Newspace parameters
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.o (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(24.8502001097\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.219615408.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{5}\cdot 3 \) |
Twist minimal: | no (minimal twist has level 228) |
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 \( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{4} + 2\nu^{3} - 9\nu^{2} + 8\nu + 7 ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( 2\nu^{5} - 5\nu^{4} + 32\nu^{3} - 43\nu^{2} + 98\nu - 42 ) / 19 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{4} + 2\nu^{3} - 13\nu^{2} + 12\nu - 17 ) / 2 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{5} - 5\nu^{4} + 108\nu^{3} - 157\nu^{2} + 934\nu - 441 ) / 38 \) |
\(\beta_{5}\) | \(=\) | \( ( -6\nu^{5} + 15\nu^{4} - 172\nu^{3} + 243\nu^{2} - 902\nu + 411 ) / 38 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} + \beta_{4} + \beta_{2} + 3 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + \beta_{4} - 3\beta_{3} + \beta_{2} + 3\beta _1 - 33 ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( -19\beta_{5} - 13\beta_{4} - 9\beta_{3} - 22\beta_{2} + 9\beta _1 - 102 ) / 12 \) |
\(\nu^{4}\) | \(=\) | \( ( -20\beta_{5} - 14\beta_{4} + 18\beta_{3} - 23\beta_{2} - 30\beta _1 + 261 ) / 6 \) |
\(\nu^{5}\) | \(=\) | \( ( 149\beta_{5} + 83\beta_{4} + 105\beta_{3} + 296\beta_{2} - 165\beta _1 + 1476 ) / 12 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
\(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
721.1 |
|
0 | − | 1.73205i | 0 | −7.39180 | 0 | −3.28502 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||
721.2 | 0 | − | 1.73205i | 0 | 0.556406 | 0 | 9.52587 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||
721.3 | 0 | − | 1.73205i | 0 | 5.83539 | 0 | −5.24085 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||
721.4 | 0 | 1.73205i | 0 | −7.39180 | 0 | −3.28502 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||
721.5 | 0 | 1.73205i | 0 | 0.556406 | 0 | 9.52587 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||
721.6 | 0 | 1.73205i | 0 | 5.83539 | 0 | −5.24085 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.3.o.c | 6 | |
3.b | odd | 2 | 1 | 2736.3.o.m | 6 | ||
4.b | odd | 2 | 1 | 228.3.h.a | ✓ | 6 | |
12.b | even | 2 | 1 | 684.3.h.e | 6 | ||
19.b | odd | 2 | 1 | inner | 912.3.o.c | 6 | |
57.d | even | 2 | 1 | 2736.3.o.m | 6 | ||
76.d | even | 2 | 1 | 228.3.h.a | ✓ | 6 | |
228.b | odd | 2 | 1 | 684.3.h.e | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
228.3.h.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
228.3.h.a | ✓ | 6 | 76.d | even | 2 | 1 | |
684.3.h.e | 6 | 12.b | even | 2 | 1 | ||
684.3.h.e | 6 | 228.b | odd | 2 | 1 | ||
912.3.o.c | 6 | 1.a | even | 1 | 1 | trivial | |
912.3.o.c | 6 | 19.b | odd | 2 | 1 | inner | |
2736.3.o.m | 6 | 3.b | odd | 2 | 1 | ||
2736.3.o.m | 6 | 57.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + T_{5}^{2} - 44T_{5} + 24 \)
acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( (T^{2} + 3)^{3} \)
$5$
\( (T^{3} + T^{2} - 44 T + 24)^{2} \)
$7$
\( (T^{3} - T^{2} - 64 T - 164)^{2} \)
$11$
\( (T^{3} - 13 T^{2} - 8 T + 12)^{2} \)
$13$
\( T^{6} + 792 T^{4} + 98064 T^{2} + \cdots + 2495232 \)
$17$
\( (T^{3} + 25 T^{2} - 32 T - 108)^{2} \)
$19$
\( T^{6} - 10 T^{5} + 407 T^{4} + \cdots + 47045881 \)
$23$
\( (T^{3} + 14 T^{2} - 560 T - 5136)^{2} \)
$29$
\( T^{6} + 3168 T^{4} + \cdots + 322486272 \)
$31$
\( T^{6} + 3192 T^{4} + 1009296 T^{2} + \cdots + 62208 \)
$37$
\( T^{6} + 5976 T^{4} + \cdots + 832267008 \)
$41$
\( T^{6} + 10896 T^{4} + \cdots + 29104386048 \)
$43$
\( (T^{3} - 105 T^{2} + 3432 T - 35764)^{2} \)
$47$
\( (T^{3} + 11 T^{2} - 2084 T - 25272)^{2} \)
$53$
\( T^{6} + 4272 T^{4} + \cdots + 63700992 \)
$59$
\( T^{6} + 19536 T^{4} + \cdots + 207873257472 \)
$61$
\( (T^{3} - 107 T^{2} - 64 T + 104276)^{2} \)
$67$
\( T^{6} + 4320 T^{4} + \cdots + 511377408 \)
$71$
\( T^{6} + 11472 T^{4} + \cdots + 48612814848 \)
$73$
\( (T^{3} - 51 T^{2} - 7896 T + 115492)^{2} \)
$79$
\( T^{6} + 19848 T^{4} + \cdots + 60723965952 \)
$83$
\( (T^{3} - 202 T^{2} + 12892 T - 259992)^{2} \)
$89$
\( T^{6} + 3408 T^{4} + \cdots + 956510208 \)
$97$
\( T^{6} + 8400 T^{4} + \cdots + 168210432 \)
show more
show less