Properties

Label 676.2.a.h
Level $676$
Weight $2$
Character orbit 676.a
Self dual yes
Analytic conductor $5.398$
Analytic rank $0$
Dimension $3$
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] = [676,2,Mod(1,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("676.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.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: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} + \beta_1 + 2) q^{5} + (\beta_{2} - 3 \beta_1 + 1) q^{7} + (2 \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} + \beta_1 + 2) q^{5} + (\beta_{2} - 3 \beta_1 + 1) q^{7} + (2 \beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{2} - \beta_1 + 2) q^{11} + ( - 4 \beta_{2} - \beta_1 - 1) q^{15} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{17} + ( - 3 \beta_{2} + 4 \beta_1 - 1) q^{19} + (5 \beta_{2} - 2 \beta_1 + 7) q^{21} + (4 \beta_{2} - 2 \beta_1 + 3) q^{23} + ( - 6 \beta_{2} + 5 \beta_1) q^{25} + ( - \beta_{2} - \beta_1 - 7) q^{27} + (\beta_{2} - 3 \beta_1 + 1) q^{29} + (3 \beta_{2} - \beta_1 + 3) q^{31} + ( - \beta_1 + 5) q^{33} + (3 \beta_{2} - 6 \beta_1 - 1) q^{35} + ( - \beta_{2} + 6 \beta_1 - 1) q^{37} + (2 \beta_{2} + 9) q^{41} + (7 \beta_{2} + 4) q^{43} + (6 \beta_{2} + 2 \beta_1 + 5) q^{45} + (3 \beta_{2} + \beta_1 - 2) q^{47} + (4 \beta_{2} - 5 \beta_1 + 7) q^{49} + ( - \beta_{2} + 7 \beta_1 - 5) q^{51} + ( - \beta_{2} - 4 \beta_1) q^{53} + ( - 6 \beta_{2} + \beta_1 + 3) q^{55} + ( - 7 \beta_{2} + 4 \beta_1 - 6) q^{57} + (6 \beta_{2} + 5) q^{59} + ( - 5 \beta_{2} + 6 \beta_1 - 1) q^{61} + ( - 6 \beta_{2} - 3 \beta_1 - 7) q^{63} + (3 \beta_{2} - 2) q^{67} + (\beta_{2} - 7 \beta_1 - 2) q^{69} + ( - 2 \beta_{2} + \beta_1 - 4) q^{71} + ( - 3 \beta_{2} - 2 \beta_1) q^{73} + ( - 10 \beta_{2} + 6 \beta_1 - 3) q^{75} + (7 \beta_{2} - 8 \beta_1 + 9) q^{77} + ( - 6 \beta_{2} + 9 \beta_1 - 2) q^{79} + (3 \beta_{2} + 5 \beta_1 - 1) q^{81} + ( - \beta_1 + 4) q^{83} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{85} + (5 \beta_{2} - 2 \beta_1 + 7) q^{87} + (4 \beta_{2} - \beta_1 + 10) q^{89} + ( - \beta_{2} - 6 \beta_1 - 3) q^{93} + ( - 11 \beta_{2} + 10 \beta_1 + 2) q^{95} + ( - \beta_{2} - 5 \beta_1 - 3) q^{97} + ( - 2 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{5} - q^{7} + 5 q^{9} + 6 q^{11} - 10 q^{17} + 4 q^{19} + 14 q^{21} + 3 q^{23} + 11 q^{25} - 21 q^{27} - q^{29} + 5 q^{31} + 14 q^{33} - 12 q^{35} + 4 q^{37} + 25 q^{41} + 5 q^{43} + 11 q^{45} - 8 q^{47} + 12 q^{49} - 7 q^{51} - 3 q^{53} + 16 q^{55} - 7 q^{57} + 9 q^{59} + 8 q^{61} - 18 q^{63} - 9 q^{67} - 14 q^{69} - 9 q^{71} + q^{73} + 7 q^{75} + 12 q^{77} + 9 q^{79} - q^{81} + 11 q^{83} - 15 q^{85} + 14 q^{87} + 25 q^{89} - 14 q^{93} + 27 q^{95} - 13 q^{97} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
0 −3.04892 0 2.55496 0 −3.15883 0 6.29590 0
1.2 0 1.35690 0 4.24698 0 −2.13706 0 −1.15883 0
1.3 0 1.69202 0 1.19806 0 4.29590 0 −0.137063 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 676.2.a.h yes 3
3.b odd 2 1 6084.2.a.x 3
4.b odd 2 1 2704.2.a.y 3
13.b even 2 1 676.2.a.g 3
13.c even 3 2 676.2.e.g 6
13.d odd 4 2 676.2.d.e 6
13.e even 6 2 676.2.e.f 6
13.f odd 12 4 676.2.h.e 12
39.d odd 2 1 6084.2.a.bc 3
39.f even 4 2 6084.2.b.p 6
52.b odd 2 1 2704.2.a.x 3
52.f even 4 2 2704.2.f.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
676.2.a.g 3 13.b even 2 1
676.2.a.h yes 3 1.a even 1 1 trivial
676.2.d.e 6 13.d odd 4 2
676.2.e.f 6 13.e even 6 2
676.2.e.g 6 13.c even 3 2
676.2.h.e 12 13.f odd 12 4
2704.2.a.x 3 52.b odd 2 1
2704.2.a.y 3 4.b odd 2 1
2704.2.f.n 6 52.f even 4 2
6084.2.a.x 3 3.b odd 2 1
6084.2.a.bc 3 39.d odd 2 1
6084.2.b.p 6 39.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(676))\):

\( T_{3}^{3} - 7T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{3} - 8T_{5}^{2} + 19T_{5} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 7T + 7 \) Copy content Toggle raw display
$5$ \( T^{3} - 8 T^{2} + 19 T - 13 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T - 29 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + 5 T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 10 T^{2} + 17 T + 1 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 25 T + 71 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 25 T + 83 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 16 T - 29 \) Copy content Toggle raw display
$31$ \( T^{3} - 5 T^{2} - 8 T + 41 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} - 67 T + 239 \) Copy content Toggle raw display
$41$ \( T^{3} - 25 T^{2} + 199 T - 503 \) Copy content Toggle raw display
$43$ \( T^{3} - 5 T^{2} - 106 T + 97 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 9 T - 113 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} - 46 T + 1 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 57 T + 169 \) Copy content Toggle raw display
$61$ \( T^{3} - 8 T^{2} - 51 T + 239 \) Copy content Toggle raw display
$67$ \( T^{3} + 9 T^{2} + 6 T - 43 \) Copy content Toggle raw display
$71$ \( T^{3} + 9 T^{2} + 20 T - 1 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 44 T + 127 \) Copy content Toggle raw display
$79$ \( T^{3} - 9 T^{2} - 120 T + 911 \) Copy content Toggle raw display
$83$ \( T^{3} - 11 T^{2} + 38 T - 41 \) Copy content Toggle raw display
$89$ \( T^{3} - 25 T^{2} + 178 T - 293 \) Copy content Toggle raw display
$97$ \( T^{3} + 13 T^{2} - 16 T - 167 \) Copy content Toggle raw display
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