Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(139.738672144\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} + 631153566 x^{6} + \cdots + 19\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{4}\cdot 5^{14}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 631153566 x^{6} + \cdots + 19\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 10\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} + 1577883915\nu^{2} + 6982554807927600 ) / 396194261281 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 631153566 \nu^{5} + \cdots + 10\!\cdots\!04 \nu ) / 11\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 500 \nu^{6} - 306199678000 \nu^{4} + \cdots - 27\!\cdots\!50 ) / 24\!\cdots\!17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2500 \nu^{6} + 1647911763380 \nu^{4} + \cdots + 54\!\cdots\!00 ) / 27\!\cdots\!87 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 65686337405 \nu^{7} + \cdots - 19\!\cdots\!20 \nu ) / 25\!\cdots\!16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4049789151445 \nu^{7} + \cdots + 50\!\cdots\!40 \nu ) / 28\!\cdots\!82 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{5} + 5\beta_{4} - 37668\beta_{2} - 15778839150 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 289289\beta_{7} - 489262\beta_{6} - 594291391921500\beta_{3} - 30672888032\beta_1 ) / 1000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3471344613\beta_{5} - 1577883915\beta_{4} + 19811031487664\beta_{2} + 4839784202272902450 ) / 100 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 178455629459107 \beta_{7} + 114780301876046 \beta_{6} + \cdots + 98\!\cdots\!16 \beta_1 ) / 1000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 55\!\cdots\!04 \beta_{5} + \cdots - 77\!\cdots\!50 ) / 500 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 83\!\cdots\!61 \beta_{7} + \cdots - 32\!\cdots\!68 \beta_1 ) / 1000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 1024.00i | − | 163894.i | −1.04858e6 | 0 | −1.67827e8 | − | 1.27262e8i | 1.07374e9i | −1.64008e10 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 1024.00i | 12234.5i | −1.04858e6 | 0 | 1.25282e7 | − | 9.06115e8i | 1.07374e9i | 1.03107e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 1024.00i | 62614.8i | −1.04858e6 | 0 | 6.41176e7 | 1.35870e9i | 1.07374e9i | 6.53974e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 1024.00i | 185808.i | −1.04858e6 | 0 | 1.90268e8 | − | 4.88079e8i | 1.07374e9i | −2.40644e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.5 | 1024.00i | − | 185808.i | −1.04858e6 | 0 | 1.90268e8 | 4.88079e8i | − | 1.07374e9i | −2.40644e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.6 | 1024.00i | − | 62614.8i | −1.04858e6 | 0 | 6.41176e7 | − | 1.35870e9i | − | 1.07374e9i | 6.53974e9 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.7 | 1024.00i | − | 12234.5i | −1.04858e6 | 0 | 1.25282e7 | 9.06115e8i | − | 1.07374e9i | 1.03107e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.8 | 1024.00i | 163894.i | −1.04858e6 | 0 | −1.67827e8 | 1.27262e8i | − | 1.07374e9i | −1.64008e10 | 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 | 50.22.b.h | 8 | |
| 5.b | even | 2 | 1 | inner | 50.22.b.h | 8 | |
| 5.c | odd | 4 | 1 | 50.22.a.i | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 50.22.a.j | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.22.a.i | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 50.22.a.j | yes | 4 | 5.c | odd | 4 | 1 | |
| 50.22.b.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 50.22.b.h | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 65456174524 T_{3}^{6} + \cdots + 54\!\cdots\!21 \)
acting on \(S_{22}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1048576)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 54\!\cdots\!21 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 58\!\cdots\!76 \)
|
| $11$ |
\( (T^{4} + \cdots - 48\!\cdots\!59)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 74\!\cdots\!76 \)
|
| $17$ |
\( T^{8} + \cdots + 30\!\cdots\!41 \)
|
| $19$ |
\( (T^{4} + \cdots - 34\!\cdots\!75)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 37\!\cdots\!36 \)
|
| $29$ |
\( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 88\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 37\!\cdots\!36 \)
|
| $41$ |
\( (T^{4} + \cdots + 61\!\cdots\!61)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 18\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 45\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 13\!\cdots\!56 \)
|
| $59$ |
\( (T^{4} + \cdots - 84\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 19\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 70\!\cdots\!81 \)
|
| $71$ |
\( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 72\!\cdots\!21 \)
|
| $79$ |
\( (T^{4} + \cdots - 20\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 86\!\cdots\!21 \)
|
| $89$ |
\( (T^{4} + \cdots + 53\!\cdots\!25)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| show more | |
| show less |