Newspace parameters
| Level: | \( N \) | \(=\) | \( 1925 = 5^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1925.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.3712023891\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 13x^{5} + 10x^{4} + 47x^{3} - 25x^{2} - 35x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 13x^{5} + 10x^{4} + 47x^{3} - 25x^{2} - 35x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 15\nu^{4} - 3\nu^{3} + 60\nu^{2} + 21\nu - 36 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 15\nu^{4} - 3\nu^{3} + 64\nu^{2} + 21\nu - 52 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 13\nu^{4} - \nu^{3} + 46\nu^{2} + 7\nu - 30 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 13\nu^{4} - 3\nu^{3} + 46\nu^{2} + 19\nu - 28 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 2\nu^{5} + 39\nu^{4} - 11\nu^{3} - 138\nu^{2} - 17\nu + 80 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} - 9\beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{6} - 7\beta_{5} + 10\beta_{4} + 40\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{5} + 3\beta_{4} + 45\beta_{3} - 71\beta_{2} + 12\beta _1 + 159 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.78276 | 3.17748 | 5.74378 | 0 | −8.84217 | −1.00000 | −10.4180 | 7.09637 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.47176 | −1.33124 | 4.10959 | 0 | 3.29051 | −1.00000 | −5.21439 | −1.22779 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.815390 | −3.26294 | −1.33514 | 0 | 2.66057 | −1.00000 | 2.71944 | 7.64680 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.633891 | 0.425088 | −1.59818 | 0 | −0.269459 | −1.00000 | 2.28085 | −2.81930 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.06204 | 1.17047 | −0.872076 | 0 | 1.24309 | −1.00000 | −3.05025 | −1.63000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.02168 | −2.31630 | 2.08720 | 0 | −4.68282 | −1.00000 | 0.176288 | 2.36524 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.62008 | 2.13745 | 4.86484 | 0 | 5.60029 | −1.00000 | 7.50611 | 1.56868 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1925.2.a.ba | ✓ | 7 |
| 5.b | even | 2 | 1 | 1925.2.a.bc | yes | 7 | |
| 5.c | odd | 4 | 2 | 1925.2.b.q | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1925.2.a.ba | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1925.2.a.bc | yes | 7 | 5.b | even | 2 | 1 | |
| 1925.2.b.q | 14 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1925))\):
|
\( T_{2}^{7} + T_{2}^{6} - 13T_{2}^{5} - 10T_{2}^{4} + 47T_{2}^{3} + 25T_{2}^{2} - 35T_{2} - 20 \)
|
|
\( T_{3}^{7} - 17T_{3}^{5} + 2T_{3}^{4} + 77T_{3}^{3} - 20T_{3}^{2} - 85T_{3} + 34 \)
|
|
\( T_{13}^{7} - 3T_{13}^{6} - 49T_{13}^{5} + 75T_{13}^{4} + 668T_{13}^{3} - 460T_{13}^{2} - 2404T_{13} + 1324 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 20 \)
|
| $3$ |
\( T^{7} - 17 T^{5} + \cdots + 34 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( (T + 1)^{7} \)
|
| $13$ |
\( T^{7} - 3 T^{6} + \cdots + 1324 \)
|
| $17$ |
\( T^{7} - 80 T^{5} + \cdots + 509 \)
|
| $19$ |
\( T^{7} - 18 T^{6} + \cdots + 64912 \)
|
| $23$ |
\( T^{7} - 7 T^{6} + \cdots + 9088 \)
|
| $29$ |
\( T^{7} + 2 T^{6} + \cdots - 256 \)
|
| $31$ |
\( T^{7} - 24 T^{6} + \cdots - 9680 \)
|
| $37$ |
\( T^{7} + 21 T^{6} + \cdots + 2092 \)
|
| $41$ |
\( T^{7} + 10 T^{6} + \cdots - 1836 \)
|
| $43$ |
\( T^{7} - 2 T^{6} + \cdots - 3548 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots + 680 \)
|
| $53$ |
\( T^{7} - 11 T^{6} + \cdots + 101363 \)
|
| $59$ |
\( T^{7} - T^{6} + \cdots + 9112 \)
|
| $61$ |
\( T^{7} - 28 T^{6} + \cdots + 59060 \)
|
| $67$ |
\( T^{7} - 206 T^{5} + \cdots + 128080 \)
|
| $71$ |
\( T^{7} + 10 T^{6} + \cdots - 358928 \)
|
| $73$ |
\( T^{7} - 11 T^{6} + \cdots + 18148 \)
|
| $79$ |
\( T^{7} - 19 T^{6} + \cdots + 777320 \)
|
| $83$ |
\( T^{7} - 19 T^{6} + \cdots + 371072 \)
|
| $89$ |
\( T^{7} + 2 T^{6} + \cdots - 702976 \)
|
| $97$ |
\( T^{7} + 8 T^{6} + \cdots - 131008 \)
|
| show more | |
| show less |