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Properties

Label 728.2.q.c
Level $728$
Weight $2$
Character orbit 728.q
Analytic conductor $5.813$
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] = [728,2,Mod(289,728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("728.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.81310926715\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{3} + (\beta_{5} - \beta_1) q^{5} + ( - 3 \beta_{6} + 1) q^{7} + (\beta_{7} + \beta_{6} + 2 \beta_{4} + \cdots - 1) q^{9} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{11}+ \cdots + (4 \beta_{7} - 2 \beta_{5} + 7 \beta_{4} + \cdots - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} + q^{5} - 4 q^{7} - 5 q^{9} + 4 q^{11} + 10 q^{13} + 11 q^{15} + 12 q^{17} - q^{19} - 12 q^{21} + 10 q^{23} - 3 q^{25} + 24 q^{27} + 8 q^{29} - 12 q^{31} + 10 q^{33} + 4 q^{35} + 2 q^{37}+ \cdots - 114 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{7} + 844\nu^{6} - 1100\nu^{5} + 6429\nu^{4} + 791\nu^{3} + 37851\nu^{2} - 16973\nu - 1253 ) / 4143 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -352\nu^{7} - 725\nu^{6} - 1876\nu^{5} - 9605\nu^{4} - 21671\nu^{3} - 49186\nu^{2} + 1822\nu - 6295 ) / 4143 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -371\nu^{7} + 252\nu^{6} - 2809\nu^{5} - 2014\nu^{4} - 19196\nu^{3} - 689\nu^{2} - 159\nu - 793 ) / 4143 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 634\nu^{7} - 1056\nu^{6} + 5984\nu^{5} - 1885\nu^{4} + 34144\nu^{3} - 26048\nu^{2} + 36375\nu - 2464 ) / 4143 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -793\nu^{7} + 1164\nu^{6} - 6596\nu^{5} + 430\nu^{4} - 37636\nu^{3} + 28712\nu^{2} - 8034\nu + 2716 ) / 4143 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2716\nu^{7} - 1923\nu^{6} + 20564\nu^{5} + 14744\nu^{4} + 135370\nu^{3} + 5044\nu^{2} + 1164\nu + 9323 ) / 4143 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{6} + \beta_{3} - \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{5} - 8\beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} - 22\beta_{6} - 7\beta_{5} - 8\beta_{4} - 16\beta_{3} - 8\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} - 30\beta_{6} + 8\beta_{5} + 53\beta_{4} - 13\beta_{3} + 13\beta_{2} - 66\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{7} + 66\beta_{5} + 140\beta_{4} + 66\beta_{3} + 132\beta_{2} - 66\beta _1 + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140\beta_{7} + 341\beta_{6} + 74\beta_{5} + 140\beta_{4} + 280\beta_{3} + 140\beta_{2} + 428\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/728\mathbb{Z}\right)^\times\).

\(n\) \(183\) \(365\) \(521\) \(561\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.15777 + 2.00531i
−0.132681 + 0.229810i
0.267083 − 0.462601i
1.52336 − 2.63854i
−1.15777 − 2.00531i
−0.132681 − 0.229810i
0.267083 + 0.462601i
1.52336 + 2.63854i
0 −1.65777 + 2.87134i 0 1.44183 − 2.49733i 0 −0.500000 + 2.59808i 0 −3.99638 − 6.92193i 0
289.2 0 −0.632681 + 1.09584i 0 −1.25154 + 2.16772i 0 −0.500000 + 2.59808i 0 0.699429 + 1.21145i 0
289.3 0 −0.232917 + 0.403424i 0 1.16896 − 2.02469i 0 −0.500000 + 2.59808i 0 1.39150 + 2.41015i 0
289.4 0 1.02336 − 1.77252i 0 −0.859254 + 1.48827i 0 −0.500000 + 2.59808i 0 −0.594550 − 1.02979i 0
529.1 0 −1.65777 − 2.87134i 0 1.44183 + 2.49733i 0 −0.500000 − 2.59808i 0 −3.99638 + 6.92193i 0
529.2 0 −0.632681 − 1.09584i 0 −1.25154 − 2.16772i 0 −0.500000 − 2.59808i 0 0.699429 − 1.21145i 0
529.3 0 −0.232917 − 0.403424i 0 1.16896 + 2.02469i 0 −0.500000 − 2.59808i 0 1.39150 − 2.41015i 0
529.4 0 1.02336 + 1.77252i 0 −0.859254 − 1.48827i 0 −0.500000 − 2.59808i 0 −0.594550 + 1.02979i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 728.2.q.c ✓ 8
7.c even 3 1 728.2.t.c yes 8
13.c even 3 1 728.2.t.c yes 8
91.h even 3 1 inner 728.2.q.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.q.c ✓ 8 1.a even 1 1 trivial
728.2.q.c ✓ 8 91.h even 3 1 inner
728.2.t.c yes 8 7.c even 3 1
728.2.t.c yes 8 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 3T_{3}^{7} + 13T_{3}^{6} + 10T_{3}^{5} + 53T_{3}^{4} + 68T_{3}^{3} + 105T_{3}^{2} + 44T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 7056 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} + \cdots - 487)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} + \cdots + 245)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots + 877969 \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} - 29 T^{2} + \cdots + 105)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + \cdots + 5612161 \) Copy content Toggle raw display
$43$ \( T^{8} - 8 T^{7} + \cdots + 859329 \) Copy content Toggle raw display
$47$ \( T^{8} + 6 T^{7} + \cdots + 5968249 \) Copy content Toggle raw display
$53$ \( T^{8} - 6 T^{7} + \cdots + 310249 \) Copy content Toggle raw display
$59$ \( (T^{4} - 182 T^{2} + \cdots + 7629)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 3 T^{7} + \cdots + 114276100 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 1687401 \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + \cdots + 54037201 \) Copy content Toggle raw display
$79$ \( T^{8} - 2 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + \cdots - 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + 5 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 104182849 \) Copy content Toggle raw display
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