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Properties

Label 2695.1.q.g
Level $2695$
Weight $1$
Character orbit 2695.q
Analytic conductor $1.345$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2695,1,Mod(2419,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2695.2419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2695.q (of order \(6\), degree \(2\), 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: \(1.34498020905\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1037575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{7} - \zeta_{24}) q^{3} + \zeta_{24}^{4} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{24}^{7} - \zeta_{24}) q^{3} + \zeta_{24}^{4} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24}^{8} q^{9} - \zeta_{24}^{4} q^{11} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{12} + (\zeta_{24}^{6} + 1) q^{15} + \zeta_{24}^{8} q^{16} - \zeta_{24}^{9} q^{20} + \zeta_{24}^{2} q^{23} + \zeta_{24}^{10} q^{25} - \zeta_{24}^{3} q^{27} + (\zeta_{24}^{7} + \zeta_{24}) q^{31} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{33} + q^{36} - \zeta_{24}^{8} q^{44} - \zeta_{24} q^{45} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{47} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{48} - \zeta_{24}^{10} q^{53} + \zeta_{24}^{9} q^{55} + (\zeta_{24}^{7} + \zeta_{24}) q^{59} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{60} - q^{64} + \zeta_{24}^{10} q^{67} + (2 \zeta_{24}^{9} - 2 \zeta_{24}^{3}) q^{69} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{75} + \zeta_{24} q^{80} + \zeta_{24}^{4} q^{81} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{89} + 2 \zeta_{24}^{6} q^{92} - \zeta_{24}^{2} q^{93} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{11} + 8 q^{15} - 4 q^{16} + 8 q^{36} + 4 q^{44} + 4 q^{60} - 8 q^{64} + 4 q^{81} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2695\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1816\) \(2157\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2419.1
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0 −1.22474 − 0.707107i 0.500000 − 0.866025i −0.965926 − 0.258819i 0 0 0 0.500000 + 0.866025i 0
2419.2 0 −1.22474 − 0.707107i 0.500000 − 0.866025i −0.258819 + 0.965926i 0 0 0 0.500000 + 0.866025i 0
2419.3 0 1.22474 + 0.707107i 0.500000 − 0.866025i 0.258819 − 0.965926i 0 0 0 0.500000 + 0.866025i 0
2419.4 0 1.22474 + 0.707107i 0.500000 − 0.866025i 0.965926 + 0.258819i 0 0 0 0.500000 + 0.866025i 0
2529.1 0 −1.22474 + 0.707107i 0.500000 + 0.866025i −0.965926 + 0.258819i 0 0 0 0.500000 − 0.866025i 0
2529.2 0 −1.22474 + 0.707107i 0.500000 + 0.866025i −0.258819 − 0.965926i 0 0 0 0.500000 − 0.866025i 0
2529.3 0 1.22474 − 0.707107i 0.500000 + 0.866025i 0.258819 + 0.965926i 0 0 0 0.500000 − 0.866025i 0
2529.4 0 1.22474 − 0.707107i 0.500000 + 0.866025i 0.965926 − 0.258819i 0 0 0 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2419.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
55.d odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner
385.h even 2 1 inner
385.o even 6 1 inner
385.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2695.1.q.g 8
5.b even 2 1 inner 2695.1.q.g 8
7.b odd 2 1 inner 2695.1.q.g 8
7.c even 3 1 2695.1.g.i ✓ 4
7.c even 3 1 inner 2695.1.q.g 8
7.d odd 6 1 2695.1.g.i ✓ 4
7.d odd 6 1 inner 2695.1.q.g 8
11.b odd 2 1 CM 2695.1.q.g 8
35.c odd 2 1 inner 2695.1.q.g 8
35.i odd 6 1 2695.1.g.i ✓ 4
35.i odd 6 1 inner 2695.1.q.g 8
35.j even 6 1 2695.1.g.i ✓ 4
35.j even 6 1 inner 2695.1.q.g 8
55.d odd 2 1 inner 2695.1.q.g 8
77.b even 2 1 inner 2695.1.q.g 8
77.h odd 6 1 2695.1.g.i ✓ 4
77.h odd 6 1 inner 2695.1.q.g 8
77.i even 6 1 2695.1.g.i ✓ 4
77.i even 6 1 inner 2695.1.q.g 8
385.h even 2 1 inner 2695.1.q.g 8
385.o even 6 1 2695.1.g.i ✓ 4
385.o even 6 1 inner 2695.1.q.g 8
385.q odd 6 1 2695.1.g.i ✓ 4
385.q odd 6 1 inner 2695.1.q.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2695.1.g.i ✓ 4 7.c even 3 1
2695.1.g.i ✓ 4 7.d odd 6 1
2695.1.g.i ✓ 4 35.i odd 6 1
2695.1.g.i ✓ 4 35.j even 6 1
2695.1.g.i ✓ 4 77.h odd 6 1
2695.1.g.i ✓ 4 77.i even 6 1
2695.1.g.i ✓ 4 385.o even 6 1
2695.1.g.i ✓ 4 385.q odd 6 1
2695.1.q.g 8 1.a even 1 1 trivial
2695.1.q.g 8 5.b even 2 1 inner
2695.1.q.g 8 7.b odd 2 1 inner
2695.1.q.g 8 7.c even 3 1 inner
2695.1.q.g 8 7.d odd 6 1 inner
2695.1.q.g 8 11.b odd 2 1 CM
2695.1.q.g 8 35.c odd 2 1 inner
2695.1.q.g 8 35.i odd 6 1 inner
2695.1.q.g 8 35.j even 6 1 inner
2695.1.q.g 8 55.d odd 2 1 inner
2695.1.q.g 8 77.b even 2 1 inner
2695.1.q.g 8 77.h odd 6 1 inner
2695.1.q.g 8 77.i even 6 1 inner
2695.1.q.g 8 385.h even 2 1 inner
2695.1.q.g 8 385.o even 6 1 inner
2695.1.q.g 8 385.q odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2695, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{31}^{4} + 2T_{31}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
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