LMFDB - Embedded newform 4840.2.a.bc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,6,0,-4,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.08589$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35095 q^{3} +1.00000 q^{5} +3.00679 q^{7} -1.17493 q^{9} -5.69877 q^{13} +1.35095 q^{15} -4.70191 q^{17} +3.69877 q^{19} +4.06204 q^{21} -6.83866 q^{23} +1.00000 q^{25} -5.64013 q^{27} -10.6956 q^{29} -3.50491 q^{31} +3.00679 q^{35} +5.79409 q^{37} -7.69877 q^{39} -8.39893 q^{41} -2.80716 q^{43} -1.17493 q^{45} +5.14504 q^{47} +2.04081 q^{49} -6.35205 q^{51} -12.2204 q^{53} +4.99686 q^{57} +3.30528 q^{59} -10.7990 q^{61} -3.53276 q^{63} -5.69877 q^{65} +11.5328 q^{67} -9.23871 q^{69} +2.05440 q^{71} -3.29589 q^{73} +1.35095 q^{75} +4.67984 q^{79} -4.09477 q^{81} +12.5247 q^{83} -4.70191 q^{85} -14.4493 q^{87} -4.32366 q^{89} -17.1350 q^{91} -4.73497 q^{93} +3.69877 q^{95} +2.94235 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

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
$3$	1.35095	0.779973	0.389987	−	0.920821i	$-0.372480\pi$
$3$	1.35095	0.779973	0.389987	+	0.920821i	$0.372480\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00679	1.13646	0.568231	−	0.822869i	$-0.307628\pi$
$7$	3.00679	1.13646	0.568231	+	0.822869i	$0.307628\pi$
$8$	0	0
$9$	−1.17493	−0.391642
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.69877	−1.58055	−0.790277	−	0.612750i	$-0.790063\pi$
$13$	−5.69877	−1.58055	−0.790277	+	0.612750i	$0.790063\pi$
$14$	0	0
$15$	1.35095	0.348815
$16$	0	0
$17$	−4.70191	−1.14038	−0.570190	−	0.821513i	$-0.693130\pi$
$17$	−4.70191	−1.14038	−0.570190	+	0.821513i	$0.693130\pi$
$18$	0	0
$19$	3.69877	0.848556	0.424278	−	0.905532i	$-0.360528\pi$
$19$	3.69877	0.848556	0.424278	+	0.905532i	$0.360528\pi$
$20$	0	0
$21$	4.06204	0.886409
$22$	0	0
$23$	−6.83866	−1.42596	−0.712980	−	0.701185i	$-0.752655\pi$
$23$	−6.83866	−1.42596	−0.712980	+	0.701185i	$0.752655\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.64013	−1.08544
$28$	0	0
$29$	−10.6956	−1.98613	−0.993064	−	0.117573i	$-0.962489\pi$
$29$	−10.6956	−1.98613	−0.993064	+	0.117573i	$0.962489\pi$
$30$	0	0
$31$	−3.50491	−0.629501	−0.314750	−	0.949174i	$-0.601921\pi$
$31$	−3.50491	−0.629501	−0.314750	+	0.949174i	$0.601921\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00679	0.508241
$36$	0	0
$37$	5.79409	0.952542	0.476271	−	0.879298i	$-0.341988\pi$
$37$	5.79409	0.952542	0.476271	+	0.879298i	$0.341988\pi$
$38$	0	0
$39$	−7.69877	−1.23279
$40$	0	0
$41$	−8.39893	−1.31169	−0.655846	−	0.754895i	$-0.727688\pi$
$41$	−8.39893	−1.31169	−0.655846	+	0.754895i	$0.727688\pi$
$42$	0	0
$43$	−2.80716	−0.428088	−0.214044	−	0.976824i	$-0.568664\pi$
$43$	−2.80716	−0.428088	−0.214044	+	0.976824i	$0.568664\pi$
$44$	0	0
$45$	−1.17493	−0.175148
$46$	0	0
$47$	5.14504	0.750481	0.375241	−	0.926927i	$-0.377560\pi$
$47$	5.14504	0.750481	0.375241	+	0.926927i	$0.377560\pi$
$48$	0	0
$49$	2.04081	0.291544
$50$	0	0
$51$	−6.35205	−0.889466
$52$	0	0
$53$	−12.2204	−1.67860	−0.839301	−	0.543667i	$-0.817035\pi$
$53$	−12.2204	−1.67860	−0.839301	+	0.543667i	$0.817035\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.99686	0.661851
$58$	0	0
$59$	3.30528	0.430311	0.215155	−	0.976580i	$-0.430974\pi$
$59$	3.30528	0.430311	0.215155	+	0.976580i	$0.430974\pi$
$60$	0	0
$61$	−10.7990	−1.38267	−0.691334	−	0.722536i	$-0.742976\pi$
$61$	−10.7990	−1.38267	−0.691334	+	0.722536i	$0.742976\pi$
$62$	0	0
$63$	−3.53276	−0.445086
$64$	0	0
$65$	−5.69877	−0.706845
$66$	0	0
$67$	11.5328	1.40895	0.704475	−	0.709729i	$-0.251183\pi$
$67$	11.5328	1.40895	0.704475	+	0.709729i	$0.251183\pi$
$68$	0	0
$69$	−9.23871	−1.11221
$70$	0	0
$71$	2.05440	0.243812	0.121906	−	0.992542i	$-0.461099\pi$
$71$	2.05440	0.243812	0.121906	+	0.992542i	$0.461099\pi$
$72$	0	0
$73$	−3.29589	−0.385755	−0.192877	−	0.981223i	$-0.561782\pi$
$73$	−3.29589	−0.385755	−0.192877	+	0.981223i	$0.561782\pi$
$74$	0	0
$75$	1.35095	0.155995
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.67984	0.526523	0.263261	−	0.964725i	$-0.415202\pi$
$79$	4.67984	0.526523	0.263261	+	0.964725i	$0.415202\pi$
$80$	0	0
$81$	−4.09477	−0.454975
$82$	0	0
$83$	12.5247	1.37477	0.687385	−	0.726293i	$-0.258759\pi$
$83$	12.5247	1.37477	0.687385	+	0.726293i	$0.258759\pi$
$84$	0	0
$85$	−4.70191	−0.509993
$86$	0	0
$87$	−14.4493	−1.54913
$88$	0	0
$89$	−4.32366	−0.458307	−0.229153	−	0.973390i	$-0.573596\pi$
$89$	−4.32366	−0.458307	−0.229153	+	0.973390i	$0.573596\pi$
$90$	0	0
$91$	−17.1350	−1.79624
$92$	0	0
$93$	−4.73497	−0.490994
$94$	0	0
$95$	3.69877	0.379486
$96$	0	0
$97$	2.94235	0.298751	0.149375	−	0.988781i	$-0.452274\pi$
$97$	2.94235	0.298751	0.149375	+	0.988781i	$0.452274\pi$
$98$	0	0
$99$	0	0

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	4840.2.a.bc.1.5	✓	6
4.3	odd	2		9680.2.a.db.1.2		6
11.10	odd	2		4840.2.a.bd.1.5	yes	6
44.43	even	2		9680.2.a.da.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.5	✓	6	1.1	even	1	trivial
4840.2.a.bd.1.5	yes	6	11.10	odd	2
9680.2.a.da.1.2		6	44.43	even	2
9680.2.a.db.1.2		6	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35095 q^{3} +1.00000 q^{5} +3.00679 q^{7} -1.17493 q^{9} -5.69877 q^{13} +1.35095 q^{15} -4.70191 q^{17} +3.69877 q^{19} +4.06204 q^{21} -6.83866 q^{23} +1.00000 q^{25} -5.64013 q^{27} -10.6956 q^{29} -3.50491 q^{31} +3.00679 q^{35} +5.79409 q^{37} -7.69877 q^{39} -8.39893 q^{41} -2.80716 q^{43} -1.17493 q^{45} +5.14504 q^{47} +2.04081 q^{49} -6.35205 q^{51} -12.2204 q^{53} +4.99686 q^{57} +3.30528 q^{59} -10.7990 q^{61} -3.53276 q^{63} -5.69877 q^{65} +11.5328 q^{67} -9.23871 q^{69} +2.05440 q^{71} -3.29589 q^{73} +1.35095 q^{75} +4.67984 q^{79} -4.09477 q^{81} +12.5247 q^{83} -4.70191 q^{85} -14.4493 q^{87} -4.32366 q^{89} -17.1350 q^{91} -4.73497 q^{93} +3.69877 q^{95} +2.94235 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

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