Newspace parameters
| Level: | \( N \) | \(=\) | \( 878 = 2 \cdot 439 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 878.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.01086529747\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.139439152.1 |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 17x^{3} - 26x^{2} + 10x - 1 \)
|
| 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
\( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 17x^{3} - 26x^{2} + 10x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} + \nu^{3} + 18\nu^{2} - 8\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 9\nu^{4} - 9\nu^{3} - 18\nu^{2} + 21\nu - 5 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 9\nu^{4} + 10\nu^{3} + 18\nu^{2} - 26\nu + 6 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - \nu^{5} - 18\nu^{4} + 10\nu^{3} + 37\nu^{2} - 29\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( -4\nu^{6} + 2\nu^{5} + 37\nu^{4} - 22\nu^{3} - 78\nu^{2} + 67\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} - \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 6\beta_{5} + 2\beta_{4} + 6\beta_{3} - 4\beta_{2} + \beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( 8\beta_{4} + 9\beta_{3} + \beta_{2} + 27\beta _1 - 4 \)
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| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 36\beta_{5} + 17\beta_{4} + 35\beta_{3} - 17\beta_{2} + 12\beta _1 + 81 \)
|
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 |
|
1.00000 | −2.62365 | 1.00000 | 0.827725 | −2.62365 | 2.76633 | 1.00000 | 3.88354 | 0.827725 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.488866 | 1.00000 | −4.43611 | −0.488866 | 3.29140 | 1.00000 | −2.76101 | −4.43611 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.204221 | 1.00000 | 0.597188 | −0.204221 | 0.0164134 | 1.00000 | −2.95829 | 0.597188 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.612666 | 1.00000 | 2.71706 | 0.612666 | 4.31071 | 1.00000 | −2.62464 | 2.71706 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.26973 | 1.00000 | 2.59734 | 1.26973 | 0.266543 | 1.00000 | −1.38779 | 2.59734 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.12634 | 1.00000 | −1.35513 | 2.12634 | 4.61147 | 1.00000 | 1.52132 | −1.35513 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.30800 | 1.00000 | 3.05193 | 2.30800 | −1.26288 | 1.00000 | 2.32687 | 3.05193 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(439\) | \( +1 \) |
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 | 878.2.a.i | ✓ | 7 |
| 3.b | odd | 2 | 1 | 7902.2.a.t | 7 | ||
| 4.b | odd | 2 | 1 | 7024.2.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 878.2.a.i | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 7024.2.a.m | 7 | 4.b | odd | 2 | 1 | ||
| 7902.2.a.t | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 3T_{3}^{6} - 5T_{3}^{5} + 22T_{3}^{4} - 13T_{3}^{3} - 8T_{3}^{2} + 4T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(878))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
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| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 1 \)
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| $5$ |
\( T^{7} - 4 T^{6} + \cdots - 64 \)
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| $7$ |
\( T^{7} - 14 T^{6} + \cdots + 1 \)
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| $11$ |
\( T^{7} - T^{6} + \cdots + 64 \)
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| $13$ |
\( T^{7} - 2 T^{6} + \cdots + 832 \)
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| $17$ |
\( T^{7} - 2 T^{6} + \cdots - 1072 \)
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| $19$ |
\( T^{7} - 14 T^{6} + \cdots - 80 \)
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| $23$ |
\( T^{7} + 2 T^{6} + \cdots + 128 \)
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| $29$ |
\( T^{7} - 9 T^{6} + \cdots + 1280 \)
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| $31$ |
\( T^{7} - 119 T^{5} + \cdots - 2864 \)
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| $37$ |
\( T^{7} - 7 T^{6} + \cdots - 871 \)
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| $41$ |
\( T^{7} - 53 T^{5} + \cdots + 16 \)
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| $43$ |
\( T^{7} - 9 T^{6} + \cdots - 18433 \)
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| $47$ |
\( T^{7} + 3 T^{6} + \cdots - 2672 \)
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| $53$ |
\( T^{7} + 6 T^{6} + \cdots + 65728 \)
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| $59$ |
\( T^{7} + T^{6} + \cdots + 102640 \)
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| $61$ |
\( T^{7} - 22 T^{6} + \cdots - 244784 \)
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| $67$ |
\( T^{7} - 20 T^{6} + \cdots - 17104 \)
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| $71$ |
\( T^{7} - 13 T^{6} + \cdots - 67 \)
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| $73$ |
\( T^{7} + 10 T^{6} + \cdots + 127 \)
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| $79$ |
\( T^{7} - 25 T^{6} + \cdots + 26480 \)
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| $83$ |
\( T^{7} + 9 T^{6} + \cdots - 2626496 \)
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| $89$ |
\( T^{7} + 19 T^{6} + \cdots - 281360 \)
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| $97$ |
\( T^{7} + 7 T^{6} + \cdots + 338704 \)
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