Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(36.0863594579\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
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|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 5) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 2.00000i | 0 | 28.0000 | 0 | 0 | 192.000i | − | 120.000i | 0 | 0 | ||||||||||||||||||||||
| 199.2 | 2.00000i | 0 | 28.0000 | 0 | 0 | − | 192.000i | 120.000i | 0 | 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 | 225.6.b.e | 2 | |
| 3.b | odd | 2 | 1 | 25.6.b.a | 2 | ||
| 5.b | even | 2 | 1 | inner | 225.6.b.e | 2 | |
| 5.c | odd | 4 | 1 | 45.6.a.b | 1 | ||
| 5.c | odd | 4 | 1 | 225.6.a.f | 1 | ||
| 12.b | even | 2 | 1 | 400.6.c.j | 2 | ||
| 15.d | odd | 2 | 1 | 25.6.b.a | 2 | ||
| 15.e | even | 4 | 1 | 5.6.a.a | ✓ | 1 | |
| 15.e | even | 4 | 1 | 25.6.a.a | 1 | ||
| 20.e | even | 4 | 1 | 720.6.a.a | 1 | ||
| 60.h | even | 2 | 1 | 400.6.c.j | 2 | ||
| 60.l | odd | 4 | 1 | 80.6.a.e | 1 | ||
| 60.l | odd | 4 | 1 | 400.6.a.g | 1 | ||
| 105.k | odd | 4 | 1 | 245.6.a.b | 1 | ||
| 120.q | odd | 4 | 1 | 320.6.a.g | 1 | ||
| 120.w | even | 4 | 1 | 320.6.a.j | 1 | ||
| 165.l | odd | 4 | 1 | 605.6.a.a | 1 | ||
| 195.s | even | 4 | 1 | 845.6.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.6.a.a | ✓ | 1 | 15.e | even | 4 | 1 | |
| 25.6.a.a | 1 | 15.e | even | 4 | 1 | ||
| 25.6.b.a | 2 | 3.b | odd | 2 | 1 | ||
| 25.6.b.a | 2 | 15.d | odd | 2 | 1 | ||
| 45.6.a.b | 1 | 5.c | odd | 4 | 1 | ||
| 80.6.a.e | 1 | 60.l | odd | 4 | 1 | ||
| 225.6.a.f | 1 | 5.c | odd | 4 | 1 | ||
| 225.6.b.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 225.6.b.e | 2 | 5.b | even | 2 | 1 | inner | |
| 245.6.a.b | 1 | 105.k | odd | 4 | 1 | ||
| 320.6.a.g | 1 | 120.q | odd | 4 | 1 | ||
| 320.6.a.j | 1 | 120.w | even | 4 | 1 | ||
| 400.6.a.g | 1 | 60.l | odd | 4 | 1 | ||
| 400.6.c.j | 2 | 12.b | even | 2 | 1 | ||
| 400.6.c.j | 2 | 60.h | even | 2 | 1 | ||
| 605.6.a.a | 1 | 165.l | odd | 4 | 1 | ||
| 720.6.a.a | 1 | 20.e | even | 4 | 1 | ||
| 845.6.a.b | 1 | 195.s | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\):
|
\( T_{2}^{2} + 4 \)
|
|
\( T_{7}^{2} + 36864 \)
|
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\( T_{11} - 148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 36864 \)
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| $11$ |
\( (T - 148)^{2} \)
|
| $13$ |
\( T^{2} + 81796 \)
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| $17$ |
\( T^{2} + 2815684 \)
|
| $19$ |
\( (T + 1060)^{2} \)
|
| $23$ |
\( T^{2} + 8856576 \)
|
| $29$ |
\( (T + 3410)^{2} \)
|
| $31$ |
\( (T + 2448)^{2} \)
|
| $37$ |
\( T^{2} + 33124 \)
|
| $41$ |
\( (T - 9398)^{2} \)
|
| $43$ |
\( T^{2} + 1547536 \)
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| $47$ |
\( T^{2} + 146119744 \)
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| $53$ |
\( T^{2} + 568631716 \)
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| $59$ |
\( (T + 20020)^{2} \)
|
| $61$ |
\( (T - 32302)^{2} \)
|
| $67$ |
\( T^{2} + 3717584784 \)
|
| $71$ |
\( (T - 32648)^{2} \)
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| $73$ |
\( T^{2} + 1503423076 \)
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| $79$ |
\( (T - 33360)^{2} \)
|
| $83$ |
\( T^{2} + 279424656 \)
|
| $89$ |
\( (T - 101370)^{2} \)
|
| $97$ |
\( T^{2} + 14170045444 \)
|
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