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Properties

Label 234.8.h.c
Level $234$
Weight $8$
Character orbit 234.h
Analytic conductor $73.098$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [234,8,Mod(55,234)] 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("234.55"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 234.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-32,0,-256,386] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.0980959633\)
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: \( x^{8} - 3 x^{7} - 14114 x^{6} + 42351 x^{5} + 205543918 x^{4} + 13390412127 x^{3} + \cdots + 28150431267204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 78)
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.

\(f(q)\) \(=\) \( q + ( - 8 \beta_1 - 8) q^{2} + 64 \beta_1 q^{4} + (\beta_{4} + \beta_{2} + 48) q^{5} + ( - \beta_{6} - 189 \beta_1) q^{7} + 512 q^{8} + ( - 8 \beta_{4} - 384 \beta_1 - 384) q^{10} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \cdots - 1130) q^{11}+ \cdots + ( - 1488 \beta_{7} + \cdots - 1059168 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} - 256 q^{4} + 386 q^{5} + 757 q^{7} + 4096 q^{8} - 1544 q^{10} - 4524 q^{11} + 13272 q^{13} - 12112 q^{14} - 16384 q^{16} + 12775 q^{17} + 38646 q^{19} - 12352 q^{20} - 36192 q^{22} - 24428 q^{23}+ \cdots + 4224696 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3 x^{7} - 14114 x^{6} + 42351 x^{5} + 205543918 x^{4} + 13390412127 x^{3} + \cdots + 28150431267204 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!43 \nu^{7} + \cdots + 18\!\cdots\!38 ) / 44\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!68 \nu^{7} + \cdots - 22\!\cdots\!78 ) / 30\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37\!\cdots\!43 \nu^{7} + \cdots - 15\!\cdots\!52 ) / 90\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!07 \nu^{7} + \cdots - 15\!\cdots\!92 ) / 30\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!37 \nu^{7} + \cdots + 27\!\cdots\!52 ) / 18\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!52 \nu^{7} + \cdots + 35\!\cdots\!02 ) / 90\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\!\cdots\!47 \nu^{7} + \cdots + 20\!\cdots\!78 ) / 20\!\cdots\!47 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 20\beta_{6} - \beta_{5} + 13\beta_{4} + 10\beta_{3} + 26\beta_{2} - 44\beta _1 + 44 ) / 180 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 23\beta_{7} + 80\beta_{6} + 23\beta_{5} + 411\beta_{2} - 141108\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4561 \beta_{7} + 73375 \beta_{6} - 9122 \beta_{5} - 111088 \beta_{4} - 73375 \beta_{3} + \cdots - 22284344 ) / 90 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 108129\beta_{5} - 3623123\beta_{4} - 1116100\beta_{3} - 1058814544\beta _1 - 1058814544 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 66686011 \beta_{7} - 2372463970 \beta_{6} - 66686011 \beta_{5} - 8880375206 \beta_{4} + \cdots - 2100008436088 ) / 180 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 438265541 \beta_{7} - 9638713896 \beta_{6} - 25874321249 \beta_{4} - 9638713896 \beta_{3} + \cdots - 7138089139324 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 354700794104 \beta_{7} - 41056359865910 \beta_{6} + 177350397052 \beta_{5} - 42822569250241 \beta_{4} + \cdots - 10\!\cdots\!68 ) / 90 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−21.4559 + 12.3876i
−14.5673 + 8.41041i
−80.7401 + 46.6153i
118.263 − 68.2794i
−21.4559 − 12.3876i
−14.5673 − 8.41041i
−80.7401 − 46.6153i
118.263 + 68.2794i
−4.00000 − 6.92820i 0 −32.0000 + 55.4256i −443.927 0 −163.897 + 283.878i 512.000 0 1775.71 + 3075.62i
55.2 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −76.2027 0 237.214 − 410.867i 512.000 0 304.811 + 527.947i
55.3 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 292.365 0 682.601 − 1182.30i 512.000 0 −1169.46 − 2025.56i
55.4 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 420.765 0 −377.418 + 653.707i 512.000 0 −1683.06 − 2915.15i
217.1 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −443.927 0 −163.897 − 283.878i 512.000 0 1775.71 − 3075.62i
217.2 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −76.2027 0 237.214 + 410.867i 512.000 0 304.811 − 527.947i
217.3 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 292.365 0 682.601 + 1182.30i 512.000 0 −1169.46 + 2025.56i
217.4 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 420.765 0 −377.418 − 653.707i 512.000 0 −1683.06 + 2915.15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.8.h.c 8
3.b odd 2 1 78.8.e.a ✓ 8
13.c even 3 1 inner 234.8.h.c 8
39.i odd 6 1 78.8.e.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.8.e.a ✓ 8 3.b odd 2 1
78.8.e.a ✓ 8 39.i odd 6 1
234.8.h.c 8 1.a even 1 1 trivial
234.8.h.c 8 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 193T_{5}^{3} - 214075T_{5}^{2} + 39860725T_{5} + 4161471750 \) acting on \(S_{8}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 193 T^{3} + \cdots + 4161471750)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots - 43\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 16\!\cdots\!75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 36\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
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