Properties

Label 45.4.a.c
Level $45$
Weight $4$
Character orbit 45.a
Self dual yes
Analytic conductor $2.655$
Analytic rank $1$
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] = [45,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(2.65508595026\)
Analytic rank: \(1\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} - 5 q^{5} - 24 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} - 5 q^{5} - 24 q^{7} + 15 q^{8} + 5 q^{10} - 52 q^{11} + 22 q^{13} + 24 q^{14} + 41 q^{16} + 14 q^{17} - 20 q^{19} + 35 q^{20} + 52 q^{22} + 168 q^{23} + 25 q^{25} - 22 q^{26} + 168 q^{28} - 230 q^{29} - 288 q^{31} - 161 q^{32} - 14 q^{34} + 120 q^{35} - 34 q^{37} + 20 q^{38} - 75 q^{40} - 122 q^{41} - 188 q^{43} + 364 q^{44} - 168 q^{46} - 256 q^{47} + 233 q^{49} - 25 q^{50} - 154 q^{52} + 338 q^{53} + 260 q^{55} - 360 q^{56} + 230 q^{58} - 100 q^{59} + 742 q^{61} + 288 q^{62} - 167 q^{64} - 110 q^{65} - 84 q^{67} - 98 q^{68} - 120 q^{70} + 328 q^{71} - 38 q^{73} + 34 q^{74} + 140 q^{76} + 1248 q^{77} - 240 q^{79} - 205 q^{80} + 122 q^{82} - 1212 q^{83} - 70 q^{85} + 188 q^{86} - 780 q^{88} - 330 q^{89} - 528 q^{91} - 1176 q^{92} + 256 q^{94} + 100 q^{95} + 866 q^{97} - 233 q^{98}+O(q^{100}) \) 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
−1.00000 0 −7.00000 −5.00000 0 −24.0000 15.0000 0 5.00000
\(n\): 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.c 1
3.b odd 2 1 15.4.a.a 1
4.b odd 2 1 720.4.a.n 1
5.b even 2 1 225.4.a.f 1
5.c odd 4 2 225.4.b.e 2
7.b odd 2 1 2205.4.a.l 1
9.c even 3 2 405.4.e.i 2
9.d odd 6 2 405.4.e.g 2
12.b even 2 1 240.4.a.e 1
15.d odd 2 1 75.4.a.b 1
15.e even 4 2 75.4.b.b 2
21.c even 2 1 735.4.a.e 1
24.f even 2 1 960.4.a.ba 1
24.h odd 2 1 960.4.a.b 1
33.d even 2 1 1815.4.a.e 1
60.h even 2 1 1200.4.a.t 1
60.l odd 4 2 1200.4.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 3.b odd 2 1
45.4.a.c 1 1.a even 1 1 trivial
75.4.a.b 1 15.d odd 2 1
75.4.b.b 2 15.e even 4 2
225.4.a.f 1 5.b even 2 1
225.4.b.e 2 5.c odd 4 2
240.4.a.e 1 12.b even 2 1
405.4.e.g 2 9.d odd 6 2
405.4.e.i 2 9.c even 3 2
720.4.a.n 1 4.b odd 2 1
735.4.a.e 1 21.c even 2 1
960.4.a.b 1 24.h odd 2 1
960.4.a.ba 1 24.f even 2 1
1200.4.a.t 1 60.h even 2 1
1200.4.f.b 2 60.l odd 4 2
1815.4.a.e 1 33.d even 2 1
2205.4.a.l 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 24 \) Copy content Toggle raw display
$11$ \( T + 52 \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T - 14 \) Copy content Toggle raw display
$19$ \( T + 20 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T + 230 \) Copy content Toggle raw display
$31$ \( T + 288 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T + 122 \) Copy content Toggle raw display
$43$ \( T + 188 \) Copy content Toggle raw display
$47$ \( T + 256 \) Copy content Toggle raw display
$53$ \( T - 338 \) Copy content Toggle raw display
$59$ \( T + 100 \) Copy content Toggle raw display
$61$ \( T - 742 \) Copy content Toggle raw display
$67$ \( T + 84 \) Copy content Toggle raw display
$71$ \( T - 328 \) Copy content Toggle raw display
$73$ \( T + 38 \) Copy content Toggle raw display
$79$ \( T + 240 \) Copy content Toggle raw display
$83$ \( T + 1212 \) Copy content Toggle raw display
$89$ \( T + 330 \) Copy content Toggle raw display
$97$ \( T - 866 \) Copy content Toggle raw display
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