LMFDB - Embedded newform 7440.2.a.bd.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,-2,0,1,0,2,0,-5,0,12,0,2,0,-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(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.53113$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -3.53113 q^{7} +1.00000 q^{9} +1.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} +3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} -1.53113 q^{33} +3.53113 q^{35} +9.06226 q^{37} -6.00000 q^{39} -9.06226 q^{41} +0.468871 q^{43} -1.00000 q^{45} -11.0623 q^{47} +5.46887 q^{49} +4.00000 q^{51} +5.53113 q^{53} -1.53113 q^{55} -3.53113 q^{57} -7.06226 q^{59} -11.0623 q^{61} -3.53113 q^{63} -6.00000 q^{65} +11.0623 q^{67} -1.53113 q^{69} +4.46887 q^{71} -0.468871 q^{73} -1.00000 q^{75} -5.40661 q^{77} +0.468871 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} +1.53113 q^{89} -21.1868 q^{91} +1.00000 q^{93} -3.53113 q^{95} -16.1245 q^{97} +1.53113 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * 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
$7$	−3.53113	−1.33464	−0.667321	−	0.744771i	$-0.732559\pi$
$7$	−3.53113	−1.33464	−0.667321	+	0.744771i	$0.732559\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.53113	0.461653	0.230826	−	0.972995i	$-0.425857\pi$
$11$	1.53113	0.461653	0.230826	+	0.972995i	$0.425857\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	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
$19$	3.53113	0.810097	0.405048	−	0.914295i	$-0.367255\pi$
$19$	3.53113	0.810097	0.405048	+	0.914295i	$0.367255\pi$
$20$	0	0
$21$	3.53113	0.770555
$22$	0	0
$23$	1.53113	0.319262	0.159631	−	0.987177i	$-0.448969\pi$
$23$	1.53113	0.319262	0.159631	+	0.987177i	$0.448969\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.53113	−0.266535
$34$	0	0
$35$	3.53113	0.596870
$36$	0	0
$37$	9.06226	1.48983	0.744913	−	0.667162i	$-0.232491\pi$
$37$	9.06226	1.48983	0.744913	+	0.667162i	$0.232491\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−9.06226	−1.41529	−0.707643	−	0.706570i	$-0.750242\pi$
$41$	−9.06226	−1.41529	−0.707643	+	0.706570i	$0.750242\pi$
$42$	0	0
$43$	0.468871	0.0715022	0.0357511	−	0.999361i	$-0.488618\pi$
$43$	0.468871	0.0715022	0.0357511	+	0.999361i	$0.488618\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−11.0623	−1.61360	−0.806798	−	0.590827i	$-0.798801\pi$
$47$	−11.0623	−1.61360	−0.806798	+	0.590827i	$0.798801\pi$
$48$	0	0
$49$	5.46887	0.781267
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	5.53113	0.759759	0.379879	−	0.925036i	$-0.375965\pi$
$53$	5.53113	0.759759	0.379879	+	0.925036i	$0.375965\pi$
$54$	0	0
$55$	−1.53113	−0.206457
$56$	0	0
$57$	−3.53113	−0.467709
$58$	0	0
$59$	−7.06226	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$59$	−7.06226	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$60$	0	0
$61$	−11.0623	−1.41638	−0.708188	−	0.706023i	$-0.750487\pi$
$61$	−11.0623	−1.41638	−0.708188	+	0.706023i	$0.750487\pi$
$62$	0	0
$63$	−3.53113	−0.444880
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	11.0623	1.35147	0.675735	−	0.737145i	$-0.263826\pi$
$67$	11.0623	1.35147	0.675735	+	0.737145i	$0.263826\pi$
$68$	0	0
$69$	−1.53113	−0.184326
$70$	0	0
$71$	4.46887	0.530357	0.265179	−	0.964199i	$-0.414569\pi$
$71$	4.46887	0.530357	0.265179	+	0.964199i	$0.414569\pi$
$72$	0	0
$73$	−0.468871	−0.0548772	−0.0274386	−	0.999623i	$-0.508735\pi$
$73$	−0.468871	−0.0548772	−0.0274386	+	0.999623i	$0.508735\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.40661	−0.616141
$78$	0	0
$79$	0.468871	0.0527521	0.0263761	−	0.999652i	$-0.491603\pi$
$79$	0.468871	0.0527521	0.0263761	+	0.999652i	$0.491603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.53113	0.162299	0.0811497	−	0.996702i	$-0.474141\pi$
$89$	1.53113	0.162299	0.0811497	+	0.996702i	$0.474141\pi$
$90$	0	0
$91$	−21.1868	−2.22098
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−3.53113	−0.362286
$96$	0	0
$97$	−16.1245	−1.63720	−0.818598	−	0.574367i	$-0.805248\pi$
$97$	−16.1245	−1.63720	−0.818598	+	0.574367i	$0.805248\pi$
$98$	0	0
$99$	1.53113	0.153884

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.bd.1.1		2
4.3	odd	2		930.2.a.q.1.2	✓	2
12.11	even	2		2790.2.a.bf.1.2		2
20.3	even	4		4650.2.d.bg.3349.1		4
20.7	even	4		4650.2.d.bg.3349.4		4
20.19	odd	2		4650.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.2	✓	2	4.3	odd	2
2790.2.a.bf.1.2		2	12.11	even	2
4650.2.a.bz.1.1		2	20.19	odd	2
4650.2.d.bg.3349.1		4	20.3	even	4
4650.2.d.bg.3349.4		4	20.7	even	4
7440.2.a.bd.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.53113 q^{7} +1.00000 q^{9} +1.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} +3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} -1.53113 q^{33} +3.53113 q^{35} +9.06226 q^{37} -6.00000 q^{39} -9.06226 q^{41} +0.468871 q^{43} -1.00000 q^{45} -11.0623 q^{47} +5.46887 q^{49} +4.00000 q^{51} +5.53113 q^{53} -1.53113 q^{55} -3.53113 q^{57} -7.06226 q^{59} -11.0623 q^{61} -3.53113 q^{63} -6.00000 q^{65} +11.0623 q^{67} -1.53113 q^{69} +4.46887 q^{71} -0.468871 q^{73} -1.00000 q^{75} -5.40661 q^{77} +0.468871 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} +1.53113 q^{89} -21.1868 q^{91} +1.00000 q^{93} -3.53113 q^{95} -16.1245 q^{97} +1.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * q^99

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