Newspace parameters
| Level: | \( N \) | \(=\) | \( 2325 = 3 \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2325.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.5652184699\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{2} + 5\beta _1 - 7 \)
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| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times\).
| \(n\) | \(652\) | \(776\) | \(1801\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1024.1 |
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− | 2.21432i | − | 1.00000i | −2.90321 | 0 | −2.21432 | 0.903212i | 2.00000i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1024.2 | − | 1.67513i | 1.00000i | −0.806063 | 0 | 1.67513 | 1.19394i | − | 2.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 1024.3 | − | 0.539189i | 1.00000i | 1.70928 | 0 | 0.539189 | 3.70928i | − | 2.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 1024.4 | 0.539189i | − | 1.00000i | 1.70928 | 0 | 0.539189 | − | 3.70928i | 2.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 1024.5 | 1.67513i | − | 1.00000i | −0.806063 | 0 | 1.67513 | − | 1.19394i | 2.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 1024.6 | 2.21432i | 1.00000i | −2.90321 | 0 | −2.21432 | − | 0.903212i | − | 2.00000i | −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 | 2325.2.c.m | 6 | |
| 5.b | even | 2 | 1 | inner | 2325.2.c.m | 6 | |
| 5.c | odd | 4 | 1 | 2325.2.a.t | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 2325.2.a.u | yes | 3 | |
| 15.e | even | 4 | 1 | 6975.2.a.bc | 3 | ||
| 15.e | even | 4 | 1 | 6975.2.a.bd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2325.2.a.t | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 2325.2.a.u | yes | 3 | 5.c | odd | 4 | 1 | |
| 2325.2.c.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 2325.2.c.m | 6 | 5.b | even | 2 | 1 | inner | |
| 6975.2.a.bc | 3 | 15.e | even | 4 | 1 | ||
| 6975.2.a.bd | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2325, [\chi])\):
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\( T_{2}^{6} + 8T_{2}^{4} + 16T_{2}^{2} + 4 \)
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\( T_{7}^{6} + 16T_{7}^{4} + 32T_{7}^{2} + 16 \)
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\( T_{11}^{3} + 7T_{11}^{2} + 3T_{11} - 37 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 8 T^{4} + \cdots + 4 \)
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| $3$ |
\( (T^{2} + 1)^{3} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 16 T^{4} + \cdots + 16 \)
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| $11$ |
\( (T^{3} + 7 T^{2} + 3 T - 37)^{2} \)
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| $13$ |
\( T^{6} + 68 T^{4} + \cdots + 3844 \)
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| $17$ |
\( T^{6} + 63 T^{4} + \cdots + 3481 \)
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| $19$ |
\( (T^{3} - 11 T^{2} + 27 T - 1)^{2} \)
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| $23$ |
\( T^{6} + 139 T^{4} + \cdots + 57121 \)
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| $29$ |
\( (T^{3} - 5 T^{2} + 5 T + 1)^{2} \)
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| $31$ |
\( (T + 1)^{6} \)
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| $37$ |
\( T^{6} + 232 T^{4} + \cdots + 386884 \)
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| $41$ |
\( (T^{3} + 8 T^{2} + \cdots - 130)^{2} \)
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| $43$ |
\( T^{6} + 208 T^{4} + \cdots + 135424 \)
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| $47$ |
\( T^{6} + 140 T^{4} + \cdots + 64 \)
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| $53$ |
\( T^{6} + 199 T^{4} + \cdots + 625 \)
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| $59$ |
\( (T^{3} - 14 T^{2} + \cdots + 302)^{2} \)
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| $61$ |
\( (T^{3} - 12 T^{2} + \cdots + 64)^{2} \)
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| $67$ |
\( T^{6} + 323 T^{4} + \cdots + 72361 \)
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| $71$ |
\( (T^{3} + 26 T^{2} + \cdots + 248)^{2} \)
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| $73$ |
\( T^{6} + 80 T^{4} + \cdots + 256 \)
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| $79$ |
\( (T^{3} + 10 T^{2} + \cdots - 34)^{2} \)
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| $83$ |
\( T^{6} + 107 T^{4} + \cdots + 32041 \)
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| $89$ |
\( (T^{3} - 9 T^{2} + \cdots + 1523)^{2} \)
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| $97$ |
\( T^{6} + 379 T^{4} + \cdots + 312481 \)
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