Newspace parameters
| Level: | \( N \) | \(=\) | \( 235 = 5 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 235.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.87648444750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 28x^{3} - 17x^{2} - 19x + 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 28x^{3} - 17x^{2} - 19x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 8\nu^{4} - 2\nu^{3} + 12\nu^{2} + 5\nu + 2 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 8\nu^{4} + 4\nu^{3} - 14\nu^{2} - 13\nu + 2 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 10\nu^{4} - 24\nu^{2} + 3\nu + 6 ) / 2 \)
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| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 10\nu^{4} - 18\nu^{3} + 22\nu^{2} + 31\nu - 4 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta_{3} + 7\beta_{2} + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 10\beta_{2} + 19\beta _1 + 11 \)
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| \(\nu^{6}\) | \(=\) |
\( 8\beta_{5} + 10\beta_{4} + 20\beta_{3} + 46\beta_{2} + 3\beta _1 + 84 \)
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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 |
|
−2.24729 | −2.14783 | 3.05029 | 1.00000 | 4.82679 | −4.71377 | −2.36031 | 1.61318 | −2.24729 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.89749 | 1.57213 | 1.60046 | 1.00000 | −2.98310 | 0.327488 | 0.758120 | −0.528410 | −1.89749 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.730051 | 3.14616 | −1.46703 | 1.00000 | −2.29686 | −0.964297 | 2.53111 | 6.89835 | −0.730051 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0980806 | −3.03215 | −1.99038 | 1.00000 | −0.297395 | 0.786854 | −0.391379 | 6.19391 | 0.0980806 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.49903 | 2.14267 | 0.247101 | 1.00000 | 3.21194 | 0.914748 | −2.62765 | 1.59105 | 1.49903 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.68601 | −0.335153 | 0.842624 | 1.00000 | −0.565071 | 4.20242 | −1.95135 | −2.88767 | 1.68601 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.59170 | −0.345837 | 4.71692 | 1.00000 | −0.896307 | −3.55345 | 7.04146 | −2.88040 | 2.59170 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(47\) | \( +1 \) |
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 | 235.2.a.e | ✓ | 7 |
| 3.b | odd | 2 | 1 | 2115.2.a.v | 7 | ||
| 4.b | odd | 2 | 1 | 3760.2.a.bi | 7 | ||
| 5.b | even | 2 | 1 | 1175.2.a.h | 7 | ||
| 5.c | odd | 4 | 2 | 1175.2.c.g | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 235.2.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1175.2.a.h | 7 | 5.b | even | 2 | 1 | ||
| 1175.2.c.g | 14 | 5.c | odd | 4 | 2 | ||
| 2115.2.a.v | 7 | 3.b | odd | 2 | 1 | ||
| 3760.2.a.bi | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(235))\):
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\( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 8T_{2}^{4} + 28T_{2}^{3} - 17T_{2}^{2} - 19T_{2} + 2 \)
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\( T_{11}^{7} - T_{11}^{6} - 46T_{11}^{5} + 40T_{11}^{4} + 512T_{11}^{3} - 80T_{11}^{2} - 1408T_{11} - 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} - 10 T^{5} + \cdots + 2 \)
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| $3$ |
\( T^{7} - T^{6} - 15 T^{5} + \cdots - 8 \)
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| $5$ |
\( (T - 1)^{7} \)
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| $7$ |
\( T^{7} + 3 T^{6} + \cdots + 16 \)
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| $11$ |
\( T^{7} - T^{6} + \cdots - 256 \)
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| $13$ |
\( T^{7} - 2 T^{6} + \cdots + 32 \)
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| $17$ |
\( T^{7} - 12 T^{6} + \cdots - 3424 \)
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| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 4352 \)
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| $23$ |
\( T^{7} - T^{6} + \cdots - 152576 \)
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| $29$ |
\( T^{7} - 26 T^{6} + \cdots + 1024 \)
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| $31$ |
\( T^{7} + 5 T^{6} + \cdots + 1024 \)
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| $37$ |
\( T^{7} - 201 T^{5} + \cdots - 397984 \)
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| $41$ |
\( T^{7} - 12 T^{6} + \cdots - 54784 \)
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| $43$ |
\( T^{7} + 33 T^{6} + \cdots + 65536 \)
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| $47$ |
\( (T + 1)^{7} \)
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| $53$ |
\( T^{7} - 4 T^{6} + \cdots - 16448 \)
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| $59$ |
\( T^{7} - 13 T^{6} + \cdots - 191656 \)
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| $61$ |
\( T^{7} - 20 T^{6} + \cdots - 6218 \)
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| $67$ |
\( T^{7} + 32 T^{6} + \cdots + 200704 \)
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| $71$ |
\( T^{7} - 11 T^{6} + \cdots + 550688 \)
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| $73$ |
\( T^{7} + 8 T^{6} + \cdots + 31648 \)
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| $79$ |
\( T^{7} - 17 T^{6} + \cdots + 1921952 \)
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| $83$ |
\( T^{7} - 19 T^{6} + \cdots - 2048 \)
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| $89$ |
\( T^{7} - 13 T^{6} + \cdots + 14252 \)
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| $97$ |
\( T^{7} + 12 T^{6} + \cdots + 473984 \)
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