Properties

Label 1344.4.a.bi
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{337}) \)
Defining polynomial: \(x^{2} - x - 84\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{337}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( 3 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( 3 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 13 + \beta ) q^{11} + ( -48 - 2 \beta ) q^{13} + ( -9 - 3 \beta ) q^{15} + ( 39 - 3 \beta ) q^{17} + 20 q^{19} -21 q^{21} + ( -11 + 5 \beta ) q^{23} + ( 221 + 6 \beta ) q^{25} -27 q^{27} -102 q^{29} + ( -48 + 16 \beta ) q^{31} + ( -39 - 3 \beta ) q^{33} + ( 21 + 7 \beta ) q^{35} + ( -252 + 2 \beta ) q^{37} + ( 144 + 6 \beta ) q^{39} + ( -51 + 11 \beta ) q^{41} + ( 148 + 16 \beta ) q^{43} + ( 27 + 9 \beta ) q^{45} + ( 390 - 10 \beta ) q^{47} + 49 q^{49} + ( -117 + 9 \beta ) q^{51} + ( -96 - 18 \beta ) q^{53} + ( 376 + 16 \beta ) q^{55} -60 q^{57} + ( 106 + 14 \beta ) q^{59} + ( 50 + 16 \beta ) q^{61} + 63 q^{63} + ( -818 - 54 \beta ) q^{65} + ( 106 - 2 \beta ) q^{67} + ( 33 - 15 \beta ) q^{69} + ( 267 + 11 \beta ) q^{71} + ( 564 + 10 \beta ) q^{73} + ( -663 - 18 \beta ) q^{75} + ( 91 + 7 \beta ) q^{77} + ( -234 - 58 \beta ) q^{79} + 81 q^{81} + ( -412 - 48 \beta ) q^{83} + ( -894 + 30 \beta ) q^{85} + 306 q^{87} + ( -1059 + 27 \beta ) q^{89} + ( -336 - 14 \beta ) q^{91} + ( 144 - 48 \beta ) q^{93} + ( 60 + 20 \beta ) q^{95} + ( 200 + 70 \beta ) q^{97} + ( 117 + 9 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 6q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 6q^{5} + 14q^{7} + 18q^{9} + 26q^{11} - 96q^{13} - 18q^{15} + 78q^{17} + 40q^{19} - 42q^{21} - 22q^{23} + 442q^{25} - 54q^{27} - 204q^{29} - 96q^{31} - 78q^{33} + 42q^{35} - 504q^{37} + 288q^{39} - 102q^{41} + 296q^{43} + 54q^{45} + 780q^{47} + 98q^{49} - 234q^{51} - 192q^{53} + 752q^{55} - 120q^{57} + 212q^{59} + 100q^{61} + 126q^{63} - 1636q^{65} + 212q^{67} + 66q^{69} + 534q^{71} + 1128q^{73} - 1326q^{75} + 182q^{77} - 468q^{79} + 162q^{81} - 824q^{83} - 1788q^{85} + 612q^{87} - 2118q^{89} - 672q^{91} + 288q^{93} + 120q^{95} + 400q^{97} + 234q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.67878
9.67878
0 −3.00000 0 −15.3576 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 21.3576 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.bi 2
4.b odd 2 1 1344.4.a.bq 2
8.b even 2 1 336.4.a.o 2
8.d odd 2 1 168.4.a.g 2
24.f even 2 1 504.4.a.n 2
24.h odd 2 1 1008.4.a.bg 2
56.e even 2 1 1176.4.a.w 2
56.h odd 2 1 2352.4.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.g 2 8.d odd 2 1
336.4.a.o 2 8.b even 2 1
504.4.a.n 2 24.f even 2 1
1008.4.a.bg 2 24.h odd 2 1
1176.4.a.w 2 56.e even 2 1
1344.4.a.bi 2 1.a even 1 1 trivial
1344.4.a.bq 2 4.b odd 2 1
2352.4.a.br 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} - 6 T_{5} - 328 \)
\( T_{11}^{2} - 26 T_{11} - 168 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -328 - 6 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -168 - 26 T + T^{2} \)
$13$ \( 956 + 96 T + T^{2} \)
$17$ \( -1512 - 78 T + T^{2} \)
$19$ \( ( -20 + T )^{2} \)
$23$ \( -8304 + 22 T + T^{2} \)
$29$ \( ( 102 + T )^{2} \)
$31$ \( -83968 + 96 T + T^{2} \)
$37$ \( 62156 + 504 T + T^{2} \)
$41$ \( -38176 + 102 T + T^{2} \)
$43$ \( -64368 - 296 T + T^{2} \)
$47$ \( 118400 - 780 T + T^{2} \)
$53$ \( -99972 + 192 T + T^{2} \)
$59$ \( -54816 - 212 T + T^{2} \)
$61$ \( -83772 - 100 T + T^{2} \)
$67$ \( 9888 - 212 T + T^{2} \)
$71$ \( 30512 - 534 T + T^{2} \)
$73$ \( 284396 - 1128 T + T^{2} \)
$79$ \( -1078912 + 468 T + T^{2} \)
$83$ \( -606704 + 824 T + T^{2} \)
$89$ \( 875808 + 2118 T + T^{2} \)
$97$ \( -1611300 - 400 T + T^{2} \)
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