Newspace parameters
Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 361.f (of order \(18\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.83653754341\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{18})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{3} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 19) |
Sato-Tate group: | $\mathrm{U}(1)[D_{18}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\zeta_{18}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
116.1 |
|
0 | 0 | −3.75877 | − | 1.36808i | 8.45723 | − | 3.07818i | 0 | 2.50000 | + | 4.33013i | 0 | 1.56283 | + | 8.86327i | 0 | ||||||||||||||||||||||||||||
127.1 | 0 | 0 | 0.694593 | − | 3.93923i | −1.56283 | − | 8.86327i | 0 | 2.50000 | − | 4.33013i | 0 | 6.89440 | + | 5.78509i | 0 | |||||||||||||||||||||||||||||
262.1 | 0 | 0 | 3.06418 | + | 2.57115i | −6.89440 | + | 5.78509i | 0 | 2.50000 | − | 4.33013i | 0 | −8.45723 | + | 3.07818i | 0 | |||||||||||||||||||||||||||||
299.1 | 0 | 0 | 3.06418 | − | 2.57115i | −6.89440 | − | 5.78509i | 0 | 2.50000 | + | 4.33013i | 0 | −8.45723 | − | 3.07818i | 0 | |||||||||||||||||||||||||||||
307.1 | 0 | 0 | 0.694593 | + | 3.93923i | −1.56283 | + | 8.86327i | 0 | 2.50000 | + | 4.33013i | 0 | 6.89440 | − | 5.78509i | 0 | |||||||||||||||||||||||||||||
333.1 | 0 | 0 | −3.75877 | + | 1.36808i | 8.45723 | + | 3.07818i | 0 | 2.50000 | − | 4.33013i | 0 | 1.56283 | − | 8.86327i | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
19.c | even | 3 | 2 | inner |
19.d | odd | 6 | 2 | inner |
19.e | even | 9 | 3 | inner |
19.f | odd | 18 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 361.3.f.a | 6 | |
19.b | odd | 2 | 1 | CM | 361.3.f.a | 6 | |
19.c | even | 3 | 2 | inner | 361.3.f.a | 6 | |
19.d | odd | 6 | 2 | inner | 361.3.f.a | 6 | |
19.e | even | 9 | 1 | 19.3.b.a | ✓ | 1 | |
19.e | even | 9 | 2 | 361.3.d.a | 2 | ||
19.e | even | 9 | 3 | inner | 361.3.f.a | 6 | |
19.f | odd | 18 | 1 | 19.3.b.a | ✓ | 1 | |
19.f | odd | 18 | 2 | 361.3.d.a | 2 | ||
19.f | odd | 18 | 3 | inner | 361.3.f.a | 6 | |
57.j | even | 18 | 1 | 171.3.c.a | 1 | ||
57.l | odd | 18 | 1 | 171.3.c.a | 1 | ||
76.k | even | 18 | 1 | 304.3.e.a | 1 | ||
76.l | odd | 18 | 1 | 304.3.e.a | 1 | ||
95.o | odd | 18 | 1 | 475.3.c.a | 1 | ||
95.p | even | 18 | 1 | 475.3.c.a | 1 | ||
95.q | odd | 36 | 2 | 475.3.d.a | 2 | ||
95.r | even | 36 | 2 | 475.3.d.a | 2 | ||
152.s | odd | 18 | 1 | 1216.3.e.a | 1 | ||
152.t | even | 18 | 1 | 1216.3.e.a | 1 | ||
152.u | odd | 18 | 1 | 1216.3.e.b | 1 | ||
152.v | even | 18 | 1 | 1216.3.e.b | 1 | ||
228.u | odd | 18 | 1 | 2736.3.o.a | 1 | ||
228.v | even | 18 | 1 | 2736.3.o.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.3.b.a | ✓ | 1 | 19.e | even | 9 | 1 | |
19.3.b.a | ✓ | 1 | 19.f | odd | 18 | 1 | |
171.3.c.a | 1 | 57.j | even | 18 | 1 | ||
171.3.c.a | 1 | 57.l | odd | 18 | 1 | ||
304.3.e.a | 1 | 76.k | even | 18 | 1 | ||
304.3.e.a | 1 | 76.l | odd | 18 | 1 | ||
361.3.d.a | 2 | 19.e | even | 9 | 2 | ||
361.3.d.a | 2 | 19.f | odd | 18 | 2 | ||
361.3.f.a | 6 | 1.a | even | 1 | 1 | trivial | |
361.3.f.a | 6 | 19.b | odd | 2 | 1 | CM | |
361.3.f.a | 6 | 19.c | even | 3 | 2 | inner | |
361.3.f.a | 6 | 19.d | odd | 6 | 2 | inner | |
361.3.f.a | 6 | 19.e | even | 9 | 3 | inner | |
361.3.f.a | 6 | 19.f | odd | 18 | 3 | inner | |
475.3.c.a | 1 | 95.o | odd | 18 | 1 | ||
475.3.c.a | 1 | 95.p | even | 18 | 1 | ||
475.3.d.a | 2 | 95.q | odd | 36 | 2 | ||
475.3.d.a | 2 | 95.r | even | 36 | 2 | ||
1216.3.e.a | 1 | 152.s | odd | 18 | 1 | ||
1216.3.e.a | 1 | 152.t | even | 18 | 1 | ||
1216.3.e.b | 1 | 152.u | odd | 18 | 1 | ||
1216.3.e.b | 1 | 152.v | even | 18 | 1 | ||
2736.3.o.a | 1 | 228.u | odd | 18 | 1 | ||
2736.3.o.a | 1 | 228.v | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} \)
$5$
\( T^{6} - 729 T^{3} + 531441 \)
$7$
\( (T^{2} - 5 T + 25)^{3} \)
$11$
\( (T^{2} + 3 T + 9)^{3} \)
$13$
\( T^{6} \)
$17$
\( T^{6} + 3375 T^{3} + \cdots + 11390625 \)
$19$
\( T^{6} \)
$23$
\( T^{6} - 27000 T^{3} + \cdots + 729000000 \)
$29$
\( T^{6} \)
$31$
\( T^{6} \)
$37$
\( T^{6} \)
$41$
\( T^{6} \)
$43$
\( T^{6} - 614125 T^{3} + \cdots + 377149515625 \)
$47$
\( T^{6} + 421875 T^{3} + \cdots + 177978515625 \)
$53$
\( T^{6} \)
$59$
\( T^{6} \)
$61$
\( T^{6} + 1092727 T^{3} + \cdots + 1194052296529 \)
$67$
\( T^{6} \)
$71$
\( T^{6} \)
$73$
\( T^{6} - 15625 T^{3} + \cdots + 244140625 \)
$79$
\( T^{6} \)
$83$
\( (T^{2} + 90 T + 8100)^{3} \)
$89$
\( T^{6} \)
$97$
\( T^{6} \)
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