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Properties

Label 3800.1.ck.c
Level $3800$
Weight $1$
Character orbit 3800.ck
Analytic conductor $1.896$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -40
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(107,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 3, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.ck (of order \(12\), 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: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.31694067200.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + (\zeta_{24}^{5} + \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + (\zeta_{24}^{5} + \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} - q^{11} - \zeta_{24}^{11} q^{13} + ( - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{14} + \zeta_{24}^{4} q^{16} + \zeta_{24}^{3} q^{18} + \zeta_{24}^{10} q^{19} + \zeta_{24} q^{22} + (\zeta_{24}^{7} + \zeta_{24}^{3}) q^{23} - 2 q^{26} + (\zeta_{24}^{7} + \zeta_{24}^{3}) q^{28} - \zeta_{24}^{5} q^{32} - \zeta_{24}^{4} q^{36} + \zeta_{24}^{9} q^{37} - \zeta_{24}^{11} q^{38} + ( - \zeta_{24}^{8} + 1) q^{41} - \zeta_{24}^{2} q^{44} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{46} + (\zeta_{24}^{10} + \cdots + \zeta_{24}^{2}) q^{49} + \cdots + \zeta_{24}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{11} + 4 q^{16} - 16 q^{26} - 4 q^{36} + 12 q^{41} - 8 q^{76} + 4 q^{81} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-\zeta_{24}^{8}\) \(-1\) \(-1\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0 0 1.22474 − 1.22474i 0.707107 − 0.707107i 0.866025 + 0.500000i 0
107.2 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0 0 −1.22474 + 1.22474i −0.707107 + 0.707107i 0.866025 + 0.500000i 0
1243.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0 0 1.22474 + 1.22474i 0.707107 + 0.707107i 0.866025 − 0.500000i 0
1243.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.22474 − 1.22474i −0.707107 − 0.707107i 0.866025 − 0.500000i 0
2307.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0 0 1.22474 − 1.22474i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
2307.2 0.965926 − 0.258819i 0 0.866025 − 0.500000i 0 0 −1.22474 + 1.22474i 0.707107 − 0.707107i −0.866025 + 0.500000i 0
2843.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0 0 1.22474 + 1.22474i −0.707107 − 0.707107i −0.866025 − 0.500000i 0
2843.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −1.22474 − 1.22474i 0.707107 + 0.707107i −0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
19.d odd 6 1 inner
40.k even 4 2 inner
95.h odd 6 1 inner
95.l even 12 2 inner
152.o even 6 1 inner
760.bf even 6 1 inner
760.bu odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.ck.c ✓ 8
5.b even 2 1 inner 3800.1.ck.c ✓ 8
5.c odd 4 2 inner 3800.1.ck.c ✓ 8
8.d odd 2 1 inner 3800.1.ck.c ✓ 8
19.d odd 6 1 inner 3800.1.ck.c ✓ 8
40.e odd 2 1 CM 3800.1.ck.c ✓ 8
40.k even 4 2 inner 3800.1.ck.c ✓ 8
95.h odd 6 1 inner 3800.1.ck.c ✓ 8
95.l even 12 2 inner 3800.1.ck.c ✓ 8
152.o even 6 1 inner 3800.1.ck.c ✓ 8
760.bf even 6 1 inner 3800.1.ck.c ✓ 8
760.bu odd 12 2 inner 3800.1.ck.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.ck.c ✓ 8 1.a even 1 1 trivial
3800.1.ck.c ✓ 8 5.b even 2 1 inner
3800.1.ck.c ✓ 8 5.c odd 4 2 inner
3800.1.ck.c ✓ 8 8.d odd 2 1 inner
3800.1.ck.c ✓ 8 19.d odd 6 1 inner
3800.1.ck.c ✓ 8 40.e odd 2 1 CM
3800.1.ck.c ✓ 8 40.k even 4 2 inner
3800.1.ck.c ✓ 8 95.h odd 6 1 inner
3800.1.ck.c ✓ 8 95.l even 12 2 inner
3800.1.ck.c ✓ 8 152.o even 6 1 inner
3800.1.ck.c ✓ 8 760.bf even 6 1 inner
3800.1.ck.c ✓ 8 760.bu odd 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - T^{4} + 1 \) 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
$71$ \( T^{8} \) 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
$89$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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