Properties

Label 1344.2.q.c
Level $1344$
Weight $2$
Character orbit 1344.q
Analytic conductor $10.732$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (of order \(3\), degree \(2\), not 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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - q^{13} + 2 q^{15} - \zeta_{6} q^{19} + (2 \zeta_{6} - 3) q^{21} + ( - \zeta_{6} + 1) q^{25} + q^{27} - 4 q^{29} + ( - 9 \zeta_{6} + 9) q^{31} + 2 \zeta_{6} q^{33} + ( - 6 \zeta_{6} + 2) q^{35} + 3 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} - 10 q^{41} + 5 q^{43} + (2 \zeta_{6} - 2) q^{45} - 6 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + ( - 12 \zeta_{6} + 12) q^{53} - 4 q^{55} + q^{57} + ( - 12 \zeta_{6} + 12) q^{59} + 10 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 1) q^{63} + 2 \zeta_{6} q^{65} + ( - 5 \zeta_{6} + 5) q^{67} + 6 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} + \zeta_{6} q^{75} + ( - 4 \zeta_{6} + 6) q^{77} - \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 6 q^{83} + ( - 4 \zeta_{6} + 4) q^{87} - 16 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} + 9 \zeta_{6} q^{93} + (2 \zeta_{6} - 2) q^{95} - 6 q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} - q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} - q^{19} - 4 q^{21} + q^{25} + 2 q^{27} - 8 q^{29} + 9 q^{31} + 2 q^{33} - 2 q^{35} + 3 q^{37} + q^{39} - 20 q^{41} + 10 q^{43} - 2 q^{45} - 6 q^{47} + 11 q^{49} + 12 q^{53} - 8 q^{55} + 2 q^{57} + 12 q^{59} + 10 q^{61} - q^{63} + 2 q^{65} + 5 q^{67} + 12 q^{71} + 3 q^{73} + q^{75} + 8 q^{77} - q^{79} - q^{81} + 12 q^{83} + 4 q^{87} - 16 q^{89} - 5 q^{91} + 9 q^{93} - 2 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.q.c 2
4.b odd 2 1 1344.2.q.m 2
7.c even 3 1 inner 1344.2.q.c 2
7.c even 3 1 9408.2.a.cv 1
7.d odd 6 1 9408.2.a.k 1
8.b even 2 1 336.2.q.f 2
8.d odd 2 1 21.2.e.a 2
24.f even 2 1 63.2.e.b 2
24.h odd 2 1 1008.2.s.d 2
28.f even 6 1 9408.2.a.bz 1
28.g odd 6 1 1344.2.q.m 2
28.g odd 6 1 9408.2.a.bg 1
40.e odd 2 1 525.2.i.e 2
40.k even 4 2 525.2.r.e 4
56.e even 2 1 147.2.e.a 2
56.h odd 2 1 2352.2.q.c 2
56.j odd 6 1 2352.2.a.w 1
56.j odd 6 1 2352.2.q.c 2
56.k odd 6 1 21.2.e.a 2
56.k odd 6 1 147.2.a.c 1
56.m even 6 1 147.2.a.b 1
56.m even 6 1 147.2.e.a 2
56.p even 6 1 336.2.q.f 2
56.p even 6 1 2352.2.a.d 1
72.l even 6 1 567.2.g.f 2
72.l even 6 1 567.2.h.a 2
72.p odd 6 1 567.2.g.a 2
72.p odd 6 1 567.2.h.f 2
168.e odd 2 1 441.2.e.e 2
168.s odd 6 1 1008.2.s.d 2
168.s odd 6 1 7056.2.a.bp 1
168.v even 6 1 63.2.e.b 2
168.v even 6 1 441.2.a.b 1
168.ba even 6 1 7056.2.a.m 1
168.be odd 6 1 441.2.a.a 1
168.be odd 6 1 441.2.e.e 2
280.ba even 6 1 3675.2.a.c 1
280.bi odd 6 1 525.2.i.e 2
280.bi odd 6 1 3675.2.a.a 1
280.br even 12 2 525.2.r.e 4
504.ba odd 6 1 567.2.h.f 2
504.bt even 6 1 567.2.g.f 2
504.ce odd 6 1 567.2.g.a 2
504.cy even 6 1 567.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 8.d odd 2 1
21.2.e.a 2 56.k odd 6 1
63.2.e.b 2 24.f even 2 1
63.2.e.b 2 168.v even 6 1
147.2.a.b 1 56.m even 6 1
147.2.a.c 1 56.k odd 6 1
147.2.e.a 2 56.e even 2 1
147.2.e.a 2 56.m even 6 1
336.2.q.f 2 8.b even 2 1
336.2.q.f 2 56.p even 6 1
441.2.a.a 1 168.be odd 6 1
441.2.a.b 1 168.v even 6 1
441.2.e.e 2 168.e odd 2 1
441.2.e.e 2 168.be odd 6 1
525.2.i.e 2 40.e odd 2 1
525.2.i.e 2 280.bi odd 6 1
525.2.r.e 4 40.k even 4 2
525.2.r.e 4 280.br even 12 2
567.2.g.a 2 72.p odd 6 1
567.2.g.a 2 504.ce odd 6 1
567.2.g.f 2 72.l even 6 1
567.2.g.f 2 504.bt even 6 1
567.2.h.a 2 72.l even 6 1
567.2.h.a 2 504.cy even 6 1
567.2.h.f 2 72.p odd 6 1
567.2.h.f 2 504.ba odd 6 1
1008.2.s.d 2 24.h odd 2 1
1008.2.s.d 2 168.s odd 6 1
1344.2.q.c 2 1.a even 1 1 trivial
1344.2.q.c 2 7.c even 3 1 inner
1344.2.q.m 2 4.b odd 2 1
1344.2.q.m 2 28.g odd 6 1
2352.2.a.d 1 56.p even 6 1
2352.2.a.w 1 56.j odd 6 1
2352.2.q.c 2 56.h odd 2 1
2352.2.q.c 2 56.j odd 6 1
3675.2.a.a 1 280.bi odd 6 1
3675.2.a.c 1 280.ba even 6 1
7056.2.a.m 1 168.ba even 6 1
7056.2.a.bp 1 168.s odd 6 1
9408.2.a.k 1 7.d odd 6 1
9408.2.a.bg 1 28.g odd 6 1
9408.2.a.bz 1 28.f even 6 1
9408.2.a.cv 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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