Properties

Label 3700.2.a.e
Level $3700$
Weight $2$
Character orbit 3700.a
Self dual yes
Analytic conductor $29.545$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3700,2,Mod(1,3700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3700.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.5446487479\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 148)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} + 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 q^{7} - 2 q^{9} + 5 q^{11} + 6 q^{17} + 2 q^{19} + 3 q^{21} + 6 q^{23} - 5 q^{27} - 6 q^{29} + 4 q^{31} + 5 q^{33} - q^{37} - 9 q^{41} - 4 q^{43} + 7 q^{47} + 2 q^{49} + 6 q^{51} - 9 q^{53} + 2 q^{57} - 4 q^{59} - 8 q^{61} - 6 q^{63} + 12 q^{67} + 6 q^{69} + 3 q^{71} + 5 q^{73} + 15 q^{77} + 6 q^{79} + q^{81} + q^{83} - 6 q^{87} + 2 q^{89} + 4 q^{93} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 1.00000 0 0 0 3.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(37\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.2.a.e 1
5.b even 2 1 148.2.a.a 1
5.c odd 4 2 3700.2.d.e 2
15.d odd 2 1 1332.2.a.e 1
20.d odd 2 1 592.2.a.b 1
35.c odd 2 1 7252.2.a.b 1
40.e odd 2 1 2368.2.a.h 1
40.f even 2 1 2368.2.a.o 1
60.h even 2 1 5328.2.a.x 1
185.d even 2 1 5476.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.a.a 1 5.b even 2 1
592.2.a.b 1 20.d odd 2 1
1332.2.a.e 1 15.d odd 2 1
2368.2.a.h 1 40.e odd 2 1
2368.2.a.o 1 40.f even 2 1
3700.2.a.e 1 1.a even 1 1 trivial
3700.2.d.e 2 5.c odd 4 2
5328.2.a.x 1 60.h even 2 1
5476.2.a.a 1 185.d even 2 1
7252.2.a.b 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3700))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 7 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T - 5 \) Copy content Toggle raw display
$79$ \( T - 6 \) Copy content Toggle raw display
$83$ \( T - 1 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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