Properties

Label 2475.1.b.a
Level $2475$
Weight $1$
Character orbit 2475.b
Self dual yes
Analytic conductor $1.235$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -11, -55, 5
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.23518590627\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)
Artin image: $D_4$
Artin field: Galois closure of 4.2.12375.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} + q^{11} + q^{16} - 2 q^{31} + q^{44} + q^{49} + 2 q^{59} + q^{64} - 2 q^{71} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0
0 0 1.00000 0 0 0 0 0 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
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.b.a 1
3.b odd 2 1 275.1.c.a 1
5.b even 2 1 RM 2475.1.b.a 1
5.c odd 4 2 495.1.h.a 1
11.b odd 2 1 CM 2475.1.b.a 1
15.d odd 2 1 275.1.c.a 1
15.e even 4 2 55.1.d.a 1
33.d even 2 1 275.1.c.a 1
33.f even 10 4 3025.1.x.a 4
33.h odd 10 4 3025.1.x.a 4
55.d odd 2 1 CM 2475.1.b.a 1
55.e even 4 2 495.1.h.a 1
60.l odd 4 2 880.1.i.a 1
105.k odd 4 2 2695.1.g.c 1
105.w odd 12 4 2695.1.q.b 2
105.x even 12 4 2695.1.q.c 2
120.q odd 4 2 3520.1.i.a 1
120.w even 4 2 3520.1.i.b 1
165.d even 2 1 275.1.c.a 1
165.l odd 4 2 55.1.d.a 1
165.o odd 10 4 3025.1.x.a 4
165.r even 10 4 3025.1.x.a 4
165.u odd 20 8 605.1.h.a 4
165.v even 20 8 605.1.h.a 4
660.q even 4 2 880.1.i.a 1
1155.t even 4 2 2695.1.g.c 1
1155.cg odd 12 4 2695.1.q.c 2
1155.cj even 12 4 2695.1.q.b 2
1320.bn odd 4 2 3520.1.i.b 1
1320.bt even 4 2 3520.1.i.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 15.e even 4 2
55.1.d.a 1 165.l odd 4 2
275.1.c.a 1 3.b odd 2 1
275.1.c.a 1 15.d odd 2 1
275.1.c.a 1 33.d even 2 1
275.1.c.a 1 165.d even 2 1
495.1.h.a 1 5.c odd 4 2
495.1.h.a 1 55.e even 4 2
605.1.h.a 4 165.u odd 20 8
605.1.h.a 4 165.v even 20 8
880.1.i.a 1 60.l odd 4 2
880.1.i.a 1 660.q even 4 2
2475.1.b.a 1 1.a even 1 1 trivial
2475.1.b.a 1 5.b even 2 1 RM
2475.1.b.a 1 11.b odd 2 1 CM
2475.1.b.a 1 55.d odd 2 1 CM
2695.1.g.c 1 105.k odd 4 2
2695.1.g.c 1 1155.t even 4 2
2695.1.q.b 2 105.w odd 12 4
2695.1.q.b 2 1155.cj even 12 4
2695.1.q.c 2 105.x even 12 4
2695.1.q.c 2 1155.cg odd 12 4
3025.1.x.a 4 33.f even 10 4
3025.1.x.a 4 33.h odd 10 4
3025.1.x.a 4 165.o odd 10 4
3025.1.x.a 4 165.r even 10 4
3520.1.i.a 1 120.q odd 4 2
3520.1.i.a 1 1320.bt even 4 2
3520.1.i.b 1 120.w even 4 2
3520.1.i.b 1 1320.bn odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{89} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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