Properties

Label 45.4.b.a
Level $45$
Weight $4$
Character orbit 45.b
Analytic conductor $2.655$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.65508595026\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{4} + 5 \beta q^{5} + 11 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 3 q^{4} + 5 \beta q^{5} + 11 \beta q^{8} -25 q^{10} -31 q^{16} -62 \beta q^{17} + 164 q^{19} + 15 \beta q^{20} -44 \beta q^{23} -125 q^{25} -232 q^{31} + 57 \beta q^{32} + 310 q^{34} + 164 \beta q^{38} -275 q^{40} + 220 q^{46} + 244 \beta q^{47} + 343 q^{49} -125 \beta q^{50} -278 \beta q^{53} -358 q^{61} -232 \beta q^{62} -533 q^{64} -186 \beta q^{68} + 492 q^{76} -304 q^{79} -155 \beta q^{80} + 568 \beta q^{83} + 1550 q^{85} -132 \beta q^{92} -1220 q^{94} + 820 \beta q^{95} + 343 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} + O(q^{10}) \) \( 2 q + 6 q^{4} - 50 q^{10} - 62 q^{16} + 328 q^{19} - 250 q^{25} - 464 q^{31} + 620 q^{34} - 550 q^{40} + 440 q^{46} + 686 q^{49} - 716 q^{61} - 1066 q^{64} + 984 q^{76} - 608 q^{79} + 3100 q^{85} - 2440 q^{94} + O(q^{100}) \)

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 11.1803i 0 0 24.5967i 0 −25.0000
19.2 2.23607i 0 3.00000 11.1803i 0 0 24.5967i 0 −25.0000
\(n\): 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 45.4.b.a 2
3.b odd 2 1 inner 45.4.b.a 2
4.b odd 2 1 720.4.f.d 2
5.b even 2 1 inner 45.4.b.a 2
5.c odd 4 2 225.4.a.k 2
12.b even 2 1 720.4.f.d 2
15.d odd 2 1 CM 45.4.b.a 2
15.e even 4 2 225.4.a.k 2
20.d odd 2 1 720.4.f.d 2
60.h even 2 1 720.4.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.b.a 2 1.a even 1 1 trivial
45.4.b.a 2 3.b odd 2 1 inner
45.4.b.a 2 5.b even 2 1 inner
45.4.b.a 2 15.d odd 2 1 CM
225.4.a.k 2 5.c odd 4 2
225.4.a.k 2 15.e even 4 2
720.4.f.d 2 4.b odd 2 1
720.4.f.d 2 12.b even 2 1
720.4.f.d 2 20.d odd 2 1
720.4.f.d 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 125 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 19220 + T^{2} \)
$19$ \( ( -164 + T )^{2} \)
$23$ \( 9680 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 232 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 297680 + T^{2} \)
$53$ \( 386420 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 358 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 304 + T )^{2} \)
$83$ \( 1613120 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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