Newspace parameters
| Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 975.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.78541419707\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
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|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 195) |
| 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 \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 2.00000i | 1.00000i | −2.00000 | 0 | 2.00000 | 3.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||
| 274.2 | 2.00000i | − | 1.00000i | −2.00000 | 0 | 2.00000 | − | 3.00000i | 0 | −1.00000 | 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 | 975.2.c.b | 2 | |
| 3.b | odd | 2 | 1 | 2925.2.c.d | 2 | ||
| 5.b | even | 2 | 1 | inner | 975.2.c.b | 2 | |
| 5.c | odd | 4 | 1 | 195.2.a.d | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 975.2.a.b | 1 | ||
| 15.d | odd | 2 | 1 | 2925.2.c.d | 2 | ||
| 15.e | even | 4 | 1 | 585.2.a.a | 1 | ||
| 15.e | even | 4 | 1 | 2925.2.a.t | 1 | ||
| 20.e | even | 4 | 1 | 3120.2.a.n | 1 | ||
| 35.f | even | 4 | 1 | 9555.2.a.t | 1 | ||
| 60.l | odd | 4 | 1 | 9360.2.a.w | 1 | ||
| 65.h | odd | 4 | 1 | 2535.2.a.b | 1 | ||
| 195.s | even | 4 | 1 | 7605.2.a.v | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.a.d | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 585.2.a.a | 1 | 15.e | even | 4 | 1 | ||
| 975.2.a.b | 1 | 5.c | odd | 4 | 1 | ||
| 975.2.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 975.2.c.b | 2 | 5.b | even | 2 | 1 | inner | |
| 2535.2.a.b | 1 | 65.h | odd | 4 | 1 | ||
| 2925.2.a.t | 1 | 15.e | even | 4 | 1 | ||
| 2925.2.c.d | 2 | 3.b | odd | 2 | 1 | ||
| 2925.2.c.d | 2 | 15.d | odd | 2 | 1 | ||
| 3120.2.a.n | 1 | 20.e | even | 4 | 1 | ||
| 7605.2.a.v | 1 | 195.s | even | 4 | 1 | ||
| 9360.2.a.w | 1 | 60.l | odd | 4 | 1 | ||
| 9555.2.a.t | 1 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):
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\( T_{2}^{2} + 4 \)
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\( T_{7}^{2} + 9 \)
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\( T_{11} + 5 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4 \)
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| $3$ |
\( T^{2} + 1 \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} + 9 \)
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| $11$ |
\( (T + 5)^{2} \)
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| $13$ |
\( T^{2} + 1 \)
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| $17$ |
\( T^{2} + 25 \)
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| $19$ |
\( (T + 2)^{2} \)
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| $23$ |
\( T^{2} + 1 \)
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| $29$ |
\( (T + 10)^{2} \)
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| $31$ |
\( (T + 2)^{2} \)
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| $37$ |
\( T^{2} + 9 \)
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| $41$ |
\( (T + 9)^{2} \)
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| $43$ |
\( T^{2} + 16 \)
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| $47$ |
\( T^{2} + 100 \)
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| $53$ |
\( T^{2} + 81 \)
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| $59$ |
\( T^{2} \)
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| $61$ |
\( (T + 11)^{2} \)
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| $67$ |
\( T^{2} + 16 \)
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| $71$ |
\( (T - 15)^{2} \)
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| $73$ |
\( T^{2} + 36 \)
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| $79$ |
\( (T - 11)^{2} \)
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| $83$ |
\( T^{2} + 64 \)
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| $89$ |
\( (T - 11)^{2} \)
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| $97$ |
\( T^{2} + 81 \)
|
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