Newspace parameters
| Level: | \( N \) | \(=\) | \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2300.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(135.704393013\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.96668224.1 |
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|
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| Defining polynomial: |
\( x^{6} + 15x^{4} + 61x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 92) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 15x^{4} + 61x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} - 23\nu ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + 13\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 35\nu ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 7\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 9\nu^{2} + 11 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} + \beta_{2} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{5} + 9\beta_{4} + 68 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41\beta_{3} - 6\beta_{2} + 59\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1151\) | \(1201\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1749.1 |
|
0 | − | 6.13366i | 0 | 0 | 0 | 19.8565i | 0 | −10.6218 | 0 | |||||||||||||||||||||||||||||||||||
| 1749.2 | 0 | − | 3.31427i | 0 | 0 | 0 | − | 35.2976i | 0 | 16.0156 | 0 | |||||||||||||||||||||||||||||||||||
| 1749.3 | 0 | − | 1.18060i | 0 | 0 | 0 | − | 9.15411i | 0 | 25.6062 | 0 | |||||||||||||||||||||||||||||||||||
| 1749.4 | 0 | 1.18060i | 0 | 0 | 0 | 9.15411i | 0 | 25.6062 | 0 | |||||||||||||||||||||||||||||||||||||
| 1749.5 | 0 | 3.31427i | 0 | 0 | 0 | 35.2976i | 0 | 16.0156 | 0 | |||||||||||||||||||||||||||||||||||||
| 1749.6 | 0 | 6.13366i | 0 | 0 | 0 | − | 19.8565i | 0 | −10.6218 | 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 | 2300.4.c.b | 6 | |
| 5.b | even | 2 | 1 | inner | 2300.4.c.b | 6 | |
| 5.c | odd | 4 | 1 | 92.4.a.a | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 2300.4.a.b | 3 | ||
| 15.e | even | 4 | 1 | 828.4.a.f | 3 | ||
| 20.e | even | 4 | 1 | 368.4.a.k | 3 | ||
| 40.i | odd | 4 | 1 | 1472.4.a.w | 3 | ||
| 40.k | even | 4 | 1 | 1472.4.a.p | 3 | ||
| 115.e | even | 4 | 1 | 2116.4.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 92.4.a.a | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 368.4.a.k | 3 | 20.e | even | 4 | 1 | ||
| 828.4.a.f | 3 | 15.e | even | 4 | 1 | ||
| 1472.4.a.p | 3 | 40.k | even | 4 | 1 | ||
| 1472.4.a.w | 3 | 40.i | odd | 4 | 1 | ||
| 2116.4.a.a | 3 | 115.e | even | 4 | 1 | ||
| 2300.4.a.b | 3 | 5.c | odd | 4 | 1 | ||
| 2300.4.c.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 2300.4.c.b | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 50T_{3}^{4} + 481T_{3}^{2} + 576 \)
acting on \(S_{4}^{\mathrm{new}}(2300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 50 T^{4} + \cdots + 576 \)
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| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1724 T^{4} + \cdots + 41165056 \)
|
| $11$ |
\( (T^{3} + 64 T^{2} + \cdots - 88184)^{2} \)
|
| $13$ |
\( T^{6} + 5414 T^{4} + \cdots + 11115556 \)
|
| $17$ |
\( T^{6} + \cdots + 2616293310016 \)
|
| $19$ |
\( (T^{3} - 94 T^{2} + \cdots + 184984)^{2} \)
|
| $23$ |
\( (T^{2} + 529)^{3} \)
|
| $29$ |
\( (T^{3} + 308 T^{2} + \cdots + 1008698)^{2} \)
|
| $31$ |
\( (T^{3} + 140 T^{2} + \cdots - 1074768)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 21834910310656 \)
|
| $41$ |
\( (T^{3} - 584 T^{2} + \cdots + 21405186)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 208570901856256 \)
|
| $47$ |
\( T^{6} + \cdots + 671132441938176 \)
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| $53$ |
\( T^{6} + \cdots + 3992217856 \)
|
| $59$ |
\( (T^{3} + 144 T^{2} + \cdots + 15495744)^{2} \)
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| $61$ |
\( (T^{3} + 1052 T^{2} + \cdots - 55378432)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 52\!\cdots\!24 \)
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| $71$ |
\( (T^{3} - 360 T^{2} + \cdots + 7542816)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!56 \)
|
| $79$ |
\( (T^{3} - 720 T^{2} + \cdots + 932658192)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 21\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + 534 T^{2} + \cdots - 77599776)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 45\!\cdots\!56 \)
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