Properties

Label 2475.1.b.b
Level $2475$
Weight $1$
Character orbit 2475.b
Analytic conductor $1.235$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -55
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: no
Analytic conductor: \(1.23518590627\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2475.1
Artin image: $\SD_{16}$
Artin field: Galois closure of 8.2.15160921875.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{4} -\beta q^{7} +O(q^{10})\) \( q + \beta q^{2} - q^{4} -\beta q^{7} - q^{11} -\beta q^{13} + 2 q^{14} - q^{16} -\beta q^{17} -\beta q^{22} + 2 q^{26} + \beta q^{28} -\beta q^{32} + 2 q^{34} -\beta q^{43} + q^{44} - q^{49} + \beta q^{52} + q^{64} + \beta q^{68} + \beta q^{73} + \beta q^{77} -\beta q^{83} + 2 q^{86} + 2 q^{89} -2 q^{91} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} - 2 q^{11} + 4 q^{14} - 2 q^{16} + 4 q^{26} + 4 q^{34} + 2 q^{44} - 2 q^{49} + 2 q^{64} + 4 q^{86} + 4 q^{89} - 4 q^{91} + O(q^{100}) \)

Character values

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

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

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
1.41421i
1.41421i
1.41421i 0 −1.00000 0 0 1.41421i 0 0 0
901.2 1.41421i 0 −1.00000 0 0 1.41421i 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
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.b.b 2
3.b odd 2 1 2475.1.b.c 2
5.b even 2 1 inner 2475.1.b.b 2
5.c odd 4 2 495.1.h.c yes 2
11.b odd 2 1 inner 2475.1.b.b 2
15.d odd 2 1 2475.1.b.c 2
15.e even 4 2 495.1.h.b 2
33.d even 2 1 2475.1.b.c 2
55.d odd 2 1 CM 2475.1.b.b 2
55.e even 4 2 495.1.h.c yes 2
165.d even 2 1 2475.1.b.c 2
165.l odd 4 2 495.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.1.h.b 2 15.e even 4 2
495.1.h.b 2 165.l odd 4 2
495.1.h.c yes 2 5.c odd 4 2
495.1.h.c yes 2 55.e even 4 2
2475.1.b.b 2 1.a even 1 1 trivial
2475.1.b.b 2 5.b even 2 1 inner
2475.1.b.b 2 11.b odd 2 1 inner
2475.1.b.b 2 55.d odd 2 1 CM
2475.1.b.c 2 3.b odd 2 1
2475.1.b.c 2 15.d odd 2 1
2475.1.b.c 2 33.d even 2 1
2475.1.b.c 2 165.d even 2 1

Hecke kernels

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

\( T_{2}^{2} + 2 \)
\( T_{89} - 2 \)

Hecke characteristic polynomials

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