Properties

Label 1520.1.m.b
Level $1520$
Weight $1$
Character orbit 1520.m
Self dual yes
Analytic conductor $0.759$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -95
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1520.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.475.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.1097440000.5

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{3} - q^{5} + q^{9} +O(q^{10})\) \( q -\beta q^{3} - q^{5} + q^{9} -\beta q^{13} + \beta q^{15} + q^{19} + q^{25} + \beta q^{37} + 2 q^{39} - q^{45} + q^{49} + \beta q^{53} -\beta q^{57} + \beta q^{65} + \beta q^{67} -\beta q^{75} - q^{81} - q^{95} + \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{9} + 2 q^{19} + 2 q^{25} + 4 q^{39} - 2 q^{45} + 2 q^{49} - 2 q^{81} - 2 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1329.1
1.41421
−1.41421
0 −1.41421 0 −1.00000 0 0 0 1.00000 0
1329.2 0 1.41421 0 −1.00000 0 0 0 1.00000 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
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.1.m.b 2
4.b odd 2 1 95.1.d.b 2
5.b even 2 1 inner 1520.1.m.b 2
12.b even 2 1 855.1.g.c 2
19.b odd 2 1 inner 1520.1.m.b 2
20.d odd 2 1 95.1.d.b 2
20.e even 4 2 475.1.c.b 2
60.h even 2 1 855.1.g.c 2
76.d even 2 1 95.1.d.b 2
76.f even 6 2 1805.1.h.b 4
76.g odd 6 2 1805.1.h.b 4
76.k even 18 6 1805.1.o.b 12
76.l odd 18 6 1805.1.o.b 12
95.d odd 2 1 CM 1520.1.m.b 2
228.b odd 2 1 855.1.g.c 2
380.d even 2 1 95.1.d.b 2
380.j odd 4 2 475.1.c.b 2
380.p odd 6 2 1805.1.h.b 4
380.s even 6 2 1805.1.h.b 4
380.ba odd 18 6 1805.1.o.b 12
380.bb even 18 6 1805.1.o.b 12
1140.p odd 2 1 855.1.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 4.b odd 2 1
95.1.d.b 2 20.d odd 2 1
95.1.d.b 2 76.d even 2 1
95.1.d.b 2 380.d even 2 1
475.1.c.b 2 20.e even 4 2
475.1.c.b 2 380.j odd 4 2
855.1.g.c 2 12.b even 2 1
855.1.g.c 2 60.h even 2 1
855.1.g.c 2 228.b odd 2 1
855.1.g.c 2 1140.p odd 2 1
1520.1.m.b 2 1.a even 1 1 trivial
1520.1.m.b 2 5.b even 2 1 inner
1520.1.m.b 2 19.b odd 2 1 inner
1520.1.m.b 2 95.d odd 2 1 CM
1805.1.h.b 4 76.f even 6 2
1805.1.h.b 4 76.g odd 6 2
1805.1.h.b 4 380.p odd 6 2
1805.1.h.b 4 380.s even 6 2
1805.1.o.b 12 76.k even 18 6
1805.1.o.b 12 76.l odd 18 6
1805.1.o.b 12 380.ba odd 18 6
1805.1.o.b 12 380.bb even 18 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{2} - 2 \)
\( T_{7} \)

Hecke characteristic polynomials

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