LMFDB - Embedded newform 352.2.e.a.351.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(351,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, 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("352.351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.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:	$2.81073415115$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.5
Root			$-1.16947 - 0.795191i$ of defining polynomial
Character	$\chi$	$=$	352.351
Dual form			352.2.e.a.351.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.529317i q^{3} -2.77846 q^{5} -3.18077 q^{7} +2.71982 q^{9} +O(q^{10})$	$q-0.529317i q^{3} -2.77846 q^{5} -3.18077 q^{7} +2.71982 q^{9} +(-0.665647 + 3.24914i) q^{11} +6.00919i q^{13} +1.47068i q^{15} +4.67790i q^{17} -7.50633 q^{19} +1.68363i q^{21} -5.02760i q^{23} +2.71982 q^{25} -3.02760i q^{27} -5.03024i q^{29} +3.96896i q^{31} +(1.71982 + 0.352338i) q^{33} +8.83762 q^{35} -2.77846 q^{37} +3.18077 q^{39} -7.34049i q^{41} -1.33129 q^{43} -7.55691 q^{45} +8.61555i q^{47} +3.11727 q^{49} +2.47609 q^{51} +0.117266 q^{53} +(1.84947 - 9.02760i) q^{55} +3.97322i q^{57} -6.41205i q^{59} +4.32556i q^{61} -8.65112 q^{63} -16.6963i q^{65} +10.5845i q^{67} -2.66119 q^{69} -1.91377i q^{71} +7.34049i q^{73} -1.43965i q^{75} +(2.11727 - 10.3348i) q^{77} +11.3137 q^{79} +6.55691 q^{81} +7.69282 q^{83} -12.9973i q^{85} -2.66259 q^{87} +1.95436 q^{89} -19.1138i q^{91} +2.10084 q^{93} +20.8560 q^{95} -13.3940 q^{97} +(-1.81044 + 8.83709i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9}+O(q^{10}) $ 12 * q - 4 * q^9	$ 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	0.529317i		−	0.305601i	−0.988257	−	0.152801i	$-0.951171\pi$
$3$		−	0.529317i		−	0.305601i	0.988257	−	0.152801i	$-0.0488292\pi$
$4$		0			0
$5$	−2.77846			−1.24256			−0.621282	−	0.783587i	$-0.713388\pi$
$5$	−2.77846			−1.24256			−0.621282	+	0.783587i	$0.713388\pi$
$6$		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
$9$	2.71982			0.906608
$10$		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	$0.313566\pi$
$13$			6.00919i			1.66665i	−0.552783	+	0.833325i	$0.686434\pi$
$14$		0			0
$15$			1.47068i			0.379729i
$16$		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
$21$			1.68363i			0.367399i
$22$		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
$27$		−	3.02760i		−	0.582661i
$28$		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	$0.116004\pi$
$31$			3.96896i			0.712847i	−0.934324	+	0.356424i	$0.883996\pi$
$32$		0			0
$33$	1.71982	+	0.352338i	0.299383	+	0.0613342i
$34$		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
$39$	3.18077			0.509330
$40$		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
$45$	−7.55691			−1.12652
$46$		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
$51$	2.47609			0.346722
$52$		0			0
$53$	0.117266			0.0161078			0.00805388	−	0.999968i	$-0.497436\pi$
$53$	0.117266			0.0161078			0.00805388	+	0.999968i	$0.497436\pi$
$54$		0			0
$55$	1.84947	−	9.02760i	0.249383	−	1.21728i
$56$		0			0
$57$			3.97322i			0.526266i
$58$		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	$0.0893123\pi$
$61$			4.32556i			0.553831i	−0.960894	+	0.276916i	$0.910688\pi$
$62$		0			0
$63$	−8.65112			−1.08994
$64$		0			0
$65$		−	16.6963i		−	2.07092i
$66$		0			0
$67$			10.5845i			1.29310i	0.762870	+	0.646552i	$0.223790\pi$
$67$			10.5845i			1.29310i	−0.762870	+	0.646552i	$0.776210\pi$
$68$		0			0
$69$	−2.66119			−0.320370
$70$		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	$0.141335\pi$
$73$			7.34049i			0.859139i	−0.903034	+	0.429569i	$0.858665\pi$
$74$		0			0
$75$		−	1.43965i		−	0.166236i
$76$		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
$81$	6.55691			0.728546
$82$		0			0
$83$	7.69282			0.844397			0.422199	−	0.906503i	$-0.361259\pi$
$83$	7.69282			0.844397			0.422199	+	0.906503i	$0.361259\pi$
$84$		0			0
$85$		−	12.9973i		−	1.40976i
$86$		0			0
$87$	−2.66259			−0.285459
$88$		0			0
$89$	1.95436			0.207161			0.103581	−	0.994621i	$-0.466970\pi$
$89$	1.95436			0.207161			0.103581	+	0.994621i	$0.466970\pi$
$90$		0			0
$91$		−	19.1138i		−	2.00367i
$92$		0			0
$93$	2.10084			0.217847
$94$		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
$99$	−1.81044	+	8.83709i	−0.181956	+	0.888161i
$100$		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	$0.0873710\pi$
$103$			5.50172i			0.542100i	−0.962565	+	0.271050i	$0.912629\pi$
$104$		0			0
$105$		−	4.67790i		−	0.456516i
$106$		0			0
$107$	2.55415			0.246919			0.123459	−	0.992350i	$-0.460601\pi$
$107$	2.55415			0.246919			0.123459	+	0.992350i	$0.460601\pi$
$108$		0			0
$109$		−	6.98815i		−	0.669343i	−0.942335	−	0.334672i	$-0.891375\pi$
$109$		−	6.98815i		−	0.669343i	0.942335	−	0.334672i	$-0.108625\pi$
$110$		0			0
$111$			1.47068i			0.139591i
$112$		0			0
$113$	−11.2767			−1.06083			−0.530413	−	0.847739i	$-0.677963\pi$
$113$	−11.2767			−1.06083			−0.530413	+	0.847739i	$0.677963\pi$
$114$		0			0
$115$			13.9690i			1.30261i
$116$		0			0
$117$			16.3439i			1.51100i
$118$		0			0
$119$		−	14.8793i		−	1.36398i
$120$		0			0
$121$	−10.1138	−	4.32556i	−0.919439	−	0.393233i
$122$		0			0
$123$	−3.88544			−0.350338
$124$		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
$129$			0.704676i			0.0620432i
$130$		0			0
$131$	−7.50633			−0.655831			−0.327915	−	0.944707i	$-0.606346\pi$
$131$	−7.50633			−0.655831			−0.327915	+	0.944707i	$0.606346\pi$
$132$		0			0
$133$	23.8759			2.07030
$134$		0			0
$135$			8.41205i			0.723994i
$136$		0			0
$137$	−0.162910			−0.0139183			−0.00695916	−	0.999976i	$-0.502215\pi$
$137$	−0.162910			−0.0139183			−0.00695916	+	0.999976i	$0.502215\pi$
$138$		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
$141$	4.56035			0.384051
$142$		0			0
$143$	−19.5247	−	4.00000i	−1.63274	−	0.334497i
$144$		0			0
$145$			13.9763i			1.16067i
$146$		0			0
$147$		−	1.65002i		−	0.136091i
$148$		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
$153$			12.7231i			1.02860i
$154$		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
$159$		−	0.0620710i		−	0.00492255i
$160$		0			0
$161$			15.9916i			1.26032i
$162$		0			0
$163$			10.2637i			0.803919i	0.915657	+	0.401959i	$0.131671\pi$
$163$			10.2637i			0.803919i	−0.915657	+	0.401959i	$0.868329\pi$
$164$		0			0
$165$	−4.77846	−	0.978956i	−0.372002	−	0.0762116i
$166$		0			0
$167$	14.4945			1.12162			0.560808	−	0.827946i	$-0.310491\pi$
$167$	14.4945			1.12162			0.560808	+	0.827946i	$0.310491\pi$
$168$		0			0
$169$	−23.1104			−1.77772
$170$		0			0
$171$	−20.4159			−1.56124
$172$		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
$177$	−3.39400			−0.255109
$178$		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
$183$	2.28959			0.169251
$184$		0			0
$185$	7.71982			0.567573
$186$		0			0
$187$	−15.1991	−	3.11383i	−1.11147	−	0.227706i
$188$		0			0
$189$			9.63008i			0.700485i
$190$		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	$0.0224489\pi$
$193$			1.95791i			0.140934i	−0.997514	+	0.0704668i	$0.977551\pi$
$194$		0			0
$195$	−8.83762			−0.632875
$196$		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	$0.0496283\pi$
$199$			4.38101i			0.310562i	−0.987870	+	0.155281i	$0.950372\pi$
$200$		0			0
$201$	5.60256			0.395174
$202$		0			0
$203$			16.0000i			1.12298i
$204$		0			0
$205$			20.3952i			1.42446i
$206$		0			0
$207$		−	13.6742i		−	0.950421i
$208$		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
$213$	−1.01299			−0.0694089
$214$		0			0
$215$	3.69894			0.252266
$216$		0			0
$217$		−	12.6243i		−	0.856996i
$218$		0			0
$219$	3.88544			0.262554
$220$		0			0
$221$	−28.1104			−1.89091
$222$		0			0
$223$			19.9690i			1.33722i	0.743613	+	0.668610i	$0.233111\pi$
$223$			19.9690i			1.33722i	−0.743613	+	0.668610i	$0.766889\pi$
$224$		0			0
$225$	7.39744			0.493163
$226$		0			0
$227$	−2.55415			−0.169525			−0.0847624	−	0.996401i	$-0.527013\pi$
$227$	−2.55415			−0.169525			−0.0847624	+	0.996401i	$0.527013\pi$
$228$		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
$231$	−5.47036	−	1.12070i	−0.359923	−	0.0737369i
$232$		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
$237$		−	5.98853i		−	0.388997i
$238$		0			0
$239$	25.0731			1.62185			0.810923	−	0.585153i	$-0.198966\pi$
$239$	25.0731			1.62185			0.810923	+	0.585153i	$0.198966\pi$
$240$		0			0
$241$			22.0789i			1.42222i	0.703079	+	0.711112i	$0.251808\pi$
$241$			22.0789i			1.42222i	−0.703079	+	0.711112i	$0.748192\pi$
$242$		0			0
$243$		−	12.5535i		−	0.805306i
$244$		0			0
$245$	−8.66119			−0.553343
$246$		0			0
$247$		−	45.1070i		−	2.87009i
$248$		0			0
$249$		−	4.07194i		−	0.258049i
$250$		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
$255$	−6.87971			−0.430824
$256$		0			0
$257$	22.1104			1.37921			0.689604	−	0.724187i	$-0.257785\pi$
$257$	22.1104			1.37921			0.689604	+	0.724187i	$0.257785\pi$
$258$		0			0
$259$	8.83762			0.549143
$260$		0			0
$261$		−	13.6814i		−	0.846855i
$262$		0			0
$263$	−9.16930			−0.565403			−0.282702	−	0.959208i	$-0.591231\pi$
$263$	−9.16930			−0.565403			−0.282702	+	0.959208i	$0.591231\pi$
$264$		0			0
$265$	−0.325819			−0.0200149
$266$		0			0
$267$		−	1.03447i		−	0.0633087i
$268$		0			0
$269$	−7.88273			−0.480619			−0.240309	−	0.970696i	$-0.577249\pi$
$269$	−7.88273			−0.480619			−0.240309	+	0.970696i	$0.577249\pi$
$270$		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
$273$	−10.1173			−0.612325
$274$		0			0
$275$	−1.81044	+	8.83709i	−0.109174	+	0.532897i
$276$		0			0
$277$			6.98815i			0.419877i	0.977715	+	0.209939i	$0.0673264\pi$
$277$			6.98815i			0.419877i	−0.977715	+	0.209939i	$0.932674\pi$
$278$		0			0
$279$			10.7949i			0.646273i
$280$		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
$285$		−	11.0394i		−	0.653919i
$286$		0			0
$287$			23.3484i			1.37821i
$288$		0			0
$289$	−4.88273			−0.287220
$290$		0			0
$291$			7.08967i			0.415604i
$292$		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
$297$	9.83709	+	2.01531i	0.570806	+	0.116940i
$298$		0			0
$299$	30.2118			1.74719
$300$		0			0
$301$	4.23453			0.244074
$302$		0			0
$303$	2.80777			0.161302
$304$		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
$309$	2.91215			0.165666
$310$		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
$315$	24.0368			1.35432
$316$		0			0
$317$	−13.8923			−0.780268			−0.390134	−	0.920758i	$-0.627571\pi$
$317$	−13.8923			−0.780268			−0.390134	+	0.920758i	$0.627571\pi$
$318$		0			0
$319$	16.3439	+	3.34836i	0.915085	+	0.187472i
$320$		0			0
$321$		−	1.35195i		−	0.0754586i
$322$		0			0
$323$		−	35.1138i		−	1.95379i
$324$		0			0
$325$			16.3439i			0.906599i
$326$		0			0
$327$	−3.69894			−0.204552
$328$		0			0
$329$		−	27.4040i		−	1.51083i
$330$		0			0
$331$		−	2.64658i		−	0.145469i	−0.997351	−	0.0727347i	$-0.976827\pi$
$331$		−	2.64658i		−	0.145469i	0.997351	−	0.0727347i	$-0.0231726\pi$
$332$		0			0
$333$	−7.55691			−0.414116
$334$		0			0
$335$		−	29.4086i		−	1.60676i
$336$		0			0
$337$		−	28.7147i		−	1.56419i	−0.623161	−	0.782094i	$-0.714152\pi$
$337$		−	28.7147i		−	1.56419i	0.623161	−	0.782094i	$-0.285848\pi$
$338$		0			0
$339$			5.96896i			0.324190i
$340$		0			0
$341$	−12.8957	−	2.64193i	−0.698343	−	0.143068i
$342$		0			0
$343$	12.3501			0.666841
$344$		0			0
$345$	7.39400			0.398080
$346$		0			0
$347$	26.5909			1.42748			0.713738	−	0.700413i	$-0.247001\pi$
$347$	26.5909			1.42748			0.713738	+	0.700413i	$0.247001\pi$
$348$		0			0
$349$		−	36.7392i		−	1.96660i	−0.181985	−	0.983301i	$-0.558252\pi$
$349$		−	36.7392i		−	1.96660i	0.181985	−	0.983301i	$-0.441748\pi$
$350$		0			0
$351$	18.1934			0.971093
$352$		0			0
$353$	2.60600			0.138703			0.0693516	−	0.997592i	$-0.477907\pi$
$353$	2.60600			0.138703			0.0693516	+	0.997592i	$0.477907\pi$
$354$		0			0
$355$			5.31733i			0.282214i
$356$		0			0
$357$	−7.87586			−0.416835
$358$		0			0
$359$	−4.21712			−0.222571			−0.111286	−	0.993788i	$-0.535497\pi$
$359$	−4.21712			−0.222571			−0.111286	+	0.993788i	$0.535497\pi$
$360$		0			0
$361$	37.3449			1.96552
$362$		0			0
$363$	−2.28959	+	5.35342i	−0.120172	+	0.280982i
$364$		0			0
$365$		−	20.3952i		−	1.06753i
$366$		0			0
$367$			15.7344i			0.821331i	0.911786	+	0.410665i	$0.134704\pi$
$367$			15.7344i			0.821331i	−0.911786	+	0.410665i	$0.865296\pi$
$368$		0			0
$369$		−	19.9648i		−	1.03933i
$370$		0			0
$371$	−0.372997			−0.0193650
$372$		0			0
$373$		−	0.0206587i		−	0.00106967i	−1.00000		0.000534833i	$-0.999830\pi$
$373$		−	0.0206587i		−	0.00106967i	1.00000		0.000534833i	$-0.000170243\pi$
$374$		0			0
$375$		−	3.35342i		−	0.173170i
$376$		0			0
$377$	30.2277			1.55680
$378$		0			0
$379$			31.8156i			1.63426i	0.576455	+	0.817129i	$0.304436\pi$
$379$			31.8156i			1.63426i	−0.576455	+	0.817129i	$0.695564\pi$
$380$		0			0
$381$			3.64154i			0.186562i
$382$		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
$387$	−3.62088			−0.184060
$388$		0			0
$389$	−26.6543			−1.35143			−0.675714	−	0.737164i	$-0.736164\pi$
$389$	−26.6543			−1.35143			−0.675714	+	0.737164i	$0.736164\pi$
$390$		0			0
$391$	23.5186			1.18939
$392$		0			0
$393$			3.97322i			0.200423i
$394$		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
$399$		−	12.6379i		−	0.632686i
$400$		0			0
$401$	33.8759			1.69168			0.845840	−	0.533437i	$-0.179100\pi$
$401$	33.8759			1.69168			0.845840	+	0.533437i	$0.179100\pi$
$402$		0			0
$403$	−23.8503			−1.18807
$404$		0			0
$405$	−18.2181			−0.905265
$406$		0			0
$407$	1.84947	−	9.02760i	0.0916749	−	0.447481i
$408$		0			0
$409$		−	0.704676i		−	0.0348440i	−0.999848	−	0.0174220i	$-0.994454\pi$
$409$		−	0.704676i		−	0.0348440i	0.999848	−	0.0174220i	$-0.00554587\pi$
$410$		0			0
$411$			0.0862308i			0.00425345i
$412$		0			0
$413$			20.3952i			1.00358i
$414$		0			0
$415$	−21.3742			−1.04922
$416$		0			0
$417$		−	7.34049i		−	0.359465i
$418$		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
$423$			23.4328i			1.13934i
$424$		0			0
$425$			12.7231i			0.617159i
$426$		0			0
$427$		−	13.7586i		−	0.665825i
$428$		0			0
$429$	−2.11727	+	10.3348i	−0.102223	+	0.498967i
$430$		0			0
$431$	−17.1571			−0.826427			−0.413213	−	0.910634i	$-0.635594\pi$
$431$	−17.1571			−0.826427			−0.413213	+	0.910634i	$0.635594\pi$
$432$		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
$435$	7.39789			0.354701
$436$		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
$441$	8.47842			0.403734
$442$		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
$447$	−3.18077			−0.150445
$448$		0			0
$449$	−15.0422			−0.709886			−0.354943	−	0.934888i	$-0.615500\pi$
$449$	−15.0422			−0.709886			−0.354943	+	0.934888i	$0.615500\pi$
$450$		0			0
$451$	23.8503	+	4.88617i	1.12307	+	0.230081i
$452$		0			0
$453$			10.7651i			0.505791i
$454$		0			0
$455$			53.1070i			2.48969i
$456$		0			0
$457$			32.6879i			1.52907i	0.644579	+	0.764537i	$0.277033\pi$
$457$			32.6879i			1.52907i	−0.644579	+	0.764537i	$0.722967\pi$
$458$		0			0
$459$	14.1628			0.661063
$460$		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	$-0.885529\pi$
$463$		−	15.1449i		−	0.703842i	0.936030	−	0.351921i	$-0.114471\pi$
$464$		0			0
$465$	−5.83709			−0.270689
$466$		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
$471$		−	7.52588i		−	0.346774i
$472$		0			0
$473$	0.886172	−	4.32556i	0.0407462	−	0.198889i
$474$		0			0
$475$	−20.4159			−0.936745
$476$		0			0
$477$	0.318944			0.0146034
$478$		0			0
$479$	−16.7841			−0.766883			−0.383442	−	0.923565i	$-0.625261\pi$
$479$	−16.7841			−0.766883			−0.383442	+	0.923565i	$0.625261\pi$
$480$		0			0
$481$		−	16.6963i		−	0.761285i
$482$		0			0
$483$	8.46462			0.385154
$484$		0			0
$485$	37.2147			1.68983
$486$		0			0
$487$			28.9655i			1.31255i	0.754520	+	0.656277i	$0.227870\pi$
$487$			28.9655i			1.31255i	−0.754520	+	0.656277i	$0.772130\pi$
$488$		0			0
$489$	5.43277			0.245678
$490$		0			0
$491$	−26.4044			−1.19161			−0.595807	−	0.803127i	$-0.703168\pi$
$491$	−26.4044			−1.19161			−0.595807	+	0.803127i	$0.703168\pi$
$492$		0			0
$493$	23.5309			1.05978
$494$		0			0
$495$	5.03024	−	24.5535i	0.226092	−	1.10360i
$496$		0			0
$497$			6.08725i			0.273050i
$498$		0			0
$499$		−	27.4948i		−	1.23084i	−0.788200	−	0.615419i	$-0.788987\pi$
$499$		−	27.4948i		−	1.23084i	0.788200	−	0.615419i	$-0.211013\pi$
$500$		0			0
$501$		−	7.67217i		−	0.342767i
$502$		0			0
$503$	−3.69894			−0.164928			−0.0824638	−	0.996594i	$-0.526279\pi$
$503$	−3.69894			−0.164928			−0.0824638	+	0.996594i	$0.526279\pi$
$504$		0			0
$505$		−	14.7384i		−	0.655849i
$506$		0			0
$507$			12.2327i			0.543274i
$508$		0			0
$509$	2.10771			0.0934228			0.0467114	−	0.998908i	$-0.485126\pi$
$509$	2.10771			0.0934228			0.0467114	+	0.998908i	$0.485126\pi$
$510$		0			0
$511$		−	23.3484i		−	1.03287i
$512$		0			0
$513$			22.7261i			1.00338i
$514$		0			0
$515$		−	15.2863i		−	0.673594i
$516$		0			0
$517$	−27.9931	−	5.73491i	−1.23114	−	0.252221i
$518$		0			0
$519$	3.03558			0.133247
$520$		0			0
$521$	−22.8596			−1.00150			−0.500749	−	0.865592i	$-0.666942\pi$
$521$	−22.8596			−1.00150			−0.500749	+	0.865592i	$0.666942\pi$
$522$		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
$525$			4.57918i			0.199852i
$526$		0			0
$527$	−18.5664			−0.808766
$528$		0			0
$529$	−2.27674			−0.0989886
$530$		0			0
$531$		−	17.4396i		−	0.756816i
$532$		0			0
$533$	44.1104			1.91063
$534$		0			0
$535$	−7.09659			−0.306812
$536$		0			0
$537$	−2.27330			−0.0981002
$538$		0			0
$539$	−2.07500	+	10.1284i	−0.0893765	+	0.436263i
$540$		0			0
$541$			16.7743i			0.721185i	0.932723	+	0.360593i	$0.117426\pi$
$541$			16.7743i			0.721185i	−0.932723	+	0.360593i	$0.882574\pi$
$542$		0			0
$543$		−	1.64315i		−	0.0705141i
$544$		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
$549$			11.7648i			0.502108i
$550$		0			0
$551$			37.7586i			1.60857i
$552$		0			0
$553$	−35.9862			−1.53029
$554$		0			0
$555$		−	4.08623i		−	0.173451i
$556$		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
$561$	−1.64820	+	8.04516i	−0.0695871	+	0.339667i
$562$		0			0
$563$	−45.7059			−1.92627			−0.963137	−	0.269012i	$-0.913303\pi$
$563$	−45.7059			−1.92627			−0.963137	+	0.269012i	$0.913303\pi$
$564$		0			0
$565$	31.3319			1.31814
$566$		0			0
$567$	−20.8560			−0.875870
$568$		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
$573$	10.5699			0.441564
$574$		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
$579$	1.03636			0.0430694
$580$		0			0
$581$	−24.4691			−1.01515
$582$		0			0
$583$	−0.0780580	+	0.381015i	−0.00323283	+	0.0157800i
$584$		0			0
$585$		−	45.4109i		−	1.87751i
$586$		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
$591$	−4.07194			−0.167497
$592$		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
$597$	2.31894			0.0949080
$598$		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	$-0.885598\pi$
$601$		−	17.2448i		−	0.703432i	0.936107	−	0.351716i	$-0.114402\pi$
$602$		0			0
$603$			28.7880i			1.17234i
$604$		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
$609$	8.46907			0.343184
$610$		0			0
$611$	−51.7725			−2.09449
$612$		0			0
$613$			20.3746i			0.822921i	0.911428	+	0.411460i	$0.134981\pi$
$613$			20.3746i			0.822921i	−0.911428	+	0.411460i	$0.865019\pi$
$614$		0			0
$615$	10.7955			0.435318
$616$		0			0
$617$	−16.1173			−0.648857			−0.324428	−	0.945910i	$-0.605172\pi$
$617$	−16.1173			−0.648857			−0.324428	+	0.945910i	$0.605172\pi$
$618$		0			0
$619$		−	32.8742i		−	1.32133i	−0.750682	−	0.660664i	$-0.770275\pi$
$619$		−	32.8742i		−	1.32133i	0.750682	−	0.660664i	$-0.229725\pi$
$620$		0			0
$621$	−15.2215			−0.610819
$622$		0			0
$623$	−6.21635			−0.249053
$624$		0			0
$625$	−31.2017			−1.24807
$626$		0			0
$627$	−12.9096	−	2.64476i	−0.515558	−	0.105622i
$628$		0			0
$629$		−	12.9973i		−	0.518238i
$630$		0			0
$631$		−	19.9069i		−	0.792481i	−0.918147	−	0.396240i	$-0.870315\pi$
$631$		−	19.9069i		−	0.792481i	0.918147	−	0.396240i	$-0.129685\pi$
$632$		0			0
$633$		−	6.69321i		−	0.266031i
$634$		0			0
$635$	19.1150			0.758555
$636$		0			0
$637$			18.7323i			0.742199i
$638$		0			0
$639$		−	5.20512i		−	0.205911i
$640$		0			0
$641$	6.71639			0.265281			0.132641	−	0.991164i	$-0.457654\pi$
$641$	6.71639			0.265281			0.132641	+	0.991164i	$0.457654\pi$
$642$		0			0
$643$			2.58451i			0.101923i	0.998701	+	0.0509616i	$0.0162286\pi$
$643$			2.58451i			0.101923i	−0.998701	+	0.0509616i	$0.983771\pi$
$644$		0			0
$645$		−	1.95791i		−	0.0770927i
$646$		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
$651$	−6.68227			−0.261899
$652$		0			0
$653$	24.3354			0.952317			0.476158	−	0.879360i	$-0.342029\pi$
$653$	24.3354			0.952317			0.476158	+	0.879360i	$0.342029\pi$
$654$		0			0
$655$	20.8560			0.814911
$656$		0			0
$657$			19.9648i			0.778902i
$658$		0			0
$659$	26.4044			1.02857			0.514285	−	0.857619i	$-0.328057\pi$
$659$	26.4044			1.02857			0.514285	+	0.857619i	$0.328057\pi$
$660$		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
$663$			14.8793i			0.577864i
$664$		0			0
$665$	−66.3380			−2.57248
$666$		0			0
$667$	−25.2900			−0.979233
$668$		0			0
$669$	10.5699			0.408656
$670$		0			0
$671$	−14.0544	−	2.87930i	−0.542562	−	0.111154i
$672$		0			0
$673$		−	48.7369i		−	1.87867i	−0.343002	−	0.939335i	$-0.611444\pi$
$673$		−	48.7369i		−	1.87867i	0.343002	−	0.939335i	$-0.388556\pi$
$674$		0			0
$675$		−	8.23453i		−	0.316947i
$676$		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
$681$			1.35195i			0.0518069i
$682$		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
$687$		−	4.93955i		−	0.188455i
$688$		0			0
$689$			0.704676i			0.0268460i
$690$		0			0
$691$		−	23.8777i		−	0.908350i	−0.890913	−	0.454175i	$-0.849934\pi$
$691$		−	23.8777i		−	0.908350i	0.890913	−	0.454175i	$-0.150066\pi$
$692$		0			0
$693$	5.75859	−	28.1087i	0.218751	−	1.06776i
$694$		0			0
$695$	−38.5312			−1.46157
$696$		0			0
$697$	34.3380			1.30065
$698$		0			0
$699$	8.14388			0.308030
$700$		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
$705$	−12.6707			−0.477208
$706$		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
$711$	30.7713			1.15401
$712$		0			0
$713$	19.9544			0.747297
$714$		0			0
$715$	54.2486	+	11.1138i	2.02878	+	0.415634i
$716$		0			0
$717$		−	13.2716i		−	0.495638i
$718$		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
$723$	11.6867			0.434633
$724$		0			0
$725$		−	13.6814i		−	0.508113i
$726$		0			0
$727$		−	42.1346i		−	1.56268i	−0.624103	−	0.781342i	$-0.714535\pi$
$727$		−	42.1346i		−	1.56268i	0.624103	−	0.781342i	$-0.285465\pi$
$728$		0			0
$729$	13.0260			0.482444
$730$		0			0
$731$		−	6.22766i		−	0.230338i
$732$		0			0
$733$		−	40.9293i		−	1.51176i	−0.654712	−	0.755879i	$-0.727210\pi$
$733$		−	40.9293i		−	1.51176i	0.654712	−	0.755879i	$-0.272790\pi$
$734$		0			0
$735$			4.58451i			0.169102i
$736$		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
$741$	−23.8759			−0.877102
$742$		0			0
$743$	−28.7721			−1.05554			−0.527772	−	0.849386i	$-0.676973\pi$
$743$	−28.7721			−1.05554			−0.527772	+	0.849386i	$0.676973\pi$
$744$		0			0
$745$			16.6963i			0.611704i
$746$		0			0
$747$	20.9231			0.765537
$748$		0			0
$749$	−8.12414			−0.296850
$750$		0			0
$751$			53.7795i			1.96244i	0.192886	+	0.981221i	$0.438215\pi$
$751$			53.7795i			1.96244i	−0.192886	+	0.981221i	$0.561785\pi$
$752$		0			0
$753$	10.9578			0.399324
$754$		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
$759$	1.77141	−	8.64658i	0.0642982	−	0.313851i
$760$		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
$765$		−	35.3505i		−	1.27810i
$766$		0			0
$767$	38.5312			1.39128
$768$		0			0
$769$		−	28.7147i		−	1.03548i	−0.855539	−	0.517739i	$-0.826774\pi$
$769$		−	28.7147i		−	1.03548i	0.855539	−	0.517739i	$-0.173226\pi$
$770$		0			0
$771$		−	11.7034i		−	0.421488i
$772$		0			0
$773$	19.7586			0.710667			0.355334	−	0.934740i	$-0.384367\pi$
$773$	19.7586			0.710667			0.355334	+	0.934740i	$0.384367\pi$
$774$		0			0
$775$			10.7949i			0.387764i
$776$		0			0
$777$		−	4.67790i		−	0.167819i
$778$		0			0
$779$			55.1001i			1.97417i
$780$		0			0
$781$	6.21811	+	1.27389i	0.222501	+	0.0455835i
$782$		0			0
$783$	−15.2295			−0.544259
$784$		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
$789$			4.85346i			0.172788i
$790$		0			0
$791$	35.8687			1.27534
$792$		0			0
$793$	−25.9931			−0.923043
$794$		0			0
$795$			0.172462i			0.00611658i
$796$		0			0
$797$	−49.5336			−1.75457			−0.877285	−	0.479969i	$-0.840648\pi$
$797$	−49.5336			−1.75457			−0.877285	+	0.479969i	$0.840648\pi$
$798$		0			0
$799$	−40.3027			−1.42581
$800$		0			0
$801$	5.31551			0.187814
$802$		0			0
$803$	−23.8503	−	4.88617i	−0.841658	−	0.172429i
$804$		0			0
$805$		−	44.4320i		−	1.56602i
$806$		0			0
$807$			4.17246i			0.146878i
$808$		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
$813$			2.89555i			0.101551i
$814$		0			0
$815$		−	28.5174i		−	0.998920i
$816$		0			0
$817$	9.99312			0.349615
$818$		0			0
$819$		−	51.9862i		−	1.81655i
$820$		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	$-0.929759\pi$
$823$		−	12.5585i		−	0.437763i	0.975751	−	0.218881i	$-0.0702408\pi$
$824$		0			0
$825$	4.67762	+	0.958297i	0.162854	+	0.0333636i
$826$		0			0
$827$	−46.5557			−1.61890			−0.809451	−	0.587188i	$-0.800235\pi$
$827$	−46.5557			−1.61890			−0.809451	+	0.587188i	$0.800235\pi$
$828$		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
$831$	3.69894			0.128315
$832$		0			0
$833$			14.5823i			0.505245i
$834$		0			0
$835$	−40.2723			−1.39368
$836$		0			0
$837$	12.0164			0.415348
$838$		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
$843$	−1.40935			−0.0485407
$844$		0			0
$845$	64.2112			2.20893
$846$		0			0
$847$	32.1697	+	13.7586i	1.10536	+	0.472751i
$848$		0			0
$849$			10.0605i			0.345275i
$850$		0			0
$851$			13.9690i			0.478850i
$852$		0			0
$853$			51.6944i			1.76998i	0.465607	+	0.884992i	$0.345836\pi$
$853$			51.6944i			1.76998i	−0.465607	+	0.884992i	$0.654164\pi$
$854$		0			0
$855$	56.7247			1.93994
$856$		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	$-0.994261\pi$
$859$		−	1.05681i		−	0.0360580i	0.999837	−	0.0180290i	$-0.00573912\pi$
$860$		0			0
$861$	12.3587			0.421183
$862$		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
$867$			2.58451i			0.0877746i
$868$		0			0
$869$	−7.53093	+	36.7598i	−0.255469	+	1.24699i
$870$		0			0
$871$	−63.6044			−2.15515
$872$		0			0
$873$	−36.4293			−1.23295
$874$		0			0
$875$	−20.1513			−0.681239
$876$		0			0
$877$			0.0206587i			0.000697594i	1.00000		0.000348797i	$0.000111026\pi$
$877$			0.0206587i			0.000697594i	−1.00000		0.000348797i	$0.999889\pi$
$878$		0			0
$879$	−2.14441			−0.0723292
$880$		0			0
$881$	45.0682			1.51839			0.759193	−	0.650866i	$-0.225594\pi$
$881$	45.0682			1.51839			0.759193	+	0.650866i	$0.225594\pi$
$882$		0			0
$883$		−	30.6087i		−	1.03006i	−0.857171	−	0.515032i	$-0.827780\pi$
$883$		−	30.6087i		−	1.03006i	0.857171	−	0.515032i	$-0.172220\pi$
$884$		0			0
$885$	9.43010			0.316989
$886$		0			0
$887$	−13.6033			−0.456754			−0.228377	−	0.973573i	$-0.573342\pi$
$887$	−13.6033			−0.456754			−0.228377	+	0.973573i	$0.573342\pi$
$888$		0			0
$889$	21.8827			0.733923
$890$		0			0
$891$	−4.36459	+	21.3043i	−0.146219	+	0.713722i
$892$		0			0
$893$		−	64.6711i		−	2.16414i
$894$		0			0
$895$			11.9329i			0.398872i
$896$		0			0
$897$		−	15.9916i		−	0.533944i
$898$		0			0
$899$	19.9648			0.665864
$900$		0			0
$901$			0.548560i			0.0182752i
$902$		0			0
$903$		−	2.24141i		−	0.0745894i
$904$		0			0
$905$	−8.62510			−0.286708
$906$		0			0
$907$		−	1.50172i		−	0.0498638i	−0.999689	−	0.0249319i	$-0.992063\pi$
$907$		−	1.50172i		−	0.0498638i	0.999689	−	0.0249319i	$-0.00793689\pi$
$908$		0			0
$909$			14.4274i			0.478525i
$910$		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
$915$	−6.36153			−0.210306
$916$		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
$921$			0.605959i			0.0199670i
$922$		0			0
$923$	11.5002			0.378534
$924$		0			0
$925$	−7.55691			−0.248470
$926$		0			0
$927$			14.9637i			0.491473i
$928$		0			0
$929$	19.6482			0.644637			0.322318	−	0.946631i	$-0.395538\pi$
$929$	19.6482			0.644637			0.322318	+	0.946631i	$0.395538\pi$
$930$		0			0
$931$	−23.3992			−0.766878
$932$		0			0
$933$	−9.44652			−0.309265
$934$		0			0
$935$	42.2302	+	8.65164i	1.38107	+	0.282939i
$936$		0			0
$937$			6.04593i			0.197512i	0.995112	+	0.0987560i	$0.0314863\pi$
$937$			6.04593i			0.197512i	−0.995112	+	0.0987560i	$0.968514\pi$
$938$		0			0
$939$			17.2069i			0.561527i
$940$		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
$945$		−	26.7568i		−	0.870397i
$946$		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
$951$			7.35342i			0.238451i
$952$		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
$957$	1.77234	−	8.65112i	0.0572917	−	0.279651i
$958$		0			0
$959$	0.518178			0.0167328
$960$		0			0
$961$	15.2473			0.491849
$962$		0			0
$963$	6.94683			0.223858
$964$		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
$969$	−18.5863			−0.597079
$970$		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
$975$	8.65112			0.277058
$976$		0			0
$977$	18.8268			0.602322			0.301161	−	0.953573i	$-0.402626\pi$
$977$	18.8268			0.602322			0.301161	+	0.953573i	$0.402626\pi$
$978$		0			0
$979$	−1.30091	+	6.34998i	−0.0415773	+	0.202946i
$980$		0			0
$981$		−	19.0065i		−	0.606832i
$982$		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
$987$	−14.5054			−0.461712
$988$		0			0
$989$			6.69321i			0.212832i
$990$		0			0
$991$		−	5.14304i		−	0.163374i	−0.996658	−	0.0816871i	$-0.973969\pi$
$991$		−	5.14304i		−	0.163374i	0.996658	−	0.0816871i	$-0.0260308\pi$
$992$		0			0
$993$	−1.40088			−0.0444556
$994$		0			0
$995$		−	12.1725i		−	0.385893i
$996$		0			0
$997$		−	10.5884i		−	0.335337i	−0.985843	−	0.167669i	$-0.946376\pi$
$997$		−	10.5884i		−	0.335337i	0.985843	−	0.167669i	$-0.0536239\pi$
$998$		0			0
$999$			8.41205i			0.266145i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.e.a.351.5	✓	12
3.2	odd	2		3168.2.o.e.703.9		12
4.3	odd	2	inner	352.2.e.a.351.8	yes	12
8.3	odd	2		704.2.e.d.703.6		12
8.5	even	2		704.2.e.d.703.7		12
11.10	odd	2	inner	352.2.e.a.351.6	yes	12
12.11	even	2		3168.2.o.e.703.12		12
16.3	odd	4		2816.2.g.d.1407.7		12
16.5	even	4		2816.2.g.d.1407.6		12
16.11	odd	4		2816.2.g.i.1407.5		12
16.13	even	4		2816.2.g.i.1407.8		12
33.32	even	2		3168.2.o.e.703.11		12
44.43	even	2	inner	352.2.e.a.351.7	yes	12
88.21	odd	2		704.2.e.d.703.8		12
88.43	even	2		704.2.e.d.703.5		12
132.131	odd	2		3168.2.o.e.703.10		12
176.21	odd	4		2816.2.g.d.1407.5		12
176.43	even	4		2816.2.g.i.1407.6		12
176.109	odd	4		2816.2.g.i.1407.7		12
176.131	even	4		2816.2.g.d.1407.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.5	✓	12	1.1	even	1	trivial
352.2.e.a.351.6	yes	12	11.10	odd	2	inner
352.2.e.a.351.7	yes	12	44.43	even	2	inner
352.2.e.a.351.8	yes	12	4.3	odd	2	inner
704.2.e.d.703.5		12	88.43	even	2
704.2.e.d.703.6		12	8.3	odd	2
704.2.e.d.703.7		12	8.5	even	2
704.2.e.d.703.8		12	88.21	odd	2
2816.2.g.d.1407.5		12	176.21	odd	4
2816.2.g.d.1407.6		12	16.5	even	4
2816.2.g.d.1407.7		12	16.3	odd	4
2816.2.g.d.1407.8		12	176.131	even	4
2816.2.g.i.1407.5		12	16.11	odd	4
2816.2.g.i.1407.6		12	176.43	even	4
2816.2.g.i.1407.7		12	176.109	odd	4
2816.2.g.i.1407.8		12	16.13	even	4
3168.2.o.e.703.9		12	3.2	odd	2
3168.2.o.e.703.10		12	132.131	odd	2
3168.2.o.e.703.11		12	33.32	even	2
3168.2.o.e.703.12		12	12.11	even	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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.529317i q^{3} -2.77846 q^{5} -3.18077 q^{7} +2.71982 q^{9} +O(q^{10})\)	\(q-0.529317i q^{3} -2.77846 q^{5} -3.18077 q^{7} +2.71982 q^{9} +(-0.665647 + 3.24914i) q^{11} +6.00919i q^{13} +1.47068i q^{15} +4.67790i q^{17} -7.50633 q^{19} +1.68363i q^{21} -5.02760i q^{23} +2.71982 q^{25} -3.02760i q^{27} -5.03024i q^{29} +3.96896i q^{31} +(1.71982 + 0.352338i) q^{33} +8.83762 q^{35} -2.77846 q^{37} +3.18077 q^{39} -7.34049i q^{41} -1.33129 q^{43} -7.55691 q^{45} +8.61555i q^{47} +3.11727 q^{49} +2.47609 q^{51} +0.117266 q^{53} +(1.84947 - 9.02760i) q^{55} +3.97322i q^{57} -6.41205i q^{59} +4.32556i q^{61} -8.65112 q^{63} -16.6963i q^{65} +10.5845i q^{67} -2.66119 q^{69} -1.91377i q^{71} +7.34049i q^{73} -1.43965i q^{75} +(2.11727 - 10.3348i) q^{77} +11.3137 q^{79} +6.55691 q^{81} +7.69282 q^{83} -12.9973i q^{85} -2.66259 q^{87} +1.95436 q^{89} -19.1138i q^{91} +2.10084 q^{93} +20.8560 q^{95} -13.3940 q^{97} +(-1.81044 + 8.83709i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9}+O(q^{10}) \) 12 * q - 4 * q^9	\( 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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