Properties

Label 608.4.a.a
Level $608$
Weight $4$
Character orbit 608.a
Self dual yes
Analytic conductor $35.873$
Analytic rank $0$
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] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, 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("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.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: \(35.8731612835\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 8 q^{5} - 17 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 8 q^{5} - 17 q^{7} - 26 q^{9} - 70 q^{11} - 61 q^{13} + 8 q^{15} + 83 q^{17} + 19 q^{19} + 17 q^{21} + 115 q^{23} - 61 q^{25} + 53 q^{27} + 279 q^{29} - 72 q^{31} + 70 q^{33} + 136 q^{35} - 34 q^{37} + 61 q^{39} + 108 q^{41} - 192 q^{43} + 208 q^{45} - 392 q^{47} - 54 q^{49} - 83 q^{51} + 131 q^{53} + 560 q^{55} - 19 q^{57} - 609 q^{59} + 338 q^{61} + 442 q^{63} + 488 q^{65} - 461 q^{67} - 115 q^{69} + 750 q^{71} + 1177 q^{73} + 61 q^{75} + 1190 q^{77} - 22 q^{79} + 649 q^{81} - 810 q^{83} - 664 q^{85} - 279 q^{87} - 476 q^{89} + 1037 q^{91} + 72 q^{93} - 152 q^{95} + 1426 q^{97} + 1820 q^{99}+O(q^{100}) \) 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 −1.00000 0 −8.00000 0 −17.0000 0 −26.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 608.4.a.a 1
4.b odd 2 1 608.4.a.b yes 1
8.b even 2 1 1216.4.a.d 1
8.d odd 2 1 1216.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.a 1 1.a even 1 1 trivial
608.4.a.b yes 1 4.b odd 2 1
1216.4.a.c 1 8.d odd 2 1
1216.4.a.d 1 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(608))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 8 \) Copy content Toggle raw display
$7$ \( T + 17 \) Copy content Toggle raw display
$11$ \( T + 70 \) Copy content Toggle raw display
$13$ \( T + 61 \) Copy content Toggle raw display
$17$ \( T - 83 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T - 115 \) Copy content Toggle raw display
$29$ \( T - 279 \) Copy content Toggle raw display
$31$ \( T + 72 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T - 108 \) Copy content Toggle raw display
$43$ \( T + 192 \) Copy content Toggle raw display
$47$ \( T + 392 \) Copy content Toggle raw display
$53$ \( T - 131 \) Copy content Toggle raw display
$59$ \( T + 609 \) Copy content Toggle raw display
$61$ \( T - 338 \) Copy content Toggle raw display
$67$ \( T + 461 \) Copy content Toggle raw display
$71$ \( T - 750 \) Copy content Toggle raw display
$73$ \( T - 1177 \) Copy content Toggle raw display
$79$ \( T + 22 \) Copy content Toggle raw display
$83$ \( T + 810 \) Copy content Toggle raw display
$89$ \( T + 476 \) Copy content Toggle raw display
$97$ \( T - 1426 \) Copy content Toggle raw display
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