Properties

Label 3477.1.gk.a
Level $3477$
Weight $1$
Character orbit 3477.gk
Analytic conductor $1.735$
Analytic rank $0$
Dimension $24$
Projective image $D_{90}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3477,1,Mod(161,3477)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3477.161"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3477, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 40, 69])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3477 = 3 \cdot 19 \cdot 61 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3477.gk (of order \(90\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.73524904892\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{90}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{90} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{90}^{44} q^{3} - \zeta_{90}^{28} q^{4} + ( - \zeta_{90}^{7} + \zeta_{90}^{5}) q^{7} - \zeta_{90}^{43} q^{9} + \zeta_{90}^{27} q^{12} + ( - \zeta_{90}^{24} + \zeta_{90}^{11}) q^{13} - \zeta_{90}^{11} q^{16} + \cdots + ( - \zeta_{90}^{32} - \zeta_{90}^{20}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{12} - 3 q^{13} - 3 q^{19} + 3 q^{21} + 3 q^{27} + 3 q^{28} - 6 q^{43} - 3 q^{49} + 3 q^{52} - 12 q^{61} - 3 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 12 q^{75} - 24 q^{79} + 6 q^{91} + 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3477\mathbb{Z}\right)^\times\).

\(n\) \(856\) \(1160\) \(1465\)
\(\chi(n)\) \(\zeta_{90}^{33}\) \(-1\) \(\zeta_{90}^{40}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.374607 + 0.927184i
0.0348995 0.999391i
−0.615661 + 0.788011i
0.0348995 + 0.999391i
0.961262 0.275637i
−0.882948 0.469472i
0.559193 0.829038i
−0.882948 + 0.469472i
0.559193 + 0.829038i
0.848048 + 0.529919i
0.961262 + 0.275637i
0.848048 0.529919i
0.438371 + 0.898794i
0.990268 + 0.139173i
−0.719340 + 0.694658i
−0.374607 0.927184i
−0.997564 0.0697565i
−0.719340 0.694658i
−0.241922 0.970296i
−0.615661 0.788011i
0 −0.374607 0.927184i 0.241922 + 0.970296i 0 0 1.37806 + 1.24081i 0 −0.719340 + 0.694658i 0
263.1 0 0.0348995 + 0.999391i −0.559193 0.829038i 0 0 −0.415570 + 1.95510i 0 −0.997564 + 0.0697565i 0
890.1 0 −0.615661 0.788011i −0.961262 + 0.275637i 0 0 −1.17121 + 1.05456i 0 −0.241922 + 0.970296i 0
899.1 0 0.0348995 0.999391i −0.559193 + 0.829038i 0 0 −0.415570 1.95510i 0 −0.997564 0.0697565i 0
1346.1 0 0.961262 + 0.275637i −0.0348995 + 0.999391i 0 0 −0.548255 + 0.0576239i 0 0.848048 + 0.529919i 0
1448.1 0 −0.882948 + 0.469472i −0.438371 0.898794i 0 0 0.195217 + 0.918425i 0 0.559193 0.829038i 0
1469.1 0 0.559193 + 0.829038i 0.615661 + 0.788011i 0 0 0.674400 1.51473i 0 −0.374607 + 0.927184i 0
1544.1 0 −0.882948 0.469472i −0.438371 + 0.898794i 0 0 0.195217 0.918425i 0 0.559193 + 0.829038i 0
1574.1 0 0.559193 0.829038i 0.615661 0.788011i 0 0 0.674400 + 1.51473i 0 −0.374607 0.927184i 0
1727.1 0 0.848048 0.529919i 0.997564 0.0697565i 0 0 0.220353 1.03668i 0 0.438371 0.898794i 0
1754.1 0 0.961262 0.275637i −0.0348995 0.999391i 0 0 −0.548255 0.0576239i 0 0.848048 0.529919i 0
1814.1 0 0.848048 + 0.529919i 0.997564 + 0.0697565i 0 0 0.220353 + 1.03668i 0 0.438371 + 0.898794i 0
2018.1 0 0.438371 0.898794i −0.990268 + 0.139173i 0 0 −0.731145 + 1.64218i 0 −0.615661 0.788011i 0
2171.1 0 0.990268 0.139173i 0.719340 + 0.694658i 0 0 −0.206852 + 0.186250i 0 0.961262 0.275637i 0
2303.1 0 −0.719340 0.694658i 0.882948 + 0.469472i 0 0 −1.38171 0.145223i 0 0.0348995 + 0.999391i 0
2354.1 0 −0.374607 + 0.927184i 0.241922 0.970296i 0 0 1.37806 1.24081i 0 −0.719340 0.694658i 0
2384.1 0 −0.997564 + 0.0697565i 0.374607 0.927184i 0 0 0.0567450 0.127451i 0 0.990268 0.139173i 0
2627.1 0 −0.719340 + 0.694658i 0.882948 0.469472i 0 0 −1.38171 + 0.145223i 0 0.0348995 0.999391i 0
2669.1 0 −0.241922 + 0.970296i −0.848048 + 0.529919i 0 0 1.92996 + 0.202847i 0 −0.882948 0.469472i 0
2723.1 0 −0.615661 + 0.788011i −0.961262 0.275637i 0 0 −1.17121 1.05456i 0 −0.241922 0.970296i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
1159.de even 90 1 inner
3477.gk odd 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3477.1.gk.a 24
3.b odd 2 1 CM 3477.1.gk.a 24
19.e even 9 1 3477.1.gw.a yes 24
57.l odd 18 1 3477.1.gw.a yes 24
61.k even 30 1 3477.1.gw.a yes 24
183.v odd 30 1 3477.1.gw.a yes 24
1159.de even 90 1 inner 3477.1.gk.a 24
3477.gk odd 90 1 inner 3477.1.gk.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3477.1.gk.a 24 1.a even 1 1 trivial
3477.1.gk.a 24 3.b odd 2 1 CM
3477.1.gk.a 24 1159.de even 90 1 inner
3477.1.gk.a 24 3477.gk odd 90 1 inner
3477.1.gw.a yes 24 19.e even 9 1
3477.1.gw.a yes 24 57.l odd 18 1
3477.1.gw.a yes 24 61.k even 30 1
3477.1.gw.a yes 24 183.v odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3477, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} - T^{21} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} + 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} + 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( T^{24} - 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{24} - 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} + 6 T^{23} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{12} \) Copy content Toggle raw display
$67$ \( T^{24} - 6 T^{23} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{24} + 24 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( T^{24} - 6 T^{21} + \cdots + 1 \) Copy content Toggle raw display
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