Properties

Label 1344.2.q.m
Level $1344$
Weight $2$
Character orbit 1344.q
Analytic conductor $10.732$
Analytic rank $1$
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: \(1\)
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 21)
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.

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

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 −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.m 2
4.b odd 2 1 1344.2.q.c 2
7.c even 3 1 inner 1344.2.q.m 2
7.c even 3 1 9408.2.a.bg 1
7.d odd 6 1 9408.2.a.bz 1
8.b even 2 1 21.2.e.a 2
8.d odd 2 1 336.2.q.f 2
24.f even 2 1 1008.2.s.d 2
24.h odd 2 1 63.2.e.b 2
28.f even 6 1 9408.2.a.k 1
28.g odd 6 1 1344.2.q.c 2
28.g odd 6 1 9408.2.a.cv 1
40.f even 2 1 525.2.i.e 2
40.i odd 4 2 525.2.r.e 4
56.e even 2 1 2352.2.q.c 2
56.h odd 2 1 147.2.e.a 2
56.j odd 6 1 147.2.a.b 1
56.j odd 6 1 147.2.e.a 2
56.k odd 6 1 336.2.q.f 2
56.k odd 6 1 2352.2.a.d 1
56.m even 6 1 2352.2.a.w 1
56.m even 6 1 2352.2.q.c 2
56.p even 6 1 21.2.e.a 2
56.p even 6 1 147.2.a.c 1
72.j odd 6 1 567.2.g.f 2
72.j odd 6 1 567.2.h.a 2
72.n even 6 1 567.2.g.a 2
72.n even 6 1 567.2.h.f 2
168.i even 2 1 441.2.e.e 2
168.s odd 6 1 63.2.e.b 2
168.s odd 6 1 441.2.a.b 1
168.v even 6 1 1008.2.s.d 2
168.v even 6 1 7056.2.a.bp 1
168.ba even 6 1 441.2.a.a 1
168.ba even 6 1 441.2.e.e 2
168.be odd 6 1 7056.2.a.m 1
280.bf even 6 1 525.2.i.e 2
280.bf even 6 1 3675.2.a.a 1
280.bk odd 6 1 3675.2.a.c 1
280.bt odd 12 2 525.2.r.e 4
504.w even 6 1 567.2.h.f 2
504.bi odd 6 1 567.2.g.f 2
504.cq even 6 1 567.2.g.a 2
504.db odd 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.b even 2 1
21.2.e.a 2 56.p even 6 1
63.2.e.b 2 24.h odd 2 1
63.2.e.b 2 168.s odd 6 1
147.2.a.b 1 56.j odd 6 1
147.2.a.c 1 56.p even 6 1
147.2.e.a 2 56.h odd 2 1
147.2.e.a 2 56.j odd 6 1
336.2.q.f 2 8.d odd 2 1
336.2.q.f 2 56.k odd 6 1
441.2.a.a 1 168.ba even 6 1
441.2.a.b 1 168.s odd 6 1
441.2.e.e 2 168.i even 2 1
441.2.e.e 2 168.ba even 6 1
525.2.i.e 2 40.f even 2 1
525.2.i.e 2 280.bf even 6 1
525.2.r.e 4 40.i odd 4 2
525.2.r.e 4 280.bt odd 12 2
567.2.g.a 2 72.n even 6 1
567.2.g.a 2 504.cq even 6 1
567.2.g.f 2 72.j odd 6 1
567.2.g.f 2 504.bi odd 6 1
567.2.h.a 2 72.j odd 6 1
567.2.h.a 2 504.db odd 6 1
567.2.h.f 2 72.n even 6 1
567.2.h.f 2 504.w even 6 1
1008.2.s.d 2 24.f even 2 1
1008.2.s.d 2 168.v even 6 1
1344.2.q.c 2 4.b odd 2 1
1344.2.q.c 2 28.g odd 6 1
1344.2.q.m 2 1.a even 1 1 trivial
1344.2.q.m 2 7.c even 3 1 inner
2352.2.a.d 1 56.k odd 6 1
2352.2.a.w 1 56.m even 6 1
2352.2.q.c 2 56.e even 2 1
2352.2.q.c 2 56.m even 6 1
3675.2.a.a 1 280.bf even 6 1
3675.2.a.c 1 280.bk odd 6 1
7056.2.a.m 1 168.be odd 6 1
7056.2.a.bp 1 168.v even 6 1
9408.2.a.k 1 28.f even 6 1
9408.2.a.bg 1 7.c even 3 1
9408.2.a.bz 1 7.d odd 6 1
9408.2.a.cv 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 \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} + 1 \)