Properties

Label 3700.2.d.g
Level $3700$
Weight $2$
Character orbit 3700.d
Analytic conductor $29.545$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3700,2,Mod(149,3700)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3700.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(29.5446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 740)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 1) q^{9} + (2 \beta_{3} - 2) q^{11} - 4 \beta_1 q^{13} - 4 \beta_1 q^{17} + (3 \beta_{3} - 1) q^{19} + 2 \beta_{3} q^{21} + ( - 4 \beta_{2} + 2 \beta_1) q^{23} - 4 \beta_1 q^{27} + (2 \beta_{3} - 4) q^{29} + ( - 3 \beta_{3} + 5) q^{31} + (4 \beta_{2} - 8 \beta_1) q^{33} + \beta_1 q^{37} + ( - 4 \beta_{3} + 4) q^{39} + ( - 4 \beta_{3} - 2) q^{41} - 6 \beta_1 q^{43} + (\beta_{2} + 3 \beta_1) q^{47} + ( - 6 \beta_{3} - 5) q^{49} + ( - 4 \beta_{3} + 4) q^{51} + (4 \beta_{2} - 6 \beta_1) q^{53} + (4 \beta_{2} - 10 \beta_1) q^{57} + (5 \beta_{3} + 5) q^{59} + 14 q^{61} + (5 \beta_{2} + 3 \beta_1) q^{63} + ( - \beta_{2} - 7 \beta_1) q^{67} + (6 \beta_{3} - 14) q^{69} + 8 q^{71} - 6 \beta_{2} q^{73} + 4 \beta_{2} q^{77} + (3 \beta_{3} - 1) q^{79} + (2 \beta_{3} + 1) q^{81} + (5 \beta_{2} + 7 \beta_1) q^{83} + (6 \beta_{2} - 10 \beta_1) q^{87} + 2 q^{89} + (4 \beta_{3} + 12) q^{91} + ( - 8 \beta_{2} + 14 \beta_1) q^{93} - 14 \beta_1 q^{97} + ( - 6 \beta_{3} + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{11} - 4 q^{19} - 16 q^{29} + 20 q^{31} + 16 q^{39} - 8 q^{41} - 20 q^{49} + 16 q^{51} + 20 q^{59} + 56 q^{61} - 56 q^{69} + 32 q^{71} - 4 q^{79} + 4 q^{81} + 8 q^{89} + 48 q^{91} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 2.73205i 0 0 0 1.26795i 0 −4.46410 0
149.2 0 0.732051i 0 0 0 4.73205i 0 2.46410 0
149.3 0 0.732051i 0 0 0 4.73205i 0 2.46410 0
149.4 0 2.73205i 0 0 0 1.26795i 0 −4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.2.d.g 4
5.b even 2 1 inner 3700.2.d.g 4
5.c odd 4 1 740.2.a.d 2
5.c odd 4 1 3700.2.a.h 2
15.e even 4 1 6660.2.a.i 2
20.e even 4 1 2960.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.d 2 5.c odd 4 1
2960.2.a.p 2 20.e even 4 1
3700.2.a.h 2 5.c odd 4 1
3700.2.d.g 4 1.a even 1 1 trivial
3700.2.d.g 4 5.b even 2 1 inner
6660.2.a.i 2 15.e even 4 1

Hecke kernels

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

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 50)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 248T^{2} + 676 \) Copy content Toggle raw display
$89$ \( (T - 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
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