Newspace parameters
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.35357252674\) |
| 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: | \( 2^{8}\cdot 5^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 68121\nu^{5} + 149814\nu^{4} + 329476\nu^{3} - 39101454\nu^{2} + 4554477405\nu - 10394598718 ) / 10016360270 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + \cdots - 18141291569772 ) / 30049080810 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 60844529 \nu^{5} - 325695636 \nu^{4} - 716280824 \nu^{3} + 85006560996 \nu^{2} + \cdots + 822290385952 ) / 30049080810 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -31840\nu^{5} - 738610\nu^{4} - 8310240\nu^{3} + 18276160\nu^{2} - 20819537159 ) / 10470063 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 517677579 \nu^{5} - 1138493986 \nu^{4} + 16684615276 \nu^{3} + 797964943846 \nu^{2} + \cdots + 67978379651882 ) / 30049080810 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 10\beta_{3} + 3\beta _1 + 3 ) / 400 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 25\beta_{3} - 25\beta_{2} + 17383\beta_1 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 261\beta_{5} - 261\beta_{4} + 2610\beta_{2} - 228817\beta _1 + 228817 ) / 400 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{4} - 3980\beta_{3} - 3980\beta_{2} - 2268051 ) / 50 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -13165\beta_{5} - 13165\beta_{4} + 159202\beta_{3} + 19926521\beta _1 + 19926521 ) / 80 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−16.0000 | + | 16.0000i | −266.441 | − | 266.441i | − | 512.000i | 2583.39 | + | 1758.32i | 8526.10 | −13021.6 | + | 13021.6i | 8192.00 | + | 8192.00i | 82932.3i | −69467.5 | + | 13201.2i | |||||||||||||||||||||||
| 3.2 | −16.0000 | + | 16.0000i | 4.29207 | + | 4.29207i | − | 512.000i | −2978.33 | + | 946.124i | −137.346 | 21284.7 | − | 21284.7i | 8192.00 | + | 8192.00i | − | 59012.2i | 32515.4 | − | 62791.3i | |||||||||||||||||||||||
| 3.3 | −16.0000 | + | 16.0000i | 326.149 | + | 326.149i | − | 512.000i | 3124.94 | − | 19.4459i | −10436.8 | −1507.13 | + | 1507.13i | 8192.00 | + | 8192.00i | 153697.i | −49687.9 | + | 50310.2i | ||||||||||||||||||||||||
| 7.1 | −16.0000 | − | 16.0000i | −266.441 | + | 266.441i | 512.000i | 2583.39 | − | 1758.32i | 8526.10 | −13021.6 | − | 13021.6i | 8192.00 | − | 8192.00i | − | 82932.3i | −69467.5 | − | 13201.2i | ||||||||||||||||||||||||
| 7.2 | −16.0000 | − | 16.0000i | 4.29207 | − | 4.29207i | 512.000i | −2978.33 | − | 946.124i | −137.346 | 21284.7 | + | 21284.7i | 8192.00 | − | 8192.00i | 59012.2i | 32515.4 | + | 62791.3i | |||||||||||||||||||||||||
| 7.3 | −16.0000 | − | 16.0000i | 326.149 | − | 326.149i | 512.000i | 3124.94 | + | 19.4459i | −10436.8 | −1507.13 | − | 1507.13i | 8192.00 | − | 8192.00i | − | 153697.i | −49687.9 | − | 50310.2i | ||||||||||||||||||||||||
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 | 10.11.c.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.11.g.c | 6 | ||
| 4.b | odd | 2 | 1 | 80.11.p.c | 6 | ||
| 5.b | even | 2 | 1 | 50.11.c.e | 6 | ||
| 5.c | odd | 4 | 1 | inner | 10.11.c.c | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.11.c.e | 6 | ||
| 15.e | even | 4 | 1 | 90.11.g.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 | 1.a | even | 1 | 1 | trivial |
| 10.11.c.c | ✓ | 6 | 5.c | odd | 4 | 1 | inner |
| 50.11.c.e | 6 | 5.b | even | 2 | 1 | ||
| 50.11.c.e | 6 | 5.c | odd | 4 | 1 | ||
| 80.11.p.c | 6 | 4.b | odd | 2 | 1 | ||
| 80.11.p.c | 6 | 20.e | even | 4 | 1 | ||
| 90.11.g.c | 6 | 3.b | 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 \(S_{11}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 512)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 1112900707872 \)
|
| $5$ |
\( T^{6} + \cdots + 93\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 13\!\cdots\!12 \)
|
| $11$ |
\( (T^{3} + \cdots + 26\!\cdots\!52)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 77\!\cdots\!52 \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!32 \)
|
| $19$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 71\!\cdots\!92 \)
|
| $29$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 10\!\cdots\!88)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 28\!\cdots\!52 \)
|
| $41$ |
\( (T^{3} + \cdots + 19\!\cdots\!72)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 31\!\cdots\!52 \)
|
| $47$ |
\( T^{6} + \cdots + 89\!\cdots\!52 \)
|
| $53$ |
\( T^{6} + \cdots + 85\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 15\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 29\!\cdots\!12 \)
|
| $71$ |
\( (T^{3} + \cdots + 58\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 16\!\cdots\!12 \)
|
| $79$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 11\!\cdots\!32 \)
|
| show more | |
| show less |