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Properties

Label 2640.2.d.h
Level $2640$
Weight $2$
Character orbit 2640.d
Analytic conductor $21.081$
Analytic rank $0$
Dimension $6$
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] = [2640,2,Mod(529,2640)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2640.529");
 
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.d (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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) 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 165)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{4} - \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{4} - \beta_1) q^{7} - q^{9} - q^{11} + (\beta_{4} + \beta_1) q^{13} + (\beta_{3} - \beta_{2}) q^{15} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{17} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} - 2) q^{19} + ( - \beta_{2} + 1) q^{21} - 4 \beta_{4} q^{23} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \cdots - 1) q^{25}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} - 6 q^{9} - 6 q^{11} - 12 q^{19} + 8 q^{21} - 2 q^{25} - 16 q^{29} + 16 q^{31} - 12 q^{35} + 4 q^{39} + 16 q^{41} - 2 q^{45} + 10 q^{49} - 2 q^{55} + 16 q^{59} - 44 q^{61} + 12 q^{65} - 24 q^{69} + 24 q^{71} - 8 q^{75} - 20 q^{79} + 6 q^{81} - 4 q^{85} - 36 q^{89} + 16 q^{91} + 4 q^{95} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.403032 + 0.403032i
1.45161 + 1.45161i
−0.854638 − 0.854638i
0.403032 − 0.403032i
1.45161 − 1.45161i
−0.854638 + 0.854638i
0 − 1.00000i 0 −1.48119 + 1.67513i 0 1.19394i 0 −1.00000 0
529.2 0 − 1.00000i 0 0.311108 − 2.21432i 0 − 0.903212i 0 −1.00000 0
529.3 0 − 1.00000i 0 2.17009 + 0.539189i 0 3.70928i 0 −1.00000 0
529.4 0 1.00000i 0 −1.48119 − 1.67513i 0 − 1.19394i 0 −1.00000 0
529.5 0 1.00000i 0 0.311108 + 2.21432i 0 0.903212i 0 −1.00000 0
529.6 0 1.00000i 0 2.17009 − 0.539189i 0 − 3.70928i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.d.h 6
4.b odd 2 1 165.2.c.b ✓ 6
5.b even 2 1 inner 2640.2.d.h 6
12.b even 2 1 495.2.c.e 6
20.d odd 2 1 165.2.c.b ✓ 6
20.e even 4 1 825.2.a.j 3
20.e even 4 1 825.2.a.l 3
44.c even 2 1 1815.2.c.e 6
60.h even 2 1 495.2.c.e 6
60.l odd 4 1 2475.2.a.ba 3
60.l odd 4 1 2475.2.a.bc 3
220.g even 2 1 1815.2.c.e 6
220.i odd 4 1 9075.2.a.cg 3
220.i odd 4 1 9075.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b ✓ 6 4.b odd 2 1
165.2.c.b ✓ 6 20.d odd 2 1
495.2.c.e 6 12.b even 2 1
495.2.c.e 6 60.h even 2 1
825.2.a.j 3 20.e even 4 1
825.2.a.l 3 20.e even 4 1
1815.2.c.e 6 44.c even 2 1
1815.2.c.e 6 220.g even 2 1
2475.2.a.ba 3 60.l odd 4 1
2475.2.a.bc 3 60.l odd 4 1
2640.2.d.h 6 1.a even 1 1 trivial
2640.2.d.h 6 5.b even 2 1 inner
9075.2.a.cg 3 220.i odd 4 1
9075.2.a.ch 3 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 16T_{7}^{4} + 32T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 16 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{6} + 56 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 4 T - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} + \cdots + 928)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 144 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$47$ \( T^{6} + 160 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{6} + 192 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 64 T - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 22 T^{2} + 108 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 176 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} + \cdots + 944)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 188 T^{4} + \cdots + 150544 \) Copy content Toggle raw display
$79$ \( (T^{3} + 10 T^{2} + \cdots - 1720)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 220 T^{4} + \cdots + 364816 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + \cdots - 520)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 576 T^{4} + \cdots + 5914624 \) Copy content Toggle raw display
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