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Properties

Label 726.2.b.c
Level $726$
Weight $2$
Character orbit 726.b
Analytic conductor $5.797$
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] = [726,2,Mod(725,726)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(726, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) 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("726.725"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.185640625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 66)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \cdots + \beta_1) q^{5} + \beta_1 q^{6} + (\beta_{5} + \beta_{3} + \beta_1) q^{7} - q^{8} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 1) q^{9}+ \cdots + (\beta_{6} - 3 \beta_{4} - \beta_{3} + \cdots - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 3 q^{3} + 8 q^{4} + 3 q^{6} - 8 q^{8} + 7 q^{9} - 3 q^{12} + 4 q^{15} + 8 q^{16} - 10 q^{17} - 7 q^{18} + 12 q^{21} + 3 q^{24} + 14 q^{25} - 15 q^{27} - 2 q^{29} - 4 q^{30} - 22 q^{31} - 8 q^{32}+ \cdots - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 57\nu^{2} - 54\nu + 27 ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 5\nu^{6} + 6\nu^{5} + 2\nu^{4} - 10\nu^{3} - 19\nu^{2} - 69\nu + 144 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\nu^{7} - 15\nu^{6} - 17\nu^{5} - 2\nu^{4} + 46\nu^{3} + 108\nu^{2} + 153\nu - 567 ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{7} - 18\nu^{6} - 7\nu^{5} - 28\nu^{4} + 32\nu^{3} + 93\nu^{2} + 252\nu - 567 ) / 108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -20\nu^{7} + 21\nu^{6} + 25\nu^{5} + 22\nu^{4} - 2\nu^{3} - 126\nu^{2} - 423\nu + 783 ) / 108 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 4\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 4\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 2\beta_{6} + 3\beta_{5} + 10\beta_{4} + 4\beta_{3} - 2\beta_{2} + 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{7} + 9\beta_{5} + 14\beta_{4} - 8\beta_{3} - 8\beta_{2} + \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{7} - 2\beta_{6} + 10\beta_{5} + 23\beta_{4} - 17\beta_{2} - 17\beta _1 + 34 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(607\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
725.1
1.71634 + 0.232753i
1.71634 − 0.232753i
1.55470 + 0.763481i
1.55470 − 0.763481i
−0.245684 + 1.71454i
−0.245684 − 1.71454i
−1.52536 + 0.820539i
−1.52536 − 0.820539i
−1.00000 −1.71634 − 0.232753i 1.00000 − 2.65532i 1.71634 + 0.232753i 1.64108i −1.00000 2.89165 + 0.798968i 2.65532i
725.2 −1.00000 −1.71634 + 0.232753i 1.00000 2.65532i 1.71634 − 0.232753i − 1.64108i −1.00000 2.89165 − 0.798968i − 2.65532i
725.3 −1.00000 −1.55470 − 0.763481i 1.00000 2.11929i 1.55470 + 0.763481i 3.42908i −1.00000 1.83419 + 2.37397i − 2.11929i
725.4 −1.00000 −1.55470 + 0.763481i 1.00000 − 2.11929i 1.55470 − 0.763481i − 3.42908i −1.00000 1.83419 − 2.37397i 2.11929i
725.5 −1.00000 0.245684 − 1.71454i 1.00000 0.943715i −0.245684 + 1.71454i 1.52696i −1.00000 −2.87928 − 0.842471i − 0.943715i
725.6 −1.00000 0.245684 + 1.71454i 1.00000 − 0.943715i −0.245684 − 1.71454i − 1.52696i −1.00000 −2.87928 + 0.842471i 0.943715i
725.7 −1.00000 1.52536 − 0.820539i 1.00000 − 0.753205i −1.52536 + 0.820539i 0.465507i −1.00000 1.65343 − 2.50323i 0.753205i
725.8 −1.00000 1.52536 + 0.820539i 1.00000 0.753205i −1.52536 − 0.820539i − 0.465507i −1.00000 1.65343 + 2.50323i − 0.753205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 725.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.b.c 8
3.b odd 2 1 726.2.b.e 8
11.b odd 2 1 726.2.b.e 8
11.c even 5 1 66.2.h.b yes 8
11.c even 5 1 726.2.h.f 8
11.c even 5 1 726.2.h.h 8
11.c even 5 1 726.2.h.j 8
11.d odd 10 1 66.2.h.a ✓ 8
11.d odd 10 1 726.2.h.a 8
11.d odd 10 1 726.2.h.c 8
11.d odd 10 1 726.2.h.d 8
33.d even 2 1 inner 726.2.b.c 8
33.f even 10 1 66.2.h.b yes 8
33.f even 10 1 726.2.h.f 8
33.f even 10 1 726.2.h.h 8
33.f even 10 1 726.2.h.j 8
33.h odd 10 1 66.2.h.a ✓ 8
33.h odd 10 1 726.2.h.a 8
33.h odd 10 1 726.2.h.c 8
33.h odd 10 1 726.2.h.d 8
44.g even 10 1 528.2.bn.b 8
44.h odd 10 1 528.2.bn.a 8
132.n odd 10 1 528.2.bn.a 8
132.o even 10 1 528.2.bn.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.h.a ✓ 8 11.d odd 10 1
66.2.h.a ✓ 8 33.h odd 10 1
66.2.h.b yes 8 11.c even 5 1
66.2.h.b yes 8 33.f even 10 1
528.2.bn.a 8 44.h odd 10 1
528.2.bn.a 8 132.n odd 10 1
528.2.bn.b 8 44.g even 10 1
528.2.bn.b 8 132.o even 10 1
726.2.b.c 8 1.a even 1 1 trivial
726.2.b.c 8 33.d even 2 1 inner
726.2.b.e 8 3.b odd 2 1
726.2.b.e 8 11.b odd 2 1
726.2.h.a 8 11.d odd 10 1
726.2.h.a 8 33.h odd 10 1
726.2.h.c 8 11.d odd 10 1
726.2.h.c 8 33.h odd 10 1
726.2.h.d 8 11.d odd 10 1
726.2.h.d 8 33.h odd 10 1
726.2.h.f 8 11.c even 5 1
726.2.h.f 8 33.f even 10 1
726.2.h.h 8 11.c even 5 1
726.2.h.h 8 33.f even 10 1
726.2.h.j 8 11.c even 5 1
726.2.h.j 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\):

\( T_{5}^{8} + 13T_{5}^{6} + 49T_{5}^{4} + 52T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 5T_{17}^{3} - 35T_{17}^{2} - 250T_{17} - 380 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 13 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 17 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{4} + 5 T^{3} + \cdots - 380)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 85 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( (T^{4} + T^{3} - 79 T^{2} + \cdots - 124)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 11 T^{3} + \cdots - 1084)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} + \cdots + 176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - T^{3} - 59 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 113 T^{6} + \cdots + 430336 \) Copy content Toggle raw display
$47$ \( T^{8} + 212 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{8} + 185 T^{6} + \cdots + 2310400 \) Copy content Toggle raw display
$59$ \( T^{8} + 202 T^{6} + \cdots + 844561 \) Copy content Toggle raw display
$61$ \( T^{8} + 212 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 149 T^{2} + \cdots + 3076)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 100 T^{2} + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 90 T^{6} + \cdots + 24025 \) Copy content Toggle raw display
$79$ \( T^{8} + 157 T^{6} + \cdots + 55696 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots - 859)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 513 T^{6} + \cdots + 12702096 \) Copy content Toggle raw display
$97$ \( (T^{4} + 8 T^{3} + \cdots - 1439)^{2} \) Copy content Toggle raw display
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