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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} + 61918633x^{4} + 958456664523288x^{2} + 127494184747700656656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{4}\cdot 5^{8} \) |
| 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
\( x^{6} + 61918633x^{4} + 958456664523288x^{2} + 127494184747700656656 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 11353250917\nu^{3} + 350530392197213316\nu ) / 127843756671942014400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 11353250917\nu^{3} + 383881800408023256516\nu ) / 12784375667194201440 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 30959317\nu^{2} + 15065429484 ) / 377409720 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2741\nu^{4} + 141470945897\nu^{2} + 1168393402823422356 ) / 1698343740 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -523702391\nu^{5} - 27032212340008547\nu^{3} - 331690373831286936495756\nu ) / 177560773155475020 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 10\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{4} + 5482\beta_{3} - 6191863300 ) / 300 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{5} - 309598642\beta_{2} + 3396687480100\beta_1 ) / 300 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -278633853\beta_{4} - 282941891794\beta_{3} + 191700378354211300 ) / 300 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -102179258253\beta_{5} + 9647144216321554\beta_{2} - 175265207405418600100\beta_1 ) / 300 \)
|
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 | − | 156690.i | −1.04858e6 | 0 | −1.60450e8 | − | 4.70592e7i | 1.07374e9i | −1.40914e10 | 0 | |||||||||||||||||||||||||||||||||
| 49.2 | − | 1024.00i | 26460.2i | −1.04858e6 | 0 | 2.70953e7 | − | 3.30980e8i | 1.07374e9i | 9.76021e9 | 0 | |||||||||||||||||||||||||||||||||||
| 49.3 | − | 1024.00i | 176613.i | −1.04858e6 | 0 | 1.80851e8 | 1.28981e9i | 1.07374e9i | −2.07317e10 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.4 | 1024.00i | − | 176613.i | −1.04858e6 | 0 | 1.80851e8 | − | 1.28981e9i | − | 1.07374e9i | −2.07317e10 | 0 | ||||||||||||||||||||||||||||||||||
| 49.5 | 1024.00i | − | 26460.2i | −1.04858e6 | 0 | 2.70953e7 | 3.30980e8i | − | 1.07374e9i | 9.76021e9 | 0 | |||||||||||||||||||||||||||||||||||
| 49.6 | 1024.00i | 156690.i | −1.04858e6 | 0 | −1.60450e8 | 4.70592e7i | − | 1.07374e9i | −1.40914e10 | 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.g | 6 | |
| 5.b | even | 2 | 1 | inner | 50.22.b.g | 6 | |
| 5.c | odd | 4 | 1 | 50.22.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 50.22.a.h | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.22.a.g | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 50.22.a.h | yes | 3 | 5.c | odd | 4 | 1 | |
| 50.22.b.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 50.22.b.g | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 56443896963T_{3}^{4} + 804846675395843920323T_{3}^{2} + 536182095566877790016470103361 \)
acting on \(S_{22}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1048576)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 53\!\cdots\!61 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 40\!\cdots\!04 \)
|
| $11$ |
\( (T^{3} + \cdots - 77\!\cdots\!13)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 19\!\cdots\!16 \)
|
| $17$ |
\( T^{6} + \cdots + 13\!\cdots\!89 \)
|
| $19$ |
\( (T^{3} + \cdots + 89\!\cdots\!25)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!16 \)
|
| $29$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots - 31\!\cdots\!88)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 10\!\cdots\!24 \)
|
| $41$ |
\( (T^{3} + \cdots - 23\!\cdots\!93)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 16\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!76 \)
|
| $59$ |
\( (T^{3} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 35\!\cdots\!68)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 26\!\cdots\!29 \)
|
| $71$ |
\( (T^{3} + \cdots - 16\!\cdots\!88)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 52\!\cdots\!81 \)
|
| $79$ |
\( (T^{3} + \cdots - 48\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 53\!\cdots\!41 \)
|
| $89$ |
\( (T^{3} + \cdots + 19\!\cdots\!75)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!64 \)
|
| show more | |
| show less |