Properties

Label 3762.2.a.bb
Level $3762$
Weight $2$
Character orbit 3762.a
Self dual yes
Analytic conductor $30.040$
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] = [3762,2,Mod(1,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, 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("3762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.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: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + (\beta - 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{5} + (\beta - 1) q^{7} + q^{8} + \beta q^{10} - q^{11} + ( - \beta - 1) q^{13} + (\beta - 1) q^{14} + q^{16} + 2 \beta q^{17} + q^{19} + \beta q^{20} - q^{22} + ( - 2 \beta + 6) q^{23} + (\beta - 2) q^{25} + ( - \beta - 1) q^{26} + (\beta - 1) q^{28} + (\beta + 6) q^{29} + ( - \beta - 1) q^{31} + q^{32} + 2 \beta q^{34} + 3 q^{35} + (4 \beta - 4) q^{37} + q^{38} + \beta q^{40} + ( - 3 \beta + 3) q^{41} + ( - 3 \beta + 8) q^{43} - q^{44} + ( - 2 \beta + 6) q^{46} + 6 q^{47} + ( - \beta - 3) q^{49} + (\beta - 2) q^{50} + ( - \beta - 1) q^{52} + (2 \beta + 6) q^{53} - \beta q^{55} + (\beta - 1) q^{56} + (\beta + 6) q^{58} + (4 \beta + 2) q^{61} + ( - \beta - 1) q^{62} + q^{64} + ( - 2 \beta - 3) q^{65} + ( - 5 \beta + 5) q^{67} + 2 \beta q^{68} + 3 q^{70} + ( - \beta + 12) q^{71} + ( - 2 \beta + 2) q^{73} + (4 \beta - 4) q^{74} + q^{76} + ( - \beta + 1) q^{77} + (6 \beta + 2) q^{79} + \beta q^{80} + ( - 3 \beta + 3) q^{82} + ( - \beta - 12) q^{83} + (2 \beta + 6) q^{85} + ( - 3 \beta + 8) q^{86} - q^{88} - 2 \beta q^{89} + ( - \beta - 2) q^{91} + ( - 2 \beta + 6) q^{92} + 6 q^{94} + \beta q^{95} + 8 q^{97} + ( - \beta - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - q^{7} + 2 q^{8} + q^{10} - 2 q^{11} - 3 q^{13} - q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} + q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{25} - 3 q^{26} - q^{28} + 13 q^{29} - 3 q^{31} + 2 q^{32} + 2 q^{34} + 6 q^{35} - 4 q^{37} + 2 q^{38} + q^{40} + 3 q^{41} + 13 q^{43} - 2 q^{44} + 10 q^{46} + 12 q^{47} - 7 q^{49} - 3 q^{50} - 3 q^{52} + 14 q^{53} - q^{55} - q^{56} + 13 q^{58} + 8 q^{61} - 3 q^{62} + 2 q^{64} - 8 q^{65} + 5 q^{67} + 2 q^{68} + 6 q^{70} + 23 q^{71} + 2 q^{73} - 4 q^{74} + 2 q^{76} + q^{77} + 10 q^{79} + q^{80} + 3 q^{82} - 25 q^{83} + 14 q^{85} + 13 q^{86} - 2 q^{88} - 2 q^{89} - 5 q^{91} + 10 q^{92} + 12 q^{94} + q^{95} + 16 q^{97} - 7 q^{98}+O(q^{100}) \) 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
−1.30278
2.30278
1.00000 0 1.00000 −1.30278 0 −2.30278 1.00000 0 −1.30278
1.2 1.00000 0 1.00000 2.30278 0 1.30278 1.00000 0 2.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)
\(19\) \( -1 \)

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 3762.2.a.bb 2
3.b odd 2 1 418.2.a.d 2
12.b even 2 1 3344.2.a.o 2
33.d even 2 1 4598.2.a.bc 2
57.d even 2 1 7942.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.d 2 3.b odd 2 1
3344.2.a.o 2 12.b even 2 1
3762.2.a.bb 2 1.a even 1 1 trivial
4598.2.a.bc 2 33.d even 2 1
7942.2.a.bb 2 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\):

\( T_{5}^{2} - T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 39 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 13 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T - 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 23T + 129 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$83$ \( T^{2} + 25T + 153 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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