Properties

Label 45.2.b.a
Level $45$
Weight $2$
Character orbit 45.b
Analytic conductor $0.359$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -3 q^{4} -\beta q^{5} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -3 q^{4} -\beta q^{5} -\beta q^{8} + 5 q^{10} - q^{16} -2 \beta q^{17} -4 q^{19} + 3 \beta q^{20} + 4 \beta q^{23} -5 q^{25} + 8 q^{31} -3 \beta q^{32} + 10 q^{34} -4 \beta q^{38} -5 q^{40} -20 q^{46} + 4 \beta q^{47} + 7 q^{49} -5 \beta q^{50} -2 \beta q^{53} + 2 q^{61} + 8 \beta q^{62} + 13 q^{64} + 6 \beta q^{68} + 12 q^{76} -16 q^{79} + \beta q^{80} -8 \beta q^{83} -10 q^{85} -12 \beta q^{92} -20 q^{94} + 4 \beta q^{95} + 7 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{4} + O(q^{10}) \) \( 2q - 6q^{4} + 10q^{10} - 2q^{16} - 8q^{19} - 10q^{25} + 16q^{31} + 20q^{34} - 10q^{40} - 40q^{46} + 14q^{49} + 4q^{61} + 26q^{64} + 24q^{76} - 32q^{79} - 20q^{85} - 40q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
2.23607i
2.23607i
2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 5.00000
19.2 2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.b.a 2
3.b odd 2 1 inner 45.2.b.a 2
4.b odd 2 1 720.2.f.d 2
5.b even 2 1 inner 45.2.b.a 2
5.c odd 4 2 225.2.a.f 2
7.b odd 2 1 2205.2.d.a 2
8.b even 2 1 2880.2.f.k 2
8.d odd 2 1 2880.2.f.j 2
9.c even 3 2 405.2.j.c 4
9.d odd 6 2 405.2.j.c 4
12.b even 2 1 720.2.f.d 2
15.d odd 2 1 CM 45.2.b.a 2
15.e even 4 2 225.2.a.f 2
20.d odd 2 1 720.2.f.d 2
20.e even 4 2 3600.2.a.bs 2
21.c even 2 1 2205.2.d.a 2
24.f even 2 1 2880.2.f.j 2
24.h odd 2 1 2880.2.f.k 2
35.c odd 2 1 2205.2.d.a 2
40.e odd 2 1 2880.2.f.j 2
40.f even 2 1 2880.2.f.k 2
45.h odd 6 2 405.2.j.c 4
45.j even 6 2 405.2.j.c 4
60.h even 2 1 720.2.f.d 2
60.l odd 4 2 3600.2.a.bs 2
105.g even 2 1 2205.2.d.a 2
120.i odd 2 1 2880.2.f.k 2
120.m even 2 1 2880.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 1.a even 1 1 trivial
45.2.b.a 2 3.b odd 2 1 inner
45.2.b.a 2 5.b even 2 1 inner
45.2.b.a 2 15.d odd 2 1 CM
225.2.a.f 2 5.c odd 4 2
225.2.a.f 2 15.e even 4 2
405.2.j.c 4 9.c even 3 2
405.2.j.c 4 9.d odd 6 2
405.2.j.c 4 45.h odd 6 2
405.2.j.c 4 45.j even 6 2
720.2.f.d 2 4.b odd 2 1
720.2.f.d 2 12.b even 2 1
720.2.f.d 2 20.d odd 2 1
720.2.f.d 2 60.h even 2 1
2205.2.d.a 2 7.b odd 2 1
2205.2.d.a 2 21.c even 2 1
2205.2.d.a 2 35.c odd 2 1
2205.2.d.a 2 105.g even 2 1
2880.2.f.j 2 8.d odd 2 1
2880.2.f.j 2 24.f even 2 1
2880.2.f.j 2 40.e odd 2 1
2880.2.f.j 2 120.m even 2 1
2880.2.f.k 2 8.b even 2 1
2880.2.f.k 2 24.h odd 2 1
2880.2.f.k 2 40.f even 2 1
2880.2.f.k 2 120.i odd 2 1
3600.2.a.bs 2 20.e even 4 2
3600.2.a.bs 2 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 20 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 80 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 80 + T^{2} \)
$53$ \( 20 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 320 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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