Newspace parameters
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(61.2274649949\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 341x^{3} - 912x^{2} + 1764x + 4860 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 28) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 341x^{3} - 912x^{2} + 1764x + 4860 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -47\nu^{4} + 161\nu^{3} + 15517\nu^{2} + 4764\nu - 74628 ) / 351 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 43\nu^{4} - 217\nu^{3} - 14057\nu^{2} + 17040\nu + 97452 ) / 117 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2969\nu^{4} + 10031\nu^{3} + 987043\nu^{2} + 415812\nu - 5574366 ) / 351 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 473\nu^{4} - 1659\nu^{3} - 157539\nu^{2} - 32416\nu + 966789 ) / 39 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} - 3\beta_{2} - 44\beta _1 + 117 ) / 504 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -11\beta_{4} + 5\beta_{3} + 15\beta_{2} - 1271\beta _1 + 69363 ) / 504 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 95\beta_{4} + 208\beta_{3} - 579\beta_{2} - 6124\beta _1 + 128535 ) / 168 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2554\beta_{4} + 3991\beta_{3} - 1302\beta_{2} - 490777\beta _1 + 23432616 ) / 504 \)
|
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 |
|
0 | −84.0979 | 0 | 116.330 | 0 | 0 | 0 | 4885.46 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −18.4587 | 0 | −176.232 | 0 | 0 | 0 | −1846.28 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.00117 | 0 | 178.056 | 0 | 0 | 0 | −2161.99 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 58.7379 | 0 | −418.382 | 0 | 0 | 0 | 1263.14 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 75.8199 | 0 | 549.228 | 0 | 0 | 0 | 3561.66 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 196.8.a.e | 5 | |
| 7.b | odd | 2 | 1 | 196.8.a.d | 5 | ||
| 7.c | even | 3 | 2 | 196.8.e.f | 10 | ||
| 7.d | odd | 6 | 2 | 28.8.e.a | ✓ | 10 | |
| 21.g | even | 6 | 2 | 252.8.k.c | 10 | ||
| 28.f | even | 6 | 2 | 112.8.i.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.8.e.a | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 112.8.i.d | 10 | 28.f | even | 6 | 2 | ||
| 196.8.a.d | 5 | 7.b | odd | 2 | 1 | ||
| 196.8.a.e | 5 | 1.a | even | 1 | 1 | trivial | |
| 196.8.e.f | 10 | 7.c | even | 3 | 2 | ||
| 252.8.k.c | 10 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 27T_{3}^{4} - 7954T_{3}^{3} + 208878T_{3}^{2} + 8152929T_{3} + 34574877 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(196))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 27 T^{4} + \cdots + 34574877 \)
|
| $5$ |
\( T^{5} + \cdots - 838797481125 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 15\!\cdots\!47 \)
|
| $13$ |
\( T^{5} + \cdots + 14\!\cdots\!56 \)
|
| $17$ |
\( T^{5} + \cdots + 71\!\cdots\!79 \)
|
| $19$ |
\( T^{5} + \cdots - 32\!\cdots\!49 \)
|
| $23$ |
\( T^{5} + \cdots + 79\!\cdots\!89 \)
|
| $29$ |
\( T^{5} + \cdots - 28\!\cdots\!16 \)
|
| $31$ |
\( T^{5} + \cdots + 79\!\cdots\!73 \)
|
| $37$ |
\( T^{5} + \cdots + 11\!\cdots\!57 \)
|
| $41$ |
\( T^{5} + \cdots - 90\!\cdots\!56 \)
|
| $43$ |
\( T^{5} + \cdots + 47\!\cdots\!12 \)
|
| $47$ |
\( T^{5} + \cdots + 96\!\cdots\!67 \)
|
| $53$ |
\( T^{5} + \cdots - 55\!\cdots\!51 \)
|
| $59$ |
\( T^{5} + \cdots + 87\!\cdots\!01 \)
|
| $61$ |
\( T^{5} + \cdots + 52\!\cdots\!19 \)
|
| $67$ |
\( T^{5} + \cdots + 17\!\cdots\!59 \)
|
| $71$ |
\( T^{5} + \cdots - 21\!\cdots\!20 \)
|
| $73$ |
\( T^{5} + \cdots + 88\!\cdots\!87 \)
|
| $79$ |
\( T^{5} + \cdots - 18\!\cdots\!67 \)
|
| $83$ |
\( T^{5} + \cdots + 47\!\cdots\!04 \)
|
| $89$ |
\( T^{5} + \cdots + 24\!\cdots\!55 \)
|
| $97$ |
\( T^{5} + \cdots + 95\!\cdots\!40 \)
|
| show more | |
| show less |