Newspace parameters
| 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
| Self dual: | no |
| Analytic conductor: | \(1.86849940730\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 12.0.157365759791601.1 |
|
|
|
| Defining polynomial: |
\( x^{12} - 3x^{11} + x^{10} + 11x^{8} - 6x^{7} - 17x^{6} - 12x^{5} + 44x^{4} + 16x^{2} - 96x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 3x^{11} + x^{10} + 11x^{8} - 6x^{7} - 17x^{6} - 12x^{5} + 44x^{4} + 16x^{2} - 96x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5 \nu^{11} - 3 \nu^{10} - 7 \nu^{9} - 20 \nu^{8} + 23 \nu^{7} + 46 \nu^{6} + 11 \nu^{5} - 96 \nu^{4} + \cdots - 128 ) / 32 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3 \nu^{11} - 9 \nu^{10} - 9 \nu^{9} + 4 \nu^{8} + 53 \nu^{7} + 14 \nu^{6} - 87 \nu^{5} - 124 \nu^{4} + \cdots - 256 ) / 32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{11} + 5 \nu^{10} + 7 \nu^{9} + 12 \nu^{8} - 25 \nu^{7} - 32 \nu^{6} + 11 \nu^{5} + 82 \nu^{4} + \cdots + 128 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5 \nu^{11} + 11 \nu^{10} + 3 \nu^{9} - 51 \nu^{7} - 6 \nu^{6} + 81 \nu^{5} + 128 \nu^{4} + \cdots + 352 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5 \nu^{11} + 10 \nu^{10} + 6 \nu^{9} + 3 \nu^{8} - 55 \nu^{7} - 21 \nu^{6} + 79 \nu^{5} + 141 \nu^{4} + \cdots + 320 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 7 \nu^{11} + 21 \nu^{10} + \nu^{9} - 8 \nu^{8} - 101 \nu^{7} + 10 \nu^{6} + 191 \nu^{5} + 212 \nu^{4} + \cdots + 736 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5 \nu^{11} - 12 \nu^{10} - 6 \nu^{9} + \nu^{8} + 61 \nu^{7} + 11 \nu^{6} - 93 \nu^{5} - 139 \nu^{4} + \cdots - 352 ) / 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 19 \nu^{11} + 25 \nu^{10} + 29 \nu^{9} + 40 \nu^{8} - 153 \nu^{7} - 134 \nu^{6} + 131 \nu^{5} + \cdots + 736 ) / 32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 4 \nu^{11} + 7 \nu^{10} + 5 \nu^{9} + 7 \nu^{8} - 36 \nu^{7} - 25 \nu^{6} + 36 \nu^{5} + 103 \nu^{4} + \cdots + 184 ) / 8 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 17 \nu^{11} + 29 \nu^{10} + 21 \nu^{9} + 30 \nu^{8} - 151 \nu^{7} - 108 \nu^{6} + 145 \nu^{5} + \cdots + 768 ) / 32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 7 \nu^{11} - 10 \nu^{10} - 10 \nu^{9} - 15 \nu^{8} + 55 \nu^{7} + 49 \nu^{6} - 45 \nu^{5} - 165 \nu^{4} + \cdots - 256 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{11} - 2\beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + \cdots + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{3} + \beta_{2} + 3\beta _1 + 6 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + 6 \beta_{4} + \cdots - 2 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 6 \beta_{11} - 5 \beta_{10} - \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots + 3 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -3\beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{7} - 3\beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta _1 - 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6 \beta_{11} - 4 \beta_{10} + 13 \beta_{9} + 3 \beta_{8} + 16 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + \cdots - 2 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 17 \beta_{11} - 10 \beta_{10} - 7 \beta_{9} - 11 \beta_{8} + 2 \beta_{7} + 14 \beta_{6} - 16 \beta_{5} + \cdots - 25 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 17 \beta_{11} - 13 \beta_{10} + 27 \beta_{9} + 9 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 19 \beta_{5} + \cdots - 8 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - \beta_{11} + 10 \beta_{10} + 13 \beta_{9} - 4 \beta_{8} + 25 \beta_{7} + 22 \beta_{6} - 47 \beta_{5} + \cdots + 4 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 16 \beta_{11} + 12 \beta_{10} - 4 \beta_{9} - 46 \beta_{8} + 11 \beta_{7} + 9 \beta_{6} - 15 \beta_{5} + \cdots - 25 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) | \(\beta_{1}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 133.1 |
|
1.00000 | −1.66659 | − | 0.471659i | 1.00000 | 1.09436 | + | 1.89549i | −1.66659 | − | 0.471659i | 0.790433 | + | 1.36907i | 1.00000 | 2.55508 | + | 1.57213i | 1.09436 | + | 1.89549i | ||||||||||||||||||||||||||||||||||||||||||
| 133.2 | 1.00000 | −1.65432 | + | 0.513065i | 1.00000 | −1.57078 | − | 2.72066i | −1.65432 | + | 0.513065i | −2.01009 | − | 3.48158i | 1.00000 | 2.47353 | − | 1.69754i | −1.57078 | − | 2.72066i | |||||||||||||||||||||||||||||||||||||||||||
| 133.3 | 1.00000 | −0.801583 | + | 1.53540i | 1.00000 | 0.864869 | + | 1.49800i | −0.801583 | + | 1.53540i | 0.163072 | + | 0.282449i | 1.00000 | −1.71493 | − | 2.46151i | 0.864869 | + | 1.49800i | |||||||||||||||||||||||||||||||||||||||||||
| 133.4 | 1.00000 | 1.15754 | − | 1.28845i | 1.00000 | 0.635090 | + | 1.10001i | 1.15754 | − | 1.28845i | 0.919212 | + | 1.59212i | 1.00000 | −0.320191 | − | 2.98286i | 0.635090 | + | 1.10001i | |||||||||||||||||||||||||||||||||||||||||||
| 133.5 | 1.00000 | 1.25033 | + | 1.19862i | 1.00000 | 1.44068 | + | 2.49532i | 1.25033 | + | 1.19862i | −2.39007 | − | 4.13973i | 1.00000 | 0.126635 | + | 2.99733i | 1.44068 | + | 2.49532i | |||||||||||||||||||||||||||||||||||||||||||
| 133.6 | 1.00000 | 1.71463 | + | 0.245071i | 1.00000 | −1.96422 | − | 3.40213i | 1.71463 | + | 0.245071i | 0.0274495 | + | 0.0475438i | 1.00000 | 2.87988 | + | 0.840411i | −1.96422 | − | 3.40213i | |||||||||||||||||||||||||||||||||||||||||||
| 139.1 | 1.00000 | −1.66659 | + | 0.471659i | 1.00000 | 1.09436 | − | 1.89549i | −1.66659 | + | 0.471659i | 0.790433 | − | 1.36907i | 1.00000 | 2.55508 | − | 1.57213i | 1.09436 | − | 1.89549i | |||||||||||||||||||||||||||||||||||||||||||
| 139.2 | 1.00000 | −1.65432 | − | 0.513065i | 1.00000 | −1.57078 | + | 2.72066i | −1.65432 | − | 0.513065i | −2.01009 | + | 3.48158i | 1.00000 | 2.47353 | + | 1.69754i | −1.57078 | + | 2.72066i | |||||||||||||||||||||||||||||||||||||||||||
| 139.3 | 1.00000 | −0.801583 | − | 1.53540i | 1.00000 | 0.864869 | − | 1.49800i | −0.801583 | − | 1.53540i | 0.163072 | − | 0.282449i | 1.00000 | −1.71493 | + | 2.46151i | 0.864869 | − | 1.49800i | |||||||||||||||||||||||||||||||||||||||||||
| 139.4 | 1.00000 | 1.15754 | + | 1.28845i | 1.00000 | 0.635090 | − | 1.10001i | 1.15754 | + | 1.28845i | 0.919212 | − | 1.59212i | 1.00000 | −0.320191 | + | 2.98286i | 0.635090 | − | 1.10001i | |||||||||||||||||||||||||||||||||||||||||||
| 139.5 | 1.00000 | 1.25033 | − | 1.19862i | 1.00000 | 1.44068 | − | 2.49532i | 1.25033 | − | 1.19862i | −2.39007 | + | 4.13973i | 1.00000 | 0.126635 | − | 2.99733i | 1.44068 | − | 2.49532i | |||||||||||||||||||||||||||||||||||||||||||
| 139.6 | 1.00000 | 1.71463 | − | 0.245071i | 1.00000 | −1.96422 | + | 3.40213i | 1.71463 | − | 0.245071i | 0.0274495 | − | 0.0475438i | 1.00000 | 2.87988 | − | 0.840411i | −1.96422 | + | 3.40213i | |||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 12 |
| 3.b | odd | 2 | 1 | 702.2.f.c | 12 | ||
| 9.c | even | 3 | 1 | 234.2.g.c | yes | 12 | |
| 9.d | odd | 6 | 1 | 702.2.g.d | 12 | ||
| 13.c | even | 3 | 1 | 234.2.g.c | yes | 12 | |
| 39.i | odd | 6 | 1 | 702.2.g.d | 12 | ||
| 117.h | even | 3 | 1 | inner | 234.2.f.d | ✓ | 12 |
| 117.k | odd | 6 | 1 | 702.2.f.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.2.f.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 234.2.f.d | ✓ | 12 | 117.h | even | 3 | 1 | inner |
| 234.2.g.c | yes | 12 | 9.c | even | 3 | 1 | |
| 234.2.g.c | yes | 12 | 13.c | even | 3 | 1 | |
| 702.2.f.c | 12 | 3.b | odd | 2 | 1 | ||
| 702.2.f.c | 12 | 117.k | odd | 6 | 1 | ||
| 702.2.g.d | 12 | 9.d | odd | 6 | 1 | ||
| 702.2.g.d | 12 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - T_{5}^{11} + 22 T_{5}^{10} - 55 T_{5}^{9} + 385 T_{5}^{8} - 883 T_{5}^{7} + 3487 T_{5}^{6} + \cdots + 29241 \)
acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{12} \)
|
| $3$ |
\( T^{12} - 6 T^{10} + \cdots + 729 \)
|
| $5$ |
\( T^{12} - T^{11} + \cdots + 29241 \)
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| $7$ |
\( T^{12} + 5 T^{11} + \cdots + 1 \)
|
| $11$ |
\( (T^{6} + 8 T^{5} + 6 T^{4} + \cdots + 9)^{2} \)
|
| $13$ |
\( T^{12} + T^{11} + \cdots + 4826809 \)
|
| $17$ |
\( T^{12} - 3 T^{11} + \cdots + 69605649 \)
|
| $19$ |
\( T^{12} + 7 T^{11} + \cdots + 77841 \)
|
| $23$ |
\( T^{12} - T^{11} + \cdots + 6561 \)
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| $29$ |
\( (T^{6} + T^{5} - 84 T^{4} + \cdots - 4113)^{2} \)
|
| $31$ |
\( T^{12} + 14 T^{11} + \cdots + 657721 \)
|
| $37$ |
\( T^{12} + 7 T^{11} + \cdots + 81 \)
|
| $41$ |
\( T^{12} - 9 T^{11} + \cdots + 5517801 \)
|
| $43$ |
\( T^{12} + \cdots + 324684361 \)
|
| $47$ |
\( T^{12} + \cdots + 189692478369 \)
|
| $53$ |
\( (T^{6} + 6 T^{5} + \cdots + 8667)^{2} \)
|
| $59$ |
\( (T^{6} + 4 T^{5} + \cdots + 927)^{2} \)
|
| $61$ |
\( T^{12} - 28 T^{11} + \cdots + 97871449 \)
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| $67$ |
\( T^{12} + \cdots + 5143614961 \)
|
| $71$ |
\( T^{12} + 13 T^{11} + \cdots + 431649 \)
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| $73$ |
\( (T^{6} - 27 T^{5} + \cdots - 1403)^{2} \)
|
| $79$ |
\( T^{12} + 6 T^{11} + \cdots + 28270489 \)
|
| $83$ |
\( T^{12} + \cdots + 3799612881 \)
|
| $89$ |
\( T^{12} + \cdots + 89771545161 \)
|
| $97$ |
\( T^{12} + \cdots + 3329636209 \)
|
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