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Properties

Label 1080.2.b.d
Level $1080$
Weight $2$
Character orbit 1080.b
Analytic conductor $8.624$
Analytic rank $0$
Dimension $16$
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] = [1080,2,Mod(971,1080)] 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("1080.971"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 6 x^{10} - 8 x^{9} + 32 x^{8} - 16 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{6} q^{4} - q^{5} - \beta_{4} q^{7} + \beta_{7} q^{8} + \beta_{3} q^{10} - \beta_{13} q^{11} + (\beta_{11} - \beta_{3}) q^{13} + (\beta_{12} + \beta_{11} + \cdots + \beta_1) q^{14}+ \cdots + (\beta_{15} + \beta_{14} + \beta_{13} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 2 q^{4} - 16 q^{5} - 4 q^{8} - 2 q^{10} + 6 q^{14} + 10 q^{16} - 8 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 16 q^{25} - 16 q^{26} + 20 q^{28} - 8 q^{32} + 18 q^{34} - 14 q^{38} + 4 q^{40}+ \cdots - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 6 x^{10} - 8 x^{9} + 32 x^{8} - 16 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{12} - \nu^{10} + 2\nu^{8} + 2\nu^{6} + 4\nu^{5} + 8\nu^{3} - 48\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + \nu^{13} + 2 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} - 6 \nu^{9} - 8 \nu^{8} + \cdots - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{14} + 2 \nu^{13} - \nu^{12} - 2 \nu^{11} - 2 \nu^{10} - 8 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 6 \nu^{12} - 2 \nu^{11} + 10 \nu^{9} - 16 \nu^{8} + 8 \nu^{7} + \cdots - 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - \nu^{14} - \nu^{13} + 3 \nu^{12} - 4 \nu^{11} + 2 \nu^{10} + 2 \nu^{9} - 14 \nu^{8} + \cdots - 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} + \nu^{13} - 2 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} + 10 \nu^{9} + 16 \nu^{8} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} - 2 \nu^{14} + \nu^{13} + 2 \nu^{12} + 6 \nu^{11} - 8 \nu^{10} - 22 \nu^{9} + 24 \nu^{8} + \cdots - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{15} + 6 \nu^{14} - 3 \nu^{13} - 6 \nu^{12} + 10 \nu^{11} - 8 \nu^{10} + 26 \nu^{9} + \cdots + 640 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{15} + 4 \nu^{14} + \nu^{13} - 8 \nu^{12} + 6 \nu^{11} - 12 \nu^{10} + 10 \nu^{9} + \cdots + 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{15} - \nu^{14} + 5 \nu^{13} - \nu^{12} + 4 \nu^{11} - 2 \nu^{10} - 6 \nu^{9} + 18 \nu^{8} + \cdots + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2 \nu^{15} - \nu^{14} - 2 \nu^{13} - \nu^{12} - 4 \nu^{11} + 6 \nu^{10} - 18 \nu^{8} + 28 \nu^{7} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2 \nu^{15} - \nu^{14} - 4 \nu^{13} - \nu^{12} - 6 \nu^{11} + 6 \nu^{10} + 12 \nu^{9} - 18 \nu^{8} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2 \nu^{15} + \nu^{14} + 2 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} - 10 \nu^{10} + 26 \nu^{8} - 28 \nu^{7} + \cdots + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5 \nu^{15} - 12 \nu^{14} - 3 \nu^{13} + 16 \nu^{12} - 14 \nu^{11} + 44 \nu^{10} - 6 \nu^{9} + \cdots - 896 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} - \beta_{12} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{14} + 2 \beta_{13} - \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{6} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} + 6 \beta_{10} + 2 \beta_{8} + 6 \beta_{6} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5 \beta_{14} + 6 \beta_{13} + \beta_{12} + 10 \beta_{11} + 6 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + \cdots - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6 \beta_{15} - 6 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 6 \beta_{10} + 9 \beta_{9} - 6 \beta_{8} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7 \beta_{14} - 6 \beta_{13} + 19 \beta_{12} + 14 \beta_{11} - 14 \beta_{10} + 16 \beta_{9} + \cdots - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 6 \beta_{15} - 4 \beta_{14} + 14 \beta_{13} - 8 \beta_{12} + 14 \beta_{11} - 6 \beta_{10} - \beta_{9} + \cdots + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8 \beta_{15} + 23 \beta_{14} - 26 \beta_{13} + 5 \beta_{12} + 2 \beta_{11} + 30 \beta_{10} + 4 \beta_{9} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 18 \beta_{15} - 8 \beta_{14} - 46 \beta_{13} + 20 \beta_{12} - 30 \beta_{11} + 14 \beta_{10} + \cdots + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 64 \beta_{15} - 79 \beta_{14} + 50 \beta_{13} - 13 \beta_{12} + 70 \beta_{11} - 118 \beta_{10} + \cdots - 78 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
971.1
−1.41035 + 0.104529i
−1.41035 − 0.104529i
−1.23596 + 0.687323i
−1.23596 − 0.687323i
−0.437001 + 1.34500i
−0.437001 − 1.34500i
−0.119860 + 1.40913i
−0.119860 − 1.40913i
0.710396 + 1.22284i
0.710396 − 1.22284i
0.794441 + 1.16998i
0.794441 − 1.16998i
1.31981 + 0.508037i
1.31981 − 0.508037i
1.37852 + 0.315744i
1.37852 − 0.315744i
−1.41035 − 0.104529i 0 1.97815 + 0.294844i −1.00000 0 − 3.66736i −2.75905 − 0.622605i 0 1.41035 + 0.104529i
971.2 −1.41035 + 0.104529i 0 1.97815 − 0.294844i −1.00000 0 3.66736i −2.75905 + 0.622605i 0 1.41035 − 0.104529i
971.3 −1.23596 − 0.687323i 0 1.05517 + 1.69900i −1.00000 0 0.920206i −0.136384 − 2.82514i 0 1.23596 + 0.687323i
971.4 −1.23596 + 0.687323i 0 1.05517 − 1.69900i −1.00000 0 − 0.920206i −0.136384 + 2.82514i 0 1.23596 − 0.687323i
971.5 −0.437001 − 1.34500i 0 −1.61806 + 1.17554i −1.00000 0 − 0.738395i 2.28819 + 1.66258i 0 0.437001 + 1.34500i
971.6 −0.437001 + 1.34500i 0 −1.61806 − 1.17554i −1.00000 0 0.738395i 2.28819 − 1.66258i 0 0.437001 − 1.34500i
971.7 −0.119860 − 1.40913i 0 −1.97127 + 0.337796i −1.00000 0 − 1.81970i 0.712273 + 2.73727i 0 0.119860 + 1.40913i
971.8 −0.119860 + 1.40913i 0 −1.97127 − 0.337796i −1.00000 0 1.81970i 0.712273 − 2.73727i 0 0.119860 − 1.40913i
971.9 0.710396 − 1.22284i 0 −0.990676 − 1.73740i −1.00000 0 1.47745i −2.82834 − 0.0228030i 0 −0.710396 + 1.22284i
971.10 0.710396 + 1.22284i 0 −0.990676 + 1.73740i −1.00000 0 − 1.47745i −2.82834 + 0.0228030i 0 −0.710396 − 1.22284i
971.11 0.794441 − 1.16998i 0 −0.737726 − 1.85897i −1.00000 0 4.51932i −2.76104 − 0.613714i 0 −0.794441 + 1.16998i
971.12 0.794441 + 1.16998i 0 −0.737726 + 1.85897i −1.00000 0 − 4.51932i −2.76104 + 0.613714i 0 −0.794441 − 1.16998i
971.13 1.31981 − 0.508037i 0 1.48380 − 1.34103i −1.00000 0 − 3.32511i 1.27704 − 2.52372i 0 −1.31981 + 0.508037i
971.14 1.31981 + 0.508037i 0 1.48380 + 1.34103i −1.00000 0 3.32511i 1.27704 + 2.52372i 0 −1.31981 − 0.508037i
971.15 1.37852 − 0.315744i 0 1.80061 − 0.870516i −1.00000 0 2.86072i 2.20731 − 1.76855i 0 −1.37852 + 0.315744i
971.16 1.37852 + 0.315744i 0 1.80061 + 0.870516i −1.00000 0 − 2.86072i 2.20731 + 1.76855i 0 −1.37852 − 0.315744i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 971.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.b.d yes 16
3.b odd 2 1 1080.2.b.a ✓ 16
4.b odd 2 1 4320.2.b.b 16
8.b even 2 1 4320.2.b.d 16
8.d odd 2 1 1080.2.b.a ✓ 16
12.b even 2 1 4320.2.b.d 16
24.f even 2 1 inner 1080.2.b.d yes 16
24.h odd 2 1 4320.2.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.b.a ✓ 16 3.b odd 2 1
1080.2.b.a ✓ 16 8.d odd 2 1
1080.2.b.d yes 16 1.a even 1 1 trivial
1080.2.b.d yes 16 24.f even 2 1 inner
4320.2.b.b 16 4.b odd 2 1
4320.2.b.b 16 24.h odd 2 1
4320.2.b.d 16 8.b even 2 1
4320.2.b.d 16 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{16} + 60 T_{7}^{14} + 1398 T_{7}^{12} + 16180 T_{7}^{10} + 98625 T_{7}^{8} + 312216 T_{7}^{6} + \cdots + 82944 \) Copy content Toggle raw display
\( T_{23}^{8} + 4T_{23}^{7} - 90T_{23}^{6} - 372T_{23}^{5} + 1805T_{23}^{4} + 9080T_{23}^{3} + 7192T_{23}^{2} - 9696T_{23} - 10992 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 60 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$11$ \( T^{16} + 96 T^{14} + \cdots + 30976 \) Copy content Toggle raw display
$13$ \( T^{16} + 116 T^{14} + \cdots + 7080921 \) Copy content Toggle raw display
$17$ \( T^{16} + 128 T^{14} + \cdots + 937024 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 44176)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots - 10992)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 110 T^{6} + \cdots + 63096)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 8074100736 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 5492588544 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 2099838976 \) Copy content Toggle raw display
$43$ \( (T^{8} - 246 T^{6} + \cdots + 1503568)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots + 2904)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 16 T^{7} + \cdots + 1536)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 97863860224 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 823273134336 \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots - 1652144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 12 T^{7} + \cdots - 70272)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} + \cdots + 15616)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 26506321168041 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 3115196760064 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 21034771431424 \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{7} + \cdots - 2312192)^{2} \) Copy content Toggle raw display
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