Properties

Label 3267.1.bf.a
Level $3267$
Weight $1$
Character orbit 3267.bf
Analytic conductor $1.630$
Analytic rank $0$
Dimension $24$
Projective image $D_{9}$
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(40,3267)]
 
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([40, 63]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3267.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.bf (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: no (minimal twist has level 297)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{90}^{34} q^{3} - \zeta_{90}^{31} q^{4} + (\zeta_{90}^{42} + \zeta_{90}^{32}) q^{5} - \zeta_{90}^{23} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{90}^{34} q^{3} - \zeta_{90}^{31} q^{4} + (\zeta_{90}^{42} + \zeta_{90}^{32}) q^{5} - \zeta_{90}^{23} q^{9} + \zeta_{90}^{20} q^{12} + ( - \zeta_{90}^{31} - \zeta_{90}^{21}) q^{15} - \zeta_{90}^{17} q^{16} + (\zeta_{90}^{28} + \zeta_{90}^{18}) q^{20} - \zeta_{90}^{40} q^{23} + ( - \zeta_{90}^{39} + \cdots - \zeta_{90}^{19}) q^{25} + \cdots + ( - \zeta_{90}^{43} + \zeta_{90}^{18}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 3 q^{5} + 3 q^{15} - 6 q^{20} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} + 3 q^{64} + 24 q^{67} - 6 q^{75} + 12 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(-\zeta_{90}^{36}\) \(\zeta_{90}^{40}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−0.374607 + 0.927184i
−0.997564 0.0697565i
−0.719340 + 0.694658i
0.848048 + 0.529919i
−0.241922 + 0.970296i
0.559193 + 0.829038i
0.0348995 + 0.999391i
0.990268 + 0.139173i
−0.615661 0.788011i
0.559193 0.829038i
−0.241922 0.970296i
−0.882948 + 0.469472i
−0.719340 0.694658i
−0.997564 + 0.0697565i
−0.882948 0.469472i
−0.615661 + 0.788011i
0.990268 0.139173i
0.438371 + 0.898794i
0.961262 + 0.275637i
0.0348995 0.999391i
0 −0.882948 0.469472i −0.615661 0.788011i 1.87481 + 0.131099i 0 0 0 0.559193 + 0.829038i 0
94.1 0 −0.719340 + 0.694658i 0.559193 0.829038i −1.59381 + 0.995922i 0 0 0 0.0348995 0.999391i 0
112.1 0 0.559193 0.829038i −0.241922 0.970296i 1.51718 0.213226i 0 0 0 −0.374607 0.927184i 0
403.1 0 0.990268 + 0.139173i 0.0348995 0.999391i 0.454664 1.82356i 0 0 0 0.961262 + 0.275637i 0
457.1 0 0.438371 0.898794i 0.961262 0.275637i 0.704030 + 1.74254i 0 0 0 −0.615661 0.788011i 0
475.1 0 −0.241922 + 0.970296i 0.438371 0.898794i 0.0121205 0.347085i 0 0 0 −0.882948 0.469472i 0
481.1 0 −0.374607 + 0.927184i −0.882948 0.469472i 0.333843 + 0.0957278i 0 0 0 −0.719340 0.694658i 0
844.1 0 0.0348995 0.999391i −0.374607 0.927184i 0.671624 1.37703i 0 0 0 −0.997564 0.0697565i 0
1129.1 0 0.848048 0.529919i 0.990268 0.139173i 0.194206 0.287922i 0 0 0 0.438371 0.898794i 0
1183.1 0 −0.241922 0.970296i 0.438371 + 0.898794i 0.0121205 + 0.347085i 0 0 0 −0.882948 + 0.469472i 0
1201.1 0 0.438371 + 0.898794i 0.961262 + 0.275637i 0.704030 1.74254i 0 0 0 −0.615661 + 0.788011i 0
1492.1 0 −0.615661 + 0.788011i 0.848048 + 0.529919i −1.10209 1.06428i 0 0 0 −0.241922 0.970296i 0
1546.1 0 0.559193 + 0.829038i −0.241922 + 0.970296i 1.51718 + 0.213226i 0 0 0 −0.374607 + 0.927184i 0
1564.1 0 −0.719340 0.694658i 0.559193 + 0.829038i −1.59381 0.995922i 0 0 0 0.0348995 + 0.999391i 0
1570.1 0 −0.615661 0.788011i 0.848048 0.529919i −1.10209 + 1.06428i 0 0 0 −0.241922 + 0.970296i 0
1933.1 0 0.848048 + 0.529919i 0.990268 + 0.139173i 0.194206 + 0.287922i 0 0 0 0.438371 + 0.898794i 0
2218.1 0 0.0348995 + 0.999391i −0.374607 + 0.927184i 0.671624 + 1.37703i 0 0 0 −0.997564 + 0.0697565i 0
2272.1 0 0.961262 + 0.275637i −0.997564 0.0697565i −1.35275 0.719272i 0 0 0 0.848048 + 0.529919i 0
2290.1 0 −0.997564 0.0697565i −0.719340 + 0.694658i −0.213817 0.273673i 0 0 0 0.990268 + 0.139173i 0
2581.1 0 −0.374607 0.927184i −0.882948 + 0.469472i 0.333843 0.0957278i 0 0 0 −0.719340 + 0.694658i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.1
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.e even 9 1 inner
297.q odd 18 1 inner
297.u even 45 3 inner
297.w odd 90 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.bf.a 24
11.b odd 2 1 CM 3267.1.bf.a 24
11.c even 5 1 297.1.q.a 6
11.c even 5 3 inner 3267.1.bf.a 24
11.d odd 10 1 297.1.q.a 6
11.d odd 10 3 inner 3267.1.bf.a 24
27.e even 9 1 inner 3267.1.bf.a 24
33.f even 10 1 891.1.q.a 6
33.h odd 10 1 891.1.q.a 6
99.m even 15 1 2673.1.q.b 6
99.m even 15 1 2673.1.q.d 6
99.n odd 30 1 2673.1.q.a 6
99.n odd 30 1 2673.1.q.c 6
99.o odd 30 1 2673.1.q.b 6
99.o odd 30 1 2673.1.q.d 6
99.p even 30 1 2673.1.q.a 6
99.p even 30 1 2673.1.q.c 6
297.q odd 18 1 inner 3267.1.bf.a 24
297.u even 45 1 297.1.q.a 6
297.u even 45 1 2673.1.q.b 6
297.u even 45 1 2673.1.q.d 6
297.u even 45 3 inner 3267.1.bf.a 24
297.v odd 90 1 891.1.q.a 6
297.v odd 90 1 2673.1.q.a 6
297.v odd 90 1 2673.1.q.c 6
297.w odd 90 1 297.1.q.a 6
297.w odd 90 1 2673.1.q.b 6
297.w odd 90 1 2673.1.q.d 6
297.w odd 90 3 inner 3267.1.bf.a 24
297.x even 90 1 891.1.q.a 6
297.x even 90 1 2673.1.q.a 6
297.x even 90 1 2673.1.q.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.1.q.a 6 11.c even 5 1
297.1.q.a 6 11.d odd 10 1
297.1.q.a 6 297.u even 45 1
297.1.q.a 6 297.w odd 90 1
891.1.q.a 6 33.f even 10 1
891.1.q.a 6 33.h odd 10 1
891.1.q.a 6 297.v odd 90 1
891.1.q.a 6 297.x even 90 1
2673.1.q.a 6 99.n odd 30 1
2673.1.q.a 6 99.p even 30 1
2673.1.q.a 6 297.v odd 90 1
2673.1.q.a 6 297.x even 90 1
2673.1.q.b 6 99.m even 15 1
2673.1.q.b 6 99.o odd 30 1
2673.1.q.b 6 297.u even 45 1
2673.1.q.b 6 297.w odd 90 1
2673.1.q.c 6 99.n odd 30 1
2673.1.q.c 6 99.p even 30 1
2673.1.q.c 6 297.v odd 90 1
2673.1.q.c 6 297.x even 90 1
2673.1.q.d 6 99.m even 15 1
2673.1.q.d 6 99.o odd 30 1
2673.1.q.d 6 297.u even 45 1
2673.1.q.d 6 297.w odd 90 1
3267.1.bf.a 24 1.a even 1 1 trivial
3267.1.bf.a 24 11.b odd 2 1 CM
3267.1.bf.a 24 11.c even 5 3 inner
3267.1.bf.a 24 11.d odd 10 3 inner
3267.1.bf.a 24 27.e even 9 1 inner
3267.1.bf.a 24 297.q odd 18 1 inner
3267.1.bf.a 24 297.u even 45 3 inner
3267.1.bf.a 24 297.w odd 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
$3$ \( T^{24} - T^{21} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{24} - 3 T^{23} + \cdots + 1 \) 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
$23$ \( (T^{6} - T^{3} + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{24} - 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{12} + 3 T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{24} \) Copy content Toggle raw display
$67$ \( (T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{24} - 3 T^{22} + \cdots + 1 \) 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
$89$ \( (T^{2} - T + 1)^{12} \) Copy content Toggle raw display
$97$ \( T^{24} + 6 T^{23} + \cdots + 1 \) Copy content Toggle raw display
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