Properties

Label 1344.2.q.e
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + 5 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + q^{27} + 8 q^{29} + ( - 3 \zeta_{6} + 3) q^{31} - 2 \zeta_{6} q^{33} - 9 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} + 2 q^{41} - q^{43} - 8 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + 2 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} - 5 q^{57} + ( - 6 \zeta_{6} + 6) q^{59} - 2 \zeta_{6} q^{61} + (3 \zeta_{6} - 1) q^{63} + (5 \zeta_{6} - 5) q^{67} + 6 q^{69} + 4 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + 5 \zeta_{6} q^{75} + ( - 4 \zeta_{6} + 6) q^{77} + 5 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + (8 \zeta_{6} - 8) q^{87} - 12 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} + 3 \zeta_{6} q^{93} + 18 q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 5 q^{7} - q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} + 5 q^{19} + 4 q^{21} - 6 q^{23} + 5 q^{25} + 2 q^{27} + 16 q^{29} + 3 q^{31} - 2 q^{33} - 9 q^{37} + q^{39} + 4 q^{41} - 2 q^{43} - 8 q^{47} + 11 q^{49} + 2 q^{51} + 6 q^{53} - 10 q^{57} + 6 q^{59} - 2 q^{61} + q^{63} - 5 q^{67} + 12 q^{69} + 8 q^{71} + 11 q^{73} + 5 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} - 8 q^{87} - 12 q^{89} + 5 q^{91} + 3 q^{93} + 36 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0 −2.50000 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 0 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.e 2
4.b odd 2 1 1344.2.q.q 2
7.c even 3 1 inner 1344.2.q.e 2
7.c even 3 1 9408.2.a.cn 1
7.d odd 6 1 9408.2.a.x 1
8.b even 2 1 672.2.q.h yes 2
8.d odd 2 1 672.2.q.d 2
24.f even 2 1 2016.2.s.h 2
24.h odd 2 1 2016.2.s.e 2
28.f even 6 1 9408.2.a.ci 1
28.g odd 6 1 1344.2.q.q 2
28.g odd 6 1 9408.2.a.t 1
56.j odd 6 1 4704.2.a.y 1
56.k odd 6 1 672.2.q.d 2
56.k odd 6 1 4704.2.a.ba 1
56.m even 6 1 4704.2.a.j 1
56.p even 6 1 672.2.q.h yes 2
56.p even 6 1 4704.2.a.g 1
168.s odd 6 1 2016.2.s.e 2
168.v even 6 1 2016.2.s.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 8.d odd 2 1
672.2.q.d 2 56.k odd 6 1
672.2.q.h yes 2 8.b even 2 1
672.2.q.h yes 2 56.p even 6 1
1344.2.q.e 2 1.a even 1 1 trivial
1344.2.q.e 2 7.c even 3 1 inner
1344.2.q.q 2 4.b odd 2 1
1344.2.q.q 2 28.g odd 6 1
2016.2.s.e 2 24.h odd 2 1
2016.2.s.e 2 168.s odd 6 1
2016.2.s.h 2 24.f even 2 1
2016.2.s.h 2 168.v even 6 1
4704.2.a.g 1 56.p even 6 1
4704.2.a.j 1 56.m even 6 1
4704.2.a.y 1 56.j odd 6 1
4704.2.a.ba 1 56.k odd 6 1
9408.2.a.t 1 28.g odd 6 1
9408.2.a.x 1 7.d odd 6 1
9408.2.a.ci 1 28.f even 6 1
9408.2.a.cn 1 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 \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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