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: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{12} + 373 x^{10} - 3270 x^{9} + 108613 x^{8} - 867771 x^{7} + 15135417 x^{6} - 142511676 x^{5} + \cdots + 291450979044 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 373 x^{10} - 3270 x^{9} + 108613 x^{8} - 867771 x^{7} + 15135417 x^{6} - 142511676 x^{5} + \cdots + 291450979044 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 80\!\cdots\!37 \nu^{11} + \cdots - 33\!\cdots\!52 ) / 84\!\cdots\!72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 30\!\cdots\!87 \nu^{11} + \cdots - 83\!\cdots\!80 ) / 88\!\cdots\!85 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60\!\cdots\!01 \nu^{11} + \cdots - 69\!\cdots\!92 ) / 16\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!48 \nu^{11} + \cdots - 67\!\cdots\!11 ) / 32\!\cdots\!35 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!25 \nu^{11} + \cdots + 40\!\cdots\!88 ) / 35\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 47\!\cdots\!09 \nu^{11} + \cdots + 56\!\cdots\!28 ) / 11\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\!\cdots\!86 \nu^{11} + \cdots + 77\!\cdots\!27 ) / 32\!\cdots\!35 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 25\!\cdots\!07 \nu^{11} + \cdots - 30\!\cdots\!44 ) / 58\!\cdots\!90 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 48\!\cdots\!91 \nu^{11} + \cdots - 14\!\cdots\!76 ) / 58\!\cdots\!90 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 17\!\cdots\!07 \nu^{11} + \cdots - 17\!\cdots\!38 ) / 88\!\cdots\!85 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 17\!\cdots\!38 \nu^{11} + \cdots - 83\!\cdots\!22 ) / 88\!\cdots\!85 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9\beta_{11} + 6\beta_{10} + 2\beta_{8} - 3\beta_{6} - 6\beta_{5} + \beta_{3} - 3\beta_{2} + 248\beta _1 - 242 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 110 \beta_{11} - 122 \beta_{10} + 26 \beta_{9} - 26 \beta_{8} - 12 \beta_{7} - 12 \beta_{5} + 75 \beta_{4} + \cdots + 711 \)
|
| \(\nu^{4}\) | \(=\) |
\( 524 \beta_{11} + 524 \beta_{10} - 673 \beta_{9} - 251 \beta_{7} + 251 \beta_{6} + 1732 \beta_{5} + \cdots - 26472 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 35650 \beta_{11} + 29025 \beta_{10} + 12308 \beta_{8} + 3066 \beta_{6} - 29025 \beta_{5} + \cdots - 265095 \)
|
| \(\nu^{6}\) | \(=\) |
\( 409928 \beta_{11} - 579226 \beta_{10} + 263023 \beta_{9} - 263023 \beta_{8} + 28229 \beta_{7} + \cdots + 6287165 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2574799 \beta_{11} + 2574799 \beta_{10} - 4639907 \beta_{9} + 753915 \beta_{7} - 753915 \beta_{6} + \cdots - 97671319 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 187590205 \beta_{11} + 133505345 \beta_{10} + 90448213 \beta_{8} - 142190 \beta_{6} + \cdots - 1774875452 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2547118350 \beta_{11} - 3440525779 \beta_{10} + 1597523108 \beta_{9} - 1597523108 \beta_{8} + \cdots + 27762633675 \)
|
| \(\nu^{10}\) | \(=\) |
\( 17225570189 \beta_{11} + 17225570189 \beta_{10} - 29759264272 \beta_{9} + 1385785225 \beta_{7} + \cdots - 590299334586 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1086451922686 \beta_{11} + 790596020571 \beta_{10} + 527685083963 \beta_{8} + 56894186790 \beta_{6} + \cdots - 8847107033901 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −8.91731 | + | 15.4452i | 6.00000 | 2.44803 | − | 18.3578i | 8.00000 | −4.50000 | + | 7.79423i | −17.8346 | − | 30.8905i | ||||||||||||||||||||||||||||||||||||||||
| 79.2 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −5.84896 | + | 10.1307i | 6.00000 | 16.9054 | + | 7.56350i | 8.00000 | −4.50000 | + | 7.79423i | −11.6979 | − | 20.2614i | |||||||||||||||||||||||||||||||||||||||||
| 79.3 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −2.67198 | + | 4.62800i | 6.00000 | −3.62709 | + | 18.1616i | 8.00000 | −4.50000 | + | 7.79423i | −5.34396 | − | 9.25600i | |||||||||||||||||||||||||||||||||||||||||
| 79.4 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 1.03417 | − | 1.79124i | 6.00000 | −15.1072 | − | 10.7131i | 8.00000 | −4.50000 | + | 7.79423i | 2.06835 | + | 3.58249i | |||||||||||||||||||||||||||||||||||||||||
| 79.5 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 4.10101 | − | 7.10316i | 6.00000 | 12.2270 | − | 13.9105i | 8.00000 | −4.50000 | + | 7.79423i | 8.20202 | + | 14.2063i | |||||||||||||||||||||||||||||||||||||||||
| 79.6 | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 8.80306 | − | 15.2474i | 6.00000 | −18.3461 | + | 2.53379i | 8.00000 | −4.50000 | + | 7.79423i | 17.6061 | + | 30.4947i | |||||||||||||||||||||||||||||||||||||||||
| 235.1 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −8.91731 | − | 15.4452i | 6.00000 | 2.44803 | + | 18.3578i | 8.00000 | −4.50000 | − | 7.79423i | −17.8346 | + | 30.8905i | |||||||||||||||||||||||||||||||||||||||||
| 235.2 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −5.84896 | − | 10.1307i | 6.00000 | 16.9054 | − | 7.56350i | 8.00000 | −4.50000 | − | 7.79423i | −11.6979 | + | 20.2614i | |||||||||||||||||||||||||||||||||||||||||
| 235.3 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −2.67198 | − | 4.62800i | 6.00000 | −3.62709 | − | 18.1616i | 8.00000 | −4.50000 | − | 7.79423i | −5.34396 | + | 9.25600i | |||||||||||||||||||||||||||||||||||||||||
| 235.4 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 1.03417 | + | 1.79124i | 6.00000 | −15.1072 | + | 10.7131i | 8.00000 | −4.50000 | − | 7.79423i | 2.06835 | − | 3.58249i | |||||||||||||||||||||||||||||||||||||||||
| 235.5 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 4.10101 | + | 7.10316i | 6.00000 | 12.2270 | + | 13.9105i | 8.00000 | −4.50000 | − | 7.79423i | 8.20202 | − | 14.2063i | |||||||||||||||||||||||||||||||||||||||||
| 235.6 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 8.80306 | + | 15.2474i | 6.00000 | −18.3461 | − | 2.53379i | 8.00000 | −4.50000 | − | 7.79423i | 17.6061 | − | 30.4947i | |||||||||||||||||||||||||||||||||||||||||
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.e | ✓ | 12 |
| 7.c | even | 3 | 1 | inner | 546.4.i.e | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.i.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 546.4.i.e | ✓ | 12 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 7 T_{5}^{11} + 457 T_{5}^{10} + 2146 T_{5}^{9} + 152989 T_{5}^{8} + 697720 T_{5}^{7} + \cdots + 110889000000 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{6} \)
|
| $5$ |
\( T^{12} + \cdots + 110889000000 \)
|
| $7$ |
\( T^{12} + \cdots + 16\!\cdots\!49 \)
|
| $11$ |
\( T^{12} + \cdots + 37\!\cdots\!61 \)
|
| $13$ |
\( (T + 13)^{12} \)
|
| $17$ |
\( T^{12} + \cdots + 41\!\cdots\!89 \)
|
| $19$ |
\( T^{12} + \cdots + 45066235233321 \)
|
| $23$ |
\( T^{12} + \cdots + 24\!\cdots\!96 \)
|
| $29$ |
\( (T^{6} + \cdots + 2481780157353)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 17\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 31\!\cdots\!16 \)
|
| $41$ |
\( (T^{6} + \cdots - 107775128854008)^{2} \)
|
| $43$ |
\( (T^{6} + \cdots + 29145825344040)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 27\!\cdots\!49 \)
|
| $59$ |
\( T^{12} + \cdots + 73\!\cdots\!61 \)
|
| $61$ |
\( T^{12} + \cdots + 13\!\cdots\!09 \)
|
| $67$ |
\( T^{12} + \cdots + 10\!\cdots\!89 \)
|
| $71$ |
\( (T^{6} + \cdots + 91\!\cdots\!93)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 54\!\cdots\!84 \)
|
| $79$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $83$ |
\( (T^{6} + \cdots - 11\!\cdots\!84)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 61\!\cdots\!84 \)
|
| $97$ |
\( (T^{6} + \cdots + 97\!\cdots\!04)^{2} \)
|
| show more | |
| show less |