Newspace parameters
Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 880.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.02683537787\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.3317760000.9 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{6} - 3\nu^{5} + 3\nu^{4} - \nu^{3} - 9\nu^{2} + 9\nu + 170 ) / 33 \) |
\(\beta_{2}\) | \(=\) | \( ( 4\nu^{6} - 12\nu^{5} - 10\nu^{4} + 40\nu^{3} + 118\nu^{2} - 140\nu + 328 ) / 33 \) |
\(\beta_{3}\) | \(=\) | \( ( -2\nu^{6} + 6\nu^{5} + 6\nu^{4} - 22\nu^{3} - 54\nu^{2} + 66\nu - 160 ) / 3 \) |
\(\beta_{4}\) | \(=\) | \( ( 16\nu^{7} - 56\nu^{6} + 4\nu^{5} + 130\nu^{4} + 104\nu^{3} - 314\nu^{2} + 2636\nu - 1260 ) / 507 \) |
\(\beta_{5}\) | \(=\) | \( ( -518\nu^{7} + 1813\nu^{6} + 969\nu^{5} - 6955\nu^{4} - 12493\nu^{3} + 26601\nu^{2} - 42499\nu + 16541 ) / 5577 \) |
\(\beta_{6}\) | \(=\) | \( ( 62\nu^{7} - 217\nu^{6} - 69\nu^{5} + 715\nu^{4} + 1417\nu^{3} - 2949\nu^{2} + 6919\nu - 2939 ) / 507 \) |
\(\beta_{7}\) | \(=\) | \( ( 62\nu^{7} - 217\nu^{6} - 69\nu^{5} + 715\nu^{4} + 1417\nu^{3} - 2949\nu^{2} + 5905\nu - 2432 ) / 507 \) |
\(\nu\) | \(=\) | \( ( -\beta_{7} + \beta_{6} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{7} + 2\beta_{6} + \beta_{3} + 6\beta_{2} - 2\beta _1 + 6 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -6\beta_{7} + 19\beta_{6} + 11\beta_{5} - 18\beta_{4} + 3\beta_{3} + 18\beta_{2} - 6\beta _1 + 16 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( -4\beta_{7} + 17\beta_{6} + 11\beta_{5} - 18\beta_{4} + 10\beta_{3} + 54\beta_{2} + 4\beta _1 - 22 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( 146\beta_{7} + 27\beta_{6} + 187\beta_{5} - 120\beta_{4} + 45\beta_{3} + 240\beta_{2} + 30\beta _1 - 136 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( 228\beta_{7} - \beta_{6} + 253\beta_{5} - 135\beta_{4} + 48\beta_{3} + 261\beta_{2} + 144\beta _1 - 774 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 1122\beta_{7} - 497\beta_{6} + 737\beta_{5} - 126\beta_{4} + 91\beta_{3} + 504\beta_{2} + 448\beta _1 - 2462 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).
\(n\) | \(111\) | \(177\) | \(321\) | \(661\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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879.1 |
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0 | −2.44949 | 0 | 1.00000 | − | 2.00000i | 0 | − | 4.47214i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
879.2 | 0 | −2.44949 | 0 | 1.00000 | − | 2.00000i | 0 | 4.47214i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
879.3 | 0 | −2.44949 | 0 | 1.00000 | + | 2.00000i | 0 | − | 4.47214i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
879.4 | 0 | −2.44949 | 0 | 1.00000 | + | 2.00000i | 0 | 4.47214i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
879.5 | 0 | 2.44949 | 0 | 1.00000 | − | 2.00000i | 0 | − | 4.47214i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
879.6 | 0 | 2.44949 | 0 | 1.00000 | − | 2.00000i | 0 | 4.47214i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
879.7 | 0 | 2.44949 | 0 | 1.00000 | + | 2.00000i | 0 | − | 4.47214i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
879.8 | 0 | 2.44949 | 0 | 1.00000 | + | 2.00000i | 0 | 4.47214i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
44.c | even | 2 | 1 | inner |
55.d | odd | 2 | 1 | inner |
220.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 880.2.m.g | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
5.b | even | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
11.b | odd | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
20.d | odd | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
44.c | even | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
55.d | odd | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
220.g | even | 2 | 1 | inner | 880.2.m.g | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
880.2.m.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
880.2.m.g | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 5.b | even | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 44.c | even | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 55.d | odd | 2 | 1 | inner |
880.2.m.g | ✓ | 8 | 220.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} - 6)^{4} \)
$5$
\( (T^{2} - 2 T + 5)^{4} \)
$7$
\( (T^{2} + 20)^{4} \)
$11$
\( (T^{4} + 2 T^{2} + 121)^{2} \)
$13$
\( (T^{2} - 30)^{4} \)
$17$
\( (T^{2} - 30)^{4} \)
$19$
\( (T^{2} - 20)^{4} \)
$23$
\( (T^{2} - 6)^{4} \)
$29$
\( T^{8} \)
$31$
\( (T^{2} + 24)^{4} \)
$37$
\( (T^{2} + 64)^{4} \)
$41$
\( (T^{2} + 120)^{4} \)
$43$
\( (T^{2} + 20)^{4} \)
$47$
\( (T^{2} - 54)^{4} \)
$53$
\( (T^{2} + 16)^{4} \)
$59$
\( (T^{2} + 96)^{4} \)
$61$
\( (T^{2} + 120)^{4} \)
$67$
\( (T^{2} - 54)^{4} \)
$71$
\( T^{8} \)
$73$
\( (T^{2} - 30)^{4} \)
$79$
\( (T^{2} - 20)^{4} \)
$83$
\( (T^{2} + 20)^{4} \)
$89$
\( (T - 4)^{8} \)
$97$
\( (T^{2} + 144)^{4} \)
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