LMFDB - Embedded newform 7440.2.a.be.1.1

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 = [2,0,-2,0,-2,0,4,0,2,0,0,0,-8,0,2,0,-8] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +0.585786 q^{7} +1.00000 q^{9} +2.82843 q^{11} -2.58579 q^{13} +1.00000 q^{15} -4.00000 q^{17} -2.82843 q^{19} -0.585786 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.24264 q^{29} -1.00000 q^{31} -2.82843 q^{33} -0.585786 q^{35} -1.41421 q^{37} +2.58579 q^{39} -0.828427 q^{41} +11.3137 q^{43} -1.00000 q^{45} +4.82843 q^{47} -6.65685 q^{49} +4.00000 q^{51} -4.00000 q^{53} -2.82843 q^{55} +2.82843 q^{57} +0.242641 q^{59} -10.4853 q^{61} +0.585786 q^{63} +2.58579 q^{65} -3.89949 q^{67} -6.00000 q^{69} -9.89949 q^{71} -5.89949 q^{73} -1.00000 q^{75} +1.65685 q^{77} +14.4853 q^{79} +1.00000 q^{81} +0.343146 q^{83} +4.00000 q^{85} -2.24264 q^{87} +5.07107 q^{89} -1.51472 q^{91} +1.00000 q^{93} +2.82843 q^{95} +15.6569 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 8 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{31} - 4 q^{35} + 8 q^{39} + 4 q^{41} - 2 q^{45} + 4 q^{47} - 2 q^{49}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 8 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 2 * q^31 - 4 * q^35 + 8 * q^39 + 4 * q^41 - 2 * q^45 + 4 * q^47 - 2 * q^49 + 8 * q^51 - 8 * q^53 - 8 * q^59 - 4 * q^61 + 4 * q^63 + 8 * q^65 + 12 * q^67 - 12 * q^69 + 8 * q^73 - 2 * q^75 - 8 * q^77 + 12 * q^79 + 2 * q^81 + 12 * q^83 + 8 * q^85 + 4 * q^87 - 4 * q^89 - 20 * q^91 + 2 * q^93 + 20 * q^97

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
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−0.585786	−0.127829
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.24264	0.416448	0.208224	−	0.978081i	$-0.433232\pi$
$29$	2.24264	0.416448	0.208224	+	0.978081i	$0.433232\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	0	0
$39$	2.58579	0.414057
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	0	0
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	0.242641	0.0315891	0.0157946	−	0.999875i	$-0.494972\pi$
$59$	0.242641	0.0315891	0.0157946	+	0.999875i	$0.494972\pi$
$60$	0	0
$61$	−10.4853	−1.34250	−0.671251	−	0.741230i	$-0.734243\pi$
$61$	−10.4853	−1.34250	−0.671251	+	0.741230i	$0.734243\pi$
$62$	0	0
$63$	0.585786	0.0738022
$64$	0	0
$65$	2.58579	0.320727
$66$	0	0
$67$	−3.89949	−0.476399	−0.238200	−	0.971216i	$-0.576557\pi$
$67$	−3.89949	−0.476399	−0.238200	+	0.971216i	$0.576557\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−9.89949	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$71$	−9.89949	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$72$	0	0
$73$	−5.89949	−0.690484	−0.345242	−	0.938514i	$-0.612203\pi$
$73$	−5.89949	−0.690484	−0.345242	+	0.938514i	$0.612203\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.65685	0.188816
$78$	0	0
$79$	14.4853	1.62972	0.814861	−	0.579657i	$-0.196813\pi$
$79$	14.4853	1.62972	0.814861	+	0.579657i	$0.196813\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.343146	0.0376651	0.0188326	−	0.999823i	$-0.494005\pi$
$83$	0.343146	0.0376651	0.0188326	+	0.999823i	$0.494005\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	−2.24264	−0.240436
$88$	0	0
$89$	5.07107	0.537532	0.268766	−	0.963205i	$-0.413384\pi$
$89$	5.07107	0.537532	0.268766	+	0.963205i	$0.413384\pi$
$90$	0	0
$91$	−1.51472	−0.158786
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	15.6569	1.58971	0.794856	−	0.606798i	$-0.207546\pi$
$97$	15.6569	1.58971	0.794856	+	0.606798i	$0.207546\pi$
$98$	0	0
$99$	2.82843	0.284268

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.be.1.1		2
4.3	odd	2		465.2.a.c.1.1	✓	2
12.11	even	2		1395.2.a.g.1.2		2
20.3	even	4		2325.2.c.i.1024.4		4
20.7	even	4		2325.2.c.i.1024.1		4
20.19	odd	2		2325.2.a.n.1.2		2
60.59	even	2		6975.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.c.1.1	✓	2	4.3	odd	2
1395.2.a.g.1.2		2	12.11	even	2
2325.2.a.n.1.2		2	20.19	odd	2
2325.2.c.i.1024.1		4	20.7	even	4
2325.2.c.i.1024.4		4	20.3	even	4
6975.2.a.u.1.1		2	60.59	even	2
7440.2.a.be.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +0.585786 q^{7} +1.00000 q^{9} +2.82843 q^{11} -2.58579 q^{13} +1.00000 q^{15} -4.00000 q^{17} -2.82843 q^{19} -0.585786 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.24264 q^{29} -1.00000 q^{31} -2.82843 q^{33} -0.585786 q^{35} -1.41421 q^{37} +2.58579 q^{39} -0.828427 q^{41} +11.3137 q^{43} -1.00000 q^{45} +4.82843 q^{47} -6.65685 q^{49} +4.00000 q^{51} -4.00000 q^{53} -2.82843 q^{55} +2.82843 q^{57} +0.242641 q^{59} -10.4853 q^{61} +0.585786 q^{63} +2.58579 q^{65} -3.89949 q^{67} -6.00000 q^{69} -9.89949 q^{71} -5.89949 q^{73} -1.00000 q^{75} +1.65685 q^{77} +14.4853 q^{79} +1.00000 q^{81} +0.343146 q^{83} +4.00000 q^{85} -2.24264 q^{87} +5.07107 q^{89} -1.51472 q^{91} +1.00000 q^{93} +2.82843 q^{95} +15.6569 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 8 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{31} - 4 q^{35} + 8 q^{39} + 4 q^{41} - 2 q^{45} + 4 q^{47} - 2 q^{49}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 8 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 2 * q^31 - 4 * q^35 + 8 * q^39 + 4 * q^41 - 2 * q^45 + 4 * q^47 - 2 * q^49 + 8 * q^51 - 8 * q^53 - 8 * q^59 - 4 * q^61 + 4 * q^63 + 8 * q^65 + 12 * q^67 - 12 * q^69 + 8 * q^73 - 2 * q^75 - 8 * q^77 + 12 * q^79 + 2 * q^81 + 12 * q^83 + 8 * q^85 + 4 * q^87 - 4 * q^89 - 20 * q^91 + 2 * q^93 + 20 * q^97

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