Newspace parameters
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.0897435397\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 76.1 |
|
0 | − | 17.0000i | −16.0000 | 0 | 0 | − | 49.0000i | 0 | −208.000 | 0 | ||||||||||||||||||||||
| 76.2 | 0 | 17.0000i | −16.0000 | 0 | 0 | 49.0000i | 0 | −208.000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 35.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-35}) \) |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.5.d.c | 2 | |
| 5.b | even | 2 | 1 | inner | 175.5.d.c | 2 | |
| 5.c | odd | 4 | 1 | 35.5.c.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 35.5.c.b | yes | 1 | |
| 7.b | odd | 2 | 1 | inner | 175.5.d.c | 2 | |
| 15.e | even | 4 | 1 | 315.5.e.a | 1 | ||
| 15.e | even | 4 | 1 | 315.5.e.b | 1 | ||
| 20.e | even | 4 | 1 | 560.5.p.a | 1 | ||
| 20.e | even | 4 | 1 | 560.5.p.b | 1 | ||
| 35.c | odd | 2 | 1 | CM | 175.5.d.c | 2 | |
| 35.f | even | 4 | 1 | 35.5.c.a | ✓ | 1 | |
| 35.f | even | 4 | 1 | 35.5.c.b | yes | 1 | |
| 105.k | odd | 4 | 1 | 315.5.e.a | 1 | ||
| 105.k | odd | 4 | 1 | 315.5.e.b | 1 | ||
| 140.j | odd | 4 | 1 | 560.5.p.a | 1 | ||
| 140.j | odd | 4 | 1 | 560.5.p.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.5.c.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 35.5.c.a | ✓ | 1 | 35.f | even | 4 | 1 | |
| 35.5.c.b | yes | 1 | 5.c | odd | 4 | 1 | |
| 35.5.c.b | yes | 1 | 35.f | even | 4 | 1 | |
| 175.5.d.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 175.5.d.c | 2 | 5.b | even | 2 | 1 | inner | |
| 175.5.d.c | 2 | 7.b | odd | 2 | 1 | inner | |
| 175.5.d.c | 2 | 35.c | odd | 2 | 1 | CM | |
| 315.5.e.a | 1 | 15.e | even | 4 | 1 | ||
| 315.5.e.a | 1 | 105.k | odd | 4 | 1 | ||
| 315.5.e.b | 1 | 15.e | even | 4 | 1 | ||
| 315.5.e.b | 1 | 105.k | odd | 4 | 1 | ||
| 560.5.p.a | 1 | 20.e | even | 4 | 1 | ||
| 560.5.p.a | 1 | 140.j | odd | 4 | 1 | ||
| 560.5.p.b | 1 | 20.e | even | 4 | 1 | ||
| 560.5.p.b | 1 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 289 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 2401 \)
|
| $11$ |
\( (T + 73)^{2} \)
|
| $13$ |
\( T^{2} + 529 \)
|
| $17$ |
\( T^{2} + 69169 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( (T - 1153)^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( T^{2} \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( T^{2} + 11950849 \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( T^{2} \)
|
| $67$ |
\( T^{2} \)
|
| $71$ |
\( (T + 10078)^{2} \)
|
| $73$ |
\( T^{2} + 90288004 \)
|
| $79$ |
\( (T + 12167)^{2} \)
|
| $83$ |
\( T^{2} + 40729924 \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( T^{2} + 11444689 \)
|
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