Properties

Label 495.1.h.c
Level $495$
Weight $1$
Character orbit 495.h
Self dual yes
Analytic conductor $0.247$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -55
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 495.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.247037181253\)
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: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2475.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.606436875.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{2} + q^{4} + q^{5} + \beta q^{7} +O(q^{10})\) \( q -\beta q^{2} + q^{4} + q^{5} + \beta q^{7} -\beta q^{10} - q^{11} -\beta q^{13} -2 q^{14} - q^{16} + \beta q^{17} + q^{20} + \beta q^{22} + q^{25} + 2 q^{26} + \beta q^{28} + \beta q^{32} -2 q^{34} + \beta q^{35} -\beta q^{43} - q^{44} + q^{49} -\beta q^{50} -\beta q^{52} - q^{55} - q^{64} -\beta q^{65} + \beta q^{68} -2 q^{70} + \beta q^{73} -\beta q^{77} - q^{80} -\beta q^{83} + \beta q^{85} + 2 q^{86} -2 q^{89} -2 q^{91} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{5} - 2q^{11} - 4q^{14} - 2q^{16} + 2q^{20} + 2q^{25} + 4q^{26} - 4q^{34} - 2q^{44} + 2q^{49} - 2q^{55} - 2q^{64} - 4q^{70} - 2q^{80} + 4q^{86} - 4q^{89} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.41421
−1.41421
−1.41421 0 1.00000 1.00000 0 1.41421 0 0 −1.41421
109.2 1.41421 0 1.00000 1.00000 0 −1.41421 0 0 1.41421
\(n\): 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 495.1.h.c yes 2
3.b odd 2 1 495.1.h.b 2
5.b even 2 1 inner 495.1.h.c yes 2
5.c odd 4 2 2475.1.b.b 2
11.b odd 2 1 inner 495.1.h.c yes 2
15.d odd 2 1 495.1.h.b 2
15.e even 4 2 2475.1.b.c 2
33.d even 2 1 495.1.h.b 2
55.d odd 2 1 CM 495.1.h.c yes 2
55.e even 4 2 2475.1.b.b 2
165.d even 2 1 495.1.h.b 2
165.l odd 4 2 2475.1.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.1.h.b 2 3.b odd 2 1
495.1.h.b 2 15.d odd 2 1
495.1.h.b 2 33.d even 2 1
495.1.h.b 2 165.d even 2 1
495.1.h.c yes 2 1.a even 1 1 trivial
495.1.h.c yes 2 5.b even 2 1 inner
495.1.h.c yes 2 11.b odd 2 1 inner
495.1.h.c yes 2 55.d odd 2 1 CM
2475.1.b.b 2 5.c odd 4 2
2475.1.b.b 2 55.e even 4 2
2475.1.b.c 2 15.e even 4 2
2475.1.b.c 2 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(495, [\chi])\):

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

Hecke characteristic polynomials

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