Properties

Label 308.4.a.a
Level $308$
Weight $4$
Character orbit 308.a
Self dual yes
Analytic conductor $18.173$
Analytic rank $1$
Dimension $1$
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] = [308,4,Mod(1,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 308.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: \(18.1725882818\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 7 q^{3} - q^{5} + 7 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 7 q^{3} - q^{5} + 7 q^{7} + 22 q^{9} + 11 q^{11} + 12 q^{13} + 7 q^{15} + 2 q^{17} + 82 q^{19} - 49 q^{21} + 7 q^{23} - 124 q^{25} + 35 q^{27} - 102 q^{29} - 171 q^{31} - 77 q^{33} - 7 q^{35} - 357 q^{37} - 84 q^{39} - 114 q^{41} - 344 q^{43} - 22 q^{45} + 96 q^{47} + 49 q^{49} - 14 q^{51} - 430 q^{53} - 11 q^{55} - 574 q^{57} - 201 q^{59} - 2 q^{61} + 154 q^{63} - 12 q^{65} + 313 q^{67} - 49 q^{69} - 579 q^{71} - 438 q^{73} + 868 q^{75} + 77 q^{77} + 494 q^{79} - 839 q^{81} + 748 q^{83} - 2 q^{85} + 714 q^{87} + 457 q^{89} + 84 q^{91} + 1197 q^{93} - 82 q^{95} - 1037 q^{97} + 242 q^{99}+O(q^{100}) \) 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
0
0 −7.00000 0 −1.00000 0 7.00000 0 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 308.4.a.a 1
4.b odd 2 1 1232.4.a.h 1
7.b odd 2 1 2156.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.a.a 1 1.a even 1 1 trivial
1232.4.a.h 1 4.b odd 2 1
2156.4.a.c 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 12 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 82 \) Copy content Toggle raw display
$23$ \( T - 7 \) Copy content Toggle raw display
$29$ \( T + 102 \) Copy content Toggle raw display
$31$ \( T + 171 \) Copy content Toggle raw display
$37$ \( T + 357 \) Copy content Toggle raw display
$41$ \( T + 114 \) Copy content Toggle raw display
$43$ \( T + 344 \) Copy content Toggle raw display
$47$ \( T - 96 \) Copy content Toggle raw display
$53$ \( T + 430 \) Copy content Toggle raw display
$59$ \( T + 201 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 313 \) Copy content Toggle raw display
$71$ \( T + 579 \) Copy content Toggle raw display
$73$ \( T + 438 \) Copy content Toggle raw display
$79$ \( T - 494 \) Copy content Toggle raw display
$83$ \( T - 748 \) Copy content Toggle raw display
$89$ \( T - 457 \) Copy content Toggle raw display
$97$ \( T + 1037 \) Copy content Toggle raw display
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