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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,6,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

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 45.6.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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 192 \) Copy content Toggle raw display
$11$ \( T - 148 \) Copy content Toggle raw display
$13$ \( T - 286 \) Copy content Toggle raw display
$17$ \( T - 1678 \) Copy content Toggle raw display
$19$ \( T - 1060 \) Copy content Toggle raw display
$23$ \( T + 2976 \) Copy content Toggle raw display
$29$ \( T - 3410 \) Copy content Toggle raw display
$31$ \( T + 2448 \) Copy content Toggle raw display
$37$ \( T - 182 \) Copy content Toggle raw display
$41$ \( T - 9398 \) Copy content Toggle raw display
$43$ \( T + 1244 \) Copy content Toggle raw display
$47$ \( T - 12088 \) Copy content Toggle raw display
$53$ \( T + 23846 \) Copy content Toggle raw display
$59$ \( T - 20020 \) Copy content Toggle raw display
$61$ \( T - 32302 \) Copy content Toggle raw display
$67$ \( T - 60972 \) Copy content Toggle raw display
$71$ \( T - 32648 \) Copy content Toggle raw display
$73$ \( T + 38774 \) Copy content Toggle raw display
$79$ \( T + 33360 \) Copy content Toggle raw display
$83$ \( T + 16716 \) Copy content Toggle raw display
$89$ \( T + 101370 \) Copy content Toggle raw display
$97$ \( T + 119038 \) Copy content Toggle raw display
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