Properties

Label 3330.2.a.bj
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
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] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
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 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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + (\beta_{2} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + (\beta_{2} + 1) q^{7} - q^{8} + q^{10} + (\beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{2} - 1) q^{14} + q^{16} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{3} - \beta_{2} + 1) q^{19} - q^{20} + ( - \beta_1 + 1) q^{22} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} + q^{25} + (\beta_{3} - \beta_1 + 1) q^{26} + (\beta_{2} + 1) q^{28} + (\beta_{3} + \beta_{2} + \beta_1) q^{29} + (\beta_{3} - \beta_{2} + \beta_1) q^{31} - q^{32} + (2 \beta_{3} - \beta_{2} + 1) q^{34} + ( - \beta_{2} - 1) q^{35} - q^{37} + (\beta_{3} + \beta_{2} - 1) q^{38} + q^{40} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{43} + (\beta_1 - 1) q^{44} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{46} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{47} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{49} - q^{50} + ( - \beta_{3} + \beta_1 - 1) q^{52} + (2 \beta_{3} - \beta_{2} + 5) q^{53} + ( - \beta_1 + 1) q^{55} + ( - \beta_{2} - 1) q^{56} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{58} + (2 \beta_{2} + 2) q^{59} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{61} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{62} + q^{64} + (\beta_{3} - \beta_1 + 1) q^{65} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{67} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{68} + (\beta_{2} + 1) q^{70} + ( - 2 \beta_{3} - 2 \beta_1 + 6) q^{71} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{73} + q^{74} + ( - \beta_{3} - \beta_{2} + 1) q^{76} + ( - 3 \beta_{2} + 1) q^{77} + 4 q^{79} - q^{80} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{82} + ( - \beta_{3} + 3 \beta_1 - 1) q^{83} + (2 \beta_{3} - \beta_{2} + 1) q^{85} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{86} + ( - \beta_1 + 1) q^{88} + ( - \beta_{3} - 3 \beta_1 + 3) q^{89} + ( - 3 \beta_{3} - 4 \beta_{2} + \cdots - 3) q^{91}+ \cdots + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 7\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 11\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 7\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{2} + 11\beta _1 + 4 ) / 2 \) 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
0.655762
3.36007
−0.339102
−2.67673
−1.00000 0 1.00000 −1.00000 0 −3.46569 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 0.487359 −1.00000 0 1.00000
1.3 −1.00000 0 1.00000 −1.00000 0 1.84556 −1.00000 0 1.00000
1.4 −1.00000 0 1.00000 −1.00000 0 5.13277 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 3330.2.a.bj 4
3.b odd 2 1 1110.2.a.s 4
12.b even 2 1 8880.2.a.cg 4
15.d odd 2 1 5550.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.s 4 3.b odd 2 1
3330.2.a.bj 4 1.a even 1 1 trivial
5550.2.a.cj 4 15.d odd 2 1
8880.2.a.cg 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7}^{4} - 4T_{7}^{3} - 13T_{7}^{2} + 40T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 23T_{11}^{2} - 68T_{11} - 40 \) Copy content Toggle raw display
\( T_{13}^{4} + 3T_{13}^{3} - 38T_{13}^{2} - 44T_{13} + 328 \) Copy content Toggle raw display
\( T_{17}^{4} + 6T_{17}^{3} - 47T_{17}^{2} - 412T_{17} - 764 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots - 40 \) Copy content Toggle raw display
$13$ \( T^{4} + 3 T^{3} + \cdots + 328 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots - 764 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + \cdots - 80 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{3} + \cdots + 1280 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 1208 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 11 T^{3} + \cdots - 80 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 1280 \) Copy content Toggle raw display
$47$ \( T^{4} - 88 T^{2} + \cdots + 1280 \) Copy content Toggle raw display
$53$ \( T^{4} - 22 T^{3} + \cdots - 2524 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$61$ \( T^{4} + 5 T^{3} + \cdots + 1096 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$73$ \( T^{4} + 5 T^{3} + \cdots - 2336 \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + \cdots + 5344 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} + \cdots - 3208 \) Copy content Toggle raw display
$97$ \( T^{4} - 7 T^{3} + \cdots + 9344 \) Copy content Toggle raw display
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