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Properties

Label 2640.2.f.b
Level $2640$
Weight $2$
Character orbit 2640.f
Analytic conductor $21.081$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2640,2,Mod(1121,2640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("2640.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.f (of order \(2\), degree \(1\), 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: \(21.0805061336\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2051727616.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 330)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{5} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_1 + 2) q^{11}+ \cdots + ( - \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{9} + 6 q^{11} - 4 q^{15} - 4 q^{17} + 8 q^{21} - 8 q^{25} + 22 q^{27} + 4 q^{29} + 4 q^{31} + 18 q^{33} - 12 q^{37} - 8 q^{39} + 16 q^{41} - 4 q^{49} - 28 q^{51} + 6 q^{55} - 14 q^{57}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 9\nu^{5} + 14\nu^{4} + 27\nu^{3} + 18\nu^{2} + 30\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 4\nu^{4} + 47\nu^{3} + 20\nu^{2} + 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 9\nu^{5} - 14\nu^{4} - 19\nu^{3} - 18\nu^{2} + 2\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 14\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} + 9\nu^{4} + 14\nu^{3} + 23\nu^{2} + 22\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} + 4\nu^{4} - 47\nu^{3} + 20\nu^{2} - 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} - 9\nu^{4} + 14\nu^{3} - 23\nu^{2} + 22\nu - 10 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} + 7\beta_{6} - 5\beta_{5} - 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 5\beta _1 + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} + 17\beta_{6} + 19\beta_{5} + 17\beta_{4} - 31\beta_{3} - 17\beta_{2} - 31\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{7} - 20\beta_{6} + 13\beta_{5} + 11\beta_{4} + 11\beta_{3} - 20\beta_{2} - 11\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -93\beta_{7} - 75\beta_{6} - 93\beta_{5} - 67\beta_{4} + 155\beta_{3} + 75\beta_{2} + 155\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
1.18994i
− 1.18994i
0.356500i
− 0.356500i
− 2.25619i
2.25619i
2.08963i
− 2.08963i
0 −1.38699 − 1.03743i 0 1.00000i 0 0.394100i 0 0.847487 + 2.87781i 0
1121.2 0 −1.38699 + 1.03743i 0 − 1.00000i 0 − 0.394100i 0 0.847487 − 2.87781i 0
1121.3 0 −1.25820 − 1.19035i 0 − 1.00000i 0 − 3.22941i 0 0.166154 + 2.99540i 0
1121.4 0 −1.25820 + 1.19035i 0 1.00000i 0 3.22941i 0 0.166154 − 2.99540i 0
1121.5 0 −0.0828988 − 1.73007i 0 − 1.00000i 0 4.34658i 0 −2.98626 + 0.286841i 0
1121.6 0 −0.0828988 + 1.73007i 0 1.00000i 0 − 4.34658i 0 −2.98626 − 0.286841i 0
1121.7 0 1.72809 − 0.117016i 0 − 1.00000i 0 − 0.723074i 0 2.97261 − 0.404430i 0
1121.8 0 1.72809 + 0.117016i 0 1.00000i 0 0.723074i 0 2.97261 + 0.404430i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2640.2.f.b 8
3.b odd 2 1 2640.2.f.a 8
4.b odd 2 1 330.2.d.a ✓ 8
11.b odd 2 1 2640.2.f.a 8
12.b even 2 1 330.2.d.b yes 8
20.d odd 2 1 1650.2.d.f 8
20.e even 4 1 1650.2.f.c 8
20.e even 4 1 1650.2.f.e 8
33.d even 2 1 inner 2640.2.f.b 8
44.c even 2 1 330.2.d.b yes 8
60.h even 2 1 1650.2.d.c 8
60.l odd 4 1 1650.2.f.d 8
60.l odd 4 1 1650.2.f.f 8
132.d odd 2 1 330.2.d.a ✓ 8
220.g even 2 1 1650.2.d.c 8
220.i odd 4 1 1650.2.f.d 8
220.i odd 4 1 1650.2.f.f 8
660.g odd 2 1 1650.2.d.f 8
660.q even 4 1 1650.2.f.c 8
660.q even 4 1 1650.2.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.d.a ✓ 8 4.b odd 2 1
330.2.d.a ✓ 8 132.d odd 2 1
330.2.d.b yes 8 12.b even 2 1
330.2.d.b yes 8 44.c even 2 1
1650.2.d.c 8 60.h even 2 1
1650.2.d.c 8 220.g even 2 1
1650.2.d.f 8 20.d odd 2 1
1650.2.d.f 8 660.g odd 2 1
1650.2.f.c 8 20.e even 4 1
1650.2.f.c 8 660.q even 4 1
1650.2.f.d 8 60.l odd 4 1
1650.2.f.d 8 220.i odd 4 1
1650.2.f.e 8 20.e even 4 1
1650.2.f.e 8 660.q even 4 1
1650.2.f.f 8 60.l odd 4 1
1650.2.f.f 8 220.i odd 4 1
2640.2.f.a 8 3.b odd 2 1
2640.2.f.a 8 11.b odd 2 1
2640.2.f.b 8 1.a even 1 1 trivial
2640.2.f.b 8 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\):

\( T_{7}^{8} + 30T_{7}^{6} + 217T_{7}^{4} + 136T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 31T_{17}^{2} - 104T_{17} - 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 30 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 31 T^{2} + \cdots - 68)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 98 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$23$ \( T^{8} + 104 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots + 452)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + \cdots + 2000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6 T^{3} + \cdots - 1388)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 12 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 160 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{8} + 256 T^{6} + \cdots + 2715904 \) Copy content Toggle raw display
$53$ \( T^{8} + 58 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$59$ \( T^{8} + 188 T^{6} + \cdots + 640000 \) Copy content Toggle raw display
$61$ \( T^{8} + 310 T^{6} + \cdots + 712336 \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots + 3824)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 170 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
$73$ \( T^{8} + 380 T^{6} + \cdots + 1183744 \) Copy content Toggle raw display
$79$ \( T^{8} + 316 T^{6} + \cdots + 173056 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + \cdots - 2944)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 262 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 11408)^{2} \) Copy content Toggle raw display
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