Newspace parameters
Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 882.k (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(52.0396846251\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 16.0.721389578983833600000000.8 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 625x^{8} + 390625 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 2^{22}\cdot 3^{12} \) |
Twist minimal: | no (minimal twist has level 126) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 625x^{8} + 390625 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{4} ) / 25 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{8} ) / 625 \) |
\(\beta_{3}\) | \(=\) | \( ( 2\nu^{12} ) / 15625 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{15} + 250\nu^{9} + 1250\nu^{7} + 156250\nu ) / 78125 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{14} + 46875\nu^{2} ) / 78125 \) |
\(\beta_{6}\) | \(=\) | \( ( -2\nu^{15} - 250\nu^{9} + 2500\nu^{7} + 312500\nu ) / 78125 \) |
\(\beta_{7}\) | \(=\) | \( ( -3\nu^{14} + 75\nu^{10} + 1875\nu^{6} ) / 78125 \) |
\(\beta_{8}\) | \(=\) | \( ( 6\nu^{14} + 93750\nu^{2} ) / 78125 \) |
\(\beta_{9}\) | \(=\) | \( ( -6\nu^{10} + 150\nu^{6} + 3750\nu^{2} ) / 3125 \) |
\(\beta_{10}\) | \(=\) | \( ( -4\nu^{13} - 40\nu^{11} - 2500\nu^{5} + 12500\nu^{3} ) / 15625 \) |
\(\beta_{11}\) | \(=\) | \( ( 4\nu^{13} - 20\nu^{11} - 5000\nu^{5} - 12500\nu^{3} ) / 15625 \) |
\(\beta_{12}\) | \(=\) | \( ( 6\nu^{15} + 10\nu^{13} + 100\nu^{11} - 750\nu^{9} - 7500\nu^{7} + 6250\nu^{5} - 31250\nu^{3} + 937500\nu ) / 78125 \) |
\(\beta_{13}\) | \(=\) | \( ( 12\nu^{15} + 10\nu^{13} - 50\nu^{11} - 1500\nu^{9} - 3750\nu^{7} - 12500\nu^{5} - 31250\nu^{3} + 468750\nu ) / 78125 \) |
\(\beta_{14}\) | \(=\) | \( ( 12\nu^{13} + 60\nu^{11} - 15000\nu^{5} + 37500\nu^{3} ) / 15625 \) |
\(\beta_{15}\) | \(=\) | \( ( 12\nu^{13} - 120\nu^{11} + 7500\nu^{5} + 37500\nu^{3} ) / 15625 \) |
\(\nu\) | \(=\) | \( ( -2\beta_{13} + 4\beta_{12} + \beta_{11} + 2\beta_{10} + 6\beta_{6} + 6\beta_{4} ) / 72 \) |
\(\nu^{2}\) | \(=\) | \( ( 5\beta_{8} + 10\beta_{5} ) / 12 \) |
\(\nu^{3}\) | \(=\) | \( ( 5\beta_{15} + 10\beta_{14} - 30\beta_{11} + 15\beta_{10} ) / 72 \) |
\(\nu^{4}\) | \(=\) | \( ( 25\beta_1 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 25\beta_{15} - 25\beta_{14} - 75\beta_{11} - 75\beta_{10} ) / 72 \) |
\(\nu^{6}\) | \(=\) | \( ( 125\beta_{9} + 250\beta_{7} - 250\beta_{5} ) / 12 \) |
\(\nu^{7}\) | \(=\) | \( ( 250\beta_{13} - 500\beta_{12} - 125\beta_{11} - 250\beta_{10} + 750\beta_{6} + 750\beta_{4} ) / 72 \) |
\(\nu^{8}\) | \(=\) | \( 625\beta_{2} \) |
\(\nu^{9}\) | \(=\) | \( ( -2500\beta_{13} + 1250\beta_{12} + 1250\beta_{11} + 625\beta_{10} - 3750\beta_{6} + 7500\beta_{4} ) / 72 \) |
\(\nu^{10}\) | \(=\) | \( ( -3125\beta_{9} + 3125\beta_{8} + 6250\beta_{7} ) / 12 \) |
\(\nu^{11}\) | \(=\) | \( ( -3125\beta_{15} + 3125\beta_{14} - 9375\beta_{11} - 9375\beta_{10} ) / 72 \) |
\(\nu^{12}\) | \(=\) | \( ( 15625\beta_{3} ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 31250\beta_{15} + 15625\beta_{14} + 46875\beta_{11} - 93750\beta_{10} ) / 72 \) |
\(\nu^{14}\) | \(=\) | \( ( 78125\beta_{8} - 156250\beta_{5} ) / 12 \) |
\(\nu^{15}\) | \(=\) | \( ( 312500\beta_{13} - 156250\beta_{12} - 156250\beta_{11} - 78125\beta_{10} - 468750\beta_{6} + 937500\beta_{4} ) / 72 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
\(n\) | \(199\) | \(785\) |
\(\chi(n)\) | \(1 - \beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
215.1 |
|
−1.73205 | − | 1.00000i | 0 | 2.00000 | + | 3.46410i | −7.15634 | + | 12.3951i | 0 | 0 | − | 8.00000i | 0 | 24.7903 | − | 14.3127i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.2 | −1.73205 | − | 1.00000i | 0 | 2.00000 | + | 3.46410i | −2.96425 | + | 5.13424i | 0 | 0 | − | 8.00000i | 0 | 10.2685 | − | 5.92851i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.3 | −1.73205 | − | 1.00000i | 0 | 2.00000 | + | 3.46410i | 2.96425 | − | 5.13424i | 0 | 0 | − | 8.00000i | 0 | −10.2685 | + | 5.92851i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.4 | −1.73205 | − | 1.00000i | 0 | 2.00000 | + | 3.46410i | 7.15634 | − | 12.3951i | 0 | 0 | − | 8.00000i | 0 | −24.7903 | + | 14.3127i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.5 | 1.73205 | + | 1.00000i | 0 | 2.00000 | + | 3.46410i | −7.15634 | + | 12.3951i | 0 | 0 | 8.00000i | 0 | −24.7903 | + | 14.3127i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.6 | 1.73205 | + | 1.00000i | 0 | 2.00000 | + | 3.46410i | −2.96425 | + | 5.13424i | 0 | 0 | 8.00000i | 0 | −10.2685 | + | 5.92851i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.7 | 1.73205 | + | 1.00000i | 0 | 2.00000 | + | 3.46410i | 2.96425 | − | 5.13424i | 0 | 0 | 8.00000i | 0 | 10.2685 | − | 5.92851i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.8 | 1.73205 | + | 1.00000i | 0 | 2.00000 | + | 3.46410i | 7.15634 | − | 12.3951i | 0 | 0 | 8.00000i | 0 | 24.7903 | − | 14.3127i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.1 | −1.73205 | + | 1.00000i | 0 | 2.00000 | − | 3.46410i | −7.15634 | − | 12.3951i | 0 | 0 | 8.00000i | 0 | 24.7903 | + | 14.3127i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.2 | −1.73205 | + | 1.00000i | 0 | 2.00000 | − | 3.46410i | −2.96425 | − | 5.13424i | 0 | 0 | 8.00000i | 0 | 10.2685 | + | 5.92851i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.3 | −1.73205 | + | 1.00000i | 0 | 2.00000 | − | 3.46410i | 2.96425 | + | 5.13424i | 0 | 0 | 8.00000i | 0 | −10.2685 | − | 5.92851i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.4 | −1.73205 | + | 1.00000i | 0 | 2.00000 | − | 3.46410i | 7.15634 | + | 12.3951i | 0 | 0 | 8.00000i | 0 | −24.7903 | − | 14.3127i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.5 | 1.73205 | − | 1.00000i | 0 | 2.00000 | − | 3.46410i | −7.15634 | − | 12.3951i | 0 | 0 | − | 8.00000i | 0 | −24.7903 | − | 14.3127i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.6 | 1.73205 | − | 1.00000i | 0 | 2.00000 | − | 3.46410i | −2.96425 | − | 5.13424i | 0 | 0 | − | 8.00000i | 0 | −10.2685 | − | 5.92851i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.7 | 1.73205 | − | 1.00000i | 0 | 2.00000 | − | 3.46410i | 2.96425 | + | 5.13424i | 0 | 0 | − | 8.00000i | 0 | 10.2685 | + | 5.92851i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.8 | 1.73205 | − | 1.00000i | 0 | 2.00000 | − | 3.46410i | 7.15634 | + | 12.3951i | 0 | 0 | − | 8.00000i | 0 | 24.7903 | + | 14.3127i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.c | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
21.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 882.4.k.c | 16 | |
3.b | odd | 2 | 1 | inner | 882.4.k.c | 16 | |
7.b | odd | 2 | 1 | inner | 882.4.k.c | 16 | |
7.c | even | 3 | 1 | 126.4.d.a | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 882.4.k.c | 16 | |
7.d | odd | 6 | 1 | 126.4.d.a | ✓ | 8 | |
7.d | odd | 6 | 1 | inner | 882.4.k.c | 16 | |
21.c | even | 2 | 1 | inner | 882.4.k.c | 16 | |
21.g | even | 6 | 1 | 126.4.d.a | ✓ | 8 | |
21.g | even | 6 | 1 | inner | 882.4.k.c | 16 | |
21.h | odd | 6 | 1 | 126.4.d.a | ✓ | 8 | |
21.h | odd | 6 | 1 | inner | 882.4.k.c | 16 | |
28.f | even | 6 | 1 | 1008.4.k.c | 8 | ||
28.g | odd | 6 | 1 | 1008.4.k.c | 8 | ||
84.j | odd | 6 | 1 | 1008.4.k.c | 8 | ||
84.n | even | 6 | 1 | 1008.4.k.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
126.4.d.a | ✓ | 8 | 7.c | even | 3 | 1 | |
126.4.d.a | ✓ | 8 | 7.d | odd | 6 | 1 | |
126.4.d.a | ✓ | 8 | 21.g | even | 6 | 1 | |
126.4.d.a | ✓ | 8 | 21.h | odd | 6 | 1 | |
882.4.k.c | 16 | 1.a | even | 1 | 1 | trivial | |
882.4.k.c | 16 | 3.b | odd | 2 | 1 | inner | |
882.4.k.c | 16 | 7.b | odd | 2 | 1 | inner | |
882.4.k.c | 16 | 7.c | even | 3 | 1 | inner | |
882.4.k.c | 16 | 7.d | odd | 6 | 1 | inner | |
882.4.k.c | 16 | 21.c | even | 2 | 1 | inner | |
882.4.k.c | 16 | 21.g | even | 6 | 1 | inner | |
882.4.k.c | 16 | 21.h | odd | 6 | 1 | inner | |
1008.4.k.c | 8 | 28.f | even | 6 | 1 | ||
1008.4.k.c | 8 | 28.g | odd | 6 | 1 | ||
1008.4.k.c | 8 | 84.j | odd | 6 | 1 | ||
1008.4.k.c | 8 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 240T_{5}^{6} + 50400T_{5}^{4} + 1728000T_{5}^{2} + 51840000 \)
acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 4 T^{2} + 16)^{4} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} + 240 T^{6} + 50400 T^{4} + \cdots + 51840000)^{2} \)
$7$
\( T^{16} \)
$11$
\( (T^{8} - 3564 T^{6} + 12701772 T^{4} + \cdots + 104976)^{2} \)
$13$
\( (T^{4} + 8160 T^{2} + 15235200)^{4} \)
$17$
\( (T^{8} + 20400 T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
$19$
\( (T^{8} - 13920 T^{6} + \cdots + 23\!\cdots\!00)^{2} \)
$23$
\( (T^{8} - 4428 T^{6} + \cdots + 2981133747216)^{2} \)
$29$
\( (T^{4} + 37476 T^{2} + \cdots + 133125444)^{4} \)
$31$
\( (T^{8} - 17280 T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
$37$
\( (T^{4} - 40 T^{3} + 94512 T^{2} + \cdots + 8632639744)^{4} \)
$41$
\( (T^{4} - 89520 T^{2} + \cdots + 1999648800)^{4} \)
$43$
\( (T^{2} - 260 T - 3908)^{8} \)
$47$
\( (T^{8} + 265920 T^{6} + \cdots + 24\!\cdots\!00)^{2} \)
$53$
\( (T^{8} - 73188 T^{6} + \cdots + 17\!\cdots\!96)^{2} \)
$59$
\( (T^{8} + 568320 T^{6} + \cdots + 24\!\cdots\!00)^{2} \)
$61$
\( (T^{8} - 407040 T^{6} + \cdots + 45\!\cdots\!00)^{2} \)
$67$
\( (T^{4} - 920 T^{3} + \cdots + 19937440000)^{4} \)
$71$
\( (T^{4} + 1100844 T^{2} + \cdots + 913369284)^{4} \)
$73$
\( (T^{8} - 255840 T^{6} + \cdots + 29\!\cdots\!00)^{2} \)
$79$
\( (T^{4} + 112 T^{3} + \cdots + 336474244096)^{4} \)
$83$
\( (T^{4} - 945600 T^{2} + \cdots + 205927948800)^{4} \)
$89$
\( (T^{8} + 1721520 T^{6} + \cdots + 28\!\cdots\!00)^{2} \)
$97$
\( (T^{4} + 2605920 T^{2} + \cdots + 409114396800)^{4} \)
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