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Properties

Label 6440.2.a.z
Level $6440$
Weight $2$
Character orbit 6440.a
Self dual yes
Analytic conductor $51.424$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6440,2,Mod(1,6440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6440.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6440 = 2^{3} \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6440.a (trivial)

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: yes
Analytic conductor: \(51.4236589017\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 15x^{5} + 10x^{4} + 55x^{3} - 10x^{2} - 60x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{6} - \beta_1) q^{11} + (\beta_{3} - \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{6} - \beta_1 - 2) q^{17} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{19}+ \cdots + (\beta_{6} - 2 \beta_{5} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9} - 4 q^{11} + 2 q^{13} - q^{15} - 12 q^{17} - 12 q^{19} + q^{21} - 7 q^{23} + 7 q^{25} + 10 q^{27} + 13 q^{29} - 14 q^{31} - 25 q^{33} - 7 q^{35} - 29 q^{37} - 24 q^{39} + 5 q^{41} + 2 q^{43} - 10 q^{45} - 3 q^{47} + 7 q^{49} - 39 q^{51} - 3 q^{53} + 4 q^{55} + q^{57} + 29 q^{59} - 15 q^{61} + 10 q^{63} - 2 q^{65} + 7 q^{67} - q^{69} - 38 q^{71} - 15 q^{73} + q^{75} - 4 q^{77} - 37 q^{79} + 35 q^{81} + 7 q^{83} + 12 q^{85} - 3 q^{87} - 13 q^{89} + 2 q^{91} - 9 q^{93} + 12 q^{95} - 44 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 15x^{5} + 10x^{4} + 55x^{3} - 10x^{2} - 60x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 3\nu^{5} + 24\nu^{4} - 32\nu^{3} - 40\nu^{2} + 49\nu + 10 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} + 21\nu^{5} + 51\nu^{4} - 251\nu^{3} - 28\nu^{2} + 442\nu + 25 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -10\nu^{6} + 15\nu^{5} + 129\nu^{4} - 151\nu^{3} - 299\nu^{2} + 155\nu + 131 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{6} - 12\nu^{5} - 114\nu^{4} + 137\nu^{3} + 358\nu^{2} - 259\nu - 256 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{5} - 2\beta_{3} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - 3\beta_{3} + 11\beta_{2} + \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{6} + 26\beta_{5} + 2\beta_{4} - 29\beta_{3} + 2\beta_{2} + 91\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} + 19\beta_{5} + 3\beta_{4} - 52\beta_{3} + 115\beta_{2} + 29\beta _1 + 328 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.11016
−1.45344
−1.08917
−0.306632
1.50655
2.09494
3.35792
0 −3.11016 0 −1.00000 0 1.00000 0 6.67310 0
1.2 0 −1.45344 0 −1.00000 0 1.00000 0 −0.887499 0
1.3 0 −1.08917 0 −1.00000 0 1.00000 0 −1.81371 0
1.4 0 −0.306632 0 −1.00000 0 1.00000 0 −2.90598 0
1.5 0 1.50655 0 −1.00000 0 1.00000 0 −0.730315 0
1.6 0 2.09494 0 −1.00000 0 1.00000 0 1.38879 0
1.7 0 3.35792 0 −1.00000 0 1.00000 0 8.27561 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6440.2.a.z ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6440.2.a.z ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6440))\):

\( T_{3}^{7} - T_{3}^{6} - 15T_{3}^{5} + 10T_{3}^{4} + 55T_{3}^{3} - 10T_{3}^{2} - 60T_{3} - 16 \) Copy content Toggle raw display
\( T_{11}^{7} + 4T_{11}^{6} - 43T_{11}^{5} - 239T_{11}^{4} + 173T_{11}^{3} + 3128T_{11}^{2} + 6464T_{11} + 4096 \) Copy content Toggle raw display
\( T_{13}^{7} - 2T_{13}^{6} - 37T_{13}^{5} + 97T_{13}^{4} + 185T_{13}^{3} - 310T_{13}^{2} - 484T_{13} - 152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - T^{6} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( (T - 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 4 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{7} - 2 T^{6} + \cdots - 152 \) Copy content Toggle raw display
$17$ \( T^{7} + 12 T^{6} + \cdots + 1864 \) Copy content Toggle raw display
$19$ \( T^{7} + 12 T^{6} + \cdots + 2048 \) Copy content Toggle raw display
$23$ \( (T + 1)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 13 T^{6} + \cdots + 19392 \) Copy content Toggle raw display
$31$ \( T^{7} + 14 T^{6} + \cdots - 42208 \) Copy content Toggle raw display
$37$ \( T^{7} + 29 T^{6} + \cdots + 245984 \) Copy content Toggle raw display
$41$ \( T^{7} - 5 T^{6} + \cdots - 1735456 \) Copy content Toggle raw display
$43$ \( T^{7} - 2 T^{6} + \cdots - 145536 \) Copy content Toggle raw display
$47$ \( T^{7} + 3 T^{6} + \cdots - 15424 \) Copy content Toggle raw display
$53$ \( T^{7} + 3 T^{6} + \cdots + 28656 \) Copy content Toggle raw display
$59$ \( T^{7} - 29 T^{6} + \cdots + 532096 \) Copy content Toggle raw display
$61$ \( T^{7} + 15 T^{6} + \cdots - 3568 \) Copy content Toggle raw display
$67$ \( T^{7} - 7 T^{6} + \cdots + 221696 \) Copy content Toggle raw display
$71$ \( T^{7} + 38 T^{6} + \cdots - 429056 \) Copy content Toggle raw display
$73$ \( T^{7} + 15 T^{6} + \cdots + 310512 \) Copy content Toggle raw display
$79$ \( T^{7} + 37 T^{6} + \cdots + 1665824 \) Copy content Toggle raw display
$83$ \( T^{7} - 7 T^{6} + \cdots - 56064 \) Copy content Toggle raw display
$89$ \( T^{7} + 13 T^{6} + \cdots + 1893552 \) Copy content Toggle raw display
$97$ \( T^{7} + 44 T^{6} + \cdots + 16312 \) Copy content Toggle raw display
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