Properties

Label 1344.2.q.r
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 672)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + q^{15} + ( 8 - 8 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} + 4 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} - q^{27} + 5 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( -1 - 2 \zeta_{6} ) q^{35} + 8 \zeta_{6} q^{37} + 4 q^{41} + 10 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} -8 \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{53} - q^{55} + 4 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + ( -2 + 2 \zeta_{6} ) q^{67} + 4 q^{69} -6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( 2 - 3 \zeta_{6} ) q^{77} -9 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + 8 q^{85} + ( 5 - 5 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} -7 \zeta_{6} q^{93} + ( -4 + 4 \zeta_{6} ) q^{95} - q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} - 5q^{7} - q^{9} - q^{11} + 2q^{15} + 8q^{17} + 4q^{19} - q^{21} + 4q^{23} + 4q^{25} - 2q^{27} + 10q^{29} + 7q^{31} + q^{33} - 4q^{35} + 8q^{37} + 8q^{41} + 20q^{43} + q^{45} + 6q^{47} + 11q^{49} - 8q^{51} - q^{53} - 2q^{55} + 8q^{57} - 9q^{59} - 2q^{61} + 4q^{63} - 2q^{67} + 8q^{69} - 12q^{71} - 2q^{73} - 4q^{75} + q^{77} - 9q^{79} - q^{81} - 6q^{83} + 16q^{85} + 5q^{87} + 6q^{89} - 7q^{93} - 4q^{95} - 2q^{97} + 2q^{99} + O(q^{100}) \)

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 0.500000 + 0.866025i 0 −2.50000 + 0.866025i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 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.r 2
4.b odd 2 1 1344.2.q.h 2
7.c even 3 1 inner 1344.2.q.r 2
7.c even 3 1 9408.2.a.o 1
7.d odd 6 1 9408.2.a.cp 1
8.b even 2 1 672.2.q.b 2
8.d odd 2 1 672.2.q.g yes 2
24.f even 2 1 2016.2.s.j 2
24.h odd 2 1 2016.2.s.i 2
28.f even 6 1 9408.2.a.bb 1
28.g odd 6 1 1344.2.q.h 2
28.g odd 6 1 9408.2.a.cg 1
56.j odd 6 1 4704.2.a.f 1
56.k odd 6 1 672.2.q.g yes 2
56.k odd 6 1 4704.2.a.l 1
56.m even 6 1 4704.2.a.w 1
56.p even 6 1 672.2.q.b 2
56.p even 6 1 4704.2.a.bc 1
168.s odd 6 1 2016.2.s.i 2
168.v even 6 1 2016.2.s.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.b 2 8.b even 2 1
672.2.q.b 2 56.p even 6 1
672.2.q.g yes 2 8.d odd 2 1
672.2.q.g yes 2 56.k odd 6 1
1344.2.q.h 2 4.b odd 2 1
1344.2.q.h 2 28.g odd 6 1
1344.2.q.r 2 1.a even 1 1 trivial
1344.2.q.r 2 7.c even 3 1 inner
2016.2.s.i 2 24.h odd 2 1
2016.2.s.i 2 168.s odd 6 1
2016.2.s.j 2 24.f even 2 1
2016.2.s.j 2 168.v even 6 1
4704.2.a.f 1 56.j odd 6 1
4704.2.a.l 1 56.k odd 6 1
4704.2.a.w 1 56.m even 6 1
4704.2.a.bc 1 56.p even 6 1
9408.2.a.o 1 7.c even 3 1
9408.2.a.bb 1 28.f even 6 1
9408.2.a.cg 1 28.g odd 6 1
9408.2.a.cp 1 7.d odd 6 1

Hecke kernels

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

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