Newspace parameters
Level: | \( N \) | \(=\) | \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2268.k (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.1100711784\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 8.0.310217769.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 3^{3} \) |
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 \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( 14\nu^{7} + 23\nu^{6} - 92\nu^{5} - 14\nu^{4} - 391\nu^{3} + 437\nu^{2} - 1586\nu + 92 ) / 289 \) |
\(\beta_{2}\) | \(=\) | \( ( 64\nu^{7} - 60\nu^{6} + 240\nu^{5} - 353\nu^{4} + 1020\nu^{3} - 1140\nu^{2} + 305\nu - 240 ) / 289 \) |
\(\beta_{3}\) | \(=\) | \( ( 125\nu^{7} - 63\nu^{6} + 541\nu^{5} - 414\nu^{4} + 2227\nu^{3} - 1197\nu^{2} + 1693\nu + 326 ) / 289 \) |
\(\beta_{4}\) | \(=\) | \( ( -237\nu^{7} - 121\nu^{6} - 961\nu^{5} - 52\nu^{4} - 3434\nu^{3} - 854\nu^{2} - 854\nu - 773 ) / 289 \) |
\(\beta_{5}\) | \(=\) | \( ( 273\nu^{7} + 15\nu^{6} + 1096\nu^{5} - 562\nu^{4} + 4080\nu^{3} - 1160\nu^{2} + 1730\nu - 229 ) / 289 \) |
\(\beta_{6}\) | \(=\) | \( ( -344\nu^{7} - 111\nu^{6} - 1290\nu^{5} + 344\nu^{4} - 4471\nu^{3} - 86\nu^{2} - 86\nu - 444 ) / 289 \) |
\(\beta_{7}\) | \(=\) | \( ( -412\nu^{7} + 25\nu^{6} - 1545\nu^{5} + 990\nu^{4} - 5916\nu^{3} + 1920\nu^{2} - 970\nu - 189 ) / 289 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{7} - \beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 6\beta_{2} - \beta _1 - 6 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{7} + 5\beta_{6} + \beta_{5} - 3\beta_{4} - 4\beta_{3} + 3 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 9\beta_{7} + \beta_{5} - 8\beta_{4} + \beta_{3} + 21\beta_{2} + 5\beta_1 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( -6\beta_{7} - 20\beta_{6} - 17\beta_{5} + 17\beta_{4} + 6\beta_{3} - 18\beta_{2} - 20\beta _1 - 18 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( -7\beta_{7} + 24\beta_{6} + 31\beta_{5} - 7\beta_{4} - 38\beta_{3} + 81 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 73\beta_{7} + 42\beta_{5} - 31\beta_{4} + 42\beta_{3} + 90\beta_{2} + 80\beta_1 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(1135\) | \(1541\) |
\(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1297.1 |
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0 | 0 | 0 | −1.25300 | + | 2.17026i | 0 | −2.60333 | + | 0.471863i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
1297.2 | 0 | 0 | 0 | −0.705299 | + | 1.22161i | 0 | 2.57934 | + | 0.589053i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1297.3 | 0 | 0 | 0 | 0.951526 | − | 1.64809i | 0 | 1.17915 | + | 2.36846i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1297.4 | 0 | 0 | 0 | 2.00677 | − | 3.47583i | 0 | −0.655163 | − | 2.56335i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1621.1 | 0 | 0 | 0 | −1.25300 | − | 2.17026i | 0 | −2.60333 | − | 0.471863i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1621.2 | 0 | 0 | 0 | −0.705299 | − | 1.22161i | 0 | 2.57934 | − | 0.589053i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1621.3 | 0 | 0 | 0 | 0.951526 | + | 1.64809i | 0 | 1.17915 | − | 2.36846i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1621.4 | 0 | 0 | 0 | 2.00677 | + | 3.47583i | 0 | −0.655163 | + | 2.56335i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
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 | 2268.2.k.d | yes | 8 |
3.b | odd | 2 | 1 | 2268.2.k.c | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 2268.2.k.d | yes | 8 |
9.c | even | 3 | 1 | 2268.2.i.m | 8 | ||
9.c | even | 3 | 1 | 2268.2.l.l | 8 | ||
9.d | odd | 6 | 1 | 2268.2.i.l | 8 | ||
9.d | odd | 6 | 1 | 2268.2.l.m | 8 | ||
21.h | odd | 6 | 1 | 2268.2.k.c | ✓ | 8 | |
63.g | even | 3 | 1 | 2268.2.i.m | 8 | ||
63.h | even | 3 | 1 | 2268.2.l.l | 8 | ||
63.j | odd | 6 | 1 | 2268.2.l.m | 8 | ||
63.n | odd | 6 | 1 | 2268.2.i.l | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2268.2.i.l | 8 | 9.d | odd | 6 | 1 | ||
2268.2.i.l | 8 | 63.n | odd | 6 | 1 | ||
2268.2.i.m | 8 | 9.c | even | 3 | 1 | ||
2268.2.i.m | 8 | 63.g | even | 3 | 1 | ||
2268.2.k.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
2268.2.k.c | ✓ | 8 | 21.h | odd | 6 | 1 | |
2268.2.k.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
2268.2.k.d | yes | 8 | 7.c | even | 3 | 1 | inner |
2268.2.l.l | 8 | 9.c | even | 3 | 1 | ||
2268.2.l.l | 8 | 63.h | even | 3 | 1 | ||
2268.2.l.m | 8 | 9.d | odd | 6 | 1 | ||
2268.2.l.m | 8 | 63.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2T_{5}^{7} + 16T_{5}^{6} + 6T_{5}^{5} + 135T_{5}^{4} + 405T_{5}^{2} + 243T_{5} + 729 \)
acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 2 T^{7} + 16 T^{6} + 6 T^{5} + \cdots + 729 \)
$7$
\( T^{8} - T^{7} - 2 T^{6} + 7 T^{5} + \cdots + 2401 \)
$11$
\( T^{8} + 5 T^{7} + 31 T^{6} + 42 T^{5} + \cdots + 729 \)
$13$
\( (T^{4} - 3 T^{3} - 12 T^{2} + 16 T - 3)^{2} \)
$17$
\( T^{8} + 2 T^{7} + 37 T^{6} + \cdots + 35721 \)
$19$
\( T^{8} + 8 T^{7} + 94 T^{6} + \cdots + 97969 \)
$23$
\( T^{8} + 2 T^{7} + 58 T^{6} + \cdots + 35721 \)
$29$
\( (T^{4} + 2 T^{3} - 60 T^{2} - 18 T + 27)^{2} \)
$31$
\( T^{8} + 75 T^{6} + 680 T^{5} + \cdots + 178929 \)
$37$
\( T^{8} - 4 T^{7} + 46 T^{6} - 70 T^{5} + \cdots + 361 \)
$41$
\( (T^{4} + 3 T^{3} - 72 T^{2} - 270 T - 243)^{2} \)
$43$
\( (T^{4} - 5 T^{3} - 30 T^{2} + 94 T + 199)^{2} \)
$47$
\( T^{8} + 15 T^{7} + 180 T^{6} + \cdots + 6561 \)
$53$
\( T^{8} - 24 T^{7} + 504 T^{6} + \cdots + 66928761 \)
$59$
\( T^{8} + 10 T^{7} + 190 T^{6} + \cdots + 5774409 \)
$61$
\( T^{8} + 12 T^{7} + 249 T^{6} + \cdots + 20223009 \)
$67$
\( T^{8} + 7 T^{7} + 301 T^{6} + \cdots + 156925729 \)
$71$
\( (T^{4} - 11 T^{3} - 141 T^{2} + 2295 T - 7803)^{2} \)
$73$
\( T^{8} + 10 T^{7} + 190 T^{6} + \cdots + 2411809 \)
$79$
\( T^{8} + 120 T^{6} - 454 T^{5} + \cdots + 531441 \)
$83$
\( (T^{4} - 35 T^{3} + 384 T^{2} - 1350 T + 27)^{2} \)
$89$
\( T^{8} - 18 T^{7} + \cdots + 344065401 \)
$97$
\( (T^{4} - 19 T^{3} - 54 T^{2} + 1780 T - 4639)^{2} \)
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