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Properties

Label 810.4.e.bh
Level $810$
Weight $4$
Character orbit 810.e
Analytic conductor $47.792$
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] = [810,4,Mod(271,810)] 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("810.271"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,0,-16,-20,0,-2,-64,0,-80,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.7915471046\)
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: \( x^{8} + 254x^{6} + 23581x^{4} + 947376x^{2} + 13883076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{10} \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_1 + 2) q^{2} + 4 \beta_1 q^{4} + 5 \beta_1 q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{2}) q^{7} - 8 q^{8} - 10 q^{10} + (\beta_{4} - 9 \beta_1 - 9) q^{11} + ( - \beta_{7} - 2 \beta_{6} + \cdots + 7 \beta_1) q^{13}+ \cdots + ( - 2 \beta_{5} + 24 \beta_{3} + \cdots - 668) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{4} - 20 q^{5} - 2 q^{7} - 64 q^{8} - 80 q^{10} - 36 q^{11} - 32 q^{13} + 4 q^{14} - 64 q^{16} - 180 q^{17} + 328 q^{19} - 80 q^{20} + 72 q^{22} + 42 q^{23} - 100 q^{25} - 128 q^{26}+ \cdots - 5352 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 254x^{6} + 23581x^{4} + 947376x^{2} + 13883076 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 254\nu^{5} + 19855\nu^{3} + 474174\nu - 67068 ) / 134136 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 191\nu^{4} - 11368\nu^{2} - 208332 ) / 360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 371\nu^{4} + 34228\nu^{2} + 879012 ) / 1080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31 \nu^{7} + 69 \nu^{6} - 6011 \nu^{5} + 25599 \nu^{4} - 375178 \nu^{3} + 2361732 \nu^{2} + \cdots + 60651828 ) / 149040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 191\nu^{4} - 11773\nu^{2} - 233982 ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83 \nu^{7} + 1242 \nu^{6} - 15493 \nu^{5} + 237222 \nu^{4} - 871094 \nu^{3} + 14622066 \nu^{2} + \cdots + 290605644 ) / 335340 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31 \nu^{7} + 69 \nu^{6} + 6011 \nu^{5} + 13179 \nu^{4} + 375178 \nu^{3} + 784392 \nu^{2} + \cdots + 14374908 ) / 49680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 6\beta_{4} - 3\beta_{3} + \beta_{2} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 8\beta_{2} - 570 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -56\beta_{7} + 162\beta_{6} + 81\beta_{5} - 384\beta_{4} + 192\beta_{3} - 28\beta_{2} - 648\beta _1 - 324 ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 381\beta_{5} + 54\beta_{3} - 998\beta_{2} + 38856 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1204 \beta_{7} - 20898 \beta_{6} - 10449 \beta_{5} + 26412 \beta_{4} - 13206 \beta_{3} + \cdots + 71928 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -38667\beta_{5} - 10314\beta_{3} + 96434\beta_{2} - 2816724 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 469348 \beta_{7} + 2091582 \beta_{6} + 1045791 \beta_{5} - 1929372 \beta_{4} + 964686 \beta_{3} + \cdots - 10025856 ) / 27 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
− 7.40373i
9.37910i
6.40373i
− 8.37910i
7.40373i
− 9.37910i
− 6.40373i
8.37910i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 −15.8018 − 27.3694i −8.00000 0 −10.0000
271.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 −8.37250 − 14.5016i −8.00000 0 −10.0000
271.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 4.90945 + 8.50341i −8.00000 0 −10.0000
271.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 18.2648 + 31.6356i −8.00000 0 −10.0000
541.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.50000 − 4.33013i 0 −15.8018 + 27.3694i −8.00000 0 −10.0000
541.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.50000 − 4.33013i 0 −8.37250 + 14.5016i −8.00000 0 −10.0000
541.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.50000 − 4.33013i 0 4.90945 − 8.50341i −8.00000 0 −10.0000
541.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.50000 − 4.33013i 0 18.2648 − 31.6356i −8.00000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.4
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 810.4.e.bh 8
3.b odd 2 1 810.4.e.bg 8
9.c even 3 1 810.4.a.t ✓ 4
9.c even 3 1 inner 810.4.e.bh 8
9.d odd 6 1 810.4.a.u yes 4
9.d odd 6 1 810.4.e.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.4.a.t ✓ 4 9.c even 3 1
810.4.a.u yes 4 9.d odd 6 1
810.4.e.bg 8 3.b odd 2 1
810.4.e.bg 8 9.d odd 6 1
810.4.e.bh 8 1.a even 1 1 trivial
810.4.e.bh 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{8} + 2 T_{7}^{7} + 1357 T_{7}^{6} + 11666 T_{7}^{5} + 1655167 T_{7}^{4} + 8963402 T_{7}^{3} + \cdots + 36029354596 \) Copy content Toggle raw display
\( T_{11}^{8} + 36 T_{11}^{7} + 4347 T_{11}^{6} + 37908 T_{11}^{5} + 9830565 T_{11}^{4} + \cdots + 4568598455184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 36029354596 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 4568598455184 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13888501585984 \) Copy content Toggle raw display
$17$ \( (T^{4} + 90 T^{3} + \cdots + 769176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 82 T - 2207)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1557623810304 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{4} - 686 T^{3} + \cdots - 6422144)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 94\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + 168 T^{3} + \cdots + 1171497654)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + 348 T^{3} + \cdots - 247505868)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 94 T^{3} + \cdots - 38907781064)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1128 T^{3} + \cdots + 211674255696)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
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