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Properties

Label 1040.2.da.c
Level $1040$
Weight $2$
Character orbit 1040.da
Analytic conductor $8.304$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(641,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.641");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.da (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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + 2 \beta_{6} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + 2 \beta_{6} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{7} - 3 \beta_{5} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 6 q^{7} - 4 q^{9} - 12 q^{11} - 8 q^{13} + 6 q^{15} + 6 q^{17} + 6 q^{23} - 8 q^{25} - 4 q^{27} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{41} - 10 q^{43} - 4 q^{49} + 24 q^{53} + 24 q^{59} - 4 q^{61} - 24 q^{63} + 54 q^{67} - 24 q^{69} + 36 q^{71} - 2 q^{75} + 12 q^{77} + 16 q^{79} + 8 q^{81} + 18 q^{85} + 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.40994 + 0.109843i
0.665665 + 1.24775i
−1.27597 − 0.609843i
1.20036 − 0.747754i
1.40994 − 0.109843i
0.665665 − 1.24775i
−1.27597 + 0.609843i
1.20036 + 0.747754i
0 −1.16612 + 2.01978i 0 − 1.00000i 0 −0.346241 + 0.199902i 0 −1.21969 − 2.11256i 0
641.2 0 −0.0473938 + 0.0820885i 0 1.00000i 0 −0.716063 + 0.413419i 0 1.49551 + 2.59030i 0
641.3 0 0.800098 − 1.38581i 0 − 1.00000i 0 −3.75184 + 2.16612i 0 0.219687 + 0.380509i 0
641.4 0 1.41342 − 2.44811i 0 1.00000i 0 1.81414 − 1.04739i 0 −2.49551 − 4.32235i 0
881.1 0 −1.16612 − 2.01978i 0 1.00000i 0 −0.346241 − 0.199902i 0 −1.21969 + 2.11256i 0
881.2 0 −0.0473938 − 0.0820885i 0 − 1.00000i 0 −0.716063 − 0.413419i 0 1.49551 − 2.59030i 0
881.3 0 0.800098 + 1.38581i 0 1.00000i 0 −3.75184 − 2.16612i 0 0.219687 − 0.380509i 0
881.4 0 1.41342 + 2.44811i 0 − 1.00000i 0 1.81414 + 1.04739i 0 −2.49551 + 4.32235i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.da.c 8
4.b odd 2 1 260.2.x.a ✓ 8
12.b even 2 1 2340.2.dj.d 8
13.e even 6 1 inner 1040.2.da.c 8
20.d odd 2 1 1300.2.y.b 8
20.e even 4 1 1300.2.ba.b 8
20.e even 4 1 1300.2.ba.c 8
52.i odd 6 1 260.2.x.a ✓ 8
52.i odd 6 1 3380.2.f.i 8
52.j odd 6 1 3380.2.f.i 8
52.l even 12 1 3380.2.a.p 4
52.l even 12 1 3380.2.a.q 4
156.r even 6 1 2340.2.dj.d 8
260.w odd 6 1 1300.2.y.b 8
260.bg even 12 1 1300.2.ba.b 8
260.bg even 12 1 1300.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a ✓ 8 4.b odd 2 1
260.2.x.a ✓ 8 52.i odd 6 1
1040.2.da.c 8 1.a even 1 1 trivial
1040.2.da.c 8 13.e even 6 1 inner
1300.2.y.b 8 20.d odd 2 1
1300.2.y.b 8 260.w odd 6 1
1300.2.ba.b 8 20.e even 4 1
1300.2.ba.b 8 260.bg even 12 1
1300.2.ba.c 8 20.e even 4 1
1300.2.ba.c 8 260.bg even 12 1
2340.2.dj.d 8 12.b even 2 1
2340.2.dj.d 8 156.r even 6 1
3380.2.a.p 4 52.l even 12 1
3380.2.a.q 4 52.l even 12 1
3380.2.f.i 8 52.i odd 6 1
3380.2.f.i 8 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 10T_{3}^{6} - 8T_{3}^{5} + 55T_{3}^{4} - 56T_{3}^{3} + 106T_{3}^{2} + 10T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{8} - 30 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{8} + 42 T^{6} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 2112)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 42849 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 10 T^{7} + \cdots + 5031049 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots - 624)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 24 T^{7} + \cdots + 558009 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 942841 \) Copy content Toggle raw display
$67$ \( T^{8} - 54 T^{7} + \cdots + 1083681 \) Copy content Toggle raw display
$71$ \( T^{8} - 36 T^{7} + \cdots + 45198729 \) Copy content Toggle raw display
$73$ \( T^{8} + 264 T^{6} + \cdots + 2509056 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots - 368)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 480 T^{6} + \cdots + 331776 \) Copy content Toggle raw display
$89$ \( T^{8} + 24 T^{7} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{8} + 30 T^{7} + \cdots + 12981609 \) Copy content Toggle raw display
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