Properties

Label 1428.1.bu.a
Level $1428$
Weight $1$
Character orbit 1428.bu
Analytic conductor $0.713$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1428,1,Mod(83,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.83");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1428.bu (of order \(8\), degree \(4\), 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: \(0.712664838040\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.2918512474101504.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16}^{6} q^{2} - \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{5} + \zeta_{16}^{7} q^{6} - \zeta_{16}^{3} q^{7} - \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16}^{6} q^{2} - \zeta_{16} q^{3} - \zeta_{16}^{4} q^{4} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{5} + \zeta_{16}^{7} q^{6} - \zeta_{16}^{3} q^{7} - \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} + (\zeta_{16}^{5} + \zeta_{16}) q^{10} + (\zeta_{16}^{6} - 1) q^{11} + \zeta_{16}^{5} q^{12} - \zeta_{16} q^{14} + ( - \zeta_{16}^{4} + 1) q^{15} - q^{16} - \zeta_{16} q^{17} + q^{18} + (\zeta_{16}^{3} - \zeta_{16}) q^{19} + ( - \zeta_{16}^{7} + \zeta_{16}^{3}) q^{20} + \zeta_{16}^{4} q^{21} + (\zeta_{16}^{6} + \zeta_{16}^{4}) q^{22} + (\zeta_{16}^{4} - \zeta_{16}^{2}) q^{23} + \zeta_{16}^{3} q^{24} - \zeta_{16}^{2} q^{25} - \zeta_{16}^{3} q^{27} + \zeta_{16}^{7} q^{28} + ( - \zeta_{16}^{6} - \zeta_{16}^{2}) q^{30} + ( - \zeta_{16}^{7} + \zeta_{16}^{3}) q^{31} + \zeta_{16}^{6} q^{32} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{33} + \zeta_{16}^{7} q^{34} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{35} - \zeta_{16}^{6} q^{36} + ( - \zeta_{16}^{2} - 1) q^{37} + (\zeta_{16}^{7} + \zeta_{16}) q^{38} + ( - \zeta_{16}^{5} + \zeta_{16}) q^{40} + \zeta_{16}^{2} q^{42} + (\zeta_{16}^{4} + \zeta_{16}^{2}) q^{44} + (\zeta_{16}^{5} - \zeta_{16}) q^{45} + (\zeta_{16}^{2} - 1) q^{46} + \zeta_{16} q^{48} + \zeta_{16}^{6} q^{49} - q^{50} + \zeta_{16}^{2} q^{51} - \zeta_{16} q^{54} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16}) q^{55} + \zeta_{16}^{5} q^{56} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{57} + ( - \zeta_{16}^{4} - 1) q^{60} + ( - \zeta_{16}^{5} + \zeta_{16}) q^{62} - \zeta_{16}^{5} q^{63} + \zeta_{16}^{4} q^{64} + ( - \zeta_{16}^{7} - \zeta_{16}^{5}) q^{66} + \zeta_{16}^{5} q^{68} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{69} + ( - \zeta_{16}^{4} + 1) q^{70} + ( - \zeta_{16}^{6} + \zeta_{16}^{4}) q^{71} - \zeta_{16}^{4} q^{72} + (\zeta_{16}^{6} - 1) q^{74} + \zeta_{16}^{3} q^{75} + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{76} + (\zeta_{16}^{3} + \zeta_{16}) q^{77} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{80} + \zeta_{16}^{4} q^{81} + q^{84} + ( - \zeta_{16}^{4} + 1) q^{85} + (\zeta_{16}^{2} + 1) q^{88} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{89} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{90} + (\zeta_{16}^{6} + 1) q^{92} + ( - \zeta_{16}^{4} - 1) q^{93} + (\zeta_{16}^{6} - \zeta_{16}^{4} - \zeta_{16}^{2} + 1) q^{95} - \zeta_{16}^{7} q^{96} + \zeta_{16}^{4} q^{98} + ( - \zeta_{16}^{2} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{11} + 8 q^{15} - 8 q^{16} + 8 q^{18} - 8 q^{37} - 8 q^{46} - 8 q^{50} - 8 q^{60} + 8 q^{70} - 8 q^{74} + 8 q^{84} + 8 q^{85} + 8 q^{88} + 8 q^{92} - 8 q^{93} + 8 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(715\) \(953\) \(1261\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{16}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
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
−0.923880 + 0.382683i
−0.707107 0.707107i −0.382683 0.923880i 1.00000i −1.30656 + 0.541196i −0.382683 + 0.923880i 0.923880 + 0.382683i 0.707107 0.707107i −0.707107 + 0.707107i 1.30656 + 0.541196i
83.2 −0.707107 0.707107i 0.382683 + 0.923880i 1.00000i 1.30656 0.541196i 0.382683 0.923880i −0.923880 0.382683i 0.707107 0.707107i −0.707107 + 0.707107i −1.30656 0.541196i
587.1 0.707107 0.707107i −0.923880 0.382683i 1.00000i −0.541196 + 1.30656i −0.923880 + 0.382683i −0.382683 0.923880i −0.707107 0.707107i 0.707107 + 0.707107i 0.541196 + 1.30656i
587.2 0.707107 0.707107i 0.923880 + 0.382683i 1.00000i 0.541196 1.30656i 0.923880 0.382683i 0.382683 + 0.923880i −0.707107 0.707107i 0.707107 + 0.707107i −0.541196 1.30656i
671.1 −0.707107 + 0.707107i −0.382683 + 0.923880i 1.00000i −1.30656 0.541196i −0.382683 0.923880i 0.923880 0.382683i 0.707107 + 0.707107i −0.707107 0.707107i 1.30656 0.541196i
671.2 −0.707107 + 0.707107i 0.382683 0.923880i 1.00000i 1.30656 + 0.541196i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.707107 + 0.707107i −0.707107 0.707107i −1.30656 + 0.541196i
1175.1 0.707107 + 0.707107i −0.923880 + 0.382683i 1.00000i −0.541196 1.30656i −0.923880 0.382683i −0.382683 + 0.923880i −0.707107 + 0.707107i 0.707107 0.707107i 0.541196 1.30656i
1175.2 0.707107 + 0.707107i 0.923880 0.382683i 1.00000i 0.541196 + 1.30656i 0.923880 + 0.382683i 0.382683 0.923880i −0.707107 + 0.707107i 0.707107 0.707107i −0.541196 + 1.30656i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
7.b odd 2 1 inner
12.b even 2 1 inner
17.d even 8 1 inner
119.l odd 8 1 inner
204.p even 8 1 inner
1428.bu odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.1.bu.a 8
3.b odd 2 1 1428.1.bu.b yes 8
4.b odd 2 1 1428.1.bu.b yes 8
7.b odd 2 1 inner 1428.1.bu.a 8
12.b even 2 1 inner 1428.1.bu.a 8
17.d even 8 1 inner 1428.1.bu.a 8
21.c even 2 1 1428.1.bu.b yes 8
28.d even 2 1 1428.1.bu.b yes 8
51.g odd 8 1 1428.1.bu.b yes 8
68.g odd 8 1 1428.1.bu.b yes 8
84.h odd 2 1 CM 1428.1.bu.a 8
119.l odd 8 1 inner 1428.1.bu.a 8
204.p even 8 1 inner 1428.1.bu.a 8
357.w even 8 1 1428.1.bu.b yes 8
476.w even 8 1 1428.1.bu.b yes 8
1428.bu odd 8 1 inner 1428.1.bu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.1.bu.a 8 1.a even 1 1 trivial
1428.1.bu.a 8 7.b odd 2 1 inner
1428.1.bu.a 8 12.b even 2 1 inner
1428.1.bu.a 8 17.d even 8 1 inner
1428.1.bu.a 8 84.h odd 2 1 CM
1428.1.bu.a 8 119.l odd 8 1 inner
1428.1.bu.a 8 204.p even 8 1 inner
1428.1.bu.a 8 1428.bu odd 8 1 inner
1428.1.bu.b yes 8 3.b odd 2 1
1428.1.bu.b yes 8 4.b odd 2 1
1428.1.bu.b yes 8 21.c even 2 1
1428.1.bu.b yes 8 28.d even 2 1
1428.1.bu.b yes 8 51.g odd 8 1
1428.1.bu.b yes 8 68.g odd 8 1
1428.1.bu.b yes 8 357.w even 8 1
1428.1.bu.b yes 8 476.w even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 4T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1428, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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