Properties

Label 2475.1.y.a
Level $2475$
Weight $1$
Character orbit 2475.y
Analytic conductor $1.235$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -11
Inner twists $4$

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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: \(\Q(\zeta_{6})\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{11} -\zeta_{6}^{2} q^{12} + \zeta_{6}^{2} q^{16} + 2 \zeta_{6} q^{23} - q^{27} + \zeta_{6} q^{31} - q^{33} + q^{36} + q^{37} + q^{44} + \zeta_{6}^{2} q^{47} - q^{48} -\zeta_{6} q^{49} + q^{53} + \zeta_{6} q^{59} + q^{64} -\zeta_{6} q^{67} + 2 \zeta_{6}^{2} q^{69} - q^{71} -\zeta_{6} q^{81} + 2 q^{89} -2 \zeta_{6}^{2} q^{92} + \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{97} -\zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{4} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{4} - q^{9} - q^{11} + q^{12} - q^{16} + 2q^{23} - 2q^{27} + q^{31} - 2q^{33} + 2q^{36} + 2q^{37} + 2q^{44} - q^{47} - 2q^{48} - q^{49} + 2q^{53} + q^{59} + 2q^{64} - q^{67} - 2q^{69} - 2q^{71} - q^{81} + 4q^{89} + 2q^{92} - q^{93} - q^{97} - q^{99} + O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \( T_{23}^{2} - 2 T_{23} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 4 - 2 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 1 + T + T^{2} \)
$53$ \( ( -1 + T )^{2} \)
$59$ \( 1 - T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 1 + T + T^{2} \)
$71$ \( ( 1 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( 1 + T + T^{2} \)
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