Properties

Label 168.4.q.d
Level $168$
Weight $4$
Character orbit 168.q
Analytic conductor $9.912$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
Defining polynomial: \(x^{4} - x^{3} + 127 x^{2} + 126 x + 15876\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 5 \beta_{2} ) q^{5} + ( 8 - 17 \beta_{2} + \beta_{3} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 5 \beta_{2} ) q^{5} + ( 8 - 17 \beta_{2} + \beta_{3} ) q^{7} -9 \beta_{2} q^{9} + ( -5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{11} + ( -9 - \beta_{3} ) q^{13} + ( -12 - 3 \beta_{3} ) q^{15} + ( 4 - 4 \beta_{1} - 4 \beta_{3} ) q^{17} + ( 7 \beta_{1} - 62 \beta_{2} ) q^{19} + ( -27 + 3 \beta_{1} - 27 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -4 \beta_{1} - 72 \beta_{2} ) q^{23} + ( -17 - 9 \beta_{1} + 26 \beta_{2} - 9 \beta_{3} ) q^{25} -27 q^{27} + ( -50 + 5 \beta_{3} ) q^{29} + ( 179 + 2 \beta_{1} - 181 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -15 \beta_{1} + 15 \beta_{2} ) q^{33} + ( -68 - 4 \beta_{1} - 86 \beta_{2} - 17 \beta_{3} ) q^{35} + ( \beta_{1} + 26 \beta_{2} ) q^{37} + ( -27 - 3 \beta_{1} + 30 \beta_{2} - 3 \beta_{3} ) q^{39} + ( -94 + 18 \beta_{3} ) q^{41} + ( 215 - 27 \beta_{3} ) q^{43} + ( -36 - 9 \beta_{1} + 45 \beta_{2} - 9 \beta_{3} ) q^{45} + ( 36 \beta_{1} - 202 \beta_{2} ) q^{47} + ( -99 + 34 \beta_{1} - 17 \beta_{2} + 17 \beta_{3} ) q^{49} -12 \beta_{1} q^{51} + ( -346 - 5 \beta_{1} + 351 \beta_{2} - 5 \beta_{3} ) q^{53} + ( 630 + 25 \beta_{3} ) q^{55} + ( -165 - 21 \beta_{3} ) q^{57} + ( 306 - 27 \beta_{1} - 279 \beta_{2} - 27 \beta_{3} ) q^{59} + ( -28 \beta_{1} + 594 \beta_{2} ) q^{61} + ( -153 + 9 \beta_{1} + 72 \beta_{2} ) q^{63} + ( -14 \beta_{1} + 176 \beta_{2} ) q^{65} + ( -153 + 73 \beta_{1} + 80 \beta_{2} + 73 \beta_{3} ) q^{67} + ( -228 + 12 \beta_{3} ) q^{69} + ( 270 + 76 \beta_{3} ) q^{71} + ( 377 + 63 \beta_{1} - 440 \beta_{2} + 63 \beta_{3} ) q^{73} + ( -27 \beta_{1} + 78 \beta_{2} ) q^{75} + ( -630 - 45 \beta_{1} + 675 \beta_{2} + 40 \beta_{3} ) q^{77} + ( 48 \beta_{1} + 377 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( -108 - 67 \beta_{3} ) q^{83} + ( 488 + 16 \beta_{3} ) q^{85} + ( -150 + 15 \beta_{1} + 135 \beta_{2} + 15 \beta_{3} ) q^{87} + ( 22 \beta_{1} + 918 \beta_{2} ) q^{89} + ( -198 - 17 \beta_{1} + 170 \beta_{2} - 18 \beta_{3} ) q^{91} + ( 6 \beta_{1} - 543 \beta_{2} ) q^{93} + ( -1102 - 90 \beta_{1} + 1192 \beta_{2} - 90 \beta_{3} ) q^{95} + ( 892 - 55 \beta_{3} ) q^{97} + 45 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} - 9q^{5} - 18q^{9} + O(q^{10}) \) \( 4q + 6q^{3} - 9q^{5} - 18q^{9} - 5q^{11} - 38q^{13} - 54q^{15} + 4q^{17} - 117q^{19} - 153q^{21} - 148q^{23} - 43q^{25} - 108q^{27} - 190q^{29} + 360q^{31} + 15q^{33} - 482q^{35} + 53q^{37} - 57q^{39} - 340q^{41} + 806q^{43} - 81q^{45} - 368q^{47} - 362q^{49} - 12q^{51} - 697q^{53} + 2570q^{55} - 702q^{57} + 585q^{59} + 1160q^{61} - 459q^{63} + 338q^{65} - 233q^{67} - 888q^{69} + 1232q^{71} + 817q^{73} + 129q^{75} - 1135q^{77} + 802q^{79} - 162q^{81} - 566q^{83} + 1984q^{85} - 285q^{87} + 1858q^{89} - 505q^{91} - 1080q^{93} - 2294q^{95} + 3458q^{97} + 90q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 127 x^{2} + 126 x + 15876\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 127 \nu^{2} - 127 \nu + 15876 \)\()/16002\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 253 \)\()/127\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 126 \beta_{2} + \beta_{1} - 127\)
\(\nu^{3}\)\(=\)\(127 \beta_{3} - 253\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−5.36805 9.29774i
5.86805 + 10.1638i
−5.36805 + 9.29774i
5.86805 10.1638i
0 1.50000 2.59808i 0 −7.86805 13.6279i 0 11.2361 14.7224i 0 −4.50000 7.79423i 0
25.2 0 1.50000 2.59808i 0 3.36805 + 5.83364i 0 −11.2361 14.7224i 0 −4.50000 7.79423i 0
121.1 0 1.50000 + 2.59808i 0 −7.86805 + 13.6279i 0 11.2361 + 14.7224i 0 −4.50000 + 7.79423i 0
121.2 0 1.50000 + 2.59808i 0 3.36805 5.83364i 0 −11.2361 + 14.7224i 0 −4.50000 + 7.79423i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.q.d 4
3.b odd 2 1 504.4.s.f 4
4.b odd 2 1 336.4.q.g 4
7.c even 3 1 inner 168.4.q.d 4
7.c even 3 1 1176.4.a.r 2
7.d odd 6 1 1176.4.a.u 2
21.h odd 6 1 504.4.s.f 4
28.f even 6 1 2352.4.a.bo 2
28.g odd 6 1 336.4.q.g 4
28.g odd 6 1 2352.4.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 1.a even 1 1 trivial
168.4.q.d 4 7.c even 3 1 inner
336.4.q.g 4 4.b odd 2 1
336.4.q.g 4 28.g odd 6 1
504.4.s.f 4 3.b odd 2 1
504.4.s.f 4 21.h odd 6 1
1176.4.a.r 2 7.c even 3 1
1176.4.a.u 2 7.d odd 6 1
2352.4.a.bo 2 28.f even 6 1
2352.4.a.cc 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 9 T_{5}^{3} + 187 T_{5}^{2} - 954 T_{5} + 11236 \) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( 11236 - 954 T + 187 T^{2} + 9 T^{3} + T^{4} \)
$7$ \( 117649 + 181 T^{2} + T^{4} \)
$11$ \( 9922500 - 15750 T + 3175 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( ( -36 + 19 T + T^{2} )^{2} \)
$17$ \( 4064256 + 8064 T + 2032 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 7639696 - 323388 T + 16453 T^{2} + 117 T^{3} + T^{4} \)
$23$ \( 11943936 + 511488 T + 18448 T^{2} + 148 T^{3} + T^{4} \)
$29$ \( ( -900 + 95 T + T^{2} )^{2} \)
$31$ \( 1017291025 - 11482200 T + 97705 T^{2} - 360 T^{3} + T^{4} \)
$37$ \( 331776 - 30528 T + 2233 T^{2} - 53 T^{3} + T^{4} \)
$41$ \( ( -33680 + 170 T + T^{2} )^{2} \)
$43$ \( ( -51434 - 403 T + T^{2} )^{2} \)
$47$ \( 16838695696 - 47753152 T + 265188 T^{2} + 368 T^{3} + T^{4} \)
$53$ \( 13993943616 + 82452312 T + 367513 T^{2} + 697 T^{3} + T^{4} \)
$59$ \( 41990400 + 3790800 T + 348705 T^{2} - 585 T^{3} + T^{4} \)
$61$ \( 56368256400 - 275407200 T + 1108180 T^{2} - 1160 T^{3} + T^{4} \)
$67$ \( 434563097796 - 153596862 T + 713503 T^{2} + 233 T^{3} + T^{4} \)
$71$ \( ( -634356 - 616 T + T^{2} )^{2} \)
$73$ \( 111698997796 + 273052838 T + 1001703 T^{2} - 817 T^{3} + T^{4} \)
$79$ \( 16920546241 + 104323358 T + 773283 T^{2} - 802 T^{3} + T^{4} \)
$83$ \( ( -546714 + 283 T + T^{2} )^{2} \)
$89$ \( 643101348096 - 1489997088 T + 2650228 T^{2} - 1858 T^{3} + T^{4} \)
$97$ \( ( 365454 - 1729 T + T^{2} )^{2} \)
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