Newspace parameters
Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 361.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.83653754341\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 13x^{2} + 169 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 19) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 169 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 13 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 13 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 13\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 13\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
69.1 |
|
−3.12250 | + | 1.80278i | 3.12250 | − | 1.80278i | 4.50000 | − | 7.79423i | −2.00000 | − | 3.46410i | −6.50000 | + | 11.2583i | −5.00000 | 18.0278i | 2.00000 | − | 3.46410i | 12.4900 | + | 7.21110i | ||||||||||||||||
69.2 | 3.12250 | − | 1.80278i | −3.12250 | + | 1.80278i | 4.50000 | − | 7.79423i | −2.00000 | − | 3.46410i | −6.50000 | + | 11.2583i | −5.00000 | − | 18.0278i | 2.00000 | − | 3.46410i | −12.4900 | − | 7.21110i | ||||||||||||||||
293.1 | −3.12250 | − | 1.80278i | 3.12250 | + | 1.80278i | 4.50000 | + | 7.79423i | −2.00000 | + | 3.46410i | −6.50000 | − | 11.2583i | −5.00000 | − | 18.0278i | 2.00000 | + | 3.46410i | 12.4900 | − | 7.21110i | ||||||||||||||||
293.2 | 3.12250 | + | 1.80278i | −3.12250 | − | 1.80278i | 4.50000 | + | 7.79423i | −2.00000 | + | 3.46410i | −6.50000 | − | 11.2583i | −5.00000 | 18.0278i | 2.00000 | + | 3.46410i | −12.4900 | + | 7.21110i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
19.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 361.3.d.b | 4 | |
19.b | odd | 2 | 1 | inner | 361.3.d.b | 4 | |
19.c | even | 3 | 1 | 19.3.b.b | ✓ | 2 | |
19.c | even | 3 | 1 | inner | 361.3.d.b | 4 | |
19.d | odd | 6 | 1 | 19.3.b.b | ✓ | 2 | |
19.d | odd | 6 | 1 | inner | 361.3.d.b | 4 | |
19.e | even | 9 | 6 | 361.3.f.d | 12 | ||
19.f | odd | 18 | 6 | 361.3.f.d | 12 | ||
57.f | even | 6 | 1 | 171.3.c.b | 2 | ||
57.h | odd | 6 | 1 | 171.3.c.b | 2 | ||
76.f | even | 6 | 1 | 304.3.e.d | 2 | ||
76.g | odd | 6 | 1 | 304.3.e.d | 2 | ||
95.h | odd | 6 | 1 | 475.3.c.b | 2 | ||
95.i | even | 6 | 1 | 475.3.c.b | 2 | ||
95.l | even | 12 | 2 | 475.3.d.b | 4 | ||
95.m | odd | 12 | 2 | 475.3.d.b | 4 | ||
152.k | odd | 6 | 1 | 1216.3.e.h | 2 | ||
152.l | odd | 6 | 1 | 1216.3.e.g | 2 | ||
152.o | even | 6 | 1 | 1216.3.e.h | 2 | ||
152.p | even | 6 | 1 | 1216.3.e.g | 2 | ||
228.m | even | 6 | 1 | 2736.3.o.d | 2 | ||
228.n | odd | 6 | 1 | 2736.3.o.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.3.b.b | ✓ | 2 | 19.c | even | 3 | 1 | |
19.3.b.b | ✓ | 2 | 19.d | odd | 6 | 1 | |
171.3.c.b | 2 | 57.f | even | 6 | 1 | ||
171.3.c.b | 2 | 57.h | odd | 6 | 1 | ||
304.3.e.d | 2 | 76.f | even | 6 | 1 | ||
304.3.e.d | 2 | 76.g | odd | 6 | 1 | ||
361.3.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
361.3.d.b | 4 | 19.b | odd | 2 | 1 | inner | |
361.3.d.b | 4 | 19.c | even | 3 | 1 | inner | |
361.3.d.b | 4 | 19.d | odd | 6 | 1 | inner | |
361.3.f.d | 12 | 19.e | even | 9 | 6 | ||
361.3.f.d | 12 | 19.f | odd | 18 | 6 | ||
475.3.c.b | 2 | 95.h | odd | 6 | 1 | ||
475.3.c.b | 2 | 95.i | even | 6 | 1 | ||
475.3.d.b | 4 | 95.l | even | 12 | 2 | ||
475.3.d.b | 4 | 95.m | odd | 12 | 2 | ||
1216.3.e.g | 2 | 152.l | odd | 6 | 1 | ||
1216.3.e.g | 2 | 152.p | even | 6 | 1 | ||
1216.3.e.h | 2 | 152.k | odd | 6 | 1 | ||
1216.3.e.h | 2 | 152.o | even | 6 | 1 | ||
2736.3.o.d | 2 | 228.m | even | 6 | 1 | ||
2736.3.o.d | 2 | 228.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 13T_{2}^{2} + 169 \)
acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 13T^{2} + 169 \)
$3$
\( T^{4} - 13T^{2} + 169 \)
$5$
\( (T^{2} + 4 T + 16)^{2} \)
$7$
\( (T + 5)^{4} \)
$11$
\( (T + 10)^{4} \)
$13$
\( T^{4} - 13T^{2} + 169 \)
$17$
\( (T^{2} + 15 T + 225)^{2} \)
$19$
\( T^{4} \)
$23$
\( (T^{2} + 35 T + 1225)^{2} \)
$29$
\( T^{4} - 325 T^{2} + 105625 \)
$31$
\( (T^{2} + 1300)^{2} \)
$37$
\( (T^{2} + 468)^{2} \)
$41$
\( T^{4} - 1300 T^{2} + \cdots + 1690000 \)
$43$
\( (T^{2} - 20 T + 400)^{2} \)
$47$
\( (T^{2} + 10 T + 100)^{2} \)
$53$
\( T^{4} - 5733 T^{2} + \cdots + 32867289 \)
$59$
\( T^{4} - 325 T^{2} + 105625 \)
$61$
\( (T^{2} - 40 T + 1600)^{2} \)
$67$
\( T^{4} - 1573 T^{2} + \cdots + 2474329 \)
$71$
\( T^{4} - 11700 T^{2} + \cdots + 136890000 \)
$73$
\( (T^{2} + 105 T + 11025)^{2} \)
$79$
\( T^{4} - 1300 T^{2} + \cdots + 1690000 \)
$83$
\( (T + 40)^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} - 15028 T^{2} + \cdots + 225840784 \)
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