Newspace parameters
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(28.0671684673\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{65})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 33x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 33x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 17\nu ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 17 \)
|
| \(\nu^{3}\) | \(=\) |
\( 16\beta_{2} - 17\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 4.53113i | − | 10.5934i | 11.4689 | 0 | −48.0000 | 49.0000i | − | 196.963i | 130.780 | 0 | |||||||||||||||||||||||||||
| 99.2 | − | 3.53113i | − | 13.5934i | 19.5311 | 0 | −48.0000 | − | 49.0000i | − | 181.963i | 58.2198 | 0 | |||||||||||||||||||||||||||
| 99.3 | 3.53113i | 13.5934i | 19.5311 | 0 | −48.0000 | 49.0000i | 181.963i | 58.2198 | 0 | |||||||||||||||||||||||||||||||
| 99.4 | 4.53113i | 10.5934i | 11.4689 | 0 | −48.0000 | − | 49.0000i | 196.963i | 130.780 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.6.b.d | 4 | |
| 5.b | even | 2 | 1 | inner | 175.6.b.d | 4 | |
| 5.c | odd | 4 | 1 | 35.6.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 175.6.a.d | 2 | ||
| 15.e | even | 4 | 1 | 315.6.a.c | 2 | ||
| 20.e | even | 4 | 1 | 560.6.a.l | 2 | ||
| 35.f | even | 4 | 1 | 245.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.a.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 175.6.a.d | 2 | 5.c | odd | 4 | 1 | ||
| 175.6.b.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 175.6.b.d | 4 | 5.b | even | 2 | 1 | inner | |
| 245.6.a.c | 2 | 35.f | even | 4 | 1 | ||
| 315.6.a.c | 2 | 15.e | even | 4 | 1 | ||
| 560.6.a.l | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 33T_{2}^{2} + 256 \)
acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 33T^{2} + 256 \)
|
| $3$ |
\( T^{4} + 297 T^{2} + 20736 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2401)^{2} \)
|
| $11$ |
\( (T^{2} + 601 T - 62596)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 1412707396 \)
|
| $17$ |
\( T^{4} + \cdots + 1047237035716 \)
|
| $19$ |
\( (T^{2} + 630 T - 20960)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 173507485106176 \)
|
| $29$ |
\( (T^{2} + 5885 T - 5853350)^{2} \)
|
| $31$ |
\( (T^{2} + 396 T - 13834656)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 35738827414416 \)
|
| $41$ |
\( (T^{2} - 1774 T - 236091496)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots + 53\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 59\!\cdots\!16 \)
|
| $59$ |
\( (T^{2} + 59600 T + 451339840)^{2} \)
|
| $61$ |
\( (T^{2} + 51846 T + 142927344)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 30\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} - 80744 T + 1172746624)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 57\!\cdots\!56 \)
|
| $79$ |
\( (T^{2} - 51795 T - 4382959400)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 76\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} - 37650 T - 177665240)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 70\!\cdots\!56 \)
|
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