LMFDB - Embedded newform 7440.2.a.bs.1.3

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			1.3
Root			$2.51414$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.485863 q^{7} +1.00000 q^{9} -5.02827 q^{11} +3.51414 q^{13} -1.00000 q^{15} -1.32088 q^{17} +6.64177 q^{19} -0.485863 q^{21} -0.292611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.86330 q^{29} -1.00000 q^{31} -5.02827 q^{33} +0.485863 q^{35} +5.51414 q^{37} +3.51414 q^{39} -7.02827 q^{41} +1.02827 q^{43} -1.00000 q^{45} +6.93438 q^{47} -6.76394 q^{49} -1.32088 q^{51} -1.70739 q^{53} +5.02827 q^{55} +6.64177 q^{57} +2.19325 q^{59} -2.00000 q^{61} -0.485863 q^{63} -3.51414 q^{65} -9.12763 q^{67} -0.292611 q^{69} -13.4768 q^{71} +12.5424 q^{73} +1.00000 q^{75} +2.44305 q^{77} +0.349158 q^{79} +1.00000 q^{81} -10.9344 q^{83} +1.32088 q^{85} -9.86330 q^{87} +5.03374 q^{89} -1.70739 q^{91} -1.00000 q^{93} -6.64177 q^{95} +10.4431 q^{97} -5.02827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.485863	−0.183639	−0.0918195	−	0.995776i	$-0.529268\pi$
$7$	−0.485863	−0.183639	−0.0918195	+	0.995776i	$0.529268\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.02827	−1.51608	−0.758041	−	0.652207i	$-0.773843\pi$
$11$	−5.02827	−1.51608	−0.758041	+	0.652207i	$0.773843\pi$
$12$	0	0
$13$	3.51414	0.974646	0.487323	−	0.873222i	$-0.337973\pi$
$13$	3.51414	0.974646	0.487323	+	0.873222i	$0.337973\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.32088	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$17$	−1.32088	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$18$	0	0
$19$	6.64177	1.52373	0.761863	−	0.647738i	$-0.224285\pi$
$19$	6.64177	1.52373	0.761863	+	0.647738i	$0.224285\pi$
$20$	0	0
$21$	−0.485863	−0.106024
$22$	0	0
$23$	−0.292611	−0.0610135	−0.0305068	−	0.999535i	$-0.509712\pi$
$23$	−0.292611	−0.0610135	−0.0305068	+	0.999535i	$0.509712\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.86330	−1.83157	−0.915784	−	0.401671i	$-0.868429\pi$
$29$	−9.86330	−1.83157	−0.915784	+	0.401671i	$0.868429\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−5.02827	−0.875310
$34$	0	0
$35$	0.485863	0.0821258
$36$	0	0
$37$	5.51414	0.906519	0.453259	−	0.891379i	$-0.350261\pi$
$37$	5.51414	0.906519	0.453259	+	0.891379i	$0.350261\pi$
$38$	0	0
$39$	3.51414	0.562712
$40$	0	0
$41$	−7.02827	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$41$	−7.02827	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$42$	0	0
$43$	1.02827	0.156810	0.0784051	−	0.996922i	$-0.475017\pi$
$43$	1.02827	0.156810	0.0784051	+	0.996922i	$0.475017\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.93438	1.01148	0.505742	−	0.862685i	$-0.331219\pi$
$47$	6.93438	1.01148	0.505742	+	0.862685i	$0.331219\pi$
$48$	0	0
$49$	−6.76394	−0.966277
$50$	0	0
$51$	−1.32088	−0.184961
$52$	0	0
$53$	−1.70739	−0.234528	−0.117264	−	0.993101i	$-0.537412\pi$
$53$	−1.70739	−0.234528	−0.117264	+	0.993101i	$0.537412\pi$
$54$	0	0
$55$	5.02827	0.678012
$56$	0	0
$57$	6.64177	0.879724
$58$	0	0
$59$	2.19325	0.285537	0.142769	−	0.989756i	$-0.454400\pi$
$59$	2.19325	0.285537	0.142769	+	0.989756i	$0.454400\pi$
$60$	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
$63$	−0.485863	−0.0612130
$64$	0	0
$65$	−3.51414	−0.435875
$66$	0	0
$67$	−9.12763	−1.11512	−0.557559	−	0.830137i	$-0.688262\pi$
$67$	−9.12763	−1.11512	−0.557559	+	0.830137i	$0.688262\pi$
$68$	0	0
$69$	−0.292611	−0.0352262
$70$	0	0
$71$	−13.4768	−1.59940	−0.799700	−	0.600399i	$-0.795008\pi$
$71$	−13.4768	−1.59940	−0.799700	+	0.600399i	$0.795008\pi$
$72$	0	0
$73$	12.5424	1.46798	0.733989	−	0.679161i	$-0.237656\pi$
$73$	12.5424	1.46798	0.733989	+	0.679161i	$0.237656\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.44305	0.278412
$78$	0	0
$79$	0.349158	0.0392834	0.0196417	−	0.999807i	$-0.493747\pi$
$79$	0.349158	0.0392834	0.0196417	+	0.999807i	$0.493747\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.9344	−1.20020	−0.600102	−	0.799923i	$-0.704873\pi$
$83$	−10.9344	−1.20020	−0.600102	+	0.799923i	$0.704873\pi$
$84$	0	0
$85$	1.32088	0.143270
$86$	0	0
$87$	−9.86330	−1.05746
$88$	0	0
$89$	5.03374	0.533575	0.266788	−	0.963755i	$-0.414038\pi$
$89$	5.03374	0.533575	0.266788	+	0.963755i	$0.414038\pi$
$90$	0	0
$91$	−1.70739	−0.178983
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−6.64177	−0.681431
$96$	0	0
$97$	10.4431	1.06033	0.530166	−	0.847894i	$-0.322130\pi$
$97$	10.4431	1.06033	0.530166	+	0.847894i	$0.322130\pi$
$98$	0	0
$99$	−5.02827	−0.505361

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.1.3		3
4.3	odd	2		465.2.a.e.1.3	✓	3
12.11	even	2		1395.2.a.j.1.1		3
20.3	even	4		2325.2.c.k.1024.1		6
20.7	even	4		2325.2.c.k.1024.6		6
20.19	odd	2		2325.2.a.r.1.1		3
60.59	even	2		6975.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.3	✓	3	4.3	odd	2
1395.2.a.j.1.1		3	12.11	even	2
2325.2.a.r.1.1		3	20.19	odd	2
2325.2.c.k.1024.1		6	20.3	even	4
2325.2.c.k.1024.6		6	20.7	even	4
6975.2.a.bf.1.3		3	60.59	even	2
7440.2.a.bs.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.485863 q^{7} +1.00000 q^{9} -5.02827 q^{11} +3.51414 q^{13} -1.00000 q^{15} -1.32088 q^{17} +6.64177 q^{19} -0.485863 q^{21} -0.292611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.86330 q^{29} -1.00000 q^{31} -5.02827 q^{33} +0.485863 q^{35} +5.51414 q^{37} +3.51414 q^{39} -7.02827 q^{41} +1.02827 q^{43} -1.00000 q^{45} +6.93438 q^{47} -6.76394 q^{49} -1.32088 q^{51} -1.70739 q^{53} +5.02827 q^{55} +6.64177 q^{57} +2.19325 q^{59} -2.00000 q^{61} -0.485863 q^{63} -3.51414 q^{65} -9.12763 q^{67} -0.292611 q^{69} -13.4768 q^{71} +12.5424 q^{73} +1.00000 q^{75} +2.44305 q^{77} +0.349158 q^{79} +1.00000 q^{81} -10.9344 q^{83} +1.32088 q^{85} -9.86330 q^{87} +5.03374 q^{89} -1.70739 q^{91} -1.00000 q^{93} -6.64177 q^{95} +10.4431 q^{97} -5.02827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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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