LMFDB - Embedded newform 3720.2.a.v.1.5

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			1.5
Root			$0.504009$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +3.74598 q^{7} +1.00000 q^{9} -0.896022 q^{11} -1.29437 q^{13} -1.00000 q^{15} -4.92835 q^{17} -3.04035 q^{19} +3.74598 q^{21} +1.90404 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.22272 q^{29} +1.00000 q^{31} -0.896022 q^{33} -3.74598 q^{35} +8.78632 q^{37} -1.29437 q^{39} +10.0647 q^{41} +5.02431 q^{43} -1.00000 q^{45} -1.00802 q^{47} +7.03233 q^{49} -4.92835 q^{51} -2.01629 q^{53} +0.896022 q^{55} -3.04035 q^{57} +2.28635 q^{59} +9.93637 q^{61} +3.74598 q^{63} +1.29437 q^{65} +11.1025 q^{67} +1.90404 q^{69} -2.16525 q^{71} +4.17436 q^{73} +1.00000 q^{75} -3.35648 q^{77} -3.73224 q^{79} +1.00000 q^{81} +2.80006 q^{83} +4.92835 q^{85} +8.22272 q^{87} -12.6107 q^{89} -4.84869 q^{91} +1.00000 q^{93} +3.04035 q^{95} +1.42729 q^{97} -0.896022 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.74598	1.41585	0.707923	−	0.706290i	$-0.249632\pi$
$7$	3.74598	1.41585	0.707923	+	0.706290i	$0.249632\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.896022	−0.270161	−0.135080	−	0.990835i	$-0.543129\pi$
$11$	−0.896022	−0.270161	−0.135080	+	0.990835i	$0.543129\pi$
$12$	0	0
$13$	−1.29437	−0.358994	−0.179497	−	0.983758i	$-0.557447\pi$
$13$	−1.29437	−0.358994	−0.179497	+	0.983758i	$0.557447\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.92835	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$17$	−4.92835	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$18$	0	0
$19$	−3.04035	−0.697503	−0.348752	−	0.937215i	$-0.613394\pi$
$19$	−3.04035	−0.697503	−0.348752	+	0.937215i	$0.613394\pi$
$20$	0	0
$21$	3.74598	0.817439
$22$	0	0
$23$	1.90404	0.397020	0.198510	−	0.980099i	$-0.436390\pi$
$23$	1.90404	0.397020	0.198510	+	0.980099i	$0.436390\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.22272	1.52692	0.763461	−	0.645854i	$-0.223499\pi$
$29$	8.22272	1.52692	0.763461	+	0.645854i	$0.223499\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−0.896022	−0.155977
$34$	0	0
$35$	−3.74598	−0.633185
$36$	0	0
$37$	8.78632	1.44446	0.722231	−	0.691652i	$-0.243117\pi$
$37$	8.78632	1.44446	0.722231	+	0.691652i	$0.243117\pi$
$38$	0	0
$39$	−1.29437	−0.207265
$40$	0	0
$41$	10.0647	1.57184	0.785918	−	0.618331i	$-0.212191\pi$
$41$	10.0647	1.57184	0.785918	+	0.618331i	$0.212191\pi$
$42$	0	0
$43$	5.02431	0.766200	0.383100	−	0.923707i	$-0.374856\pi$
$43$	5.02431	0.766200	0.383100	+	0.923707i	$0.374856\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.00802	−0.147034	−0.0735172	−	0.997294i	$-0.523422\pi$
$47$	−1.00802	−0.147034	−0.0735172	+	0.997294i	$0.523422\pi$
$48$	0	0
$49$	7.03233	1.00462
$50$	0	0
$51$	−4.92835	−0.690107
$52$	0	0
$53$	−2.01629	−0.276959	−0.138480	−	0.990365i	$-0.544222\pi$
$53$	−2.01629	−0.276959	−0.138480	+	0.990365i	$0.544222\pi$
$54$	0	0
$55$	0.896022	0.120820
$56$	0	0
$57$	−3.04035	−0.402704
$58$	0	0
$59$	2.28635	0.297658	0.148829	−	0.988863i	$-0.452450\pi$
$59$	2.28635	0.297658	0.148829	+	0.988863i	$0.452450\pi$
$60$	0	0
$61$	9.93637	1.27222	0.636111	−	0.771598i	$-0.280542\pi$
$61$	9.93637	1.27222	0.636111	+	0.771598i	$0.280542\pi$
$62$	0	0
$63$	3.74598	0.471949
$64$	0	0
$65$	1.29437	0.160547
$66$	0	0
$67$	11.1025	1.35638	0.678190	−	0.734886i	$-0.262765\pi$
$67$	11.1025	1.35638	0.678190	+	0.734886i	$0.262765\pi$
$68$	0	0
$69$	1.90404	0.229219
$70$	0	0
$71$	−2.16525	−0.256968	−0.128484	−	0.991712i	$-0.541011\pi$
$71$	−2.16525	−0.256968	−0.128484	+	0.991712i	$0.541011\pi$
$72$	0	0
$73$	4.17436	0.488572	0.244286	−	0.969703i	$-0.421446\pi$
$73$	4.17436	0.488572	0.244286	+	0.969703i	$0.421446\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−3.35648	−0.382506
$78$	0	0
$79$	−3.73224	−0.419909	−0.209955	−	0.977711i	$-0.567332\pi$
$79$	−3.73224	−0.419909	−0.209955	+	0.977711i	$0.567332\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.80006	0.307347	0.153673	−	0.988122i	$-0.450890\pi$
$83$	2.80006	0.307347	0.153673	+	0.988122i	$0.450890\pi$
$84$	0	0
$85$	4.92835	0.534555
$86$	0	0
$87$	8.22272	0.881569
$88$	0	0
$89$	−12.6107	−1.33673	−0.668366	−	0.743833i	$-0.733006\pi$
$89$	−12.6107	−1.33673	−0.668366	+	0.743833i	$0.733006\pi$
$90$	0	0
$91$	−4.84869	−0.508280
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	3.04035	0.311933
$96$	0	0
$97$	1.42729	0.144919	0.0724597	−	0.997371i	$-0.476915\pi$
$97$	1.42729	0.144919	0.0724597	+	0.997371i	$0.476915\pi$
$98$	0	0
$99$	−0.896022	−0.0900536

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +3.74598 q^{7} +1.00000 q^{9} -0.896022 q^{11} -1.29437 q^{13} -1.00000 q^{15} -4.92835 q^{17} -3.04035 q^{19} +3.74598 q^{21} +1.90404 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.22272 q^{29} +1.00000 q^{31} -0.896022 q^{33} -3.74598 q^{35} +8.78632 q^{37} -1.29437 q^{39} +10.0647 q^{41} +5.02431 q^{43} -1.00000 q^{45} -1.00802 q^{47} +7.03233 q^{49} -4.92835 q^{51} -2.01629 q^{53} +0.896022 q^{55} -3.04035 q^{57} +2.28635 q^{59} +9.93637 q^{61} +3.74598 q^{63} +1.29437 q^{65} +11.1025 q^{67} +1.90404 q^{69} -2.16525 q^{71} +4.17436 q^{73} +1.00000 q^{75} -3.35648 q^{77} -3.73224 q^{79} +1.00000 q^{81} +2.80006 q^{83} +4.92835 q^{85} +8.22272 q^{87} -12.6107 q^{89} -4.84869 q^{91} +1.00000 q^{93} +3.04035 q^{95} +1.42729 q^{97} -0.896022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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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