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Properties

Label 3267.1.be.a
Level $3267$
Weight $1$
Character orbit 3267.be
Analytic conductor $1.630$
Analytic rank $0$
Dimension $24$
Projective image $D_{18}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3267,1,Mod(245,3267)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3267, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([5, 72]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3267.245");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.be (of order \(90\), degree \(24\), not 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: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
comment: defining polynomial
 
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: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9} + \zeta_{90}^{25} q^{12} + (\zeta_{90}^{24} - \zeta_{90}^{14}) q^{15} + \zeta_{90}^{28} q^{16} + (\zeta_{90}^{27} - \zeta_{90}^{17}) q^{20} + ( - \zeta_{90}^{35} - \zeta_{90}^{20}) q^{23} + (\zeta_{90}^{26} - \zeta_{90}^{16} + \zeta_{90}^{6}) q^{25} + \zeta_{90}^{33} q^{27} + ( - \zeta_{90}^{12} + \zeta_{90}^{7}) q^{31} + \zeta_{90}^{36} q^{36} + (\zeta_{90}^{29} + \zeta_{90}^{19}) q^{37} + (\zeta_{90}^{35} - \zeta_{90}^{25}) q^{45} + ( - \zeta_{90}^{41} + \zeta_{90}^{21}) q^{47} + \zeta_{90}^{39} q^{48} - \zeta_{90}^{8} q^{49} + ( - \zeta_{90}^{7} - \zeta_{90}^{2}) q^{53} + ( - \zeta_{90}^{24} - \zeta_{90}^{19}) q^{59} + (\zeta_{90}^{38} - \zeta_{90}^{28}) q^{60} + \zeta_{90}^{42} q^{64} + ( - \zeta_{90}^{40} - 1) q^{67} + ( - \zeta_{90}^{31} + \zeta_{90}) q^{69} + ( - \zeta_{90}^{43} + \zeta_{90}^{23}) q^{71} + (\zeta_{90}^{37} - \zeta_{90}^{27} + \zeta_{90}^{17}) q^{75} + (\zeta_{90}^{41} - \zeta_{90}^{31}) q^{80} + \zeta_{90}^{44} q^{81} + ( - \zeta_{90}^{15} - 1) q^{89} + ( - \zeta_{90}^{34} + \zeta_{90}^{4}) q^{92} + ( - \zeta_{90}^{23} + \zeta_{90}^{18}) q^{93} + ( - \zeta_{90}^{27} + \zeta_{90}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(-\zeta_{90}^{9}\) \(\zeta_{90}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−0.374607 + 0.927184i
0.848048 + 0.529919i
0.438371 + 0.898794i
0.961262 + 0.275637i
0.0348995 + 0.999391i
0.961262 − 0.275637i
0.438371 − 0.898794i
0.990268 + 0.139173i
−0.615661 − 0.788011i
−0.882948 + 0.469472i
−0.997564 − 0.0697565i
−0.719340 + 0.694658i
−0.882948 − 0.469472i
−0.241922 + 0.970296i
0.559193 + 0.829038i
−0.615661 + 0.788011i
0.990268 − 0.139173i
0.0348995 − 0.999391i
0.559193 − 0.829038i
−0.241922 − 0.970296i
0 0.882948 − 0.469472i −0.615661 + 0.788011i −0.0477162 − 0.682374i 0 0 0 0.559193 − 0.829038i 0
608.1 0 −0.990268 + 0.139173i 0.0348995 + 0.999391i −0.663721 + 0.165484i 0 0 0 0.961262 − 0.275637i 0
614.1 0 −0.961262 + 0.275637i −0.997564 + 0.0697565i −0.603541 − 1.13510i 0 0 0 0.848048 − 0.529919i 0
632.1 0 0.997564 − 0.0697565i −0.719340 − 0.694658i 1.55208 + 1.21262i 0 0 0 0.990268 − 0.139173i 0
686.1 0 0.374607 + 0.927184i −0.882948 + 0.469472i −0.542900 − 1.89332i 0 0 0 −0.719340 + 0.694658i 0
977.1 0 0.997564 + 0.0697565i −0.719340 + 0.694658i 1.55208 − 1.21262i 0 0 0 0.990268 + 0.139173i 0
995.1 0 −0.961262 − 0.275637i −0.997564 − 0.0697565i −0.603541 + 1.13510i 0 0 0 0.848048 + 0.529919i 0
1049.1 0 −0.0348995 − 0.999391i −0.374607 + 0.927184i 1.15547 − 0.563559i 0 0 0 −0.997564 + 0.0697565i 0
1334.1 0 −0.848048 − 0.529919i 0.990268 + 0.139173i 1.63289 − 1.10140i 0 0 0 0.438371 + 0.898794i 0
1697.1 0 0.615661 + 0.788011i 0.848048 − 0.529919i 0.893036 + 0.924765i 0 0 0 −0.241922 + 0.970296i 0
1703.1 0 0.719340 + 0.694658i 0.559193 + 0.829038i −0.362486 + 0.580099i 0 0 0 0.0348995 + 0.999391i 0
1721.1 0 −0.559193 − 0.829038i −0.241922 + 0.970296i −0.178917 + 1.27306i 0 0 0 −0.374607 + 0.927184i 0
1775.1 0 0.615661 − 0.788011i 0.848048 + 0.529919i 0.893036 − 0.924765i 0 0 0 −0.241922 − 0.970296i 0
2066.1 0 −0.438371 − 0.898794i 0.961262 + 0.275637i 0.634231 + 0.256246i 0 0 0 −0.615661 + 0.788011i 0
2084.1 0 0.241922 + 0.970296i 0.438371 + 0.898794i −1.96842 + 0.0687386i 0 0 0 −0.882948 + 0.469472i 0
2138.1 0 −0.848048 + 0.529919i 0.990268 − 0.139173i 1.63289 + 1.10140i 0 0 0 0.438371 − 0.898794i 0
2423.1 0 −0.0348995 + 0.999391i −0.374607 − 0.927184i 1.15547 + 0.563559i 0 0 0 −0.997564 − 0.0697565i 0
2786.1 0 0.374607 − 0.927184i −0.882948 − 0.469472i −0.542900 + 1.89332i 0 0 0 −0.719340 − 0.694658i 0
2792.1 0 0.241922 − 0.970296i 0.438371 − 0.898794i −1.96842 − 0.0687386i 0 0 0 −0.882948 − 0.469472i 0
2810.1 0 −0.438371 + 0.898794i 0.961262 − 0.275637i 0.634231 − 0.256246i 0 0 0 −0.615661 − 0.788011i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3227.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
11.c even 5 3 inner
11.d odd 10 3 inner
27.f odd 18 1 inner
297.o even 18 1 inner
297.v odd 90 3 inner
297.x even 90 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.be.a 24
11.b odd 2 1 CM 3267.1.be.a 24
11.c even 5 1 3267.1.q.a ✓ 6
11.c even 5 3 inner 3267.1.be.a 24
11.d odd 10 1 3267.1.q.a ✓ 6
11.d odd 10 3 inner 3267.1.be.a 24
27.f odd 18 1 inner 3267.1.be.a 24
297.o even 18 1 inner 3267.1.be.a 24
297.v odd 90 1 3267.1.q.a ✓ 6
297.v odd 90 3 inner 3267.1.be.a 24
297.x even 90 1 3267.1.q.a ✓ 6
297.x even 90 3 inner 3267.1.be.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.q.a ✓ 6 11.c even 5 1
3267.1.q.a ✓ 6 11.d odd 10 1
3267.1.q.a ✓ 6 297.v odd 90 1
3267.1.q.a ✓ 6 297.x even 90 1
3267.1.be.a 24 1.a even 1 1 trivial
3267.1.be.a 24 11.b odd 2 1 CM
3267.1.be.a 24 11.c even 5 3 inner
3267.1.be.a 24 11.d odd 10 3 inner
3267.1.be.a 24 27.f odd 18 1 inner
3267.1.be.a 24 297.o even 18 1 inner
3267.1.be.a 24 297.v odd 90 3 inner
3267.1.be.a 24 297.x even 90 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3267, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} + T^{21} - T^{15} - T^{12} - T^{9} + T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} - 3 T^{23} + 3 T^{22} + 3 T^{21} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( (T^{6} + 9 T^{3} + 27)^{4} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( T^{24} + 3 T^{23} + 3 T^{22} - T^{21} - 9 T^{20} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{24} - 3 T^{22} + 2 T^{21} + 6 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} + 3 T^{23} + 3 T^{22} - 3 T^{21} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{24} - 6 T^{22} + 27 T^{20} - 111 T^{18} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{24} + 3 T^{23} + 3 T^{22} - 3 T^{21} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{24} \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{24} + 3 T^{22} - 36 T^{19} - 30 T^{18} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( T^{24} \) Copy content Toggle raw display
$79$ \( T^{24} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{12} \) Copy content Toggle raw display
$97$ \( T^{24} + 6 T^{23} + 21 T^{22} + 55 T^{21} + \cdots + 1 \) Copy content Toggle raw display
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