Newspace parameters
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.59513806569\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
1.00000 | − | 2.00000i | −1.00000 | 0 | − | 2.00000i | −5.00000 | −3.00000 | −1.00000 | 0 | ||||||||||||||||||||||
| 324.2 | 1.00000 | 2.00000i | −1.00000 | 0 | 2.00000i | −5.00000 | −3.00000 | −1.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.2.d.c | 2 | |
| 5.b | even | 2 | 1 | 325.2.d.b | 2 | ||
| 5.c | odd | 4 | 1 | 325.2.c.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 325.2.c.f | yes | 2 | |
| 13.b | even | 2 | 1 | 325.2.d.b | 2 | ||
| 65.d | even | 2 | 1 | inner | 325.2.d.c | 2 | |
| 65.f | even | 4 | 1 | 4225.2.a.d | 1 | ||
| 65.f | even | 4 | 1 | 4225.2.a.f | 1 | ||
| 65.h | odd | 4 | 1 | 325.2.c.a | ✓ | 2 | |
| 65.h | odd | 4 | 1 | 325.2.c.f | yes | 2 | |
| 65.k | even | 4 | 1 | 4225.2.a.l | 1 | ||
| 65.k | even | 4 | 1 | 4225.2.a.n | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.2.c.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 325.2.c.a | ✓ | 2 | 65.h | odd | 4 | 1 | |
| 325.2.c.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 325.2.c.f | yes | 2 | 65.h | odd | 4 | 1 | |
| 325.2.d.b | 2 | 5.b | even | 2 | 1 | ||
| 325.2.d.b | 2 | 13.b | even | 2 | 1 | ||
| 325.2.d.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 325.2.d.c | 2 | 65.d | even | 2 | 1 | inner | |
| 4225.2.a.d | 1 | 65.f | even | 4 | 1 | ||
| 4225.2.a.f | 1 | 65.f | even | 4 | 1 | ||
| 4225.2.a.l | 1 | 65.k | even | 4 | 1 | ||
| 4225.2.a.n | 1 | 65.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
|
| $3$ |
\( T^{2} + 4 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T + 5)^{2} \)
|
| $11$ |
\( T^{2} + 9 \)
|
| $13$ |
\( T^{2} - 4T + 13 \)
|
| $17$ |
\( T^{2} + 25 \)
|
| $19$ |
\( T^{2} + 16 \)
|
| $23$ |
\( T^{2} + 16 \)
|
| $29$ |
\( (T - 1)^{2} \)
|
| $31$ |
\( T^{2} + 1 \)
|
| $37$ |
\( (T - 4)^{2} \)
|
| $41$ |
\( T^{2} + 64 \)
|
| $43$ |
\( T^{2} + 16 \)
|
| $47$ |
\( (T + 7)^{2} \)
|
| $53$ |
\( T^{2} + 9 \)
|
| $59$ |
\( T^{2} + 9 \)
|
| $61$ |
\( (T - 1)^{2} \)
|
| $67$ |
\( (T - 3)^{2} \)
|
| $71$ |
\( T^{2} + 64 \)
|
| $73$ |
\( (T + 4)^{2} \)
|
| $79$ |
\( (T + 10)^{2} \)
|
| $83$ |
\( (T - 9)^{2} \)
|
| $89$ |
\( T^{2} + 324 \)
|
| $97$ |
\( (T + 14)^{2} \)
|
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