Properties

Label 68.4.a.a
Level $68$
Weight $4$
Character orbit 68.a
Self dual yes
Analytic conductor $4.012$
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] = [68,4,Mod(1,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("68.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 68.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: \(4.01212988039\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} - 8 q^{5} - 12 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - 8 q^{5} - 12 q^{7} - 23 q^{9} - 10 q^{11} - 38 q^{13} + 16 q^{15} - 17 q^{17} + 4 q^{19} + 24 q^{21} + 120 q^{23} - 61 q^{25} + 100 q^{27} + 56 q^{29} + 164 q^{31} + 20 q^{33} + 96 q^{35} - 236 q^{37} + 76 q^{39} + 70 q^{41} - 144 q^{43} + 184 q^{45} + 48 q^{47} - 199 q^{49} + 34 q^{51} - 366 q^{53} + 80 q^{55} - 8 q^{57} - 504 q^{59} - 460 q^{61} + 276 q^{63} + 304 q^{65} - 768 q^{67} - 240 q^{69} + 72 q^{71} - 734 q^{73} + 122 q^{75} + 120 q^{77} + 736 q^{79} + 421 q^{81} + 856 q^{83} + 136 q^{85} - 112 q^{87} + 906 q^{89} + 456 q^{91} - 328 q^{93} - 32 q^{95} + 46 q^{97} + 230 q^{99}+O(q^{100}) \) 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 −2.00000 0 −8.00000 0 −12.0000 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( +1 \)

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 68.4.a.a 1
3.b odd 2 1 612.4.a.c 1
4.b odd 2 1 272.4.a.b 1
5.b even 2 1 1700.4.a.b 1
5.c odd 4 2 1700.4.e.c 2
8.b even 2 1 1088.4.a.h 1
8.d odd 2 1 1088.4.a.e 1
12.b even 2 1 2448.4.a.l 1
17.b even 2 1 1156.4.a.a 1
17.c even 4 2 1156.4.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.a.a 1 1.a even 1 1 trivial
272.4.a.b 1 4.b odd 2 1
612.4.a.c 1 3.b odd 2 1
1088.4.a.e 1 8.d odd 2 1
1088.4.a.h 1 8.b even 2 1
1156.4.a.a 1 17.b even 2 1
1156.4.b.b 2 17.c even 4 2
1700.4.a.b 1 5.b even 2 1
1700.4.e.c 2 5.c odd 4 2
2448.4.a.l 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(68))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 8 \) Copy content Toggle raw display
$7$ \( T + 12 \) Copy content Toggle raw display
$11$ \( T + 10 \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T - 56 \) Copy content Toggle raw display
$31$ \( T - 164 \) Copy content Toggle raw display
$37$ \( T + 236 \) Copy content Toggle raw display
$41$ \( T - 70 \) Copy content Toggle raw display
$43$ \( T + 144 \) Copy content Toggle raw display
$47$ \( T - 48 \) Copy content Toggle raw display
$53$ \( T + 366 \) Copy content Toggle raw display
$59$ \( T + 504 \) Copy content Toggle raw display
$61$ \( T + 460 \) Copy content Toggle raw display
$67$ \( T + 768 \) Copy content Toggle raw display
$71$ \( T - 72 \) Copy content Toggle raw display
$73$ \( T + 734 \) Copy content Toggle raw display
$79$ \( T - 736 \) Copy content Toggle raw display
$83$ \( T - 856 \) Copy content Toggle raw display
$89$ \( T - 906 \) Copy content Toggle raw display
$97$ \( T - 46 \) Copy content Toggle raw display
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