Properties

Label 1344.2.q.w
Level $1344$
Weight $2$
Character orbit 1344.q
Analytic conductor $10.732$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{11} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{15} + ( 4 - 4 \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{21} -4 \beta_{2} q^{23} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{25} + q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} ) q^{31} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{33} + ( 11 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{45} + 6 \beta_{2} q^{47} + ( -4 + 3 \beta_{1} - 3 \beta_{3} ) q^{49} + 4 \beta_{2} q^{51} + ( 6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 15 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{59} + 10 \beta_{2} q^{61} + ( \beta_{2} - \beta_{3} ) q^{63} + ( 2 - 4 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{67} + 4 q^{69} + 2 q^{71} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( -1 + 2 \beta_{1} - 9 \beta_{2} - \beta_{3} ) q^{75} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 2 - 4 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{83} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -3 + 4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{91} + \beta_{2} q^{93} + ( 14 - 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 13 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - q^{5} - 6q^{7} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - q^{5} - 6q^{7} - 2q^{9} - q^{11} - 10q^{13} + 2q^{15} + 8q^{17} - 5q^{19} + 3q^{21} - 8q^{23} - 19q^{25} + 4q^{27} - 6q^{29} + 2q^{31} - q^{33} + 30q^{35} + 3q^{37} + 5q^{39} + 12q^{41} + 14q^{43} - q^{45} + 12q^{47} - 10q^{49} + 8q^{51} + 11q^{53} + 58q^{55} + 10q^{57} + 5q^{59} + 20q^{61} + 3q^{63} - 26q^{65} + 7q^{67} + 16q^{69} + 8q^{71} + q^{73} - 19q^{75} - 27q^{77} - 8q^{79} - 2q^{81} - 14q^{83} - 8q^{85} + 3q^{87} + 6q^{89} + 15q^{91} + 2q^{93} + 26q^{95} + 50q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\) \(1\)

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
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0 −0.500000 + 0.866025i 0 −2.13746 3.70219i 0 −1.50000 + 2.17945i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 1.63746 + 2.83616i 0 −1.50000 2.17945i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 −2.13746 + 3.70219i 0 −1.50000 2.17945i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 1.63746 2.83616i 0 −1.50000 + 2.17945i 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.w 4
4.b odd 2 1 1344.2.q.x 4
7.c even 3 1 inner 1344.2.q.w 4
7.c even 3 1 9408.2.a.ec 2
7.d odd 6 1 9408.2.a.dj 2
8.b even 2 1 168.2.q.c 4
8.d odd 2 1 336.2.q.g 4
24.f even 2 1 1008.2.s.r 4
24.h odd 2 1 504.2.s.i 4
28.f even 6 1 9408.2.a.dw 2
28.g odd 6 1 1344.2.q.x 4
28.g odd 6 1 9408.2.a.dp 2
56.e even 2 1 2352.2.q.bf 4
56.h odd 2 1 1176.2.q.l 4
56.j odd 6 1 1176.2.a.n 2
56.j odd 6 1 1176.2.q.l 4
56.k odd 6 1 336.2.q.g 4
56.k odd 6 1 2352.2.a.bf 2
56.m even 6 1 2352.2.a.ba 2
56.m even 6 1 2352.2.q.bf 4
56.p even 6 1 168.2.q.c 4
56.p even 6 1 1176.2.a.k 2
168.i even 2 1 3528.2.s.bk 4
168.s odd 6 1 504.2.s.i 4
168.s odd 6 1 3528.2.a.bk 2
168.v even 6 1 1008.2.s.r 4
168.v even 6 1 7056.2.a.cu 2
168.ba even 6 1 3528.2.a.bd 2
168.ba even 6 1 3528.2.s.bk 4
168.be odd 6 1 7056.2.a.ch 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 8.b even 2 1
168.2.q.c 4 56.p even 6 1
336.2.q.g 4 8.d odd 2 1
336.2.q.g 4 56.k odd 6 1
504.2.s.i 4 24.h odd 2 1
504.2.s.i 4 168.s odd 6 1
1008.2.s.r 4 24.f even 2 1
1008.2.s.r 4 168.v even 6 1
1176.2.a.k 2 56.p even 6 1
1176.2.a.n 2 56.j odd 6 1
1176.2.q.l 4 56.h odd 2 1
1176.2.q.l 4 56.j odd 6 1
1344.2.q.w 4 1.a even 1 1 trivial
1344.2.q.w 4 7.c even 3 1 inner
1344.2.q.x 4 4.b odd 2 1
1344.2.q.x 4 28.g odd 6 1
2352.2.a.ba 2 56.m even 6 1
2352.2.a.bf 2 56.k odd 6 1
2352.2.q.bf 4 56.e even 2 1
2352.2.q.bf 4 56.m even 6 1
3528.2.a.bd 2 168.ba even 6 1
3528.2.a.bk 2 168.s odd 6 1
3528.2.s.bk 4 168.i even 2 1
3528.2.s.bk 4 168.ba even 6 1
7056.2.a.ch 2 168.be odd 6 1
7056.2.a.cu 2 168.v even 6 1
9408.2.a.dj 2 7.d odd 6 1
9408.2.a.dp 2 28.g odd 6 1
9408.2.a.dw 2 28.f even 6 1
9408.2.a.ec 2 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 \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196 \)
\( T_{11}^{4} + T_{11}^{3} + 15 T_{11}^{2} - 14 T_{11} + 196 \)
\( T_{13}^{2} + 5 T_{13} - 8 \)