Properties

Label 2475.1.t.a
Level $2475$
Weight $1$
Character orbit 2475.t
Analytic conductor $1.235$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -11
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2475.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.23518590627\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{2} q^{11} + \zeta_{12}^{5} q^{12} - \zeta_{12}^{2} q^{16} - \zeta_{12} q^{23} - \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{31} + \zeta_{12}^{3} q^{33} + q^{36} + \zeta_{12}^{3} q^{37} - q^{44} - \zeta_{12}^{5} q^{47} + \zeta_{12}^{3} q^{48} - \zeta_{12}^{4} q^{49} - \zeta_{12}^{3} q^{53} + \zeta_{12}^{4} q^{59} - q^{64} - \zeta_{12} q^{67} + 2 \zeta_{12}^{2} q^{69} - q^{71} + \zeta_{12}^{4} q^{81} - q^{89} + 2 \zeta_{12}^{5} q^{92} + \zeta_{12}^{5} q^{93} - \zeta_{12}^{5} q^{97} - \zeta_{12}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{31} + 4 q^{36} - 4 q^{44} + 2 q^{49} - 2 q^{59} - 4 q^{64} + 4 q^{69} - 4 q^{71} - 2 q^{81} - 8 q^{89} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 0.500000i 0.500000 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
274.2 0 0.866025 + 0.500000i 0.500000 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
1924.1 0 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 0 0 0.500000 0.866025i 0
1924.2 0 0.866025 0.500000i 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 \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner
55.d odd 2 1 inner
99.h odd 6 1 inner
495.o 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.t.a 4
5.b even 2 1 inner 2475.1.t.a 4
5.c odd 4 1 99.1.h.a 2
5.c odd 4 1 2475.1.y.a 2
9.c even 3 1 inner 2475.1.t.a 4
11.b odd 2 1 CM 2475.1.t.a 4
15.e even 4 1 297.1.h.a 2
20.e even 4 1 1584.1.bf.b 2
45.j even 6 1 inner 2475.1.t.a 4
45.k odd 12 1 99.1.h.a 2
45.k odd 12 1 891.1.c.a 1
45.k odd 12 1 2475.1.y.a 2
45.l even 12 1 297.1.h.a 2
45.l even 12 1 891.1.c.b 1
55.d odd 2 1 inner 2475.1.t.a 4
55.e even 4 1 99.1.h.a 2
55.e even 4 1 2475.1.y.a 2
55.k odd 20 4 1089.1.s.a 8
55.l even 20 4 1089.1.s.a 8
99.h odd 6 1 inner 2475.1.t.a 4
165.l odd 4 1 297.1.h.a 2
165.u odd 20 4 3267.1.w.a 8
165.v even 20 4 3267.1.w.a 8
180.x even 12 1 1584.1.bf.b 2
220.i odd 4 1 1584.1.bf.b 2
495.o odd 6 1 inner 2475.1.t.a 4
495.bd odd 12 1 297.1.h.a 2
495.bd odd 12 1 891.1.c.b 1
495.bf even 12 1 99.1.h.a 2
495.bf even 12 1 891.1.c.a 1
495.bf even 12 1 2475.1.y.a 2
495.bs even 60 4 1089.1.s.a 8
495.bt odd 60 4 1089.1.s.a 8
495.bu odd 60 4 3267.1.w.a 8
495.bv even 60 4 3267.1.w.a 8
1980.cf odd 12 1 1584.1.bf.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 5.c odd 4 1
99.1.h.a 2 45.k odd 12 1
99.1.h.a 2 55.e even 4 1
99.1.h.a 2 495.bf even 12 1
297.1.h.a 2 15.e even 4 1
297.1.h.a 2 45.l even 12 1
297.1.h.a 2 165.l odd 4 1
297.1.h.a 2 495.bd odd 12 1
891.1.c.a 1 45.k odd 12 1
891.1.c.a 1 495.bf even 12 1
891.1.c.b 1 45.l even 12 1
891.1.c.b 1 495.bd odd 12 1
1089.1.s.a 8 55.k odd 20 4
1089.1.s.a 8 55.l even 20 4
1089.1.s.a 8 495.bs even 60 4
1089.1.s.a 8 495.bt odd 60 4
1584.1.bf.b 2 20.e even 4 1
1584.1.bf.b 2 180.x even 12 1
1584.1.bf.b 2 220.i odd 4 1
1584.1.bf.b 2 1980.cf odd 12 1
2475.1.t.a 4 1.a even 1 1 trivial
2475.1.t.a 4 5.b even 2 1 inner
2475.1.t.a 4 9.c even 3 1 inner
2475.1.t.a 4 11.b odd 2 1 CM
2475.1.t.a 4 45.j even 6 1 inner
2475.1.t.a 4 55.d odd 2 1 inner
2475.1.t.a 4 99.h odd 6 1 inner
2475.1.t.a 4 495.o odd 6 1 inner
2475.1.y.a 2 5.c odd 4 1
2475.1.y.a 2 45.k odd 12 1
2475.1.y.a 2 55.e even 4 1
2475.1.y.a 2 495.bf even 12 1
3267.1.w.a 8 165.u odd 20 4
3267.1.w.a 8 165.v even 20 4
3267.1.w.a 8 495.bu odd 60 4
3267.1.w.a 8 495.bv even 60 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
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