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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{70})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 1225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 5^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 1225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 35\nu ) / 35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 35\nu ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 5\beta_{2} ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{3} + 35\beta_{2} ) / 2 \)
|
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 |
|
− | 8.36660i | 0 | −38.0000 | 0 | 0 | − | 45.0000i | 50.1996i | 0 | 0 | ||||||||||||||||||||||||||||
| 199.2 | − | 8.36660i | 0 | −38.0000 | 0 | 0 | 45.0000i | 50.1996i | 0 | 0 | ||||||||||||||||||||||||||||||
| 199.3 | 8.36660i | 0 | −38.0000 | 0 | 0 | − | 45.0000i | − | 50.1996i | 0 | 0 | |||||||||||||||||||||||||||||
| 199.4 | 8.36660i | 0 | −38.0000 | 0 | 0 | 45.0000i | − | 50.1996i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 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.h | 4 | |
| 3.b | odd | 2 | 1 | inner | 225.6.b.h | 4 | |
| 5.b | even | 2 | 1 | inner | 225.6.b.h | 4 | |
| 5.c | odd | 4 | 1 | 225.6.a.o | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 225.6.a.p | yes | 2 | |
| 15.d | odd | 2 | 1 | inner | 225.6.b.h | 4 | |
| 15.e | even | 4 | 1 | 225.6.a.o | ✓ | 2 | |
| 15.e | even | 4 | 1 | 225.6.a.p | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 225.6.a.o | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 225.6.a.o | ✓ | 2 | 15.e | even | 4 | 1 | |
| 225.6.a.p | yes | 2 | 5.c | odd | 4 | 1 | |
| 225.6.a.p | yes | 2 | 15.e | even | 4 | 1 | |
| 225.6.b.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.6.b.h | 4 | 3.b | odd | 2 | 1 | inner | |
| 225.6.b.h | 4 | 5.b | even | 2 | 1 | inner | |
| 225.6.b.h | 4 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\):
|
\( T_{2}^{2} + 70 \)
|
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\( T_{7}^{2} + 2025 \)
|
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\( T_{11}^{2} - 112000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 70)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2025)^{2} \)
|
| $11$ |
\( (T^{2} - 112000)^{2} \)
|
| $13$ |
\( (T^{2} + 1092025)^{2} \)
|
| $17$ |
\( (T^{2} + 1617280)^{2} \)
|
| $19$ |
\( (T + 1159)^{4} \)
|
| $23$ |
\( (T^{2} + 14555520)^{2} \)
|
| $29$ |
\( (T^{2} - 13552000)^{2} \)
|
| $31$ |
\( (T - 3633)^{4} \)
|
| $37$ |
\( (T^{2} + 9672100)^{2} \)
|
| $41$ |
\( (T^{2} - 314608000)^{2} \)
|
| $43$ |
\( (T^{2} + 107226025)^{2} \)
|
| $47$ |
\( (T^{2} + 182801920)^{2} \)
|
| $53$ |
\( (T^{2} + 99460480)^{2} \)
|
| $59$ |
\( (T^{2} - 828352000)^{2} \)
|
| $61$ |
\( (T + 7613)^{4} \)
|
| $67$ |
\( (T^{2} + 2544698025)^{2} \)
|
| $71$ |
\( (T^{2} - 6505072000)^{2} \)
|
| $73$ |
\( (T^{2} + 5581584100)^{2} \)
|
| $79$ |
\( (T + 38316)^{4} \)
|
| $83$ |
\( (T^{2} + 1277498880)^{2} \)
|
| $89$ |
\( (T^{2} - 14515200000)^{2} \)
|
| $97$ |
\( (T^{2} + 5148780025)^{2} \)
|
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