Newspace parameters
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
Defining polynomial: |
\( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( -304\nu^{7} - 31\nu^{6} - 2546\nu^{5} - 1710\nu^{4} - 23348\nu^{3} - 1178\nu^{2} - 380\nu + 48866 ) / 13629 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 536\nu^{7} - 304\nu^{6} + 4489\nu^{5} + 3015\nu^{4} + 36145\nu^{3} + 2077\nu^{2} + 670\nu + 3506 ) / 13629 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 1753\nu^{7} - 2825\nu^{6} + 16385\nu^{5} - 5472\nu^{4} + 107915\nu^{3} - 107350\nu^{2} + 39671\nu - 18080 ) / 27258 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 1418 \nu^{7} - 2635 \nu^{6} + 15283 \nu^{5} - 9060 \nu^{4} + 100657 \nu^{3} - 100130 \nu^{2} + 131248 \nu - 16864 ) / 13629 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 3274 \nu^{7} - 5315 \nu^{6} + 30827 \nu^{5} - 12249 \nu^{4} + 203033 \nu^{3} - 201970 \nu^{2} + 65423 \nu - 34016 ) / 13629 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -1328\nu^{7} + 821\nu^{6} - 11122\nu^{5} - 7470\nu^{4} - 84061\nu^{3} - 5146\nu^{2} - 1660\nu - 16552 ) / 4543 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{6} - 4\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 4 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} + 8\beta_{3} + \beta_{2} - 2 \)
|
\(\nu^{4}\) | \(=\) |
\( -9\beta_{6} + \beta_{5} + 32\beta_{4} - 13\beta_1 \)
|
\(\nu^{5}\) | \(=\) |
\( -9\beta_{7} - 14\beta_{6} + 9\beta_{5} + 36\beta_{4} - 72\beta_{3} - 14\beta_{2} - 72\beta _1 + 36 \)
|
\(\nu^{6}\) | \(=\) |
\( -14\beta_{7} - 150\beta_{3} - 81\beta_{2} + 278 \)
|
\(\nu^{7}\) | \(=\) |
\( 164\beta_{6} - 81\beta_{5} - 466\beta_{4} + 671\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(77\) | \(191\) |
\(\chi(n)\) | \(-1 - \beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
121.1 |
|
0 | −1.58253 | − | 2.74101i | 0 | −0.500000 | − | 0.866025i | 0 | −1.53315 | 0 | −3.50877 | + | 6.07738i | 0 | ||||||||||||||||||||||||||||||||||||
121.2 | 0 | −0.354609 | − | 0.614201i | 0 | −0.500000 | − | 0.866025i | 0 | 3.11079 | 0 | 1.24850 | − | 2.16247i | 0 | |||||||||||||||||||||||||||||||||||||
121.3 | 0 | 0.176725 | + | 0.306096i | 0 | −0.500000 | − | 0.866025i | 0 | −4.30507 | 0 | 1.43754 | − | 2.48989i | 0 | |||||||||||||||||||||||||||||||||||||
121.4 | 0 | 1.26041 | + | 2.18309i | 0 | −0.500000 | − | 0.866025i | 0 | 2.72743 | 0 | −1.67727 | + | 2.90511i | 0 | |||||||||||||||||||||||||||||||||||||
201.1 | 0 | −1.58253 | + | 2.74101i | 0 | −0.500000 | + | 0.866025i | 0 | −1.53315 | 0 | −3.50877 | − | 6.07738i | 0 | |||||||||||||||||||||||||||||||||||||
201.2 | 0 | −0.354609 | + | 0.614201i | 0 | −0.500000 | + | 0.866025i | 0 | 3.11079 | 0 | 1.24850 | + | 2.16247i | 0 | |||||||||||||||||||||||||||||||||||||
201.3 | 0 | 0.176725 | − | 0.306096i | 0 | −0.500000 | + | 0.866025i | 0 | −4.30507 | 0 | 1.43754 | + | 2.48989i | 0 | |||||||||||||||||||||||||||||||||||||
201.4 | 0 | 1.26041 | − | 2.18309i | 0 | −0.500000 | + | 0.866025i | 0 | 2.72743 | 0 | −1.67727 | − | 2.90511i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.i.c | ✓ | 8 |
3.b | odd | 2 | 1 | 3420.2.t.w | 8 | ||
4.b | odd | 2 | 1 | 1520.2.q.m | 8 | ||
5.b | even | 2 | 1 | 1900.2.i.d | 8 | ||
5.c | odd | 4 | 2 | 1900.2.s.d | 16 | ||
19.c | even | 3 | 1 | inner | 380.2.i.c | ✓ | 8 |
19.c | even | 3 | 1 | 7220.2.a.r | 4 | ||
19.d | odd | 6 | 1 | 7220.2.a.p | 4 | ||
57.h | odd | 6 | 1 | 3420.2.t.w | 8 | ||
76.g | odd | 6 | 1 | 1520.2.q.m | 8 | ||
95.i | even | 6 | 1 | 1900.2.i.d | 8 | ||
95.m | odd | 12 | 2 | 1900.2.s.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.i.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
380.2.i.c | ✓ | 8 | 19.c | even | 3 | 1 | inner |
1520.2.q.m | 8 | 4.b | odd | 2 | 1 | ||
1520.2.q.m | 8 | 76.g | odd | 6 | 1 | ||
1900.2.i.d | 8 | 5.b | even | 2 | 1 | ||
1900.2.i.d | 8 | 95.i | even | 6 | 1 | ||
1900.2.s.d | 16 | 5.c | odd | 4 | 2 | ||
1900.2.s.d | 16 | 95.m | odd | 12 | 2 | ||
3420.2.t.w | 8 | 3.b | odd | 2 | 1 | ||
3420.2.t.w | 8 | 57.h | odd | 6 | 1 | ||
7220.2.a.p | 4 | 19.d | odd | 6 | 1 | ||
7220.2.a.r | 4 | 19.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} + 9T_{3}^{6} - 2T_{3}^{5} + 65T_{3}^{4} + 20T_{3}^{3} + 25T_{3}^{2} - 6T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + T^{7} + 9 T^{6} - 2 T^{5} + 65 T^{4} + \cdots + 4 \)
$5$
\( (T^{2} + T + 1)^{4} \)
$7$
\( (T^{4} - 19 T^{2} + 11 T + 56)^{2} \)
$11$
\( (T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267)^{2} \)
$13$
\( T^{8} - 9 T^{7} + 86 T^{6} + \cdots + 2704 \)
$17$
\( T^{8} - T^{7} + 35 T^{6} + 52 T^{5} + \cdots + 36864 \)
$19$
\( T^{8} - 3 T^{7} - 43 T^{6} + \cdots + 130321 \)
$23$
\( T^{8} + 39 T^{6} + 198 T^{5} + \cdots + 2916 \)
$29$
\( T^{8} - 5 T^{7} + 84 T^{6} + \cdots + 74529 \)
$31$
\( (T^{4} + 10 T^{3} - 29 T^{2} - 303 T - 277)^{2} \)
$37$
\( (T^{4} + 26 T^{3} + 217 T^{2} + 609 T + 414)^{2} \)
$41$
\( T^{8} + 8 T^{7} + 59 T^{6} + 106 T^{5} + \cdots + 324 \)
$43$
\( T^{8} - 7 T^{7} + 81 T^{6} + \cdots + 31684 \)
$47$
\( T^{8} - 16 T^{7} + 281 T^{6} + \cdots + 1382976 \)
$53$
\( T^{8} - 5 T^{7} + 143 T^{6} + \cdots + 1483524 \)
$59$
\( T^{8} - 11 T^{7} + 176 T^{6} + \cdots + 2518569 \)
$61$
\( T^{8} - 12 T^{7} + 166 T^{6} + \cdots + 841 \)
$67$
\( T^{8} + 124 T^{6} + \cdots + 10863616 \)
$71$
\( T^{8} - 14 T^{7} + 197 T^{6} + \cdots + 5049009 \)
$73$
\( T^{8} + 4 T^{7} + 153 T^{6} + \cdots + 311364 \)
$79$
\( T^{8} - 13 T^{7} + 297 T^{6} + \cdots + 9096256 \)
$83$
\( (T^{4} - 5 T^{3} - 82 T^{2} + 669 T - 1302)^{2} \)
$89$
\( T^{8} - 5 T^{7} + 147 T^{6} + \cdots + 1382976 \)
$97$
\( T^{8} + T^{7} + 123 T^{6} + \cdots + 13351716 \)
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