Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(172.877215626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{1801})\) |
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|
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| Defining polynomial: |
\( x^{4} + 901x^{2} + 202500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} + 901x^{2} + 202500 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 451\nu ) / 225 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 451 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 451 \)
|
| \(\nu^{3}\) | \(=\) |
\( 225\beta_{2} - 451\beta_1 \)
|
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 |
|
− | 27.7191i | 0 | 1279.65 | 0 | 0 | 72157.2i | − | 92239.5i | 0 | 0 | ||||||||||||||||||||||||||||
| 199.2 | − | 14.7191i | 0 | 1831.35 | 0 | 0 | 79941.2i | − | 57100.5i | 0 | 0 | |||||||||||||||||||||||||||||
| 199.3 | 14.7191i | 0 | 1831.35 | 0 | 0 | − | 79941.2i | 57100.5i | 0 | 0 | ||||||||||||||||||||||||||||||
| 199.4 | 27.7191i | 0 | 1279.65 | 0 | 0 | − | 72157.2i | 92239.5i | 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 | 225.12.b.i | 4 | |
| 3.b | odd | 2 | 1 | 75.12.b.d | 4 | ||
| 5.b | even | 2 | 1 | inner | 225.12.b.i | 4 | |
| 5.c | odd | 4 | 1 | 45.12.a.c | 2 | ||
| 5.c | odd | 4 | 1 | 225.12.a.i | 2 | ||
| 15.d | odd | 2 | 1 | 75.12.b.d | 4 | ||
| 15.e | even | 4 | 1 | 15.12.a.c | ✓ | 2 | |
| 15.e | even | 4 | 1 | 75.12.a.c | 2 | ||
| 60.l | odd | 4 | 1 | 240.12.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.12.a.c | ✓ | 2 | 15.e | even | 4 | 1 | |
| 45.12.a.c | 2 | 5.c | odd | 4 | 1 | ||
| 75.12.a.c | 2 | 15.e | even | 4 | 1 | ||
| 75.12.b.d | 4 | 3.b | odd | 2 | 1 | ||
| 75.12.b.d | 4 | 15.d | odd | 2 | 1 | ||
| 225.12.a.i | 2 | 5.c | odd | 4 | 1 | ||
| 225.12.b.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.12.b.i | 4 | 5.b | even | 2 | 1 | inner | |
| 240.12.a.m | 2 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(225, [\chi])\):
|
\( T_{2}^{4} + 985T_{2}^{2} + 166464 \)
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\( T_{11}^{2} + 295568T_{11} - 410180426688 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 985 T^{2} + 166464 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
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| $11$ |
\( (T^{2} + 295568 T - 410180426688)^{2} \)
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| $13$ |
\( T^{4} + \cdots + 49\!\cdots\!64 \)
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| $17$ |
\( T^{4} + \cdots + 24\!\cdots\!76 \)
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| $19$ |
\( (T^{2} + \cdots + 69890598801680)^{2} \)
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| $23$ |
\( T^{4} + \cdots + 79\!\cdots\!00 \)
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| $29$ |
\( (T^{2} + \cdots + 21\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{2} + \cdots - 17\!\cdots\!00)^{2} \)
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| $37$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots - 12\!\cdots\!80)^{2} \)
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| $43$ |
\( T^{4} + \cdots + 64\!\cdots\!36 \)
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| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
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| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
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| $59$ |
\( (T^{2} + \cdots - 25\!\cdots\!20)^{2} \)
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| $61$ |
\( (T^{2} + \cdots + 10\!\cdots\!16)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!16 \)
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| $71$ |
\( (T^{2} + \cdots - 20\!\cdots\!16)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
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| $79$ |
\( (T^{2} + \cdots - 60\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{4} + \cdots + 18\!\cdots\!44 \)
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| $89$ |
\( (T^{2} + \cdots + 33\!\cdots\!20)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
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