Properties

Label 168.4.a.h
Level $168$
Weight $4$
Character orbit 168.a
Self dual yes
Analytic conductor $9.912$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 7 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 7 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 9 + 3 \beta ) q^{11} + ( 24 - 2 \beta ) q^{13} + ( 21 - 3 \beta ) q^{15} + ( 17 + 9 \beta ) q^{17} + ( -8 + 4 \beta ) q^{19} + 21 q^{21} + ( 55 - 7 \beta ) q^{23} + ( 101 - 14 \beta ) q^{25} + 27 q^{27} + ( 106 - 4 \beta ) q^{29} + ( -68 + 4 \beta ) q^{31} + ( 27 + 9 \beta ) q^{33} + ( 49 - 7 \beta ) q^{35} + ( -12 + 26 \beta ) q^{37} + ( 72 - 6 \beta ) q^{39} + ( 347 - 9 \beta ) q^{41} + ( -292 - 8 \beta ) q^{43} + ( 63 - 9 \beta ) q^{45} + ( -158 - 2 \beta ) q^{47} + 49 q^{49} + ( 51 + 27 \beta ) q^{51} + ( 280 + 6 \beta ) q^{53} + ( -468 + 12 \beta ) q^{55} + ( -24 + 12 \beta ) q^{57} + ( -246 + 26 \beta ) q^{59} + ( -302 - 28 \beta ) q^{61} + 63 q^{63} + ( 522 - 38 \beta ) q^{65} + ( -510 + 10 \beta ) q^{67} + ( 165 - 21 \beta ) q^{69} + ( -855 - 25 \beta ) q^{71} + ( -656 - 6 \beta ) q^{73} + ( 303 - 42 \beta ) q^{75} + ( 63 + 21 \beta ) q^{77} + ( -278 - 42 \beta ) q^{79} + 81 q^{81} + ( -132 + 32 \beta ) q^{83} + ( -1474 + 46 \beta ) q^{85} + ( 318 - 12 \beta ) q^{87} + ( 35 + 95 \beta ) q^{89} + ( 168 - 14 \beta ) q^{91} + ( -204 + 12 \beta ) q^{93} + ( -764 + 36 \beta ) q^{95} + ( -68 - 10 \beta ) q^{97} + ( 81 + 27 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + 18q^{11} + 48q^{13} + 42q^{15} + 34q^{17} - 16q^{19} + 42q^{21} + 110q^{23} + 202q^{25} + 54q^{27} + 212q^{29} - 136q^{31} + 54q^{33} + 98q^{35} - 24q^{37} + 144q^{39} + 694q^{41} - 584q^{43} + 126q^{45} - 316q^{47} + 98q^{49} + 102q^{51} + 560q^{53} - 936q^{55} - 48q^{57} - 492q^{59} - 604q^{61} + 126q^{63} + 1044q^{65} - 1020q^{67} + 330q^{69} - 1710q^{71} - 1312q^{73} + 606q^{75} + 126q^{77} - 556q^{79} + 162q^{81} - 264q^{83} - 2948q^{85} + 636q^{87} + 70q^{89} + 336q^{91} - 408q^{93} - 1528q^{95} - 136q^{97} + 162q^{99} + O(q^{100}) \)

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
7.15207
−6.15207
0 3.00000 0 −6.30413 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 20.3041 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 168.4.a.h 2
3.b odd 2 1 504.4.a.j 2
4.b odd 2 1 336.4.a.n 2
7.b odd 2 1 1176.4.a.p 2
8.b even 2 1 1344.4.a.bd 2
8.d odd 2 1 1344.4.a.bl 2
12.b even 2 1 1008.4.a.y 2
28.d even 2 1 2352.4.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.h 2 1.a even 1 1 trivial
336.4.a.n 2 4.b odd 2 1
504.4.a.j 2 3.b odd 2 1
1008.4.a.y 2 12.b even 2 1
1176.4.a.p 2 7.b odd 2 1
1344.4.a.bd 2 8.b even 2 1
1344.4.a.bl 2 8.d odd 2 1
2352.4.a.bv 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(168))\):

\( T_{5}^{2} - 14 T_{5} - 128 \)
\( T_{11}^{2} - 18 T_{11} - 1512 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( -128 - 14 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1512 - 18 T + T^{2} \)
$13$ \( -132 - 48 T + T^{2} \)
$17$ \( -14048 - 34 T + T^{2} \)
$19$ \( -2768 + 16 T + T^{2} \)
$23$ \( -5648 - 110 T + T^{2} \)
$29$ \( 8404 - 212 T + T^{2} \)
$31$ \( 1792 + 136 T + T^{2} \)
$37$ \( -119508 + 24 T + T^{2} \)
$41$ \( 106072 - 694 T + T^{2} \)
$43$ \( 73936 + 584 T + T^{2} \)
$47$ \( 24256 + 316 T + T^{2} \)
$53$ \( 72028 - 560 T + T^{2} \)
$59$ \( -59136 + 492 T + T^{2} \)
$61$ \( -47564 + 604 T + T^{2} \)
$67$ \( 242400 + 1020 T + T^{2} \)
$71$ \( 620400 + 1710 T + T^{2} \)
$73$ \( 423964 + 1312 T + T^{2} \)
$79$ \( -234944 + 556 T + T^{2} \)
$83$ \( -163824 + 264 T + T^{2} \)
$89$ \( -1596200 - 70 T + T^{2} \)
$97$ \( -13076 + 136 T + T^{2} \)
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