Properties

Label 1344.4.b.b
Level $1344$
Weight $4$
Character orbit 1344.b
Analytic conductor $79.299$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \(x^{2} + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + \beta q^{5} + ( 17 - 3 \beta ) q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + \beta q^{5} + ( 17 - 3 \beta ) q^{7} + 9 q^{9} -23 \beta q^{11} -30 \beta q^{13} -3 \beta q^{15} -21 \beta q^{17} -10 q^{19} + ( -51 + 9 \beta ) q^{21} -47 \beta q^{23} + 119 q^{25} -27 q^{27} -126 q^{29} + 8 q^{31} + 69 \beta q^{33} + ( 18 + 17 \beta ) q^{35} -244 q^{37} + 90 \beta q^{39} + 151 \beta q^{41} -66 \beta q^{43} + 9 \beta q^{45} -180 q^{47} + ( 235 - 102 \beta ) q^{49} + 63 \beta q^{51} -594 q^{53} + 138 q^{55} + 30 q^{57} + 540 q^{59} -144 \beta q^{61} + ( 153 - 27 \beta ) q^{63} + 180 q^{65} + 432 \beta q^{67} + 141 \beta q^{69} -73 \beta q^{71} + 270 \beta q^{73} -357 q^{75} + ( -414 - 391 \beta ) q^{77} + 420 \beta q^{79} + 81 q^{81} + 864 q^{83} + 126 q^{85} + 378 q^{87} -527 \beta q^{89} + ( -540 - 510 \beta ) q^{91} -24 q^{93} -10 \beta q^{95} -330 \beta q^{97} -207 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 34q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 34q^{7} + 18q^{9} - 20q^{19} - 102q^{21} + 238q^{25} - 54q^{27} - 252q^{29} + 16q^{31} + 36q^{35} - 488q^{37} - 360q^{47} + 470q^{49} - 1188q^{53} + 276q^{55} + 60q^{57} + 1080q^{59} + 306q^{63} + 360q^{65} - 714q^{75} - 828q^{77} + 162q^{81} + 1728q^{83} + 252q^{85} + 756q^{87} - 1080q^{91} - 48q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
2.44949i
2.44949i
0 −3.00000 0 2.44949i 0 17.0000 + 7.34847i 0 9.00000 0
895.2 0 −3.00000 0 2.44949i 0 17.0000 7.34847i 0 9.00000 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.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.b 2
4.b odd 2 1 1344.4.b.c 2
7.b odd 2 1 1344.4.b.c 2
8.b even 2 1 336.4.b.d yes 2
8.d odd 2 1 336.4.b.a 2
24.f even 2 1 1008.4.b.a 2
24.h odd 2 1 1008.4.b.f 2
28.d even 2 1 inner 1344.4.b.b 2
56.e even 2 1 336.4.b.d yes 2
56.h odd 2 1 336.4.b.a 2
168.e odd 2 1 1008.4.b.f 2
168.i even 2 1 1008.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.a 2 8.d odd 2 1
336.4.b.a 2 56.h odd 2 1
336.4.b.d yes 2 8.b even 2 1
336.4.b.d yes 2 56.e even 2 1
1008.4.b.a 2 24.f even 2 1
1008.4.b.a 2 168.i even 2 1
1008.4.b.f 2 24.h odd 2 1
1008.4.b.f 2 168.e odd 2 1
1344.4.b.b 2 1.a even 1 1 trivial
1344.4.b.b 2 28.d even 2 1 inner
1344.4.b.c 2 4.b odd 2 1
1344.4.b.c 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{2} + 6 \)
\( T_{19} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( 6 + T^{2} \)
$7$ \( 343 - 34 T + T^{2} \)
$11$ \( 3174 + T^{2} \)
$13$ \( 5400 + T^{2} \)
$17$ \( 2646 + T^{2} \)
$19$ \( ( 10 + T )^{2} \)
$23$ \( 13254 + T^{2} \)
$29$ \( ( 126 + T )^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( ( 244 + T )^{2} \)
$41$ \( 136806 + T^{2} \)
$43$ \( 26136 + T^{2} \)
$47$ \( ( 180 + T )^{2} \)
$53$ \( ( 594 + T )^{2} \)
$59$ \( ( -540 + T )^{2} \)
$61$ \( 124416 + T^{2} \)
$67$ \( 1119744 + T^{2} \)
$71$ \( 31974 + T^{2} \)
$73$ \( 437400 + T^{2} \)
$79$ \( 1058400 + T^{2} \)
$83$ \( ( -864 + T )^{2} \)
$89$ \( 1666374 + T^{2} \)
$97$ \( 653400 + T^{2} \)
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