Properties

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

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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 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{2} + 8 q^{4} + 5 q^{5} + 6 q^{7} + O(q^{10}) \) \( q + 4 q^{2} + 8 q^{4} + 5 q^{5} + 6 q^{7} + 20 q^{10} - 32 q^{11} - 38 q^{13} + 24 q^{14} - 64 q^{16} - 26 q^{17} + 100 q^{19} + 40 q^{20} - 128 q^{22} + 78 q^{23} + 25 q^{25} - 152 q^{26} + 48 q^{28} + 50 q^{29} - 108 q^{31} - 256 q^{32} - 104 q^{34} + 30 q^{35} + 266 q^{37} + 400 q^{38} - 22 q^{41} + 442 q^{43} - 256 q^{44} + 312 q^{46} + 514 q^{47} - 307 q^{49} + 100 q^{50} - 304 q^{52} - 2 q^{53} - 160 q^{55} + 200 q^{58} - 500 q^{59} - 518 q^{61} - 432 q^{62} - 512 q^{64} - 190 q^{65} + 126 q^{67} - 208 q^{68} + 120 q^{70} - 412 q^{71} - 878 q^{73} + 1064 q^{74} + 800 q^{76} - 192 q^{77} + 600 q^{79} - 320 q^{80} - 88 q^{82} - 282 q^{83} - 130 q^{85} + 1768 q^{86} + 150 q^{89} - 228 q^{91} + 624 q^{92} + 2056 q^{94} + 500 q^{95} + 386 q^{97} - 1228 q^{98} + O(q^{100}) \)

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
4.00000 0 8.00000 5.00000 0 6.00000 0 0 20.0000
\(n\): 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.d 1
3.b odd 2 1 5.4.a.a 1
4.b odd 2 1 720.4.a.u 1
5.b even 2 1 225.4.a.b 1
5.c odd 4 2 225.4.b.c 2
7.b odd 2 1 2205.4.a.q 1
9.c even 3 2 405.4.e.c 2
9.d odd 6 2 405.4.e.l 2
12.b even 2 1 80.4.a.d 1
15.d odd 2 1 25.4.a.c 1
15.e even 4 2 25.4.b.a 2
21.c even 2 1 245.4.a.a 1
21.g even 6 2 245.4.e.g 2
21.h odd 6 2 245.4.e.f 2
24.f even 2 1 320.4.a.h 1
24.h odd 2 1 320.4.a.g 1
33.d even 2 1 605.4.a.d 1
39.d odd 2 1 845.4.a.b 1
48.i odd 4 2 1280.4.d.e 2
48.k even 4 2 1280.4.d.l 2
51.c odd 2 1 1445.4.a.a 1
57.d even 2 1 1805.4.a.h 1
60.h even 2 1 400.4.a.m 1
60.l odd 4 2 400.4.c.k 2
105.g even 2 1 1225.4.a.k 1
120.i odd 2 1 1600.4.a.bi 1
120.m even 2 1 1600.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 3.b odd 2 1
25.4.a.c 1 15.d odd 2 1
25.4.b.a 2 15.e even 4 2
45.4.a.d 1 1.a even 1 1 trivial
80.4.a.d 1 12.b even 2 1
225.4.a.b 1 5.b even 2 1
225.4.b.c 2 5.c odd 4 2
245.4.a.a 1 21.c even 2 1
245.4.e.f 2 21.h odd 6 2
245.4.e.g 2 21.g even 6 2
320.4.a.g 1 24.h odd 2 1
320.4.a.h 1 24.f even 2 1
400.4.a.m 1 60.h even 2 1
400.4.c.k 2 60.l odd 4 2
405.4.e.c 2 9.c even 3 2
405.4.e.l 2 9.d odd 6 2
605.4.a.d 1 33.d even 2 1
720.4.a.u 1 4.b odd 2 1
845.4.a.b 1 39.d odd 2 1
1225.4.a.k 1 105.g even 2 1
1280.4.d.e 2 48.i odd 4 2
1280.4.d.l 2 48.k even 4 2
1445.4.a.a 1 51.c odd 2 1
1600.4.a.s 1 120.m even 2 1
1600.4.a.bi 1 120.i odd 2 1
1805.4.a.h 1 57.d even 2 1
2205.4.a.q 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( -5 + T \)
$7$ \( -6 + T \)
$11$ \( 32 + T \)
$13$ \( 38 + T \)
$17$ \( 26 + T \)
$19$ \( -100 + T \)
$23$ \( -78 + T \)
$29$ \( -50 + T \)
$31$ \( 108 + T \)
$37$ \( -266 + T \)
$41$ \( 22 + T \)
$43$ \( -442 + T \)
$47$ \( -514 + T \)
$53$ \( 2 + T \)
$59$ \( 500 + T \)
$61$ \( 518 + T \)
$67$ \( -126 + T \)
$71$ \( 412 + T \)
$73$ \( 878 + T \)
$79$ \( -600 + T \)
$83$ \( 282 + T \)
$89$ \( -150 + T \)
$97$ \( -386 + T \)
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