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Properties

Label 1428.1.bl.d
Level $1428$
Weight $1$
Character orbit 1428.bl
Analytic conductor $0.713$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -68
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(815,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.815");
 
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.bl (of order \(6\), degree \(2\), 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: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + \zeta_{24}^{7} q^{6} - \zeta_{24}^{3} q^{7} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + \zeta_{24}^{7} q^{6} - \zeta_{24}^{3} q^{7} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} + (\zeta_{24}^{11} + \zeta_{24}^{9}) q^{11} - \zeta_{24}^{9} q^{12} + \zeta_{24}^{6} q^{13} + \zeta_{24}^{5} q^{14} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{4} q^{17} + q^{18} + \zeta_{24}^{8} q^{21} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{22} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{23} + \zeta_{24}^{11} q^{24} - \zeta_{24}^{4} q^{25} - \zeta_{24}^{8} q^{26} + \zeta_{24}^{3} q^{27} - \zeta_{24}^{7} q^{28} + (\zeta_{24}^{7} - \zeta_{24}) q^{31} - \zeta_{24}^{10} q^{32} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{33} - \zeta_{24}^{6} q^{34} - \zeta_{24}^{2} q^{36} - \zeta_{24}^{11} q^{39} - \zeta_{24}^{10} q^{42} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{44} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{46} + \zeta_{24} q^{48} + \zeta_{24}^{6} q^{49} + \zeta_{24}^{6} q^{50} - \zeta_{24}^{9} q^{51} + \zeta_{24}^{10} q^{52} + (\zeta_{24}^{8} - 1) q^{53} - \zeta_{24}^{5} q^{54} + \zeta_{24}^{9} q^{56} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{62} + \zeta_{24} q^{63} - q^{64} + ( - \zeta_{24}^{6} - \zeta_{24}^{4}) q^{66} + \zeta_{24}^{8} q^{68} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{69} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{71} + \zeta_{24}^{4} q^{72} + \zeta_{24}^{9} q^{75} + (\zeta_{24}^{2} + 1) q^{77} - \zeta_{24} q^{78} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{79} - \zeta_{24}^{8} q^{81} - q^{84} + (\zeta_{24}^{5} + \zeta_{24}^{3}) q^{88} + (\zeta_{24}^{10} + \zeta_{24}^{6}) q^{89} - \zeta_{24}^{9} q^{91} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{92} + (\zeta_{24}^{6} + 1) q^{93} - \zeta_{24}^{3} q^{96} - \zeta_{24}^{8} q^{98} + ( - \zeta_{24}^{9} - \zeta_{24}^{7}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 4 q^{16} + 4 q^{17} + 8 q^{18} - 4 q^{21} - 4 q^{25} + 4 q^{26} + 4 q^{33} - 12 q^{53} - 8 q^{64} - 4 q^{66} - 4 q^{68} - 4 q^{69} + 4 q^{72} + 8 q^{77} + 4 q^{81} - 8 q^{84} + 8 q^{93} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(409\) \(715\) \(953\) \(1261\)
\(\chi(n)\) \(-\zeta_{24}^{8}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
815.1
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i 0 −0.258819 − 0.965926i −0.707107 + 0.707107i 1.00000i −0.866025 − 0.500000i 0
815.2 −0.866025 + 0.500000i 0.258819 − 0.965926i 0.500000 − 0.866025i 0 0.258819 + 0.965926i 0.707107 − 0.707107i 1.00000i −0.866025 − 0.500000i 0
815.3 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i 0 −0.965926 + 0.258819i 0.707107 + 0.707107i − 1.00000i 0.866025 + 0.500000i 0
815.4 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i 0 0.965926 − 0.258819i −0.707107 − 0.707107i − 1.00000i 0.866025 + 0.500000i 0
1223.1 −0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i 0 −0.258819 + 0.965926i −0.707107 − 0.707107i − 1.00000i −0.866025 + 0.500000i 0
1223.2 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 0 0.258819 − 0.965926i 0.707107 + 0.707107i − 1.00000i −0.866025 + 0.500000i 0
1223.3 0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i 0 −0.965926 − 0.258819i 0.707107 − 0.707107i 1.00000i 0.866025 − 0.500000i 0
1223.4 0.866025 + 0.500000i 0.965926 − 0.258819i 0.500000 + 0.866025i 0 0.965926 + 0.258819i −0.707107 + 0.707107i 1.00000i 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 815.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
17.b even 2 1 inner
21.g even 6 1 inner
84.j odd 6 1 inner
357.s even 6 1 inner
1428.bl odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.1.bl.d yes 8
3.b odd 2 1 1428.1.bl.c ✓ 8
4.b odd 2 1 inner 1428.1.bl.d yes 8
7.d odd 6 1 1428.1.bl.c ✓ 8
12.b even 2 1 1428.1.bl.c ✓ 8
17.b even 2 1 inner 1428.1.bl.d yes 8
21.g even 6 1 inner 1428.1.bl.d yes 8
28.f even 6 1 1428.1.bl.c ✓ 8
51.c odd 2 1 1428.1.bl.c ✓ 8
68.d odd 2 1 CM 1428.1.bl.d yes 8
84.j odd 6 1 inner 1428.1.bl.d yes 8
119.h odd 6 1 1428.1.bl.c ✓ 8
204.h even 2 1 1428.1.bl.c ✓ 8
357.s even 6 1 inner 1428.1.bl.d yes 8
476.q even 6 1 1428.1.bl.c ✓ 8
1428.bl odd 6 1 inner 1428.1.bl.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.1.bl.c ✓ 8 3.b odd 2 1
1428.1.bl.c ✓ 8 7.d odd 6 1
1428.1.bl.c ✓ 8 12.b even 2 1
1428.1.bl.c ✓ 8 28.f even 6 1
1428.1.bl.c ✓ 8 51.c odd 2 1
1428.1.bl.c ✓ 8 119.h odd 6 1
1428.1.bl.c ✓ 8 204.h even 2 1
1428.1.bl.c ✓ 8 476.q even 6 1
1428.1.bl.d yes 8 1.a even 1 1 trivial
1428.1.bl.d yes 8 4.b odd 2 1 inner
1428.1.bl.d yes 8 17.b even 2 1 inner
1428.1.bl.d yes 8 21.g even 6 1 inner
1428.1.bl.d yes 8 68.d odd 2 1 CM
1428.1.bl.d yes 8 84.j odd 6 1 inner
1428.1.bl.d yes 8 357.s even 6 1 inner
1428.1.bl.d yes 8 1428.bl odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1428, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{47} \) Copy content Toggle raw display
\( T_{53}^{2} + 3T_{53} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) 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
$53$ \( (T^{2} + 3 T + 3)^{4} \) 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
$71$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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