Properties

Label 45.2.f.a
Level $45$
Weight $2$
Character orbit 45.f
Analytic conductor $0.359$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,2,Mod(8,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.f (of order \(4\), degree \(2\), 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: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} - \zeta_{8}^{2} q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (2 \zeta_{8}^{2} - 2) q^{7} - 3 \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} - \zeta_{8}^{2} q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (2 \zeta_{8}^{2} - 2) q^{7} - 3 \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{2} - 2) q^{10} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{11} + (\zeta_{8}^{2} + 1) q^{13} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{14} + q^{16} + 4 \zeta_{8} q^{17} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{20} + ( - 2 \zeta_{8}^{2} + 2) q^{22} + 4 \zeta_{8}^{3} q^{23} + ( - 3 \zeta_{8}^{2} + 4) q^{25} + (\zeta_{8}^{3} + \zeta_{8}) q^{26} + (2 \zeta_{8}^{2} + 2) q^{28} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{29} - 4 q^{31} - 5 \zeta_{8} q^{32} + 4 \zeta_{8}^{2} q^{34} + ( - 6 \zeta_{8}^{3} - 2 \zeta_{8}) q^{35} + ( - \zeta_{8}^{2} + 1) q^{37} + (6 \zeta_{8}^{2} - 3) q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} + ( - 8 \zeta_{8}^{2} - 8) q^{43} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{44} - 4 q^{46} - 8 \zeta_{8} q^{47} - \zeta_{8}^{2} q^{49} + ( - 3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{50} + ( - \zeta_{8}^{2} + 1) q^{52} + 4 \zeta_{8}^{3} q^{53} + (6 \zeta_{8}^{2} + 2) q^{55} + (6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{56} + (3 \zeta_{8}^{2} + 3) q^{58} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{59} + 8 q^{61} - 4 \zeta_{8} q^{62} - 7 \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} - 3 \zeta_{8}) q^{65} + ( - 4 \zeta_{8}^{2} + 4) q^{67} - 4 \zeta_{8}^{3} q^{68} + ( - 2 \zeta_{8}^{2} + 6) q^{70} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{71} + (\zeta_{8}^{2} + 1) q^{73} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{74} + 8 \zeta_{8} q^{77} + 12 \zeta_{8}^{2} q^{79} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{80} + (\zeta_{8}^{2} - 1) q^{82} + 4 \zeta_{8}^{3} q^{83} + ( - 4 \zeta_{8}^{2} - 8) q^{85} + ( - 8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{86} + ( - 6 \zeta_{8}^{2} - 6) q^{88} + (9 \zeta_{8}^{3} - 9 \zeta_{8}) q^{89} - 4 q^{91} + 4 \zeta_{8} q^{92} - 8 \zeta_{8}^{2} q^{94} + (11 \zeta_{8}^{2} - 11) q^{97} - \zeta_{8}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 8 q^{10} + 4 q^{13} + 4 q^{16} + 8 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{31} + 4 q^{37} - 12 q^{40} - 32 q^{43} - 16 q^{46} + 4 q^{52} + 8 q^{55} + 12 q^{58} + 32 q^{61} + 16 q^{67} + 24 q^{70} + 4 q^{73} - 4 q^{82} - 32 q^{85} - 24 q^{88} - 16 q^{91} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 0 1.00000i 2.12132 0.707107i 0 −2.00000 + 2.00000i −2.12132 + 2.12132i 0 −2.00000 1.00000i
8.2 0.707107 + 0.707107i 0 1.00000i −2.12132 + 0.707107i 0 −2.00000 + 2.00000i 2.12132 2.12132i 0 −2.00000 1.00000i
17.1 −0.707107 + 0.707107i 0 1.00000i 2.12132 + 0.707107i 0 −2.00000 2.00000i −2.12132 2.12132i 0 −2.00000 + 1.00000i
17.2 0.707107 0.707107i 0 1.00000i −2.12132 0.707107i 0 −2.00000 2.00000i 2.12132 + 2.12132i 0 −2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.f.a 4
3.b odd 2 1 inner 45.2.f.a 4
4.b odd 2 1 720.2.w.d 4
5.b even 2 1 225.2.f.a 4
5.c odd 4 1 inner 45.2.f.a 4
5.c odd 4 1 225.2.f.a 4
8.b even 2 1 2880.2.w.b 4
8.d odd 2 1 2880.2.w.k 4
9.c even 3 2 405.2.m.a 8
9.d odd 6 2 405.2.m.a 8
12.b even 2 1 720.2.w.d 4
15.d odd 2 1 225.2.f.a 4
15.e even 4 1 inner 45.2.f.a 4
15.e even 4 1 225.2.f.a 4
20.d odd 2 1 3600.2.w.b 4
20.e even 4 1 720.2.w.d 4
20.e even 4 1 3600.2.w.b 4
24.f even 2 1 2880.2.w.k 4
24.h odd 2 1 2880.2.w.b 4
40.i odd 4 1 2880.2.w.b 4
40.k even 4 1 2880.2.w.k 4
45.k odd 12 2 405.2.m.a 8
45.l even 12 2 405.2.m.a 8
60.h even 2 1 3600.2.w.b 4
60.l odd 4 1 720.2.w.d 4
60.l odd 4 1 3600.2.w.b 4
120.q odd 4 1 2880.2.w.k 4
120.w even 4 1 2880.2.w.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.f.a 4 1.a even 1 1 trivial
45.2.f.a 4 3.b odd 2 1 inner
45.2.f.a 4 5.c odd 4 1 inner
45.2.f.a 4 15.e even 4 1 inner
225.2.f.a 4 5.b even 2 1
225.2.f.a 4 5.c odd 4 1
225.2.f.a 4 15.d odd 2 1
225.2.f.a 4 15.e even 4 1
405.2.m.a 8 9.c even 3 2
405.2.m.a 8 9.d odd 6 2
405.2.m.a 8 45.k odd 12 2
405.2.m.a 8 45.l even 12 2
720.2.w.d 4 4.b odd 2 1
720.2.w.d 4 12.b even 2 1
720.2.w.d 4 20.e even 4 1
720.2.w.d 4 60.l odd 4 1
2880.2.w.b 4 8.b even 2 1
2880.2.w.b 4 24.h odd 2 1
2880.2.w.b 4 40.i odd 4 1
2880.2.w.b 4 120.w even 4 1
2880.2.w.k 4 8.d odd 2 1
2880.2.w.k 4 24.f even 2 1
2880.2.w.k 4 40.k even 4 1
2880.2.w.k 4 120.q odd 4 1
3600.2.w.b 4 20.d odd 2 1
3600.2.w.b 4 20.e even 4 1
3600.2.w.b 4 60.h even 2 1
3600.2.w.b 4 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16 T + 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 22 T + 242)^{2} \) Copy content Toggle raw display
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