LMFDB - Embedded newform 6664.2.a.bd.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 14 x^{10} + 64 x^{9} + 59 x^{8} - 348 x^{7} - 74 x^{6} + 760 x^{5} + 27 x^{4} + \cdots - 14 $ x^12 - 4x^11 - 14x^10 + 64x^9 + 59x^8 - 348x^7 - 74x^6 + 760x^5 + 27x^4 - 616x^3 - 46x^2 + 108*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.166977$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.166977 q^{3} +2.77381 q^{5} -2.97212 q^{9} +1.58253 q^{11} -1.13818 q^{13} +0.463164 q^{15} -1.00000 q^{17} +5.44730 q^{19} -5.72774 q^{23} +2.69404 q^{25} -0.997209 q^{27} -4.61822 q^{29} -8.31516 q^{31} +0.264247 q^{33} -3.69999 q^{37} -0.190050 q^{39} +0.657025 q^{41} -7.20105 q^{43} -8.24410 q^{45} -8.00066 q^{47} -0.166977 q^{51} -5.86058 q^{53} +4.38964 q^{55} +0.909576 q^{57} +2.14390 q^{59} -0.245210 q^{61} -3.15709 q^{65} +4.90529 q^{67} -0.956404 q^{69} -5.44611 q^{71} +14.0699 q^{73} +0.449844 q^{75} +1.40559 q^{79} +8.74984 q^{81} -7.32101 q^{83} -2.77381 q^{85} -0.771138 q^{87} -6.19730 q^{89} -1.38844 q^{93} +15.1098 q^{95} -6.85909 q^{97} -4.70346 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} - 8 q^{5} + 8 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} - 12 q^{17} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 4 q^{31} - 16 q^{33} + 12 q^{37} - 16 q^{39} - 24 q^{41} - 4 q^{43} - 28 q^{45} - 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 - 8 * q^5 + 8 * q^9 - 4 * q^11 - 12 * q^13 - 8 * q^15 - 12 * q^17 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 4 * q^31 - 16 * q^33 + 12 * q^37 - 16 * q^39 - 24 * q^41 - 4 * q^43 - 28 * q^45 - 28 * q^47 - 4 * q^51 + 4 * q^55 + 12 * q^57 - 36 * q^61 + 20 * q^65 - 4 * q^69 - 40 * q^73 - 20 * q^75 - 4 * q^79 - 4 * q^81 - 12 * q^83 + 8 * q^85 - 36 * q^89 - 16 * q^93 - 8 * q^95 - 12 * q^97 - 20 * q^99

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$	0.166977	0.0964045	0.0482022	−	0.998838i	$-0.484651\pi$
$3$	0.166977	0.0964045	0.0482022	+	0.998838i	$0.484651\pi$
$4$	0	0
$5$	2.77381	1.24049	0.620243	−	0.784409i	$-0.287034\pi$
$5$	2.77381	1.24049	0.620243	+	0.784409i	$0.287034\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.97212	−0.990706
$10$	0	0
$11$	1.58253	0.477150	0.238575	−	0.971124i	$-0.423320\pi$
$11$	1.58253	0.477150	0.238575	+	0.971124i	$0.423320\pi$
$12$	0	0
$13$	−1.13818	−0.315674	−0.157837	−	0.987465i	$-0.550452\pi$
$13$	−1.13818	−0.315674	−0.157837	+	0.987465i	$0.550452\pi$
$14$	0	0
$15$	0.463164	0.119588
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.44730	1.24970	0.624848	−	0.780746i	$-0.285161\pi$
$19$	5.44730	1.24970	0.624848	+	0.780746i	$0.285161\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.72774	−1.19432	−0.597159	−	0.802123i	$-0.703704\pi$
$23$	−5.72774	−1.19432	−0.597159	+	0.802123i	$0.703704\pi$
$24$	0	0
$25$	2.69404	0.538808
$26$	0	0
$27$	−0.997209	−0.191913
$28$	0	0
$29$	−4.61822	−0.857582	−0.428791	−	0.903404i	$-0.641060\pi$
$29$	−4.61822	−0.857582	−0.428791	+	0.903404i	$0.641060\pi$
$30$	0	0
$31$	−8.31516	−1.49345	−0.746723	−	0.665135i	$-0.768374\pi$
$31$	−8.31516	−1.49345	−0.746723	+	0.665135i	$0.768374\pi$
$32$	0	0
$33$	0.264247	0.0459994
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.69999	−0.608274	−0.304137	−	0.952628i	$-0.598368\pi$
$37$	−3.69999	−0.608274	−0.304137	+	0.952628i	$0.598368\pi$
$38$	0	0
$39$	−0.190050	−0.0304324
$40$	0	0
$41$	0.657025	0.102610	0.0513050	−	0.998683i	$-0.483662\pi$
$41$	0.657025	0.102610	0.0513050	+	0.998683i	$0.483662\pi$
$42$	0	0
$43$	−7.20105	−1.09815	−0.549075	−	0.835773i	$-0.685020\pi$
$43$	−7.20105	−1.09815	−0.549075	+	0.835773i	$0.685020\pi$
$44$	0	0
$45$	−8.24410	−1.22896
$46$	0	0
$47$	−8.00066	−1.16702	−0.583508	−	0.812107i	$-0.698320\pi$
$47$	−8.00066	−1.16702	−0.583508	+	0.812107i	$0.698320\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.166977	−0.0233815
$52$	0	0
$53$	−5.86058	−0.805013	−0.402506	−	0.915417i	$-0.631861\pi$
$53$	−5.86058	−0.805013	−0.402506	+	0.915417i	$0.631861\pi$
$54$	0	0
$55$	4.38964	0.591899
$56$	0	0
$57$	0.909576	0.120476
$58$	0	0
$59$	2.14390	0.279112	0.139556	−	0.990214i	$-0.455432\pi$
$59$	2.14390	0.279112	0.139556	+	0.990214i	$0.455432\pi$
$60$	0	0
$61$	−0.245210	−0.0313959	−0.0156980	−	0.999877i	$-0.504997\pi$
$61$	−0.245210	−0.0313959	−0.0156980	+	0.999877i	$0.504997\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.15709	−0.391589
$66$	0	0
$67$	4.90529	0.599276	0.299638	−	0.954053i	$-0.403134\pi$
$67$	4.90529	0.599276	0.299638	+	0.954053i	$0.403134\pi$
$68$	0	0
$69$	−0.956404	−0.115138
$70$	0	0
$71$	−5.44611	−0.646335	−0.323167	−	0.946342i	$-0.604748\pi$
$71$	−5.44611	−0.646335	−0.323167	+	0.946342i	$0.604748\pi$
$72$	0	0
$73$	14.0699	1.64675	0.823377	−	0.567495i	$-0.192087\pi$
$73$	14.0699	1.64675	0.823377	+	0.567495i	$0.192087\pi$
$74$	0	0
$75$	0.449844	0.0519435
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.40559	0.158141	0.0790706	−	0.996869i	$-0.474805\pi$
$79$	1.40559	0.158141	0.0790706	+	0.996869i	$0.474805\pi$
$80$	0	0
$81$	8.74984	0.972205
$82$	0	0
$83$	−7.32101	−0.803585	−0.401793	−	0.915731i	$-0.631613\pi$
$83$	−7.32101	−0.803585	−0.401793	+	0.915731i	$0.631613\pi$
$84$	0	0
$85$	−2.77381	−0.300862
$86$	0	0
$87$	−0.771138	−0.0826747
$88$	0	0
$89$	−6.19730	−0.656913	−0.328456	−	0.944519i	$-0.606528\pi$
$89$	−6.19730	−0.656913	−0.328456	+	0.944519i	$0.606528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.38844	−0.143975
$94$	0	0
$95$	15.1098	1.55023
$96$	0	0
$97$	−6.85909	−0.696435	−0.348218	−	0.937414i	$-0.613213\pi$
$97$	−6.85909	−0.696435	−0.348218	+	0.937414i	$0.613213\pi$
$98$	0	0
$99$	−4.70346	−0.472716

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.bd.1.6	yes	12
7.6	odd	2		6664.2.a.bc.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6664.2.a.bc.1.7	✓	12	7.6	odd	2
6664.2.a.bd.1.6	yes	12	1.1	even	1	trivial

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Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+0.166977 q^{3} +2.77381 q^{5} -2.97212 q^{9} +1.58253 q^{11} -1.13818 q^{13} +0.463164 q^{15} -1.00000 q^{17} +5.44730 q^{19} -5.72774 q^{23} +2.69404 q^{25} -0.997209 q^{27} -4.61822 q^{29} -8.31516 q^{31} +0.264247 q^{33} -3.69999 q^{37} -0.190050 q^{39} +0.657025 q^{41} -7.20105 q^{43} -8.24410 q^{45} -8.00066 q^{47} -0.166977 q^{51} -5.86058 q^{53} +4.38964 q^{55} +0.909576 q^{57} +2.14390 q^{59} -0.245210 q^{61} -3.15709 q^{65} +4.90529 q^{67} -0.956404 q^{69} -5.44611 q^{71} +14.0699 q^{73} +0.449844 q^{75} +1.40559 q^{79} +8.74984 q^{81} -7.32101 q^{83} -2.77381 q^{85} -0.771138 q^{87} -6.19730 q^{89} -1.38844 q^{93} +15.1098 q^{95} -6.85909 q^{97} -4.70346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} - 8 q^{5} + 8 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} - 12 q^{17} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 4 q^{31} - 16 q^{33} + 12 q^{37} - 16 q^{39} - 24 q^{41} - 4 q^{43} - 28 q^{45} - 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 - 8 * q^5 + 8 * q^9 - 4 * q^11 - 12 * q^13 - 8 * q^15 - 12 * q^17 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 4 * q^31 - 16 * q^33 + 12 * q^37 - 16 * q^39 - 24 * q^41 - 4 * q^43 - 28 * q^45 - 28 * q^47 - 4 * q^51 + 4 * q^55 + 12 * q^57 - 36 * q^61 + 20 * q^65 - 4 * q^69 - 40 * q^73 - 20 * q^75 - 4 * q^79 - 4 * q^81 - 12 * q^83 + 8 * q^85 - 36 * q^89 - 16 * q^93 - 8 * q^95 - 12 * q^97 - 20 * q^99

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