Newspace parameters
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(38.4921167551\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 30) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | − | 9.00000i | 0 | −55.0000 | + | 10.0000i | 0 | 4.00000i | 0 | −81.0000 | 0 | |||||||||||||||||||||
49.2 | 0 | 9.00000i | 0 | −55.0000 | − | 10.0000i | 0 | − | 4.00000i | 0 | −81.0000 | 0 | ||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 240.6.f.a | 2 | |
3.b | odd | 2 | 1 | 720.6.f.g | 2 | ||
4.b | odd | 2 | 1 | 30.6.c.a | ✓ | 2 | |
5.b | even | 2 | 1 | inner | 240.6.f.a | 2 | |
12.b | even | 2 | 1 | 90.6.c.b | 2 | ||
15.d | odd | 2 | 1 | 720.6.f.g | 2 | ||
20.d | odd | 2 | 1 | 30.6.c.a | ✓ | 2 | |
20.e | even | 4 | 1 | 150.6.a.f | 1 | ||
20.e | even | 4 | 1 | 150.6.a.j | 1 | ||
60.h | even | 2 | 1 | 90.6.c.b | 2 | ||
60.l | odd | 4 | 1 | 450.6.a.f | 1 | ||
60.l | odd | 4 | 1 | 450.6.a.s | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.6.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
30.6.c.a | ✓ | 2 | 20.d | odd | 2 | 1 | |
90.6.c.b | 2 | 12.b | even | 2 | 1 | ||
90.6.c.b | 2 | 60.h | even | 2 | 1 | ||
150.6.a.f | 1 | 20.e | even | 4 | 1 | ||
150.6.a.j | 1 | 20.e | even | 4 | 1 | ||
240.6.f.a | 2 | 1.a | even | 1 | 1 | trivial | |
240.6.f.a | 2 | 5.b | even | 2 | 1 | inner | |
450.6.a.f | 1 | 60.l | odd | 4 | 1 | ||
450.6.a.s | 1 | 60.l | odd | 4 | 1 | ||
720.6.f.g | 2 | 3.b | odd | 2 | 1 | ||
720.6.f.g | 2 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 16 \)
acting on \(S_{6}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 81 \)
$5$
\( T^{2} + 110T + 3125 \)
$7$
\( T^{2} + 16 \)
$11$
\( (T - 500)^{2} \)
$13$
\( T^{2} + 82944 \)
$17$
\( T^{2} + 2298256 \)
$19$
\( (T + 1344)^{2} \)
$23$
\( T^{2} + 16810000 \)
$29$
\( (T - 2646)^{2} \)
$31$
\( (T - 5612)^{2} \)
$37$
\( T^{2} + 53114944 \)
$41$
\( (T + 18986)^{2} \)
$43$
\( T^{2} + 5779216 \)
$47$
\( T^{2} + 79210000 \)
$53$
\( T^{2} + 1584358416 \)
$59$
\( (T + 28300)^{2} \)
$61$
\( (T - 18290)^{2} \)
$67$
\( T^{2} + 4350193936 \)
$71$
\( (T - 28800)^{2} \)
$73$
\( T^{2} + 949132864 \)
$79$
\( (T - 60228)^{2} \)
$83$
\( T^{2} + 6091024 \)
$89$
\( (T + 22678)^{2} \)
$97$
\( T^{2} + 1366633024 \)
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