Newspace parameters
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3 \) |
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( \zeta_{12}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{12}^{3} + \zeta_{12} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{12}^{3} + 2\zeta_{12} \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 3 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).
\(n\) | \(106\) | \(131\) | \(157\) |
\(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 |
|
−0.366025 | − | 0.633975i | −0.500000 | − | 0.866025i | 0.732051 | − | 1.26795i | 1.00000 | −0.366025 | + | 0.633975i | 0.866025 | − | 1.50000i | −2.53590 | −0.500000 | + | 0.866025i | −0.366025 | − | 0.633975i | ||||||||||||||||
16.2 | 1.36603 | + | 2.36603i | −0.500000 | − | 0.866025i | −2.73205 | + | 4.73205i | 1.00000 | 1.36603 | − | 2.36603i | −0.866025 | + | 1.50000i | −9.46410 | −0.500000 | + | 0.866025i | 1.36603 | + | 2.36603i | |||||||||||||||||
61.1 | −0.366025 | + | 0.633975i | −0.500000 | + | 0.866025i | 0.732051 | + | 1.26795i | 1.00000 | −0.366025 | − | 0.633975i | 0.866025 | + | 1.50000i | −2.53590 | −0.500000 | − | 0.866025i | −0.366025 | + | 0.633975i | |||||||||||||||||
61.2 | 1.36603 | − | 2.36603i | −0.500000 | + | 0.866025i | −2.73205 | − | 4.73205i | 1.00000 | 1.36603 | + | 2.36603i | −0.866025 | − | 1.50000i | −9.46410 | −0.500000 | − | 0.866025i | 1.36603 | − | 2.36603i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 195.2.i.c | ✓ | 4 |
3.b | odd | 2 | 1 | 585.2.j.c | 4 | ||
5.b | even | 2 | 1 | 975.2.i.j | 4 | ||
5.c | odd | 4 | 1 | 975.2.bb.a | 4 | ||
5.c | odd | 4 | 1 | 975.2.bb.h | 4 | ||
13.c | even | 3 | 1 | inner | 195.2.i.c | ✓ | 4 |
13.c | even | 3 | 1 | 2535.2.a.o | 2 | ||
13.e | even | 6 | 1 | 2535.2.a.r | 2 | ||
39.h | odd | 6 | 1 | 7605.2.a.z | 2 | ||
39.i | odd | 6 | 1 | 585.2.j.c | 4 | ||
39.i | odd | 6 | 1 | 7605.2.a.bj | 2 | ||
65.n | even | 6 | 1 | 975.2.i.j | 4 | ||
65.q | odd | 12 | 1 | 975.2.bb.a | 4 | ||
65.q | odd | 12 | 1 | 975.2.bb.h | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.i.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
195.2.i.c | ✓ | 4 | 13.c | even | 3 | 1 | inner |
585.2.j.c | 4 | 3.b | odd | 2 | 1 | ||
585.2.j.c | 4 | 39.i | odd | 6 | 1 | ||
975.2.i.j | 4 | 5.b | even | 2 | 1 | ||
975.2.i.j | 4 | 65.n | even | 6 | 1 | ||
975.2.bb.a | 4 | 5.c | odd | 4 | 1 | ||
975.2.bb.a | 4 | 65.q | odd | 12 | 1 | ||
975.2.bb.h | 4 | 5.c | odd | 4 | 1 | ||
975.2.bb.h | 4 | 65.q | odd | 12 | 1 | ||
2535.2.a.o | 2 | 13.c | even | 3 | 1 | ||
2535.2.a.r | 2 | 13.e | even | 6 | 1 | ||
7605.2.a.z | 2 | 39.h | odd | 6 | 1 | ||
7605.2.a.bj | 2 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \)
$3$
\( (T^{2} + T + 1)^{2} \)
$5$
\( (T - 1)^{4} \)
$7$
\( T^{4} + 3T^{2} + 9 \)
$11$
\( (T^{2} - 2 T + 4)^{2} \)
$13$
\( T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169 \)
$17$
\( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \)
$19$
\( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \)
$23$
\( (T^{2} + 2 T + 4)^{2} \)
$29$
\( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \)
$31$
\( (T^{2} - 2 T - 11)^{2} \)
$37$
\( T^{4} + 108 T^{2} + 11664 \)
$41$
\( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \)
$43$
\( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \)
$47$
\( (T^{2} + 10 T - 2)^{2} \)
$53$
\( (T^{2} + 12 T + 24)^{2} \)
$59$
\( T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356 \)
$61$
\( T^{4} - 10 T^{3} + 183 T^{2} + \cdots + 6889 \)
$67$
\( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \)
$71$
\( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \)
$73$
\( (T^{2} - 3)^{2} \)
$79$
\( (T + 11)^{4} \)
$83$
\( (T^{2} + 8 T - 32)^{2} \)
$89$
\( T^{4} + 14 T^{3} + 150 T^{2} + \cdots + 2116 \)
$97$
\( T^{4} + 27T^{2} + 729 \)
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