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Properties

Label 45.4.b.b
Level $45$
Weight $4$
Character orbit 45.b
Analytic conductor $2.655$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,4,Mod(19,45)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8} + (\beta_{3} + 4 \beta_{2} + 10 \beta_1 + 3) q^{10} + ( - 2 \beta_{3} + 22) q^{11} + ( - 8 \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{3} - 58) q^{14} + ( - \beta_{3} + 13) q^{16} + ( - 6 \beta_{2} - 16 \beta_1) q^{17} + ( - 8 \beta_{3} - 24) q^{19} + ( - 6 \beta_{3} - 4 \beta_{2} + \cdots + 102) q^{20}+ \cdots + ( - 48 \beta_{2} - 67 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 32 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 65\beta_1 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 3.70156i
− 2.70156i
2.70156i
3.70156i
− 4.70156i 0 −14.1047 −11.1047 − 1.29844i 0 − 16.2094i 28.7016i 0 −6.10469 + 52.2094i
19.2 − 1.70156i 0 5.10469 8.10469 + 7.70156i 0 − 22.2094i − 22.2984i 0 13.1047 − 13.7906i
19.3 1.70156i 0 5.10469 8.10469 − 7.70156i 0 22.2094i 22.2984i 0 13.1047 + 13.7906i
19.4 4.70156i 0 −14.1047 −11.1047 + 1.29844i 0 16.2094i − 28.7016i 0 −6.10469 − 52.2094i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.b.b 4
3.b odd 2 1 15.4.b.a ✓ 4
4.b odd 2 1 720.4.f.j 4
5.b even 2 1 inner 45.4.b.b 4
5.c odd 4 1 225.4.a.i 2
5.c odd 4 1 225.4.a.o 2
12.b even 2 1 240.4.f.f 4
15.d odd 2 1 15.4.b.a ✓ 4
15.e even 4 1 75.4.a.c 2
15.e even 4 1 75.4.a.f 2
20.d odd 2 1 720.4.f.j 4
24.f even 2 1 960.4.f.p 4
24.h odd 2 1 960.4.f.q 4
60.h even 2 1 240.4.f.f 4
60.l odd 4 1 1200.4.a.bn 2
60.l odd 4 1 1200.4.a.bt 2
120.i odd 2 1 960.4.f.q 4
120.m even 2 1 960.4.f.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a ✓ 4 3.b odd 2 1
15.4.b.a ✓ 4 15.d odd 2 1
45.4.b.b 4 1.a even 1 1 trivial
45.4.b.b 4 5.b even 2 1 inner
75.4.a.c 2 15.e even 4 1
75.4.a.f 2 15.e even 4 1
225.4.a.i 2 5.c odd 4 1
225.4.a.o 2 5.c odd 4 1
240.4.f.f 4 12.b even 2 1
240.4.f.f 4 60.h even 2 1
720.4.f.j 4 4.b odd 2 1
720.4.f.j 4 20.d odd 2 1
960.4.f.p 4 24.f even 2 1
960.4.f.p 4 120.m even 2 1
960.4.f.q 4 24.h odd 2 1
960.4.f.q 4 120.i odd 2 1
1200.4.a.bn 2 60.l odd 4 1
1200.4.a.bt 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 756 T^{2} + 129600 \) Copy content Toggle raw display
$11$ \( (T^{2} - 42 T + 72)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3780 T^{2} + 1327104 \) Copy content Toggle raw display
$17$ \( T^{4} + 7252 T^{2} + 2483776 \) Copy content Toggle raw display
$19$ \( (T^{2} + 56 T - 5120)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2464 T^{2} + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 318 T + 7200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 52 T - 800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 106596 T^{2} + 41990400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 408 T + 40140)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 8256266496 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 6186766336 \) Copy content Toggle raw display
$53$ \( T^{4} + 218644 T^{2} + 813390400 \) Copy content Toggle raw display
$59$ \( (T^{2} + 186 T + 8280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 340 T - 65564)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 9419867136 \) Copy content Toggle raw display
$71$ \( (T^{2} - 36 T - 331776)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 104976000000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 380 T - 886400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 40558737664 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1116 T + 98820)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 196199387136 \) Copy content Toggle raw display
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