Newspace parameters
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.350464.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :
\(\beta_{1}\) | \(=\) | \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) |
\(\beta_{3}\) | \(=\) | \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) |
\(\beta_{4}\) | \(=\) | \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) |
\(\beta_{5}\) | \(=\) | \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) |
\(\nu^{5}\) | \(=\) | \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).
\(n\) | \(131\) | \(157\) | \(301\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
259.1 |
|
−1.00000 | − | 1.00000i | 1.00000 | −2.17009 | + | 0.539189i | 1.00000i | −0.630898 | −1.00000 | −1.00000 | 2.17009 | − | 0.539189i | |||||||||||||||||||||||||||||||
259.2 | −1.00000 | − | 1.00000i | 1.00000 | −0.311108 | − | 2.21432i | 1.00000i | −1.52543 | −1.00000 | −1.00000 | 0.311108 | + | 2.21432i | ||||||||||||||||||||||||||||||||
259.3 | −1.00000 | − | 1.00000i | 1.00000 | 1.48119 | + | 1.67513i | 1.00000i | 4.15633 | −1.00000 | −1.00000 | −1.48119 | − | 1.67513i | ||||||||||||||||||||||||||||||||
259.4 | −1.00000 | 1.00000i | 1.00000 | −2.17009 | − | 0.539189i | − | 1.00000i | −0.630898 | −1.00000 | −1.00000 | 2.17009 | + | 0.539189i | ||||||||||||||||||||||||||||||||
259.5 | −1.00000 | 1.00000i | 1.00000 | −0.311108 | + | 2.21432i | − | 1.00000i | −1.52543 | −1.00000 | −1.00000 | 0.311108 | − | 2.21432i | ||||||||||||||||||||||||||||||||
259.6 | −1.00000 | 1.00000i | 1.00000 | 1.48119 | − | 1.67513i | − | 1.00000i | 4.15633 | −1.00000 | −1.00000 | −1.48119 | + | 1.67513i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.f.a | ✓ | 6 |
3.b | odd | 2 | 1 | 1170.2.f.d | 6 | ||
4.b | odd | 2 | 1 | 3120.2.r.g | 6 | ||
5.b | even | 2 | 1 | 390.2.f.b | yes | 6 | |
5.c | odd | 4 | 1 | 1950.2.b.l | 6 | ||
5.c | odd | 4 | 1 | 1950.2.b.m | 6 | ||
13.b | even | 2 | 1 | 390.2.f.b | yes | 6 | |
15.d | odd | 2 | 1 | 1170.2.f.c | 6 | ||
20.d | odd | 2 | 1 | 3120.2.r.h | 6 | ||
39.d | odd | 2 | 1 | 1170.2.f.c | 6 | ||
52.b | odd | 2 | 1 | 3120.2.r.h | 6 | ||
65.d | even | 2 | 1 | inner | 390.2.f.a | ✓ | 6 |
65.h | odd | 4 | 1 | 1950.2.b.l | 6 | ||
65.h | odd | 4 | 1 | 1950.2.b.m | 6 | ||
195.e | odd | 2 | 1 | 1170.2.f.d | 6 | ||
260.g | odd | 2 | 1 | 3120.2.r.g | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.f.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
390.2.f.a | ✓ | 6 | 65.d | even | 2 | 1 | inner |
390.2.f.b | yes | 6 | 5.b | even | 2 | 1 | |
390.2.f.b | yes | 6 | 13.b | even | 2 | 1 | |
1170.2.f.c | 6 | 15.d | odd | 2 | 1 | ||
1170.2.f.c | 6 | 39.d | odd | 2 | 1 | ||
1170.2.f.d | 6 | 3.b | odd | 2 | 1 | ||
1170.2.f.d | 6 | 195.e | odd | 2 | 1 | ||
1950.2.b.l | 6 | 5.c | odd | 4 | 1 | ||
1950.2.b.l | 6 | 65.h | odd | 4 | 1 | ||
1950.2.b.m | 6 | 5.c | odd | 4 | 1 | ||
1950.2.b.m | 6 | 65.h | odd | 4 | 1 | ||
3120.2.r.g | 6 | 4.b | odd | 2 | 1 | ||
3120.2.r.g | 6 | 260.g | odd | 2 | 1 | ||
3120.2.r.h | 6 | 20.d | odd | 2 | 1 | ||
3120.2.r.h | 6 | 52.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 1)^{6} \)
$3$
\( (T^{2} + 1)^{3} \)
$5$
\( T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125 \)
$7$
\( (T^{3} - 2 T^{2} - 8 T - 4)^{2} \)
$11$
\( T^{6} + 44 T^{4} + 496 T^{2} + \cdots + 1600 \)
$13$
\( T^{6} + 8 T^{5} + 7 T^{4} - 64 T^{3} + \cdots + 2197 \)
$17$
\( T^{6} + 16 T^{4} + 32 T^{2} + 16 \)
$19$
\( T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 16 \)
$23$
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
$29$
\( (T^{3} - 2 T^{2} - 44 T + 20)^{2} \)
$31$
\( T^{6} + 108 T^{4} + 2864 T^{2} + \cdots + 1600 \)
$37$
\( (T^{3} - 12 T^{2} + 32 T + 16)^{2} \)
$41$
\( T^{6} + 76 T^{4} + 944 T^{2} + \cdots + 1600 \)
$43$
\( T^{6} + 128 T^{4} + 5376 T^{2} + \cdots + 73984 \)
$47$
\( (T^{3} + 16 T^{2} + 48 T + 32)^{2} \)
$53$
\( T^{6} + 236 T^{4} + 7600 T^{2} + \cdots + 40000 \)
$59$
\( T^{6} + 252 T^{4} + 14640 T^{2} + \cdots + 61504 \)
$61$
\( (T^{3} + 2 T^{2} - 36 T - 104)^{2} \)
$67$
\( (T^{3} - 2 T^{2} - 52 T + 184)^{2} \)
$71$
\( T^{6} + 112 T^{4} + 2624 T^{2} + \cdots + 6400 \)
$73$
\( (T^{3} - 20 T^{2} + 36 T + 548)^{2} \)
$79$
\( (T^{3} + 8 T^{2} - 216 T - 2000)^{2} \)
$83$
\( (T^{3} - 144 T - 432)^{2} \)
$89$
\( T^{6} + 252 T^{4} + 17072 T^{2} + \cdots + 287296 \)
$97$
\( (T^{3} - 100 T - 268)^{2} \)
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