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Properties

Label 3249.1.be.a
Level $3249$
Weight $1$
Character orbit 3249.be
Analytic conductor $1.621$
Analytic rank $0$
Dimension $12$
Projective image $A_{4}$
CM/RM no
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3249,1,Mod(1210,3249)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3249, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3249.1210");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3249.be (of order \(18\), degree \(6\), 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: \(1.62146222604\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.29241.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{36} q^{2} + \zeta_{36} q^{3} - \zeta_{36}^{10} q^{5} - \zeta_{36}^{2} q^{6} + q^{7} + \zeta_{36}^{3} q^{8} + \zeta_{36}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36} q^{2} + \zeta_{36} q^{3} - \zeta_{36}^{10} q^{5} - \zeta_{36}^{2} q^{6} + q^{7} + \zeta_{36}^{3} q^{8} + \zeta_{36}^{2} q^{9} + \zeta_{36}^{11} q^{10} + \zeta_{36}^{6} q^{11} - \zeta_{36}^{5} q^{13} - \zeta_{36} q^{14} - \zeta_{36}^{11} q^{15} - \zeta_{36}^{4} q^{16} - \zeta_{36}^{3} q^{18} + \zeta_{36} q^{21} - \zeta_{36}^{7} q^{22} + \zeta_{36}^{2} q^{23} + \zeta_{36}^{4} q^{24} + \zeta_{36}^{6} q^{26} + \zeta_{36}^{3} q^{27} + \zeta_{36}^{5} q^{29} + \zeta_{36}^{12} q^{30} - \zeta_{36}^{3} q^{31} + \zeta_{36}^{7} q^{33} - \zeta_{36}^{10} q^{35} - \zeta_{36}^{6} q^{39} - \zeta_{36}^{13} q^{40} + \zeta_{36}^{7} q^{41} - \zeta_{36}^{2} q^{42} + \zeta_{36}^{16} q^{43} - \zeta_{36}^{12} q^{45} - \zeta_{36}^{3} q^{46} - \zeta_{36}^{14} q^{47} - \zeta_{36}^{5} q^{48} - \zeta_{36}^{4} q^{54} - \zeta_{36}^{16} q^{55} + \zeta_{36}^{3} q^{56} - \zeta_{36}^{6} q^{58} + \zeta_{36}^{13} q^{59} - \zeta_{36}^{8} q^{61} + \zeta_{36}^{4} q^{62} + \zeta_{36}^{2} q^{63} + \zeta_{36}^{6} q^{64} + \zeta_{36}^{15} q^{65} - \zeta_{36}^{8} q^{66} + \zeta_{36}^{17} q^{67} + \zeta_{36}^{3} q^{69} + \zeta_{36}^{11} q^{70} + \zeta_{36}^{5} q^{72} + \zeta_{36}^{6} q^{77} + \zeta_{36}^{7} q^{78} + \zeta_{36}^{13} q^{79} + \zeta_{36}^{14} q^{80} + \zeta_{36}^{4} q^{81} - \zeta_{36}^{8} q^{82} - q^{83} - \zeta_{36}^{17} q^{86} + \zeta_{36}^{6} q^{87} + \zeta_{36}^{9} q^{88} + \zeta_{36}^{13} q^{90} - \zeta_{36}^{5} q^{91} - \zeta_{36}^{4} q^{93} + \zeta_{36}^{15} q^{94} - \zeta_{36} q^{97} + \zeta_{36}^{8} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} + 6 q^{11} + 6 q^{26} - 6 q^{30} - 6 q^{39} + 6 q^{45} - 6 q^{58} + 6 q^{64} + 6 q^{77} - 12 q^{83} + 6 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3249\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(2890\)
\(\chi(n)\) \(-\zeta_{36}^{6}\) \(\zeta_{36}^{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1210.1
0.342020 − 0.939693i
−0.342020 + 0.939693i
0.984808 − 0.173648i
−0.984808 + 0.173648i
0.642788 + 0.766044i
−0.642788 − 0.766044i
0.984808 + 0.173648i
−0.984808 − 0.173648i
0.342020 + 0.939693i
−0.342020 − 0.939693i
0.642788 − 0.766044i
−0.642788 + 0.766044i
−0.342020 + 0.939693i 0.342020 − 0.939693i 0 −0.939693 − 0.342020i 0.766044 + 0.642788i 1.00000 −0.866025 + 0.500000i −0.766044 − 0.642788i 0.642788 − 0.766044i
1210.2 0.342020 − 0.939693i −0.342020 + 0.939693i 0 −0.939693 − 0.342020i 0.766044 + 0.642788i 1.00000 0.866025 − 0.500000i −0.766044 − 0.642788i −0.642788 + 0.766044i
1345.1 −0.984808 + 0.173648i 0.984808 − 0.173648i 0 0.173648 + 0.984808i −0.939693 + 0.342020i 1.00000 0.866025 − 0.500000i 0.939693 − 0.342020i −0.342020 − 0.939693i
1345.2 0.984808 − 0.173648i −0.984808 + 0.173648i 0 0.173648 + 0.984808i −0.939693 + 0.342020i 1.00000 −0.866025 + 0.500000i 0.939693 − 0.342020i 0.342020 + 0.939693i
1777.1 −0.642788 − 0.766044i 0.642788 + 0.766044i 0 0.766044 − 0.642788i 0.173648 − 0.984808i 1.00000 −0.866025 + 0.500000i −0.173648 + 0.984808i −0.984808 − 0.173648i
1777.2 0.642788 + 0.766044i −0.642788 − 0.766044i 0 0.766044 − 0.642788i 0.173648 − 0.984808i 1.00000 0.866025 − 0.500000i −0.173648 + 0.984808i 0.984808 + 0.173648i
2104.1 −0.984808 − 0.173648i 0.984808 + 0.173648i 0 0.173648 − 0.984808i −0.939693 − 0.342020i 1.00000 0.866025 + 0.500000i 0.939693 + 0.342020i −0.342020 + 0.939693i
2104.2 0.984808 + 0.173648i −0.984808 − 0.173648i 0 0.173648 − 0.984808i −0.939693 − 0.342020i 1.00000 −0.866025 − 0.500000i 0.939693 + 0.342020i 0.342020 − 0.939693i
2473.1 −0.342020 − 0.939693i 0.342020 + 0.939693i 0 −0.939693 + 0.342020i 0.766044 − 0.642788i 1.00000 −0.866025 − 0.500000i −0.766044 + 0.642788i 0.642788 + 0.766044i
2473.2 0.342020 + 0.939693i −0.342020 − 0.939693i 0 −0.939693 + 0.342020i 0.766044 − 0.642788i 1.00000 0.866025 + 0.500000i −0.766044 + 0.642788i −0.642788 − 0.766044i
3004.1 −0.642788 + 0.766044i 0.642788 − 0.766044i 0 0.766044 + 0.642788i 0.173648 + 0.984808i 1.00000 −0.866025 − 0.500000i −0.173648 − 0.984808i −0.984808 + 0.173648i
3004.2 0.642788 − 0.766044i −0.642788 + 0.766044i 0 0.766044 + 0.642788i 0.173648 + 0.984808i 1.00000 0.866025 + 0.500000i −0.173648 − 0.984808i 0.984808 − 0.173648i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1210.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
171.w even 9 3 inner
171.be odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.1.be.a 12
9.c even 3 1 3249.1.bc.a 12
19.b odd 2 1 inner 3249.1.be.a 12
19.c even 3 2 inner 3249.1.be.a 12
19.d odd 6 2 inner 3249.1.be.a 12
19.e even 9 1 171.1.o.a ✓ 4
19.e even 9 1 3249.1.i.a 4
19.e even 9 1 3249.1.s.a 4
19.e even 9 3 3249.1.bc.a 12
19.f odd 18 1 171.1.o.a ✓ 4
19.f odd 18 1 3249.1.i.a 4
19.f odd 18 1 3249.1.s.a 4
19.f odd 18 3 3249.1.bc.a 12
57.j even 18 1 513.1.o.a 4
57.l odd 18 1 513.1.o.a 4
76.k even 18 1 2736.1.bs.a 4
76.l odd 18 1 2736.1.bs.a 4
171.g even 3 1 3249.1.bc.a 12
171.h even 3 1 3249.1.bc.a 12
171.i odd 6 1 3249.1.bc.a 12
171.o odd 6 1 3249.1.bc.a 12
171.s odd 6 1 3249.1.bc.a 12
171.v even 9 1 171.1.o.a ✓ 4
171.v even 9 1 3249.1.i.a 4
171.v even 9 1 3249.1.s.a 4
171.w even 9 1 1539.1.c.d 2
171.w even 9 3 inner 3249.1.be.a 12
171.x even 18 1 513.1.o.a 4
171.z odd 18 1 513.1.o.a 4
171.bc odd 18 1 171.1.o.a ✓ 4
171.bc odd 18 1 3249.1.i.a 4
171.bc odd 18 1 3249.1.s.a 4
171.bd even 18 1 1539.1.c.c 2
171.be odd 18 1 1539.1.c.d 2
171.be odd 18 3 inner 3249.1.be.a 12
171.bf odd 18 1 1539.1.c.c 2
684.cb odd 18 1 2736.1.bs.a 4
684.cc even 18 1 2736.1.bs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.o.a ✓ 4 19.e even 9 1
171.1.o.a ✓ 4 19.f odd 18 1
171.1.o.a ✓ 4 171.v even 9 1
171.1.o.a ✓ 4 171.bc odd 18 1
513.1.o.a 4 57.j even 18 1
513.1.o.a 4 57.l odd 18 1
513.1.o.a 4 171.x even 18 1
513.1.o.a 4 171.z odd 18 1
1539.1.c.c 2 171.bd even 18 1
1539.1.c.c 2 171.bf odd 18 1
1539.1.c.d 2 171.w even 9 1
1539.1.c.d 2 171.be odd 18 1
2736.1.bs.a 4 76.k even 18 1
2736.1.bs.a 4 76.l odd 18 1
2736.1.bs.a 4 684.cb odd 18 1
2736.1.bs.a 4 684.cc even 18 1
3249.1.i.a 4 19.e even 9 1
3249.1.i.a 4 19.f odd 18 1
3249.1.i.a 4 171.v even 9 1
3249.1.i.a 4 171.bc odd 18 1
3249.1.s.a 4 19.e even 9 1
3249.1.s.a 4 19.f odd 18 1
3249.1.s.a 4 171.v even 9 1
3249.1.s.a 4 171.bc odd 18 1
3249.1.bc.a 12 9.c even 3 1
3249.1.bc.a 12 19.e even 9 3
3249.1.bc.a 12 19.f odd 18 3
3249.1.bc.a 12 171.g even 3 1
3249.1.bc.a 12 171.h even 3 1
3249.1.bc.a 12 171.i odd 6 1
3249.1.bc.a 12 171.o odd 6 1
3249.1.bc.a 12 171.s odd 6 1
3249.1.be.a 12 1.a even 1 1 trivial
3249.1.be.a 12 19.b odd 2 1 inner
3249.1.be.a 12 19.c even 3 2 inner
3249.1.be.a 12 19.d odd 6 2 inner
3249.1.be.a 12 171.w even 9 3 inner
3249.1.be.a 12 171.be odd 18 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3249, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$43$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$83$ \( (T + 1)^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
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