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: |
\( x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
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
\( x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 \)
:
\(\beta_{1}\) | \(=\) |
\( ( - 1049071 \nu^{7} + 115038401 \nu^{6} - 163698896 \nu^{5} + 6652953189 \nu^{4} + 3920788192 \nu^{3} + 430332705877 \nu^{2} + \cdots + 368827486562 ) / 3033579204498 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 1696542203 \nu^{7} + 16739134507 \nu^{6} - 23819679472 \nu^{5} + 926039756877 \nu^{4} + 570510371744 \nu^{3} + \cdots + 495081144679420 ) / 63705163294458 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 863252840 \nu^{7} - 9761813387 \nu^{6} + 75308363320 \nu^{5} - 543267423074 \nu^{4} + 3215243331501 \nu^{3} + \cdots - 128122572731930 ) / 10617527215743 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 7051508536 \nu^{7} + 31228732907 \nu^{6} - 412942482080 \nu^{5} + 868174142577 \nu^{4} - 20223346800686 \nu^{3} + \cdots - 39943270058320 ) / 63705163294458 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 1208164720 \nu^{7} - 1739457099 \nu^{6} - 58942130264 \nu^{5} - 384930665533 \nu^{4} - 3607234572013 \nu^{3} + \cdots - 212042096779230 ) / 10617527215743 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 8617899467 \nu^{7} + 10568597603 \nu^{6} + 583780276750 \nu^{5} + 1638688334415 \nu^{4} + 33815120349688 \nu^{3} + \cdots + 74164049534822 ) / 63705163294458 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 9748790105 \nu^{7} + 20249178479 \nu^{6} + 500910335965 \nu^{5} + 2482301067297 \nu^{4} + 34135187416753 \nu^{3} + \cdots + 219625364013092 ) / 63705163294458 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 6 \)
|
\(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} - 5\beta_{6} - 3\beta_{5} - 2\beta_{4} + 6\beta_{3} + 197\beta_1 ) / 6 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 94\beta_{7} - 101\beta_{6} + 12\beta_{5} + 41\beta_{3} + 101\beta_{2} - 257 ) / 6 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -275\beta_{5} + 42\beta_{4} - 233\beta_{3} + 423\beta_{2} - 10311\beta _1 - 10311 ) / 6 \)
|
\(\nu^{5}\) | \(=\) |
\( ( -5158\beta_{7} + 6515\beta_{6} + 2431\beta_{5} - 5158\beta_{4} - 4862\beta_{3} - 25967\beta_1 ) / 6 \)
|
\(\nu^{6}\) | \(=\) |
\( ( -3062\beta_{7} + 31417\beta_{6} + 29096\beta_{5} - 16079\beta_{3} - 31417\beta_{2} + 605185 ) / 6 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -140789\beta_{5} + 295406\beta_{4} + 154617\beta_{3} - 407365\beta_{2} + 2144401\beta _1 + 2144401 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) |
\(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | −8.97703 | − | 15.5487i | 0 | 13.2278 | + | 12.9624i | −8.00000 | 0 | 17.9541 | − | 31.0973i | ||||||||||||||||||||||||||||||||
109.2 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.21575 | − | 3.83779i | 0 | −11.5959 | − | 14.4407i | −8.00000 | 0 | 4.43150 | − | 7.67558i | |||||||||||||||||||||||||||||||||
109.3 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | 1.18685 | + | 2.05569i | 0 | 9.15909 | − | 16.0969i | −8.00000 | 0 | −2.37371 | + | 4.11138i | |||||||||||||||||||||||||||||||||
109.4 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | 8.00592 | + | 13.8667i | 0 | 1.70902 | + | 18.4412i | −8.00000 | 0 | −16.0118 | + | 27.7333i | |||||||||||||||||||||||||||||||||
163.1 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −8.97703 | + | 15.5487i | 0 | 13.2278 | − | 12.9624i | −8.00000 | 0 | 17.9541 | + | 31.0973i | |||||||||||||||||||||||||||||||||
163.2 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.21575 | + | 3.83779i | 0 | −11.5959 | + | 14.4407i | −8.00000 | 0 | 4.43150 | + | 7.67558i | |||||||||||||||||||||||||||||||||
163.3 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 1.18685 | − | 2.05569i | 0 | 9.15909 | + | 16.0969i | −8.00000 | 0 | −2.37371 | − | 4.11138i | |||||||||||||||||||||||||||||||||
163.4 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 8.00592 | − | 13.8667i | 0 | 1.70902 | − | 18.4412i | −8.00000 | 0 | −16.0118 | − | 27.7333i | |||||||||||||||||||||||||||||||||
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.e | yes | 8 |
3.b | odd | 2 | 1 | 378.4.g.d | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 378.4.g.e | yes | 8 |
21.h | odd | 6 | 1 | 378.4.g.d | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.4.g.d | ✓ | 8 | 3.b | odd | 2 | 1 | |
378.4.g.d | ✓ | 8 | 21.h | odd | 6 | 1 | |
378.4.g.e | yes | 8 | 1.a | even | 1 | 1 | trivial |
378.4.g.e | yes | 8 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 4T_{5}^{7} + 310T_{5}^{6} + 48T_{5}^{5} + 85860T_{5}^{4} + 155736T_{5}^{3} + 1263600T_{5}^{2} - 1850688T_{5} + 9144576 \)
acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 4)^{4} \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 4 T^{7} + 310 T^{6} + \cdots + 9144576 \)
$7$
\( T^{8} - 25 T^{7} + \cdots + 13841287201 \)
$11$
\( T^{8} + 56 T^{7} + \cdots + 943105130496 \)
$13$
\( (T^{4} + 9 T^{3} - 2637 T^{2} + \cdots + 1313022)^{2} \)
$17$
\( T^{8} - 118 T^{7} + \cdots + 8822183086656 \)
$19$
\( T^{8} - 37 T^{7} + \cdots + 96\!\cdots\!84 \)
$23$
\( T^{8} + 200 T^{7} + \cdots + 10\!\cdots\!44 \)
$29$
\( (T^{4} - 262 T^{3} - 27174 T^{2} + \cdots - 737431128)^{2} \)
$31$
\( T^{8} - 276 T^{7} + \cdots + 11\!\cdots\!69 \)
$37$
\( T^{8} + 185 T^{7} + \cdots + 26\!\cdots\!16 \)
$41$
\( (T^{4} + 30 T^{3} - 329454 T^{2} + \cdots + 23111436456)^{2} \)
$43$
\( (T^{4} + 778 T^{3} + 137328 T^{2} + \cdots - 309283361)^{2} \)
$47$
\( T^{8} + 30 T^{7} + \cdots + 47\!\cdots\!36 \)
$53$
\( T^{8} + 480 T^{7} + \cdots + 21\!\cdots\!24 \)
$59$
\( T^{8} - 296 T^{7} + \cdots + 17\!\cdots\!64 \)
$61$
\( T^{8} - 474 T^{7} + \cdots + 37\!\cdots\!61 \)
$67$
\( T^{8} - 1319 T^{7} + \cdots + 35\!\cdots\!64 \)
$71$
\( (T^{4} - 926 T^{3} + \cdots - 55652552064)^{2} \)
$73$
\( T^{8} + 1423 T^{7} + \cdots + 42\!\cdots\!44 \)
$79$
\( T^{8} - 765 T^{7} + \cdots + 25\!\cdots\!36 \)
$83$
\( (T^{4} - 830 T^{3} - 10644 T^{2} + \cdots + 331327584)^{2} \)
$89$
\( T^{8} - 864 T^{7} + \cdots + 10\!\cdots\!04 \)
$97$
\( (T^{4} - 544 T^{3} + \cdots - 91533471583)^{2} \)
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