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Properties

Label 2640.2.d.j
Level $2640$
Weight $2$
Character orbit 2640.d
Analytic conductor $21.081$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 49x^{6} - 8x^{5} + 72x^{3} + 256x^{2} + 128x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1320)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{2} q^{5} - \beta_{9} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{2} q^{5} - \beta_{9} q^{7} - q^{9} - q^{11} + (\beta_{8} - \beta_{6} + \beta_{4}) q^{13} + \beta_{8} q^{15} + ( - \beta_{9} + \beta_{8} + \cdots - 2 \beta_1) q^{17}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 12 q^{19} + 4 q^{21} + 14 q^{25} - 24 q^{29} - 8 q^{31} - 16 q^{35} + 16 q^{39} - 16 q^{41} + 2 q^{45} - 26 q^{49} - 12 q^{51} + 2 q^{55} + 40 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{71} - 4 q^{75} + 60 q^{79} + 10 q^{81} + 52 q^{85} + 28 q^{89} - 20 q^{95} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 49x^{6} - 8x^{5} + 72x^{3} + 256x^{2} + 128x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 54999 \nu^{9} + 324324 \nu^{8} - 81760 \nu^{7} + 24444 \nu^{6} - 2839095 \nu^{5} + \cdots + 32166696 ) / 12684472 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 107559 \nu^{9} + 340038 \nu^{8} - 174424 \nu^{7} + 47804 \nu^{6} - 5421519 \nu^{5} + \cdots + 25494888 ) / 12684472 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 123273 \nu^{9} + 432702 \nu^{8} - 197784 \nu^{7} + 54788 \nu^{6} - 6232689 \nu^{5} + \cdots + 61866384 ) / 12684472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 104247 \nu^{9} - 26280 \nu^{8} + 7857 \nu^{7} - 46332 \nu^{6} + 5119783 \nu^{5} - 2125188 \nu^{4} + \cdots + 6907940 ) / 6342236 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 119205 \nu^{9} - 81794 \nu^{8} + 286180 \nu^{7} - 52980 \nu^{6} + 5172705 \nu^{5} - 5793430 \nu^{4} + \cdots - 4763984 ) / 6342236 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 106488 \nu^{9} + 23149 \nu^{8} + 125028 \nu^{7} + 47328 \nu^{6} - 4875854 \nu^{5} + \cdots - 3682456 ) / 3171118 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 280621 \nu^{9} - 69237 \nu^{8} - 33057 \nu^{7} + 227626 \nu^{6} + 13781201 \nu^{5} + \cdots + 16646604 ) / 6342236 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 306901 \nu^{9} - 77094 \nu^{8} + 13275 \nu^{7} + 215946 \nu^{6} + 15072413 \nu^{5} + \cdots + 19982508 ) / 6342236 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 593362 \nu^{9} - 148077 \nu^{8} - 9486 \nu^{7} + 88630 \nu^{6} + 29140550 \nu^{5} + \cdots + 37370424 ) / 6342236 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} - \beta_{8} - \beta_{7} - 6\beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} + 6\beta_{8} - 5\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 5\beta_{2} + 6\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{8} - 5\beta_{7} + 14\beta_{3} - 9\beta_{2} - 9\beta _1 - 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9 \beta_{9} + 20 \beta_{8} - 29 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} + \beta_{4} - 9 \beta_{3} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -94\beta_{9} + 65\beta_{8} + 65\beta_{7} + 174\beta_{4} + 21\beta_{2} - 21\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73 \beta_{9} - 246 \beta_{8} + 173 \beta_{7} + 65 \beta_{6} + 65 \beta_{5} - 31 \beta_{4} + 73 \beta_{3} + \cdots - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -69\beta_{8} + 69\beta_{7} - 8\beta_{5} - 630\beta_{3} + 441\beta_{2} + 449\beta _1 + 1078 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 569 \beta_{9} - 596 \beta_{8} + 1181 \beta_{7} - 441 \beta_{6} + 441 \beta_{5} + 415 \beta_{4} + \cdots - 415 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
−0.272353 + 0.272353i
1.71537 − 1.71537i
−1.87439 + 1.87439i
1.30586 − 1.30586i
−0.874489 + 0.874489i
−0.272353 − 0.272353i
1.71537 + 1.71537i
−1.87439 − 1.87439i
1.30586 + 1.30586i
−0.874489 − 0.874489i
0 − 1.00000i 0 −2.21942 + 0.272353i 0 − 4.79872i 0 −1.00000 0
529.2 0 − 1.00000i 0 −1.43440 − 1.71537i 0 − 0.264803i 0 −1.00000 0
529.3 0 − 1.00000i 0 −1.21929 + 1.87439i 0 4.68176i 0 −1.00000 0
529.4 0 − 1.00000i 0 1.81514 − 1.30586i 0 0.919829i 0 −1.00000 0
529.5 0 − 1.00000i 0 2.05798 + 0.874489i 0 1.46193i 0 −1.00000 0
529.6 0 1.00000i 0 −2.21942 − 0.272353i 0 4.79872i 0 −1.00000 0
529.7 0 1.00000i 0 −1.43440 + 1.71537i 0 0.264803i 0 −1.00000 0
529.8 0 1.00000i 0 −1.21929 − 1.87439i 0 − 4.68176i 0 −1.00000 0
529.9 0 1.00000i 0 1.81514 + 1.30586i 0 − 0.919829i 0 −1.00000 0
529.10 0 1.00000i 0 2.05798 − 0.874489i 0 − 1.46193i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.10
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.j 10
4.b odd 2 1 1320.2.d.d ✓ 10
5.b even 2 1 inner 2640.2.d.j 10
12.b even 2 1 3960.2.d.g 10
20.d odd 2 1 1320.2.d.d ✓ 10
20.e even 4 1 6600.2.a.by 5
20.e even 4 1 6600.2.a.ca 5
60.h even 2 1 3960.2.d.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.d.d ✓ 10 4.b odd 2 1
1320.2.d.d ✓ 10 20.d odd 2 1
2640.2.d.j 10 1.a even 1 1 trivial
2640.2.d.j 10 5.b even 2 1 inner
3960.2.d.g 10 12.b even 2 1
3960.2.d.g 10 60.h even 2 1
6600.2.a.by 5 20.e even 4 1
6600.2.a.ca 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 48T_{7}^{8} + 644T_{7}^{6} + 1632T_{7}^{4} + 1024T_{7}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 48 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 100 T^{8} + \cdots + 207936 \) Copy content Toggle raw display
$17$ \( T^{10} + 140 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{5} - 6 T^{4} + \cdots - 640)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 108 T^{8} + \cdots + 589824 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots - 1216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 4 T^{4} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 104 T^{8} + \cdots + 409600 \) Copy content Toggle raw display
$41$ \( (T^{5} + 8 T^{4} + \cdots - 768)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 212 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$47$ \( T^{10} + 220 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 748569600 \) Copy content Toggle raw display
$59$ \( (T^{5} - 20 T^{4} + \cdots + 28544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 6 T^{4} + \cdots - 2320)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 280 T^{8} + \cdots + 6553600 \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} + \cdots + 46240)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 536015104 \) Copy content Toggle raw display
$79$ \( (T^{5} - 30 T^{4} + \cdots + 1168)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 256 T^{8} + \cdots + 28217344 \) Copy content Toggle raw display
$89$ \( (T^{5} - 14 T^{4} + \cdots + 8224)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 504 T^{8} + \cdots + 1638400 \) Copy content Toggle raw display
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