Newspace parameters
| Level: | \( N \) | \(=\) | \( 113 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 113.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.902309542840\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.24698 | −1.55496 | 3.04892 | 2.60388 | 3.49396 | −4.80194 | −2.35690 | −0.582105 | −5.85086 | |||||||||||||||||||||||||||
| 1.2 | −0.554958 | −0.198062 | −1.69202 | −3.49396 | 0.109916 | −1.75302 | 2.04892 | −2.96077 | 1.93900 | ||||||||||||||||||||||||||||
| 1.3 | 0.801938 | −3.24698 | −1.35690 | −0.109916 | −2.60388 | −3.44504 | −2.69202 | 7.54288 | −0.0881460 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(113\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 113.2.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1017.2.a.q | 3 | ||
| 4.b | odd | 2 | 1 | 1808.2.a.i | 3 | ||
| 5.b | even | 2 | 1 | 2825.2.a.e | 3 | ||
| 7.b | odd | 2 | 1 | 5537.2.a.g | 3 | ||
| 8.b | even | 2 | 1 | 7232.2.a.t | 3 | ||
| 8.d | odd | 2 | 1 | 7232.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 113.2.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1017.2.a.q | 3 | 3.b | odd | 2 | 1 | ||
| 1808.2.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 2825.2.a.e | 3 | 5.b | even | 2 | 1 | ||
| 5537.2.a.g | 3 | 7.b | odd | 2 | 1 | ||
| 7232.2.a.o | 3 | 8.d | odd | 2 | 1 | ||
| 7232.2.a.t | 3 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(113))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2T^{2} - T - 1 \)
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| $3$ |
\( T^{3} + 5 T^{2} + \cdots + 1 \)
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| $5$ |
\( T^{3} + T^{2} - 9T - 1 \)
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| $7$ |
\( T^{3} + 10 T^{2} + \cdots + 29 \)
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| $11$ |
\( T^{3} - 2 T^{2} + \cdots - 13 \)
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| $13$ |
\( T^{3} + 8 T^{2} + \cdots - 43 \)
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| $17$ |
\( T^{3} + 2 T^{2} + \cdots + 13 \)
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| $19$ |
\( T^{3} + 4 T^{2} + \cdots - 1 \)
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| $23$ |
\( T^{3} + 6 T^{2} + \cdots - 27 \)
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| $29$ |
\( T^{3} - 5 T^{2} + \cdots + 97 \)
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| $31$ |
\( T^{3} + 15 T^{2} + \cdots - 211 \)
|
| $37$ |
\( T^{3} + 2 T^{2} + \cdots - 113 \)
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| $41$ |
\( T^{3} - T^{2} + \cdots + 29 \)
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| $43$ |
\( T^{3} + 2 T^{2} + \cdots + 13 \)
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| $47$ |
\( T^{3} + 7 T^{2} + \cdots + 7 \)
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| $53$ |
\( T^{3} + 5 T^{2} + \cdots + 29 \)
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| $59$ |
\( T^{3} - 15 T^{2} + \cdots + 169 \)
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| $61$ |
\( T^{3} + 21 T^{2} + \cdots + 301 \)
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| $67$ |
\( T^{3} - 5 T^{2} + \cdots - 43 \)
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| $71$ |
\( T^{3} - 14T^{2} + 392 \)
|
| $73$ |
\( T^{3} + 11 T^{2} + \cdots + 41 \)
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| $79$ |
\( T^{3} + 5 T^{2} + \cdots - 125 \)
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| $83$ |
\( T^{3} - 14 T^{2} + \cdots - 91 \)
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| $89$ |
\( T^{3} + 16 T^{2} + \cdots - 841 \)
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| $97$ |
\( T^{3} - 217T + 1183 \)
|
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