LMFDB - Embedded newform 2816.2.g.d.1407.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.g (of order $2$, degree $1$, not 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:	$22.4858732092$
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^{11} $
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1407.7
Root			$1.16947 - 0.795191i$ of defining polynomial
Character	$\chi$	$=$	2816.1407
Dual form			2816.2.g.d.1407.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.529317 q^{3} +2.77846i q^{5} -3.18077 q^{7} -2.71982 q^{9} +O(q^{10})$	$q-0.529317 q^{3} +2.77846i q^{5} -3.18077 q^{7} -2.71982 q^{9} +(-3.24914 - 0.665647i) q^{11} -6.00919 q^{13} -1.47068i q^{15} +4.67790i q^{17} +7.50633i q^{19} +1.68363 q^{21} -5.02760i q^{23} -2.71982 q^{25} +3.02760 q^{27} +5.03024 q^{29} -3.96896i q^{31} +(1.71982 + 0.352338i) q^{33} -8.83762i q^{35} +2.77846i q^{37} +3.18077 q^{39} +7.34049i q^{41} -1.33129i q^{43} -7.55691i q^{45} -8.61555i q^{47} +3.11727 q^{49} -2.47609i q^{51} -0.117266i q^{53} +(1.84947 - 9.02760i) q^{55} -3.97322i q^{57} +6.41205 q^{59} -4.32556 q^{61} +8.65112 q^{63} -16.6963i q^{65} +10.5845 q^{67} +2.66119i q^{69} -1.91377i q^{71} -7.34049i q^{73} +1.43965 q^{75} +(10.3348 + 2.11727i) q^{77} -11.3137 q^{79} +6.55691 q^{81} -7.69282i q^{83} -12.9973 q^{85} -2.66259 q^{87} -1.95436 q^{89} +19.1138 q^{91} +2.10084i q^{93} -20.8560 q^{95} -13.3940 q^{97} +(8.83709 + 1.81044i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} + 4 q^{9}+O(q^{10}) $ 12 * q - 8 * q^3 + 4 * q^9	$ 12 q - 8 q^{3} + 4 q^{9} - 4 q^{11} + 4 q^{25} - 32 q^{27} - 16 q^{33} + 44 q^{49} + 72 q^{59} - 8 q^{67} - 56 q^{75} + 12 q^{81} + 96 q^{91} - 64 q^{97} + 76 q^{99}+O(q^{100}) $ 12 * q - 8 * q^3 + 4 * q^9 - 4 * q^11 + 4 * q^25 - 32 * q^27 - 16 * q^33 + 44 * q^49 + 72 * q^59 - 8 * q^67 - 56 * q^75 + 12 * q^81 + 96 * q^91 - 64 * q^97 + 76 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−0.529317			−0.305601			−0.152801	−	0.988257i	$-0.548829\pi$
$3$	−0.529317			−0.305601			−0.152801	+	0.988257i	$0.548829\pi$
$4$		0			0
$5$			2.77846i			1.24256i	0.783587	+	0.621282i	$0.213388\pi$
$5$			2.77846i			1.24256i	−0.783587	+	0.621282i	$0.786612\pi$
$6$		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
$11$	−3.24914	−	0.665647i	−0.979653	−	0.200700i
$11$	−3.24914	−	0.665647i	−0.979653	−	0.200700i
$12$		0			0
$13$	−6.00919			−1.66665			−0.833325	−	0.552783i	$-0.813566\pi$
$13$	−6.00919			−1.66665			−0.833325	+	0.552783i	$0.813566\pi$
$14$		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
$19$			7.50633i			1.72207i	0.508546	+	0.861035i	$0.330183\pi$
$19$			7.50633i			1.72207i	−0.508546	+	0.861035i	$0.669817\pi$
$20$		0			0
$21$	1.68363			0.367399
$22$		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
$27$	3.02760			0.582661
$28$		0			0
$29$	5.03024			0.934091			0.467046	−	0.884233i	$-0.345318\pi$
$29$	5.03024			0.934091			0.467046	+	0.884233i	$0.345318\pi$
$30$		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
$33$	1.71982	+	0.352338i	0.299383	+	0.0613342i
$34$		0			0
$35$		−	8.83762i		−	1.49383i
$36$		0			0
$37$			2.77846i			0.456776i	0.973570	+	0.228388i	$0.0733454\pi$
$37$			2.77846i			0.456776i	−0.973570	+	0.228388i	$0.926655\pi$
$38$		0			0
$39$	3.18077			0.509330
$40$		0			0
$41$			7.34049i			1.14639i	0.819419	+	0.573196i	$0.194297\pi$
$41$			7.34049i			1.14639i	−0.819419	+	0.573196i	$0.805703\pi$
$42$		0			0
$43$		−	1.33129i		−	0.203020i	−0.994834	−	0.101510i	$-0.967633\pi$
$43$		−	1.33129i		−	0.203020i	0.994834	−	0.101510i	$-0.0323674\pi$
$44$		0			0
$45$		−	7.55691i		−	1.12652i
$46$		0			0
$47$		−	8.61555i		−	1.25671i	−0.777928	−	0.628353i	$-0.783729\pi$
$47$		−	8.61555i		−	1.25671i	0.777928	−	0.628353i	$-0.216271\pi$
$48$		0			0
$49$	3.11727			0.445324
$50$		0			0
$51$		−	2.47609i		−	0.346722i
$52$		0			0
$53$		−	0.117266i		−	0.0161078i	−0.999968	−	0.00805388i	$-0.997436\pi$
$53$		−	0.117266i		−	0.0161078i	0.999968	−	0.00805388i	$-0.00256366\pi$
$54$		0			0
$55$	1.84947	−	9.02760i	0.249383	−	1.21728i
$56$		0			0
$57$		−	3.97322i		−	0.526266i
$58$		0			0
$59$	6.41205			0.834778			0.417389	−	0.908728i	$-0.362945\pi$
$59$	6.41205			0.834778			0.417389	+	0.908728i	$0.362945\pi$
$60$		0			0
$61$	−4.32556			−0.553831			−0.276916	−	0.960894i	$-0.589312\pi$
$61$	−4.32556			−0.553831			−0.276916	+	0.960894i	$0.589312\pi$
$62$		0			0
$63$	8.65112			1.08994
$64$		0			0
$65$		−	16.6963i		−	2.07092i
$66$		0			0
$67$	10.5845			1.29310			0.646552	−	0.762870i	$-0.276210\pi$
$67$	10.5845			1.29310			0.646552	+	0.762870i	$0.276210\pi$
$68$		0			0
$69$			2.66119i			0.320370i
$70$		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
$75$	1.43965			0.166236
$76$		0			0
$77$	10.3348	+	2.11727i	1.17775	+	0.241285i
$78$		0			0
$79$	−11.3137			−1.27289			−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137			−1.27289			−0.636446	+	0.771321i	$0.719596\pi$
$80$		0			0
$81$	6.55691			0.728546
$82$		0			0
$83$		−	7.69282i		−	0.844397i	−0.906503	−	0.422199i	$-0.861259\pi$
$83$		−	7.69282i		−	0.844397i	0.906503	−	0.422199i	$-0.138741\pi$
$84$		0			0
$85$	−12.9973			−1.40976
$86$		0			0
$87$	−2.66259			−0.285459
$88$		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
$91$	19.1138			2.00367
$92$		0			0
$93$			2.10084i			0.217847i
$94$		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
$99$	8.83709	+	1.81044i	0.888161	+	0.181956i
$100$		0			0
$101$	5.30452			0.527819			0.263910	−	0.964547i	$-0.414988\pi$
$101$	5.30452			0.527819			0.263910	+	0.964547i	$0.414988\pi$
$102$		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
$107$			2.55415i			0.246919i	0.992350	+	0.123459i	$0.0393989\pi$
$107$			2.55415i			0.246919i	−0.992350	+	0.123459i	$0.960601\pi$
$108$		0			0
$109$	6.98815			0.669343			0.334672	−	0.942335i	$-0.391375\pi$
$109$	6.98815			0.669343			0.334672	+	0.942335i	$0.391375\pi$
$110$		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
$115$	13.9690			1.30261
$116$		0			0
$117$	16.3439			1.51100
$118$		0			0
$119$		−	14.8793i		−	1.36398i
$120$		0			0
$121$	10.1138	+	4.32556i	0.919439	+	0.393233i
$122$		0			0
$123$		−	3.88544i		−	0.350338i
$124$		0			0
$125$			6.33537i			0.566653i
$126$		0			0
$127$	6.87971			0.610475			0.305238	−	0.952276i	$-0.401264\pi$
$127$	6.87971			0.610475			0.305238	+	0.952276i	$0.401264\pi$
$128$		0			0
$129$			0.704676i			0.0620432i
$130$		0			0
$131$			7.50633i			0.655831i	0.944707	+	0.327915i	$0.106346\pi$
$131$			7.50633i			0.655831i	−0.944707	+	0.327915i	$0.893654\pi$
$132$		0			0
$133$		−	23.8759i		−	2.07030i
$134$		0			0
$135$			8.41205i			0.723994i
$136$		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
$139$			13.8679i			1.17626i	0.808768	+	0.588128i	$0.200135\pi$
$139$			13.8679i			1.17626i	−0.808768	+	0.588128i	$0.799865\pi$
$140$		0			0
$141$			4.56035i			0.384051i
$142$		0			0
$143$	19.5247	+	4.00000i	1.63274	+	0.334497i
$144$		0			0
$145$			13.9763i			1.16067i
$146$		0			0
$147$	−1.65002			−0.136091
$148$		0			0
$149$	−6.00919			−0.492292			−0.246146	−	0.969233i	$-0.579164\pi$
$149$	−6.00919			−0.492292			−0.246146	+	0.969233i	$0.579164\pi$
$150$		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
$155$	11.0276			0.885758
$156$		0			0
$157$			14.2181i			1.13473i	0.823467	+	0.567364i	$0.192037\pi$
$157$			14.2181i			1.13473i	−0.823467	+	0.567364i	$0.807963\pi$
$158$		0			0
$159$			0.0620710i			0.00492255i
$160$		0			0
$161$			15.9916i			1.26032i
$162$		0			0
$163$	10.2637			0.803919			0.401959	−	0.915657i	$-0.368329\pi$
$163$	10.2637			0.803919			0.401959	+	0.915657i	$0.368329\pi$
$164$		0			0
$165$	−0.978956	+	4.77846i	−0.0762116	+	0.372002i
$166$		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
$171$		−	20.4159i		−	1.56124i
$172$		0			0
$173$	−5.73491			−0.436017			−0.218009	−	0.975947i	$-0.569956\pi$
$173$	−5.73491			−0.436017			−0.218009	+	0.975947i	$0.569956\pi$
$174$		0			0
$175$	8.65112			0.653963
$176$		0			0
$177$	−3.39400			−0.255109
$178$		0			0
$179$	−4.29478			−0.321007			−0.160504	−	0.987035i	$-0.551312\pi$
$179$	−4.29478			−0.321007			−0.160504	+	0.987035i	$0.551312\pi$
$180$		0			0
$181$		−	3.10428i		−	0.230739i	−0.993323	−	0.115370i	$-0.963195\pi$
$181$		−	3.10428i		−	0.230739i	0.993323	−	0.115370i	$-0.0368052\pi$
$182$		0			0
$183$	2.28959			0.169251
$184$		0			0
$185$	−7.71982			−0.567573
$186$		0			0
$187$	3.11383	−	15.1991i	0.227706	−	1.11147i
$188$		0			0
$189$	−9.63008			−0.700485
$190$		0			0
$191$		−	19.9690i		−	1.44490i	−0.691421	−	0.722452i	$-0.743015\pi$
$191$		−	19.9690i		−	1.44490i	0.691421	−	0.722452i	$-0.256985\pi$
$192$		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
$195$			8.83762i			0.632875i
$196$		0			0
$197$	−7.69282			−0.548091			−0.274046	−	0.961717i	$-0.588362\pi$
$197$	−7.69282			−0.548091			−0.274046	+	0.961717i	$0.588362\pi$
$198$		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
$203$	−16.0000			−1.12298
$204$		0			0
$205$	−20.3952			−1.42446
$206$		0			0
$207$			13.6742i			0.950421i
$208$		0			0
$209$	4.99656	−	24.3891i	0.345619	−	1.68703i
$210$		0			0
$211$		−	12.6450i		−	0.870518i	−0.900305	−	0.435259i	$-0.856657\pi$
$211$		−	12.6450i		−	0.870518i	0.900305	−	0.435259i	$-0.143343\pi$
$212$		0			0
$213$			1.01299i			0.0694089i
$214$		0			0
$215$	3.69894			0.252266
$216$		0			0
$217$			12.6243i			0.856996i
$218$		0			0
$219$			3.88544i			0.262554i
$220$		0			0
$221$		−	28.1104i		−	1.89091i
$222$		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
$225$	7.39744			0.493163
$226$		0			0
$227$			2.55415i			0.169525i	0.996401	+	0.0847624i	$0.0270131\pi$
$227$			2.55415i			0.169525i	−0.996401	+	0.0847624i	$0.972987\pi$
$228$		0			0
$229$		−	9.33193i		−	0.616672i	−0.951278	−	0.308336i	$-0.900228\pi$
$229$		−	9.33193i		−	0.616672i	0.951278	−	0.308336i	$-0.0997720\pi$
$230$		0			0
$231$	−5.47036	−	1.12070i	−0.359923	−	0.0737369i
$232$		0			0
$233$		−	15.3856i		−	1.00795i	−0.863719	−	0.503974i	$-0.831871\pi$
$233$		−	15.3856i		−	1.00795i	0.863719	−	0.503974i	$-0.168129\pi$
$234$		0			0
$235$	23.9379			1.56154
$236$		0			0
$237$	5.98853			0.388997
$238$		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	$0.251808\pi$
$241$			22.0789i			1.42222i	−0.703079	+	0.711112i	$0.748192\pi$
$242$		0			0
$243$	−12.5535			−0.805306
$244$		0			0
$245$			8.66119i			0.553343i
$246$		0			0
$247$		−	45.1070i		−	2.87009i
$248$		0			0
$249$			4.07194i			0.258049i
$250$		0			0
$251$	−20.7018			−1.30668			−0.653342	−	0.757063i	$-0.726634\pi$
$251$	−20.7018			−1.30668			−0.653342	+	0.757063i	$0.726634\pi$
$252$		0			0
$253$	−3.34660	+	16.3354i	−0.210399	+	1.02700i
$254$		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
$259$		−	8.83762i		−	0.549143i
$260$		0			0
$261$	−13.6814			−0.846855
$262$		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
$267$	1.03447			0.0633087
$268$		0			0
$269$		−	7.88273i		−	0.480619i	−0.970696	−	0.240309i	$-0.922751\pi$
$269$		−	7.88273i		−	0.480619i	0.970696	−	0.240309i	$-0.0772489\pi$
$270$		0			0
$271$	5.47036			0.332300			0.166150	−	0.986100i	$-0.446866\pi$
$271$	5.47036			0.332300			0.166150	+	0.986100i	$0.446866\pi$
$272$		0			0
$273$	−10.1173			−0.612325
$274$		0			0
$275$	8.83709	+	1.81044i	0.532897	+	0.109174i
$276$		0			0
$277$	6.98815			0.419877			0.209939	−	0.977715i	$-0.432674\pi$
$277$	6.98815			0.419877			0.209939	+	0.977715i	$0.432674\pi$
$278$		0			0
$279$			10.7949i			0.646273i
$280$		0			0
$281$			2.66259i			0.158837i	0.996841	+	0.0794183i	$0.0253063\pi$
$281$			2.66259i			0.158837i	−0.996841	+	0.0794183i	$0.974694\pi$
$282$		0			0
$283$		−	19.0065i		−	1.12982i	−0.825152	−	0.564911i	$-0.808911\pi$
$283$		−	19.0065i		−	1.12982i	0.825152	−	0.564911i	$-0.191089\pi$
$284$		0			0
$285$	11.0394			0.653919
$286$		0			0
$287$		−	23.3484i		−	1.37821i
$288$		0			0
$289$	−4.88273			−0.287220
$290$		0			0
$291$	7.08967			0.415604
$292$		0			0
$293$	−4.05128			−0.236678			−0.118339	−	0.992973i	$-0.537757\pi$
$293$	−4.05128			−0.236678			−0.118339	+	0.992973i	$0.537757\pi$
$294$		0			0
$295$			17.8156i			1.03726i
$296$		0			0
$297$	−9.83709	−	2.01531i	−0.570806	−	0.116940i
$298$		0			0
$299$			30.2118i			1.74719i
$300$		0			0
$301$			4.23453i			0.244074i
$302$		0			0
$303$	−2.80777			−0.161302
$304$		0			0
$305$		−	12.0184i		−	0.688171i
$306$		0			0
$307$			1.14480i			0.0653369i	0.999466	+	0.0326685i	$0.0104005\pi$
$307$			1.14480i			0.0653369i	−0.999466	+	0.0326685i	$0.989599\pi$
$308$		0			0
$309$		−	2.91215i		−	0.165666i
$310$		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	$-0.129219\pi$
$313$	32.5078			1.83745			0.918726	+	0.394896i	$0.129219\pi$
$314$		0			0
$315$			24.0368i			1.35432i
$316$		0			0
$317$		−	13.8923i		−	0.780268i	−0.920758	−	0.390134i	$-0.872429\pi$
$317$		−	13.8923i		−	0.780268i	0.920758	−	0.390134i	$-0.127571\pi$
$318$		0			0
$319$	−16.3439	−	3.34836i	−0.915085	−	0.187472i
$320$		0			0
$321$		−	1.35195i		−	0.0754586i
$322$		0			0
$323$	−35.1138			−1.95379
$324$		0			0
$325$	16.3439			0.906599
$326$		0			0
$327$	−3.69894			−0.204552
$328$		0			0
$329$			27.4040i			1.51083i
$330$		0			0
$331$	2.64658			0.145469			0.0727347	−	0.997351i	$-0.476827\pi$
$331$	2.64658			0.145469			0.0727347	+	0.997351i	$0.476827\pi$
$332$		0			0
$333$		−	7.55691i		−	0.414116i
$334$		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
$339$	5.96896			0.324190
$340$		0			0
$341$	−2.64193	+	12.8957i	−0.143068	+	0.698343i
$342$		0			0
$343$	12.3501			0.666841
$344$		0			0
$345$	−7.39400			−0.398080
$346$		0			0
$347$			26.5909i			1.42748i	0.700413	+	0.713738i	$0.252999\pi$
$347$			26.5909i			1.42748i	−0.700413	+	0.713738i	$0.747001\pi$
$348$		0			0
$349$	36.7392			1.96660			0.983301	−	0.181985i	$-0.0582522\pi$
$349$	36.7392			1.96660			0.983301	+	0.181985i	$0.0582522\pi$
$350$		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
$355$	5.31733			0.282214
$356$		0			0
$357$			7.87586i			0.416835i
$358$		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
$363$	−5.35342	−	2.28959i	−0.280982	−	0.120172i
$364$		0			0
$365$	20.3952			1.06753
$366$		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
$369$		−	19.9648i		−	1.03933i
$370$		0			0
$371$			0.372997i			0.0193650i
$372$		0			0
$373$	−0.0206587			−0.00106967			−0.000534833	−	1.00000i	$-0.500170\pi$
$373$	−0.0206587			−0.00106967			−0.000534833		1.00000i	$0.500170\pi$
$374$		0			0
$375$		−	3.35342i		−	0.173170i
$376$		0			0
$377$	−30.2277			−1.55680
$378$		0			0
$379$	−31.8156			−1.63426			−0.817129	−	0.576455i	$-0.804436\pi$
$379$	−31.8156			−1.63426			−0.817129	+	0.576455i	$0.804436\pi$
$380$		0			0
$381$	−3.64154			−0.186562
$382$		0			0
$383$			19.3794i			0.990241i	0.868824	+	0.495120i	$0.164876\pi$
$383$			19.3794i			0.990241i	−0.868824	+	0.495120i	$0.835124\pi$
$384$		0			0
$385$	−5.88273	+	28.7147i	−0.299812	+	1.46343i
$386$		0			0
$387$			3.62088i			0.184060i
$388$		0			0
$389$			26.6543i			1.35143i	0.737164	+	0.675714i	$0.236164\pi$
$389$			26.6543i			1.35143i	−0.737164	+	0.675714i	$0.763836\pi$
$390$		0			0
$391$	23.5186			1.18939
$392$		0			0
$393$		−	3.97322i		−	0.200423i
$394$		0			0
$395$		−	31.4347i		−	1.58165i
$396$		0			0
$397$		−	12.1173i		−	0.608148i	−0.952648	−	0.304074i	$-0.901653\pi$
$397$		−	12.1173i		−	0.608148i	0.952648	−	0.304074i	$-0.0983470\pi$
$398$		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
$403$			23.8503i			1.18807i
$404$		0			0
$405$			18.2181i			0.905265i
$406$		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
$411$	−0.0862308			−0.00425345
$412$		0			0
$413$	−20.3952			−1.00358
$414$		0			0
$415$	21.3742			1.04922
$416$		0			0
$417$		−	7.34049i		−	0.359465i
$418$		0			0
$419$	−27.4948			−1.34321			−0.671605	−	0.740909i	$-0.734395\pi$
$419$	−27.4948			−1.34321			−0.671605	+	0.740909i	$0.734395\pi$
$420$		0			0
$421$			4.11727i			0.200663i	0.994954	+	0.100332i	$0.0319904\pi$
$421$			4.11727i			0.200663i	−0.994954	+	0.100332i	$0.968010\pi$
$422$		0			0
$423$			23.4328i			1.13934i
$424$		0			0
$425$		−	12.7231i		−	0.617159i
$426$		0			0
$427$	13.7586			0.665825
$428$		0			0
$429$	−10.3348	−	2.11727i	−0.498967	−	0.102223i
$430$		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
$435$		−	7.39789i		−	0.354701i
$436$		0			0
$437$	37.7388			1.80529
$438$		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
$443$	−12.2327			−0.581194			−0.290597	−	0.956846i	$-0.593854\pi$
$443$	−12.2327			−0.581194			−0.290597	+	0.956846i	$0.593854\pi$
$444$		0			0
$445$		−	5.43010i		−	0.257411i
$446$		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
$451$	4.88617	−	23.8503i	0.230081	−	1.12307i
$452$		0			0
$453$	10.7651			0.505791
$454$		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
$459$			14.1628i			0.661063i
$460$		0			0
$461$	−22.3738			−1.04205			−0.521026	−	0.853541i	$-0.674450\pi$
$461$	−22.3738			−1.04205			−0.521026	+	0.853541i	$0.674450\pi$
$462$		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
$465$	−5.83709			−0.270689
$466$		0			0
$467$	19.2362			0.890143			0.445071	−	0.895495i	$-0.353178\pi$
$467$	19.2362			0.890143			0.445071	+	0.895495i	$0.353178\pi$
$468$		0			0
$469$	−33.6668			−1.55459
$470$		0			0
$471$		−	7.52588i		−	0.346774i
$472$		0			0
$473$	−0.886172	+	4.32556i	−0.0407462	+	0.198889i
$474$		0			0
$475$		−	20.4159i		−	0.936745i
$476$		0			0
$477$			0.318944i			0.0146034i
$478$		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
$483$		−	8.46462i		−	0.385154i
$484$		0			0
$485$		−	37.2147i		−	1.68983i
$486$		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
$491$		−	26.4044i		−	1.19161i	−0.803127	−	0.595807i	$-0.796832\pi$
$491$		−	26.4044i		−	1.19161i	0.803127	−	0.595807i	$-0.203168\pi$
$492$		0			0
$493$			23.5309i			1.05978i
$494$		0			0
$495$	−5.03024	+	24.5535i	−0.226092	+	1.10360i
$496$		0			0
$497$			6.08725i			0.273050i
$498$		0			0
$499$	−27.4948			−1.23084			−0.615419	−	0.788200i	$-0.711013\pi$
$499$	−27.4948			−1.23084			−0.615419	+	0.788200i	$0.711013\pi$
$500$		0			0
$501$	−7.67217			−0.342767
$502$		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
$507$	−12.2327			−0.543274
$508$		0			0
$509$			2.10771i			0.0934228i	0.998908	+	0.0467114i	$0.0148741\pi$
$509$			2.10771i			0.0934228i	−0.998908	+	0.0467114i	$0.985126\pi$
$510$		0			0
$511$			23.3484i			1.03287i
$512$		0			0
$513$			22.7261i			1.00338i
$514$		0			0
$515$	−15.2863			−0.673594
$516$		0			0
$517$	−5.73491	+	27.9931i	−0.252221	+	1.23114i
$518$		0			0
$519$	3.03558			0.133247
$520$		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
$523$		−	12.6146i		−	0.551599i	−0.961215	−	0.275799i	$-0.911057\pi$
$523$		−	12.6146i		−	0.551599i	0.961215	−	0.275799i	$-0.0889426\pi$
$524$		0			0
$525$	−4.57918			−0.199852
$526$		0			0
$527$	18.5664			0.808766
$528$		0			0
$529$	−2.27674			−0.0989886
$530$		0			0
$531$	−17.4396			−0.756816
$532$		0			0
$533$		−	44.1104i		−	1.91063i
$534$		0			0
$535$	−7.09659			−0.306812
$536$		0			0
$537$	2.27330			0.0981002
$538$		0			0
$539$	−10.1284	−	2.07500i	−0.436263	−	0.0893765i
$540$		0			0
$541$	−16.7743			−0.721185			−0.360593	−	0.932723i	$-0.617426\pi$
$541$	−16.7743			−0.721185			−0.360593	+	0.932723i	$0.617426\pi$
$542$		0			0
$543$			1.64315i			0.0705141i
$544$		0			0
$545$			19.4163i			0.831702i
$546$		0			0
$547$		−	14.0544i		−	0.600921i	−0.953794	−	0.300460i	$-0.902860\pi$
$547$		−	14.0544i		−	0.600921i	0.953794	−	0.300460i	$-0.0971403\pi$
$548$		0			0
$549$	11.7648			0.502108
$550$		0			0
$551$			37.7586i			1.60857i
$552$		0			0
$553$	35.9862			1.53029
$554$		0			0
$555$	4.08623			0.173451
$556$		0			0
$557$	10.0811			0.427151			0.213576	−	0.976927i	$-0.431489\pi$
$557$	10.0811			0.427151			0.213576	+	0.976927i	$0.431489\pi$
$558$		0			0
$559$			8.00000i			0.338364i
$560$		0			0
$561$	−1.64820	+	8.04516i	−0.0695871	+	0.339667i
$562$		0			0
$563$			45.7059i			1.92627i	0.269012	+	0.963137i	$0.413303\pi$
$563$			45.7059i			1.92627i	−0.269012	+	0.963137i	$0.586697\pi$
$564$		0			0
$565$		−	31.3319i		−	1.31814i
$566$		0			0
$567$	−20.8560			−0.875870
$568$		0			0
$569$		−	44.0016i		−	1.84464i	−0.386423	−	0.922321i	$-0.626290\pi$
$569$		−	44.0016i		−	1.84464i	0.386423	−	0.922321i	$-0.373710\pi$
$570$		0			0
$571$			35.4285i			1.48264i	0.671153	+	0.741319i	$0.265799\pi$
$571$			35.4285i			1.48264i	−0.671153	+	0.741319i	$0.734201\pi$
$572$		0			0
$573$			10.5699i			0.441564i
$574$		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
$579$		−	1.03636i		−	0.0430694i
$580$		0			0
$581$			24.4691i			1.01515i
$582$		0			0
$583$	−0.0780580	+	0.381015i	−0.00323283	+	0.0157800i
$584$		0			0
$585$			45.4109i			1.87751i
$586$		0			0
$587$	−9.61211			−0.396734			−0.198367	−	0.980128i	$-0.563564\pi$
$587$	−9.61211			−0.396734			−0.198367	+	0.980128i	$0.563564\pi$
$588$		0			0
$589$	29.7923			1.22757
$590$		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
$595$	41.3415			1.69484
$596$		0			0
$597$		−	2.31894i		−	0.0949080i
$598$		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
$603$	−28.7880			−1.17234
$604$		0			0
$605$	−12.0184	+	28.1008i	−0.488617	+	1.14246i
$606$		0			0
$607$	−12.7231			−0.516413			−0.258207	−	0.966090i	$-0.583132\pi$
$607$	−12.7231			−0.516413			−0.258207	+	0.966090i	$0.583132\pi$
$608$		0			0
$609$	8.46907			0.343184
$610$		0			0
$611$			51.7725i			2.09449i
$612$		0			0
$613$	20.3746			0.822921			0.411460	−	0.911428i	$-0.365019\pi$
$613$	20.3746			0.822921			0.411460	+	0.911428i	$0.365019\pi$
$614$		0			0
$615$	10.7955			0.435318
$616$		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
$619$	32.8742			1.32133			0.660664	−	0.750682i	$-0.270275\pi$
$619$	32.8742			1.32133			0.660664	+	0.750682i	$0.270275\pi$
$620$		0			0
$621$		−	15.2215i		−	0.610819i
$622$		0			0
$623$	6.21635			0.249053
$624$		0			0
$625$	−31.2017			−1.24807
$626$		0			0
$627$	−2.64476	+	12.9096i	−0.105622	+	0.515558i
$628$		0			0
$629$	−12.9973			−0.518238
$630$		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
$635$			19.1150i			0.758555i
$636$		0			0
$637$	−18.7323			−0.742199
$638$		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
$643$	2.58451			0.101923			0.0509616	−	0.998701i	$-0.483771\pi$
$643$	2.58451			0.101923			0.0509616	+	0.998701i	$0.483771\pi$
$644$		0			0
$645$	−1.95791			−0.0770927
$646$		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
$651$		−	6.68227i		−	0.261899i
$652$		0			0
$653$			24.3354i			0.952317i	0.879360	+	0.476158i	$0.157971\pi$
$653$			24.3354i			0.952317i	−0.879360	+	0.476158i	$0.842029\pi$
$654$		0			0
$655$	−20.8560			−0.814911
$656$		0			0
$657$			19.9648i			0.778902i
$658$		0			0
$659$		−	26.4044i		−	1.02857i	−0.857619	−	0.514285i	$-0.828057\pi$
$659$		−	26.4044i		−	1.02857i	0.857619	−	0.514285i	$-0.171943\pi$
$660$		0			0
$661$			16.5370i			0.643217i	0.946873	+	0.321608i	$0.104223\pi$
$661$			16.5370i			0.643217i	−0.946873	+	0.321608i	$0.895777\pi$
$662$		0			0
$663$			14.8793i			0.577864i
$664$		0			0
$665$	66.3380			2.57248
$666$		0			0
$667$		−	25.2900i		−	0.979233i
$668$		0			0
$669$			10.5699i			0.408656i
$670$		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
$675$	−8.23453			−0.316947
$676$		0			0
$677$	−20.6902			−0.795188			−0.397594	−	0.917562i	$-0.630155\pi$
$677$	−20.6902			−0.795188			−0.397594	+	0.917562i	$0.630155\pi$
$678$		0			0
$679$	42.6032			1.63496
$680$		0			0
$681$		−	1.35195i		−	0.0518069i
$682$		0			0
$683$	1.50172			0.0574617			0.0287308	−	0.999587i	$-0.490853\pi$
$683$	1.50172			0.0574617			0.0287308	+	0.999587i	$0.490853\pi$
$684$		0			0
$685$			0.452638i			0.0172944i
$686$		0			0
$687$			4.93955i			0.188455i
$688$		0			0
$689$			0.704676i			0.0268460i
$690$		0			0
$691$	−23.8777			−0.908350			−0.454175	−	0.890913i	$-0.650066\pi$
$691$	−23.8777			−0.908350			−0.454175	+	0.890913i	$0.650066\pi$
$692$		0			0
$693$	−28.1087	−	5.75859i	−1.06776	−	0.218751i
$694$		0			0
$695$	−38.5312			−1.46157
$696$		0			0
$697$	−34.3380			−1.30065
$698$		0			0
$699$			8.14388i			0.308030i
$700$		0			0
$701$	−39.4018			−1.48818			−0.744092	−	0.668077i	$-0.767118\pi$
$701$	−39.4018			−1.48818			−0.744092	+	0.668077i	$0.767118\pi$
$702$		0			0
$703$	−20.8560			−0.786599
$704$		0			0
$705$	−12.6707			−0.477208
$706$		0			0
$707$	−16.8724			−0.634553
$708$		0			0
$709$		−	45.5665i		−	1.71128i	−0.517568	−	0.855642i	$-0.673162\pi$
$709$		−	45.5665i		−	1.71128i	0.517568	−	0.855642i	$-0.326838\pi$
$710$		0			0
$711$	30.7713			1.15401
$712$		0			0
$713$	−19.9544			−0.747297
$714$		0			0
$715$	−11.1138	+	54.2486i	−0.415634	+	2.02878i
$716$		0			0
$717$	13.2716			0.495638
$718$		0			0
$719$			36.2518i			1.35196i	0.736918	+	0.675982i	$0.236280\pi$
$719$			36.2518i			1.35196i	−0.736918	+	0.675982i	$0.763720\pi$
$720$		0			0
$721$		−	17.4997i		−	0.651722i
$722$		0			0
$723$		−	11.6867i		−	0.434633i
$724$		0			0
$725$	−13.6814			−0.508113
$726$		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
$731$	6.22766			0.230338
$732$		0			0
$733$	40.9293			1.51176			0.755879	−	0.654712i	$-0.227210\pi$
$733$	40.9293			1.51176			0.755879	+	0.654712i	$0.227210\pi$
$734$		0			0
$735$		−	4.58451i		−	0.169102i
$736$		0			0
$737$	−34.3906	−	7.04555i	−1.26679	−	0.259526i
$738$		0			0
$739$		−	20.0429i		−	0.737290i	−0.929570	−	0.368645i	$-0.879822\pi$
$739$		−	20.0429i		−	0.737290i	0.929570	−	0.368645i	$-0.120178\pi$
$740$		0			0
$741$			23.8759i			0.877102i
$742$		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
$747$			20.9231i			0.765537i
$748$		0			0
$749$		−	8.12414i		−	0.296850i
$750$		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
$753$	10.9578			0.399324
$754$		0			0
$755$		−	56.5078i		−	2.05653i
$756$		0			0
$757$			15.8827i			0.577268i	0.957440	+	0.288634i	$0.0932010\pi$
$757$			15.8827i			0.577268i	−0.957440	+	0.288634i	$0.906799\pi$
$758$		0			0
$759$	1.77141	−	8.64658i	0.0642982	−	0.313851i
$760$		0			0
$761$		−	3.42466i		−	0.124144i	−0.998072	−	0.0620720i	$-0.980229\pi$
$761$		−	3.42466i		−	0.124144i	0.998072	−	0.0620720i	$-0.0197708\pi$
$762$		0			0
$763$	−22.2277			−0.804695
$764$		0			0
$765$	35.3505			1.27810
$766$		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
$771$	−11.7034			−0.421488
$772$		0			0
$773$		−	19.7586i		−	0.710667i	−0.934740	−	0.355334i	$-0.884367\pi$
$773$		−	19.7586i		−	0.710667i	0.934740	−	0.355334i	$-0.115633\pi$
$774$		0			0
$775$			10.7949i			0.387764i
$776$		0			0
$777$			4.67790i			0.167819i
$778$		0			0
$779$	−55.1001			−1.97417
$780$		0			0
$781$	−1.27389	+	6.21811i	−0.0455835	+	0.222501i
$782$		0			0
$783$	15.2295			0.544259
$784$		0			0
$785$	−39.5044			−1.40997
$786$		0			0
$787$		−	17.9702i		−	0.640568i	−0.947322	−	0.320284i	$-0.896222\pi$
$787$		−	17.9702i		−	0.640568i	0.947322	−	0.320284i	$-0.103778\pi$
$788$		0			0
$789$	4.85346			0.172788
$790$		0			0
$791$	35.8687			1.27534
$792$		0			0
$793$	25.9931			0.923043
$794$		0			0
$795$	−0.172462			−0.00611658
$796$		0			0
$797$		−	49.5336i		−	1.75457i	−0.479969	−	0.877285i	$-0.659352\pi$
$797$		−	49.5336i		−	1.75457i	0.479969	−	0.877285i	$-0.340648\pi$
$798$		0			0
$799$	40.3027			1.42581
$800$		0			0
$801$	5.31551			0.187814
$802$		0			0
$803$	−4.88617	+	23.8503i	−0.172429	+	0.841658i
$804$		0			0
$805$	−44.4320			−1.56602
$806$		0			0
$807$			4.17246i			0.146878i
$808$		0			0
$809$			35.9564i			1.26416i	0.774903	+	0.632080i	$0.217799\pi$
$809$			35.9564i			1.26416i	−0.774903	+	0.632080i	$0.782201\pi$
$810$		0			0
$811$			4.65724i			0.163538i	0.996651	+	0.0817689i	$0.0260569\pi$
$811$			4.65724i			0.163538i	−0.996651	+	0.0817689i	$0.973943\pi$
$812$		0			0
$813$	−2.89555			−0.101551
$814$		0			0
$815$			28.5174i			0.998920i
$816$		0			0
$817$	9.99312			0.349615
$818$		0			0
$819$	−51.9862			−1.81655
$820$		0			0
$821$	25.6997			0.896927			0.448464	−	0.893801i	$-0.351971\pi$
$821$	25.6997			0.896927			0.448464	+	0.893801i	$0.351971\pi$
$822$		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
$827$		−	46.5557i		−	1.61890i	−0.587188	−	0.809451i	$-0.699765\pi$
$827$		−	46.5557i		−	1.61890i	0.587188	−	0.809451i	$-0.300235\pi$
$828$		0			0
$829$		−	29.8923i		−	1.03820i	−0.854713	−	0.519101i	$-0.826267\pi$
$829$		−	29.8923i		−	1.03820i	0.854713	−	0.519101i	$-0.173733\pi$
$830$		0			0
$831$	−3.69894			−0.128315
$832$		0			0
$833$			14.5823i			0.505245i
$834$		0			0
$835$			40.2723i			1.39368i
$836$		0			0
$837$		−	12.0164i		−	0.415348i
$838$		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
$843$		−	1.40935i		−	0.0485407i
$844$		0			0
$845$			64.2112i			2.20893i
$846$		0			0
$847$	−32.1697	−	13.7586i	−1.10536	−	0.472751i
$848$		0			0
$849$			10.0605i			0.345275i
$850$		0			0
$851$	13.9690			0.478850
$852$		0			0
$853$	51.6944			1.76998			0.884992	−	0.465607i	$-0.154164\pi$
$853$	51.6944			1.76998			0.884992	+	0.465607i	$0.154164\pi$
$854$		0			0
$855$	56.7247			1.93994
$856$		0			0
$857$		−	30.7300i		−	1.04972i	−0.851190	−	0.524858i	$-0.824118\pi$
$857$		−	30.7300i		−	1.04972i	0.851190	−	0.524858i	$-0.175882\pi$
$858$		0			0
$859$	1.05681			0.0360580			0.0180290	−	0.999837i	$-0.494261\pi$
$859$	1.05681			0.0360580			0.0180290	+	0.999837i	$0.494261\pi$
$860$		0			0
$861$			12.3587i			0.421183i
$862$		0			0
$863$		−	0.146482i		−	0.00498631i	−0.999997	−	0.00249315i	$-0.999206\pi$
$863$		−	0.146482i		−	0.00498631i	0.999997	−	0.00249315i	$-0.000793597\pi$
$864$		0			0
$865$		−	15.9342i		−	0.541779i
$866$		0			0
$867$	2.58451			0.0877746
$868$		0			0
$869$	36.7598	+	7.53093i	1.24699	+	0.255469i
$870$		0			0
$871$	−63.6044			−2.15515
$872$		0			0
$873$	36.4293			1.23295
$874$		0			0
$875$		−	20.1513i		−	0.681239i
$876$		0			0
$877$	−0.0206587			−0.000697594			−0.000348797	−	1.00000i	$-0.500111\pi$
$877$	−0.0206587			−0.000697594			−0.000348797		1.00000i	$0.500111\pi$
$878$		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
$883$	−30.6087			−1.03006			−0.515032	−	0.857171i	$-0.672220\pi$
$883$	−30.6087			−1.03006			−0.515032	+	0.857171i	$0.672220\pi$
$884$		0			0
$885$		−	9.43010i		−	0.316989i
$886$		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
$891$	−21.3043	−	4.36459i	−0.713722	−	0.146219i
$892$		0			0
$893$	64.6711			2.16414
$894$		0			0
$895$		−	11.9329i		−	0.398872i
$896$		0			0
$897$		−	15.9916i		−	0.533944i
$898$		0			0
$899$		−	19.9648i		−	0.665864i
$900$		0			0
$901$	0.548560			0.0182752
$902$		0			0
$903$		−	2.24141i		−	0.0745894i
$904$		0			0
$905$	8.62510			0.286708
$906$		0			0
$907$	1.50172			0.0498638			0.0249319	−	0.999689i	$-0.492063\pi$
$907$	1.50172			0.0498638			0.0249319	+	0.999689i	$0.492063\pi$
$908$		0			0
$909$	−14.4274			−0.478525
$910$		0			0
$911$			13.6121i			0.450989i	0.974244	+	0.225495i	$0.0723998\pi$
$911$			13.6121i			0.450989i	−0.974244	+	0.225495i	$0.927600\pi$
$912$		0			0
$913$	−5.12070	+	24.9951i	−0.169471	+	0.827216i
$914$		0			0
$915$			6.36153i			0.210306i
$916$		0			0
$917$		−	23.8759i		−	0.788450i
$918$		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
$923$			11.5002i			0.378534i
$924$		0			0
$925$		−	7.55691i		−	0.248470i
$926$		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
$931$			23.3992i			0.766878i
$932$		0			0
$933$			9.44652i			0.309265i
$934$		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
$939$	−17.2069			−0.561527
$940$		0			0
$941$	−31.7296			−1.03436			−0.517178	−	0.855878i	$-0.673017\pi$
$941$	−31.7296			−1.03436			−0.517178	+	0.855878i	$0.673017\pi$
$942$		0			0
$943$	36.9050			1.20179
$944$		0			0
$945$		−	26.7568i		−	0.870397i
$946$		0			0
$947$	−20.7638			−0.674734			−0.337367	−	0.941373i	$-0.609536\pi$
$947$	−20.7638			−0.674734			−0.337367	+	0.941373i	$0.609536\pi$
$948$		0			0
$949$			44.1104i			1.43188i
$950$		0			0
$951$			7.35342i			0.238451i
$952$		0			0
$953$			20.7269i			0.671410i	0.941967	+	0.335705i	$0.108975\pi$
$953$			20.7269i			0.671410i	−0.941967	+	0.335705i	$0.891025\pi$
$954$		0			0
$955$	55.4829			1.79538
$956$		0			0
$957$	8.65112	+	1.77234i	0.279651	+	0.0572917i
$958$		0			0
$959$	−0.518178			−0.0167328
$960$		0			0
$961$	15.2473			0.491849
$962$		0			0
$963$		−	6.94683i		−	0.223858i
$964$		0			0
$965$	−5.43997			−0.175119
$966$		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
$971$	−29.3534			−0.941996			−0.470998	−	0.882134i	$-0.656106\pi$
$971$	−29.3534			−0.941996			−0.470998	+	0.882134i	$0.656106\pi$
$972$		0			0
$973$		−	44.1104i		−	1.41411i
$974$		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
$979$	6.34998	+	1.30091i	0.202946	+	0.0415773i
$980$		0			0
$981$	−19.0065			−0.606832
$982$		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
$987$		−	14.5054i		−	0.461712i
$988$		0			0
$989$	−6.69321			−0.212832
$990$		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
$993$	−1.40088			−0.0444556
$994$		0			0
$995$	−12.1725			−0.385893
$996$		0			0
$997$	−10.5884			−0.335337			−0.167669	−	0.985843i	$-0.553624\pi$
$997$	−10.5884			−0.335337			−0.167669	+	0.985843i	$0.553624\pi$
$998$		0			0
$999$			8.41205i			0.266145i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.5	✓	12	16.11	odd	4
352.2.e.a.351.6	yes	12	176.43	even	4
352.2.e.a.351.7	yes	12	176.21	odd	4
352.2.e.a.351.8	yes	12	16.5	even	4
704.2.e.d.703.5		12	176.109	odd	4
704.2.e.d.703.6		12	16.13	even	4
704.2.e.d.703.7		12	16.3	odd	4
704.2.e.d.703.8		12	176.131	even	4
2816.2.g.d.1407.5		12	88.43	even	2	inner
2816.2.g.d.1407.6		12	8.3	odd	2	inner
2816.2.g.d.1407.7		12	1.1	even	1	trivial
2816.2.g.d.1407.8		12	11.10	odd	2	inner
2816.2.g.i.1407.5		12	8.5	even	2
2816.2.g.i.1407.6		12	88.21	odd	2
2816.2.g.i.1407.7		12	44.43	even	2
2816.2.g.i.1407.8		12	4.3	odd	2
3168.2.o.e.703.9		12	48.11	even	4
3168.2.o.e.703.10		12	528.197	even	4
3168.2.o.e.703.11		12	528.395	odd	4
3168.2.o.e.703.12		12	48.5	odd	4

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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.529317 q^{3} +2.77846i q^{5} -3.18077 q^{7} -2.71982 q^{9} +O(q^{10})\)	\(q-0.529317 q^{3} +2.77846i q^{5} -3.18077 q^{7} -2.71982 q^{9} +(-3.24914 - 0.665647i) q^{11} -6.00919 q^{13} -1.47068i q^{15} +4.67790i q^{17} +7.50633i q^{19} +1.68363 q^{21} -5.02760i q^{23} -2.71982 q^{25} +3.02760 q^{27} +5.03024 q^{29} -3.96896i q^{31} +(1.71982 + 0.352338i) q^{33} -8.83762i q^{35} +2.77846i q^{37} +3.18077 q^{39} +7.34049i q^{41} -1.33129i q^{43} -7.55691i q^{45} -8.61555i q^{47} +3.11727 q^{49} -2.47609i q^{51} -0.117266i q^{53} +(1.84947 - 9.02760i) q^{55} -3.97322i q^{57} +6.41205 q^{59} -4.32556 q^{61} +8.65112 q^{63} -16.6963i q^{65} +10.5845 q^{67} +2.66119i q^{69} -1.91377i q^{71} -7.34049i q^{73} +1.43965 q^{75} +(10.3348 + 2.11727i) q^{77} -11.3137 q^{79} +6.55691 q^{81} -7.69282i q^{83} -12.9973 q^{85} -2.66259 q^{87} -1.95436 q^{89} +19.1138 q^{91} +2.10084i q^{93} -20.8560 q^{95} -13.3940 q^{97} +(8.83709 + 1.81044i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} + 4 q^{9}+O(q^{10}) \) 12 * q - 8 * q^3 + 4 * q^9	\( 12 q - 8 q^{3} + 4 q^{9} - 4 q^{11} + 4 q^{25} - 32 q^{27} - 16 q^{33} + 44 q^{49} + 72 q^{59} - 8 q^{67} - 56 q^{75} + 12 q^{81} + 96 q^{91} - 64 q^{97} + 76 q^{99}+O(q^{100}) \) 12 * q - 8 * q^3 + 4 * q^9 - 4 * q^11 + 4 * q^25 - 32 * q^27 - 16 * q^33 + 44 * q^49 + 72 * q^59 - 8 * q^67 - 56 * q^75 + 12 * q^81 + 96 * q^91 - 64 * q^97 + 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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