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Properties

Label 234.2.f.c
Level $234$
Weight $2$
Character orbit 234.f
Analytic conductor $1.868$
Analytic rank $0$
Dimension $12$
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] = [234,2,Mod(133,234)] 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("234.133"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12] 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 4 x^{10} - 6 x^{9} + 22 x^{8} - 45 x^{7} + 75 x^{6} - 135 x^{5} + 198 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + q^{4} + ( - \beta_{8} + \beta_{6} + \cdots + \beta_1) q^{5} + (\beta_{6} + \beta_{3}) q^{6} + ( - \beta_{11} + \beta_{4}) q^{7} - q^{8} + \beta_{10} q^{9}+ \cdots + (\beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 8 q^{11} - 5 q^{13} - q^{14} + 11 q^{15} + 12 q^{16} - q^{17} - 4 q^{18} + 9 q^{19} - 3 q^{20} + 8 q^{21} + 8 q^{22} + 7 q^{23}+ \cdots + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 4 x^{10} - 6 x^{9} + 22 x^{8} - 45 x^{7} + 75 x^{6} - 135 x^{5} + 198 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4 \nu^{11} + 63 \nu^{10} - 236 \nu^{9} + 465 \nu^{8} - 875 \nu^{7} + 1578 \nu^{6} - 2697 \nu^{5} + \cdots + 7047 ) / 3159 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8 \nu^{11} + 9 \nu^{10} - 4 \nu^{9} - 240 \nu^{8} + 473 \nu^{7} - 471 \nu^{6} + 1392 \nu^{5} + \cdots - 4860 ) / 3159 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 7 \nu^{10} - 19 \nu^{9} + 4 \nu^{8} + 14 \nu^{7} - 11 \nu^{6} + 21 \nu^{5} - 105 \nu^{4} + \cdots - 2025 ) / 351 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} - 6 \nu^{8} + 22 \nu^{7} - 45 \nu^{6} + 75 \nu^{5} - 135 \nu^{4} + \cdots - 729 ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} - 33 \nu^{8} + 103 \nu^{7} - 153 \nu^{6} + 237 \nu^{5} - 486 \nu^{4} + \cdots - 2430 ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25 \nu^{11} - 84 \nu^{10} + 163 \nu^{9} - 321 \nu^{8} + 586 \nu^{7} - 999 \nu^{6} + 1776 \nu^{5} + \cdots - 972 ) / 3159 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29 \nu^{11} - 138 \nu^{10} + 278 \nu^{9} - 324 \nu^{8} + 413 \nu^{7} - 942 \nu^{6} + 1185 \nu^{5} + \cdots + 6075 ) / 3159 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29 \nu^{11} - 99 \nu^{10} - 73 \nu^{9} + 534 \nu^{8} - 757 \nu^{7} + 1320 \nu^{6} - 2559 \nu^{5} + \cdots + 9234 ) / 3159 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 6 \nu^{8} - 11 \nu^{7} - 27 \nu^{6} - 6 \nu^{5} + \cdots + 243 \nu ) / 243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} - 4 \beta_{9} + 4 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{11} + 3 \beta_{10} + \beta_{9} - 7 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 6 \beta_{5} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - 8 \beta_{8} + 9 \beta_{7} - 13 \beta_{6} - 10 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6 \beta_{11} - 6 \beta_{9} - 21 \beta_{8} + 3 \beta_{7} + 21 \beta_{6} - 9 \beta_{5} - 6 \beta_{4} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{11} - 12 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 6 \beta_{7} + 4 \beta_{6} - 27 \beta_{5} + \cdots - 58 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17 \beta_{11} - 45 \beta_{10} - \beta_{9} - 35 \beta_{8} + 19 \beta_{7} + 37 \beta_{6} - 48 \beta_{5} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18 \beta_{11} - 71 \beta_{10} + 59 \beta_{9} - 59 \beta_{8} + 33 \beta_{7} + 147 \beta_{6} + \cdots + 80 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1
−1.26151 + 1.18685i
−0.141176 + 1.72629i
1.41495 + 0.998953i
−1.29053 − 1.15522i
1.62550 − 0.598110i
1.15275 − 1.29273i
−1.26151 − 1.18685i
−0.141176 − 1.72629i
1.41495 − 0.998953i
−1.29053 + 1.15522i
1.62550 + 0.598110i
1.15275 + 1.29273i
−1.00000 −1.65859 + 0.499074i 1.00000 −0.102915 − 0.178255i 1.65859 − 0.499074i 0.779577 + 1.35027i −1.00000 2.50185 − 1.65552i 0.102915 + 0.178255i
133.2 −1.00000 −1.56560 − 0.740882i 1.00000 0.924421 + 1.60114i 1.56560 + 0.740882i −1.51269 − 2.62006i −1.00000 1.90219 + 2.31985i −0.924421 − 1.60114i
133.3 −1.00000 −0.157642 − 1.72486i 1.00000 1.07260 + 1.85779i 0.157642 + 1.72486i 2.14998 + 3.72388i −1.00000 −2.95030 + 0.543822i −1.07260 − 1.85779i
133.4 −1.00000 0.355184 + 1.69524i 1.00000 −2.14571 − 3.71649i −0.355184 − 1.69524i −0.751135 − 1.30100i −1.00000 −2.74769 + 1.20424i 2.14571 + 3.71649i
133.5 −1.00000 1.33073 − 1.10867i 1.00000 −0.205226 − 0.355461i −1.33073 + 1.10867i −2.22222 − 3.84900i −1.00000 0.541686 − 2.95069i 0.205226 + 0.355461i
133.6 −1.00000 1.69592 − 0.351949i 1.00000 −1.04316 − 1.80681i −1.69592 + 0.351949i 2.05649 + 3.56194i −1.00000 2.75226 − 1.19375i 1.04316 + 1.80681i
139.1 −1.00000 −1.65859 − 0.499074i 1.00000 −0.102915 + 0.178255i 1.65859 + 0.499074i 0.779577 − 1.35027i −1.00000 2.50185 + 1.65552i 0.102915 − 0.178255i
139.2 −1.00000 −1.56560 + 0.740882i 1.00000 0.924421 − 1.60114i 1.56560 − 0.740882i −1.51269 + 2.62006i −1.00000 1.90219 − 2.31985i −0.924421 + 1.60114i
139.3 −1.00000 −0.157642 + 1.72486i 1.00000 1.07260 − 1.85779i 0.157642 − 1.72486i 2.14998 − 3.72388i −1.00000 −2.95030 − 0.543822i −1.07260 + 1.85779i
139.4 −1.00000 0.355184 − 1.69524i 1.00000 −2.14571 + 3.71649i −0.355184 + 1.69524i −0.751135 + 1.30100i −1.00000 −2.74769 − 1.20424i 2.14571 − 3.71649i
139.5 −1.00000 1.33073 + 1.10867i 1.00000 −0.205226 + 0.355461i −1.33073 − 1.10867i −2.22222 + 3.84900i −1.00000 0.541686 + 2.95069i 0.205226 − 0.355461i
139.6 −1.00000 1.69592 + 0.351949i 1.00000 −1.04316 + 1.80681i −1.69592 − 0.351949i 2.05649 − 3.56194i −1.00000 2.75226 + 1.19375i 1.04316 − 1.80681i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.f.c ✓ 12
3.b odd 2 1 702.2.f.d 12
9.c even 3 1 234.2.g.d yes 12
9.d odd 6 1 702.2.g.c 12
13.c even 3 1 234.2.g.d yes 12
39.i odd 6 1 702.2.g.c 12
117.h even 3 1 inner 234.2.f.c ✓ 12
117.k odd 6 1 702.2.f.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.f.c ✓ 12 1.a even 1 1 trivial
234.2.f.c ✓ 12 117.h even 3 1 inner
234.2.g.d yes 12 9.c even 3 1
234.2.g.d yes 12 13.c even 3 1
702.2.f.d 12 3.b odd 2 1
702.2.f.d 12 117.k odd 6 1
702.2.g.c 12 9.d odd 6 1
702.2.g.c 12 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 3 T_{5}^{11} + 20 T_{5}^{10} + 3 T_{5}^{9} + 147 T_{5}^{8} + 51 T_{5}^{7} + 575 T_{5}^{6} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 310249 \) Copy content Toggle raw display
$11$ \( (T^{6} + 4 T^{5} - 24 T^{4} + \cdots + 63)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 5 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 592094889 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots + 33074001 \) Copy content Toggle raw display
$29$ \( (T^{6} + T^{5} - 50 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 8 T^{11} + \cdots + 1062961 \) Copy content Toggle raw display
$37$ \( T^{12} - 15 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 7784003529 \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{11} + \cdots + 41209 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 8579390625 \) Copy content Toggle raw display
$53$ \( (T^{6} + 2 T^{5} - 101 T^{4} + \cdots - 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 4 T^{5} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 19140625 \) Copy content Toggle raw display
$67$ \( T^{12} - 14 T^{11} + \cdots + 1934881 \) Copy content Toggle raw display
$71$ \( T^{12} + 15 T^{11} + \cdots + 45369 \) Copy content Toggle raw display
$73$ \( (T^{6} + 45 T^{5} + \cdots - 507731)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24891057361 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1780924401 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 44348569281 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 7628799649 \) Copy content Toggle raw display
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