	−0.999018	−	0.0443007i	$-0.985894\pi$
$439$		−	1.85641i		−	0.0886014i	0.999018	−	0.0443007i	$-0.0141060\pi$
$440$		0			0
$441$	0.732051	−	2.90931i	0.0348596	−	0.138539i
$442$		0			0
$443$	15.4790			0.735431			0.367715	−	0.929938i	$-0.380140\pi$
$443$	15.4790			0.735431			0.367715	+	0.929938i	$0.380140\pi$
$444$		0			0
$445$	28.1051			1.33231
$446$		0			0
$447$	29.6638	−	23.1244i	1.40305	−	1.09374i
$448$		0			0
$449$			14.7554i			0.696352i	0.937429	+	0.348176i	$0.113199\pi$
$449$			14.7554i			0.696352i	−0.937429	+	0.348176i	$0.886801\pi$
$450$		0			0
$451$			9.07180i			0.427174i
$452$		0			0
$453$	−0.732051	+	0.570669i	−0.0343947	+	0.0268124i
$454$		0			0
$455$	8.93682			0.418964
$456$		0			0
$457$	−28.3923			−1.32814			−0.664068	−	0.747672i	$-0.731171\pi$
$457$	−28.3923			−1.32814			−0.664068	+	0.747672i	$0.731171\pi$
$458$		0			0
$459$	−12.2079	−	27.6603i	−0.569817	−	1.29107i
$460$		0			0
$461$			18.0265i			0.839580i	0.907621	+	0.419790i	$0.137896\pi$
$461$			18.0265i			0.839580i	−0.907621	+	0.419790i	$0.862104\pi$
$462$		0			0
$463$		−	31.7128i		−	1.47382i	−0.675991	−	0.736910i	$-0.736284\pi$
$463$		−	31.7128i		−	1.47382i	0.675991	−	0.736910i	$-0.263716\pi$
$464$		0			0
$465$	5.75129	+	7.37772i	0.266710	+	0.342134i
$466$		0			0
$467$	16.8852			0.781354			0.390677	−	0.920528i	$-0.372241\pi$
$467$	16.8852			0.781354			0.390677	+	0.920528i	$0.372241\pi$
$468$		0			0
$469$	3.46410			0.159957
$470$		0			0
$471$	1.35022	+	1.73205i	0.0622147	+	0.0798087i
$472$		0			0
$473$			11.6373i			0.535081i
$474$		0			0
$475$		−	1.26795i		−	0.0581775i
$476$		0			0
$477$	4.53590	+	1.14134i	0.207685	+	0.0522583i
$478$		0			0
$479$	3.11819			0.142474			0.0712369	−	0.997459i	$-0.477305\pi$
$479$	3.11819			0.142474			0.0712369	+	0.997459i	$0.477305\pi$
$480$		0			0
$481$	48.1051			2.19340
$482$		0			0
$483$	5.81863	−	4.53590i	0.264757	−	0.206391i
$484$		0			0
$485$		−	10.4959i		−	0.476595i
$486$		0			0
$487$		−	37.3205i		−	1.69115i	−0.533854	−	0.845577i	$-0.679257\pi$
$487$		−	37.3205i		−	1.69115i	0.533854	−	0.845577i	$-0.320743\pi$
$488$		0			0
$489$	−28.5885	+	22.2861i	−1.29281	+	1.00781i
$490$		0			0
$491$	−21.7154			−0.980003			−0.490001	−	0.871722i	$-0.663004\pi$
$491$	−21.7154			−0.980003			−0.490001	+	0.871722i	$0.663004\pi$
$492$		0			0
$493$	33.8564			1.52482
$494$		0			0
$495$	−36.0531	−	9.07180i	−1.62047	−	0.407747i
$496$		0			0
$497$			1.55910i			0.0699350i
$498$		0			0
$499$		−	9.32051i		−	0.417243i	−0.977996	−	0.208622i	$-0.933102\pi$
$499$		−	9.32051i		−	0.417243i	0.977996	−	0.208622i	$-0.0668977\pi$
$500$		0			0
$501$	1.21539	+	1.55910i	0.0542996	+	0.0696553i
$502$		0			0
$503$	−12.7786			−0.569769			−0.284885	−	0.958562i	$-0.591955\pi$
$503$	−12.7786			−0.569769			−0.284885	+	0.958562i	$0.591955\pi$
$504$		0			0
$505$	20.2487			0.901056
$506$		0			0
$507$	−4.90665	−	6.29423i	−0.217912	−	0.279537i
$508$		0			0
$509$		−	9.50749i		−	0.421412i	−0.977549	−	0.210706i	$-0.932424\pi$
$509$		−	9.50749i		−	0.421412i	0.977549	−	0.210706i	$-0.0675763\pi$
$510$		0			0
$511$			11.4641i			0.507142i
$512$		0			0
$513$	−5.73205	−	12.9875i	−0.253076	−	0.573411i
$514$		0			0
$515$	−2.28268			−0.100587
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−8.72794	+	6.80385i	−0.383114	+	0.298656i
$520$		0			0
$521$			1.55910i			0.0683052i	0.999417	+	0.0341526i	$0.0108732\pi$
$521$			1.55910i			0.0683052i	−0.999417	+	0.0341526i	$0.989127\pi$
$522$		0			0
$523$			23.1244i			1.01116i	0.862780	+	0.505579i	$0.168721\pi$
$523$			23.1244i			1.01116i	−0.862780	+	0.505579i	$0.831279\pi$
$524$		0			0
$525$	−0.633975	+	0.494214i	−0.0276689	+	0.0215693i
$526$		0			0
$527$	14.7554			0.642757
$528$		0			0
$529$	−4.85641			−0.211148
$530$		0			0
$531$	6.95996	−	27.6603i	0.302037	−	1.20035i
$532$		0			0
$533$			6.54220i			0.283374i
$534$		0			0
$535$			3.32051i			0.143558i
$536$		0			0
$537$	23.1244	+	29.6638i	0.997890	+	1.28009i
$538$		0			0
$539$	−5.81863			−0.250626
$540$		0			0
$541$	3.07180			0.132067			0.0660334	−	0.997817i	$-0.478966\pi$
$541$	3.07180			0.132067			0.0660334	+	0.997817i	$0.478966\pi$
$542$		0			0
$543$	11.4284	+	14.6603i	0.490438	+	0.629132i
$544$		0			0
$545$		−	30.6522i		−	1.31300i
$546$		0			0
$547$		−	20.9282i		−	0.894825i	−0.894328	−	0.447413i	$-0.852346\pi$
$547$		−	20.9282i		−	0.894825i	0.894328	−	0.447413i	$-0.147654\pi$
$548$		0			0
$549$	0.928203	−	3.68886i	0.0396147	−	0.157437i
$550$		0			0
$551$	15.8968			0.677226
$552$		0			0
$553$	12.0000			0.510292
$554$		0			0
$555$	33.3527	−	26.0000i	1.41574	−	1.10364i
$556$		0			0
$557$			41.8717i			1.77416i	0.461614	+	0.887081i	$0.347270\pi$
$557$			41.8717i			1.77416i	−0.461614	+	0.887081i	$0.652730\pi$
$558$		0			0
$559$			8.39230i			0.354957i
$560$		0			0
$561$	−46.2487	+	36.0531i	−1.95262	+	1.52216i
$562$		0			0
$563$	−0.988427			−0.0416572			−0.0208286	−	0.999783i	$-0.506630\pi$
$563$	−0.988427			−0.0416572			−0.0208286	+	0.999783i	$0.506630\pi$
$564$		0			0
$565$	30.5359			1.28465
$566$		0			0
$567$	−4.25953	+	7.92820i	−0.178883	+	0.332953i
$568$		0			0
$569$		−	32.2113i		−	1.35037i	−0.737649	−	0.675184i	$-0.764064\pi$
$569$		−	32.2113i		−	1.35037i	0.737649	−	0.675184i	$-0.235936\pi$
$570$		0			0
$571$			26.0000i			1.08807i	0.839064	+	0.544033i	$0.183103\pi$
$571$			26.0000i			1.08807i	−0.839064	+	0.544033i	$0.816897\pi$
$572$		0			0
$573$	10.7321	+	13.7670i	0.448338	+	0.575125i
$574$		0			0
$575$	−1.97685			−0.0824405
$576$		0			0
$577$	−3.07180			−0.127881			−0.0639403	−	0.997954i	$-0.520367\pi$
$577$	−3.07180			−0.127881			−0.0639403	+	0.997954i	$0.520367\pi$
$578$		0			0
$579$	2.70043	+	3.46410i	0.112226	+	0.143963i
$580$		0			0
$581$			9.50749i			0.394437i
$582$		0			0
$583$		−	9.07180i		−	0.375715i
$584$		0			0
$585$	−26.0000	−	6.54220i	−1.07497	−	0.270487i
$586$		0			0
$587$	−12.6257			−0.521118			−0.260559	−	0.965458i	$-0.583907\pi$
$587$	−12.6257			−0.521118			−0.260559	+	0.965458i	$0.583907\pi$
$588$		0			0
$589$	6.92820			0.285472
$590$		0			0
$591$	13.7670	−	10.7321i	0.566299	−	0.441458i
$592$		0			0
$593$			20.5741i			0.844876i	0.906392	+	0.422438i	$0.138826\pi$
$593$			20.5741i			0.844876i	−0.906392	+	0.422438i	$0.861174\pi$
$594$		0			0
$595$		−	12.3923i		−	0.508035i
$596$		0			0
$597$	−10.9282	+	8.51906i	−0.447262	+	0.348662i
$598$		0			0
$599$	10.0782			0.411782			0.205891	−	0.978575i	$-0.433991\pi$
$599$	10.0782			0.411782			0.205891	+	0.978575i	$0.433991\pi$
$600$		0			0
$601$	20.2487			0.825962			0.412981	−	0.910740i	$-0.364488\pi$
$601$	20.2487			0.825962			0.412981	+	0.910740i	$0.364488\pi$
$602$		0			0
$603$	−10.0782	−	2.53590i	−0.410414	−	0.103270i
$604$		0			0
$605$			48.6788i			1.97907i
$606$		0			0
$607$			12.3923i			0.502988i	0.967859	+	0.251494i	$0.0809219\pi$
$607$			12.3923i			0.502988i	−0.967859	+	0.251494i	$0.919078\pi$
$608$		0			0
$609$	−6.19615	−	7.94839i	−0.251081	−	0.322085i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.39230			0.0966243			0.0483121	−	0.998832i	$-0.484616\pi$
$613$	2.39230			0.0966243			0.0483121	+	0.998832i	$0.484616\pi$
$614$		0			0
$615$	3.53595	+	4.53590i	0.142583	+	0.182905i
$616$		0			0
$617$		−	25.9749i		−	1.04571i	−0.852421	−	0.522856i	$-0.824867\pi$
$617$		−	25.9749i		−	1.04571i	0.852421	−	0.522856i	$-0.175133\pi$
$618$		0			0
$619$			30.0526i			1.20791i	0.797017	+	0.603957i	$0.206410\pi$
$619$			30.0526i			1.20791i	−0.797017	+	0.603957i	$0.793590\pi$
$620$		0			0
$621$	−20.2487	+	8.93682i	−0.812553	+	0.358622i
$622$		0			0
$623$	−13.1963			−0.528701
$624$		0			0
$625$	−22.4641			−0.898564
$626$		0			0
$627$	−21.7154	+	16.9282i	−0.867230	+	0.676047i
$628$		0			0
$629$		−	66.7053i		−	2.65972i
$630$		0			0
$631$		−	5.07180i		−	0.201905i	−0.994891	−	0.100953i	$-0.967811\pi$
$631$		−	5.07180i		−	0.201905i	0.994891	−	0.100953i	$-0.0321890\pi$
$632$		0			0
$633$	0.732051	−	0.570669i	0.0290964	−	0.0226820i
$634$		0			0
$635$	30.6522			1.21640
$636$		0			0
$637$	−4.19615			−0.166258
$638$		0			0
$639$	1.14134	−	4.53590i	0.0451506	−	0.179437i
$640$		0			0
$641$		−	8.93682i		−	0.352983i	−0.984302	−	0.176492i	$-0.943525\pi$
$641$		−	8.93682i		−	0.352983i	0.984302	−	0.176492i	$-0.0564748\pi$
$642$		0			0
$643$			5.94744i			0.234544i	0.993100	+	0.117272i	$0.0374150\pi$
$643$			5.94744i			0.234544i	−0.993100	+	0.117272i	$0.962585\pi$
$644$		0			0
$645$	4.53590	+	5.81863i	0.178601	+	0.229108i
$646$		0			0
$647$	−10.4959			−0.412637			−0.206318	−	0.978485i	$-0.566148\pi$
$647$	−10.4959			−0.412637			−0.206318	+	0.978485i	$0.566148\pi$
$648$		0			0
$649$	−55.3205			−2.17152
$650$		0			0
$651$	−2.70043	−	3.46410i	−0.105838	−	0.135769i
$652$		0			0
$653$			34.4940i			1.34985i	0.737884	+	0.674927i	$0.235825\pi$
$653$			34.4940i			1.34985i	−0.737884	+	0.674927i	$0.764175\pi$
$654$		0			0
$655$			38.3923i			1.50011i
$656$		0			0
$657$	8.39230	−	33.3527i	0.327415	−	1.30121i
$658$		0			0
$659$	−19.7386			−0.768905			−0.384452	−	0.923145i	$-0.625610\pi$
$659$	−19.7386			−0.768905			−0.384452	+	0.923145i	$0.625610\pi$
$660$		0			0
$661$	38.4449			1.49533			0.747666	−	0.664075i	$-0.231174\pi$
$661$	38.4449			1.49533			0.747666	+	0.664075i	$0.231174\pi$
$662$		0			0
$663$	−33.3527	+	26.0000i	−1.29531	+	1.00976i
$664$		0			0
$665$		−	5.81863i		−	0.225637i
$666$		0			0
$667$		−	24.7846i		−	0.959664i
$668$		0			0
$669$	37.3205	−	29.0931i	1.44289	−	1.12481i
$670$		0			0
$671$	−7.37772			−0.284814
$672$		0			0
$673$	−17.7128			−0.682779			−0.341389	−	0.939922i	$-0.610897\pi$
$673$	−17.7128			−0.682779			−0.341389	+	0.939922i	$0.610897\pi$
$674$		0			0
$675$	2.20622	−	0.973721i	0.0849174	−	0.0374785i
$676$		0			0
$677$		−	26.5456i		−	1.02023i	−0.860106	−	0.510115i	$-0.829603\pi$
$677$		−	26.5456i		−	1.02023i	0.860106	−	0.510115i	$-0.170397\pi$
$678$		0			0
$679$			4.92820i			0.189127i
$680$		0			0
$681$	−17.9808	−	23.0656i	−0.689024	−	0.883877i
$682$		0			0
$683$	41.8717			1.60218			0.801088	−	0.598546i	$-0.204255\pi$
$683$	41.8717			1.60218			0.801088	+	0.598546i	$0.204255\pi$
$684$		0			0
$685$	−18.1436			−0.693231
$686$		0			0
$687$	−7.16884	−	9.19615i	−0.273508	−	0.350855i
$688$		0			0
$689$		−	6.54220i		−	0.249238i
$690$		0			0
$691$		−	17.6603i		−	0.671828i	−0.941893	−	0.335914i	$-0.890955\pi$
$691$		−	17.6603i		−	0.671828i	0.941893	−	0.335914i	$-0.109045\pi$
$692$		0			0
$693$	16.9282	+	4.25953i	0.643049	+	0.161806i
$694$		0			0
$695$	−23.6923			−0.898699
$696$		0			0
$697$	9.07180			0.343619
$698$		0			0
$699$	−11.6373	+	9.07180i	−0.440161	+	0.343127i
$700$		0			0
$701$		−	46.1312i		−	1.74235i	−0.490970	−	0.871177i	$-0.663357\pi$
$701$		−	46.1312i		−	1.74235i	0.490970	−	0.871177i	$-0.336643\pi$
$702$		0			0
$703$		−	31.3205i		−	1.18128i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.50749			−0.357566
$708$		0			0
$709$	−8.53590			−0.320572			−0.160286	−	0.987071i	$-0.551242\pi$
$709$	−8.53590			−0.320572			−0.160286	+	0.987071i	$0.551242\pi$
$710$		0			0
$711$	−34.9118	−	8.78461i	−1.30929	−	0.329449i
$712$		0			0
$713$		−	10.8017i		−	0.404528i
$714$		0			0
$715$			52.0000i			1.94469i
$716$		0			0
$717$	−1.21539	−	1.55910i	−0.0453896	−	0.0582255i
$718$		0			0
$719$	11.6373			0.433996			0.216998	−	0.976172i	$-0.430373\pi$
$719$	11.6373			0.433996			0.216998	+	0.976172i	$0.430373\pi$
$720$		0			0
$721$	1.07180			0.0399158
$722$		0			0
$723$	14.9084	+	19.1244i	0.554448	+	0.711242i
$724$		0			0
$725$			2.70043i			0.100292i
$726$		0			0
$727$		−	10.9282i		−	0.405305i	−0.979251	−	0.202652i	$-0.935044\pi$
$727$		−	10.9282i		−	0.405305i	0.979251	−	0.202652i	$-0.0649561\pi$
$728$		0			0
$729$	18.1962	−	19.9474i	0.673932	−	0.738794i
$730$		0			0
$731$	11.6373			0.430419
$732$		0			0
$733$	17.2679			0.637806			0.318903	−	0.947787i	$-0.396686\pi$
$733$	17.2679			0.637806			0.318903	+	0.947787i	$0.396686\pi$
$734$		0			0
$735$	−2.90931	+	2.26795i	−0.107312	+	0.0836546i
$736$		0			0
$737$			20.1563i			0.742467i
$738$		0			0
$739$			19.0718i			0.701567i	0.936457	+	0.350784i	$0.114085\pi$
$739$			19.0718i			0.701567i	−0.936457	+	0.350784i	$0.885915\pi$
$740$		0			0
$741$	−15.6603	+	12.2079i	−0.575294	+	0.448469i
$742$		0			0
$743$	−33.7704			−1.23892			−0.619458	−	0.785030i	$-0.712648\pi$
$743$	−33.7704			−1.23892			−0.619458	+	0.785030i	$0.712648\pi$
$744$		0			0
$745$	−46.2487			−1.69442
$746$		0			0
$747$	6.95996	−	27.6603i	0.254652	−	1.01204i
$748$		0			0
$749$		−	1.55910i		−	0.0569681i
$750$		0			0
$751$			32.2487i			1.17677i	0.808580	+	0.588386i	$0.200236\pi$
$751$			32.2487i			1.17677i	−0.808580	+	0.588386i	$0.799764\pi$
$752$		0			0
$753$	−14.6603	−	18.8061i	−0.534249	−	0.685332i
$754$		0			0
$755$	1.14134			0.0415375
$756$		0			0
$757$	−45.0333			−1.63676			−0.818382	−	0.574675i	$-0.805129\pi$
$757$	−45.0333			−1.63676			−0.818382	+	0.574675i	$0.805129\pi$
$758$		0			0
$759$	26.3927	+	33.8564i	0.957994	+	1.22891i
$760$		0			0
$761$			36.4709i			1.32207i	0.750356	+	0.661034i	$0.229882\pi$
$761$			36.4709i			1.32207i	−0.750356	+	0.661034i	$0.770118\pi$
$762$		0			0
$763$			14.3923i			0.521036i
$764$		0			0
$765$	−9.07180	+	36.0531i	−0.327992	+	1.30350i
$766$		0			0
$767$	−39.8949			−1.44052
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$	19.5856	−	15.2679i	0.705360	−	0.549862i
$772$		0			0
$773$			2.12976i			0.0766023i	0.999266	+	0.0383012i	$0.0121946\pi$
$773$			2.12976i			0.0766023i	−0.999266	+	0.0383012i	$0.987805\pi$
$774$		0			0
$775$			1.17691i			0.0422760i
$776$		0			0
$777$	−15.6603	+	12.2079i	−0.561808	+	0.437957i
$778$		0			0
$779$	4.25953			0.152613
$780$		0			0
$781$	−9.07180			−0.324614
$782$		0			0
$783$	12.2079	+	27.6603i	0.436275	+	0.988497i
$784$		0			0
$785$		−	2.70043i		−	0.0963826i
$786$		0			0
$787$			0.588457i			0.0209762i	0.999945	+	0.0104881i	$0.00333853\pi$
$787$			0.588457i			0.0209762i	−0.999945	+	0.0104881i	$0.996661\pi$
$788$		0			0
$789$	−7.41154	−	9.50749i	−0.263858	−	0.338475i
$790$		0			0
$791$	−14.3377			−0.509789
$792$		0			0
$793$	−5.32051			−0.188937
$794$		0			0
$795$	−3.53595	−	4.53590i	−0.125407	−	0.160872i
$796$		0			0
$797$			38.1829i			1.35251i	0.736669	+	0.676253i	$0.236397\pi$
$797$			38.1829i			1.35251i	−0.736669	+	0.676253i	$0.763603\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	38.3923	+	9.66040i	1.35653	+	0.341333i
$802$		0			0
$803$	−66.7053			−2.35398
$804$		0			0
$805$	−9.07180			−0.319739
$806$		0			0
$807$	−20.3652	+	15.8756i	−0.716889	+	0.558849i
$808$		0			0
$809$		−	34.4940i		−	1.21274i	−0.795181	−	0.606372i	$-0.792624\pi$
$809$		−	34.4940i		−	1.21274i	0.795181	−	0.606372i	$-0.207376\pi$
$810$		0			0
$811$			29.6603i			1.04151i	0.853705	+	0.520756i	$0.174350\pi$
$811$			29.6603i			1.04151i	−0.853705	+	0.520756i	$0.825650\pi$
$812$		0			0
$813$	20.9282	−	16.3145i	0.733984	−	0.572176i
$814$		0			0
$815$	44.5722			1.56129
$816$		0			0
$817$	5.46410			0.191165
$818$		0			0
$819$	12.2079	+	3.07180i	0.426579	+	0.107337i
$820$		0			0
$821$			7.79548i			0.272064i	0.990704	+	0.136032i	$0.0434350\pi$
$821$			7.79548i			0.272064i	−0.990704	+	0.136032i	$0.956565\pi$
$822$		0			0
$823$		−	48.0000i		−	1.67317i	−0.547833	−	0.836587i	$-0.684547\pi$
$823$		−	48.0000i		−	1.67317i	0.547833	−	0.836587i	$-0.315453\pi$
$824$		0			0
$825$	−2.87564	−	3.68886i	−0.100117	−	0.128430i
$826$		0			0
$827$	−10.0782			−0.350452			−0.175226	−	0.984528i	$-0.556066\pi$
$827$	−10.0782			−0.350452			−0.175226	+	0.984528i	$0.556066\pi$
$828$		0			0
$829$	30.7321			1.06737			0.533684	−	0.845684i	$-0.320807\pi$
$829$	30.7321			1.06737			0.533684	+	0.845684i	$0.320807\pi$
$830$		0			0
$831$	18.0265	+	23.1244i	0.625334	+	0.802175i
$832$		0			0
$833$			5.81863i			0.201603i
$834$		0			0
$835$		−	2.43078i		−	0.0841206i
$836$		0			0
$837$	5.32051	+	12.0550i	0.183904	+	0.416682i
$838$		0			0
$839$	−19.0150			−0.656470			−0.328235	−	0.944596i	$-0.606454\pi$
$839$	−19.0150			−0.656470			−0.328235	+	0.944596i	$0.606454\pi$
$840$		0			0
$841$	−4.85641			−0.167462
$842$		0			0
$843$	23.2745	−	18.1436i	0.801616	−	0.624899i
$844$		0			0
$845$			9.81331i			0.337588i
$846$		0			0
$847$		−	22.8564i		−	0.785355i
$848$		0			0
$849$	40.5167	−	31.5847i	1.39053	−	1.08398i
$850$		0			0
$851$	−48.8317			−1.67393
$852$		0			0
$853$	5.94744			0.203637			0.101818	−	0.994803i	$-0.467534\pi$
$853$	5.94744			0.203637			0.101818	+	0.994803i	$0.467534\pi$
$854$		0			0
$855$	−4.25953	+	16.9282i	−0.145673	+	0.578932i
$856$		0			0
$857$		−	13.1963i		−	0.450779i	−0.974269	−	0.225389i	$-0.927635\pi$
$857$		−	13.1963i		−	0.450779i	0.974269	−	0.225389i	$-0.0723654\pi$
$858$		0			0
$859$		−	26.0526i		−	0.888902i	−0.895803	−	0.444451i	$-0.853399\pi$
$859$		−	26.0526i		−	0.888902i	0.895803	−	0.444451i	$-0.146601\pi$
$860$		0			0
$861$	−1.66025	−	2.12976i	−0.0565813	−	0.0725822i
$862$		0			0
$863$	48.1081			1.63762			0.818809	−	0.574065i	$-0.194634\pi$
$863$	48.1081			1.63762			0.818809	+	0.574065i	$0.194634\pi$
$864$		0			0
$865$	13.6077			0.462676
$866$		0			0
$867$	17.9501	+	23.0263i	0.609617	+	0.782013i
$868$		0			0
$869$			69.8235i			2.36860i
$870$		0			0
$871$			14.5359i			0.492530i
$872$		0			0
$873$	3.60770	−	14.3377i	0.122102	−	0.485257i
$874$		0			0
$875$	11.6373			0.393411
$876$		0			0
$877$	5.60770			0.189358			0.0946792	−	0.995508i	$-0.469817\pi$
$877$	5.60770			0.189358			0.0946792	+	0.995508i	$0.469817\pi$
$878$		0			0
$879$	−7.16884	+	5.58846i	−0.241799	+	0.188494i
$880$		0			0
$881$			22.8567i			0.770063i	0.922903	+	0.385032i	$0.125809\pi$
$881$			22.8567i			0.770063i	−0.922903	+	0.385032i	$0.874191\pi$
$882$		0			0
$883$			27.4641i			0.924241i	0.886817	+	0.462120i	$0.152911\pi$
$883$			27.4641i			0.924241i	−0.886817	+	0.462120i	$0.847089\pi$
$884$		0			0
$885$	−27.6603	+	21.5625i	−0.929789	+	0.724815i
$886$		0			0
$887$	15.8968			0.533762			0.266881	−	0.963730i	$-0.414007\pi$
$887$	15.8968			0.533762			0.266881	+	0.963730i	$0.414007\pi$
$888$		0			0
$889$	−14.3923			−0.482702
$890$		0			0
$891$	−46.1312	−	24.7846i	−1.54545	−	0.830316i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	46.2487i		−	1.54592i
$896$		0			0
$897$	19.0333	+	24.4158i	0.635504	+	0.815221i
$898$		0			0
$899$	−14.7554			−0.492122
$900$		0			0
$901$	−9.07180			−0.302225
$902$		0			0
$903$	−2.12976	−	2.73205i	−0.0708741	−	0.0909170i
$904$		0			0
$905$		−	22.8567i		−	0.759784i
$906$		0			0
$907$		−	20.2487i		−	0.672347i	−0.941800	−	0.336174i	$-0.890867\pi$
$907$		−	20.2487i		−	0.672347i	0.941800	−	0.336174i	$-0.109133\pi$
$908$		0			0
$909$	27.6603	+	6.95996i	0.917433	+	0.230847i
$910$		0			0
$911$	30.6522			1.01555			0.507777	−	0.861489i	$-0.330467\pi$
$911$	30.6522			1.01555			0.507777	+	0.861489i	$0.330467\pi$
$912$		0			0
$913$	−55.3205			−1.83084
$914$		0			0
$915$	−3.68886	+	2.87564i	−0.121950	+	0.0950659i
$916$		0			0
$917$		−	18.0265i		−	0.595289i
$918$		0			0
$919$		−	35.7128i		−	1.17806i	−0.808112	−	0.589028i	$-0.799511\pi$
$919$		−	35.7128i		−	1.17806i	0.808112	−	0.589028i	$-0.200489\pi$
$920$		0			0
$921$	−12.1244	+	9.45152i	−0.399511	+	0.311438i
$922$		0			0
$923$	−6.54220			−0.215339
$924$		0			0
$925$	5.32051			0.174937
$926$		0			0
$927$	−3.11819	−	0.784610i	−0.102415	−	0.0257700i
$928$		0			0
$929$			11.2195i			0.368100i	0.982917	+	0.184050i	$0.0589208\pi$
$929$			11.2195i			0.368100i	−0.982917	+	0.184050i	$0.941079\pi$
$930$		0			0
$931$			2.73205i			0.0895393i
$932$		0			0
$933$	29.3205	+	37.6122i	0.959910	+	1.23137i
$934$		0			0
$935$	72.1062			2.35812
$936$		0			0
$937$	−11.1769			−0.365134			−0.182567	−	0.983193i	$-0.558441\pi$
$937$	−11.1769			−0.365134			−0.182567	+	0.983193i	$0.558441\pi$
$938$		0			0
$939$	−9.08973	−	11.6603i	−0.296632	−	0.380518i
$940$		0			0
$941$		−	56.0565i		−	1.82739i	−0.406401	−	0.913695i	$-0.633216\pi$
$941$		−	56.0565i		−	1.82739i	0.406401	−	0.913695i	$-0.366784\pi$
$942$		0			0
$943$		−	6.64102i		−	0.216261i
$944$		0			0
$945$	10.1244	−	4.46841i	0.329345	−	0.145357i
$946$		0			0
$947$	12.0550			0.391735			0.195868	−	0.980630i	$-0.437248\pi$
$947$	12.0550			0.391735			0.195868	+	0.980630i	$0.437248\pi$
$948$		0			0
$949$	−48.1051			−1.56156
$950$		0			0
$951$	−33.9233	+	26.4449i	−1.10004	+	0.857533i
$952$		0			0
$953$		−	9.35458i		−	0.303024i	−0.988455	−	0.151512i	$-0.951586\pi$
$953$		−	9.35458i		−	0.303024i	0.988455	−	0.151512i	$-0.0484143\pi$
$954$		0			0
$955$		−	21.4641i		−	0.694562i
$956$		0			0
$957$	46.2487	−	36.0531i	1.49501	−	1.16543i
$958$		0			0
$959$	8.51906			0.275095
$960$		0			0
$961$	24.5692			0.792555
$962$		0			0
$963$	−1.14134	+	4.53590i	−0.0367791	+	0.146167i
$964$		0			0
$965$		−	5.40087i		−	0.173860i
$966$		0			0
$967$		−	36.5359i		−	1.17492i	−0.809255	−	0.587458i	$-0.800129\pi$
$967$		−	36.5359i		−	1.17492i	0.809255	−	0.587458i	$-0.199871\pi$
$968$		0			0
$969$	16.9282	+	21.7154i	0.543812	+	0.697599i
$970$		0			0
$971$	30.8051			0.988584			0.494292	−	0.869296i	$-0.335427\pi$
$971$	30.8051			0.988584			0.494292	+	0.869296i	$0.335427\pi$
$972$		0			0
$973$	11.1244			0.356630
$974$		0			0
$975$	−2.07380	−	2.66025i	−0.0664146	−	0.0851963i
$976$		0			0
$977$		−	3.11819i		−	0.0997598i	−0.998755	−	0.0498799i	$-0.984116\pi$
$977$		−	3.11819i		−	0.0997598i	0.998755	−	0.0498799i	$-0.0158839\pi$
$978$		0			0
$979$		−	76.7846i		−	2.45405i
$980$		0			0
$981$	10.5359	−	41.8717i	0.336385	−	1.33686i
$982$		0			0
$983$	−1.14134			−0.0364030			−0.0182015	−	0.999834i	$-0.505794\pi$
$983$	−1.14134			−0.0364030			−0.0182015	+	0.999834i	$0.505794\pi$
$984$		0			0
$985$	−21.4641			−0.683903
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	8.51906i		−	0.270890i
$990$		0			0
$991$		−	33.8564i		−	1.07548i	−0.843109	−	0.537742i	$-0.819277\pi$
$991$		−	33.8564i		−	1.07548i	0.843109	−	0.537742i	$-0.180723\pi$
$992$		0			0
$993$	−14.7321	+	11.4843i	−0.467508	+	0.364445i
$994$		0			0
$995$	17.0381			0.540145
$996$		0			0
$997$	−31.9090			−1.01057			−0.505284	−	0.862953i	$-0.668612\pi$
$997$	−31.9090			−1.01057			−0.505284	+	0.862953i	$0.668612\pi$
$998$		0			0
$999$	54.4974	−	24.0526i	1.72422	−	0.760989i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.h.b.239.3	✓	8
3.2	odd	2	inner	336.2.h.b.239.5	yes	8
4.3	odd	2	inner	336.2.h.b.239.6	yes	8
7.6	odd	2		2352.2.h.o.2255.6		8
8.3	odd	2		1344.2.h.g.575.3		8
8.5	even	2		1344.2.h.g.575.6		8
12.11	even	2	inner	336.2.h.b.239.4	yes	8
21.20	even	2		2352.2.h.o.2255.4		8
24.5	odd	2		1344.2.h.g.575.4		8
24.11	even	2		1344.2.h.g.575.5		8
28.27	even	2		2352.2.h.o.2255.3		8
84.83	odd	2		2352.2.h.o.2255.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.h.b.239.3	✓	8	1.1	even	1	trivial
336.2.h.b.239.4	yes	8	12.11	even	2	inner
336.2.h.b.239.5	yes	8	3.2	odd	2	inner
336.2.h.b.239.6	yes	8	4.3	odd	2	inner
1344.2.h.g.575.3		8	8.3	odd	2
1344.2.h.g.575.4		8	24.5	odd	2
1344.2.h.g.575.5		8	24.11	even	2
1344.2.h.g.575.6		8	8.5	even	2
2352.2.h.o.2255.3		8	28.27	even	2
2352.2.h.o.2255.4		8	21.20	even	2
2352.2.h.o.2255.5		8	84.83	odd	2
2352.2.h.o.2255.6		8	7.6	odd	2

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.06488 - 1.36603i) q^{3} +2.12976i q^{5} -1.00000i q^{7} +(-0.732051 + 2.90931i) q^{9} +O(q^{10})\)	\(q+(-1.06488 - 1.36603i) q^{3} +2.12976i q^{5} -1.00000i q^{7} +(-0.732051 + 2.90931i) q^{9} +5.81863 q^{11} +4.19615 q^{13} +(2.90931 - 2.26795i) q^{15} -5.81863i q^{17} -2.73205i q^{19} +(-1.36603 + 1.06488i) q^{21} -4.25953 q^{23} +0.464102 q^{25} +(4.75374 - 2.09808i) q^{27} +5.81863i q^{29} +2.53590i q^{31} +(-6.19615 - 7.94839i) q^{33} +2.12976 q^{35} +11.4641 q^{37} +(-4.46841 - 5.73205i) q^{39} +1.55910i q^{41} +2.00000i q^{43} +(-6.19615 - 1.55910i) q^{45} -1.00000 q^{49} +(-7.94839 + 6.19615i) q^{51} -1.55910i q^{53} +12.3923i q^{55} +(-3.73205 + 2.90931i) q^{57} -9.50749 q^{59} -1.26795 q^{61} +(2.90931 + 0.732051i) q^{63} +8.93682i q^{65} +3.46410i q^{67} +(4.53590 + 5.81863i) q^{69} -1.55910 q^{71} -11.4641 q^{73} +(-0.494214 - 0.633975i) q^{75} -5.81863i q^{77} +12.0000i q^{79} +(-7.92820 - 4.25953i) q^{81} -9.50749 q^{83} +12.3923 q^{85} +(7.94839 - 6.19615i) q^{87} -13.1963i q^{89} -4.19615i q^{91} +(3.46410 - 2.70043i) q^{93} +5.81863 q^{95} -4.92820 q^{97} +(-4.25953 + 16.9282i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 - 8 * q^13 - 4 * q^21 - 24 * q^25 - 8 * q^33 + 64 * q^37 - 8 * q^45 - 8 * q^49 - 16 * q^57 - 24 * q^61 + 64 * q^69 - 64 * q^73 - 8 * q^81 + 16 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more