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Properties

Label 3240.2.q.bh
Level $3240$
Weight $2$
Character orbit 3240.q
Analytic conductor $25.872$
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] = [3240,2,Mod(1081,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1081");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), 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: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.3887771904.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} - 2 q^{7} + 4 q^{11} + 4 q^{17} + 10 q^{23} - 4 q^{25} - 4 q^{29} - 14 q^{31} - 4 q^{35} + 28 q^{37} + 4 q^{41} - 22 q^{43} - 10 q^{49} + 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} - 56 q^{71} + 4 q^{73} - 30 q^{77} - 16 q^{79} + 6 q^{83} + 2 q^{85} + 16 q^{89} + 60 q^{91} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 26\nu^{5} + 193\nu^{3} + 360\nu + 96 ) / 192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 20\nu^{4} - 91\nu^{2} - 48 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 26\nu^{5} + 104\nu^{4} + 241\nu^{3} + 772\nu^{2} + 984\nu + 1344 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 4\nu^{6} - 106\nu^{5} + 80\nu^{4} - 605\nu^{3} + 364\nu^{2} - 888\nu + 192 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 20\nu^{5} + 40\nu^{4} - 91\nu^{3} + 206\nu^{2} - 48\nu + 240 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 26\nu^{4} - 193\nu^{2} - 360 ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 20\nu^{4} + 103\nu^{2} + 132 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} - 7\beta_{6} + 26\beta_{5} - 26\beta_{4} - 14\beta_{3} - 13\beta_{2} - 12\beta _1 + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{7} - 4\beta_{6} - 30\beta_{2} + 67 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 157\beta_{7} + 67\beta_{6} - 314\beta_{5} + 338\beta_{4} + 134\beta_{3} + 169\beta_{2} + 300\beta _1 - 240 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 249\beta_{7} + 80\beta_{6} + 394\beta_{2} - 751 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1933 \beta_{7} - 751 \beta_{6} + 3866 \beta_{5} - 4490 \beta_{4} - 1502 \beta_{3} - 2245 \beta_{2} + \cdots + 3636 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1081.1
− 1.16910i
2.61675i
2.16910i
− 3.61675i
1.16910i
− 2.61675i
− 2.16910i
3.61675i
0 0 0 0.500000 − 0.866025i 0 −1.95058 − 3.37850i 0 0 0
1081.2 0 0 0 0.500000 − 0.866025i 0 −0.942347 − 1.63219i 0 0 0
1081.3 0 0 0 0.500000 − 0.866025i 0 −0.281475 − 0.487529i 0 0 0
1081.4 0 0 0 0.500000 − 0.866025i 0 2.17440 + 3.76617i 0 0 0
2161.1 0 0 0 0.500000 + 0.866025i 0 −1.95058 + 3.37850i 0 0 0
2161.2 0 0 0 0.500000 + 0.866025i 0 −0.942347 + 1.63219i 0 0 0
2161.3 0 0 0 0.500000 + 0.866025i 0 −0.281475 + 0.487529i 0 0 0
2161.4 0 0 0 0.500000 + 0.866025i 0 2.17440 − 3.76617i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1081.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.2.q.bh 8
3.b odd 2 1 3240.2.q.bg 8
9.c even 3 1 3240.2.a.t ✓ 4
9.c even 3 1 inner 3240.2.q.bh 8
9.d odd 6 1 3240.2.a.v yes 4
9.d odd 6 1 3240.2.q.bg 8
36.f odd 6 1 6480.2.a.by 4
36.h even 6 1 6480.2.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.t ✓ 4 9.c even 3 1
3240.2.a.v yes 4 9.d odd 6 1
3240.2.q.bg 8 3.b odd 2 1
3240.2.q.bg 8 9.d odd 6 1
3240.2.q.bh 8 1.a even 1 1 trivial
3240.2.q.bh 8 9.c even 3 1 inner
6480.2.a.by 4 36.f odd 6 1
6480.2.a.ca 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{8} + 2T_{7}^{7} + 21T_{7}^{6} + 50T_{7}^{5} + 391T_{7}^{4} + 786T_{7}^{3} + 1458T_{7}^{2} + 756T_{7} + 324 \) Copy content Toggle raw display
\( T_{11}^{8} - 4T_{11}^{7} + 51T_{11}^{6} - 148T_{11}^{5} + 1909T_{11}^{4} - 5904T_{11}^{3} + 16956T_{11}^{2} - 15552T_{11} + 11664 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} - 38T_{17}^{2} + 120T_{17} - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$13$ \( T^{8} + 39 T^{6} + \cdots + 46656 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 38 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 50 T^{2} + 433)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$31$ \( T^{8} + 14 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots + 4761 \) Copy content Toggle raw display
$43$ \( T^{8} + 22 T^{7} + \cdots + 45805824 \) Copy content Toggle raw display
$47$ \( T^{8} + 113 T^{6} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + \cdots - 498)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 1327104 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + \cdots + 358875136 \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{3} + \cdots - 16428)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + \cdots - 552)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} + 60 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 6 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots + 11344)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
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