LMFDB - Embedded newform 7440.2.a.bh.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 = [2,0,-2,0,2,0,4,0,2,0,0,0,-8,0,-2,0,0] 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:	$2$
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 3720)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{9} -2.82843 q^{11} -2.58579 q^{13} -1.00000 q^{15} -2.82843 q^{17} -2.82843 q^{19} -3.41421 q^{21} +7.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.75736 q^{29} +1.00000 q^{31} +2.82843 q^{33} +3.41421 q^{35} -7.07107 q^{37} +2.58579 q^{39} +0.828427 q^{41} +1.65685 q^{43} +1.00000 q^{45} +0.343146 q^{47} +4.65685 q^{49} +2.82843 q^{51} -10.8284 q^{53} -2.82843 q^{55} +2.82843 q^{57} -2.58579 q^{59} +0.828427 q^{61} +3.41421 q^{63} -2.58579 q^{65} -5.07107 q^{67} -7.65685 q^{69} +2.58579 q^{71} -6.58579 q^{73} -1.00000 q^{75} -9.65685 q^{77} -3.17157 q^{79} +1.00000 q^{81} +2.00000 q^{83} -2.82843 q^{85} +1.75736 q^{87} +10.7279 q^{89} -8.82843 q^{91} -1.00000 q^{93} -2.82843 q^{95} -7.65685 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{13} - 2 q^{15} - 4 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{31} + 4 q^{35} + 8 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 12 q^{47} - 2 q^{49}+ \cdots - 4 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 - 4 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 12 * q^29 + 2 * q^31 + 4 * q^35 + 8 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 12 * q^47 - 2 * q^49 - 16 * q^53 - 8 * q^59 - 4 * q^61 + 4 * q^63 - 8 * q^65 + 4 * q^67 - 4 * q^69 + 8 * q^71 - 16 * q^73 - 2 * q^75 - 8 * q^77 - 12 * q^79 + 2 * q^81 + 4 * q^83 + 12 * q^87 - 4 * q^89 - 12 * q^91 - 2 * q^93 - 4 * 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
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	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
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	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
$21$	−3.41421	−0.745042
$22$	0	0
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.75736	−0.326333	−0.163167	−	0.986599i	$-0.552171\pi$
$29$	−1.75736	−0.326333	−0.163167	+	0.986599i	$0.552171\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	3.41421	0.577107
$36$	0	0
$37$	−7.07107	−1.16248	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	−7.07107	−1.16248	−0.581238	+	0.813733i	$0.697432\pi$
$38$	0	0
$39$	2.58579	0.414057
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	−10.8284	−1.48740	−0.743699	−	0.668514i	$-0.766931\pi$
$53$	−10.8284	−1.48740	−0.743699	+	0.668514i	$0.766931\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	−2.58579	−0.336641	−0.168320	−	0.985732i	$-0.553834\pi$
$59$	−2.58579	−0.336641	−0.168320	+	0.985732i	$0.553834\pi$
$60$	0	0
$61$	0.828427	0.106069	0.0530346	−	0.998593i	$-0.483111\pi$
$61$	0.828427	0.106069	0.0530346	+	0.998593i	$0.483111\pi$
$62$	0	0
$63$	3.41421	0.430150
$64$	0	0
$65$	−2.58579	−0.320727
$66$	0	0
$67$	−5.07107	−0.619530	−0.309765	−	0.950813i	$-0.600250\pi$
$67$	−5.07107	−0.619530	−0.309765	+	0.950813i	$0.600250\pi$
$68$	0	0
$69$	−7.65685	−0.921777
$70$	0	0
$71$	2.58579	0.306876	0.153438	−	0.988158i	$-0.450965\pi$
$71$	2.58579	0.306876	0.153438	+	0.988158i	$0.450965\pi$
$72$	0	0
$73$	−6.58579	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$73$	−6.58579	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−9.65685	−1.10050
$78$	0	0
$79$	−3.17157	−0.356830	−0.178415	−	0.983955i	$-0.557097\pi$
$79$	−3.17157	−0.356830	−0.178415	+	0.983955i	$0.557097\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	1.75736	0.188409
$88$	0	0
$89$	10.7279	1.13716	0.568579	−	0.822629i	$-0.307493\pi$
$89$	10.7279	1.13716	0.568579	+	0.822629i	$0.307493\pi$
$90$	0	0
$91$	−8.82843	−0.925471
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	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.bh.1.2		2
4.3	odd	2		3720.2.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.j.1.1	✓	2	4.3	odd	2
7440.2.a.bh.1.2		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.41421 q^{7} +1.00000 q^{9} -2.82843 q^{11} -2.58579 q^{13} -1.00000 q^{15} -2.82843 q^{17} -2.82843 q^{19} -3.41421 q^{21} +7.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.75736 q^{29} +1.00000 q^{31} +2.82843 q^{33} +3.41421 q^{35} -7.07107 q^{37} +2.58579 q^{39} +0.828427 q^{41} +1.65685 q^{43} +1.00000 q^{45} +0.343146 q^{47} +4.65685 q^{49} +2.82843 q^{51} -10.8284 q^{53} -2.82843 q^{55} +2.82843 q^{57} -2.58579 q^{59} +0.828427 q^{61} +3.41421 q^{63} -2.58579 q^{65} -5.07107 q^{67} -7.65685 q^{69} +2.58579 q^{71} -6.58579 q^{73} -1.00000 q^{75} -9.65685 q^{77} -3.17157 q^{79} +1.00000 q^{81} +2.00000 q^{83} -2.82843 q^{85} +1.75736 q^{87} +10.7279 q^{89} -8.82843 q^{91} -1.00000 q^{93} -2.82843 q^{95} -7.65685 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{13} - 2 q^{15} - 4 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{31} + 4 q^{35} + 8 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 12 q^{47} - 2 q^{49}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 - 8 * q^13 - 2 * q^15 - 4 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 12 * q^29 + 2 * q^31 + 4 * q^35 + 8 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 12 * q^47 - 2 * q^49 - 16 * q^53 - 8 * q^59 - 4 * q^61 + 4 * q^63 - 8 * q^65 + 4 * q^67 - 4 * q^69 + 8 * q^71 - 16 * q^73 - 2 * q^75 - 8 * q^77 - 12 * q^79 + 2 * q^81 + 4 * q^83 + 12 * q^87 - 4 * q^89 - 12 * q^91 - 2 * q^93 - 4 * q^97

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