LMFDB - Embedded newform 7440.2.a.bt.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,-4,0,3,0,2,0,4,0,-3,0,-2] 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:	$3$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +2.24482 q^{7} +1.00000 q^{9} +1.14399 q^{11} -2.24482 q^{13} -1.00000 q^{15} +2.67282 q^{17} +5.34565 q^{19} +2.24482 q^{21} -2.67282 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.715980 q^{29} +1.00000 q^{31} +1.14399 q^{33} -2.24482 q^{35} +6.53279 q^{37} -2.24482 q^{39} -3.14399 q^{41} -12.2017 q^{43} -1.00000 q^{45} +10.8745 q^{47} -1.96080 q^{49} +2.67282 q^{51} +11.5288 q^{53} -1.14399 q^{55} +5.34565 q^{57} +2.71598 q^{59} +6.00000 q^{61} +2.24482 q^{63} +2.24482 q^{65} -13.5905 q^{67} -2.67282 q^{69} +5.48568 q^{71} +3.75518 q^{73} +1.00000 q^{75} +2.56804 q^{77} -10.0185 q^{79} +1.00000 q^{81} +0.384851 q^{83} -2.67282 q^{85} +0.715980 q^{87} -2.62967 q^{89} -5.03920 q^{91} +1.00000 q^{93} -5.34565 q^{95} +16.1233 q^{97} +1.14399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{25} + 3 q^{27} + 3 q^{31} + 2 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} - 8 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 4 * q^19 - 4 * q^21 + 2 * q^23 + 3 * q^25 + 3 * q^27 + 3 * q^31 + 2 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 26 * q^53 - 2 * q^55 - 4 * q^57 + 6 * q^59 + 18 * q^61 - 4 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 + 22 * q^73 + 3 * q^75 + 12 * q^77 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 10 * q^89 - 28 * q^91 + 3 * q^93 + 4 * q^95 + 4 * 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
$7$	2.24482	0.848461	0.424231	−	0.905554i	$-0.360545\pi$
$7$	2.24482	0.848461	0.424231	+	0.905554i	$0.360545\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14399	0.344925	0.172462	−	0.985016i	$-0.444828\pi$
$11$	1.14399	0.344925	0.172462	+	0.985016i	$0.444828\pi$
$12$	0	0
$13$	−2.24482	−0.622600	−0.311300	−	0.950312i	$-0.600764\pi$
$13$	−2.24482	−0.622600	−0.311300	+	0.950312i	$0.600764\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.67282	0.648255	0.324127	−	0.946013i	$-0.394929\pi$
$17$	2.67282	0.648255	0.324127	+	0.946013i	$0.394929\pi$
$18$	0	0
$19$	5.34565	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$19$	5.34565	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$20$	0	0
$21$	2.24482	0.489859
$22$	0	0
$23$	−2.67282	−0.557322	−0.278661	−	0.960389i	$-0.589891\pi$
$23$	−2.67282	−0.557322	−0.278661	+	0.960389i	$0.589891\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.715980	0.132954	0.0664771	−	0.997788i	$-0.478824\pi$
$29$	0.715980	0.132954	0.0664771	+	0.997788i	$0.478824\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	1.14399	0.199142
$34$	0	0
$35$	−2.24482	−0.379443
$36$	0	0
$37$	6.53279	1.07398	0.536992	−	0.843587i	$-0.319561\pi$
$37$	6.53279	1.07398	0.536992	+	0.843587i	$0.319561\pi$
$38$	0	0
$39$	−2.24482	−0.359458
$40$	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
$43$	−12.2017	−1.86074	−0.930368	−	0.366627i	$-0.880512\pi$
$43$	−12.2017	−1.86074	−0.930368	+	0.366627i	$0.880512\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.8745	1.58621	0.793103	−	0.609087i	$-0.208464\pi$
$47$	10.8745	1.58621	0.793103	+	0.609087i	$0.208464\pi$
$48$	0	0
$49$	−1.96080	−0.280114
$50$	0	0
$51$	2.67282	0.374270
$52$	0	0
$53$	11.5288	1.58361	0.791804	−	0.610776i	$-0.209142\pi$
$53$	11.5288	1.58361	0.791804	+	0.610776i	$0.209142\pi$
$54$	0	0
$55$	−1.14399	−0.154255
$56$	0	0
$57$	5.34565	0.708048
$58$	0	0
$59$	2.71598	0.353590	0.176795	−	0.984248i	$-0.443427\pi$
$59$	2.71598	0.353590	0.176795	+	0.984248i	$0.443427\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	2.24482	0.282820
$64$	0	0
$65$	2.24482	0.278435
$66$	0	0
$67$	−13.5905	−1.66034	−0.830170	−	0.557511i	$-0.811757\pi$
$67$	−13.5905	−1.66034	−0.830170	+	0.557511i	$0.811757\pi$
$68$	0	0
$69$	−2.67282	−0.321770
$70$	0	0
$71$	5.48568	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$71$	5.48568	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$72$	0	0
$73$	3.75518	0.439511	0.219755	−	0.975555i	$-0.429474\pi$
$73$	3.75518	0.439511	0.219755	+	0.975555i	$0.429474\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.56804	0.292655
$78$	0	0
$79$	−10.0185	−1.12717	−0.563583	−	0.826059i	$-0.690578\pi$
$79$	−10.0185	−1.12717	−0.563583	+	0.826059i	$0.690578\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.384851	0.0422428	0.0211214	−	0.999777i	$-0.493276\pi$
$83$	0.384851	0.0422428	0.0211214	+	0.999777i	$0.493276\pi$
$84$	0	0
$85$	−2.67282	−0.289908
$86$	0	0
$87$	0.715980	0.0767611
$88$	0	0
$89$	−2.62967	−0.278744	−0.139372	−	0.990240i	$-0.544508\pi$
$89$	−2.62967	−0.278744	−0.139372	+	0.990240i	$0.544508\pi$
$90$	0	0
$91$	−5.03920	−0.528252
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−5.34565	−0.548452
$96$	0	0
$97$	16.1233	1.63707	0.818534	−	0.574458i	$-0.194787\pi$
$97$	16.1233	1.63707	0.818534	+	0.574458i	$0.194787\pi$
$98$	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.bt.1.3		3
4.3	odd	2		1860.2.a.f.1.1	✓	3
12.11	even	2		5580.2.a.l.1.1		3
20.3	even	4		9300.2.g.p.3349.3		6
20.7	even	4		9300.2.g.p.3349.4		6
20.19	odd	2		9300.2.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.f.1.1	✓	3	4.3	odd	2
5580.2.a.l.1.1		3	12.11	even	2
7440.2.a.bt.1.3		3	1.1	even	1	trivial
9300.2.a.v.1.3		3	20.19	odd	2
9300.2.g.p.3349.3		6	20.3	even	4
9300.2.g.p.3349.4		6	20.7	even	4

Overview	Random
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Classical	Maass
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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.24482 q^{7} +1.00000 q^{9} +1.14399 q^{11} -2.24482 q^{13} -1.00000 q^{15} +2.67282 q^{17} +5.34565 q^{19} +2.24482 q^{21} -2.67282 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.715980 q^{29} +1.00000 q^{31} +1.14399 q^{33} -2.24482 q^{35} +6.53279 q^{37} -2.24482 q^{39} -3.14399 q^{41} -12.2017 q^{43} -1.00000 q^{45} +10.8745 q^{47} -1.96080 q^{49} +2.67282 q^{51} +11.5288 q^{53} -1.14399 q^{55} +5.34565 q^{57} +2.71598 q^{59} +6.00000 q^{61} +2.24482 q^{63} +2.24482 q^{65} -13.5905 q^{67} -2.67282 q^{69} +5.48568 q^{71} +3.75518 q^{73} +1.00000 q^{75} +2.56804 q^{77} -10.0185 q^{79} +1.00000 q^{81} +0.384851 q^{83} -2.67282 q^{85} +0.715980 q^{87} -2.62967 q^{89} -5.03920 q^{91} +1.00000 q^{93} -5.34565 q^{95} +16.1233 q^{97} +1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{25} + 3 q^{27} + 3 q^{31} + 2 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} - 8 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 4 * q^19 - 4 * q^21 + 2 * q^23 + 3 * q^25 + 3 * q^27 + 3 * q^31 + 2 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 26 * q^53 - 2 * q^55 - 4 * q^57 + 6 * q^59 + 18 * q^61 - 4 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 + 22 * q^73 + 3 * q^75 + 12 * q^77 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 10 * q^89 - 28 * q^91 + 3 * q^93 + 4 * q^95 + 4 * q^97 + 2 * q^99

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