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Properties

Label 4320.2.f.k
Level $4320$
Weight $2$
Character orbit 4320.f
Analytic conductor $34.495$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(1729,4320)] 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("4320.1729"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,-6,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.4953736732\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 21x^{8} - 174x^{6} - 525x^{4} + 1875x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{8} q^{7} + ( - \beta_{11} + \beta_{9} - \beta_1) q^{11} - \beta_{7} q^{13} + (\beta_{10} - \beta_{5}) q^{17} + ( - \beta_{4} + 1) q^{19} + (\beta_{9} - \beta_{5} + \beta_1) q^{23}+ \cdots + ( - 4 \beta_{8} - \beta_{6} + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{19} - 6 q^{25} - 4 q^{31} - 8 q^{49} - 30 q^{55} - 8 q^{61} + 88 q^{79} - 8 q^{85} - 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 21x^{8} - 174x^{6} - 525x^{4} + 1875x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 28\nu^{8} - 179\nu^{6} + 199\nu^{4} + 3500\nu^{2} + 18125 ) / 6000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 2\nu^{9} + 89\nu^{7} - 319\nu^{5} - 3530\nu^{3} - 5375\nu ) / 15000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{10} - 43\nu^{8} + 51\nu^{6} + 2194\nu^{4} + 13125\nu^{2} - 8125 ) / 15000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{11} - 112\nu^{9} - 91\nu^{7} + 571\nu^{5} + 20200\nu^{3} + 33125\nu ) / 150000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\nu^{10} + 112\nu^{8} + 91\nu^{6} - 571\nu^{4} - 20200\nu^{2} - 33125 ) / 30000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{10} + 24\nu^{8} - 43\nu^{6} + 1383\nu^{4} + 4000\nu^{2} - 16875 ) / 15000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41\nu^{10} + 198\nu^{8} - 11\nu^{6} - 4959\nu^{4} - 16450\nu^{2} - 16875 ) / 30000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 3\nu^{9} - 21\nu^{7} - 174\nu^{5} - 525\nu^{3} + 1875\nu ) / 3125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\nu^{11} + 288\nu^{9} + 859\nu^{7} - 3379\nu^{5} - 25800\nu^{3} - 103125\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 2\nu^{9} - 11\nu^{7} - 119\nu^{5} + 70\nu^{3} + 1525\nu ) / 2000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{10} - 2\beta_{9} + 3\beta_{5} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} + 2\beta_{7} + 8\beta_{6} + 3\beta_{4} + \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{11} - 4\beta_{10} - 9\beta_{9} - 36\beta_{5} - 6\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{8} - 18\beta_{7} - 16\beta_{6} + 12\beta_{4} - 32\beta_{2} + 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 44\beta_{11} + 28\beta_{10} - 90\beta_{9} + 36\beta_{5} + 30\beta_{3} + 81\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 123\beta_{8} + 146\beta_{7} + 17\beta_{6} + 51\beta_{4} - 8\beta_{2} + 304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -118\beta_{11} + 131\beta_{10} + 108\beta_{9} + 33\beta_{5} - 363\beta_{3} + 158\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 175\beta_{8} - 468\beta_{7} + 480\beta_{6} + 521\beta_{4} + 151\beta_{2} - 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1632\beta_{11} + 24\beta_{10} - 1705\beta_{9} - 4032\beta_{5} - 900\beta_{3} + 396\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2081\) \(2431\) \(3457\) \(3781\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
−2.23265 − 0.123563i
−2.23265 + 0.123563i
−1.28187 − 1.83216i
−1.28187 + 1.83216i
−0.349411 − 2.20860i
−0.349411 + 2.20860i
0.349411 − 2.20860i
0.349411 + 2.20860i
1.28187 − 1.83216i
1.28187 + 1.83216i
2.23265 − 0.123563i
2.23265 + 0.123563i
0 0 0 −2.23265 − 0.123563i 0 2.78440i 0 0 0
1729.2 0 0 0 −2.23265 + 0.123563i 0 − 2.78440i 0 0 0
1729.3 0 0 0 −1.28187 − 1.83216i 0 3.41531i 0 0 0
1729.4 0 0 0 −1.28187 + 1.83216i 0 − 3.41531i 0 0 0
1729.5 0 0 0 −0.349411 − 2.20860i 0 1.89283i 0 0 0
1729.6 0 0 0 −0.349411 + 2.20860i 0 − 1.89283i 0 0 0
1729.7 0 0 0 0.349411 − 2.20860i 0 − 1.89283i 0 0 0
1729.8 0 0 0 0.349411 + 2.20860i 0 1.89283i 0 0 0
1729.9 0 0 0 1.28187 − 1.83216i 0 − 3.41531i 0 0 0
1729.10 0 0 0 1.28187 + 1.83216i 0 3.41531i 0 0 0
1729.11 0 0 0 2.23265 − 0.123563i 0 − 2.78440i 0 0 0
1729.12 0 0 0 2.23265 + 0.123563i 0 2.78440i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1729.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4320.2.f.k yes 12
3.b odd 2 1 inner 4320.2.f.k yes 12
4.b odd 2 1 4320.2.f.h ✓ 12
5.b even 2 1 inner 4320.2.f.k yes 12
12.b even 2 1 4320.2.f.h ✓ 12
15.d odd 2 1 inner 4320.2.f.k yes 12
20.d odd 2 1 4320.2.f.h ✓ 12
60.h even 2 1 4320.2.f.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4320.2.f.h ✓ 12 4.b odd 2 1
4320.2.f.h ✓ 12 12.b even 2 1
4320.2.f.h ✓ 12 20.d odd 2 1
4320.2.f.h ✓ 12 60.h even 2 1
4320.2.f.k yes 12 1.a even 1 1 trivial
4320.2.f.k yes 12 3.b odd 2 1 inner
4320.2.f.k yes 12 5.b even 2 1 inner
4320.2.f.k yes 12 15.d odd 2 1 inner

Hecke kernels

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

\( T_{7}^{6} + 23T_{7}^{4} + 160T_{7}^{2} + 324 \) Copy content Toggle raw display
\( T_{11}^{6} - 39T_{11}^{4} + 360T_{11}^{2} - 324 \) Copy content Toggle raw display
\( T_{19}^{3} - 4T_{19}^{2} - 17T_{19} + 66 \) Copy content Toggle raw display
\( T_{29}^{6} - 124T_{29}^{4} + 4044T_{29}^{2} - 36864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 23 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 39 T^{4} + \cdots - 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 36 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 46 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 17 T + 66)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 38 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 124 T^{4} + \cdots - 36864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 35 T - 39)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 92 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 40 T^{4} + \cdots - 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 200 T^{4} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 284 T^{4} + \cdots + 97344)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 79 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 192 T^{4} + \cdots - 219024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} - 111 T - 36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 156 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 356 T^{4} + \cdots - 576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 295 T^{4} + \cdots + 695556)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 22 T^{2} + \cdots + 648)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 363 T^{4} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 456 T^{4} + \cdots - 524176)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 435 T^{4} + \cdots + 2050624)^{2} \) Copy content Toggle raw display
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