Properties

Label 1344.2.q.n
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + 3 q^{13} -2 q^{15} + ( -8 + 8 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} - q^{27} -4 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} + ( 6 - 2 \zeta_{6} ) q^{35} -\zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} + 6 q^{41} + 11 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 8 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} + 4 q^{55} + q^{57} + ( -4 + 4 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} + ( 3 - \zeta_{6} ) q^{63} -6 \zeta_{6} q^{65} + ( -13 + 13 \zeta_{6} ) q^{67} + 8 q^{69} + 10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( -2 - 4 \zeta_{6} ) q^{77} -3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 2 q^{83} + 16 q^{85} + ( -4 + 4 \zeta_{6} ) q^{87} + ( -6 + 9 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + 10 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 2q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 2q^{5} - q^{7} - q^{9} - 2q^{11} + 6q^{13} - 4q^{15} - 8q^{17} + q^{19} + 4q^{21} + 8q^{23} + q^{25} - 2q^{27} - 8q^{29} + 3q^{31} + 2q^{33} + 10q^{35} - q^{37} + 3q^{39} + 12q^{41} + 22q^{43} - 2q^{45} + 6q^{47} - 13q^{49} + 8q^{51} - 12q^{53} + 8q^{55} + 2q^{57} - 4q^{59} - 6q^{61} + 5q^{63} - 6q^{65} - 13q^{67} + 16q^{69} + 20q^{71} + 11q^{73} - q^{75} - 8q^{77} - 3q^{79} - q^{81} + 4q^{83} + 32q^{85} - 4q^{87} - 3q^{91} - 3q^{93} + 2q^{95} + 20q^{97} + 4q^{99} + O(q^{100}) \)

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.

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 −0.500000 + 2.59808i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 −0.500000 2.59808i 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.n 2
4.b odd 2 1 1344.2.q.b 2
7.c even 3 1 inner 1344.2.q.n 2
7.c even 3 1 9408.2.a.bi 1
7.d odd 6 1 9408.2.a.bx 1
8.b even 2 1 336.2.q.c 2
8.d odd 2 1 84.2.i.a 2
24.f even 2 1 252.2.k.a 2
24.h odd 2 1 1008.2.s.c 2
28.f even 6 1 9408.2.a.i 1
28.g odd 6 1 1344.2.q.b 2
28.g odd 6 1 9408.2.a.cx 1
40.e odd 2 1 2100.2.q.b 2
40.k even 4 2 2100.2.bc.a 4
56.e even 2 1 588.2.i.b 2
56.h odd 2 1 2352.2.q.q 2
56.j odd 6 1 2352.2.a.k 1
56.j odd 6 1 2352.2.q.q 2
56.k odd 6 1 84.2.i.a 2
56.k odd 6 1 588.2.a.a 1
56.m even 6 1 588.2.a.f 1
56.m even 6 1 588.2.i.b 2
56.p even 6 1 336.2.q.c 2
56.p even 6 1 2352.2.a.o 1
72.l even 6 1 2268.2.i.b 2
72.l even 6 1 2268.2.l.g 2
72.p odd 6 1 2268.2.i.g 2
72.p odd 6 1 2268.2.l.b 2
168.e odd 2 1 1764.2.k.j 2
168.s odd 6 1 1008.2.s.c 2
168.s odd 6 1 7056.2.a.bs 1
168.v even 6 1 252.2.k.a 2
168.v even 6 1 1764.2.a.h 1
168.ba even 6 1 7056.2.a.o 1
168.be odd 6 1 1764.2.a.c 1
168.be odd 6 1 1764.2.k.j 2
280.bi odd 6 1 2100.2.q.b 2
280.br even 12 2 2100.2.bc.a 4
504.ba odd 6 1 2268.2.i.g 2
504.bt even 6 1 2268.2.l.g 2
504.ce odd 6 1 2268.2.l.b 2
504.cy even 6 1 2268.2.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 8.d odd 2 1
84.2.i.a 2 56.k odd 6 1
252.2.k.a 2 24.f even 2 1
252.2.k.a 2 168.v even 6 1
336.2.q.c 2 8.b even 2 1
336.2.q.c 2 56.p even 6 1
588.2.a.a 1 56.k odd 6 1
588.2.a.f 1 56.m even 6 1
588.2.i.b 2 56.e even 2 1
588.2.i.b 2 56.m even 6 1
1008.2.s.c 2 24.h odd 2 1
1008.2.s.c 2 168.s odd 6 1
1344.2.q.b 2 4.b odd 2 1
1344.2.q.b 2 28.g odd 6 1
1344.2.q.n 2 1.a even 1 1 trivial
1344.2.q.n 2 7.c even 3 1 inner
1764.2.a.c 1 168.be odd 6 1
1764.2.a.h 1 168.v even 6 1
1764.2.k.j 2 168.e odd 2 1
1764.2.k.j 2 168.be odd 6 1
2100.2.q.b 2 40.e odd 2 1
2100.2.q.b 2 280.bi odd 6 1
2100.2.bc.a 4 40.k even 4 2
2100.2.bc.a 4 280.br even 12 2
2268.2.i.b 2 72.l even 6 1
2268.2.i.b 2 504.cy even 6 1
2268.2.i.g 2 72.p odd 6 1
2268.2.i.g 2 504.ba odd 6 1
2268.2.l.b 2 72.p odd 6 1
2268.2.l.b 2 504.ce odd 6 1
2268.2.l.g 2 72.l even 6 1
2268.2.l.g 2 504.bt even 6 1
2352.2.a.k 1 56.j odd 6 1
2352.2.a.o 1 56.p even 6 1
2352.2.q.q 2 56.h odd 2 1
2352.2.q.q 2 56.j odd 6 1
7056.2.a.o 1 168.ba even 6 1
7056.2.a.bs 1 168.s odd 6 1
9408.2.a.i 1 28.f even 6 1
9408.2.a.bi 1 7.c even 3 1
9408.2.a.bx 1 7.d odd 6 1
9408.2.a.cx 1 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 3 \)