Properties

Label 3700.2.a.h
Level $3700$
Weight $2$
Character orbit 3700.a
Self dual yes
Analytic conductor $29.545$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (\beta - 3) q^{7} + (2 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + (\beta - 3) q^{7} + (2 \beta + 1) q^{9} + ( - 2 \beta - 2) q^{11} - 4 q^{13} + 4 q^{17} + (3 \beta + 1) q^{19} - 2 \beta q^{21} + (4 \beta + 2) q^{23} + 4 q^{27} + (2 \beta + 4) q^{29} + (3 \beta + 5) q^{31} + ( - 4 \beta - 8) q^{33} - q^{37} + ( - 4 \beta - 4) q^{39} + (4 \beta - 2) q^{41} - 6 q^{43} + (\beta - 3) q^{47} + ( - 6 \beta + 5) q^{49} + (4 \beta + 4) q^{51} + ( - 4 \beta - 6) q^{53} + (4 \beta + 10) q^{57} + (5 \beta - 5) q^{59} + 14 q^{61} + ( - 5 \beta + 3) q^{63} + ( - \beta + 7) q^{67} + (6 \beta + 14) q^{69} + 8 q^{71} + 6 \beta q^{73} + 4 \beta q^{77} + (3 \beta + 1) q^{79} + ( - 2 \beta + 1) q^{81} + ( - 5 \beta + 7) q^{83} + (6 \beta + 10) q^{87} - 2 q^{89} + ( - 4 \beta + 12) q^{91} + (8 \beta + 14) q^{93} + 14 q^{97} + ( - 6 \beta - 14) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 6 q^{7} + 2 q^{9} - 4 q^{11} - 8 q^{13} + 8 q^{17} + 2 q^{19} + 4 q^{23} + 8 q^{27} + 8 q^{29} + 10 q^{31} - 16 q^{33} - 2 q^{37} - 8 q^{39} - 4 q^{41} - 12 q^{43} - 6 q^{47} + 10 q^{49} + 8 q^{51} - 12 q^{53} + 20 q^{57} - 10 q^{59} + 28 q^{61} + 6 q^{63} + 14 q^{67} + 28 q^{69} + 16 q^{71} + 2 q^{79} + 2 q^{81} + 14 q^{83} + 20 q^{87} - 4 q^{89} + 24 q^{91} + 28 q^{93} + 28 q^{97} - 28 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −0.732051 0 0 0 −4.73205 0 −2.46410 0
1.2 0 2.73205 0 0 0 −1.26795 0 4.46410 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.h 2
5.b even 2 1 740.2.a.d 2
5.c odd 4 2 3700.2.d.g 4
15.d odd 2 1 6660.2.a.i 2
20.d odd 2 1 2960.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.d 2 5.b even 2 1
2960.2.a.p 2 20.d odd 2 1
3700.2.a.h 2 1.a even 1 1 trivial
3700.2.d.g 4 5.c odd 4 2
6660.2.a.i 2 15.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T - 50 \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 46 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 108 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T - 26 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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