↑

Properties

Label 1008.2.ch.c
Level $1008$
Weight $2$
Character orbit 1008.ch
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(239,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.ch (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} + 6 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} + 6 q^{5} + 2 q^{9} + 6 q^{11} - 6 q^{15} + 2 q^{21} + 6 q^{25} - 18 q^{29} - 18 q^{31} + 22 q^{33} - 12 q^{37} + 36 q^{39} + 36 q^{41} + 34 q^{45} + 30 q^{47} + 12 q^{49} - 36 q^{51} - 16 q^{57} - 6 q^{59} - 12 q^{63} - 18 q^{65} + 18 q^{67} - 8 q^{69} - 60 q^{71} + 24 q^{73} - 36 q^{75} - 18 q^{79} - 2 q^{81} + 6 q^{83} + 6 q^{85} - 24 q^{87} + 6 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.72260 − 0.180708i 0 2.51988 + 1.45485i 0 −0.866025 + 0.500000i 0 2.93469 + 0.622575i 0
239.2 0 −1.36515 + 1.06600i 0 0.961425 + 0.555079i 0 0.866025 − 0.500000i 0 0.727288 − 2.91051i 0
239.3 0 −1.10890 − 1.33054i 0 0.278486 + 0.160784i 0 0.866025 − 0.500000i 0 −0.540673 + 2.95088i 0
239.4 0 −1.07623 − 1.35710i 0 −2.20505 − 1.27309i 0 −0.866025 + 0.500000i 0 −0.683442 + 2.92111i 0
239.5 0 0.223819 + 1.71753i 0 −0.656809 − 0.379209i 0 −0.866025 + 0.500000i 0 −2.89981 + 0.768831i 0
239.6 0 0.266900 + 1.71136i 0 2.77074 + 1.59969i 0 0.866025 − 0.500000i 0 −2.85753 + 0.913526i 0
239.7 0 0.618491 + 1.61786i 0 −3.14350 − 1.81490i 0 0.866025 − 0.500000i 0 −2.23494 + 2.00126i 0
239.8 0 0.843273 − 1.51291i 0 −0.502939 − 0.290372i 0 −0.866025 + 0.500000i 0 −1.57778 − 2.55159i 0
239.9 0 1.36133 − 1.07087i 0 2.04693 + 1.18179i 0 0.866025 − 0.500000i 0 0.706458 − 2.91563i 0
239.10 0 1.55618 + 0.760457i 0 3.42015 + 1.97463i 0 −0.866025 + 0.500000i 0 1.84341 + 2.36682i 0
239.11 0 1.67556 + 0.438756i 0 −1.07524 − 0.620788i 0 −0.866025 + 0.500000i 0 2.61499 + 1.47032i 0
239.12 0 1.72733 − 0.127784i 0 −1.41408 − 0.816419i 0 0.866025 − 0.500000i 0 2.96734 − 0.441449i 0
911.1 0 −1.72260 + 0.180708i 0 2.51988 − 1.45485i 0 −0.866025 − 0.500000i 0 2.93469 − 0.622575i 0
911.2 0 −1.36515 − 1.06600i 0 0.961425 − 0.555079i 0 0.866025 + 0.500000i 0 0.727288 + 2.91051i 0
911.3 0 −1.10890 + 1.33054i 0 0.278486 − 0.160784i 0 0.866025 + 0.500000i 0 −0.540673 − 2.95088i 0
911.4 0 −1.07623 + 1.35710i 0 −2.20505 + 1.27309i 0 −0.866025 − 0.500000i 0 −0.683442 − 2.92111i 0
911.5 0 0.223819 − 1.71753i 0 −0.656809 + 0.379209i 0 −0.866025 − 0.500000i 0 −2.89981 − 0.768831i 0
911.6 0 0.266900 − 1.71136i 0 2.77074 − 1.59969i 0 0.866025 + 0.500000i 0 −2.85753 − 0.913526i 0
911.7 0 0.618491 − 1.61786i 0 −3.14350 + 1.81490i 0 0.866025 + 0.500000i 0 −2.23494 − 2.00126i 0
911.8 0 0.843273 + 1.51291i 0 −0.502939 + 0.290372i 0 −0.866025 − 0.500000i 0 −1.57778 + 2.55159i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.ch.c yes 24
3.b odd 2 1 3024.2.ch.a 24
4.b odd 2 1 1008.2.ch.a ✓ 24
9.c even 3 1 3024.2.ch.b 24
9.d odd 6 1 1008.2.ch.a ✓ 24
12.b even 2 1 3024.2.ch.b 24
36.f odd 6 1 3024.2.ch.a 24
36.h even 6 1 inner 1008.2.ch.c yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.ch.a ✓ 24 4.b odd 2 1
1008.2.ch.a ✓ 24 9.d odd 6 1
1008.2.ch.c yes 24 1.a even 1 1 trivial
1008.2.ch.c yes 24 36.h even 6 1 inner
3024.2.ch.a 24 3.b odd 2 1
3024.2.ch.a 24 36.f odd 6 1
3024.2.ch.b 24 9.c even 3 1
3024.2.ch.b 24 12.b even 2 1