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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(979,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.979");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.bn (of order \(16\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) 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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + ( - \zeta_{16}^{5} + \zeta_{16}^{2}) q^{5} - \zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + ( - \zeta_{16}^{5} + \zeta_{16}^{2}) q^{5} - \zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} + (\zeta_{16}^{7} + \zeta_{16}^{2}) q^{10} + (\zeta_{16}^{4} + 1) q^{13} + \zeta_{16}^{4} q^{16} + \zeta_{16}^{5} q^{17} - q^{18} + (\zeta_{16}^{7} - \zeta_{16}^{4}) q^{20} + ( - \zeta_{16}^{7} + \zeta_{16}^{4} - \zeta_{16}^{2}) q^{25} + (\zeta_{16}^{5} - \zeta_{16}) q^{26} + (\zeta_{16}^{6} - \zeta_{16}) q^{29} - \zeta_{16} q^{32} - \zeta_{16}^{2} q^{34} - \zeta_{16}^{5} q^{36} + ( - \zeta_{16}^{3} + 1) q^{37} + ( - \zeta_{16}^{4} + \zeta_{16}) q^{40} + ( - \zeta_{16}^{5} + \zeta_{16}^{4}) q^{41} + (\zeta_{16}^{5} + 1) q^{45} + ( - \zeta_{16}^{7} + \zeta_{16}^{4} - \zeta_{16}) q^{50} + ( - \zeta_{16}^{6} - \zeta_{16}^{2}) q^{52} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{53} + ( - \zeta_{16}^{6} - \zeta_{16}^{3}) q^{58} + (\zeta_{16}^{5} + \zeta_{16}^{4}) q^{61} - \zeta_{16}^{6} q^{64} + (\zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{2} + \zeta_{16}) q^{65} - \zeta_{16}^{7} q^{68} + \zeta_{16}^{2} q^{72} + (\zeta_{16}^{6} - \zeta_{16}) q^{73} + (\zeta_{16}^{5} + 1) q^{74} + (\zeta_{16}^{6} + \zeta_{16}) q^{80} + \zeta_{16}^{6} q^{81} + (\zeta_{16}^{2} - \zeta_{16}) q^{82} + (\zeta_{16}^{7} + \zeta_{16}^{2}) q^{85} - \zeta_{16}^{6} q^{89} + (\zeta_{16}^{5} - \zeta_{16}^{2}) q^{90} + (\zeta_{16}^{5} + \zeta_{16}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} - 8 q^{18} + 8 q^{37} + 8 q^{45} + 8 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{16}^{3}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
979.1
−0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 − 0.382683i
−0.382683 − 0.923880i
0.382683 − 0.923880i
0.923880 + 0.382683i
0.382683 − 0.923880i 0 −0.707107 − 0.707107i 0.324423 + 1.63099i 0 0 −0.923880 + 0.382683i −0.382683 − 0.923880i 1.63099 + 0.324423i
1371.1 −0.923880 − 0.382683i 0 0.707107 + 0.707107i 0.216773 − 0.324423i 0 0 −0.382683 − 0.923880i 0.923880 − 0.382683i −0.324423 + 0.216773i
1567.1 0.923880 − 0.382683i 0 0.707107 − 0.707107i −1.63099 + 1.08979i 0 0 0.382683 − 0.923880i −0.923880 − 0.382683i −1.08979 + 1.63099i
1763.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0.324423 − 1.63099i 0 0 −0.923880 − 0.382683i −0.382683 + 0.923880i 1.63099 − 0.324423i
2351.1 −0.382683 − 0.923880i 0 −0.707107 + 0.707107i 1.08979 + 0.216773i 0 0 0.923880 + 0.382683i 0.382683 − 0.923880i −0.216773 − 1.08979i
2547.1 −0.923880 + 0.382683i 0 0.707107 − 0.707107i 0.216773 + 0.324423i 0 0 −0.382683 + 0.923880i 0.923880 + 0.382683i −0.324423 − 0.216773i
2743.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i −1.63099 − 1.08979i 0 0 0.382683 + 0.923880i −0.923880 + 0.382683i −1.08979 − 1.63099i
3135.1 −0.382683 + 0.923880i 0 −0.707107 − 0.707107i 1.08979 − 0.216773i 0 0 0.923880 − 0.382683i 0.382683 + 0.923880i −0.216773 + 1.08979i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 979.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
119.p even 16 1 inner
476.bf odd 16 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2T_{5}^{4} - 16T_{5}^{3} + 20T_{5}^{2} - 8T_{5} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{4} - 16 T^{3} + 20 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{5} + 2 T^{4} + 12 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + 28 T^{6} - 56 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{6} + 6 T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{6} + 6 T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{5} + 2 T^{4} + 12 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 2 T^{4} - 16 T^{3} + 20 T^{2} + \cdots + 2 \) Copy content Toggle raw display
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