Newspace parameters
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(32.2150428631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 149x^{6} + 684x^{5} + 20666x^{4} + 28425x^{3} + 33734x^{2} + 6895x + 1225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} - 2x^{7} + 149x^{6} + 684x^{5} + 20666x^{4} + 28425x^{3} + 33734x^{2} + 6895x + 1225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3194930 \nu^{7} + 10656195 \nu^{6} - 485497156 \nu^{5} - 1537025738 \nu^{4} + \cdots - 126846201566 ) / 105258945561 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 88111249 \nu^{7} + 192197148 \nu^{6} - 13181857076 \nu^{5} - 57840608536 \nu^{4} + \cdots - 605872859305 ) / 526294727805 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9973897508 \nu^{7} - 8419936866 \nu^{6} + 1447279171147 \nu^{5} + 8573903319452 \nu^{4} + \cdots + 69963752904710 ) / 20525494384395 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2192677675 \nu^{7} + 7394548293 \nu^{6} - 333196275110 \nu^{5} - 1054859424655 \nu^{4} + \cdots + 10233834735467 ) / 4105098876879 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3177787520 \nu^{7} + 10675452744 \nu^{6} - 482892208384 \nu^{5} - 1528778786432 \nu^{4} + \cdots + 70442063783917 ) / 4105098876879 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 61011123913 \nu^{7} + 113981641851 \nu^{6} - 9063863150657 \nu^{5} + \cdots - 421504317991960 ) / 20525494384395 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -5\beta_{7} - 7\beta_{4} + 69\beta_{3} - 3\beta_{2} + 3\beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 17\beta_{6} - 16\beta_{5} - 152\beta_{2} - 435 \)
|
| \(\nu^{4}\) | \(=\) |
\( 759\beta_{7} + 759\beta_{6} - 1047\beta_{5} + 1047\beta_{4} - 10606\beta_{3} - 936\beta _1 - 10606 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4968\beta_{7} + 5793\beta_{4} - 75840\beta_{3} + 24538\beta_{2} - 24538\beta _1 + 24538 \)
|
| \(\nu^{6}\) | \(=\) |
\( -123515\beta_{6} + 166798\beta_{5} + 214845\beta_{2} + 1962561 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1117508 \beta_{7} - 1117508 \beta_{6} + 1380400 \beta_{5} - 1380400 \beta_{4} + 16591519 \beta_{3} + \cdots + 16591519 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −9.97935 | + | 17.2847i | −6.00000 | 17.7041 | − | 5.43748i | 8.00000 | −4.50000 | + | 7.79423i | −19.9587 | − | 34.5695i | ||||||||||||||||||||||||||||
| 79.2 | −1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −9.83247 | + | 17.0303i | −6.00000 | −0.135629 | + | 18.5198i | 8.00000 | −4.50000 | + | 7.79423i | −19.6649 | − | 34.0607i | |||||||||||||||||||||||||||||
| 79.3 | −1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −0.502223 | + | 0.869875i | −6.00000 | −14.8884 | + | 11.0152i | 8.00000 | −4.50000 | + | 7.79423i | −1.00445 | − | 1.73975i | |||||||||||||||||||||||||||||
| 79.4 | −1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 3.31404 | − | 5.74009i | −6.00000 | 14.8200 | − | 11.1071i | 8.00000 | −4.50000 | + | 7.79423i | 6.62809 | + | 11.4802i | |||||||||||||||||||||||||||||
| 235.1 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −9.97935 | − | 17.2847i | −6.00000 | 17.7041 | + | 5.43748i | 8.00000 | −4.50000 | − | 7.79423i | −19.9587 | + | 34.5695i | |||||||||||||||||||||||||||||
| 235.2 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −9.83247 | − | 17.0303i | −6.00000 | −0.135629 | − | 18.5198i | 8.00000 | −4.50000 | − | 7.79423i | −19.6649 | + | 34.0607i | |||||||||||||||||||||||||||||
| 235.3 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −0.502223 | − | 0.869875i | −6.00000 | −14.8884 | − | 11.0152i | 8.00000 | −4.50000 | − | 7.79423i | −1.00445 | + | 1.73975i | |||||||||||||||||||||||||||||
| 235.4 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 3.31404 | + | 5.74009i | −6.00000 | 14.8200 | + | 11.1071i | 8.00000 | −4.50000 | − | 7.79423i | 6.62809 | − | 11.4802i | |||||||||||||||||||||||||||||
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 | 546.4.i.b | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 546.4.i.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.i.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 546.4.i.b | ✓ | 8 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 34 T_{5}^{7} + 993 T_{5}^{6} + 10484 T_{5}^{5} + 113196 T_{5}^{4} - 225089 T_{5}^{3} + \cdots + 6827769 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{4} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $5$ |
\( T^{8} + 34 T^{7} + \cdots + 6827769 \)
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| $7$ |
\( T^{8} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{8} + \cdots + 8444981300625 \)
|
| $13$ |
\( (T - 13)^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 697395668062500 \)
|
| $19$ |
\( T^{8} + \cdots + 270682518714624 \)
|
| $23$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $29$ |
\( (T^{4} - 165 T^{3} + \cdots + 87015315)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 40\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 38\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} + 274 T^{3} + \cdots - 2016555996)^{2} \)
|
| $43$ |
\( (T^{4} - 568 T^{3} + \cdots - 1242360668)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 62\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 28\!\cdots\!25 \)
|
| $59$ |
\( T^{8} + \cdots + 44\!\cdots\!21 \)
|
| $61$ |
\( T^{8} + \cdots + 44\!\cdots\!16 \)
|
| $67$ |
\( T^{8} + \cdots + 37\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + 777 T^{3} + \cdots - 1264108314)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
|
| $79$ |
\( T^{8} + \cdots + 26\!\cdots\!21 \)
|
| $83$ |
\( (T^{4} + \cdots + 1380578506041)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 38\!\cdots\!96 \)
|
| $97$ |
\( (T^{4} - 630 T^{3} + \cdots + 43414203455)^{2} \)
|
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