Properties

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

Related objects

Downloads

Learn more about

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 42)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 q^{13} -3 q^{15} + 4 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} + ( -4 + 4 \zeta_{6} ) q^{25} + q^{27} -9 q^{29} + ( -1 + \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( -3 - 6 \zeta_{6} ) q^{35} + 8 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -10 q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} -4 q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + 12 \zeta_{6} q^{65} + ( 10 - 10 \zeta_{6} ) q^{67} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( 6 - 9 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 q^{83} + ( 9 - 9 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -12 + 4 \zeta_{6} ) q^{91} -\zeta_{6} q^{93} + ( -12 + 12 \zeta_{6} ) q^{95} - q^{97} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 3q^{5} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 3q^{5} - 5q^{7} - q^{9} - 3q^{11} + 8q^{13} - 6q^{15} + 4q^{19} + q^{21} - 4q^{25} + 2q^{27} - 18q^{29} - q^{31} - 3q^{33} - 12q^{35} + 8q^{37} - 4q^{39} - 20q^{43} + 3q^{45} - 6q^{47} + 11q^{49} - 3q^{53} - 18q^{55} - 8q^{57} - 3q^{59} - 10q^{61} + 4q^{63} + 12q^{65} + 10q^{67} + 12q^{71} - 2q^{73} - 4q^{75} + 3q^{77} - q^{79} - q^{81} - 18q^{83} + 9q^{87} - 6q^{89} - 20q^{91} - q^{93} - 12q^{95} - 2q^{97} + 6q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i 0 −2.50000 + 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 1.50000 2.59808i 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.j 2
4.b odd 2 1 1344.2.q.v 2
7.c even 3 1 inner 1344.2.q.j 2
7.c even 3 1 9408.2.a.bu 1
7.d odd 6 1 9408.2.a.bm 1
8.b even 2 1 336.2.q.d 2
8.d odd 2 1 42.2.e.b 2
24.f even 2 1 126.2.g.b 2
24.h odd 2 1 1008.2.s.n 2
28.f even 6 1 9408.2.a.db 1
28.g odd 6 1 1344.2.q.v 2
28.g odd 6 1 9408.2.a.d 1
40.e odd 2 1 1050.2.i.e 2
40.k even 4 2 1050.2.o.b 4
56.e even 2 1 294.2.e.f 2
56.h odd 2 1 2352.2.q.m 2
56.j odd 6 1 2352.2.a.n 1
56.j odd 6 1 2352.2.q.m 2
56.k odd 6 1 42.2.e.b 2
56.k odd 6 1 294.2.a.d 1
56.m even 6 1 294.2.a.a 1
56.m even 6 1 294.2.e.f 2
56.p even 6 1 336.2.q.d 2
56.p even 6 1 2352.2.a.m 1
72.l even 6 1 1134.2.e.p 2
72.l even 6 1 1134.2.h.a 2
72.p odd 6 1 1134.2.e.a 2
72.p odd 6 1 1134.2.h.p 2
168.e odd 2 1 882.2.g.b 2
168.s odd 6 1 1008.2.s.n 2
168.s odd 6 1 7056.2.a.g 1
168.v even 6 1 126.2.g.b 2
168.v even 6 1 882.2.a.g 1
168.ba even 6 1 7056.2.a.bz 1
168.be odd 6 1 882.2.a.k 1
168.be odd 6 1 882.2.g.b 2
280.ba even 6 1 7350.2.a.cw 1
280.bi odd 6 1 1050.2.i.e 2
280.bi odd 6 1 7350.2.a.ce 1
280.br even 12 2 1050.2.o.b 4
504.ba odd 6 1 1134.2.e.a 2
504.bt even 6 1 1134.2.h.a 2
504.ce odd 6 1 1134.2.h.p 2
504.cy even 6 1 1134.2.e.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 8.d odd 2 1
42.2.e.b 2 56.k odd 6 1
126.2.g.b 2 24.f even 2 1
126.2.g.b 2 168.v even 6 1
294.2.a.a 1 56.m even 6 1
294.2.a.d 1 56.k odd 6 1
294.2.e.f 2 56.e even 2 1
294.2.e.f 2 56.m even 6 1
336.2.q.d 2 8.b even 2 1
336.2.q.d 2 56.p even 6 1
882.2.a.g 1 168.v even 6 1
882.2.a.k 1 168.be odd 6 1
882.2.g.b 2 168.e odd 2 1
882.2.g.b 2 168.be odd 6 1
1008.2.s.n 2 24.h odd 2 1
1008.2.s.n 2 168.s odd 6 1
1050.2.i.e 2 40.e odd 2 1
1050.2.i.e 2 280.bi odd 6 1
1050.2.o.b 4 40.k even 4 2
1050.2.o.b 4 280.br even 12 2
1134.2.e.a 2 72.p odd 6 1
1134.2.e.a 2 504.ba odd 6 1
1134.2.e.p 2 72.l even 6 1
1134.2.e.p 2 504.cy even 6 1
1134.2.h.a 2 72.l even 6 1
1134.2.h.a 2 504.bt even 6 1
1134.2.h.p 2 72.p odd 6 1
1134.2.h.p 2 504.ce odd 6 1
1344.2.q.j 2 1.a even 1 1 trivial
1344.2.q.j 2 7.c even 3 1 inner
1344.2.q.v 2 4.b odd 2 1
1344.2.q.v 2 28.g odd 6 1
2352.2.a.m 1 56.p even 6 1
2352.2.a.n 1 56.j odd 6 1
2352.2.q.m 2 56.h odd 2 1
2352.2.q.m 2 56.j odd 6 1
7056.2.a.g 1 168.s odd 6 1
7056.2.a.bz 1 168.ba even 6 1
7350.2.a.ce 1 280.bi odd 6 1
7350.2.a.cw 1 280.ba even 6 1
9408.2.a.d 1 28.g odd 6 1
9408.2.a.bm 1 7.d odd 6 1
9408.2.a.bu 1 7.c even 3 1
9408.2.a.db 1 28.f even 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 9 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + T + 97 T^{2} )^{2} \)
show more
show less