Newspace parameters
Level: | \( N \) | \(=\) | \( 204 = 2^{2} \cdot 3 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 204.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.101809262577\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.204.1 |
Artin image: | $S_3$ |
Artin field: | Galois closure of 3.1.204.1 |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/204\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(103\) | \(137\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 |
|
0 | 1.00000 | 0 | −1.00000 | 0 | 0 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
51.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-51}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 204.1.g.b | yes | 1 |
3.b | odd | 2 | 1 | 204.1.g.a | ✓ | 1 | |
4.b | odd | 2 | 1 | 816.1.m.a | 1 | ||
8.b | even | 2 | 1 | 3264.1.m.b | 1 | ||
8.d | odd | 2 | 1 | 3264.1.m.d | 1 | ||
12.b | even | 2 | 1 | 816.1.m.b | 1 | ||
17.b | even | 2 | 1 | 204.1.g.a | ✓ | 1 | |
17.c | even | 4 | 2 | 3468.1.d.a | 2 | ||
17.d | even | 8 | 4 | 3468.1.i.a | 4 | ||
17.e | odd | 16 | 8 | 3468.1.m.a | 8 | ||
24.f | even | 2 | 1 | 3264.1.m.a | 1 | ||
24.h | odd | 2 | 1 | 3264.1.m.c | 1 | ||
51.c | odd | 2 | 1 | CM | 204.1.g.b | yes | 1 |
51.f | odd | 4 | 2 | 3468.1.d.a | 2 | ||
51.g | odd | 8 | 4 | 3468.1.i.a | 4 | ||
51.i | even | 16 | 8 | 3468.1.m.a | 8 | ||
68.d | odd | 2 | 1 | 816.1.m.b | 1 | ||
136.e | odd | 2 | 1 | 3264.1.m.a | 1 | ||
136.h | even | 2 | 1 | 3264.1.m.c | 1 | ||
204.h | even | 2 | 1 | 816.1.m.a | 1 | ||
408.b | odd | 2 | 1 | 3264.1.m.b | 1 | ||
408.h | even | 2 | 1 | 3264.1.m.d | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
204.1.g.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
204.1.g.a | ✓ | 1 | 17.b | even | 2 | 1 | |
204.1.g.b | yes | 1 | 1.a | even | 1 | 1 | trivial |
204.1.g.b | yes | 1 | 51.c | odd | 2 | 1 | CM |
816.1.m.a | 1 | 4.b | odd | 2 | 1 | ||
816.1.m.a | 1 | 204.h | even | 2 | 1 | ||
816.1.m.b | 1 | 12.b | even | 2 | 1 | ||
816.1.m.b | 1 | 68.d | odd | 2 | 1 | ||
3264.1.m.a | 1 | 24.f | even | 2 | 1 | ||
3264.1.m.a | 1 | 136.e | odd | 2 | 1 | ||
3264.1.m.b | 1 | 8.b | even | 2 | 1 | ||
3264.1.m.b | 1 | 408.b | odd | 2 | 1 | ||
3264.1.m.c | 1 | 24.h | odd | 2 | 1 | ||
3264.1.m.c | 1 | 136.h | even | 2 | 1 | ||
3264.1.m.d | 1 | 8.d | odd | 2 | 1 | ||
3264.1.m.d | 1 | 408.h | even | 2 | 1 | ||
3468.1.d.a | 2 | 17.c | even | 4 | 2 | ||
3468.1.d.a | 2 | 51.f | odd | 4 | 2 | ||
3468.1.i.a | 4 | 17.d | even | 8 | 4 | ||
3468.1.i.a | 4 | 51.g | odd | 8 | 4 | ||
3468.1.m.a | 8 | 17.e | odd | 16 | 8 | ||
3468.1.m.a | 8 | 51.i | even | 16 | 8 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(204, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 1 \)
$5$
\( T + 1 \)
$7$
\( T \)
$11$
\( T + 1 \)
$13$
\( T + 1 \)
$17$
\( T - 1 \)
$19$
\( T + 1 \)
$23$
\( T + 1 \)
$29$
\( T - 2 \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T + 1 \)
$43$
\( T + 1 \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T \)
$67$
\( T - 2 \)
$71$
\( T - 2 \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T \)
$97$
\( T \)
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