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Properties

Label 2640.2.q.b
Level $2640$
Weight $2$
Character orbit 2640.q
Analytic conductor $21.081$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2640,2,Mod(1759,2640)]
 
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([1, 0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2640.1759");
 
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.q (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16x^{10} + 109x^{8} + 52x^{6} - 1646x^{4} - 4350x^{2} + 18225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + q^{9} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 4 q^{5} + 12 q^{9} - 4 q^{15} + 12 q^{25} + 12 q^{27} - 4 q^{45} + 4 q^{49} - 8 q^{55} - 24 q^{67} + 12 q^{75} + 12 q^{81} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 109x^{8} + 52x^{6} - 1646x^{4} - 4350x^{2} + 18225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 279737 \nu^{10} - 7521542 \nu^{8} - 116419838 \nu^{6} - 619039094 \nu^{4} + \cdots + 10082058420 ) / 2524934355 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 949328 \nu^{11} + 405803 \nu^{9} - 294912703 \nu^{7} - 2804363494 \nu^{5} + \cdots + 39896565690 \nu ) / 22724409195 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1044377 \nu^{10} - 23423642 \nu^{8} - 183321938 \nu^{6} - 478061084 \nu^{4} + \cdots + 1689501180 ) / 2524934355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1410134 \nu^{11} - 52546859 \nu^{9} - 515169671 \nu^{7} - 1764251378 \nu^{5} + \cdots + 54191938020 \nu ) / 22724409195 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 82904\nu^{10} + 1407138\nu^{8} + 8416024\nu^{6} - 8258240\nu^{4} - 254690400\nu^{2} - 268045638 ) / 168328957 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1576384 \nu^{10} - 11227564 \nu^{8} + 7473464 \nu^{6} + 710564882 \nu^{4} + \cdots - 5540649150 ) / 2524934355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4416289 \nu^{11} - 18682038 \nu^{10} + 68505484 \nu^{9} - 295090623 \nu^{8} + \cdots + 65654934075 ) / 22724409195 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4416289 \nu^{11} + 18682038 \nu^{10} + 68505484 \nu^{9} + 295090623 \nu^{8} + \cdots - 65654934075 ) / 22724409195 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4416289 \nu^{11} - 18682038 \nu^{10} - 68505484 \nu^{9} - 295090623 \nu^{8} + \cdots + 65654934075 ) / 22724409195 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2441377 \nu^{11} + 25531522 \nu^{9} + 123803533 \nu^{7} - 304411511 \nu^{5} + \cdots - 7370443560 \nu ) / 7574803065 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15774196 \nu^{11} - 18682038 \nu^{10} - 287001541 \nu^{9} - 295090623 \nu^{8} + \cdots + 65654934075 ) / 22724409195 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + 2\beta_{7} - 3\beta_{6} - 2\beta_{5} - 5\beta_{3} - 3\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - 3\beta_{10} - 5\beta_{9} - 4\beta_{8} + 2\beta_{7} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -26\beta_{9} - 26\beta_{7} + 32\beta_{6} + 43\beta_{3} + 45\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{11} + 31\beta_{10} + 32\beta_{9} - 21\beta_{8} - 44\beta_{7} - 32\beta_{4} + 15\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 161\beta_{9} + 161\beta_{7} - 191\beta_{6} + 45\beta_{5} - 195\beta_{3} - 385\beta _1 + 390 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 127\beta_{11} - 135\beta_{10} + 34\beta_{9} + 496\beta_{8} + 335\beta_{7} + 146\beta_{4} - 339\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -236\beta_{9} - 236\beta_{7} + 384\beta_{6} - 647\beta_{5} - 864\beta_{3} + 1688\beta _1 - 4986 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1336\beta_{11} - 1160\beta_{10} - 2257\beta_{9} - 4652\beta_{8} - 1059\beta_{7} + 263\beta_{4} + 3209\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -8784\beta_{9} - 8784\beta_{7} + 6843\beta_{6} + 4330\beta_{5} + 23383\beta_{3} + 5699\beta _1 + 34266 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5815 \beta_{11} + 27265 \beta_{10} + 25917 \beta_{9} + 21521 \beta_{8} - 10211 \beta_{7} + \cdots - 14498 \beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1759.1
0.763465 − 2.24782i
−0.763465 + 2.24782i
−0.763465 − 2.24782i
0.763465 + 2.24782i
1.65693 + 0.223979i
−1.65693 − 0.223979i
−1.65693 + 0.223979i
1.65693 − 0.223979i
1.08245 + 2.71977i
−1.08245 − 2.71977i
−1.08245 + 2.71977i
1.08245 − 2.71977i
0 1.00000 0 −2.19869 − 0.407132i 0 − 3.75045i 0 1.00000 0
1759.2 0 1.00000 0 −2.19869 − 0.407132i 0 3.75045i 0 1.00000 0
1759.3 0 1.00000 0 −2.19869 + 0.407132i 0 − 3.75045i 0 1.00000 0
1759.4 0 1.00000 0 −2.19869 + 0.407132i 0 3.75045i 0 1.00000 0
1759.5 0 1.00000 0 −0.713538 − 2.11917i 0 − 1.56376i 0 1.00000 0
1759.6 0 1.00000 0 −0.713538 − 2.11917i 0 1.56376i 0 1.00000 0
1759.7 0 1.00000 0 −0.713538 + 2.11917i 0 − 1.56376i 0 1.00000 0
1759.8 0 1.00000 0 −0.713538 + 2.11917i 0 1.56376i 0 1.00000 0
1759.9 0 1.00000 0 1.91223 − 1.15904i 0 − 1.86783i 0 1.00000 0
1759.10 0 1.00000 0 1.91223 − 1.15904i 0 1.86783i 0 1.00000 0
1759.11 0 1.00000 0 1.91223 + 1.15904i 0 − 1.86783i 0 1.00000 0
1759.12 0 1.00000 0 1.91223 + 1.15904i 0 1.86783i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1759.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
20.d odd 2 1 inner
220.g 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.q.b yes 12
4.b odd 2 1 2640.2.q.a ✓ 12
5.b even 2 1 2640.2.q.a ✓ 12
11.b odd 2 1 inner 2640.2.q.b yes 12
20.d odd 2 1 inner 2640.2.q.b yes 12
44.c even 2 1 2640.2.q.a ✓ 12
55.d odd 2 1 2640.2.q.a ✓ 12
220.g even 2 1 inner 2640.2.q.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2640.2.q.a ✓ 12 4.b odd 2 1
2640.2.q.a ✓ 12 5.b even 2 1
2640.2.q.a ✓ 12 44.c even 2 1
2640.2.q.a ✓ 12 55.d odd 2 1
2640.2.q.b yes 12 1.a even 1 1 trivial
2640.2.q.b yes 12 11.b odd 2 1 inner
2640.2.q.b yes 12 20.d odd 2 1 inner
2640.2.q.b yes 12 220.g 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 \(S_{2}^{\mathrm{new}}(2640, [\chi])\):

\( T_{7}^{6} + 20T_{7}^{4} + 92T_{7}^{2} + 120 \) Copy content Toggle raw display
\( T_{23}^{3} - 20T_{23} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 2 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 20 T^{4} + \cdots + 120)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} - 34 T^{4} + \cdots - 1080)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 56 T^{4} + \cdots - 5880)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 66 T^{4} + \cdots - 120)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 20 T - 24)^{4} \) Copy content Toggle raw display
$29$ \( (T^{6} + 130 T^{4} + \cdots + 75000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 88 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 112 T^{4} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 114 T^{4} + \cdots + 9720)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 204 T^{4} + \cdots + 9720)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 80 T + 192)^{4} \) Copy content Toggle raw display
$53$ \( (T^{6} + 204 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 144 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 160 T^{4} + \cdots + 7680)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} - 8 T - 56)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 156 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 122 T^{4} + \cdots - 120)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 162 T^{4} + \cdots - 1080)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 310 T^{4} + \cdots + 446520)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} + \cdots + 216)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 272 T^{4} + \cdots + 28224)^{2} \) Copy content Toggle raw display
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