Newspace parameters
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.07798042729\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.12745506816.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 23x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{8} + 23x^{4} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{4} + 9 ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{5} + 24\nu ) / 5 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} + 24\nu^{2} ) / 5 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} + 24\nu^{3} ) / 5 \) |
\(\beta_{6}\) | \(=\) | \( ( -3\nu^{6} - 67\nu^{2} ) / 5 \) |
\(\beta_{7}\) | \(=\) | \( -\nu^{7} - 23\nu^{3} \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + 3\beta_{4} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 5\beta_{5} \) |
\(\nu^{4}\) | \(=\) | \( 5\beta_{2} - 9 \) |
\(\nu^{5}\) | \(=\) | \( 5\beta_{3} - 24\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -24\beta_{6} - 67\beta_{4} \) |
\(\nu^{7}\) | \(=\) | \( -24\beta_{7} - 115\beta_{5} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
\(n\) | \(56\) | \(82\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 |
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−1.54779 | − | 1.54779i | 0 | 2.79129i | −1.22474 | + | 1.87083i | 0 | −2.79129 | + | 2.79129i | 1.22474 | − | 1.22474i | 0 | 4.79129 | − | 1.00000i | ||||||||||||||||||||||||||||||||
53.2 | −0.323042 | − | 0.323042i | 0 | − | 1.79129i | 1.22474 | + | 1.87083i | 0 | 1.79129 | − | 1.79129i | −1.22474 | + | 1.22474i | 0 | 0.208712 | − | 1.00000i | ||||||||||||||||||||||||||||||||
53.3 | 0.323042 | + | 0.323042i | 0 | − | 1.79129i | −1.22474 | − | 1.87083i | 0 | 1.79129 | − | 1.79129i | 1.22474 | − | 1.22474i | 0 | 0.208712 | − | 1.00000i | ||||||||||||||||||||||||||||||||
53.4 | 1.54779 | + | 1.54779i | 0 | 2.79129i | 1.22474 | − | 1.87083i | 0 | −2.79129 | + | 2.79129i | −1.22474 | + | 1.22474i | 0 | 4.79129 | − | 1.00000i | |||||||||||||||||||||||||||||||||
107.1 | −1.54779 | + | 1.54779i | 0 | − | 2.79129i | −1.22474 | − | 1.87083i | 0 | −2.79129 | − | 2.79129i | 1.22474 | + | 1.22474i | 0 | 4.79129 | + | 1.00000i | ||||||||||||||||||||||||||||||||
107.2 | −0.323042 | + | 0.323042i | 0 | 1.79129i | 1.22474 | − | 1.87083i | 0 | 1.79129 | + | 1.79129i | −1.22474 | − | 1.22474i | 0 | 0.208712 | + | 1.00000i | |||||||||||||||||||||||||||||||||
107.3 | 0.323042 | − | 0.323042i | 0 | 1.79129i | −1.22474 | + | 1.87083i | 0 | 1.79129 | + | 1.79129i | 1.22474 | + | 1.22474i | 0 | 0.208712 | + | 1.00000i | |||||||||||||||||||||||||||||||||
107.4 | 1.54779 | − | 1.54779i | 0 | − | 2.79129i | 1.22474 | + | 1.87083i | 0 | −2.79129 | − | 2.79129i | −1.22474 | − | 1.22474i | 0 | 4.79129 | + | 1.00000i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.2.f.a | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 135.2.f.a | ✓ | 8 |
4.b | odd | 2 | 1 | 2160.2.w.d | 8 | ||
5.b | even | 2 | 1 | 675.2.f.i | 8 | ||
5.c | odd | 4 | 1 | inner | 135.2.f.a | ✓ | 8 |
5.c | odd | 4 | 1 | 675.2.f.i | 8 | ||
9.c | even | 3 | 2 | 405.2.m.c | 16 | ||
9.d | odd | 6 | 2 | 405.2.m.c | 16 | ||
12.b | even | 2 | 1 | 2160.2.w.d | 8 | ||
15.d | odd | 2 | 1 | 675.2.f.i | 8 | ||
15.e | even | 4 | 1 | inner | 135.2.f.a | ✓ | 8 |
15.e | even | 4 | 1 | 675.2.f.i | 8 | ||
20.e | even | 4 | 1 | 2160.2.w.d | 8 | ||
45.k | odd | 12 | 2 | 405.2.m.c | 16 | ||
45.l | even | 12 | 2 | 405.2.m.c | 16 | ||
60.l | odd | 4 | 1 | 2160.2.w.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
135.2.f.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
135.2.f.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
135.2.f.a | ✓ | 8 | 15.e | even | 4 | 1 | inner |
405.2.m.c | 16 | 9.c | even | 3 | 2 | ||
405.2.m.c | 16 | 9.d | odd | 6 | 2 | ||
405.2.m.c | 16 | 45.k | odd | 12 | 2 | ||
405.2.m.c | 16 | 45.l | even | 12 | 2 | ||
675.2.f.i | 8 | 5.b | even | 2 | 1 | ||
675.2.f.i | 8 | 5.c | odd | 4 | 1 | ||
675.2.f.i | 8 | 15.d | odd | 2 | 1 | ||
675.2.f.i | 8 | 15.e | even | 4 | 1 | ||
2160.2.w.d | 8 | 4.b | odd | 2 | 1 | ||
2160.2.w.d | 8 | 12.b | even | 2 | 1 | ||
2160.2.w.d | 8 | 20.e | even | 4 | 1 | ||
2160.2.w.d | 8 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 23T_{2}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 23T^{4} + 1 \)
$3$
\( T^{8} \)
$5$
\( (T^{4} + 4 T^{2} + 25)^{2} \)
$7$
\( (T^{4} + 2 T^{3} + 2 T^{2} - 20 T + 100)^{2} \)
$11$
\( (T^{4} + 34 T^{2} + 100)^{2} \)
$13$
\( (T^{4} + 2 T^{3} + 2 T^{2} - 20 T + 100)^{2} \)
$17$
\( (T^{4} + 49)^{2} \)
$19$
\( (T^{2} + 9)^{4} \)
$23$
\( T^{8} + 1394T^{4} + 625 \)
$29$
\( (T^{4} - 66 T^{2} + 900)^{2} \)
$31$
\( (T + 1)^{8} \)
$37$
\( (T^{2} + 10 T + 50)^{4} \)
$41$
\( (T^{4} + 34 T^{2} + 100)^{2} \)
$43$
\( (T^{4} - 22 T^{3} + 242 T^{2} - 1100 T + 2500)^{2} \)
$47$
\( T^{8} + 368T^{4} + 256 \)
$53$
\( T^{8} + 12098T^{4} + 1 \)
$59$
\( (T^{4} - 66 T^{2} + 900)^{2} \)
$61$
\( (T + 1)^{8} \)
$67$
\( (T^{4} - 16 T^{3} + 128 T^{2} + 160 T + 100)^{2} \)
$71$
\( (T^{4} + 34 T^{2} + 100)^{2} \)
$73$
\( (T^{4} + 14 T^{3} + 98 T^{2} - 980 T + 4900)^{2} \)
$79$
\( (T^{4} + 114 T^{2} + 225)^{2} \)
$83$
\( T^{8} + 29138 T^{4} + \cdots + 141158161 \)
$89$
\( (T^{4} - 306 T^{2} + 8100)^{2} \)
$97$
\( (T^{4} - 4 T^{3} + 8 T^{2} + 160 T + 1600)^{2} \)
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