Properties

Label 1344.2.bl.f
Level $1344$
Weight $2$
Character orbit 1344.bl
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.bl (of order \(6\), 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 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 2 + 2 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{25} - q^{27} -9 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + ( -1 - 4 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + ( -6 + 12 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( 1 + \zeta_{6} ) q^{45} -12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 4 - 2 \zeta_{6} ) q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} + 3 q^{55} -2 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + ( 3 - 2 \zeta_{6} ) q^{63} + ( 8 + 8 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( 4 - 5 \zeta_{6} ) q^{77} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} -6 q^{85} + ( -9 + 9 \zeta_{6} ) q^{87} + ( -4 + 2 \zeta_{6} ) q^{89} + 5 \zeta_{6} q^{93} + ( 2 + 2 \zeta_{6} ) q^{95} + ( -11 + 22 \zeta_{6} ) q^{97} + ( -1 + 2 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 3q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 3q^{5} + q^{7} - q^{9} - 3q^{11} + 6q^{17} - 2q^{19} + 5q^{21} - 2q^{25} - 2q^{27} - 18q^{29} - 5q^{31} - 3q^{33} - 6q^{35} + 10q^{37} + 3q^{45} - 12q^{47} - 13q^{49} + 6q^{51} - 9q^{53} + 6q^{55} - 4q^{57} - 9q^{59} + 4q^{63} + 24q^{67} - 12q^{73} + 2q^{75} + 3q^{77} + 9q^{79} - q^{81} - 6q^{83} - 12q^{85} - 9q^{87} - 6q^{89} + 5q^{93} + 6q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −1.50000 + 0.866025i 0 0.500000 + 2.59808i 0 −0.500000 0.866025i 0
1279.1 0 0.500000 + 0.866025i 0 −1.50000 0.866025i 0 0.500000 2.59808i 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
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.bl.f 2
4.b odd 2 1 1344.2.bl.b 2
7.d odd 6 1 1344.2.bl.b 2
8.b even 2 1 336.2.bl.c 2
8.d odd 2 1 336.2.bl.g yes 2
24.f even 2 1 1008.2.cs.d 2
24.h odd 2 1 1008.2.cs.e 2
28.f even 6 1 inner 1344.2.bl.f 2
56.e even 2 1 2352.2.bl.b 2
56.h odd 2 1 2352.2.bl.h 2
56.j odd 6 1 336.2.bl.g yes 2
56.j odd 6 1 2352.2.b.c 2
56.k odd 6 1 2352.2.b.c 2
56.k odd 6 1 2352.2.bl.h 2
56.m even 6 1 336.2.bl.c 2
56.m even 6 1 2352.2.b.g 2
56.p even 6 1 2352.2.b.g 2
56.p even 6 1 2352.2.bl.b 2
168.s odd 6 1 7056.2.b.e 2
168.v even 6 1 7056.2.b.i 2
168.ba even 6 1 1008.2.cs.d 2
168.ba even 6 1 7056.2.b.i 2
168.be odd 6 1 1008.2.cs.e 2
168.be odd 6 1 7056.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.c 2 8.b even 2 1
336.2.bl.c 2 56.m even 6 1
336.2.bl.g yes 2 8.d odd 2 1
336.2.bl.g yes 2 56.j odd 6 1
1008.2.cs.d 2 24.f even 2 1
1008.2.cs.d 2 168.ba even 6 1
1008.2.cs.e 2 24.h odd 2 1
1008.2.cs.e 2 168.be odd 6 1
1344.2.bl.b 2 4.b odd 2 1
1344.2.bl.b 2 7.d odd 6 1
1344.2.bl.f 2 1.a even 1 1 trivial
1344.2.bl.f 2 28.f even 6 1 inner
2352.2.b.c 2 56.j odd 6 1
2352.2.b.c 2 56.k odd 6 1
2352.2.b.g 2 56.m even 6 1
2352.2.b.g 2 56.p even 6 1
2352.2.bl.b 2 56.e even 2 1
2352.2.bl.b 2 56.p even 6 1
2352.2.bl.h 2 56.h odd 2 1
2352.2.bl.h 2 56.k odd 6 1
7056.2.b.e 2 168.s odd 6 1
7056.2.b.e 2 168.be odd 6 1
7056.2.b.i 2 168.v even 6 1
7056.2.b.i 2 168.ba even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 3 \)
\( T_{11}^{2} + 3 T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 3 + 3 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 3 + 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 12 - 6 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 100 - 10 T + T^{2} \)
$41$ \( 108 + T^{2} \)
$43$ \( 12 + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( 81 + 9 T + T^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 192 - 24 T + T^{2} \)
$71$ \( 192 + T^{2} \)
$73$ \( 48 + 12 T + T^{2} \)
$79$ \( 27 - 9 T + T^{2} \)
$83$ \( ( 3 + T )^{2} \)
$89$ \( 12 + 6 T + T^{2} \)
$97$ \( 363 + T^{2} \)
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