Properties

Label 1344.4.b.d
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} + 9 q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 8 - 16 \zeta_{6} ) q^{13} + ( 24 - 48 \zeta_{6} ) q^{15} + ( -44 + 88 \zeta_{6} ) q^{17} -52 q^{19} + ( 21 + 42 \zeta_{6} ) q^{21} + ( 66 - 132 \zeta_{6} ) q^{23} -67 q^{25} + 27 q^{27} + 246 q^{29} + 116 q^{31} + ( -6 + 12 \zeta_{6} ) q^{33} + ( 280 - 224 \zeta_{6} ) q^{35} + 314 q^{37} + ( 24 - 48 \zeta_{6} ) q^{39} + ( -156 + 312 \zeta_{6} ) q^{41} + ( 218 - 436 \zeta_{6} ) q^{43} + ( 72 - 144 \zeta_{6} ) q^{45} -192 q^{47} + ( -147 + 392 \zeta_{6} ) q^{49} + ( -132 + 264 \zeta_{6} ) q^{51} + 150 q^{53} + 48 q^{55} -156 q^{57} + 204 q^{59} + ( 336 - 672 \zeta_{6} ) q^{61} + ( 63 + 126 \zeta_{6} ) q^{63} -192 q^{65} + ( -294 + 588 \zeta_{6} ) q^{67} + ( 198 - 396 \zeta_{6} ) q^{69} + ( -470 + 940 \zeta_{6} ) q^{71} + ( -72 + 144 \zeta_{6} ) q^{73} -201 q^{75} + ( -70 + 56 \zeta_{6} ) q^{77} + ( 794 - 1588 \zeta_{6} ) q^{79} + 81 q^{81} + 252 q^{83} + 1056 q^{85} + 738 q^{87} + ( 124 - 248 \zeta_{6} ) q^{89} + ( 280 - 224 \zeta_{6} ) q^{91} + 348 q^{93} + ( -416 + 832 \zeta_{6} ) q^{95} + ( 832 - 1664 \zeta_{6} ) q^{97} + ( -18 + 36 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 28q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 28q^{7} + 18q^{9} - 104q^{19} + 84q^{21} - 134q^{25} + 54q^{27} + 492q^{29} + 232q^{31} + 336q^{35} + 628q^{37} - 384q^{47} + 98q^{49} + 300q^{53} + 96q^{55} - 312q^{57} + 408q^{59} + 252q^{63} - 384q^{65} - 402q^{75} - 84q^{77} + 162q^{81} + 504q^{83} + 2112q^{85} + 1476q^{87} + 336q^{91} + 696q^{93} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 3.00000 0 13.8564i 0 14.0000 + 12.1244i 0 9.00000 0
895.2 0 3.00000 0 13.8564i 0 14.0000 12.1244i 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.d 2
4.b odd 2 1 1344.4.b.a 2
7.b odd 2 1 1344.4.b.a 2
8.b even 2 1 336.4.b.b 2
8.d odd 2 1 336.4.b.c yes 2
24.f even 2 1 1008.4.b.b 2
24.h odd 2 1 1008.4.b.e 2
28.d even 2 1 inner 1344.4.b.d 2
56.e even 2 1 336.4.b.b 2
56.h odd 2 1 336.4.b.c yes 2
168.e odd 2 1 1008.4.b.e 2
168.i even 2 1 1008.4.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.b 2 8.b even 2 1
336.4.b.b 2 56.e even 2 1
336.4.b.c yes 2 8.d odd 2 1
336.4.b.c yes 2 56.h odd 2 1
1008.4.b.b 2 24.f even 2 1
1008.4.b.b 2 168.i even 2 1
1008.4.b.e 2 24.h odd 2 1
1008.4.b.e 2 168.e odd 2 1
1344.4.b.a 2 4.b odd 2 1
1344.4.b.a 2 7.b odd 2 1
1344.4.b.d 2 1.a even 1 1 trivial
1344.4.b.d 2 28.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 192 \)
\( T_{19} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( 192 + T^{2} \)
$7$ \( 343 - 28 T + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 192 + T^{2} \)
$17$ \( 5808 + T^{2} \)
$19$ \( ( 52 + T )^{2} \)
$23$ \( 13068 + T^{2} \)
$29$ \( ( -246 + T )^{2} \)
$31$ \( ( -116 + T )^{2} \)
$37$ \( ( -314 + T )^{2} \)
$41$ \( 73008 + T^{2} \)
$43$ \( 142572 + T^{2} \)
$47$ \( ( 192 + T )^{2} \)
$53$ \( ( -150 + T )^{2} \)
$59$ \( ( -204 + T )^{2} \)
$61$ \( 338688 + T^{2} \)
$67$ \( 259308 + T^{2} \)
$71$ \( 662700 + T^{2} \)
$73$ \( 15552 + T^{2} \)
$79$ \( 1891308 + T^{2} \)
$83$ \( ( -252 + T )^{2} \)
$89$ \( 46128 + T^{2} \)
$97$ \( 2076672 + T^{2} \)
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