Newspace parameters
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.76840462631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 8\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 4\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 174.1 |
|
− | 1.00000i | −4.47214 | 3.00000 | 0 | 4.47214i | 7.00000i | − | 7.00000i | 11.0000 | 0 | ||||||||||||||||||||||||||||
| 174.2 | − | 1.00000i | 4.47214 | 3.00000 | 0 | − | 4.47214i | 7.00000i | − | 7.00000i | 11.0000 | 0 | ||||||||||||||||||||||||||||
| 174.3 | 1.00000i | −4.47214 | 3.00000 | 0 | − | 4.47214i | − | 7.00000i | 7.00000i | 11.0000 | 0 | |||||||||||||||||||||||||||||
| 174.4 | 1.00000i | 4.47214 | 3.00000 | 0 | 4.47214i | − | 7.00000i | 7.00000i | 11.0000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.3.c.d | 4 | |
| 5.b | even | 2 | 1 | inner | 175.3.c.d | 4 | |
| 5.c | odd | 4 | 1 | 35.3.d.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 175.3.d.f | 2 | ||
| 7.b | odd | 2 | 1 | inner | 175.3.c.d | 4 | |
| 15.e | even | 4 | 1 | 315.3.h.b | 2 | ||
| 20.e | even | 4 | 1 | 560.3.f.a | 2 | ||
| 35.c | odd | 2 | 1 | inner | 175.3.c.d | 4 | |
| 35.f | even | 4 | 1 | 35.3.d.a | ✓ | 2 | |
| 35.f | even | 4 | 1 | 175.3.d.f | 2 | ||
| 35.k | even | 12 | 2 | 245.3.h.b | 4 | ||
| 35.l | odd | 12 | 2 | 245.3.h.b | 4 | ||
| 105.k | odd | 4 | 1 | 315.3.h.b | 2 | ||
| 140.j | odd | 4 | 1 | 560.3.f.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.3.d.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 35.3.d.a | ✓ | 2 | 35.f | even | 4 | 1 | |
| 175.3.c.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 175.3.c.d | 4 | 5.b | even | 2 | 1 | inner | |
| 175.3.c.d | 4 | 7.b | odd | 2 | 1 | inner | |
| 175.3.c.d | 4 | 35.c | odd | 2 | 1 | inner | |
| 175.3.d.f | 2 | 5.c | odd | 4 | 1 | ||
| 175.3.d.f | 2 | 35.f | even | 4 | 1 | ||
| 245.3.h.b | 4 | 35.k | even | 12 | 2 | ||
| 245.3.h.b | 4 | 35.l | odd | 12 | 2 | ||
| 315.3.h.b | 2 | 15.e | even | 4 | 1 | ||
| 315.3.h.b | 2 | 105.k | odd | 4 | 1 | ||
| 560.3.f.a | 2 | 20.e | even | 4 | 1 | ||
| 560.3.f.a | 2 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( (T^{2} - 20)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 49)^{2} \)
$11$
\( (T - 2)^{4} \)
$13$
\( (T^{2} - 180)^{2} \)
$17$
\( (T^{2} - 720)^{2} \)
$19$
\( (T^{2} + 180)^{2} \)
$23$
\( (T^{2} + 676)^{2} \)
$29$
\( (T - 22)^{4} \)
$31$
\( (T^{2} + 2880)^{2} \)
$37$
\( (T^{2} + 196)^{2} \)
$41$
\( (T^{2} + 720)^{2} \)
$43$
\( (T^{2} + 1156)^{2} \)
$47$
\( (T^{2} - 720)^{2} \)
$53$
\( (T^{2} + 1156)^{2} \)
$59$
\( (T^{2} + 1620)^{2} \)
$61$
\( (T^{2} + 8820)^{2} \)
$67$
\( (T^{2} + 196)^{2} \)
$71$
\( (T - 62)^{4} \)
$73$
\( (T^{2} - 2880)^{2} \)
$79$
\( (T + 38)^{4} \)
$83$
\( (T^{2} - 1620)^{2} \)
$89$
\( (T^{2} + 720)^{2} \)
$97$
\( (T^{2} - 720)^{2} \)
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