Newspace parameters
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.13993817276\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5^{6} \) |
| 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
\( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 1685\nu^{3} + 587164\nu ) / 344400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} - 2450\nu^{4} + 86045\nu^{3} - 2865450\nu^{2} + 77197188\nu + 44319800 ) / 91840 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{5} - 2450\nu^{4} - 86045\nu^{3} - 2865450\nu^{2} - 77197188\nu + 44319800 ) / 91840 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 103\nu^{5} + 22750\nu^{4} + 575355\nu^{3} + 27657350\nu^{2} + 547516892\nu + 422891000 ) / 91840 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 79\nu^{5} - 9870\nu^{4} + 443075\nu^{3} - 12015990\nu^{2} + 422574076\nu - 196948920 ) / 55104 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{5} + 3\beta_{4} + 29\beta_{3} - 28\beta_{2} + 500\beta_1 ) / 1500 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -21\beta_{5} + 28\beta_{4} + 204\beta_{3} + 197\beta_{2} - 397500 ) / 500 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4684\beta_{5} - 3513\beta_{4} - 35159\beta_{3} + 33988\beta_{2} - 738500\beta_1 ) / 1500 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 24561\beta_{5} - 32748\beta_{4} - 247964\beta_{3} - 239777\beta_{2} + 473949500 ) / 500 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5543884\beta_{5} + 4157913\beta_{4} + 42215159\beta_{3} - 40829188\beta_{2} + 1467390500\beta_1 ) / 1500 \)
|
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 |
|
−32.0000 | + | 32.0000i | −864.774 | − | 864.774i | − | 2048.00i | −14634.8 | − | 5473.81i | 55345.5 | 43339.1 | − | 43339.1i | 65536.0 | + | 65536.0i | 964226.i | 643476. | − | 293152.i | |||||||||||||||||||||||
| 3.2 | −32.0000 | + | 32.0000i | 59.4453 | + | 59.4453i | − | 2048.00i | 3818.66 | + | 15151.2i | −3804.50 | −59325.1 | + | 59325.1i | 65536.0 | + | 65536.0i | − | 524374.i | −607035. | − | 362641.i | |||||||||||||||||||||||
| 3.3 | −32.0000 | + | 32.0000i | 337.328 | + | 337.328i | − | 2048.00i | 2686.16 | − | 15392.4i | −21589.0 | −6681.98 | + | 6681.98i | 65536.0 | + | 65536.0i | − | 303860.i | 406599. | + | 578513.i | |||||||||||||||||||||||
| 7.1 | −32.0000 | − | 32.0000i | −864.774 | + | 864.774i | 2048.00i | −14634.8 | + | 5473.81i | 55345.5 | 43339.1 | + | 43339.1i | 65536.0 | − | 65536.0i | − | 964226.i | 643476. | + | 293152.i | ||||||||||||||||||||||||
| 7.2 | −32.0000 | − | 32.0000i | 59.4453 | − | 59.4453i | 2048.00i | 3818.66 | − | 15151.2i | −3804.50 | −59325.1 | − | 59325.1i | 65536.0 | − | 65536.0i | 524374.i | −607035. | + | 362641.i | |||||||||||||||||||||||||
| 7.3 | −32.0000 | − | 32.0000i | 337.328 | − | 337.328i | 2048.00i | 2686.16 | + | 15392.4i | −21589.0 | −6681.98 | − | 6681.98i | 65536.0 | − | 65536.0i | 303860.i | 406599. | − | 578513.i | |||||||||||||||||||||||||
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 | 10.13.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.13.g.b | 6 | ||
| 4.b | odd | 2 | 1 | 80.13.p.b | 6 | ||
| 5.b | even | 2 | 1 | 50.13.c.d | 6 | ||
| 5.c | odd | 4 | 1 | inner | 10.13.c.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.13.c.d | 6 | ||
| 15.e | even | 4 | 1 | 90.13.g.b | 6 | ||
| 20.e | even | 4 | 1 | 80.13.p.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.13.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 10.13.c.a | ✓ | 6 | 5.c | odd | 4 | 1 | inner |
| 50.13.c.d | 6 | 5.b | even | 2 | 1 | ||
| 50.13.c.d | 6 | 5.c | odd | 4 | 1 | ||
| 80.13.p.b | 6 | 4.b | odd | 2 | 1 | ||
| 80.13.p.b | 6 | 20.e | even | 4 | 1 | ||
| 90.13.g.b | 6 | 3.b | odd | 2 | 1 | ||
| 90.13.g.b | 6 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 936 T_{3}^{5} + 438048 T_{3}^{4} - 674145288 T_{3}^{3} + 417489145956 T_{3}^{2} + \cdots + 24\!\cdots\!48 \)
acting on \(S_{13}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 64 T + 2048)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 24\!\cdots\!48 \)
|
| $5$ |
\( T^{6} + \cdots + 14\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 23\!\cdots\!48 \)
|
| $11$ |
\( (T^{3} + \cdots - 19\!\cdots\!68)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 55\!\cdots\!88 \)
|
| $17$ |
\( T^{6} + \cdots + 18\!\cdots\!48 \)
|
| $19$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 11\!\cdots\!48 \)
|
| $29$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 11\!\cdots\!68)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 34\!\cdots\!08 \)
|
| $41$ |
\( (T^{3} + \cdots - 19\!\cdots\!28)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!28 \)
|
| $47$ |
\( T^{6} + \cdots + 36\!\cdots\!68 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!88 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 23\!\cdots\!72)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 55\!\cdots\!08 \)
|
| $71$ |
\( (T^{3} + \cdots + 36\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 18\!\cdots\!88 \)
|
| $79$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 54\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 97\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 51\!\cdots\!28 \)
|
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