Newspace parameters
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(22.3027219822\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
Defining polynomial: |
\( x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
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
\( x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 \)
:
\(\beta_{1}\) | \(=\) |
\( ( - 5251 \nu^{7} + 17661 \nu^{6} - 103443 \nu^{5} + 165901 \nu^{4} - 1314483 \nu^{3} + 3812253 \nu^{2} - 811706 \nu + 2144550 ) / 2252390 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 5533 \nu^{7} + 45633 \nu^{6} - 267279 \nu^{5} + 771473 \nu^{4} - 3396399 \nu^{3} + 9850209 \nu^{2} - 43821428 \nu + 5541150 ) / 2252390 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 25642 \nu^{7} - 123347 \nu^{6} + 400691 \nu^{5} - 1182462 \nu^{4} + 3388681 \nu^{3} - 26303561 \nu^{2} - 24618118 \nu + 20095080 ) / 2252390 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 26206 \nu^{7} + 179291 \nu^{6} - 728363 \nu^{5} + 2393606 \nu^{4} - 7552513 \nu^{3} + 38379473 \nu^{2} - 34372646 \nu - 13301880 ) / 2252390 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 16807 \nu^{7} - 3492 \nu^{6} + 296256 \nu^{5} + 417823 \nu^{4} + 4902586 \nu^{3} + 3475204 \nu^{2} + 3795232 \nu + 5873470 ) / 321770 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 42509 \nu^{7} + 56798 \nu^{6} - 783152 \nu^{5} - 84149 \nu^{4} - 11885520 \nu^{3} + 6404040 \nu^{2} - 7027748 \nu + 5545466 ) / 450478 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 47855 \nu^{7} + 35959 \nu^{6} - 821980 \nu^{5} - 661525 \nu^{4} - 13230433 \nu^{3} - 636190 \nu^{2} - 394100 \nu - 3549092 ) / 450478 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta_1 ) / 12 \)
|
\(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} - \beta_{6} - 3\beta_{5} - 4\beta_{4} + 4\beta_{3} - 2\beta_{2} + 106\beta _1 - 104 ) / 12 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -14\beta_{7} + 11\beta_{6} - 9\beta_{5} - 11\beta_{4} - 9\beta_{3} + 2\beta _1 - 68 ) / 6 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 24\beta_{6} + 24\beta_{5} + \beta_{4} - 23\beta_{3} + 18\beta_{2} - 570\beta _1 - 24 ) / 4 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 502\beta_{7} - 297\beta_{6} + 421\beta_{5} + 124\beta_{4} - 124\beta_{3} + 502\beta_{2} - 3174\beta _1 + 3552 ) / 12 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 322\beta_{7} - 369\beta_{6} + 46\beta_{5} + 369\beta_{4} + 46\beta_{3} - 323\beta _1 + 8316 ) / 3 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -3072\beta_{6} - 3072\beta_{5} + 5065\beta_{4} + 8137\beta_{3} - 9326\beta_{2} + 74654\beta _1 + 3072 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) |
\(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −6.50453 | − | 11.2662i | 0 | 18.4787 | − | 1.24034i | 8.00000 | 0 | −13.0091 | + | 22.5324i | ||||||||||||||||||||||||||||||||
109.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −5.59791 | − | 9.69587i | 0 | −16.7643 | − | 7.87127i | 8.00000 | 0 | −11.1958 | + | 19.3917i | |||||||||||||||||||||||||||||||||
109.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 5.04834 | + | 8.74398i | 0 | −5.48778 | + | 17.6885i | 8.00000 | 0 | 10.0967 | − | 17.4880i | |||||||||||||||||||||||||||||||||
109.4 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 8.05411 | + | 13.9501i | 0 | 6.77345 | − | 17.2372i | 8.00000 | 0 | 16.1082 | − | 27.9003i | |||||||||||||||||||||||||||||||||
163.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −6.50453 | + | 11.2662i | 0 | 18.4787 | + | 1.24034i | 8.00000 | 0 | −13.0091 | − | 22.5324i | |||||||||||||||||||||||||||||||||
163.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −5.59791 | + | 9.69587i | 0 | −16.7643 | + | 7.87127i | 8.00000 | 0 | −11.1958 | − | 19.3917i | |||||||||||||||||||||||||||||||||
163.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 5.04834 | − | 8.74398i | 0 | −5.48778 | − | 17.6885i | 8.00000 | 0 | 10.0967 | + | 17.4880i | |||||||||||||||||||||||||||||||||
163.4 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 8.05411 | − | 13.9501i | 0 | 6.77345 | + | 17.2372i | 8.00000 | 0 | 16.1082 | + | 27.9003i | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.4.g.c | ✓ | 8 |
3.b | odd | 2 | 1 | 378.4.g.f | yes | 8 | |
7.c | even | 3 | 1 | inner | 378.4.g.c | ✓ | 8 |
21.h | odd | 6 | 1 | 378.4.g.f | yes | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.4.g.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
378.4.g.c | ✓ | 8 | 7.c | even | 3 | 1 | inner |
378.4.g.f | yes | 8 | 3.b | odd | 2 | 1 | |
378.4.g.f | yes | 8 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2 T_{5}^{7} + 330 T_{5}^{6} + 412 T_{5}^{5} + 82828 T_{5}^{4} + 55632 T_{5}^{3} + 7736688 T_{5}^{2} + 2842560 T_{5} + 561121344 \)
acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 4)^{4} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 2 T^{7} + 330 T^{6} + \cdots + 561121344 \)
$7$
\( T^{8} - 6 T^{7} + \cdots + 13841287201 \)
$11$
\( T^{8} + 32 T^{7} + \cdots + 53616328704 \)
$13$
\( (T^{4} + 2 T^{3} - 4883 T^{2} + \cdots - 513576)^{2} \)
$17$
\( T^{8} - 58 T^{7} + \cdots + 1132436505600 \)
$19$
\( T^{8} + \cdots + 543448598016256 \)
$23$
\( T^{8} + 86 T^{7} + \cdots + 2210504256 \)
$29$
\( (T^{4} - 106 T^{3} - 39170 T^{2} + \cdots - 175957920)^{2} \)
$31$
\( T^{8} + 64 T^{7} + \cdots + 139065597225 \)
$37$
\( T^{8} + 146 T^{7} + \cdots + 49\!\cdots\!00 \)
$41$
\( (T^{4} - 390 T^{3} - 61074 T^{2} + \cdots + 2132840160)^{2} \)
$43$
\( (T^{4} - 440 T^{3} + 34890 T^{2} + \cdots - 49652003)^{2} \)
$47$
\( T^{8} - 306 T^{7} + \cdots + 73\!\cdots\!36 \)
$53$
\( T^{8} - 90 T^{7} + \cdots + 17\!\cdots\!36 \)
$59$
\( T^{8} + 148 T^{7} + \cdots + 22\!\cdots\!16 \)
$61$
\( T^{8} + 364 T^{7} + \cdots + 80\!\cdots\!25 \)
$67$
\( T^{8} + 954 T^{7} + \cdots + 21\!\cdots\!00 \)
$71$
\( (T^{4} - 680 T^{3} - 431588 T^{2} + \cdots - 5339649600)^{2} \)
$73$
\( T^{8} + 54 T^{7} + \cdots + 38\!\cdots\!56 \)
$79$
\( T^{8} + 226 T^{7} + \cdots + 20\!\cdots\!56 \)
$83$
\( (T^{4} - 1568 T^{3} + \cdots - 310761735840)^{2} \)
$89$
\( T^{8} - 1458 T^{7} + \cdots + 29\!\cdots\!00 \)
$97$
\( (T^{4} + 2172 T^{3} + \cdots - 1160560203227)^{2} \)
show more
show less