Newspace parameters
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(21.8551379439\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.7600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 3\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 94.1 |
|
−12.1706 | −49.5508 | 84.1246 | 125.000 | 603.066 | 0 | −244.930 | 1726.29 | −1521.33 | ||||||||||||||||||||||||||||||
| 94.2 | −10.3863 | −21.4642 | 43.8754 | 125.000 | 222.934 | 0 | 209.020 | −268.286 | −1298.29 | |||||||||||||||||||||||||||||||
| 94.3 | 10.3863 | 21.4642 | 43.8754 | 125.000 | 222.934 | 0 | −209.020 | −268.286 | 1298.29 | |||||||||||||||||||||||||||||||
| 94.4 | 12.1706 | 49.5508 | 84.1246 | 125.000 | 603.066 | 0 | 244.930 | 1726.29 | 1521.33 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 95.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-95}) \) |
| 5.b | even | 2 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.7.d.c | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 95.7.d.c | ✓ | 4 |
| 19.b | odd | 2 | 1 | inner | 95.7.d.c | ✓ | 4 |
| 95.d | odd | 2 | 1 | CM | 95.7.d.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.7.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 95.7.d.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 95.7.d.c | ✓ | 4 | 19.b | odd | 2 | 1 | inner |
| 95.7.d.c | ✓ | 4 | 95.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 256T_{2}^{2} + 15979 \)
acting on \(S_{7}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 256 T^{2} + 15979 \)
|
| $3$ |
\( T^{4} - 2916 T^{2} + 1131184 \)
|
| $5$ |
\( (T - 125)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 6028020)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 63534282789424 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T + 6859)^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} + \cdots + 13\!\cdots\!44 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!44 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} - 170234461620)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!64 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 25\!\cdots\!44 \)
|
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