Newspace parameters
Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 36.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.980928951697\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 12) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
\(n\) | \(19\) | \(29\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.00000 | 0 | 6.92820i | −8.00000 | 0 | 2.00000 | − | 3.46410i | ||||||||||||||||||
19.2 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | 2.00000 | 0 | − | 6.92820i | −8.00000 | 0 | 2.00000 | + | 3.46410i | ||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 36.3.d.c | 2 | |
3.b | odd | 2 | 1 | 12.3.d.a | ✓ | 2 | |
4.b | odd | 2 | 1 | inner | 36.3.d.c | 2 | |
5.b | even | 2 | 1 | 900.3.c.e | 2 | ||
5.c | odd | 4 | 2 | 900.3.f.c | 4 | ||
8.b | even | 2 | 1 | 576.3.g.e | 2 | ||
8.d | odd | 2 | 1 | 576.3.g.e | 2 | ||
9.c | even | 3 | 1 | 324.3.f.a | 2 | ||
9.c | even | 3 | 1 | 324.3.f.g | 2 | ||
9.d | odd | 6 | 1 | 324.3.f.d | 2 | ||
9.d | odd | 6 | 1 | 324.3.f.j | 2 | ||
12.b | even | 2 | 1 | 12.3.d.a | ✓ | 2 | |
15.d | odd | 2 | 1 | 300.3.c.b | 2 | ||
15.e | even | 4 | 2 | 300.3.f.a | 4 | ||
16.e | even | 4 | 2 | 2304.3.b.l | 4 | ||
16.f | odd | 4 | 2 | 2304.3.b.l | 4 | ||
20.d | odd | 2 | 1 | 900.3.c.e | 2 | ||
20.e | even | 4 | 2 | 900.3.f.c | 4 | ||
21.c | even | 2 | 1 | 588.3.g.b | 2 | ||
24.f | even | 2 | 1 | 192.3.g.b | 2 | ||
24.h | odd | 2 | 1 | 192.3.g.b | 2 | ||
36.f | odd | 6 | 1 | 324.3.f.a | 2 | ||
36.f | odd | 6 | 1 | 324.3.f.g | 2 | ||
36.h | even | 6 | 1 | 324.3.f.d | 2 | ||
36.h | even | 6 | 1 | 324.3.f.j | 2 | ||
48.i | odd | 4 | 2 | 768.3.b.c | 4 | ||
48.k | even | 4 | 2 | 768.3.b.c | 4 | ||
60.h | even | 2 | 1 | 300.3.c.b | 2 | ||
60.l | odd | 4 | 2 | 300.3.f.a | 4 | ||
84.h | odd | 2 | 1 | 588.3.g.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
12.3.d.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
12.3.d.a | ✓ | 2 | 12.b | even | 2 | 1 | |
36.3.d.c | 2 | 1.a | even | 1 | 1 | trivial | |
36.3.d.c | 2 | 4.b | odd | 2 | 1 | inner | |
192.3.g.b | 2 | 24.f | even | 2 | 1 | ||
192.3.g.b | 2 | 24.h | odd | 2 | 1 | ||
300.3.c.b | 2 | 15.d | odd | 2 | 1 | ||
300.3.c.b | 2 | 60.h | even | 2 | 1 | ||
300.3.f.a | 4 | 15.e | even | 4 | 2 | ||
300.3.f.a | 4 | 60.l | odd | 4 | 2 | ||
324.3.f.a | 2 | 9.c | even | 3 | 1 | ||
324.3.f.a | 2 | 36.f | odd | 6 | 1 | ||
324.3.f.d | 2 | 9.d | odd | 6 | 1 | ||
324.3.f.d | 2 | 36.h | even | 6 | 1 | ||
324.3.f.g | 2 | 9.c | even | 3 | 1 | ||
324.3.f.g | 2 | 36.f | odd | 6 | 1 | ||
324.3.f.j | 2 | 9.d | odd | 6 | 1 | ||
324.3.f.j | 2 | 36.h | even | 6 | 1 | ||
576.3.g.e | 2 | 8.b | even | 2 | 1 | ||
576.3.g.e | 2 | 8.d | odd | 2 | 1 | ||
588.3.g.b | 2 | 21.c | even | 2 | 1 | ||
588.3.g.b | 2 | 84.h | odd | 2 | 1 | ||
768.3.b.c | 4 | 48.i | odd | 4 | 2 | ||
768.3.b.c | 4 | 48.k | even | 4 | 2 | ||
900.3.c.e | 2 | 5.b | even | 2 | 1 | ||
900.3.c.e | 2 | 20.d | odd | 2 | 1 | ||
900.3.f.c | 4 | 5.c | odd | 4 | 2 | ||
900.3.f.c | 4 | 20.e | even | 4 | 2 | ||
2304.3.b.l | 4 | 16.e | even | 4 | 2 | ||
2304.3.b.l | 4 | 16.f | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 2 \)
acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 4 \)
$3$
\( T^{2} \)
$5$
\( (T - 2)^{2} \)
$7$
\( T^{2} + 48 \)
$11$
\( T^{2} + 48 \)
$13$
\( (T - 2)^{2} \)
$17$
\( (T + 10)^{2} \)
$19$
\( T^{2} + 432 \)
$23$
\( T^{2} + 768 \)
$29$
\( (T - 26)^{2} \)
$31$
\( T^{2} + 48 \)
$37$
\( (T - 26)^{2} \)
$41$
\( (T + 58)^{2} \)
$43$
\( T^{2} + 2352 \)
$47$
\( T^{2} + 4800 \)
$53$
\( (T - 74)^{2} \)
$59$
\( T^{2} + 8112 \)
$61$
\( (T - 26)^{2} \)
$67$
\( T^{2} + 48 \)
$71$
\( T^{2} \)
$73$
\( (T + 46)^{2} \)
$79$
\( T^{2} + 13872 \)
$83$
\( T^{2} + 2352 \)
$89$
\( (T + 82)^{2} \)
$97$
\( (T - 2)^{2} \)
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