Properties

Label 45.4.a.b
Level $45$
Weight $4$
Character orbit 45.a
Self dual yes
Analytic conductor $2.655$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} + 5 q^{5} + 20 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} + 5 q^{5} + 20 q^{7} + 21 q^{8} - 15 q^{10} + 24 q^{11} + 74 q^{13} - 60 q^{14} - 71 q^{16} - 54 q^{17} - 124 q^{19} + 5 q^{20} - 72 q^{22} + 120 q^{23} + 25 q^{25} - 222 q^{26} + 20 q^{28} + 78 q^{29} + 200 q^{31} + 45 q^{32} + 162 q^{34} + 100 q^{35} - 70 q^{37} + 372 q^{38} + 105 q^{40} - 330 q^{41} + 92 q^{43} + 24 q^{44} - 360 q^{46} + 24 q^{47} + 57 q^{49} - 75 q^{50} + 74 q^{52} - 450 q^{53} + 120 q^{55} + 420 q^{56} - 234 q^{58} - 24 q^{59} - 322 q^{61} - 600 q^{62} + 433 q^{64} + 370 q^{65} - 196 q^{67} - 54 q^{68} - 300 q^{70} + 288 q^{71} - 430 q^{73} + 210 q^{74} - 124 q^{76} + 480 q^{77} - 520 q^{79} - 355 q^{80} + 990 q^{82} - 156 q^{83} - 270 q^{85} - 276 q^{86} + 504 q^{88} - 1026 q^{89} + 1480 q^{91} + 120 q^{92} - 72 q^{94} - 620 q^{95} - 286 q^{97} - 171 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−3.00000 0 1.00000 5.00000 0 20.0000 21.0000 0 −15.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.a.b 1
3.b odd 2 1 15.4.a.b 1
4.b odd 2 1 720.4.a.r 1
5.b even 2 1 225.4.a.g 1
5.c odd 4 2 225.4.b.d 2
7.b odd 2 1 2205.4.a.c 1
9.c even 3 2 405.4.e.k 2
9.d odd 6 2 405.4.e.d 2
12.b even 2 1 240.4.a.f 1
15.d odd 2 1 75.4.a.a 1
15.e even 4 2 75.4.b.a 2
21.c even 2 1 735.4.a.i 1
24.f even 2 1 960.4.a.l 1
24.h odd 2 1 960.4.a.bi 1
33.d even 2 1 1815.4.a.a 1
60.h even 2 1 1200.4.a.o 1
60.l odd 4 2 1200.4.f.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 3.b odd 2 1
45.4.a.b 1 1.a even 1 1 trivial
75.4.a.a 1 15.d odd 2 1
75.4.b.a 2 15.e even 4 2
225.4.a.g 1 5.b even 2 1
225.4.b.d 2 5.c odd 4 2
240.4.a.f 1 12.b even 2 1
405.4.e.d 2 9.d odd 6 2
405.4.e.k 2 9.c even 3 2
720.4.a.r 1 4.b odd 2 1
735.4.a.i 1 21.c even 2 1
960.4.a.l 1 24.f even 2 1
960.4.a.bi 1 24.h odd 2 1
1200.4.a.o 1 60.h even 2 1
1200.4.f.m 2 60.l odd 4 2
1815.4.a.a 1 33.d even 2 1
2205.4.a.c 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 20 \) Copy content Toggle raw display
$11$ \( T - 24 \) Copy content Toggle raw display
$13$ \( T - 74 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T + 124 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T - 78 \) Copy content Toggle raw display
$31$ \( T - 200 \) Copy content Toggle raw display
$37$ \( T + 70 \) Copy content Toggle raw display
$41$ \( T + 330 \) Copy content Toggle raw display
$43$ \( T - 92 \) Copy content Toggle raw display
$47$ \( T - 24 \) Copy content Toggle raw display
$53$ \( T + 450 \) Copy content Toggle raw display
$59$ \( T + 24 \) Copy content Toggle raw display
$61$ \( T + 322 \) Copy content Toggle raw display
$67$ \( T + 196 \) Copy content Toggle raw display
$71$ \( T - 288 \) Copy content Toggle raw display
$73$ \( T + 430 \) Copy content Toggle raw display
$79$ \( T + 520 \) Copy content Toggle raw display
$83$ \( T + 156 \) Copy content Toggle raw display
$89$ \( T + 1026 \) Copy content Toggle raw display
$97$ \( T + 286 \) Copy content Toggle raw display
show more
show less