LMFDB - Embedded newform 3720.2.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,-5,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.7043495519$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2294036.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 8x^{2} + 6x - 4 $ x^5 - x^4 - 8x^3 + 8x^2 + 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.69199$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -3.24680 q^{7} +1.00000 q^{9} -1.35637 q^{11} +1.59546 q^{13} -1.00000 q^{15} -1.89809 q^{17} +6.84226 q^{19} -3.24680 q^{21} -4.02760 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.30262 q^{29} +1.00000 q^{31} -1.35637 q^{33} +3.24680 q^{35} -8.08906 q^{37} +1.59546 q^{39} +3.08343 q^{41} +7.92569 q^{43} -1.00000 q^{45} +5.38398 q^{47} +3.54171 q^{49} -1.89809 q^{51} -11.3097 q^{53} +1.35637 q^{55} +6.84226 q^{57} +5.78851 q^{59} +0.514109 q^{61} -3.24680 q^{63} -1.59546 q^{65} -3.65067 q^{67} -4.02760 q^{69} +5.43985 q^{71} +14.5289 q^{73} +1.00000 q^{75} +4.40387 q^{77} +14.6646 q^{79} +1.00000 q^{81} -2.67123 q^{83} +1.89809 q^{85} +2.30262 q^{87} +6.83461 q^{89} -5.18015 q^{91} +1.00000 q^{93} -6.84226 q^{95} -5.57703 q^{97} -1.35637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 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
$7$	−3.24680	−1.22718	−0.613588	−	0.789627i	$-0.710274\pi$
$7$	−3.24680	−1.22718	−0.613588	+	0.789627i	$0.710274\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.35637	−0.408962	−0.204481	−	0.978871i	$-0.565551\pi$
$11$	−1.35637	−0.408962	−0.204481	+	0.978871i	$0.565551\pi$
$12$	0	0
$13$	1.59546	0.442502	0.221251	−	0.975217i	$-0.428986\pi$
$13$	1.59546	0.442502	0.221251	+	0.975217i	$0.428986\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.89809	−0.460353	−0.230177	−	0.973149i	$-0.573930\pi$
$17$	−1.89809	−0.460353	−0.230177	+	0.973149i	$0.573930\pi$
$18$	0	0
$19$	6.84226	1.56972	0.784861	−	0.619671i	$-0.212734\pi$
$19$	6.84226	1.56972	0.784861	+	0.619671i	$0.212734\pi$
$20$	0	0
$21$	−3.24680	−0.708510
$22$	0	0
$23$	−4.02760	−0.839814	−0.419907	−	0.907567i	$-0.637937\pi$
$23$	−4.02760	−0.839814	−0.419907	+	0.907567i	$0.637937\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.30262	0.427586	0.213793	−	0.976879i	$-0.431418\pi$
$29$	2.30262	0.427586	0.213793	+	0.976879i	$0.431418\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−1.35637	−0.236114
$34$	0	0
$35$	3.24680	0.548809
$36$	0	0
$37$	−8.08906	−1.32983	−0.664917	−	0.746917i	$-0.731533\pi$
$37$	−8.08906	−1.32983	−0.664917	+	0.746917i	$0.731533\pi$
$38$	0	0
$39$	1.59546	0.255478
$40$	0	0
$41$	3.08343	0.481550	0.240775	−	0.970581i	$-0.422598\pi$
$41$	3.08343	0.481550	0.240775	+	0.970581i	$0.422598\pi$
$42$	0	0
$43$	7.92569	1.20866	0.604328	−	0.796735i	$-0.293442\pi$
$43$	7.92569	1.20866	0.604328	+	0.796735i	$0.293442\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	5.38398	0.785334	0.392667	−	0.919681i	$-0.371553\pi$
$47$	5.38398	0.785334	0.392667	+	0.919681i	$0.371553\pi$
$48$	0	0
$49$	3.54171	0.505959
$50$	0	0
$51$	−1.89809	−0.265785
$52$	0	0
$53$	−11.3097	−1.55350	−0.776751	−	0.629808i	$-0.783134\pi$
$53$	−11.3097	−1.55350	−0.776751	+	0.629808i	$0.783134\pi$
$54$	0	0
$55$	1.35637	0.182893
$56$	0	0
$57$	6.84226	0.906280
$58$	0	0
$59$	5.78851	0.753600	0.376800	−	0.926295i	$-0.377024\pi$
$59$	5.78851	0.753600	0.376800	+	0.926295i	$0.377024\pi$
$60$	0	0
$61$	0.514109	0.0658250	0.0329125	−	0.999458i	$-0.489522\pi$
$61$	0.514109	0.0658250	0.0329125	+	0.999458i	$0.489522\pi$
$62$	0	0
$63$	−3.24680	−0.409058
$64$	0	0
$65$	−1.59546	−0.197893
$66$	0	0
$67$	−3.65067	−0.446000	−0.223000	−	0.974818i	$-0.571585\pi$
$67$	−3.65067	−0.446000	−0.223000	+	0.974818i	$0.571585\pi$
$68$	0	0
$69$	−4.02760	−0.484867
$70$	0	0
$71$	5.43985	0.645592	0.322796	−	0.946469i	$-0.395377\pi$
$71$	5.43985	0.645592	0.322796	+	0.946469i	$0.395377\pi$
$72$	0	0
$73$	14.5289	1.70047	0.850237	−	0.526399i	$-0.176458\pi$
$73$	14.5289	1.70047	0.850237	+	0.526399i	$0.176458\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	4.40387	0.501868
$78$	0	0
$79$	14.6646	1.64990	0.824950	−	0.565206i	$-0.191203\pi$
$79$	14.6646	1.64990	0.824950	+	0.565206i	$0.191203\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.67123	−0.293206	−0.146603	−	0.989195i	$-0.546834\pi$
$83$	−2.67123	−0.293206	−0.146603	+	0.989195i	$0.546834\pi$
$84$	0	0
$85$	1.89809	0.205876
$86$	0	0
$87$	2.30262	0.246867
$88$	0	0
$89$	6.83461	0.724467	0.362233	−	0.932087i	$-0.382014\pi$
$89$	6.83461	0.724467	0.362233	+	0.932087i	$0.382014\pi$
$90$	0	0
$91$	−5.18015	−0.543027
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−6.84226	−0.702001
$96$	0	0
$97$	−5.57703	−0.566261	−0.283131	−	0.959081i	$-0.591373\pi$
$97$	−5.57703	−0.566261	−0.283131	+	0.959081i	$0.591373\pi$
$98$	0	0
$99$	−1.35637	−0.136321

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	3720.2.a.v.1.1	✓	5
4.3	odd	2		7440.2.a.cd.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.v.1.1	✓	5	1.1	even	1	trivial
7440.2.a.cd.1.5		5	4.3	odd	2

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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.24680 q^{7} +1.00000 q^{9} -1.35637 q^{11} +1.59546 q^{13} -1.00000 q^{15} -1.89809 q^{17} +6.84226 q^{19} -3.24680 q^{21} -4.02760 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.30262 q^{29} +1.00000 q^{31} -1.35637 q^{33} +3.24680 q^{35} -8.08906 q^{37} +1.59546 q^{39} +3.08343 q^{41} +7.92569 q^{43} -1.00000 q^{45} +5.38398 q^{47} +3.54171 q^{49} -1.89809 q^{51} -11.3097 q^{53} +1.35637 q^{55} +6.84226 q^{57} +5.78851 q^{59} +0.514109 q^{61} -3.24680 q^{63} -1.59546 q^{65} -3.65067 q^{67} -4.02760 q^{69} +5.43985 q^{71} +14.5289 q^{73} +1.00000 q^{75} +4.40387 q^{77} +14.6646 q^{79} +1.00000 q^{81} -2.67123 q^{83} +1.89809 q^{85} +2.30262 q^{87} +6.83461 q^{89} -5.18015 q^{91} +1.00000 q^{93} -6.84226 q^{95} -5.57703 q^{97} -1.35637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 5 * q^99

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