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Properties

Label 3332.1.ce.c
Level $3332$
Weight $1$
Character orbit 3332.ce
Analytic conductor $1.663$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(31,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 8, 27]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.31");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.ce (of order \(48\), degree \(16\), 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: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + (\zeta_{48}^{22} - \zeta_{48}^{13}) q^{5} + \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + (\zeta_{48}^{22} - \zeta_{48}^{13}) q^{5} + \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} + (\zeta_{48}^{23} - \zeta_{48}^{14}) q^{10} + ( - \zeta_{48}^{12} + 1) q^{13} + \zeta_{48}^{4} q^{16} - \zeta_{48}^{5} q^{17} - \zeta_{48}^{8} q^{18} + ( - \zeta_{48}^{15} - 1) q^{20} + ( - \zeta_{48}^{20} + \zeta_{48}^{11} - \zeta_{48}^{2}) q^{25} + ( - \zeta_{48}^{13} + \zeta_{48}) q^{26} + (\zeta_{48}^{21} + \zeta_{48}^{6}) q^{29} + \zeta_{48}^{5} q^{32} - \zeta_{48}^{6} q^{34} - \zeta_{48}^{9} q^{36} + ( - \zeta_{48}^{16} - \zeta_{48}^{7}) q^{37} + ( - \zeta_{48}^{16} - \zeta_{48}) q^{40} + ( - \zeta_{48}^{21} - 1) q^{41} + (\zeta_{48}^{20} + \zeta_{48}^{5}) q^{45} + ( - \zeta_{48}^{21} + \zeta_{48}^{12} - \zeta_{48}^{3}) q^{50} + ( - \zeta_{48}^{14} + \zeta_{48}^{2}) q^{52} + ( - \zeta_{48}^{20} - \zeta_{48}^{14}) q^{53} + (\zeta_{48}^{22} + \zeta_{48}^{7}) q^{58} + (\zeta_{48}^{16} + \zeta_{48}^{13}) q^{61} + \zeta_{48}^{6} q^{64} + (\zeta_{48}^{22} - \zeta_{48}^{13} + \zeta_{48}^{10} - \zeta_{48}) q^{65} - \zeta_{48}^{7} q^{68} - \zeta_{48}^{10} q^{72} + ( - \zeta_{48}^{17} + \zeta_{48}^{2}) q^{73} + ( - \zeta_{48}^{17} - \zeta_{48}^{8}) q^{74} + ( - \zeta_{48}^{17} - \zeta_{48}^{2}) q^{80} + \zeta_{48}^{14} q^{81} + ( - \zeta_{48}^{22} - \zeta_{48}) q^{82} + (\zeta_{48}^{18} + \zeta_{48}^{3}) q^{85} + (\zeta_{48}^{21} + \zeta_{48}^{6}) q^{90} + ( - \zeta_{48}^{21} + \zeta_{48}^{6}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{13} - 8 q^{18} - 16 q^{20} + 8 q^{37} + 8 q^{40} - 16 q^{41} - 8 q^{61} - 8 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{48}^{15}\) \(-\zeta_{48}^{16}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.793353 − 0.608761i
−0.793353 + 0.608761i
0.130526 + 0.991445i
0.130526 − 0.991445i
0.991445 + 0.130526i
−0.991445 + 0.130526i
−0.991445 − 0.130526i
−0.130526 + 0.991445i
0.793353 − 0.608761i
0.991445 − 0.130526i
0.608761 + 0.793353i
−0.130526 − 0.991445i
0.793353 + 0.608761i
−0.608761 + 0.793353i
−0.608761 − 0.793353i
0.608761 − 0.793353i
−0.793353 − 0.608761i 0 0.258819 + 0.965926i −0.867580 + 1.75928i 0 0 0.382683 − 0.923880i −0.130526 − 0.991445i 1.75928 − 0.867580i
215.1 −0.793353 + 0.608761i 0 0.258819 − 0.965926i −0.867580 − 1.75928i 0 0 0.382683 + 0.923880i −0.130526 + 0.991445i 1.75928 + 0.867580i
227.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i −0.0255190 + 0.389345i 0 0 −0.382683 − 0.923880i 0.793353 + 0.608761i −0.389345 + 0.0255190i
411.1 0.130526 − 0.991445i 0 −0.965926 − 0.258819i −0.0255190 − 0.389345i 0 0 −0.382683 + 0.923880i 0.793353 − 0.608761i −0.389345 − 0.0255190i
607.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −0.835400 − 0.732626i 0 0 0.923880 + 0.382683i −0.608761 − 0.793353i −0.732626 − 0.835400i
619.1 −0.991445 + 0.130526i 0 0.965926 − 0.258819i −1.09645 − 1.25026i 0 0 −0.923880 + 0.382683i 0.608761 − 0.793353i 1.25026 + 1.09645i
1195.1 −0.991445 − 0.130526i 0 0.965926 + 0.258819i −1.09645 + 1.25026i 0 0 −0.923880 − 0.382683i 0.608761 + 0.793353i 1.25026 − 1.09645i
1391.1 −0.130526 + 0.991445i 0 −0.965926 − 0.258819i 1.95737 − 0.128293i 0 0 0.382683 − 0.923880i −0.793353 + 0.608761i −0.128293 + 1.95737i
1587.1 0.793353 − 0.608761i 0 0.258819 − 0.965926i 0.349942 − 0.172572i 0 0 −0.382683 − 0.923880i 0.130526 − 0.991445i 0.172572 − 0.349942i
1795.1 0.991445 − 0.130526i 0 0.965926 − 0.258819i −0.835400 + 0.732626i 0 0 0.923880 − 0.382683i −0.608761 + 0.793353i −0.732626 + 0.835400i
1979.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i −0.534534 + 1.57469i 0 0 −0.923880 + 0.382683i −0.991445 − 0.130526i −1.57469 + 0.534534i
2187.1 −0.130526 − 0.991445i 0 −0.965926 + 0.258819i 1.95737 + 0.128293i 0 0 0.382683 + 0.923880i −0.793353 − 0.608761i −0.128293 − 1.95737i
2383.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i 0.349942 + 0.172572i 0 0 −0.382683 + 0.923880i 0.130526 + 0.991445i 0.172572 + 0.349942i
2579.1 −0.608761 + 0.793353i 0 −0.258819 − 0.965926i 1.05217 − 0.357164i 0 0 0.923880 + 0.382683i 0.991445 − 0.130526i −0.357164 + 1.05217i
3155.1 −0.608761 − 0.793353i 0 −0.258819 + 0.965926i 1.05217 + 0.357164i 0 0 0.923880 − 0.382683i 0.991445 + 0.130526i −0.357164 − 1.05217i
3167.1 0.608761 − 0.793353i 0 −0.258819 − 0.965926i −0.534534 − 1.57469i 0 0 −0.923880 − 0.382683i −0.991445 + 0.130526i −1.57469 − 0.534534i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3167.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.c even 3 1 inner
28.g odd 6 1 inner
119.p even 16 1 inner
119.s even 48 1 inner
476.bf odd 16 1 inner
476.bk odd 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.ce.c 16
4.b odd 2 1 CM 3332.1.ce.c 16
7.b odd 2 1 3332.1.ce.a 16
7.c even 3 1 3332.1.bn.d yes 8
7.c even 3 1 inner 3332.1.ce.c 16
7.d odd 6 1 3332.1.bn.b ✓ 8
7.d odd 6 1 3332.1.ce.a 16
17.e odd 16 1 3332.1.ce.a 16
28.d even 2 1 3332.1.ce.a 16
28.f even 6 1 3332.1.bn.b ✓ 8
28.f even 6 1 3332.1.ce.a 16
28.g odd 6 1 3332.1.bn.d yes 8
28.g odd 6 1 inner 3332.1.ce.c 16
68.i even 16 1 3332.1.ce.a 16
119.p even 16 1 inner 3332.1.ce.c 16
119.s even 48 1 3332.1.bn.d yes 8
119.s even 48 1 inner 3332.1.ce.c 16
119.t odd 48 1 3332.1.bn.b ✓ 8
119.t odd 48 1 3332.1.ce.a 16
476.bf odd 16 1 inner 3332.1.ce.c 16
476.bk odd 48 1 3332.1.bn.d yes 8
476.bk odd 48 1 inner 3332.1.ce.c 16
476.bm even 48 1 3332.1.bn.b ✓ 8
476.bm even 48 1 3332.1.ce.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.bn.b ✓ 8 7.d odd 6 1
3332.1.bn.b ✓ 8 28.f even 6 1
3332.1.bn.b ✓ 8 119.t odd 48 1
3332.1.bn.b ✓ 8 476.bm even 48 1
3332.1.bn.d yes 8 7.c even 3 1
3332.1.bn.d yes 8 28.g odd 6 1
3332.1.bn.d yes 8 119.s even 48 1
3332.1.bn.d yes 8 476.bk odd 48 1
3332.1.ce.a 16 7.b odd 2 1
3332.1.ce.a 16 7.d odd 6 1
3332.1.ce.a 16 17.e odd 16 1
3332.1.ce.a 16 28.d even 2 1
3332.1.ce.a 16 28.f even 6 1
3332.1.ce.a 16 68.i even 16 1
3332.1.ce.a 16 119.t odd 48 1
3332.1.ce.a 16 476.bm even 48 1
3332.1.ce.c 16 1.a even 1 1 trivial
3332.1.ce.c 16 4.b odd 2 1 CM
3332.1.ce.c 16 7.c even 3 1 inner
3332.1.ce.c 16 28.g odd 6 1 inner
3332.1.ce.c 16 119.p even 16 1 inner
3332.1.ce.c 16 119.s even 48 1 inner
3332.1.ce.c 16 476.bf odd 16 1 inner
3332.1.ce.c 16 476.bk odd 48 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 16 T_{5}^{13} - 2 T_{5}^{12} + 88 T_{5}^{10} + 8 T_{5}^{9} + 2 T_{5}^{8} - 192 T_{5}^{7} + 40 T_{5}^{6} + 48 T_{5}^{5} + 140 T_{5}^{4} - 96 T_{5}^{3} + 40 T_{5}^{2} - 16 T_{5} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 16 T^{13} - 2 T^{12} + 88 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{5} + 2 T^{4} + 12 T^{2} + 8 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 8 T^{15} + 36 T^{14} - 112 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + 28 T^{6} + 56 T^{5} + \cdots + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{6} + 8 T^{5} + 2 T^{4} - 8 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + 8 T^{15} + 36 T^{14} + 112 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 2 T^{12} - 16 T^{11} + 40 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} - 8 T^{5} + 2 T^{4} + 12 T^{2} + 8 T + 2)^{2} \) Copy content Toggle raw display
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