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Properties

Label 4320.2.q.i
Level $4320$
Weight $2$
Character orbit 4320.q
Analytic conductor $34.495$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4320,2,Mod(1441,4320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4320.1441"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-4,0,-2,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.4953736732\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.3010058496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 6x^{6} + 2x^{5} - 17x^{4} + 6x^{3} + 54x^{2} - 108x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{2} - \beta_1) q^{11} - \beta_{6} q^{13} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + \cdots + 2) q^{17} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 4) q^{19}+ \cdots + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} - 2 q^{7} - 2 q^{11} + 8 q^{17} - 28 q^{19} + 6 q^{23} - 4 q^{25} - 8 q^{29} - 2 q^{31} + 4 q^{35} - 32 q^{37} + 8 q^{41} + 22 q^{43} + 8 q^{47} + 12 q^{49} + 8 q^{53} + 4 q^{55} + 12 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 6x^{6} + 2x^{5} - 17x^{4} + 6x^{3} + 54x^{2} - 108x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 44\nu^{6} - 105\nu^{5} - 7\nu^{4} + 349\nu^{3} - 324\nu^{2} - 873\nu + 1512 ) / 351 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{7} + 4\nu^{6} - 45\nu^{5} + 49\nu^{4} - 64\nu^{3} + 6\nu^{2} + 27\nu + 297 ) / 351 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 16\nu^{6} - 11\nu^{5} - 25\nu^{4} + 95\nu^{3} - 28\nu^{2} - 126\nu + 252 ) / 117 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{7} + 38\nu^{6} - 18\nu^{5} - 61\nu^{4} + 94\nu^{3} + 174\nu^{2} - 504\nu + 540 ) / 351 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 9\nu^{5} + 2\nu^{4} + 22\nu^{3} - 24\nu^{2} - 36\nu + 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\nu^{7} - 83\nu^{6} - 12\nu^{5} + 319\nu^{4} - 76\nu^{3} - 495\nu^{2} + 990\nu - 108 ) / 351 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{7} + 13\nu^{6} - 12\nu^{5} - 26\nu^{4} + 62\nu^{3} + 54\nu^{2} - 234\nu + 243 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - 4\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 6\beta_{4} + 4\beta_{3} - \beta_{2} - 3\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{6} - 8\beta_{5} + 11\beta_{4} + 2\beta_{3} + 4\beta_{2} + 4\beta _1 - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} - 16\beta_{5} + 13\beta_{4} + 3\beta_{3} - 9\beta_{2} + 11\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -13\beta_{7} + 2\beta_{6} - 6\beta_{5} + 66\beta_{4} + 16\beta_{3} + 8\beta_{2} + 6\beta _1 - 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -12\beta_{7} + 49\beta_{5} + 83\beta_{4} - 25\beta_{3} - 47\beta_{2} - 23\beta _1 + 26 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times\).

\(n\) \(2081\) \(2431\) \(3457\) \(3781\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
1.14729 + 1.29759i
1.65412 − 0.513691i
−1.54667 + 0.779618i
0.745265 − 1.56352i
1.14729 − 1.29759i
1.65412 + 0.513691i
−1.54667 − 0.779618i
0.745265 + 1.56352i
0 0 0 −0.500000 + 0.866025i 0 −1.69739 − 2.93996i 0 0 0
1441.2 0 0 0 −0.500000 + 0.866025i 0 −0.382191 − 0.661975i 0 0 0
1441.3 0 0 0 −0.500000 + 0.866025i 0 0.0981673 + 0.170031i 0 0 0
1441.4 0 0 0 −0.500000 + 0.866025i 0 0.981412 + 1.69985i 0 0 0
2881.1 0 0 0 −0.500000 − 0.866025i 0 −1.69739 + 2.93996i 0 0 0
2881.2 0 0 0 −0.500000 − 0.866025i 0 −0.382191 + 0.661975i 0 0 0
2881.3 0 0 0 −0.500000 − 0.866025i 0 0.0981673 − 0.170031i 0 0 0
2881.4 0 0 0 −0.500000 − 0.866025i 0 0.981412 − 1.69985i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1441.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4320.2.q.i 8
3.b odd 2 1 1440.2.q.n yes 8
4.b odd 2 1 4320.2.q.k 8
9.c even 3 1 inner 4320.2.q.i 8
9.d odd 6 1 1440.2.q.n yes 8
12.b even 2 1 1440.2.q.i ✓ 8
36.f odd 6 1 4320.2.q.k 8
36.h even 6 1 1440.2.q.i ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.q.i ✓ 8 12.b even 2 1
1440.2.q.i ✓ 8 36.h even 6 1
1440.2.q.n yes 8 3.b odd 2 1
1440.2.q.n yes 8 9.d odd 6 1
4320.2.q.i 8 1.a even 1 1 trivial
4320.2.q.i 8 9.c even 3 1 inner
4320.2.q.k 8 4.b odd 2 1
4320.2.q.k 8 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4320, [\chi])\):

\( T_{7}^{8} + 2T_{7}^{7} + 10T_{7}^{6} - 4T_{7}^{5} + 43T_{7}^{4} + 20T_{7}^{3} + 22T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + 2T_{11}^{7} + 28T_{11}^{6} - 16T_{11}^{5} + 508T_{11}^{4} - 16T_{11}^{3} + 2656T_{11}^{2} - 1600T_{11} + 10000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + \cdots + 7056 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} - 48 T^{2} + \cdots + 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 14 T^{3} + \cdots + 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 3316041 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 76729 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 425104 \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 1076)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + \cdots + 10426441 \) Copy content Toggle raw display
$43$ \( T^{8} - 22 T^{7} + \cdots + 309136 \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{7} + \cdots + 946729 \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + \cdots + 1888)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots + 57600 \) Copy content Toggle raw display
$61$ \( T^{8} - 4 T^{7} + \cdots + 1225 \) Copy content Toggle raw display
$67$ \( T^{8} + 54 T^{6} + \cdots + 239121 \) Copy content Toggle raw display
$71$ \( (T^{4} + 30 T^{3} + \cdots + 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 28 T^{3} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + \cdots + 628849 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 315)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} + \cdots + 120472576 \) Copy content Toggle raw display
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