LMFDB - Embedded newform 3720.2.a.v.1.2

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			1.2
Root			$2.63876$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.96303 q^{7} +1.00000 q^{9} +4.54121 q^{11} -7.02011 q^{13} -1.00000 q^{15} +5.76165 q^{17} -2.05707 q^{19} -2.96303 q^{21} +0.736304 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.25846 q^{29} +1.00000 q^{31} +4.54121 q^{33} +2.96303 q^{35} +1.09404 q^{37} -7.02011 q^{39} -0.440876 q^{41} -4.49795 q^{43} -1.00000 q^{45} -5.27751 q^{47} +1.77956 q^{49} +5.76165 q^{51} +11.7755 q^{53} -4.54121 q^{55} -2.05707 q^{57} +3.74259 q^{59} +3.51587 q^{61} -2.96303 q^{63} +7.02011 q^{65} +14.4927 q^{67} +0.736304 q^{69} +11.7257 q^{71} -4.07613 q^{73} +1.00000 q^{75} -13.4557 q^{77} +4.06409 q^{79} +1.00000 q^{81} -3.80490 q^{83} -5.76165 q^{85} +3.25846 q^{87} +11.2088 q^{89} +20.8008 q^{91} +1.00000 q^{93} +2.05707 q^{95} -1.48519 q^{97} +4.54121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.96303	−1.11992	−0.559961	−	0.828519i	$-0.689184\pi$
$7$	−2.96303	−1.11992	−0.559961	+	0.828519i	$0.689184\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.54121	1.36923	0.684613	−	0.728907i	$-0.259971\pi$
$11$	4.54121	1.36923	0.684613	+	0.728907i	$0.259971\pi$
$12$	0	0
$13$	−7.02011	−1.94703	−0.973514	−	0.228629i	$-0.926576\pi$
$13$	−7.02011	−1.94703	−0.973514	+	0.228629i	$0.926576\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.76165	1.39740	0.698702	−	0.715413i	$-0.253761\pi$
$17$	5.76165	1.39740	0.698702	+	0.715413i	$0.253761\pi$
$18$	0	0
$19$	−2.05707	−0.471925	−0.235963	−	0.971762i	$-0.575824\pi$
$19$	−2.05707	−0.471925	−0.235963	+	0.971762i	$0.575824\pi$
$20$	0	0
$21$	−2.96303	−0.646587
$22$	0	0
$23$	0.736304	0.153530	0.0767650	−	0.997049i	$-0.475541\pi$
$23$	0.736304	0.153530	0.0767650	+	0.997049i	$0.475541\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.25846	0.605081	0.302540	−	0.953137i	$-0.402165\pi$
$29$	3.25846	0.605081	0.302540	+	0.953137i	$0.402165\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	4.54121	0.790523
$34$	0	0
$35$	2.96303	0.500844
$36$	0	0
$37$	1.09404	0.179859	0.0899296	−	0.995948i	$-0.471336\pi$
$37$	1.09404	0.179859	0.0899296	+	0.995948i	$0.471336\pi$
$38$	0	0
$39$	−7.02011	−1.12412
$40$	0	0
$41$	−0.440876	−0.0688533	−0.0344266	−	0.999407i	$-0.510961\pi$
$41$	−0.440876	−0.0688533	−0.0344266	+	0.999407i	$0.510961\pi$
$42$	0	0
$43$	−4.49795	−0.685931	−0.342965	−	0.939348i	$-0.611431\pi$
$43$	−4.49795	−0.685931	−0.342965	+	0.939348i	$0.611431\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.27751	−0.769804	−0.384902	−	0.922957i	$-0.625765\pi$
$47$	−5.27751	−0.769804	−0.384902	+	0.922957i	$0.625765\pi$
$48$	0	0
$49$	1.77956	0.254223
$50$	0	0
$51$	5.76165	0.806792
$52$	0	0
$53$	11.7755	1.61748	0.808742	−	0.588164i	$-0.200149\pi$
$53$	11.7755	1.61748	0.808742	+	0.588164i	$0.200149\pi$
$54$	0	0
$55$	−4.54121	−0.612336
$56$	0	0
$57$	−2.05707	−0.272466
$58$	0	0
$59$	3.74259	0.487244	0.243622	−	0.969870i	$-0.421664\pi$
$59$	3.74259	0.487244	0.243622	+	0.969870i	$0.421664\pi$
$60$	0	0
$61$	3.51587	0.450160	0.225080	−	0.974340i	$-0.427736\pi$
$61$	3.51587	0.450160	0.225080	+	0.974340i	$0.427736\pi$
$62$	0	0
$63$	−2.96303	−0.373307
$64$	0	0
$65$	7.02011	0.870737
$66$	0	0
$67$	14.4927	1.77057	0.885283	−	0.465052i	$-0.153964\pi$
$67$	14.4927	1.77057	0.885283	+	0.465052i	$0.153964\pi$
$68$	0	0
$69$	0.736304	0.0886406
$70$	0	0
$71$	11.7257	1.39159	0.695794	−	0.718241i	$-0.255053\pi$
$71$	11.7257	1.39159	0.695794	+	0.718241i	$0.255053\pi$
$72$	0	0
$73$	−4.07613	−0.477074	−0.238537	−	0.971133i	$-0.576668\pi$
$73$	−4.07613	−0.477074	−0.238537	+	0.971133i	$0.576668\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−13.4557	−1.53342
$78$	0	0
$79$	4.06409	0.457246	0.228623	−	0.973515i	$-0.426578\pi$
$79$	4.06409	0.457246	0.228623	+	0.973515i	$0.426578\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.80490	−0.417642	−0.208821	−	0.977954i	$-0.566963\pi$
$83$	−3.80490	−0.417642	−0.208821	+	0.977954i	$0.566963\pi$
$84$	0	0
$85$	−5.76165	−0.624938
$86$	0	0
$87$	3.25846	0.349344
$88$	0	0
$89$	11.2088	1.18813	0.594066	−	0.804416i	$-0.297522\pi$
$89$	11.2088	1.18813	0.594066	+	0.804416i	$0.297522\pi$
$90$	0	0
$91$	20.8008	2.18052
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.05707	0.211051
$96$	0	0
$97$	−1.48519	−0.150798	−0.0753991	−	0.997153i	$-0.524023\pi$
$97$	−1.48519	−0.150798	−0.0753991	+	0.997153i	$0.524023\pi$
$98$	0	0
$99$	4.54121	0.456409

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.1.2	✓	5
4.3	odd	2		7440.2.a.cd.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.v.1.2	✓	5	1.1	even	1	trivial
7440.2.a.cd.1.4		5	4.3	odd	2

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.96303 q^{7} +1.00000 q^{9} +4.54121 q^{11} -7.02011 q^{13} -1.00000 q^{15} +5.76165 q^{17} -2.05707 q^{19} -2.96303 q^{21} +0.736304 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.25846 q^{29} +1.00000 q^{31} +4.54121 q^{33} +2.96303 q^{35} +1.09404 q^{37} -7.02011 q^{39} -0.440876 q^{41} -4.49795 q^{43} -1.00000 q^{45} -5.27751 q^{47} +1.77956 q^{49} +5.76165 q^{51} +11.7755 q^{53} -4.54121 q^{55} -2.05707 q^{57} +3.74259 q^{59} +3.51587 q^{61} -2.96303 q^{63} +7.02011 q^{65} +14.4927 q^{67} +0.736304 q^{69} +11.7257 q^{71} -4.07613 q^{73} +1.00000 q^{75} -13.4557 q^{77} +4.06409 q^{79} +1.00000 q^{81} -3.80490 q^{83} -5.76165 q^{85} +3.25846 q^{87} +11.2088 q^{89} +20.8008 q^{91} +1.00000 q^{93} +2.05707 q^{95} -1.48519 q^{97} +4.54121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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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