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Properties

Label 3640.2.a.ba
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
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] = [3640,2,Mod(1,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, 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("3640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.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: \(29.0655463357\)
Analytic rank: \(0\)
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: \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + q^{5} + q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + q^{5} + q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{3} + \beta_1 + 1) q^{11} + q^{13} + ( - \beta_1 + 1) q^{15} - \beta_{6} q^{17} + ( - \beta_{5} + \beta_1 + 1) q^{19} + ( - \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{2} + 2) q^{23} + q^{25} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{27}+ \cdots + ( - 2 \beta_{5} - \beta_{4} + 4 \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 15\nu^{3} + 36\nu^{2} - 33\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 6\nu^{5} + 5\nu^{4} - 59\nu^{3} - 4\nu^{2} + 125\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} - 10\nu^{4} + 22\nu^{3} + 22\nu^{2} - 33\nu - 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 10\nu^{5} + 31\nu^{4} - 85\nu^{3} - 84\nu^{2} + 159\nu + 60 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + 10\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} + \beta_{5} - 3\beta_{4} + 5\beta_{3} + 14\beta_{2} + 23\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19\beta_{6} + 2\beta_{5} - 16\beta_{4} + 33\beta_{3} + 42\beta_{2} + 125\beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 75\beta_{6} + 17\beta_{5} - 56\beta_{4} + 105\beta_{3} + 200\beta_{2} + 396\beta _1 + 479 \) 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
3.89861
2.04214
1.70423
−0.173825
−0.436282
−1.59001
−2.44487
0 −2.89861 0 1.00000 0 1.00000 0 5.40195 0
1.2 0 −1.04214 0 1.00000 0 1.00000 0 −1.91394 0
1.3 0 −0.704230 0 1.00000 0 1.00000 0 −2.50406 0
1.4 0 1.17382 0 1.00000 0 1.00000 0 −1.62214 0
1.5 0 1.43628 0 1.00000 0 1.00000 0 −0.937093 0
1.6 0 2.59001 0 1.00000 0 1.00000 0 3.70817 0
1.7 0 3.44487 0 1.00000 0 1.00000 0 8.86710 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
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 3640.2.a.ba ✓ 7
4.b odd 2 1 7280.2.a.cf 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.ba ✓ 7 1.a even 1 1 trivial
7280.2.a.cf 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{7} - 4T_{3}^{6} - 8T_{3}^{5} + 41T_{3}^{4} - 6T_{3}^{3} - 68T_{3}^{2} + 16T_{3} + 32 \) Copy content Toggle raw display
\( T_{11}^{7} - 7T_{11}^{6} - 26T_{11}^{5} + 220T_{11}^{4} + 80T_{11}^{3} - 1664T_{11}^{2} + 1280T_{11} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 4 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( (T - 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 7 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( (T - 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{7} - 13 T^{6} + \cdots + 17728 \) Copy content Toggle raw display
$23$ \( T^{7} - 11 T^{6} + \cdots + 134528 \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} + \cdots + 24224 \) Copy content Toggle raw display
$31$ \( T^{7} - 12 T^{6} + \cdots - 3008 \) Copy content Toggle raw display
$37$ \( T^{7} + 4 T^{6} + \cdots - 281104 \) Copy content Toggle raw display
$41$ \( T^{7} - 6 T^{6} + \cdots - 76624 \) Copy content Toggle raw display
$43$ \( T^{7} - 10 T^{6} + \cdots + 1688576 \) Copy content Toggle raw display
$47$ \( T^{7} - 15 T^{6} + \cdots + 131072 \) Copy content Toggle raw display
$53$ \( T^{7} - 8 T^{6} + \cdots - 77824 \) Copy content Toggle raw display
$59$ \( T^{7} - 13 T^{6} + \cdots + 148864 \) Copy content Toggle raw display
$61$ \( T^{7} + T^{6} + \cdots - 136768 \) Copy content Toggle raw display
$67$ \( T^{7} - 8 T^{6} + \cdots + 107776 \) Copy content Toggle raw display
$71$ \( T^{7} - 20 T^{6} + \cdots + 184832 \) Copy content Toggle raw display
$73$ \( T^{7} + 17 T^{6} + \cdots + 32000 \) Copy content Toggle raw display
$79$ \( T^{7} - 6 T^{6} + \cdots + 29696 \) Copy content Toggle raw display
$83$ \( T^{7} - 30 T^{6} + \cdots + 1110016 \) Copy content Toggle raw display
$89$ \( T^{7} - 5 T^{6} + \cdots - 2801056 \) Copy content Toggle raw display
$97$ \( T^{7} + 9 T^{6} + \cdots + 158528 \) Copy content Toggle raw display
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