Newspace parameters
Level: | \( N \) | \(=\) | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2475.y (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.23518590627\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 99) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.891.1 |
Artin image: | $C_6\times S_3$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).
\(n\) | \(551\) | \(2026\) | \(2377\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 |
|
0 | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0 | 0 | 0 | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||
1726.1 | 0 | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0 | 0 | 0 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
9.c | even | 3 | 1 | inner |
99.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2475.1.y.a | 2 | |
5.b | even | 2 | 1 | 99.1.h.a | ✓ | 2 | |
5.c | odd | 4 | 2 | 2475.1.t.a | 4 | ||
9.c | even | 3 | 1 | inner | 2475.1.y.a | 2 | |
11.b | odd | 2 | 1 | CM | 2475.1.y.a | 2 | |
15.d | odd | 2 | 1 | 297.1.h.a | 2 | ||
20.d | odd | 2 | 1 | 1584.1.bf.b | 2 | ||
45.h | odd | 6 | 1 | 297.1.h.a | 2 | ||
45.h | odd | 6 | 1 | 891.1.c.b | 1 | ||
45.j | even | 6 | 1 | 99.1.h.a | ✓ | 2 | |
45.j | even | 6 | 1 | 891.1.c.a | 1 | ||
45.k | odd | 12 | 2 | 2475.1.t.a | 4 | ||
55.d | odd | 2 | 1 | 99.1.h.a | ✓ | 2 | |
55.e | even | 4 | 2 | 2475.1.t.a | 4 | ||
55.h | odd | 10 | 4 | 1089.1.s.a | 8 | ||
55.j | even | 10 | 4 | 1089.1.s.a | 8 | ||
99.h | odd | 6 | 1 | inner | 2475.1.y.a | 2 | |
165.d | even | 2 | 1 | 297.1.h.a | 2 | ||
165.o | odd | 10 | 4 | 3267.1.w.a | 8 | ||
165.r | even | 10 | 4 | 3267.1.w.a | 8 | ||
180.p | odd | 6 | 1 | 1584.1.bf.b | 2 | ||
220.g | even | 2 | 1 | 1584.1.bf.b | 2 | ||
495.o | odd | 6 | 1 | 99.1.h.a | ✓ | 2 | |
495.o | odd | 6 | 1 | 891.1.c.a | 1 | ||
495.r | even | 6 | 1 | 297.1.h.a | 2 | ||
495.r | even | 6 | 1 | 891.1.c.b | 1 | ||
495.bf | even | 12 | 2 | 2475.1.t.a | 4 | ||
495.bl | even | 30 | 4 | 1089.1.s.a | 8 | ||
495.bo | even | 30 | 4 | 3267.1.w.a | 8 | ||
495.bp | odd | 30 | 4 | 3267.1.w.a | 8 | ||
495.br | odd | 30 | 4 | 1089.1.s.a | 8 | ||
1980.bk | even | 6 | 1 | 1584.1.bf.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.1.h.a | ✓ | 2 | 5.b | even | 2 | 1 | |
99.1.h.a | ✓ | 2 | 45.j | even | 6 | 1 | |
99.1.h.a | ✓ | 2 | 55.d | odd | 2 | 1 | |
99.1.h.a | ✓ | 2 | 495.o | odd | 6 | 1 | |
297.1.h.a | 2 | 15.d | odd | 2 | 1 | ||
297.1.h.a | 2 | 45.h | odd | 6 | 1 | ||
297.1.h.a | 2 | 165.d | even | 2 | 1 | ||
297.1.h.a | 2 | 495.r | even | 6 | 1 | ||
891.1.c.a | 1 | 45.j | even | 6 | 1 | ||
891.1.c.a | 1 | 495.o | odd | 6 | 1 | ||
891.1.c.b | 1 | 45.h | odd | 6 | 1 | ||
891.1.c.b | 1 | 495.r | even | 6 | 1 | ||
1089.1.s.a | 8 | 55.h | odd | 10 | 4 | ||
1089.1.s.a | 8 | 55.j | even | 10 | 4 | ||
1089.1.s.a | 8 | 495.bl | even | 30 | 4 | ||
1089.1.s.a | 8 | 495.br | odd | 30 | 4 | ||
1584.1.bf.b | 2 | 20.d | odd | 2 | 1 | ||
1584.1.bf.b | 2 | 180.p | odd | 6 | 1 | ||
1584.1.bf.b | 2 | 220.g | even | 2 | 1 | ||
1584.1.bf.b | 2 | 1980.bk | even | 6 | 1 | ||
2475.1.t.a | 4 | 5.c | odd | 4 | 2 | ||
2475.1.t.a | 4 | 45.k | odd | 12 | 2 | ||
2475.1.t.a | 4 | 55.e | even | 4 | 2 | ||
2475.1.t.a | 4 | 495.bf | even | 12 | 2 | ||
2475.1.y.a | 2 | 1.a | even | 1 | 1 | trivial | |
2475.1.y.a | 2 | 9.c | even | 3 | 1 | inner | |
2475.1.y.a | 2 | 11.b | odd | 2 | 1 | CM | |
2475.1.y.a | 2 | 99.h | odd | 6 | 1 | inner | |
3267.1.w.a | 8 | 165.o | odd | 10 | 4 | ||
3267.1.w.a | 8 | 165.r | even | 10 | 4 | ||
3267.1.w.a | 8 | 495.bo | even | 30 | 4 | ||
3267.1.w.a | 8 | 495.bp | odd | 30 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{2} - 2T_{23} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(2475, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - T + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} + T + 1 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} - 2T + 4 \)
$29$
\( T^{2} \)
$31$
\( T^{2} - T + 1 \)
$37$
\( (T - 1)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} + T + 1 \)
$53$
\( (T - 1)^{2} \)
$59$
\( T^{2} - T + 1 \)
$61$
\( T^{2} \)
$67$
\( T^{2} + T + 1 \)
$71$
\( (T + 1)^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} \)
$89$
\( (T - 2)^{2} \)
$97$
\( T^{2} + T + 1 \)
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