Newspace parameters
| Level: | \( N \) | \(=\) | \( 341 = 11 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 341.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.72289870893\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 6x^{6} + 30x^{5} + 9x^{4} - 58x^{3} - 15x^{2} + 32x + 12 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 6x^{6} + 30x^{5} + 9x^{4} - 58x^{3} - 15x^{2} + 32x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 7\nu + 8 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 6\nu^{5} + 30\nu^{4} + 7\nu^{3} - 54\nu^{2} - \nu + 18 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 6\nu^{6} - 2\nu^{5} - 38\nu^{4} + 39\nu^{3} + 48\nu^{2} - 33\nu - 14 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 6\nu^{6} - 2\nu^{5} - 38\nu^{4} + 39\nu^{3} + 52\nu^{2} - 37\nu - 26 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - \nu^{5} + 31\nu^{4} - 23\nu^{3} - 35\nu^{2} + 27\nu + 10 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 5\nu^{5} + 27\nu^{4} + 2\nu^{3} - 40\nu^{2} - 2\nu + 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 2\beta_{3} + 2\beta_{2} + 14\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{7} - 11\beta_{6} + 18\beta_{5} - 29\beta_{4} - 15\beta_{3} + 6\beta_{2} + 64\beta _1 + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37\beta_{7} - 29\beta_{6} + 88\beta_{5} - 115\beta_{4} - 43\beta_{3} + 32\beta_{2} + 171\beta _1 + 158 \)
|
| \(\nu^{7}\) | \(=\) |
\( 159\beta_{7} - 115\beta_{6} + 237\beta_{5} - 344\beta_{4} - 191\beta_{3} + 104\beta_{2} + 654\beta _1 + 401 \)
|
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 |
|
−2.49914 | −1.45657 | 4.24570 | −3.82388 | 3.64018 | −1.64018 | −5.61232 | −0.878397 | 9.55640 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.29465 | 1.42477 | 3.26544 | 0.129361 | −3.26936 | 5.26936 | −2.90374 | −0.970023 | −0.296838 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.11742 | −0.699619 | −0.751369 | 3.56228 | 0.781769 | 1.21823 | 3.07444 | −2.51053 | −3.98057 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.177928 | 3.17990 | −1.96834 | −0.963523 | −0.565793 | 2.56579 | 0.706079 | 7.11176 | 0.171438 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.231003 | −2.34769 | −1.94664 | −1.91268 | −0.542323 | 2.54232 | −0.911685 | 2.51163 | −0.441835 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.77567 | 1.82232 | 1.15300 | 1.34847 | 3.23584 | −1.23584 | −1.50399 | 0.320854 | 2.39443 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.33332 | −0.0240968 | 3.44438 | 0.530938 | −0.0562256 | 2.05623 | 3.37021 | −2.99942 | 1.23885 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.74915 | 2.10098 | 5.55783 | −3.87097 | 5.77591 | −3.77591 | 9.78100 | 1.41412 | −10.6419 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 341.2.a.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3069.2.a.k | 8 | ||
| 4.b | odd | 2 | 1 | 5456.2.a.ba | 8 | ||
| 5.b | even | 2 | 1 | 8525.2.a.g | 8 | ||
| 11.b | odd | 2 | 1 | 3751.2.a.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 341.2.a.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3069.2.a.k | 8 | 3.b | odd | 2 | 1 | ||
| 3751.2.a.h | 8 | 11.b | odd | 2 | 1 | ||
| 5456.2.a.ba | 8 | 4.b | odd | 2 | 1 | ||
| 8525.2.a.g | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 14T_{2}^{6} + 11T_{2}^{5} + 60T_{2}^{4} - 31T_{2}^{3} - 74T_{2}^{2} + 5T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(341))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 14 T^{6} + \cdots + 3 \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
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| $5$ |
\( T^{8} + 5 T^{7} + \cdots + 9 \)
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| $7$ |
\( T^{8} - 7 T^{7} + \cdots - 659 \)
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| $11$ |
\( (T + 1)^{8} \)
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| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 37 \)
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| $17$ |
\( T^{8} - 44 T^{6} + \cdots + 219 \)
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| $19$ |
\( T^{8} - 16 T^{7} + \cdots - 4996 \)
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| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 372 \)
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| $29$ |
\( T^{8} - 8 T^{7} + \cdots - 31728 \)
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| $31$ |
\( (T - 1)^{8} \)
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| $37$ |
\( T^{8} - 8 T^{7} + \cdots - 91648 \)
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| $41$ |
\( T^{8} - 3 T^{7} + \cdots + 28224 \)
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| $43$ |
\( T^{8} - 27 T^{7} + \cdots + 181312 \)
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| $47$ |
\( T^{8} + 21 T^{7} + \cdots + 852672 \)
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| $53$ |
\( T^{8} + T^{7} + \cdots - 315456 \)
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| $59$ |
\( T^{8} - 16 T^{7} + \cdots + 250368 \)
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| $61$ |
\( T^{8} - 23 T^{7} + \cdots - 138205 \)
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| $67$ |
\( T^{8} + 10 T^{7} + \cdots - 43339328 \)
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| $71$ |
\( T^{8} + 3 T^{7} + \cdots + 60864 \)
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| $73$ |
\( T^{8} - 8 T^{7} + \cdots - 442769 \)
|
| $79$ |
\( T^{8} - 32 T^{7} + \cdots - 67328192 \)
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| $83$ |
\( T^{8} + 7 T^{7} + \cdots - 960000 \)
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| $89$ |
\( T^{8} + 49 T^{7} + \cdots + 31996224 \)
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| $97$ |
\( T^{8} + 18 T^{7} + \cdots - 377968 \)
|
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