Newspace parameters
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(54.6673794597\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 16\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 10 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 8\beta_1 ) / 2 \)
|
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 |
|
− | 14.6332i | − | 54.7995i | −86.1320 | 0 | −801.895 | 343.000i | − | 612.665i | −815.985 | 0 | |||||||||||||||||||||||||||
| 99.2 | − | 1.36675i | 24.7995i | 126.132 | 0 | 33.8947 | 343.000i | − | 347.335i | 1571.98 | 0 | |||||||||||||||||||||||||||||
| 99.3 | 1.36675i | − | 24.7995i | 126.132 | 0 | 33.8947 | − | 343.000i | 347.335i | 1571.98 | 0 | |||||||||||||||||||||||||||||
| 99.4 | 14.6332i | 54.7995i | −86.1320 | 0 | −801.895 | − | 343.000i | 612.665i | −815.985 | 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 | 175.8.b.c | 4 | |
| 5.b | even | 2 | 1 | inner | 175.8.b.c | 4 | |
| 5.c | odd | 4 | 1 | 35.8.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 175.8.a.b | 2 | ||
| 15.e | even | 4 | 1 | 315.8.a.c | 2 | ||
| 20.e | even | 4 | 1 | 560.8.a.i | 2 | ||
| 35.f | even | 4 | 1 | 245.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.8.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 175.8.a.b | 2 | 5.c | odd | 4 | 1 | ||
| 175.8.b.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 175.8.b.c | 4 | 5.b | even | 2 | 1 | inner | |
| 245.8.a.b | 2 | 35.f | even | 4 | 1 | ||
| 315.8.a.c | 2 | 15.e | even | 4 | 1 | ||
| 560.8.a.i | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 216T_{2}^{2} + 400 \)
acting on \(S_{8}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 216T^{2} + 400 \)
|
| $3$ |
\( T^{4} + 3618 T^{2} + 1846881 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 117649)^{2} \)
|
| $11$ |
\( (T^{2} + 7906 T + 9272609)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 51\!\cdots\!25 \)
|
| $17$ |
\( T^{4} + \cdots + 45\!\cdots\!01 \)
|
| $19$ |
\( (T^{2} - 3612 T - 1348143980)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 24\!\cdots\!56 \)
|
| $29$ |
\( (T^{2} - 126898 T - 20838975695)^{2} \)
|
| $31$ |
\( (T^{2} - 252768 T - 6409132848)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 43\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + 111920 T - 58262616304)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!09 \)
|
| $53$ |
\( T^{4} + \cdots + 46\!\cdots\!84 \)
|
| $59$ |
\( (T^{2} - 1360120 T - 818710818800)^{2} \)
|
| $61$ |
\( (T^{2} + 1813680 T + 816706565136)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $71$ |
\( (T^{2} + \cdots - 1610731398080)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $79$ |
\( (T^{2} + \cdots + 14225767990465)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + \cdots + 77796446568160)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 25\!\cdots\!69 \)
|
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