# Properties

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

# Related objects

## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{3} + ( -7 + 7 \zeta_{6} ) q^{5} + ( -14 - 7 \zeta_{6} ) q^{7} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{3} + ( -7 + 7 \zeta_{6} ) q^{5} + ( -14 - 7 \zeta_{6} ) q^{7} + ( -9 + 9 \zeta_{6} ) q^{9} -7 \zeta_{6} q^{11} -52 q^{13} -21 q^{15} -72 \zeta_{6} q^{17} + ( -20 + 20 \zeta_{6} ) q^{19} + ( 21 - 63 \zeta_{6} ) q^{21} + ( 48 - 48 \zeta_{6} ) q^{23} + 76 \zeta_{6} q^{25} -27 q^{27} -243 q^{29} -95 \zeta_{6} q^{31} + ( 21 - 21 \zeta_{6} ) q^{33} + ( 147 - 98 \zeta_{6} ) q^{35} + ( -352 + 352 \zeta_{6} ) q^{37} -156 \zeta_{6} q^{39} -296 q^{41} + 158 q^{43} -63 \zeta_{6} q^{45} + ( 142 - 142 \zeta_{6} ) q^{47} + ( 147 + 245 \zeta_{6} ) q^{49} + ( 216 - 216 \zeta_{6} ) q^{51} + 375 \zeta_{6} q^{53} + 49 q^{55} -60 q^{57} -279 \zeta_{6} q^{59} + ( -246 + 246 \zeta_{6} ) q^{61} + ( 189 - 126 \zeta_{6} ) q^{63} + ( 364 - 364 \zeta_{6} ) q^{65} + 730 \zeta_{6} q^{67} + 144 q^{69} + 338 q^{71} + 542 \zeta_{6} q^{73} + ( -228 + 228 \zeta_{6} ) q^{75} + ( -49 + 147 \zeta_{6} ) q^{77} + ( 305 - 305 \zeta_{6} ) q^{79} -81 \zeta_{6} q^{81} + 1123 q^{83} + 504 q^{85} -729 \zeta_{6} q^{87} + ( 426 - 426 \zeta_{6} ) q^{89} + ( 728 + 364 \zeta_{6} ) q^{91} + ( 285 - 285 \zeta_{6} ) q^{93} -140 \zeta_{6} q^{95} -369 q^{97} + 63 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 7q^{5} - 35q^{7} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 7q^{5} - 35q^{7} - 9q^{9} - 7q^{11} - 104q^{13} - 42q^{15} - 72q^{17} - 20q^{19} - 21q^{21} + 48q^{23} + 76q^{25} - 54q^{27} - 486q^{29} - 95q^{31} + 21q^{33} + 196q^{35} - 352q^{37} - 156q^{39} - 592q^{41} + 316q^{43} - 63q^{45} + 142q^{47} + 539q^{49} + 216q^{51} + 375q^{53} + 98q^{55} - 120q^{57} - 279q^{59} - 246q^{61} + 252q^{63} + 364q^{65} + 730q^{67} + 288q^{69} + 676q^{71} + 542q^{73} - 228q^{75} + 49q^{77} + 305q^{79} - 81q^{81} + 2246q^{83} + 1008q^{85} - 729q^{87} + 426q^{89} + 1820q^{91} + 285q^{93} - 140q^{95} - 738q^{97} + 126q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$85$$ $$113$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 2.59808i 0 −3.50000 6.06218i 0 −17.5000 + 6.06218i 0 −4.50000 7.79423i 0
121.1 0 1.50000 + 2.59808i 0 −3.50000 + 6.06218i 0 −17.5000 6.06218i 0 −4.50000 + 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 2
3.b odd 2 1 504.4.s.e 2
4.b odd 2 1 336.4.q.a 2
7.c even 3 1 inner 168.4.q.a 2
7.c even 3 1 1176.4.a.f 1
7.d odd 6 1 1176.4.a.i 1
21.h odd 6 1 504.4.s.e 2
28.f even 6 1 2352.4.a.c 1
28.g odd 6 1 336.4.q.a 2
28.g odd 6 1 2352.4.a.bg 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 7 T_{5} + 49$$ acting on $$S_{4}^{\mathrm{new}}(168, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$49 + 7 T + T^{2}$$
$7$ $$343 + 35 T + T^{2}$$
$11$ $$49 + 7 T + T^{2}$$
$13$ $$( 52 + T )^{2}$$
$17$ $$5184 + 72 T + T^{2}$$
$19$ $$400 + 20 T + T^{2}$$
$23$ $$2304 - 48 T + T^{2}$$
$29$ $$( 243 + T )^{2}$$
$31$ $$9025 + 95 T + T^{2}$$
$37$ $$123904 + 352 T + T^{2}$$
$41$ $$( 296 + T )^{2}$$
$43$ $$( -158 + T )^{2}$$
$47$ $$20164 - 142 T + T^{2}$$
$53$ $$140625 - 375 T + T^{2}$$
$59$ $$77841 + 279 T + T^{2}$$
$61$ $$60516 + 246 T + T^{2}$$
$67$ $$532900 - 730 T + T^{2}$$
$71$ $$( -338 + T )^{2}$$
$73$ $$293764 - 542 T + T^{2}$$
$79$ $$93025 - 305 T + T^{2}$$
$83$ $$( -1123 + T )^{2}$$
$89$ $$181476 - 426 T + T^{2}$$
$97$ $$( 369 + T )^{2}$$