Properties

Label 4368.2.a.bs
Level $4368$
Weight $2$
Character orbit 4368.a
Self dual yes
Analytic conductor $34.879$
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] = [4368,2,Mod(1,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.138892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 2x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2184)
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_1 + 1) q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_1 + 1) q^{5} + q^{7} + q^{9} + ( - \beta_{2} + 1) q^{11} + q^{13} + (\beta_1 + 1) q^{15} + ( - \beta_{3} + 1) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + q^{21} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{3} - \beta_{2} + \beta_1 + 4) q^{25} + q^{27} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{29} + (\beta_{2} + \beta_1 - 2) q^{31} + ( - \beta_{2} + 1) q^{33} + (\beta_1 + 1) q^{35} + ( - \beta_{3} + 1) q^{37} + q^{39} + ( - \beta_{3} - \beta_{2} + 2) q^{41} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{43} + (\beta_1 + 1) q^{45} + ( - \beta_{3} - \beta_1 - 4) q^{47} + q^{49} + ( - \beta_{3} + 1) q^{51} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{53} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 1) q^{55} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{3} + \beta_{2} - 2) q^{59} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{61} + q^{63} + (\beta_1 + 1) q^{65} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{67} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{69} + (\beta_{3} - \beta_{2} + 2) q^{71} + (\beta_{3} + \beta_{2} - \beta_1 + 7) q^{73} + (\beta_{3} - \beta_{2} + \beta_1 + 4) q^{75} + ( - \beta_{2} + 1) q^{77} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{79} + q^{81} + (\beta_{3} + 3 \beta_1 - 2) q^{83} + (\beta_{3} - 2 \beta_1 + 3) q^{85} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{87} + (\beta_{3} + \beta_1 + 2) q^{89} + q^{91} + (\beta_{2} + \beta_1 - 2) q^{93} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{95} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{97} + ( - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 10x^{2} + 2x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + 2\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} + 5\beta_{2} + 2\beta _1 + 5 \) 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
1.21773
−2.52690
3.44214
−1.13296
0 1.00000 0 −3.70948 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.152457 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 1.69903 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 4.16291 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 4368.2.a.bs 4
4.b odd 2 1 2184.2.a.v 4
12.b even 2 1 6552.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.a.v 4 4.b odd 2 1
4368.2.a.bs 4 1.a even 1 1 trivial
6552.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4368))\):

\( T_{5}^{4} - 2T_{5}^{3} - 15T_{5}^{2} + 24T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 3T_{11}^{3} - 34T_{11}^{2} + 112T_{11} + 48 \) Copy content Toggle raw display
\( T_{17}^{4} - 3T_{17}^{3} - 36T_{17}^{2} + 20T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - 15 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} - 34 T^{2} + 112 T + 48 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} - 36 T^{2} + 20 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 45 T^{2} + 160 T - 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 41 T^{2} + 232 T + 752 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 45 T^{2} + 36 T + 108 \) Copy content Toggle raw display
$31$ \( T^{4} + 9 T^{3} - 32 T^{2} - 224 T - 256 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} - 36 T^{2} + 20 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} - 28 T^{2} + 136 T + 64 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 41 T^{2} + 232 T + 752 \) Copy content Toggle raw display
$47$ \( T^{4} + 15 T^{3} + 44 T^{2} + \cdots - 768 \) Copy content Toggle raw display
$53$ \( T^{4} - 13 T^{3} - 142 T^{2} + \cdots + 3144 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 88 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} - 9 T^{3} - 230 T^{2} + \cdots + 6056 \) Copy content Toggle raw display
$67$ \( T^{4} - 260 T^{2} + 32 T + 16064 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} - 88 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$73$ \( T^{4} - 32 T^{3} + 319 T^{2} + \cdots + 908 \) Copy content Toggle raw display
$79$ \( T^{4} - T^{3} - 118 T^{2} + 624 T - 832 \) Copy content Toggle raw display
$83$ \( T^{4} + 13 T^{3} - 78 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$89$ \( T^{4} - 7 T^{3} - 22 T^{2} + 212 T - 328 \) Copy content Toggle raw display
$97$ \( T^{4} - 19 T^{3} - 70 T^{2} + \cdots + 4328 \) Copy content Toggle raw display
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