Properties

Label 45.6.a.b
Level 45
Weight 6
Character orbit 45.a
Self dual yes
Analytic conductor 7.217
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.21727189158\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 28q^{4} - 25q^{5} + 192q^{7} + 120q^{8} + O(q^{10}) \) \( q - 2q^{2} - 28q^{4} - 25q^{5} + 192q^{7} + 120q^{8} + 50q^{10} + 148q^{11} + 286q^{13} - 384q^{14} + 656q^{16} + 1678q^{17} + 1060q^{19} + 700q^{20} - 296q^{22} - 2976q^{23} + 625q^{25} - 572q^{26} - 5376q^{28} + 3410q^{29} - 2448q^{31} - 5152q^{32} - 3356q^{34} - 4800q^{35} + 182q^{37} - 2120q^{38} - 3000q^{40} + 9398q^{41} - 1244q^{43} - 4144q^{44} + 5952q^{46} + 12088q^{47} + 20057q^{49} - 1250q^{50} - 8008q^{52} - 23846q^{53} - 3700q^{55} + 23040q^{56} - 6820q^{58} + 20020q^{59} + 32302q^{61} + 4896q^{62} - 10688q^{64} - 7150q^{65} + 60972q^{67} - 46984q^{68} + 9600q^{70} + 32648q^{71} - 38774q^{73} - 364q^{74} - 29680q^{76} + 28416q^{77} - 33360q^{79} - 16400q^{80} - 18796q^{82} - 16716q^{83} - 41950q^{85} + 2488q^{86} + 17760q^{88} - 101370q^{89} + 54912q^{91} + 83328q^{92} - 24176q^{94} - 26500q^{95} - 119038q^{97} - 40114q^{98} + O(q^{100}) \)

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
−2.00000 0 −28.0000 −25.0000 0 192.000 120.000 0 50.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 45.6.a.b 1
3.b odd 2 1 5.6.a.a 1
4.b odd 2 1 720.6.a.a 1
5.b even 2 1 225.6.a.f 1
5.c odd 4 2 225.6.b.e 2
12.b even 2 1 80.6.a.e 1
15.d odd 2 1 25.6.a.a 1
15.e even 4 2 25.6.b.a 2
21.c even 2 1 245.6.a.b 1
24.f even 2 1 320.6.a.g 1
24.h odd 2 1 320.6.a.j 1
33.d even 2 1 605.6.a.a 1
39.d odd 2 1 845.6.a.b 1
60.h even 2 1 400.6.a.g 1
60.l odd 4 2 400.6.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 3.b odd 2 1
25.6.a.a 1 15.d odd 2 1
25.6.b.a 2 15.e even 4 2
45.6.a.b 1 1.a even 1 1 trivial
80.6.a.e 1 12.b even 2 1
225.6.a.f 1 5.b even 2 1
225.6.b.e 2 5.c odd 4 2
245.6.a.b 1 21.c even 2 1
320.6.a.g 1 24.f even 2 1
320.6.a.j 1 24.h odd 2 1
400.6.a.g 1 60.h even 2 1
400.6.c.j 2 60.l odd 4 2
605.6.a.a 1 33.d even 2 1
720.6.a.a 1 4.b odd 2 1
845.6.a.b 1 39.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(45))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 32 T^{2} \)
$3$ 1
$5$ \( 1 + 25 T \)
$7$ \( 1 - 192 T + 16807 T^{2} \)
$11$ \( 1 - 148 T + 161051 T^{2} \)
$13$ \( 1 - 286 T + 371293 T^{2} \)
$17$ \( 1 - 1678 T + 1419857 T^{2} \)
$19$ \( 1 - 1060 T + 2476099 T^{2} \)
$23$ \( 1 + 2976 T + 6436343 T^{2} \)
$29$ \( 1 - 3410 T + 20511149 T^{2} \)
$31$ \( 1 + 2448 T + 28629151 T^{2} \)
$37$ \( 1 - 182 T + 69343957 T^{2} \)
$41$ \( 1 - 9398 T + 115856201 T^{2} \)
$43$ \( 1 + 1244 T + 147008443 T^{2} \)
$47$ \( 1 - 12088 T + 229345007 T^{2} \)
$53$ \( 1 + 23846 T + 418195493 T^{2} \)
$59$ \( 1 - 20020 T + 714924299 T^{2} \)
$61$ \( 1 - 32302 T + 844596301 T^{2} \)
$67$ \( 1 - 60972 T + 1350125107 T^{2} \)
$71$ \( 1 - 32648 T + 1804229351 T^{2} \)
$73$ \( 1 + 38774 T + 2073071593 T^{2} \)
$79$ \( 1 + 33360 T + 3077056399 T^{2} \)
$83$ \( 1 + 16716 T + 3939040643 T^{2} \)
$89$ \( 1 + 101370 T + 5584059449 T^{2} \)
$97$ \( 1 + 119038 T + 8587340257 T^{2} \)
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