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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3249,1,Mod(1015,3249)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3249, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3249.1015");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3249.s (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} - \zeta_{12} q^{3} + q^{5} + q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} - \zeta_{12} q^{3} + q^{5} + q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{5} q^{10} - \zeta_{12}^{4} q^{11} + \zeta_{12} q^{13} - \zeta_{12}^{3} q^{14} - \zeta_{12} q^{15} + \zeta_{12}^{2} q^{16} - \zeta_{12} q^{18} - \zeta_{12}^{5} q^{21} + \zeta_{12}^{3} q^{22} - \zeta_{12}^{4} q^{23} + \zeta_{12}^{4} q^{24} - q^{26} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{3} q^{29} + q^{30} - \zeta_{12}^{5} q^{31} + \zeta_{12} q^{32} + \zeta_{12}^{5} q^{33} + \zeta_{12}^{4} q^{35} - \zeta_{12}^{2} q^{39} - \zeta_{12}^{3} q^{40} - \zeta_{12}^{3} q^{41} + \zeta_{12}^{4} q^{42} - \zeta_{12}^{2} q^{43} + \zeta_{12}^{2} q^{45} + \zeta_{12}^{3} q^{46} + q^{47} - \zeta_{12}^{3} q^{48} + \zeta_{12}^{2} q^{54} - \zeta_{12}^{4} q^{55} + \zeta_{12} q^{56} - \zeta_{12}^{2} q^{58} + \zeta_{12}^{3} q^{59} - q^{61} + \zeta_{12}^{4} q^{62} - q^{63} - q^{64} + \zeta_{12} q^{65} - \zeta_{12}^{4} q^{66} - \zeta_{12} q^{67} + \zeta_{12}^{5} q^{69} - \zeta_{12}^{3} q^{70} - \zeta_{12}^{5} q^{72} + \zeta_{12}^{2} q^{77} + \zeta_{12} q^{78} - \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{80} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{2} q^{82} - \zeta_{12}^{4} q^{83} + \zeta_{12} q^{86} - \zeta_{12}^{4} q^{87} - \zeta_{12} q^{88} - \zeta_{12} q^{90} + \zeta_{12}^{5} q^{91} - q^{93} + \zeta_{12}^{5} q^{94} + \zeta_{12}^{5} q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{23} - 2 q^{24} - 4 q^{26} + 4 q^{30} - 2 q^{35} - 2 q^{39} - 2 q^{42} - 2 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{54} + 2 q^{55} - 2 q^{58} - 4 q^{61} - 2 q^{62} - 4 q^{63} - 4 q^{64} + 2 q^{66} + 2 q^{77} + 2 q^{80} - 2 q^{81} + 2 q^{82} + 2 q^{83} + 2 q^{87} - 4 q^{93} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(362\) \(2890\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1015.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 − 0.500000i 0 1.00000 1.00000 −0.500000 + 0.866025i − 1.00000i 0.500000 + 0.866025i −0.866025 + 0.500000i
1015.2 0.866025 − 0.500000i 0.866025 + 0.500000i 0 1.00000 1.00000 −0.500000 + 0.866025i 1.00000i 0.500000 + 0.866025i 0.866025 − 0.500000i
2596.1 −0.866025 − 0.500000i −0.866025 + 0.500000i 0 1.00000 1.00000 −0.500000 − 0.866025i 1.00000i 0.500000 − 0.866025i −0.866025 − 0.500000i
2596.2 0.866025 + 0.500000i 0.866025 − 0.500000i 0 1.00000 1.00000 −0.500000 − 0.866025i − 1.00000i 0.500000 − 0.866025i 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
171.g even 3 1 inner
171.s odd 6 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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