LMFDB - Embedded newform 2070.2.e.b.1241.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1241,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2070.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.e (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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 $ x^16 + 32x^14 + 392x^12 + 2324x^10 + 6930x^8 + 9856x^6 + 5740x^4 + 1108*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.15
Root			$-1.90023i$ of defining polynomial
Character	$\chi$	$=$	2070.1241
Dual form			2070.2.e.b.1241.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.68734i q^{7} -1.00000i q^{8} +1.00000i q^{10} +1.73106 q^{11} -2.55609 q^{13} -2.68734 q^{14} +1.00000 q^{16} +1.27312 q^{17} +1.40738i q^{19} -1.00000 q^{20} +1.73106i q^{22} +(-4.59909 + 1.35955i) q^{23} +1.00000 q^{25} -2.55609i q^{26} -2.68734i q^{28} +9.62624i q^{29} -1.47605 q^{31} +1.00000i q^{32} +1.27312i q^{34} +2.68734i q^{35} -4.20404i q^{37} -1.40738 q^{38} -1.00000i q^{40} +3.40455i q^{41} +2.35544i q^{43} -1.73106 q^{44} +(-1.35955 - 4.59909i) q^{46} +3.21211i q^{47} -0.221775 q^{49} +1.00000i q^{50} +2.55609 q^{52} -1.80473 q^{53} +1.73106 q^{55} +2.68734 q^{56} -9.62624 q^{58} +7.74331i q^{59} -6.78016i q^{61} -1.47605i q^{62} -1.00000 q^{64} -2.55609 q^{65} -7.71912i q^{67} -1.27312 q^{68} -2.68734 q^{70} +4.44631i q^{71} -1.36193 q^{73} +4.20404 q^{74} -1.40738i q^{76} +4.65193i q^{77} +10.0972i q^{79} +1.00000 q^{80} -3.40455 q^{82} -8.67571 q^{83} +1.27312 q^{85} -2.35544 q^{86} -1.73106i q^{88} -3.72626 q^{89} -6.86909i q^{91} +(4.59909 - 1.35955i) q^{92} -3.21211 q^{94} +1.40738i q^{95} -7.03405i q^{97} -0.221775i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{5} - 24 q^{11} + 16 q^{16} - 16 q^{20} - 4 q^{23} + 16 q^{25} - 8 q^{31} - 8 q^{38} + 24 q^{44} - 4 q^{46} - 16 q^{49} + 8 q^{53} - 24 q^{55} + 16 q^{58} - 16 q^{64} + 32 q^{73}+ \cdots + 4 q^{92}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^5 - 24 * q^11 + 16 * q^16 - 16 * q^20 - 4 * q^23 + 16 * q^25 - 8 * q^31 - 8 * q^38 + 24 * q^44 - 4 * q^46 - 16 * q^49 + 8 * q^53 - 24 * q^55 + 16 * q^58 - 16 * q^64 + 32 * q^73 + 16 * q^80 + 32 * q^82 - 56 * q^83 - 8 * q^86 + 40 * q^89 + 4 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			2.68734i			1.01572i	0.861440	+	0.507859i	$0.169563\pi$
$7$			2.68734i			1.01572i	−0.861440	+	0.507859i	$0.830437\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			1.00000i			0.316228i
$11$	1.73106			0.521934			0.260967	−	0.965348i	$-0.415959\pi$
$11$	1.73106			0.521934			0.260967	+	0.965348i	$0.415959\pi$
$12$		0			0
$13$	−2.55609			−0.708933			−0.354467	−	0.935069i	$-0.615338\pi$
$13$	−2.55609			−0.708933			−0.354467	+	0.935069i	$0.615338\pi$
$14$	−2.68734			−0.718221
$15$		0			0
$16$	1.00000			0.250000
$17$	1.27312			0.308778			0.154389	−	0.988010i	$-0.450659\pi$
$17$	1.27312			0.308778			0.154389	+	0.988010i	$0.450659\pi$
$18$		0			0
$19$			1.40738i			0.322875i	0.986883	+	0.161437i	$0.0516130\pi$
$19$			1.40738i			0.322875i	−0.986883	+	0.161437i	$0.948387\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$			1.73106i			0.369063i
$23$	−4.59909	+	1.35955i	−0.958976	+	0.283486i
$23$	−4.59909	+	1.35955i	−0.958976	+	0.283486i
$24$		0			0
$25$	1.00000			0.200000
$26$		−	2.55609i		−	0.501291i
$27$		0			0
$28$		−	2.68734i		−	0.507859i
$29$			9.62624i			1.78755i	0.448518	+	0.893774i	$0.351952\pi$
$29$			9.62624i			1.78755i	−0.448518	+	0.893774i	$0.648048\pi$
$30$		0			0
$31$	−1.47605			−0.265106			−0.132553	−	0.991176i	$-0.542317\pi$
$31$	−1.47605			−0.265106			−0.132553	+	0.991176i	$0.542317\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			1.27312i			0.218339i
$35$			2.68734i			0.454243i
$36$		0			0
$37$		−	4.20404i		−	0.691140i	−0.938393	−	0.345570i	$-0.887686\pi$
$37$		−	4.20404i		−	0.691140i	0.938393	−	0.345570i	$-0.112314\pi$
$38$	−1.40738			−0.228307
$39$		0			0
$40$		−	1.00000i		−	0.158114i
$41$			3.40455i			0.531701i	0.964014	+	0.265850i	$0.0856527\pi$
$41$			3.40455i			0.531701i	−0.964014	+	0.265850i	$0.914347\pi$
$42$		0			0
$43$			2.35544i			0.359202i	0.983740	+	0.179601i	$0.0574807\pi$
$43$			2.35544i			0.359202i	−0.983740	+	0.179601i	$0.942519\pi$
$44$	−1.73106			−0.260967
$45$		0			0
$46$	−1.35955	−	4.59909i	−0.200455	−	0.678099i
$47$			3.21211i			0.468534i	0.972172	+	0.234267i	$0.0752690\pi$
$47$			3.21211i			0.468534i	−0.972172	+	0.234267i	$0.924731\pi$
$48$		0			0
$49$	−0.221775			−0.0316822
$50$			1.00000i			0.141421i
$51$		0			0
$52$	2.55609			0.354467
$53$	−1.80473			−0.247899			−0.123949	−	0.992289i	$-0.539556\pi$
$53$	−1.80473			−0.247899			−0.123949	+	0.992289i	$0.539556\pi$
$54$		0			0
$55$	1.73106			0.233416
$56$	2.68734			0.359110
$57$		0			0
$58$	−9.62624			−1.26399
$59$			7.74331i			1.00809i	0.863677	+	0.504046i	$0.168156\pi$
$59$			7.74331i			1.00809i	−0.863677	+	0.504046i	$0.831844\pi$
$60$		0			0
$61$		−	6.78016i		−	0.868111i	−0.900886	−	0.434055i	$-0.857082\pi$
$61$		−	6.78016i		−	0.868111i	0.900886	−	0.434055i	$-0.142918\pi$
$62$		−	1.47605i		−	0.187458i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.55609			−0.317045
$66$		0			0
$67$		−	7.71912i		−	0.943040i	−0.881855	−	0.471520i	$-0.843705\pi$
$67$		−	7.71912i		−	0.943040i	0.881855	−	0.471520i	$-0.156295\pi$
$68$	−1.27312			−0.154389
$69$		0			0
$70$	−2.68734			−0.321198
$71$			4.44631i			0.527679i	0.964567	+	0.263840i	$0.0849890\pi$
$71$			4.44631i			0.527679i	−0.964567	+	0.263840i	$0.915011\pi$
$72$		0			0
$73$	−1.36193			−0.159401			−0.0797006	−	0.996819i	$-0.525396\pi$
$73$	−1.36193			−0.159401			−0.0797006	+	0.996819i	$0.525396\pi$
$74$	4.20404			0.488710
$75$		0			0
$76$		−	1.40738i		−	0.161437i
$77$			4.65193i			0.530137i
$78$		0			0
$79$			10.0972i			1.13602i	0.823022	+	0.568010i	$0.192286\pi$
$79$			10.0972i			1.13602i	−0.823022	+	0.568010i	$0.807714\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−3.40455			−0.375969
$83$	−8.67571			−0.952283			−0.476141	−	0.879369i	$-0.657965\pi$
$83$	−8.67571			−0.952283			−0.476141	+	0.879369i	$0.657965\pi$
$84$		0			0
$85$	1.27312			0.138090
$86$	−2.35544			−0.253994
$87$		0			0
$88$		−	1.73106i		−	0.184531i
$89$	−3.72626			−0.394983			−0.197492	−	0.980305i	$-0.563280\pi$
$89$	−3.72626			−0.394983			−0.197492	+	0.980305i	$0.563280\pi$
$90$		0			0
$91$		−	6.86909i		−	0.720076i
$92$	4.59909	−	1.35955i	0.479488	−	0.141743i
$93$		0			0
$94$	−3.21211			−0.331304
$95$			1.40738i			0.144394i
$96$		0			0
$97$		−	7.03405i		−	0.714200i	−0.934066	−	0.357100i	$-0.883766\pi$
$97$		−	7.03405i		−	0.714200i	0.934066	−	0.357100i	$-0.116234\pi$
$98$		−	0.221775i		−	0.0224027i
$99$		0			0
$100$	−1.00000			−0.100000
$101$			12.9956i			1.29311i	0.762866	+	0.646557i	$0.223792\pi$
$101$			12.9956i			1.29311i	−0.762866	+	0.646557i	$0.776208\pi$
$102$		0			0
$103$			0.230115i			0.0226739i	0.999936	+	0.0113369i	$0.00360874\pi$
$103$			0.230115i			0.0226739i	−0.999936	+	0.0113369i	$0.996391\pi$
$104$			2.55609i			0.250646i
$105$		0			0
$106$		−	1.80473i		−	0.175291i
$107$	−6.45265			−0.623801			−0.311901	−	0.950115i	$-0.600966\pi$
$107$	−6.45265			−0.623801			−0.311901	+	0.950115i	$0.600966\pi$
$108$		0			0
$109$			6.93477i			0.664231i	0.943239	+	0.332115i	$0.107762\pi$
$109$			6.93477i			0.664231i	−0.943239	+	0.332115i	$0.892238\pi$
$110$			1.73106i			0.165050i
$111$		0			0
$112$			2.68734i			0.253929i
$113$	−1.03896			−0.0977371			−0.0488685	−	0.998805i	$-0.515562\pi$
$113$	−1.03896			−0.0977371			−0.0488685	+	0.998805i	$0.515562\pi$
$114$		0			0
$115$	−4.59909	+	1.35955i	−0.428867	+	0.126779i
$116$		−	9.62624i		−	0.893774i
$117$		0			0
$118$	−7.74331			−0.712829
$119$			3.42131i			0.313631i
$120$		0			0
$121$	−8.00344			−0.727585
$122$	6.78016			0.613847
$123$		0			0
$124$	1.47605			0.132553
$125$	1.00000			0.0894427
$126$		0			0
$127$	−11.3029			−1.00297			−0.501487	−	0.865165i	$-0.667213\pi$
$127$	−11.3029			−1.00297			−0.501487	+	0.865165i	$0.667213\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		−	2.55609i		−	0.224184i
$131$			13.1993i			1.15323i	0.817017	+	0.576614i	$0.195626\pi$
$131$			13.1993i			1.15323i	−0.817017	+	0.576614i	$0.804374\pi$
$132$		0			0
$133$	−3.78210			−0.327950
$134$	7.71912			0.666830
$135$		0			0
$136$		−	1.27312i		−	0.109169i
$137$	13.6218			1.16379			0.581895	−	0.813264i	$-0.302311\pi$
$137$	13.6218			1.16379			0.581895	+	0.813264i	$0.302311\pi$
$138$		0			0
$139$	−4.08370			−0.346375			−0.173187	−	0.984889i	$-0.555407\pi$
$139$	−4.08370			−0.346375			−0.173187	+	0.984889i	$0.555407\pi$
$140$		−	2.68734i		−	0.227121i
$141$		0			0
$142$	−4.44631			−0.373126
$143$	−4.42475			−0.370016
$144$		0			0
$145$			9.62624i			0.799416i
$146$		−	1.36193i		−	0.112714i
$147$		0			0
$148$			4.20404i			0.345570i
$149$	3.31717			0.271753			0.135877	−	0.990726i	$-0.456615\pi$
$149$	3.31717			0.271753			0.135877	+	0.990726i	$0.456615\pi$
$150$		0			0
$151$	−13.1448			−1.06971			−0.534854	−	0.844944i	$-0.679634\pi$
$151$	−13.1448			−1.06971			−0.534854	+	0.844944i	$0.679634\pi$
$152$	1.40738			0.114153
$153$		0			0
$154$	−4.65193			−0.374863
$155$	−1.47605			−0.118559
$156$		0			0
$157$		−	8.31606i		−	0.663694i	−0.943333	−	0.331847i	$-0.892328\pi$
$157$		−	8.31606i		−	0.663694i	0.943333	−	0.331847i	$-0.107672\pi$
$158$	−10.0972			−0.803287
$159$		0			0
$160$			1.00000i			0.0790569i
$161$	−3.65357	−	12.3593i	−0.287942	−	0.974049i
$162$		0			0
$163$	3.77069			0.295344			0.147672	−	0.989036i	$-0.452822\pi$
$163$	3.77069			0.295344			0.147672	+	0.989036i	$0.452822\pi$
$164$		−	3.40455i		−	0.265850i
$165$		0			0
$166$		−	8.67571i		−	0.673365i
$167$			16.9307i			1.31014i	0.755569	+	0.655070i	$0.227361\pi$
$167$			16.9307i			1.31014i	−0.755569	+	0.655070i	$0.772639\pi$
$168$		0			0
$169$	−6.46638			−0.497414
$170$			1.27312i			0.0976440i
$171$		0			0
$172$		−	2.35544i		−	0.179601i
$173$		−	5.36460i		−	0.407863i	−0.978985	−	0.203931i	$-0.934628\pi$
$173$		−	5.36460i		−	0.407863i	0.978985	−	0.203931i	$-0.0653720\pi$
$174$		0			0
$175$			2.68734i			0.203144i
$176$	1.73106			0.130483
$177$		0			0
$178$		−	3.72626i		−	0.279295i
$179$		−	9.56578i		−	0.714980i	−0.933917	−	0.357490i	$-0.883633\pi$
$179$		−	9.56578i		−	0.714980i	0.933917	−	0.357490i	$-0.116367\pi$
$180$		0			0
$181$			12.4563i			0.925873i	0.886391	+	0.462937i	$0.153204\pi$
$181$			12.4563i			0.925873i	−0.886391	+	0.462937i	$0.846796\pi$
$182$	6.86909			0.509171
$183$		0			0
$184$	1.35955	+	4.59909i	0.100227	+	0.339049i
$185$		−	4.20404i		−	0.309087i
$186$		0			0
$187$	2.20385			0.161161
$188$		−	3.21211i		−	0.234267i
$189$		0			0
$190$	−1.40738			−0.102102
$191$	8.39297			0.607294			0.303647	−	0.952785i	$-0.401796\pi$
$191$	8.39297			0.607294			0.303647	+	0.952785i	$0.401796\pi$
$192$		0			0
$193$	1.01621			0.0731486			0.0365743	−	0.999331i	$-0.488355\pi$
$193$	1.01621			0.0731486			0.0365743	+	0.999331i	$0.488355\pi$
$194$	7.03405			0.505016
$195$		0			0
$196$	0.221775			0.0158411
$197$			5.60884i			0.399613i	0.979835	+	0.199806i	$0.0640313\pi$
$197$			5.60884i			0.399613i	−0.979835	+	0.199806i	$0.935969\pi$
$198$		0			0
$199$			9.76909i			0.692513i	0.938140	+	0.346256i	$0.112547\pi$
$199$			9.76909i			0.692513i	−0.938140	+	0.346256i	$0.887453\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		0			0
$202$	−12.9956			−0.914369
$203$	−25.8689			−1.81564
$204$		0			0
$205$			3.40455i			0.237784i
$206$	−0.230115			−0.0160329
$207$		0			0
$208$	−2.55609			−0.177233
$209$			2.43625i			0.168519i
$210$		0			0
$211$	8.94211			0.615600			0.307800	−	0.951451i	$-0.400407\pi$
$211$	8.94211			0.615600			0.307800	+	0.951451i	$0.400407\pi$
$212$	1.80473			0.123949
$213$		0			0
$214$		−	6.45265i		−	0.441094i
$215$			2.35544i			0.160640i
$216$		0			0
$217$		−	3.96663i		−	0.269272i
$218$	−6.93477			−0.469682
$219$		0			0
$220$	−1.73106			−0.116708
$221$	−3.25422			−0.218903
$222$		0			0
$223$	29.4525			1.97229			0.986144	−	0.165892i	$-0.0530504\pi$
$223$	29.4525			1.97229			0.986144	+	0.165892i	$0.0530504\pi$
$224$	−2.68734			−0.179555
$225$		0			0
$226$		−	1.03896i		−	0.0691106i
$227$	0.140630			0.00933396			0.00466698	−	0.999989i	$-0.498514\pi$
$227$	0.140630			0.00933396			0.00466698	+	0.999989i	$0.498514\pi$
$228$		0			0
$229$		−	22.3375i		−	1.47610i	−0.674744	−	0.738052i	$-0.735746\pi$
$229$		−	22.3375i		−	1.47610i	0.674744	−	0.738052i	$-0.264254\pi$
$230$	−1.35955	−	4.59909i	−0.0896461	−	0.303255i
$231$		0			0
$232$	9.62624			0.631994
$233$		−	20.7396i		−	1.35870i	−0.733817	−	0.679348i	$-0.762263\pi$
$233$		−	20.7396i		−	1.35870i	0.733817	−	0.679348i	$-0.237737\pi$
$234$		0			0
$235$			3.21211i			0.209535i
$236$		−	7.74331i		−	0.504046i
$237$		0			0
$238$	−3.42131			−0.221770
$239$		−	5.54170i		−	0.358463i	−0.983807	−	0.179231i	$-0.942639\pi$
$239$		−	5.54170i		−	0.358463i	0.983807	−	0.179231i	$-0.0573610\pi$
$240$		0			0
$241$		−	15.1612i		−	0.976619i	−0.872670	−	0.488310i	$-0.837614\pi$
$241$		−	15.1612i		−	0.976619i	0.872670	−	0.488310i	$-0.162386\pi$
$242$		−	8.00344i		−	0.514481i
$243$		0			0
$244$			6.78016i			0.434055i
$245$	−0.221775			−0.0141687
$246$		0			0
$247$		−	3.59739i		−	0.228897i
$248$			1.47605i			0.0937290i
$249$		0			0
$250$			1.00000i			0.0632456i
$251$	26.3921			1.66585			0.832927	−	0.553383i	$-0.186663\pi$
$251$	26.3921			1.66585			0.832927	+	0.553383i	$0.186663\pi$
$252$		0			0
$253$	−7.96129	+	2.35346i	−0.500522	+	0.147961i
$254$		−	11.3029i		−	0.709209i
$255$		0			0
$256$	1.00000			0.0625000
$257$			1.04633i			0.0652682i	0.999467	+	0.0326341i	$0.0103896\pi$
$257$			1.04633i			0.0652682i	−0.999467	+	0.0326341i	$0.989610\pi$
$258$		0			0
$259$	11.2977			0.702003
$260$	2.55609			0.158522
$261$		0			0
$262$	−13.1993			−0.815455
$263$	−10.4405			−0.643791			−0.321895	−	0.946775i	$-0.604320\pi$
$263$	−10.4405			−0.643791			−0.321895	+	0.946775i	$0.604320\pi$
$264$		0			0
$265$	−1.80473			−0.110864
$266$		−	3.78210i		−	0.231895i
$267$		0			0
$268$			7.71912i			0.471520i
$269$			2.34357i			0.142890i	0.997445	+	0.0714450i	$0.0227611\pi$
$269$			2.34357i			0.142890i	−0.997445	+	0.0714450i	$0.977239\pi$
$270$		0			0
$271$	18.7738			1.14043			0.570213	−	0.821497i	$-0.306861\pi$
$271$	18.7738			1.14043			0.570213	+	0.821497i	$0.306861\pi$
$272$	1.27312			0.0771944
$273$		0			0
$274$			13.6218i			0.822924i
$275$	1.73106			0.104387
$276$		0			0
$277$	19.4871			1.17087			0.585433	−	0.810721i	$-0.300925\pi$
$277$	19.4871			1.17087			0.585433	+	0.810721i	$0.300925\pi$
$278$		−	4.08370i		−	0.244924i
$279$		0			0
$280$	2.68734			0.160599
$281$	21.1418			1.26121			0.630607	−	0.776103i	$-0.282806\pi$
$281$	21.1418			1.26121			0.630607	+	0.776103i	$0.282806\pi$
$282$		0			0
$283$		−	8.62629i		−	0.512780i	−0.966573	−	0.256390i	$-0.917467\pi$
$283$		−	8.62629i		−	0.512780i	0.966573	−	0.256390i	$-0.0825331\pi$
$284$		−	4.44631i		−	0.263840i
$285$		0			0
$286$		−	4.42475i		−	0.261641i
$287$	−9.14916			−0.540058
$288$		0			0
$289$	−15.3792			−0.904656
$290$	−9.62624			−0.565272
$291$		0			0
$292$	1.36193			0.0797006
$293$	−23.1296			−1.35125			−0.675623	−	0.737247i	$-0.736125\pi$
$293$	−23.1296			−1.35125			−0.675623	+	0.737247i	$0.736125\pi$
$294$		0			0
$295$			7.74331i			0.450833i
$296$	−4.20404			−0.244355
$297$		0			0
$298$			3.31717i			0.192159i
$299$	11.7557	−	3.47514i	0.679850	−	0.200973i
$300$		0			0
$301$	−6.32987			−0.364848
$302$		−	13.1448i		−	0.756398i
$303$		0			0
$304$			1.40738i			0.0807187i
$305$		−	6.78016i		−	0.388231i
$306$		0			0
$307$	13.7536			0.784957			0.392479	−	0.919761i	$-0.371618\pi$
$307$	13.7536			0.784957			0.392479	+	0.919761i	$0.371618\pi$
$308$		−	4.65193i		−	0.265069i
$309$		0			0
$310$		−	1.47605i		−	0.0838337i
$311$			20.1319i			1.14157i	0.821098	+	0.570787i	$0.193362\pi$
$311$			20.1319i			1.14157i	−0.821098	+	0.570787i	$0.806638\pi$
$312$		0			0
$313$			6.80646i			0.384724i	0.981324	+	0.192362i	$0.0616148\pi$
$313$			6.80646i			0.384724i	−0.981324	+	0.192362i	$0.938385\pi$
$314$	8.31606			0.469302
$315$		0			0
$316$		−	10.0972i		−	0.568010i
$317$		−	28.1821i		−	1.58287i	−0.611255	−	0.791433i	$-0.709335\pi$
$317$		−	28.1821i		−	1.58287i	0.611255	−	0.791433i	$-0.290665\pi$
$318$		0			0
$319$			16.6636i			0.932981i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	12.3593	−	3.65357i	0.688757	−	0.203605i
$323$			1.79176i			0.0996965i
$324$		0			0
$325$	−2.55609			−0.141787
$326$			3.77069i			0.208839i
$327$		0			0
$328$	3.40455			0.187985
$329$	−8.63202			−0.475898
$330$		0			0
$331$	15.3989			0.846398			0.423199	−	0.906037i	$-0.360907\pi$
$331$	15.3989			0.846398			0.423199	+	0.906037i	$0.360907\pi$
$332$	8.67571			0.476141
$333$		0			0
$334$	−16.9307			−0.926408
$335$		−	7.71912i		−	0.421741i
$336$		0			0
$337$		−	22.3140i		−	1.21552i	−0.794121	−	0.607760i	$-0.792068\pi$
$337$		−	22.3140i		−	1.21552i	0.794121	−	0.607760i	$-0.207932\pi$
$338$		−	6.46638i		−	0.351725i
$339$		0			0
$340$	−1.27312			−0.0690448
$341$	−2.55512			−0.138367
$342$		0			0
$343$			18.2154i			0.983537i
$344$	2.35544			0.126997
$345$		0			0
$346$	5.36460			0.288403
$347$			29.2565i			1.57057i	0.619134	+	0.785285i	$0.287484\pi$
$347$			29.2565i			1.57057i	−0.619134	+	0.785285i	$0.712516\pi$
$348$		0			0
$349$	3.90447			0.209002			0.104501	−	0.994525i	$-0.466676\pi$
$349$	3.90447			0.209002			0.104501	+	0.994525i	$0.466676\pi$
$350$	−2.68734			−0.143644
$351$		0			0
$352$			1.73106i			0.0922657i
$353$			3.76521i			0.200402i	0.994967	+	0.100201i	$0.0319486\pi$
$353$			3.76521i			0.200402i	−0.994967	+	0.100201i	$0.968051\pi$
$354$		0			0
$355$			4.44631i			0.235985i
$356$	3.72626			0.197492
$357$		0			0
$358$	9.56578			0.505567
$359$	10.9238			0.576538			0.288269	−	0.957549i	$-0.406920\pi$
$359$	10.9238			0.576538			0.288269	+	0.957549i	$0.406920\pi$
$360$		0			0
$361$	17.0193			0.895752
$362$	−12.4563			−0.654691
$363$		0			0
$364$			6.86909i			0.360038i
$365$	−1.36193			−0.0712864
$366$		0			0
$367$		−	8.01924i		−	0.418601i	−0.977851	−	0.209300i	$-0.932881\pi$
$367$		−	8.01924i		−	0.418601i	0.977851	−	0.209300i	$-0.0671187\pi$
$368$	−4.59909	+	1.35955i	−0.239744	+	0.0708715i
$369$		0			0
$370$	4.20404			0.218558
$371$		−	4.84992i		−	0.251795i
$372$		0			0
$373$			19.2361i			0.996010i	0.867174	+	0.498005i	$0.165934\pi$
$373$			19.2361i			0.996010i	−0.867174	+	0.498005i	$0.834066\pi$
$374$			2.20385i			0.113958i
$375$		0			0
$376$	3.21211			0.165652
$377$		−	24.6056i		−	1.26725i
$378$		0			0
$379$		−	13.8927i		−	0.713618i	−0.934177	−	0.356809i	$-0.883865\pi$
$379$		−	13.8927i		−	0.713618i	0.934177	−	0.356809i	$-0.116135\pi$
$380$		−	1.40738i		−	0.0721970i
$381$		0			0
$382$			8.39297i			0.429421i
$383$	−4.25515			−0.217428			−0.108714	−	0.994073i	$-0.534673\pi$
$383$	−4.25515			−0.217428			−0.108714	+	0.994073i	$0.534673\pi$
$384$		0			0
$385$			4.65193i			0.237084i
$386$			1.01621i			0.0517239i
$387$		0			0
$388$			7.03405i			0.357100i
$389$	22.9092			1.16154			0.580771	−	0.814067i	$-0.302751\pi$
$389$	22.9092			1.16154			0.580771	+	0.814067i	$0.302751\pi$
$390$		0			0
$391$	−5.85520	+	1.73087i	−0.296110	+	0.0875341i
$392$			0.221775i			0.0112013i
$393$		0			0
$394$	−5.60884			−0.282569
$395$			10.0972i			0.508043i
$396$		0			0
$397$	16.8037			0.843352			0.421676	−	0.906746i	$-0.361442\pi$
$397$	16.8037			0.843352			0.421676	+	0.906746i	$0.361442\pi$
$398$	−9.76909			−0.489680
$399$		0			0
$400$	1.00000			0.0500000
$401$	22.3193			1.11457			0.557287	−	0.830320i	$-0.311842\pi$
$401$	22.3193			1.11457			0.557287	+	0.830320i	$0.311842\pi$
$402$		0			0
$403$	3.77291			0.187942
$404$		−	12.9956i		−	0.646557i
$405$		0			0
$406$		−	25.8689i		−	1.28385i
$407$		−	7.27744i		−	0.360729i
$408$		0			0
$409$	−16.1240			−0.797280			−0.398640	−	0.917107i	$-0.630518\pi$
$409$	−16.1240			−0.797280			−0.398640	+	0.917107i	$0.630518\pi$
$410$	−3.40455			−0.168139
$411$		0			0
$412$		−	0.230115i		−	0.0113369i
$413$	−20.8089			−1.02394
$414$		0			0
$415$	−8.67571			−0.425874
$416$		−	2.55609i		−	0.125323i
$417$		0			0
$418$	−2.43625			−0.119161
$419$	32.2037			1.57325			0.786627	−	0.617429i	$-0.211826\pi$
$419$	32.2037			1.57325			0.786627	+	0.617429i	$0.211826\pi$
$420$		0			0
$421$		−	16.3229i		−	0.795531i	−0.917487	−	0.397766i	$-0.869786\pi$
$421$		−	16.3229i		−	0.795531i	0.917487	−	0.397766i	$-0.130214\pi$
$422$			8.94211i			0.435295i
$423$		0			0
$424$			1.80473i			0.0876455i
$425$	1.27312			0.0617555
$426$		0			0
$427$	18.2206			0.881755
$428$	6.45265			0.311901
$429$		0			0
$430$	−2.35544			−0.113590
$431$	27.1761			1.30903			0.654514	−	0.756050i	$-0.272873\pi$
$431$	27.1761			1.30903			0.654514	+	0.756050i	$0.272873\pi$
$432$		0			0
$433$		−	2.83957i		−	0.136461i	−0.997670	−	0.0682305i	$-0.978265\pi$
$433$		−	2.83957i		−	0.136461i	0.997670	−	0.0682305i	$-0.0217353\pi$
$434$	3.96663			0.190404
$435$		0			0
$436$		−	6.93477i		−	0.332115i
$437$	−1.91340	−	6.47266i	−0.0915304	−	0.309629i
$438$		0			0
$439$	18.8418			0.899269			0.449634	−	0.893213i	$-0.351554\pi$
$439$	18.8418			0.899269			0.449634	+	0.893213i	$0.351554\pi$
$440$		−	1.73106i		−	0.0825249i
$441$		0			0
$442$		−	3.25422i		−	0.154788i
$443$			18.4455i			0.876372i	0.898884	+	0.438186i	$0.144379\pi$
$443$			18.4455i			0.876372i	−0.898884	+	0.438186i	$0.855621\pi$
$444$		0			0
$445$	−3.72626			−0.176642
$446$			29.4525i			1.39462i
$447$		0			0
$448$		−	2.68734i		−	0.126965i
$449$		−	26.3810i		−	1.24500i	−0.782620	−	0.622499i	$-0.786117\pi$
$449$		−	26.3810i		−	1.24500i	0.782620	−	0.622499i	$-0.213883\pi$
$450$		0			0
$451$			5.89347i			0.277513i
$452$	1.03896			0.0488685
$453$		0			0
$454$			0.140630i			0.00660010i
$455$		−	6.86909i		−	0.322028i
$456$		0			0
$457$			34.8995i			1.63253i	0.577679	+	0.816264i	$0.303958\pi$
$457$			34.8995i			1.63253i	−0.577679	+	0.816264i	$0.696042\pi$
$458$	22.3375			1.04376
$459$		0			0
$460$	4.59909	−	1.35955i	0.214434	−	0.0633894i
$461$		−	7.50500i		−	0.349543i	−0.984609	−	0.174771i	$-0.944081\pi$
$461$		−	7.50500i		−	0.349543i	0.984609	−	0.174771i	$-0.0559186\pi$
$462$		0			0
$463$	11.5972			0.538967			0.269483	−	0.963005i	$-0.413147\pi$
$463$	11.5972			0.538967			0.269483	+	0.963005i	$0.413147\pi$
$464$			9.62624i			0.446887i
$465$		0			0
$466$	20.7396			0.960743
$467$	32.9988			1.52700			0.763502	−	0.645805i	$-0.223478\pi$
$467$	32.9988			1.52700			0.763502	+	0.645805i	$0.223478\pi$
$468$		0			0
$469$	20.7439			0.957863
$470$	−3.21211			−0.148164
$471$		0			0
$472$	7.74331			0.356415
$473$			4.07741i			0.187480i
$474$		0			0
$475$			1.40738i			0.0645749i
$476$		−	3.42131i		−	0.156815i
$477$		0			0
$478$	5.54170			0.253471
$479$	−33.0238			−1.50890			−0.754449	−	0.656359i	$-0.772096\pi$
$479$	−33.0238			−1.50890			−0.754449	+	0.656359i	$0.772096\pi$
$480$		0			0
$481$			10.7459i			0.489972i
$482$	15.1612			0.690574
$483$		0			0
$484$	8.00344			0.363793
$485$		−	7.03405i		−	0.319400i
$486$		0			0
$487$	−0.656460			−0.0297470			−0.0148735	−	0.999889i	$-0.504735\pi$
$487$	−0.656460			−0.0297470			−0.0148735	+	0.999889i	$0.504735\pi$
$488$	−6.78016			−0.306923
$489$		0			0
$490$		−	0.221775i		−	0.0100188i
$491$			15.0228i			0.677971i	0.940792	+	0.338985i	$0.110084\pi$
$491$			15.0228i			0.677971i	−0.940792	+	0.338985i	$0.889916\pi$
$492$		0			0
$493$			12.2554i			0.551955i
$494$	3.59739			0.161854
$495$		0			0
$496$	−1.47605			−0.0662764
$497$	−11.9487			−0.535973
$498$		0			0
$499$	11.2596			0.504050			0.252025	−	0.967721i	$-0.418903\pi$
$499$	11.2596			0.504050			0.252025	+	0.967721i	$0.418903\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$			26.3921i			1.17794i
$503$	−43.0753			−1.92063			−0.960315	−	0.278916i	$-0.910025\pi$
$503$	−43.0753			−1.92063			−0.960315	+	0.278916i	$0.910025\pi$
$504$		0			0
$505$			12.9956i			0.578298i
$506$	−2.35346	−	7.96129i	−0.104624	−	0.353922i
$507$		0			0
$508$	11.3029			0.501487
$509$			15.9683i			0.707783i	0.935286	+	0.353892i	$0.115142\pi$
$509$			15.9683i			0.707783i	−0.935286	+	0.353892i	$0.884858\pi$
$510$		0			0
$511$		−	3.65995i		−	0.161907i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−1.04633			−0.0461516
$515$			0.230115i			0.0101401i
$516$		0			0
$517$			5.56034i			0.244544i
$518$			11.2977i			0.496391i
$519$		0			0
$520$			2.55609i			0.112092i
$521$	37.0312			1.62237			0.811184	−	0.584791i	$-0.198824\pi$
$521$	37.0312			1.62237			0.811184	+	0.584791i	$0.198824\pi$
$522$		0			0
$523$			7.47772i			0.326978i	0.986545	+	0.163489i	$0.0522748\pi$
$523$			7.47772i			0.326978i	−0.986545	+	0.163489i	$0.947725\pi$
$524$		−	13.1993i		−	0.576614i
$525$		0			0
$526$		−	10.4405i		−	0.455229i
$527$	−1.87919			−0.0818586
$528$		0			0
$529$	19.3032	−	12.5054i	0.839272	−	0.543713i
$530$		−	1.80473i		−	0.0783925i
$531$		0			0
$532$	3.78210			0.163975
$533$		−	8.70235i		−	0.376940i
$534$		0			0
$535$	−6.45265			−0.278972
$536$	−7.71912			−0.333415
$537$		0			0
$538$	−2.34357			−0.101039
$539$	−0.383906			−0.0165360
$540$		0			0
$541$	30.8970			1.32837			0.664183	−	0.747570i	$-0.268780\pi$
$541$	30.8970			1.32837			0.664183	+	0.747570i	$0.268780\pi$
$542$			18.7738i			0.806402i
$543$		0			0
$544$			1.27312i			0.0545847i
$545$			6.93477i			0.297053i
$546$		0			0
$547$	0.826227			0.0353269			0.0176635	−	0.999844i	$-0.494377\pi$
$547$	0.826227			0.0353269			0.0176635	+	0.999844i	$0.494377\pi$
$548$	−13.6218			−0.581895
$549$		0			0
$550$			1.73106i			0.0738125i
$551$	−13.5478			−0.577154
$552$		0			0
$553$	−27.1345			−1.15388
$554$			19.4871i			0.827927i
$555$		0			0
$556$	4.08370			0.173187
$557$	−21.0952			−0.893834			−0.446917	−	0.894576i	$-0.647478\pi$
$557$	−21.0952			−0.893834			−0.446917	+	0.894576i	$0.647478\pi$
$558$		0			0
$559$		−	6.02074i		−	0.254650i
$560$			2.68734i			0.113561i
$561$		0			0
$562$			21.1418i			0.891812i
$563$	−7.54602			−0.318027			−0.159013	−	0.987276i	$-0.550831\pi$
$563$	−7.54602			−0.318027			−0.159013	+	0.987276i	$0.550831\pi$
$564$		0			0
$565$	−1.03896			−0.0437094
$566$	8.62629			0.362590
$567$		0			0
$568$	4.44631			0.186563
$569$	−38.5319			−1.61534			−0.807671	−	0.589633i	$-0.799272\pi$
$569$	−38.5319			−1.61534			−0.807671	+	0.589633i	$0.799272\pi$
$570$		0			0
$571$			15.6650i			0.655561i	0.944754	+	0.327781i	$0.106301\pi$
$571$			15.6650i			0.655561i	−0.944754	+	0.327781i	$0.893699\pi$
$572$	4.42475			0.185008
$573$		0			0
$574$		−	9.14916i		−	0.381879i
$575$	−4.59909	+	1.35955i	−0.191795	+	0.0566972i
$576$		0			0
$577$	−18.2318			−0.759000			−0.379500	−	0.925192i	$-0.623904\pi$
$577$	−18.2318			−0.759000			−0.379500	+	0.925192i	$0.623904\pi$
$578$		−	15.3792i		−	0.639689i
$579$		0			0
$580$		−	9.62624i		−	0.399708i
$581$		−	23.3145i		−	0.967250i
$582$		0			0
$583$	−3.12409			−0.129387
$584$			1.36193i			0.0563569i
$585$		0			0
$586$		−	23.1296i		−	0.955476i
$587$		−	5.97229i		−	0.246503i	−0.992375	−	0.123251i	$-0.960668\pi$
$587$		−	5.97229i		−	0.246503i	0.992375	−	0.123251i	$-0.0393322\pi$
$588$		0			0
$589$		−	2.07735i		−	0.0855959i
$590$	−7.74331			−0.318787
$591$		0			0
$592$		−	4.20404i		−	0.172785i
$593$		−	19.0934i		−	0.784073i	−0.919950	−	0.392037i	$-0.871771\pi$
$593$		−	19.0934i		−	0.784073i	0.919950	−	0.392037i	$-0.128229\pi$
$594$		0			0
$595$			3.42131i			0.140260i
$596$	−3.31717			−0.135877
$597$		0			0
$598$	3.47514	+	11.7557i	0.142109	+	0.480727i
$599$			23.8843i			0.975887i	0.872875	+	0.487944i	$0.162253\pi$
$599$			23.8843i			0.975887i	−0.872875	+	0.487944i	$0.837747\pi$
$600$		0			0
$601$	−26.3129			−1.07333			−0.536664	−	0.843796i	$-0.680316\pi$
$601$	−26.3129			−1.07333			−0.536664	+	0.843796i	$0.680316\pi$
$602$		−	6.32987i		−	0.257986i
$603$		0			0
$604$	13.1448			0.534854
$605$	−8.00344			−0.325386
$606$		0			0
$607$	17.9243			0.727523			0.363762	−	0.931492i	$-0.381492\pi$
$607$	17.9243			0.727523			0.363762	+	0.931492i	$0.381492\pi$
$608$	−1.40738			−0.0570767
$609$		0			0
$610$	6.78016			0.274521
$611$		−	8.21045i		−	0.332159i
$612$		0			0
$613$		−	3.31763i		−	0.133998i	−0.997753	−	0.0669990i	$-0.978658\pi$
$613$		−	3.31763i		−	0.133998i	0.997753	−	0.0669990i	$-0.0213424\pi$
$614$			13.7536i			0.555048i
$615$		0			0
$616$	4.65193			0.187432
$617$	−19.1689			−0.771711			−0.385856	−	0.922559i	$-0.626094\pi$
$617$	−19.1689			−0.771711			−0.385856	+	0.922559i	$0.626094\pi$
$618$		0			0
$619$			10.8982i			0.438034i	0.975721	+	0.219017i	$0.0702850\pi$
$619$			10.8982i			0.438034i	−0.975721	+	0.219017i	$0.929715\pi$
$620$	1.47605			0.0592794
$621$		0			0
$622$	−20.1319			−0.807214
$623$		−	10.0137i		−	0.401191i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−6.80646			−0.272041
$627$		0			0
$628$			8.31606i			0.331847i
$629$		−	5.35226i		−	0.213409i
$630$		0			0
$631$			8.17920i			0.325609i	0.986658	+	0.162804i	$0.0520539\pi$
$631$			8.17920i			0.325609i	−0.986658	+	0.162804i	$0.947946\pi$
$632$	10.0972			0.401644
$633$		0			0
$634$	28.1821			1.11926
$635$	−11.3029			−0.448543
$636$		0			0
$637$	0.566879			0.0224605
$638$	−16.6636			−0.659717
$639$		0			0
$640$		−	1.00000i		−	0.0395285i
$641$	8.05801			0.318272			0.159136	−	0.987257i	$-0.449129\pi$
$641$	8.05801			0.318272			0.159136	+	0.987257i	$0.449129\pi$
$642$		0			0
$643$		−	34.3659i		−	1.35526i	−0.735404	−	0.677629i	$-0.763008\pi$
$643$		−	34.3659i		−	1.35526i	0.735404	−	0.677629i	$-0.236992\pi$
$644$	3.65357	+	12.3593i	0.143971	+	0.487025i
$645$		0			0
$646$	−1.79176			−0.0704960
$647$			12.9531i			0.509239i	0.967041	+	0.254619i	$0.0819502\pi$
$647$			12.9531i			0.509239i	−0.967041	+	0.254619i	$0.918050\pi$
$648$		0			0
$649$			13.4041i			0.526157i
$650$		−	2.55609i		−	0.100258i
$651$		0			0
$652$	−3.77069			−0.147672
$653$			0.903168i			0.0353437i	0.999844	+	0.0176718i	$0.00562541\pi$
$653$			0.903168i			0.0353437i	−0.999844	+	0.0176718i	$0.994375\pi$
$654$		0			0
$655$			13.1993i			0.515739i
$656$			3.40455i			0.132925i
$657$		0			0
$658$		−	8.63202i		−	0.336511i
$659$	16.9355			0.659712			0.329856	−	0.944031i	$-0.393000\pi$
$659$	16.9355			0.659712			0.329856	+	0.944031i	$0.393000\pi$
$660$		0			0
$661$			12.6564i			0.492278i	0.969235	+	0.246139i	$0.0791619\pi$
$661$			12.6564i			0.492278i	−0.969235	+	0.246139i	$0.920838\pi$
$662$			15.3989i			0.598493i
$663$		0			0
$664$			8.67571i			0.336683i
$665$	−3.78210			−0.146663
$666$		0			0
$667$	−13.0874	−	44.2719i	−0.506745	−	1.71422i
$668$		−	16.9307i		−	0.655070i
$669$		0			0
$670$	7.71912			0.298216
$671$		−	11.7368i		−	0.453096i
$672$		0			0
$673$	−21.1871			−0.816704			−0.408352	−	0.912825i	$-0.633896\pi$
$673$	−21.1871			−0.816704			−0.408352	+	0.912825i	$0.633896\pi$
$674$	22.3140			0.859503
$675$		0			0
$676$	6.46638			0.248707
$677$	38.9430			1.49670			0.748350	−	0.663304i	$-0.230847\pi$
$677$	38.9430			1.49670			0.748350	+	0.663304i	$0.230847\pi$
$678$		0			0
$679$	18.9029			0.725426
$680$		−	1.27312i		−	0.0488220i
$681$		0			0
$682$		−	2.55512i		−	0.0978406i
$683$		−	4.58240i		−	0.175341i	−0.996150	−	0.0876704i	$-0.972058\pi$
$683$		−	4.58240i		−	0.175341i	0.996150	−	0.0876704i	$-0.0279422\pi$
$684$		0			0
$685$	13.6218			0.520463
$686$	−18.2154			−0.695466
$687$		0			0
$688$			2.35544i			0.0898005i
$689$	4.61306			0.175744
$690$		0			0
$691$	18.5212			0.704581			0.352291	−	0.935891i	$-0.385403\pi$
$691$	18.5212			0.704581			0.352291	+	0.935891i	$0.385403\pi$
$692$			5.36460i			0.203931i
$693$		0			0
$694$	−29.2565			−1.11056
$695$	−4.08370			−0.154904
$696$		0			0
$697$			4.33441i			0.164177i
$698$			3.90447i			0.147786i
$699$		0			0
$700$		−	2.68734i		−	0.101572i
$701$	42.3964			1.60129			0.800644	−	0.599140i	$-0.204491\pi$
$701$	42.3964			1.60129			0.800644	+	0.599140i	$0.204491\pi$
$702$		0			0
$703$	5.91667			0.223152
$704$	−1.73106			−0.0652417
$705$		0			0
$706$	−3.76521			−0.141706
$707$	−34.9236			−1.31344
$708$		0			0
$709$		−	22.4911i		−	0.844672i	−0.906439	−	0.422336i	$-0.861210\pi$
$709$		−	22.4911i		−	0.844672i	0.906439	−	0.422336i	$-0.138790\pi$
$710$	−4.44631			−0.166867
$711$		0			0
$712$			3.72626i			0.139648i
$713$	6.78846	−	2.00676i	0.254230	−	0.0751537i
$714$		0			0
$715$	−4.42475			−0.165476
$716$			9.56578i			0.357490i
$717$		0			0
$718$			10.9238i			0.407674i
$719$			2.42589i			0.0904703i	0.998976	+	0.0452352i	$0.0144037\pi$
$719$			2.42589i			0.0904703i	−0.998976	+	0.0452352i	$0.985596\pi$
$720$		0			0
$721$	−0.618396			−0.0230303
$722$			17.0193i			0.633392i
$723$		0			0
$724$		−	12.4563i		−	0.462937i
$725$			9.62624i			0.357510i
$726$		0			0
$727$		−	20.2634i		−	0.751526i	−0.926716	−	0.375763i	$-0.877381\pi$
$727$		−	20.2634i		−	0.751526i	0.926716	−	0.375763i	$-0.122619\pi$
$728$	−6.86909			−0.254585
$729$		0			0
$730$		−	1.36193i		−	0.0504071i
$731$			2.99877i			0.110913i
$732$		0			0
$733$			48.1405i			1.77811i	0.457799	+	0.889056i	$0.348638\pi$
$733$			48.1405i			1.77811i	−0.457799	+	0.889056i	$0.651362\pi$
$734$	8.01924			0.295995
$735$		0			0
$736$	−1.35955	−	4.59909i	−0.0501137	−	0.169525i
$737$		−	13.3622i		−	0.492204i
$738$		0			0
$739$	20.1099			0.739753			0.369877	−	0.929081i	$-0.379400\pi$
$739$	20.1099			0.739753			0.369877	+	0.929081i	$0.379400\pi$
$740$			4.20404i			0.154544i
$741$		0			0
$742$	4.84992			0.178046
$743$	47.5425			1.74416			0.872082	−	0.489359i	$-0.162769\pi$
$743$	47.5425			1.74416			0.872082	+	0.489359i	$0.162769\pi$
$744$		0			0
$745$	3.31717			0.121532
$746$	−19.2361			−0.704285
$747$		0			0
$748$	−2.20385			−0.0805807
$749$		−	17.3404i		−	0.633606i
$750$		0			0
$751$		−	20.9244i		−	0.763542i	−0.924257	−	0.381771i	$-0.875314\pi$
$751$		−	20.9244i		−	0.763542i	0.924257	−	0.381771i	$-0.124686\pi$
$752$			3.21211i			0.117134i
$753$		0			0
$754$	24.6056			0.896082
$755$	−13.1448			−0.478388
$756$		0			0
$757$		−	39.2876i		−	1.42793i	−0.700180	−	0.713966i	$-0.746897\pi$
$757$		−	39.2876i		−	1.42793i	0.700180	−	0.713966i	$-0.253103\pi$
$758$	13.8927			0.504604
$759$		0			0
$760$	1.40738			0.0510510
$761$			2.27081i			0.0823169i	0.999153	+	0.0411584i	$0.0131048\pi$
$761$			2.27081i			0.0823169i	−0.999153	+	0.0411584i	$0.986895\pi$
$762$		0			0
$763$	−18.6361			−0.674671
$764$	−8.39297			−0.303647
$765$		0			0
$766$		−	4.25515i		−	0.153745i
$767$		−	19.7926i		−	0.714670i
$768$		0			0
$769$		−	8.59866i		−	0.310076i	−0.987909	−	0.155038i	$-0.950450\pi$
$769$		−	8.59866i		−	0.310076i	0.987909	−	0.155038i	$-0.0495499\pi$
$770$	−4.65193			−0.167644
$771$		0			0
$772$	−1.01621			−0.0365743
$773$	−2.55997			−0.0920756			−0.0460378	−	0.998940i	$-0.514659\pi$
$773$	−2.55997			−0.0920756			−0.0460378	+	0.998940i	$0.514659\pi$
$774$		0			0
$775$	−1.47605			−0.0530211
$776$	−7.03405			−0.252508
$777$		0			0
$778$			22.9092i			0.821334i
$779$	−4.79148			−0.171673
$780$		0			0
$781$			7.69681i			0.275414i
$782$	−1.73087	−	5.85520i	−0.0618959	−	0.209382i
$783$		0			0
$784$	−0.221775			−0.00792054
$785$		−	8.31606i		−	0.296813i
$786$		0			0
$787$		−	20.0596i		−	0.715046i	−0.933904	−	0.357523i	$-0.883621\pi$
$787$		−	20.0596i		−	0.715046i	0.933904	−	0.357523i	$-0.116379\pi$
$788$		−	5.60884i		−	0.199806i
$789$		0			0
$790$	−10.0972			−0.359241
$791$		−	2.79203i		−	0.0992733i
$792$		0			0
$793$			17.3307i			0.615432i
$794$			16.8037i			0.596340i
$795$		0			0
$796$		−	9.76909i		−	0.346256i
$797$	−1.42166			−0.0503578			−0.0251789	−	0.999683i	$-0.508016\pi$
$797$	−1.42166			−0.0503578			−0.0251789	+	0.999683i	$0.508016\pi$
$798$		0			0
$799$			4.08941i			0.144673i
$800$			1.00000i			0.0353553i
$801$		0			0
$802$			22.3193i			0.788123i
$803$	−2.35757			−0.0831969
$804$		0			0
$805$	−3.65357	−	12.3593i	−0.128771	−	0.435608i
$806$			3.77291i			0.132895i
$807$		0			0
$808$	12.9956			0.457185
$809$		−	17.9165i		−	0.629911i	−0.949106	−	0.314956i	$-0.898010\pi$
$809$		−	17.9165i		−	0.629911i	0.949106	−	0.314956i	$-0.101990\pi$
$810$		0			0
$811$	23.5315			0.826304			0.413152	−	0.910662i	$-0.364428\pi$
$811$	23.5315			0.826304			0.413152	+	0.910662i	$0.364428\pi$
$812$	25.8689			0.907822
$813$		0			0
$814$	7.27744			0.255074
$815$	3.77069			0.132082
$816$		0			0
$817$	−3.31500			−0.115977
$818$		−	16.1240i		−	0.563762i
$819$		0			0
$820$		−	3.40455i		−	0.118892i
$821$		−	14.0412i		−	0.490041i	−0.969518	−	0.245021i	$-0.921205\pi$
$821$		−	14.0412i		−	0.490041i	0.969518	−	0.245021i	$-0.0787947\pi$
$822$		0			0
$823$	18.0770			0.630125			0.315063	−	0.949071i	$-0.397974\pi$
$823$	18.0770			0.630125			0.315063	+	0.949071i	$0.397974\pi$
$824$	0.230115			0.00801643
$825$		0			0
$826$		−	20.8089i		−	0.724033i
$827$	5.12710			0.178287			0.0891434	−	0.996019i	$-0.471587\pi$
$827$	5.12710			0.178287			0.0891434	+	0.996019i	$0.471587\pi$
$828$		0			0
$829$	−39.1505			−1.35975			−0.679877	−	0.733326i	$-0.737967\pi$
$829$	−39.1505			−1.35975			−0.679877	+	0.733326i	$0.737967\pi$
$830$		−	8.67571i		−	0.301138i
$831$		0			0
$832$	2.55609			0.0886166
$833$	−0.282347			−0.00978275
$834$		0			0
$835$			16.9307i			0.585912i
$836$		−	2.43625i		−	0.0842596i
$837$		0			0
$838$			32.2037i			1.11246i
$839$	−0.462401			−0.0159639			−0.00798193	−	0.999968i	$-0.502541\pi$
$839$	−0.462401			−0.0159639			−0.00798193	+	0.999968i	$0.502541\pi$
$840$		0			0
$841$	−63.6645			−2.19533
$842$	16.3229			0.562526
$843$		0			0
$844$	−8.94211			−0.307800
$845$	−6.46638			−0.222450
$846$		0			0
$847$		−	21.5079i		−	0.739021i
$848$	−1.80473			−0.0619747
$849$		0			0
$850$			1.27312i			0.0436677i
$851$	5.71561	+	19.3348i	0.195928	+	0.662787i
$852$		0			0
$853$	−34.1063			−1.16778			−0.583889	−	0.811834i	$-0.698470\pi$
$853$	−34.1063			−1.16778			−0.583889	+	0.811834i	$0.698470\pi$
$854$			18.2206i			0.623495i
$855$		0			0
$856$			6.45265i			0.220547i
$857$		−	31.7411i		−	1.08425i	−0.840296	−	0.542127i	$-0.817619\pi$
$857$		−	31.7411i		−	1.08425i	0.840296	−	0.542127i	$-0.182381\pi$
$858$		0			0
$859$	−32.8985			−1.12248			−0.561241	−	0.827652i	$-0.689676\pi$
$859$	−32.8985			−1.12248			−0.561241	+	0.827652i	$0.689676\pi$
$860$		−	2.35544i		−	0.0803200i
$861$		0			0
$862$			27.1761i			0.925623i
$863$		−	44.3676i		−	1.51029i	−0.655558	−	0.755145i	$-0.727566\pi$
$863$		−	44.3676i		−	1.51029i	0.655558	−	0.755145i	$-0.272434\pi$
$864$		0			0
$865$		−	5.36460i		−	0.182402i
$866$	2.83957			0.0964925
$867$		0			0
$868$			3.96663i			0.134636i
$869$			17.4788i			0.592927i
$870$		0			0
$871$			19.7308i			0.668553i
$872$	6.93477			0.234841
$873$		0			0
$874$	6.47266	−	1.91340i	0.218941	−	0.0647218i
$875$			2.68734i			0.0908485i
$876$		0			0
$877$	−21.8841			−0.738974			−0.369487	−	0.929236i	$-0.620467\pi$
$877$	−21.8841			−0.738974			−0.369487	+	0.929236i	$0.620467\pi$
$878$			18.8418i			0.635879i
$879$		0			0
$880$	1.73106			0.0583539
$881$	54.1710			1.82507			0.912534	−	0.409002i	$-0.134123\pi$
$881$	54.1710			1.82507			0.912534	+	0.409002i	$0.134123\pi$
$882$		0			0
$883$	−42.5462			−1.43179			−0.715897	−	0.698205i	$-0.753982\pi$
$883$	−42.5462			−1.43179			−0.715897	+	0.698205i	$0.753982\pi$
$884$	3.25422			0.109451
$885$		0			0
$886$	−18.4455			−0.619689
$887$			25.0529i			0.841195i	0.907247	+	0.420597i	$0.138179\pi$
$887$			25.0529i			0.841195i	−0.907247	+	0.420597i	$0.861821\pi$
$888$		0			0
$889$		−	30.3748i		−	1.01874i
$890$		−	3.72626i		−	0.124905i
$891$		0			0
$892$	−29.4525			−0.986144
$893$	−4.52065			−0.151278
$894$		0			0
$895$		−	9.56578i		−	0.319749i
$896$	2.68734			0.0897776
$897$		0			0
$898$	26.3810			0.880347
$899$		−	14.2088i		−	0.473889i
$900$		0			0
$901$	−2.29764			−0.0765456
$902$	−5.89347			−0.196231
$903$		0			0
$904$			1.03896i			0.0345553i
$905$			12.4563i			0.414063i
$906$		0			0
$907$			44.5983i			1.48086i	0.672132	+	0.740431i	$0.265379\pi$
$907$			44.5983i			1.48086i	−0.672132	+	0.740431i	$0.734621\pi$
$908$	−0.140630			−0.00466698
$909$		0			0
$910$	6.86909			0.227708
$911$	−17.7058			−0.586618			−0.293309	−	0.956018i	$-0.594756\pi$
$911$	−17.7058			−0.586618			−0.293309	+	0.956018i	$0.594756\pi$
$912$		0			0
$913$	−15.0181			−0.497028
$914$	−34.8995			−1.15437
$915$		0			0
$916$			22.3375i			0.738052i
$917$	−35.4709			−1.17135
$918$		0			0
$919$		−	46.3021i		−	1.52737i	−0.645592	−	0.763683i	$-0.723389\pi$
$919$		−	46.3021i		−	1.52737i	0.645592	−	0.763683i	$-0.276611\pi$
$920$	1.35955	+	4.59909i	0.0448231	+	0.151627i
$921$		0			0
$922$	7.50500			0.247164
$923$		−	11.3652i		−	0.374089i
$924$		0			0
$925$		−	4.20404i		−	0.138228i
$926$			11.5972i			0.381107i
$927$		0			0
$928$	−9.62624			−0.315997
$929$		−	24.1969i		−	0.793874i	−0.917846	−	0.396937i	$-0.870073\pi$
$929$		−	24.1969i		−	0.793874i	0.917846	−	0.396937i	$-0.129927\pi$
$930$		0			0
$931$		−	0.312122i		−	0.0102294i
$932$			20.7396i			0.679348i
$933$		0			0
$934$			32.9988i			1.07976i
$935$	2.20385			0.0720736
$936$		0			0
$937$			7.81390i			0.255269i	0.991821	+	0.127634i	$0.0407384\pi$
$937$			7.81390i			0.255269i	−0.991821	+	0.127634i	$0.959262\pi$
$938$			20.7439i			0.677311i
$939$		0			0
$940$		−	3.21211i		−	0.104767i
$941$	3.53043			0.115089			0.0575444	−	0.998343i	$-0.481673\pi$
$941$	3.53043			0.115089			0.0575444	+	0.998343i	$0.481673\pi$
$942$		0			0
$943$	−4.62865	−	15.6578i	−0.150730	−	0.509889i
$944$			7.74331i			0.252023i
$945$		0			0
$946$	−4.07741			−0.132568
$947$			30.6079i			0.994624i	0.867572	+	0.497312i	$0.165680\pi$
$947$			30.6079i			0.994624i	−0.867572	+	0.497312i	$0.834320\pi$
$948$		0			0
$949$	3.48121			0.113005
$950$	−1.40738			−0.0456614
$951$		0			0
$952$	3.42131			0.110885
$953$	4.99497			0.161803			0.0809014	−	0.996722i	$-0.474220\pi$
$953$	4.99497			0.161803			0.0809014	+	0.996722i	$0.474220\pi$
$954$		0			0
$955$	8.39297			0.271590
$956$			5.54170i			0.179231i
$957$		0			0
$958$		−	33.0238i		−	1.06695i
$959$			36.6064i			1.18208i
$960$		0			0
$961$	−28.8213			−0.929719
$962$	−10.7459			−0.346463
$963$		0			0
$964$			15.1612i			0.488310i
$965$	1.01621			0.0327131
$966$		0			0
$967$	−9.11422			−0.293094			−0.146547	−	0.989204i	$-0.546816\pi$
$967$	−9.11422			−0.293094			−0.146547	+	0.989204i	$0.546816\pi$
$968$			8.00344i			0.257240i
$969$		0			0
$970$	7.03405			0.225850
$971$	28.6111			0.918175			0.459088	−	0.888391i	$-0.348176\pi$
$971$	28.6111			0.918175			0.459088	+	0.888391i	$0.348176\pi$
$972$		0			0
$973$		−	10.9743i		−	0.351819i
$974$		−	0.656460i		−	0.0210343i
$975$		0			0
$976$		−	6.78016i		−	0.217028i
$977$	55.1555			1.76458			0.882291	−	0.470705i	$-0.156000\pi$
$977$	55.1555			1.76458			0.882291	+	0.470705i	$0.156000\pi$
$978$		0			0
$979$	−6.45038			−0.206155
$980$	0.221775			0.00708435
$981$		0			0
$982$	−15.0228			−0.479398
$983$	−15.1144			−0.482074			−0.241037	−	0.970516i	$-0.577488\pi$
$983$	−15.1144			−0.482074			−0.241037	+	0.970516i	$0.577488\pi$
$984$		0			0
$985$			5.60884i			0.178712i
$986$	−12.2554			−0.390291
$987$		0			0
$988$			3.59739i			0.114448i
$989$	−3.20235	−	10.8329i	−0.101829	−	0.344466i
$990$		0			0
$991$	1.97026			0.0625875			0.0312938	−	0.999510i	$-0.490037\pi$
$991$	1.97026			0.0625875			0.0312938	+	0.999510i	$0.490037\pi$
$992$		−	1.47605i		−	0.0468645i
$993$		0			0
$994$		−	11.9487i		−	0.378990i
$995$			9.76909i			0.309701i
$996$		0			0
$997$	19.9205			0.630889			0.315445	−	0.948944i	$-0.397846\pi$
$997$	19.9205			0.630889			0.315445	+	0.948944i	$0.397846\pi$
$998$			11.2596i			0.356417i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.e.b.1241.15	yes	16
3.2	odd	2		2070.2.e.a.1241.7	✓	16
23.22	odd	2		2070.2.e.a.1241.10	yes	16
69.68	even	2	inner	2070.2.e.b.1241.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.2.e.a.1241.7	✓	16	3.2	odd	2
2070.2.e.a.1241.10	yes	16	23.22	odd	2
2070.2.e.b.1241.2	yes	16	69.68	even	2	inner
2070.2.e.b.1241.15	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.68734i q^{7} -1.00000i q^{8} +1.00000i q^{10} +1.73106 q^{11} -2.55609 q^{13} -2.68734 q^{14} +1.00000 q^{16} +1.27312 q^{17} +1.40738i q^{19} -1.00000 q^{20} +1.73106i q^{22} +(-4.59909 + 1.35955i) q^{23} +1.00000 q^{25} -2.55609i q^{26} -2.68734i q^{28} +9.62624i q^{29} -1.47605 q^{31} +1.00000i q^{32} +1.27312i q^{34} +2.68734i q^{35} -4.20404i q^{37} -1.40738 q^{38} -1.00000i q^{40} +3.40455i q^{41} +2.35544i q^{43} -1.73106 q^{44} +(-1.35955 - 4.59909i) q^{46} +3.21211i q^{47} -0.221775 q^{49} +1.00000i q^{50} +2.55609 q^{52} -1.80473 q^{53} +1.73106 q^{55} +2.68734 q^{56} -9.62624 q^{58} +7.74331i q^{59} -6.78016i q^{61} -1.47605i q^{62} -1.00000 q^{64} -2.55609 q^{65} -7.71912i q^{67} -1.27312 q^{68} -2.68734 q^{70} +4.44631i q^{71} -1.36193 q^{73} +4.20404 q^{74} -1.40738i q^{76} +4.65193i q^{77} +10.0972i q^{79} +1.00000 q^{80} -3.40455 q^{82} -8.67571 q^{83} +1.27312 q^{85} -2.35544 q^{86} -1.73106i q^{88} -3.72626 q^{89} -6.86909i q^{91} +(4.59909 - 1.35955i) q^{92} -3.21211 q^{94} +1.40738i q^{95} -7.03405i q^{97} -0.221775i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 24 q^{11} + 16 q^{16} - 16 q^{20} - 4 q^{23} + 16 q^{25} - 8 q^{31} - 8 q^{38} + 24 q^{44} - 4 q^{46} - 16 q^{49} + 8 q^{53} - 24 q^{55} + 16 q^{58} - 16 q^{64} + 32 q^{73}+ \cdots + 4 q^{92}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^5 - 24 * q^11 + 16 * q^16 - 16 * q^20 - 4 * q^23 + 16 * q^25 - 8 * q^31 - 8 * q^38 + 24 * q^44 - 4 * q^46 - 16 * q^49 + 8 * q^53 - 24 * q^55 + 16 * q^58 - 16 * q^64 + 32 * q^73 + 16 * q^80 + 32 * q^82 - 56 * q^83 - 8 * q^86 + 40 * q^89 + 4 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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