LMFDB - Embedded newform 7440.2.a.bs.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-3,0,-8,0,3,0,-2,0,4,0,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.42801 q^{7} +1.00000 q^{9} -1.14399 q^{11} +1.57199 q^{13} -1.00000 q^{15} +4.67282 q^{17} -5.34565 q^{19} -2.42801 q^{21} +1.81681 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.95684 q^{29} -1.00000 q^{31} -1.14399 q^{33} +2.42801 q^{35} +3.57199 q^{37} +1.57199 q^{39} -3.14399 q^{41} -2.85601 q^{43} -1.00000 q^{45} -7.16246 q^{47} -1.10478 q^{49} +4.67282 q^{51} -3.81681 q^{53} +1.14399 q^{55} -5.34565 q^{57} +6.24482 q^{59} -2.00000 q^{61} -2.42801 q^{63} -1.57199 q^{65} +0.917641 q^{67} +1.81681 q^{69} +6.44648 q^{71} +6.71598 q^{73} +1.00000 q^{75} +2.77761 q^{77} -9.52884 q^{79} +1.00000 q^{81} +3.16246 q^{83} -4.67282 q^{85} +1.95684 q^{87} -15.2241 q^{89} -3.81681 q^{91} -1.00000 q^{93} +5.34565 q^{95} +10.7776 q^{97} -1.14399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.42801	−0.917700	−0.458850	−	0.888514i	$-0.651739\pi$
$7$	−2.42801	−0.917700	−0.458850	+	0.888514i	$0.651739\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.14399	−0.344925	−0.172462	−	0.985016i	$-0.555172\pi$
$11$	−1.14399	−0.344925	−0.172462	+	0.985016i	$0.555172\pi$
$12$	0	0
$13$	1.57199	0.435992	0.217996	−	0.975950i	$-0.430048\pi$
$13$	1.57199	0.435992	0.217996	+	0.975950i	$0.430048\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.67282	1.13333	0.566663	−	0.823950i	$-0.308234\pi$
$17$	4.67282	1.13333	0.566663	+	0.823950i	$0.308234\pi$
$18$	0	0
$19$	−5.34565	−1.22638	−0.613188	−	0.789937i	$-0.710113\pi$
$19$	−5.34565	−1.22638	−0.613188	+	0.789937i	$0.710113\pi$
$20$	0	0
$21$	−2.42801	−0.529835
$22$	0	0
$23$	1.81681	0.378831	0.189416	−	0.981897i	$-0.439341\pi$
$23$	1.81681	0.378831	0.189416	+	0.981897i	$0.439341\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.95684	0.363377	0.181688	−	0.983356i	$-0.441844\pi$
$29$	1.95684	0.363377	0.181688	+	0.983356i	$0.441844\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.14399	−0.199142
$34$	0	0
$35$	2.42801	0.410408
$36$	0	0
$37$	3.57199	0.587232	0.293616	−	0.955923i	$-0.405141\pi$
$37$	3.57199	0.587232	0.293616	+	0.955923i	$0.405141\pi$
$38$	0	0
$39$	1.57199	0.251720
$40$	0	0
$41$	−3.14399	−0.491008	−0.245504	−	0.969396i	$-0.578953\pi$
$41$	−3.14399	−0.491008	−0.245504	+	0.969396i	$0.578953\pi$
$42$	0	0
$43$	−2.85601	−0.435538	−0.217769	−	0.976000i	$-0.569878\pi$
$43$	−2.85601	−0.435538	−0.217769	+	0.976000i	$0.569878\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−7.16246	−1.04475	−0.522376	−	0.852715i	$-0.674954\pi$
$47$	−7.16246	−1.04475	−0.522376	+	0.852715i	$0.674954\pi$
$48$	0	0
$49$	−1.10478	−0.157826
$50$	0	0
$51$	4.67282	0.654326
$52$	0	0
$53$	−3.81681	−0.524279	−0.262140	−	0.965030i	$-0.584428\pi$
$53$	−3.81681	−0.524279	−0.262140	+	0.965030i	$0.584428\pi$
$54$	0	0
$55$	1.14399	0.154255
$56$	0	0
$57$	−5.34565	−0.708048
$58$	0	0
$59$	6.24482	0.813006	0.406503	−	0.913649i	$-0.366748\pi$
$59$	6.24482	0.813006	0.406503	+	0.913649i	$0.366748\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−2.42801	−0.305900
$64$	0	0
$65$	−1.57199	−0.194982
$66$	0	0
$67$	0.917641	0.112108	0.0560538	−	0.998428i	$-0.482148\pi$
$67$	0.917641	0.112108	0.0560538	+	0.998428i	$0.482148\pi$
$68$	0	0
$69$	1.81681	0.218718
$70$	0	0
$71$	6.44648	0.765056	0.382528	−	0.923944i	$-0.375054\pi$
$71$	6.44648	0.765056	0.382528	+	0.923944i	$0.375054\pi$
$72$	0	0
$73$	6.71598	0.786046	0.393023	−	0.919529i	$-0.371429\pi$
$73$	6.71598	0.786046	0.393023	+	0.919529i	$0.371429\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.77761	0.316538
$78$	0	0
$79$	−9.52884	−1.07208	−0.536039	−	0.844193i	$-0.680080\pi$
$79$	−9.52884	−1.07208	−0.536039	+	0.844193i	$0.680080\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.16246	0.347125	0.173562	−	0.984823i	$-0.444472\pi$
$83$	3.16246	0.347125	0.173562	+	0.984823i	$0.444472\pi$
$84$	0	0
$85$	−4.67282	−0.506839
$86$	0	0
$87$	1.95684	0.209796
$88$	0	0
$89$	−15.2241	−1.61375	−0.806875	−	0.590722i	$-0.798843\pi$
$89$	−15.2241	−1.61375	−0.806875	+	0.590722i	$0.798843\pi$
$90$	0	0
$91$	−3.81681	−0.400110
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	5.34565	0.548452
$96$	0	0
$97$	10.7776	1.09430	0.547150	−	0.837035i	$-0.315713\pi$
$97$	10.7776	1.09430	0.547150	+	0.837035i	$0.315713\pi$
$98$	0	0
$99$	−1.14399	−0.114975

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	7440.2.a.bs.1.2		3
4.3	odd	2		465.2.a.e.1.2	✓	3
12.11	even	2		1395.2.a.j.1.2		3
20.3	even	4		2325.2.c.k.1024.3		6
20.7	even	4		2325.2.c.k.1024.4		6
20.19	odd	2		2325.2.a.r.1.2		3
60.59	even	2		6975.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.2	✓	3	4.3	odd	2
1395.2.a.j.1.2		3	12.11	even	2
2325.2.a.r.1.2		3	20.19	odd	2
2325.2.c.k.1024.3		6	20.3	even	4
2325.2.c.k.1024.4		6	20.7	even	4
6975.2.a.bf.1.2		3	60.59	even	2
7440.2.a.bs.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.42801 q^{7} +1.00000 q^{9} -1.14399 q^{11} +1.57199 q^{13} -1.00000 q^{15} +4.67282 q^{17} -5.34565 q^{19} -2.42801 q^{21} +1.81681 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.95684 q^{29} -1.00000 q^{31} -1.14399 q^{33} +2.42801 q^{35} +3.57199 q^{37} +1.57199 q^{39} -3.14399 q^{41} -2.85601 q^{43} -1.00000 q^{45} -7.16246 q^{47} -1.10478 q^{49} +4.67282 q^{51} -3.81681 q^{53} +1.14399 q^{55} -5.34565 q^{57} +6.24482 q^{59} -2.00000 q^{61} -2.42801 q^{63} -1.57199 q^{65} +0.917641 q^{67} +1.81681 q^{69} +6.44648 q^{71} +6.71598 q^{73} +1.00000 q^{75} +2.77761 q^{77} -9.52884 q^{79} +1.00000 q^{81} +3.16246 q^{83} -4.67282 q^{85} +1.95684 q^{87} -15.2241 q^{89} -3.81681 q^{91} -1.00000 q^{93} +5.34565 q^{95} +10.7776 q^{97} -1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * q^99

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