Newspace parameters
| Level: | \( N \) | \(=\) | \( 515 = 5 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 515.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.11229570410\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.47793675537.1 |
|
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 13x^{6} + 37x^{5} + 43x^{4} - 98x^{3} - 68x^{2} + 72x + 48 \)
|
| 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
\( x^{8} - 3x^{7} - 13x^{6} + 37x^{5} + 43x^{4} - 98x^{3} - 68x^{2} + 72x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 11\nu^{5} - 35\nu^{4} - 17\nu^{3} + 76\nu^{2} - 6\nu - 36 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{7} - 11\nu^{6} - 33\nu^{5} + 129\nu^{4} + 55\nu^{3} - 284\nu^{2} - 8\nu + 144 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 7\nu^{6} + 21\nu^{5} - 81\nu^{4} - 27\nu^{3} + 173\nu^{2} - 4\nu - 88 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 13\nu^{6} + 27\nu^{5} - 155\nu^{4} + 11\nu^{3} + 370\nu^{2} - 100\nu - 240 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 9\nu^{5} + 47\nu^{4} - 3\nu^{3} - 106\nu^{2} + 27\nu + 62 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 12\nu^{6} - 28\nu^{5} + 144\nu^{4} - 349\nu^{2} + 80\nu + 224 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} - \beta_{4} + 2\beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 9\beta_{6} + 12\beta_{5} + 10\beta_{4} + 10\beta_{3} + \beta_{2} + 10\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{7} + 2\beta_{6} + 23\beta_{5} - 10\beta_{4} - \beta_{3} + 21\beta_{2} + 61\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 26\beta_{7} + 80\beta_{6} + 120\beta_{5} + 90\beta_{4} + 88\beta_{3} + 10\beta_{2} + 93\beta _1 + 150 \)
|
| \(\nu^{7}\) | \(=\) |
\( 112\beta_{7} + 23\beta_{6} + 235\beta_{5} - 97\beta_{4} - 21\beta_{3} + 188\beta_{2} + 551\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.35284 | −1.56884 | 3.53587 | −1.00000 | 3.69123 | −1.15094 | −3.61366 | −0.538745 | 2.35284 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.35284 | 2.99947 | 3.53587 | −1.00000 | −7.05728 | −4.46272 | −3.61366 | 5.99683 | 2.35284 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.16027 | −3.03073 | −0.653767 | −1.00000 | 3.51647 | −0.572037 | 3.07909 | 6.18530 | 1.16027 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.16027 | 0.765673 | −0.653767 | −1.00000 | −0.888389 | 1.65113 | 3.07909 | −2.41374 | 1.16027 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.563729 | 0.821037 | −1.68221 | −1.00000 | 0.462842 | −0.0164538 | −2.07577 | −2.32590 | −0.563729 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.563729 | 1.37321 | −1.68221 | −1.00000 | 0.774119 | −4.05931 | −2.07577 | −1.11429 | −0.563729 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.94939 | −3.10363 | 1.80011 | −1.00000 | −6.05017 | 2.07415 | −0.389667 | 6.63252 | −1.94939 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.94939 | −1.25620 | 1.80011 | −1.00000 | −2.44882 | −4.46381 | −0.389667 | −1.42197 | −1.94939 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(103\) | \( +1 \) |
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 | 515.2.a.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 4635.2.a.h | 8 | ||
| 4.b | odd | 2 | 1 | 8240.2.a.bd | 8 | ||
| 5.b | even | 2 | 1 | 2575.2.a.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 515.2.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2575.2.a.n | 8 | 5.b | even | 2 | 1 | ||
| 4635.2.a.h | 8 | 3.b | odd | 2 | 1 | ||
| 8240.2.a.bd | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} - 5T_{2}^{2} - 3T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(515))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} - 5 T^{2} + \cdots + 3)^{2} \)
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| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 48 \)
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| $5$ |
\( (T + 1)^{8} \)
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| $7$ |
\( T^{8} + 11 T^{7} + \cdots + 3 \)
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| $11$ |
\( T^{8} - 5 T^{7} + \cdots + 1296 \)
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| $13$ |
\( T^{8} + 23 T^{7} + \cdots + 81 \)
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| $17$ |
\( T^{8} + 13 T^{7} + \cdots + 43149 \)
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| $19$ |
\( T^{8} + 12 T^{7} + \cdots + 2287 \)
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| $23$ |
\( T^{8} + 9 T^{7} + \cdots - 4313 \)
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| $29$ |
\( T^{8} - 7 T^{7} + \cdots - 705231 \)
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| $31$ |
\( T^{8} + 4 T^{7} + \cdots - 12816 \)
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| $37$ |
\( T^{8} + 27 T^{7} + \cdots - 1100592 \)
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| $41$ |
\( T^{8} - 5 T^{7} + \cdots - 273129 \)
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| $43$ |
\( T^{8} + 2 T^{7} + \cdots + 403408 \)
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| $47$ |
\( T^{8} - 5 T^{7} + \cdots - 1629072 \)
|
| $53$ |
\( T^{8} + 15 T^{7} + \cdots - 1436432 \)
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| $59$ |
\( T^{8} - 3 T^{7} + \cdots - 9210111 \)
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| $61$ |
\( T^{8} + 42 T^{7} + \cdots - 8140767 \)
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| $67$ |
\( T^{8} - 269 T^{6} + \cdots + 8423536 \)
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| $71$ |
\( T^{8} - 23 T^{7} + \cdots - 180464 \)
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| $73$ |
\( T^{8} + 31 T^{7} + \cdots + 1368208 \)
|
| $79$ |
\( T^{8} + 14 T^{7} + \cdots - 35439 \)
|
| $83$ |
\( T^{8} + 22 T^{7} + \cdots - 254313 \)
|
| $89$ |
\( T^{8} - 32 T^{7} + \cdots - 64597712 \)
|
| $97$ |
\( T^{8} + 40 T^{7} + \cdots - 55230927 \)
|
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