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Properties

Label 676.2.l.f
Level $676$
Weight $2$
Character orbit 676.l
Analytic conductor $5.398$
Analytic rank $0$
Dimension $4$
CM discriminant -52
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [676,2,Mod(19,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.l (of order \(12\), degree \(4\), not minimal)

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: no
Analytic conductor: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 3 \zeta_{12}^{3} + \cdots + 3 \zeta_{12}) q^{11} + \cdots + (9 \zeta_{12}^{3} - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{7} + 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{7} + 8 q^{8} - 6 q^{9} + 6 q^{11} + 8 q^{14} + 8 q^{16} - 12 q^{18} + 10 q^{19} - 12 q^{22} + 4 q^{28} + 16 q^{29} + 28 q^{31} - 8 q^{32} + 16 q^{34} - 24 q^{44} - 36 q^{47} - 10 q^{50} + 8 q^{53} - 16 q^{58} + 2 q^{59} - 12 q^{61} + 6 q^{63} - 22 q^{67} + 16 q^{68} + 10 q^{71} - 12 q^{72} - 20 q^{76} - 18 q^{81} + 28 q^{83} - 36 q^{94} - 10 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
1.36603 − 0.366025i 0 1.73205 − 1.00000i 0 0 1.36603 + 0.366025i 2.00000 − 2.00000i −1.50000 − 2.59808i 0
319.1 −0.366025 − 1.36603i 0 −1.73205 + 1.00000i 0 0 −0.366025 + 1.36603i 2.00000 + 2.00000i −1.50000 − 2.59808i 0
427.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i 0 0 1.36603 − 0.366025i 2.00000 + 2.00000i −1.50000 + 2.59808i 0
587.1 −0.366025 + 1.36603i 0 −1.73205 − 1.00000i 0 0 −0.366025 − 1.36603i 2.00000 − 2.00000i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
52.f even 4 1 inner
52.i odd 6 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.l.f 4
4.b odd 2 1 676.2.l.b 4
13.b even 2 1 676.2.l.b 4
13.c even 3 1 676.2.f.a ✓ 2
13.c even 3 1 inner 676.2.l.f 4
13.d odd 4 1 676.2.l.b 4
13.d odd 4 1 inner 676.2.l.f 4
13.e even 6 1 676.2.f.c yes 2
13.e even 6 1 676.2.l.b 4
13.f odd 12 1 676.2.f.a ✓ 2
13.f odd 12 1 676.2.f.c yes 2
13.f odd 12 1 676.2.l.b 4
13.f odd 12 1 inner 676.2.l.f 4
52.b odd 2 1 CM 676.2.l.f 4
52.f even 4 1 676.2.l.b 4
52.f even 4 1 inner 676.2.l.f 4
52.i odd 6 1 676.2.f.a ✓ 2
52.i odd 6 1 inner 676.2.l.f 4
52.j odd 6 1 676.2.f.c yes 2
52.j odd 6 1 676.2.l.b 4
52.l even 12 1 676.2.f.a ✓ 2
52.l even 12 1 676.2.f.c yes 2
52.l even 12 1 676.2.l.b 4
52.l even 12 1 inner 676.2.l.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
676.2.f.a ✓ 2 13.c even 3 1
676.2.f.a ✓ 2 13.f odd 12 1
676.2.f.a ✓ 2 52.i odd 6 1
676.2.f.a ✓ 2 52.l even 12 1
676.2.f.c yes 2 13.e even 6 1
676.2.f.c yes 2 13.f odd 12 1
676.2.f.c yes 2 52.j odd 6 1
676.2.f.c yes 2 52.l even 12 1
676.2.l.b 4 4.b odd 2 1
676.2.l.b 4 13.b even 2 1
676.2.l.b 4 13.d odd 4 1
676.2.l.b 4 13.e even 6 1
676.2.l.b 4 13.f odd 12 1
676.2.l.b 4 52.f even 4 1
676.2.l.b 4 52.j odd 6 1
676.2.l.b 4 52.l even 12 1
676.2.l.f 4 1.a even 1 1 trivial
676.2.l.f 4 13.c even 3 1 inner
676.2.l.f 4 13.d odd 4 1 inner
676.2.l.f 4 13.f odd 12 1 inner
676.2.l.f 4 52.b odd 2 1 CM
676.2.l.f 4 52.f even 4 1 inner
676.2.l.f 4 52.i odd 6 1 inner
676.2.l.f 4 52.l even 12 1 inner

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 16T_{17}^{2} + 256 \) Copy content Toggle raw display
\( T_{37} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + \cdots + 2500 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 18 T + 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots + 58564 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 2500 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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