LMFDB - Embedded newform 3168.2.o.e.703.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(703,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3168.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.o (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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.10
Root			$-1.16947 - 0.795191i$ of defining polynomial
Character	$\chi$	$=$	3168.703
Dual form			3168.2.o.e.703.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77846 q^{5} -3.18077 q^{7} +O(q^{10})$	$q+2.77846 q^{5} -3.18077 q^{7} +(0.665647 + 3.24914i) q^{11} -6.00919i q^{13} +4.67790i q^{17} -7.50633 q^{19} -5.02760i q^{23} +2.71982 q^{25} -5.03024i q^{29} -3.96896i q^{31} -8.83762 q^{35} -2.77846 q^{37} -7.34049i q^{41} -1.33129 q^{43} +8.61555i q^{47} +3.11727 q^{49} -0.117266 q^{53} +(1.84947 + 9.02760i) q^{55} -6.41205i q^{59} -4.32556i q^{61} -16.6963i q^{65} -10.5845i q^{67} -1.91377i q^{71} -7.34049i q^{73} +(-2.11727 - 10.3348i) q^{77} +11.3137 q^{79} -7.69282 q^{83} +12.9973i q^{85} -1.95436 q^{89} +19.1138i q^{91} -20.8560 q^{95} -13.3940 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.77846			1.24256			0.621282	−	0.783587i	$-0.286612\pi$
$5$	2.77846			1.24256			0.621282	+	0.783587i	$0.286612\pi$
$6$		0			0
$7$	−3.18077			−1.20222			−0.601108	−	0.799168i	$-0.705274\pi$
$7$	−3.18077			−1.20222			−0.601108	+	0.799168i	$0.705274\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.665647	+	3.24914i	0.200700	+	0.979653i
$11$	0.665647	+	3.24914i	0.200700	+	0.979653i
$12$		0			0
$13$		−	6.00919i		−	1.66665i	−0.552783	−	0.833325i	$-0.686434\pi$
$13$		−	6.00919i		−	1.66665i	0.552783	−	0.833325i	$-0.313566\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.67790i			1.13456i	0.823526	+	0.567279i	$0.192004\pi$
$17$			4.67790i			1.13456i	−0.823526	+	0.567279i	$0.807996\pi$
$18$		0			0
$19$	−7.50633			−1.72207			−0.861035	−	0.508546i	$-0.830183\pi$
$19$	−7.50633			−1.72207			−0.861035	+	0.508546i	$0.830183\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.02760i		−	1.04833i	−0.851618	−	0.524163i	$-0.824378\pi$
$23$		−	5.02760i		−	1.04833i	0.851618	−	0.524163i	$-0.175622\pi$
$24$		0			0
$25$	2.71982			0.543965
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.03024i		−	0.934091i	−0.884233	−	0.467046i	$-0.845318\pi$
$29$		−	5.03024i		−	0.934091i	0.884233	−	0.467046i	$-0.154682\pi$
$30$		0			0
$31$		−	3.96896i		−	0.712847i	−0.934324	−	0.356424i	$-0.883996\pi$
$31$		−	3.96896i		−	0.712847i	0.934324	−	0.356424i	$-0.116004\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.83762			−1.49383
$36$		0			0
$37$	−2.77846			−0.456776			−0.228388	−	0.973570i	$-0.573345\pi$
$37$	−2.77846			−0.456776			−0.228388	+	0.973570i	$0.573345\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.34049i		−	1.14639i	−0.819419	−	0.573196i	$-0.805703\pi$
$41$		−	7.34049i		−	1.14639i	0.819419	−	0.573196i	$-0.194297\pi$
$42$		0			0
$43$	−1.33129			−0.203020			−0.101510	−	0.994834i	$-0.532367\pi$
$43$	−1.33129			−0.203020			−0.101510	+	0.994834i	$0.532367\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.61555i			1.25671i	0.777928	+	0.628353i	$0.216271\pi$
$47$			8.61555i			1.25671i	−0.777928	+	0.628353i	$0.783729\pi$
$48$		0			0
$49$	3.11727			0.445324
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.117266			−0.0161078			−0.00805388	−	0.999968i	$-0.502564\pi$
$53$	−0.117266			−0.0161078			−0.00805388	+	0.999968i	$0.502564\pi$
$54$		0			0
$55$	1.84947	+	9.02760i	0.249383	+	1.21728i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.41205i		−	0.834778i	−0.908728	−	0.417389i	$-0.862945\pi$
$59$		−	6.41205i		−	0.834778i	0.908728	−	0.417389i	$-0.137055\pi$
$60$		0			0
$61$		−	4.32556i		−	0.553831i	−0.960894	−	0.276916i	$-0.910688\pi$
$61$		−	4.32556i		−	0.553831i	0.960894	−	0.276916i	$-0.0893123\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	16.6963i		−	2.07092i
$66$		0			0
$67$		−	10.5845i		−	1.29310i	−0.762870	−	0.646552i	$-0.776210\pi$
$67$		−	10.5845i		−	1.29310i	0.762870	−	0.646552i	$-0.223790\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.91377i		−	0.227123i	−0.993531	−	0.113561i	$-0.963774\pi$
$71$		−	1.91377i		−	0.227123i	0.993531	−	0.113561i	$-0.0362258\pi$
$72$		0			0
$73$		−	7.34049i		−	0.859139i	−0.903034	−	0.429569i	$-0.858665\pi$
$73$		−	7.34049i		−	0.859139i	0.903034	−	0.429569i	$-0.141335\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.11727	−	10.3348i	−0.241285	−	1.17775i
$78$		0			0
$79$	11.3137			1.27289			0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137			1.27289			0.636446	+	0.771321i	$0.280404\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.69282			−0.844397			−0.422199	−	0.906503i	$-0.638741\pi$
$83$	−7.69282			−0.844397			−0.422199	+	0.906503i	$0.638741\pi$
$84$		0			0
$85$			12.9973i			1.40976i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.95436			−0.207161			−0.103581	−	0.994621i	$-0.533030\pi$
$89$	−1.95436			−0.207161			−0.103581	+	0.994621i	$0.533030\pi$
$90$		0			0
$91$			19.1138i			2.00367i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−20.8560			−2.13978
$96$		0			0
$97$	−13.3940			−1.35996			−0.679978	−	0.733233i	$-0.738010\pi$
$97$	−13.3940			−1.35996			−0.679978	+	0.733233i	$0.738010\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			5.30452i			0.527819i	0.964547	+	0.263910i	$0.0850120\pi$
$101$			5.30452i			0.527819i	−0.964547	+	0.263910i	$0.914988\pi$
$102$		0			0
$103$		−	5.50172i		−	0.542100i	−0.962565	−	0.271050i	$-0.912629\pi$
$103$		−	5.50172i		−	0.542100i	0.962565	−	0.271050i	$-0.0873710\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.55415			−0.246919			−0.123459	−	0.992350i	$-0.539399\pi$
$107$	−2.55415			−0.246919			−0.123459	+	0.992350i	$0.539399\pi$
$108$		0			0
$109$			6.98815i			0.669343i	0.942335	+	0.334672i	$0.108625\pi$
$109$			6.98815i			0.669343i	−0.942335	+	0.334672i	$0.891375\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.2767			1.06083			0.530413	−	0.847739i	$-0.322037\pi$
$113$	11.2767			1.06083			0.530413	+	0.847739i	$0.322037\pi$
$114$		0			0
$115$		−	13.9690i		−	1.30261i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	14.8793i		−	1.36398i
$120$		0			0
$121$	−10.1138	+	4.32556i	−0.919439	+	0.393233i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.33537			−0.566653
$126$		0			0
$127$	−6.87971			−0.610475			−0.305238	−	0.952276i	$-0.598736\pi$
$127$	−6.87971			−0.610475			−0.305238	+	0.952276i	$0.598736\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.50633			0.655831			0.327915	−	0.944707i	$-0.393654\pi$
$131$	7.50633			0.655831			0.327915	+	0.944707i	$0.393654\pi$
$132$		0			0
$133$	23.8759			2.07030
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.162910			0.0139183			0.00695916	−	0.999976i	$-0.497785\pi$
$137$	0.162910			0.0139183			0.00695916	+	0.999976i	$0.497785\pi$
$138$		0			0
$139$	13.8679			1.17626			0.588128	−	0.808768i	$-0.299865\pi$
$139$	13.8679			1.17626			0.588128	+	0.808768i	$0.299865\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	19.5247	−	4.00000i	1.63274	−	0.334497i
$144$		0			0
$145$		−	13.9763i		−	1.16067i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.00919i		−	0.492292i	−0.969233	−	0.246146i	$-0.920836\pi$
$149$		−	6.00919i		−	0.492292i	0.969233	−	0.246146i	$-0.0791643\pi$
$150$		0			0
$151$	−20.3378			−1.65507			−0.827534	−	0.561415i	$-0.810257\pi$
$151$	−20.3378			−1.65507			−0.827534	+	0.561415i	$0.810257\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	11.0276i		−	0.885758i
$156$		0			0
$157$	14.2181			1.13473			0.567364	−	0.823467i	$-0.307963\pi$
$157$	14.2181			1.13473			0.567364	+	0.823467i	$0.307963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			15.9916i			1.26032i
$162$		0			0
$163$		−	10.2637i		−	0.803919i	−0.915657	−	0.401959i	$-0.868329\pi$
$163$		−	10.2637i		−	0.803919i	0.915657	−	0.401959i	$-0.131671\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.4945			−1.12162			−0.560808	−	0.827946i	$-0.689509\pi$
$167$	−14.4945			−1.12162			−0.560808	+	0.827946i	$0.689509\pi$
$168$		0			0
$169$	−23.1104			−1.77772
$170$		0			0
$171$		0			0
$172$		0			0
$173$			5.73491i			0.436017i	0.975947	+	0.218009i	$0.0699561\pi$
$173$			5.73491i			0.436017i	−0.975947	+	0.218009i	$0.930044\pi$
$174$		0			0
$175$	−8.65112			−0.653963
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4.29478i		−	0.321007i	−0.987035	−	0.160504i	$-0.948688\pi$
$179$		−	4.29478i		−	0.321007i	0.987035	−	0.160504i	$-0.0513118\pi$
$180$		0			0
$181$	3.10428			0.230739			0.115370	−	0.993323i	$-0.463195\pi$
$181$	3.10428			0.230739			0.115370	+	0.993323i	$0.463195\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.71982			−0.567573
$186$		0			0
$187$	−15.1991	+	3.11383i	−1.11147	+	0.227706i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			19.9690i			1.44490i	0.691421	+	0.722452i	$0.256985\pi$
$191$			19.9690i			1.44490i	−0.691421	+	0.722452i	$0.743015\pi$
$192$		0			0
$193$		−	1.95791i		−	0.140934i	−0.997514	−	0.0704668i	$-0.977551\pi$
$193$		−	1.95791i		−	0.140934i	0.997514	−	0.0704668i	$-0.0224489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.69282i		−	0.548091i	−0.961717	−	0.274046i	$-0.911638\pi$
$197$		−	7.69282i		−	0.548091i	0.961717	−	0.274046i	$-0.0883619\pi$
$198$		0			0
$199$		−	4.38101i		−	0.310562i	−0.987870	−	0.155281i	$-0.950372\pi$
$199$		−	4.38101i		−	0.310562i	0.987870	−	0.155281i	$-0.0496283\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			16.0000i			1.12298i
$204$		0			0
$205$		−	20.3952i		−	1.42446i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.99656	−	24.3891i	−0.345619	−	1.68703i
$210$		0			0
$211$	12.6450			0.870518			0.435259	−	0.900305i	$-0.356657\pi$
$211$	12.6450			0.870518			0.435259	+	0.900305i	$0.356657\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.69894			−0.252266
$216$		0			0
$217$			12.6243i			0.856996i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.1104			1.89091
$222$		0			0
$223$		−	19.9690i		−	1.33722i	−0.743613	−	0.668610i	$-0.766889\pi$
$223$		−	19.9690i		−	1.33722i	0.743613	−	0.668610i	$-0.233111\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.55415			0.169525			0.0847624	−	0.996401i	$-0.472987\pi$
$227$	2.55415			0.169525			0.0847624	+	0.996401i	$0.472987\pi$
$228$		0			0
$229$	9.33193			0.616672			0.308336	−	0.951278i	$-0.400228\pi$
$229$	9.33193			0.616672			0.308336	+	0.951278i	$0.400228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.3856i			1.00795i	0.863719	+	0.503974i	$0.168129\pi$
$233$			15.3856i			1.00795i	−0.863719	+	0.503974i	$0.831871\pi$
$234$		0			0
$235$			23.9379i			1.56154i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−25.0731			−1.62185			−0.810923	−	0.585153i	$-0.801034\pi$
$239$	−25.0731			−1.62185			−0.810923	+	0.585153i	$0.801034\pi$
$240$		0			0
$241$		−	22.0789i		−	1.42222i	−0.703079	−	0.711112i	$-0.748192\pi$
$241$		−	22.0789i		−	1.42222i	0.703079	−	0.711112i	$-0.251808\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.66119			0.553343
$246$		0			0
$247$			45.1070i			2.87009i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			20.7018i			1.30668i	0.757063	+	0.653342i	$0.226634\pi$
$251$			20.7018i			1.30668i	−0.757063	+	0.653342i	$0.773366\pi$
$252$		0			0
$253$	16.3354	−	3.34660i	1.02700	−	0.210399i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.1104			−1.37921			−0.689604	−	0.724187i	$-0.742215\pi$
$257$	−22.1104			−1.37921			−0.689604	+	0.724187i	$0.742215\pi$
$258$		0			0
$259$	8.83762			0.549143
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.16930			0.565403			0.282702	−	0.959208i	$-0.408769\pi$
$263$	9.16930			0.565403			0.282702	+	0.959208i	$0.408769\pi$
$264$		0			0
$265$	−0.325819			−0.0200149
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.88273			0.480619			0.240309	−	0.970696i	$-0.422751\pi$
$269$	7.88273			0.480619			0.240309	+	0.970696i	$0.422751\pi$
$270$		0			0
$271$	−5.47036			−0.332300			−0.166150	−	0.986100i	$-0.553134\pi$
$271$	−5.47036			−0.332300			−0.166150	+	0.986100i	$0.553134\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.81044	+	8.83709i	0.109174	+	0.532897i
$276$		0			0
$277$		−	6.98815i		−	0.419877i	−0.977715	−	0.209939i	$-0.932674\pi$
$277$		−	6.98815i		−	0.419877i	0.977715	−	0.209939i	$-0.0673264\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2.66259i		−	0.158837i	−0.996841	−	0.0794183i	$-0.974694\pi$
$281$		−	2.66259i		−	0.158837i	0.996841	−	0.0794183i	$-0.0253063\pi$
$282$		0			0
$283$	−19.0065			−1.12982			−0.564911	−	0.825152i	$-0.691089\pi$
$283$	−19.0065			−1.12982			−0.564911	+	0.825152i	$0.691089\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			23.3484i			1.37821i
$288$		0			0
$289$	−4.88273			−0.287220
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.05128i		−	0.236678i	−0.992973	−	0.118339i	$-0.962243\pi$
$293$		−	4.05128i		−	0.236678i	0.992973	−	0.118339i	$-0.0377570\pi$
$294$		0			0
$295$		−	17.8156i		−	1.03726i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−30.2118			−1.74719
$300$		0			0
$301$	4.23453			0.244074
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.0184i		−	0.688171i
$306$		0			0
$307$	−1.14480			−0.0653369			−0.0326685	−	0.999466i	$-0.510401\pi$
$307$	−1.14480			−0.0653369			−0.0326685	+	0.999466i	$0.510401\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	17.8466i		−	1.01199i	−0.862536	−	0.505995i	$-0.831125\pi$
$311$		−	17.8466i		−	1.01199i	0.862536	−	0.505995i	$-0.168875\pi$
$312$		0			0
$313$	−32.5078			−1.83745			−0.918726	−	0.394896i	$-0.870781\pi$
$313$	−32.5078			−1.83745			−0.918726	+	0.394896i	$0.870781\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.8923			0.780268			0.390134	−	0.920758i	$-0.372429\pi$
$317$	13.8923			0.780268			0.390134	+	0.920758i	$0.372429\pi$
$318$		0			0
$319$	16.3439	−	3.34836i	0.915085	−	0.187472i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	35.1138i		−	1.95379i
$324$		0			0
$325$		−	16.3439i		−	0.906599i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	27.4040i		−	1.51083i
$330$		0			0
$331$			2.64658i			0.145469i	0.997351	+	0.0727347i	$0.0231726\pi$
$331$			2.64658i			0.145469i	−0.997351	+	0.0727347i	$0.976827\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	29.4086i		−	1.60676i
$336$		0			0
$337$			28.7147i			1.56419i	0.623161	+	0.782094i	$0.285848\pi$
$337$			28.7147i			1.56419i	−0.623161	+	0.782094i	$0.714152\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.8957	−	2.64193i	0.698343	−	0.143068i
$342$		0			0
$343$	12.3501			0.666841
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.5909			−1.42748			−0.713738	−	0.700413i	$-0.752999\pi$
$347$	−26.5909			−1.42748			−0.713738	+	0.700413i	$0.752999\pi$
$348$		0			0
$349$			36.7392i			1.96660i	0.181985	+	0.983301i	$0.441748\pi$
$349$			36.7392i			1.96660i	−0.181985	+	0.983301i	$0.558252\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.60600			−0.138703			−0.0693516	−	0.997592i	$-0.522093\pi$
$353$	−2.60600			−0.138703			−0.0693516	+	0.997592i	$0.522093\pi$
$354$		0			0
$355$		−	5.31733i		−	0.282214i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.21712			0.222571			0.111286	−	0.993788i	$-0.464503\pi$
$359$	4.21712			0.222571			0.111286	+	0.993788i	$0.464503\pi$
$360$		0			0
$361$	37.3449			1.96552
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	20.3952i		−	1.06753i
$366$		0			0
$367$		−	15.7344i		−	0.821331i	−0.911786	−	0.410665i	$-0.865296\pi$
$367$		−	15.7344i		−	0.821331i	0.911786	−	0.410665i	$-0.134704\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.372997			0.0193650
$372$		0			0
$373$			0.0206587i			0.00106967i	1.00000		0.000534833i	$0.000170243\pi$
$373$			0.0206587i			0.00106967i	−1.00000		0.000534833i	$0.999830\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.2277			−1.55680
$378$		0			0
$379$		−	31.8156i		−	1.63426i	−0.576455	−	0.817129i	$-0.695564\pi$
$379$		−	31.8156i		−	1.63426i	0.576455	−	0.817129i	$-0.304436\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.3794i		−	0.990241i	−0.868824	−	0.495120i	$-0.835124\pi$
$383$		−	19.3794i		−	0.990241i	0.868824	−	0.495120i	$-0.164876\pi$
$384$		0			0
$385$	−5.88273	−	28.7147i	−0.299812	−	1.46343i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	26.6543			1.35143			0.675714	−	0.737164i	$-0.263836\pi$
$389$	26.6543			1.35143			0.675714	+	0.737164i	$0.263836\pi$
$390$		0			0
$391$	23.5186			1.18939
$392$		0			0
$393$		0			0
$394$		0			0
$395$	31.4347			1.58165
$396$		0			0
$397$	−12.1173			−0.608148			−0.304074	−	0.952648i	$-0.598347\pi$
$397$	−12.1173			−0.608148			−0.304074	+	0.952648i	$0.598347\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−33.8759			−1.69168			−0.845840	−	0.533437i	$-0.820900\pi$
$401$	−33.8759			−1.69168			−0.845840	+	0.533437i	$0.820900\pi$
$402$		0			0
$403$	−23.8503			−1.18807
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.84947	−	9.02760i	−0.0916749	−	0.447481i
$408$		0			0
$409$			0.704676i			0.0348440i	0.999848	+	0.0174220i	$0.00554587\pi$
$409$			0.704676i			0.0348440i	−0.999848	+	0.0174220i	$0.994454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			20.3952i			1.00358i
$414$		0			0
$415$	−21.3742			−1.04922
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	27.4948i		−	1.34321i	−0.740909	−	0.671605i	$-0.765605\pi$
$419$		−	27.4948i		−	1.34321i	0.740909	−	0.671605i	$-0.234395\pi$
$420$		0			0
$421$	−4.11727			−0.200663			−0.100332	−	0.994954i	$-0.531990\pi$
$421$	−4.11727			−0.200663			−0.100332	+	0.994954i	$0.531990\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			12.7231i			0.617159i
$426$		0			0
$427$			13.7586i			0.665825i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.1571			0.826427			0.413213	−	0.910634i	$-0.364406\pi$
$431$	17.1571			0.826427			0.413213	+	0.910634i	$0.364406\pi$
$432$		0			0
$433$	−5.39400			−0.259219			−0.129610	−	0.991565i	$-0.541372\pi$
$433$	−5.39400			−0.259219			−0.129610	+	0.991565i	$0.541372\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			37.7388i			1.80529i
$438$		0			0
$439$	−2.66259			−0.127078			−0.0635392	−	0.997979i	$-0.520239\pi$
$439$	−2.66259			−0.127078			−0.0635392	+	0.997979i	$0.520239\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.2327i			0.581194i	0.956846	+	0.290597i	$0.0938538\pi$
$443$			12.2327i			0.581194i	−0.956846	+	0.290597i	$0.906146\pi$
$444$		0			0
$445$	−5.43010			−0.257411
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.0422			0.709886			0.354943	−	0.934888i	$-0.384500\pi$
$449$	15.0422			0.709886			0.354943	+	0.934888i	$0.384500\pi$
$450$		0			0
$451$	23.8503	−	4.88617i	1.12307	−	0.230081i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			53.1070i			2.48969i
$456$		0			0
$457$		−	32.6879i		−	1.52907i	−0.644579	−	0.764537i	$-0.722967\pi$
$457$		−	32.6879i		−	1.52907i	0.644579	−	0.764537i	$-0.277033\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			22.3738i			1.04205i	0.853541	+	0.521026i	$0.174450\pi$
$461$			22.3738i			1.04205i	−0.853541	+	0.521026i	$0.825550\pi$
$462$		0			0
$463$			15.1449i			0.703842i	0.936030	+	0.351921i	$0.114471\pi$
$463$			15.1449i			0.703842i	−0.936030	+	0.351921i	$0.885529\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			19.2362i			0.890143i	0.895495	+	0.445071i	$0.146822\pi$
$467$			19.2362i			0.890143i	−0.895495	+	0.445071i	$0.853178\pi$
$468$		0			0
$469$			33.6668i			1.55459i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.886172	−	4.32556i	−0.0407462	−	0.198889i
$474$		0			0
$475$	−20.4159			−0.936745
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.7841			0.766883			0.383442	−	0.923565i	$-0.374739\pi$
$479$	16.7841			0.766883			0.383442	+	0.923565i	$0.374739\pi$
$480$		0			0
$481$			16.6963i			0.761285i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−37.2147			−1.68983
$486$		0			0
$487$		−	28.9655i		−	1.31255i	−0.754520	−	0.656277i	$-0.772130\pi$
$487$		−	28.9655i		−	1.31255i	0.754520	−	0.656277i	$-0.227870\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.4044			1.19161			0.595807	−	0.803127i	$-0.296832\pi$
$491$	26.4044			1.19161			0.595807	+	0.803127i	$0.296832\pi$
$492$		0			0
$493$	23.5309			1.05978
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.08725i			0.273050i
$498$		0			0
$499$			27.4948i			1.23084i	0.788200	+	0.615419i	$0.211013\pi$
$499$			27.4948i			1.23084i	−0.788200	+	0.615419i	$0.788987\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.69894			0.164928			0.0824638	−	0.996594i	$-0.473721\pi$
$503$	3.69894			0.164928			0.0824638	+	0.996594i	$0.473721\pi$
$504$		0			0
$505$			14.7384i			0.655849i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.10771			−0.0934228			−0.0467114	−	0.998908i	$-0.514874\pi$
$509$	−2.10771			−0.0934228			−0.0467114	+	0.998908i	$0.514874\pi$
$510$		0			0
$511$			23.3484i			1.03287i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	15.2863i		−	0.673594i
$516$		0			0
$517$	−27.9931	+	5.73491i	−1.23114	+	0.252221i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.8596			1.00150			0.500749	−	0.865592i	$-0.333058\pi$
$521$	22.8596			1.00150			0.500749	+	0.865592i	$0.333058\pi$
$522$		0			0
$523$	−12.6146			−0.551599			−0.275799	−	0.961215i	$-0.588943\pi$
$523$	−12.6146			−0.551599			−0.275799	+	0.961215i	$0.588943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.5664			0.808766
$528$		0			0
$529$	−2.27674			−0.0989886
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−44.1104			−1.91063
$534$		0			0
$535$	−7.09659			−0.306812
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.07500	+	10.1284i	0.0893765	+	0.436263i
$540$		0			0
$541$		−	16.7743i		−	0.721185i	−0.932723	−	0.360593i	$-0.882574\pi$
$541$		−	16.7743i		−	0.721185i	0.932723	−	0.360593i	$-0.117426\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			19.4163i			0.831702i
$546$		0			0
$547$	14.0544			0.600921			0.300460	−	0.953794i	$-0.402860\pi$
$547$	14.0544			0.600921			0.300460	+	0.953794i	$0.402860\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			37.7586i			1.60857i
$552$		0			0
$553$	−35.9862			−1.53029
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	10.0811i		−	0.427151i	−0.976927	−	0.213576i	$-0.931489\pi$
$557$		−	10.0811i		−	0.427151i	0.976927	−	0.213576i	$-0.0685110\pi$
$558$		0			0
$559$			8.00000i			0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	45.7059			1.92627			0.963137	−	0.269012i	$-0.0866972\pi$
$563$	45.7059			1.92627			0.963137	+	0.269012i	$0.0866972\pi$
$564$		0			0
$565$	31.3319			1.31814
$566$		0			0
$567$		0			0
$568$		0			0
$569$			44.0016i			1.84464i	0.386423	+	0.922321i	$0.373710\pi$
$569$			44.0016i			1.84464i	−0.386423	+	0.922321i	$0.626290\pi$
$570$		0			0
$571$	35.4285			1.48264			0.741319	−	0.671153i	$-0.234201\pi$
$571$	35.4285			1.48264			0.741319	+	0.671153i	$0.234201\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	13.6742i		−	0.570253i
$576$		0			0
$577$	4.25420			0.177105			0.0885523	−	0.996072i	$-0.471776\pi$
$577$	4.25420			0.177105			0.0885523	+	0.996072i	$0.471776\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.4691			1.01515
$582$		0			0
$583$	−0.0780580	−	0.381015i	−0.00323283	−	0.0157800i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			9.61211i			0.396734i	0.980128	+	0.198367i	$0.0635638\pi$
$587$			9.61211i			0.396734i	−0.980128	+	0.198367i	$0.936436\pi$
$588$		0			0
$589$			29.7923i			1.22757i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			37.3084i			1.53207i	0.642798	+	0.766036i	$0.277774\pi$
$593$			37.3084i			1.53207i	−0.642798	+	0.766036i	$0.722226\pi$
$594$		0			0
$595$		−	41.3415i		−	1.69484i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			10.6087i			0.433459i	0.976232	+	0.216729i	$0.0695389\pi$
$599$			10.6087i			0.433459i	−0.976232	+	0.216729i	$0.930461\pi$
$600$		0			0
$601$			17.2448i			0.703432i	0.936107	+	0.351716i	$0.114402\pi$
$601$			17.2448i			0.703432i	−0.936107	+	0.351716i	$0.885598\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−28.1008	+	12.0184i	−1.14246	+	0.488617i
$606$		0			0
$607$	12.7231			0.516413			0.258207	−	0.966090i	$-0.416868\pi$
$607$	12.7231			0.516413			0.258207	+	0.966090i	$0.416868\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	51.7725			2.09449
$612$		0			0
$613$		−	20.3746i		−	0.822921i	−0.911428	−	0.411460i	$-0.865019\pi$
$613$		−	20.3746i		−	0.822921i	0.911428	−	0.411460i	$-0.134981\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.1173			0.648857			0.324428	−	0.945910i	$-0.394828\pi$
$617$	16.1173			0.648857			0.324428	+	0.945910i	$0.394828\pi$
$618$		0			0
$619$			32.8742i			1.32133i	0.750682	+	0.660664i	$0.229725\pi$
$619$			32.8742i			1.32133i	−0.750682	+	0.660664i	$0.770275\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.21635			0.249053
$624$		0			0
$625$	−31.2017			−1.24807
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.9973i		−	0.518238i
$630$		0			0
$631$			19.9069i			0.792481i	0.918147	+	0.396240i	$0.129685\pi$
$631$			19.9069i			0.792481i	−0.918147	+	0.396240i	$0.870315\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.1150			−0.758555
$636$		0			0
$637$		−	18.7323i		−	0.742199i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.71639			−0.265281			−0.132641	−	0.991164i	$-0.542346\pi$
$641$	−6.71639			−0.265281			−0.132641	+	0.991164i	$0.542346\pi$
$642$		0			0
$643$		−	2.58451i		−	0.101923i	−0.998701	−	0.0509616i	$-0.983771\pi$
$643$		−	2.58451i		−	0.101923i	0.998701	−	0.0509616i	$-0.0162286\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14.0862i			0.553787i	0.960901	+	0.276893i	$0.0893049\pi$
$647$			14.0862i			0.553787i	−0.960901	+	0.276893i	$0.910695\pi$
$648$		0			0
$649$	20.8337	−	4.26816i	0.817792	−	0.167540i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.3354			−0.952317			−0.476158	−	0.879360i	$-0.657971\pi$
$653$	−24.3354			−0.952317			−0.476158	+	0.879360i	$0.657971\pi$
$654$		0			0
$655$	20.8560			0.814911
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−26.4044			−1.02857			−0.514285	−	0.857619i	$-0.671943\pi$
$659$	−26.4044			−1.02857			−0.514285	+	0.857619i	$0.671943\pi$
$660$		0			0
$661$	−16.5370			−0.643217			−0.321608	−	0.946873i	$-0.604223\pi$
$661$	−16.5370			−0.643217			−0.321608	+	0.946873i	$0.604223\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	66.3380			2.57248
$666$		0			0
$667$	−25.2900			−0.979233
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.0544	−	2.87930i	0.542562	−	0.111154i
$672$		0			0
$673$			48.7369i			1.87867i	0.343002	+	0.939335i	$0.388556\pi$
$673$			48.7369i			1.87867i	−0.343002	+	0.939335i	$0.611444\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	20.6902i		−	0.795188i	−0.917562	−	0.397594i	$-0.869845\pi$
$677$		−	20.6902i		−	0.795188i	0.917562	−	0.397594i	$-0.130155\pi$
$678$		0			0
$679$	42.6032			1.63496
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.50172i		−	0.0574617i	−0.999587	−	0.0287308i	$-0.990853\pi$
$683$		−	1.50172i		−	0.0574617i	0.999587	−	0.0287308i	$-0.00914657\pi$
$684$		0			0
$685$	0.452638			0.0172944
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.704676i			0.0268460i
$690$		0			0
$691$			23.8777i			0.908350i	0.890913	+	0.454175i	$0.150066\pi$
$691$			23.8777i			0.908350i	−0.890913	+	0.454175i	$0.849934\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	38.5312			1.46157
$696$		0			0
$697$	34.3380			1.30065
$698$		0			0
$699$		0			0
$700$		0			0
$701$			39.4018i			1.48818i	0.668077	+	0.744092i	$0.267118\pi$
$701$			39.4018i			1.48818i	−0.668077	+	0.744092i	$0.732882\pi$
$702$		0			0
$703$	20.8560			0.786599
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.8724i		−	0.634553i
$708$		0			0
$709$	45.5665			1.71128			0.855642	−	0.517568i	$-0.173162\pi$
$709$	45.5665			1.71128			0.855642	+	0.517568i	$0.173162\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19.9544			−0.747297
$714$		0			0
$715$	54.2486	−	11.1138i	2.02878	−	0.415634i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	36.2518i		−	1.35196i	−0.736918	−	0.675982i	$-0.763720\pi$
$719$		−	36.2518i		−	1.35196i	0.736918	−	0.675982i	$-0.236280\pi$
$720$		0			0
$721$			17.4997i			0.651722i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	13.6814i		−	0.508113i
$726$		0			0
$727$			42.1346i			1.56268i	0.624103	+	0.781342i	$0.285465\pi$
$727$			42.1346i			1.56268i	−0.624103	+	0.781342i	$0.714535\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.22766i		−	0.230338i
$732$		0			0
$733$			40.9293i			1.51176i	0.654712	+	0.755879i	$0.272790\pi$
$733$			40.9293i			1.51176i	−0.654712	+	0.755879i	$0.727210\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	34.3906	−	7.04555i	1.26679	−	0.259526i
$738$		0			0
$739$	20.0429			0.737290			0.368645	−	0.929570i	$-0.379822\pi$
$739$	20.0429			0.737290			0.368645	+	0.929570i	$0.379822\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	28.7721			1.05554			0.527772	−	0.849386i	$-0.323027\pi$
$743$	28.7721			1.05554			0.527772	+	0.849386i	$0.323027\pi$
$744$		0			0
$745$		−	16.6963i		−	0.611704i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.12414			0.296850
$750$		0			0
$751$		−	53.7795i		−	1.96244i	−0.192886	−	0.981221i	$-0.561785\pi$
$751$		−	53.7795i		−	1.96244i	0.192886	−	0.981221i	$-0.438215\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−56.5078			−2.05653
$756$		0			0
$757$	−15.8827			−0.577268			−0.288634	−	0.957440i	$-0.593201\pi$
$757$	−15.8827			−0.577268			−0.288634	+	0.957440i	$0.593201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			3.42466i			0.124144i	0.998072	+	0.0620720i	$0.0197708\pi$
$761$			3.42466i			0.124144i	−0.998072	+	0.0620720i	$0.980229\pi$
$762$		0			0
$763$		−	22.2277i		−	0.804695i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−38.5312			−1.39128
$768$		0			0
$769$			28.7147i			1.03548i	0.855539	+	0.517739i	$0.173226\pi$
$769$			28.7147i			1.03548i	−0.855539	+	0.517739i	$0.826774\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.7586			−0.710667			−0.355334	−	0.934740i	$-0.615633\pi$
$773$	−19.7586			−0.710667			−0.355334	+	0.934740i	$0.615633\pi$
$774$		0			0
$775$		−	10.7949i		−	0.387764i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			55.1001i			1.97417i
$780$		0			0
$781$	6.21811	−	1.27389i	0.222501	−	0.0455835i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	39.5044			1.40997
$786$		0			0
$787$	17.9702			0.640568			0.320284	−	0.947322i	$-0.396222\pi$
$787$	17.9702			0.640568			0.320284	+	0.947322i	$0.396222\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−35.8687			−1.27534
$792$		0			0
$793$	−25.9931			−0.923043
$794$		0			0
$795$		0			0
$796$		0			0
$797$	49.5336			1.75457			0.877285	−	0.479969i	$-0.159352\pi$
$797$	49.5336			1.75457			0.877285	+	0.479969i	$0.159352\pi$
$798$		0			0
$799$	−40.3027			−1.42581
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.8503	−	4.88617i	0.841658	−	0.172429i
$804$		0			0
$805$			44.4320i			1.56602i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	35.9564i		−	1.26416i	−0.774903	−	0.632080i	$-0.782201\pi$
$809$		−	35.9564i		−	1.26416i	0.774903	−	0.632080i	$-0.217799\pi$
$810$		0			0
$811$	4.65724			0.163538			0.0817689	−	0.996651i	$-0.473943\pi$
$811$	4.65724			0.163538			0.0817689	+	0.996651i	$0.473943\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	28.5174i		−	0.998920i
$816$		0			0
$817$	9.99312			0.349615
$818$		0			0
$819$		0			0
$820$		0			0
$821$			25.6997i			0.896927i	0.893801	+	0.448464i	$0.148029\pi$
$821$			25.6997i			0.896927i	−0.893801	+	0.448464i	$0.851971\pi$
$822$		0			0
$823$			12.5585i			0.437763i	0.975751	+	0.218881i	$0.0702408\pi$
$823$			12.5585i			0.437763i	−0.975751	+	0.218881i	$0.929759\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	46.5557			1.61890			0.809451	−	0.587188i	$-0.199765\pi$
$827$	46.5557			1.61890			0.809451	+	0.587188i	$0.199765\pi$
$828$		0			0
$829$	−29.8923			−1.03820			−0.519101	−	0.854713i	$-0.673733\pi$
$829$	−29.8923			−1.03820			−0.519101	+	0.854713i	$0.673733\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			14.5823i			0.505245i
$834$		0			0
$835$	−40.2723			−1.39368
$836$		0			0
$837$		0			0
$838$		0			0
$839$			19.1932i			0.662622i	0.943522	+	0.331311i	$0.107491\pi$
$839$			19.1932i			0.662622i	−0.943522	+	0.331311i	$0.892509\pi$
$840$		0			0
$841$	3.69672			0.127473
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−64.2112			−2.20893
$846$		0			0
$847$	32.1697	−	13.7586i	1.10536	−	0.472751i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			13.9690i			0.478850i
$852$		0			0
$853$		−	51.6944i		−	1.76998i	−0.465607	−	0.884992i	$-0.654164\pi$
$853$		−	51.6944i		−	1.76998i	0.465607	−	0.884992i	$-0.345836\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			30.7300i			1.04972i	0.851190	+	0.524858i	$0.175882\pi$
$857$			30.7300i			1.04972i	−0.851190	+	0.524858i	$0.824118\pi$
$858$		0			0
$859$			1.05681i			0.0360580i	0.999837	+	0.0180290i	$0.00573912\pi$
$859$			1.05681i			0.0360580i	−0.999837	+	0.0180290i	$0.994261\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			0.146482i			0.00498631i	0.999997	+	0.00249315i	$0.000793597\pi$
$863$			0.146482i			0.00498631i	−0.999997	+	0.00249315i	$0.999206\pi$
$864$		0			0
$865$			15.9342i			0.541779i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.53093	+	36.7598i	0.255469	+	1.24699i
$870$		0			0
$871$	−63.6044			−2.15515
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20.1513			0.681239
$876$		0			0
$877$		−	0.0206587i		−	0.000697594i	−1.00000		0.000348797i	$-0.999889\pi$
$877$		−	0.0206587i		−	0.000697594i	1.00000		0.000348797i	$-0.000111026\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−45.0682			−1.51839			−0.759193	−	0.650866i	$-0.774406\pi$
$881$	−45.0682			−1.51839			−0.759193	+	0.650866i	$0.774406\pi$
$882$		0			0
$883$			30.6087i			1.03006i	0.857171	+	0.515032i	$0.172220\pi$
$883$			30.6087i			1.03006i	−0.857171	+	0.515032i	$0.827780\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.6033			0.456754			0.228377	−	0.973573i	$-0.426658\pi$
$887$	13.6033			0.456754			0.228377	+	0.973573i	$0.426658\pi$
$888$		0			0
$889$	21.8827			0.733923
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	64.6711i		−	2.16414i
$894$		0			0
$895$		−	11.9329i		−	0.398872i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−19.9648			−0.665864
$900$		0			0
$901$		−	0.548560i		−	0.0182752i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.62510			0.286708
$906$		0			0
$907$			1.50172i			0.0498638i	0.999689	+	0.0249319i	$0.00793689\pi$
$907$			1.50172i			0.0498638i	−0.999689	+	0.0249319i	$0.992063\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13.6121i		−	0.450989i	−0.974244	−	0.225495i	$-0.927600\pi$
$911$		−	13.6121i		−	0.450989i	0.974244	−	0.225495i	$-0.0723998\pi$
$912$		0			0
$913$	−5.12070	−	24.9951i	−0.169471	−	0.827216i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−23.8759			−0.788450
$918$		0			0
$919$	−15.3856			−0.507526			−0.253763	−	0.967266i	$-0.581668\pi$
$919$	−15.3856			−0.507526			−0.253763	+	0.967266i	$0.581668\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.5002			−0.378534
$924$		0			0
$925$	−7.55691			−0.248470
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.6482			−0.644637			−0.322318	−	0.946631i	$-0.604462\pi$
$929$	−19.6482			−0.644637			−0.322318	+	0.946631i	$0.604462\pi$
$930$		0			0
$931$	−23.3992			−0.766878
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−42.2302	+	8.65164i	−1.38107	+	0.282939i
$936$		0			0
$937$		−	6.04593i		−	0.197512i	−0.995112	−	0.0987560i	$-0.968514\pi$
$937$		−	6.04593i		−	0.197512i	0.995112	−	0.0987560i	$-0.0314863\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			31.7296i			1.03436i	0.855878	+	0.517178i	$0.173017\pi$
$941$			31.7296i			1.03436i	−0.855878	+	0.517178i	$0.826983\pi$
$942$		0			0
$943$	−36.9050			−1.20179
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.7638i		−	0.674734i	−0.941373	−	0.337367i	$-0.890464\pi$
$947$		−	20.7638i		−	0.674734i	0.941373	−	0.337367i	$-0.109536\pi$
$948$		0			0
$949$	−44.1104			−1.43188
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20.7269i		−	0.671410i	−0.941967	−	0.335705i	$-0.891025\pi$
$953$		−	20.7269i		−	0.671410i	0.941967	−	0.335705i	$-0.108975\pi$
$954$		0			0
$955$			55.4829i			1.79538i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.518178			−0.0167328
$960$		0			0
$961$	15.2473			0.491849
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	5.43997i		−	0.175119i
$966$		0			0
$967$	8.65112			0.278201			0.139101	−	0.990278i	$-0.455579\pi$
$967$	8.65112			0.278201			0.139101	+	0.990278i	$0.455579\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.3534i			0.941996i	0.882134	+	0.470998i	$0.156106\pi$
$971$			29.3534i			0.941996i	−0.882134	+	0.470998i	$0.843894\pi$
$972$		0			0
$973$	−44.1104			−1.41411
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.8268			−0.602322			−0.301161	−	0.953573i	$-0.597374\pi$
$977$	−18.8268			−0.602322			−0.301161	+	0.953573i	$0.597374\pi$
$978$		0			0
$979$	−1.30091	−	6.34998i	−0.0415773	−	0.202946i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	11.2553i		−	0.358987i	−0.983759	−	0.179493i	$-0.942554\pi$
$983$		−	11.2553i		−	0.358987i	0.983759	−	0.179493i	$-0.0574459\pi$
$984$		0			0
$985$		−	21.3742i		−	0.681038i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.69321i			0.212832i
$990$		0			0
$991$			5.14304i			0.163374i	0.996658	+	0.0816871i	$0.0260308\pi$
$991$			5.14304i			0.163374i	−0.996658	+	0.0816871i	$0.973969\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	12.1725i		−	0.385893i
$996$		0			0
$997$			10.5884i			0.335337i	0.985843	+	0.167669i	$0.0536239\pi$
$997$			10.5884i			0.335337i	−0.985843	+	0.167669i	$0.946376\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.o.e.703.10		12
3.2	odd	2		352.2.e.a.351.7	yes	12
4.3	odd	2	inner	3168.2.o.e.703.11		12
11.10	odd	2	inner	3168.2.o.e.703.12		12
12.11	even	2		352.2.e.a.351.6	yes	12
24.5	odd	2		704.2.e.d.703.5		12
24.11	even	2		704.2.e.d.703.8		12
33.32	even	2		352.2.e.a.351.8	yes	12
44.43	even	2	inner	3168.2.o.e.703.9		12
48.5	odd	4		2816.2.g.i.1407.6		12
48.11	even	4		2816.2.g.d.1407.5		12
48.29	odd	4		2816.2.g.d.1407.8		12
48.35	even	4		2816.2.g.i.1407.7		12
132.131	odd	2		352.2.e.a.351.5	✓	12
264.131	odd	2		704.2.e.d.703.7		12
264.197	even	2		704.2.e.d.703.6		12
528.131	odd	4		2816.2.g.i.1407.8		12
528.197	even	4		2816.2.g.i.1407.5		12
528.395	odd	4		2816.2.g.d.1407.6		12
528.461	even	4		2816.2.g.d.1407.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.5	✓	12	132.131	odd	2
352.2.e.a.351.6	yes	12	12.11	even	2
352.2.e.a.351.7	yes	12	3.2	odd	2
352.2.e.a.351.8	yes	12	33.32	even	2
704.2.e.d.703.5		12	24.5	odd	2
704.2.e.d.703.6		12	264.197	even	2
704.2.e.d.703.7		12	264.131	odd	2
704.2.e.d.703.8		12	24.11	even	2
2816.2.g.d.1407.5		12	48.11	even	4
2816.2.g.d.1407.6		12	528.395	odd	4
2816.2.g.d.1407.7		12	528.461	even	4
2816.2.g.d.1407.8		12	48.29	odd	4
2816.2.g.i.1407.5		12	528.197	even	4
2816.2.g.i.1407.6		12	48.5	odd	4
2816.2.g.i.1407.7		12	48.35	even	4
2816.2.g.i.1407.8		12	528.131	odd	4
3168.2.o.e.703.9		12	44.43	even	2	inner
3168.2.o.e.703.10		12	1.1	even	1	trivial
3168.2.o.e.703.11		12	4.3	odd	2	inner
3168.2.o.e.703.12		12	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.77846 q^{5} -3.18077 q^{7} +O(q^{10})\)	\(q+2.77846 q^{5} -3.18077 q^{7} +(0.665647 + 3.24914i) q^{11} -6.00919i q^{13} +4.67790i q^{17} -7.50633 q^{19} -5.02760i q^{23} +2.71982 q^{25} -5.03024i q^{29} -3.96896i q^{31} -8.83762 q^{35} -2.77846 q^{37} -7.34049i q^{41} -1.33129 q^{43} +8.61555i q^{47} +3.11727 q^{49} -0.117266 q^{53} +(1.84947 + 9.02760i) q^{55} -6.41205i q^{59} -4.32556i q^{61} -16.6963i q^{65} -10.5845i q^{67} -1.91377i q^{71} -7.34049i q^{73} +(-2.11727 - 10.3348i) q^{77} +11.3137 q^{79} -7.69282 q^{83} +12.9973i q^{85} -1.95436 q^{89} +19.1138i q^{91} -20.8560 q^{95} -13.3940 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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