Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(31.7678626337\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5^{8} \) |
| Twist minimal: | no (minimal twist has level 10) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -68121\nu^{5} - 149814\nu^{4} - 329476\nu^{3} + 39101454\nu^{2} - 4554477405\nu + 10394598718 ) / 10016360270 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + \cdots - 18141291569772 ) / 30049080810 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60844529 \nu^{5} + 325695636 \nu^{4} + 716280824 \nu^{3} - 85006560996 \nu^{2} + \cdots - 822290385952 ) / 30049080810 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2252984719 \nu^{5} - 5023143454 \nu^{4} - 201396288436 \nu^{3} - 3897466898906 \nu^{2} + \cdots - 675057305384532 ) / 30049080810 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3167389679 \nu^{5} - 16943813586 \nu^{4} - 37263406124 \nu^{3} + 4422335345946 \nu^{2} + \cdots + 42778390230352 ) / 30049080810 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 52\beta_{3} - 17\beta _1 + 17 ) / 2000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 248\beta_{3} - 248\beta_{2} - 173834\beta_1 ) / 1000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -261\beta_{4} + 13572\beta_{2} + 1143563\beta _1 + 1143563 ) / 2000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{5} + 13\beta_{4} + 39826\beta_{3} - 39826\beta_{2} - 22680458 ) / 500 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2633\beta_{5} - 164468\beta_{3} - 19921255\beta _1 + 19921255 ) / 80 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
16.0000 | + | 16.0000i | −326.149 | + | 326.149i | 512.000i | 0 | −10436.8 | 1507.13 | + | 1507.13i | −8192.00 | + | 8192.00i | − | 153697.i | 0 | |||||||||||||||||||||||||||
| 7.2 | 16.0000 | + | 16.0000i | −4.29207 | + | 4.29207i | 512.000i | 0 | −137.346 | −21284.7 | − | 21284.7i | −8192.00 | + | 8192.00i | 59012.2i | 0 | |||||||||||||||||||||||||||||
| 7.3 | 16.0000 | + | 16.0000i | 266.441 | − | 266.441i | 512.000i | 0 | 8526.10 | 13021.6 | + | 13021.6i | −8192.00 | + | 8192.00i | − | 82932.3i | 0 | ||||||||||||||||||||||||||||
| 43.1 | 16.0000 | − | 16.0000i | −326.149 | − | 326.149i | − | 512.000i | 0 | −10436.8 | 1507.13 | − | 1507.13i | −8192.00 | − | 8192.00i | 153697.i | 0 | ||||||||||||||||||||||||||||
| 43.2 | 16.0000 | − | 16.0000i | −4.29207 | − | 4.29207i | − | 512.000i | 0 | −137.346 | −21284.7 | + | 21284.7i | −8192.00 | − | 8192.00i | − | 59012.2i | 0 | |||||||||||||||||||||||||||
| 43.3 | 16.0000 | − | 16.0000i | 266.441 | + | 266.441i | − | 512.000i | 0 | 8526.10 | 13021.6 | − | 13021.6i | −8192.00 | − | 8192.00i | 82932.3i | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.11.c.e | 6 | |
| 5.b | even | 2 | 1 | 10.11.c.c | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 10.11.c.c | ✓ | 6 | |
| 5.c | odd | 4 | 1 | inner | 50.11.c.e | 6 | |
| 15.d | odd | 2 | 1 | 90.11.g.c | 6 | ||
| 15.e | even | 4 | 1 | 90.11.g.c | 6 | ||
| 20.d | odd | 2 | 1 | 80.11.p.c | 6 | ||
| 20.e | even | 4 | 1 | 80.11.p.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.11.c.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 10.11.c.c | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 50.11.c.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 50.11.c.e | 6 | 5.c | odd | 4 | 1 | inner | |
| 80.11.p.c | 6 | 20.d | odd | 2 | 1 | ||
| 80.11.p.c | 6 | 20.e | even | 4 | 1 | ||
| 90.11.g.c | 6 | 15.d | odd | 2 | 1 | ||
| 90.11.g.c | 6 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 128T_{3}^{5} + 8192T_{3}^{4} - 20688696T_{3}^{3} + 30028037796T_{3}^{2} + 258527462832T_{3} + 1112900707872 \)
acting on \(S_{11}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 32 T + 512)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 1112900707872 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 13\!\cdots\!12 \)
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| $11$ |
\( (T^{3} + \cdots + 26\!\cdots\!52)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 77\!\cdots\!52 \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!32 \)
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| $19$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 71\!\cdots\!92 \)
|
| $29$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 10\!\cdots\!88)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 28\!\cdots\!52 \)
|
| $41$ |
\( (T^{3} + \cdots + 19\!\cdots\!72)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 31\!\cdots\!52 \)
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| $47$ |
\( T^{6} + \cdots + 89\!\cdots\!52 \)
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| $53$ |
\( T^{6} + \cdots + 85\!\cdots\!92 \)
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| $59$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
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| $61$ |
\( (T^{3} + \cdots + 15\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 29\!\cdots\!12 \)
|
| $71$ |
\( (T^{3} + \cdots + 58\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 16\!\cdots\!12 \)
|
| $79$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 11\!\cdots\!32 \)
|
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