# Properties

 Label 168.2.i.e Level 168 Weight 2 Character orbit 168.i Analytic conductor 1.341 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 168.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34148675396$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{4} - \beta_{7} ) q^{6} + ( 1 - \beta_{6} + 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} - \beta_{5} ) q^{8} + ( 2 + \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{4} - \beta_{7} ) q^{6} + ( 1 - \beta_{6} + 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} - \beta_{5} ) q^{8} + ( 2 + \beta_{3} + \beta_{5} ) q^{9} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{10} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{11} + ( -\beta_{1} + \beta_{6} - 2 \beta_{7} ) q^{12} + ( 2 \beta_{4} + \beta_{6} ) q^{13} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{14} + ( 1 - \beta_{3} - \beta_{5} ) q^{15} + ( -3 + \beta_{2} ) q^{16} + ( -3 - \beta_{2} + 2 \beta_{5} ) q^{18} + ( 2 \beta_{4} + \beta_{6} ) q^{19} + ( 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{20} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{21} + ( -2 + 2 \beta_{2} ) q^{22} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{23} + ( -3 \beta_{1} - \beta_{4} - 4 \beta_{6} + 3 \beta_{7} ) q^{24} + 3 q^{25} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} + ( -\beta_{4} - 4 \beta_{6} ) q^{27} + ( -1 - \beta_{2} - 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{28} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{29} + ( 3 + \beta_{2} + \beta_{5} ) q^{30} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{31} + ( -2 \beta_{3} - 3 \beta_{5} ) q^{32} + ( -4 \beta_{1} - 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{33} + ( 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{35} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{36} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{38} + ( -5 - \beta_{3} - \beta_{5} ) q^{39} + ( -\beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{40} + ( 8 \beta_{1} + 4 \beta_{4} + 6 \beta_{6} - 4 \beta_{7} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{42} + ( -2 - 4 \beta_{2} ) q^{43} + ( -4 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 \beta_{4} + \beta_{6} ) q^{45} + ( 6 + 2 \beta_{2} ) q^{46} + ( -8 \beta_{1} - 4 \beta_{4} - 6 \beta_{6} + 4 \beta_{7} ) q^{47} + ( \beta_{1} + 4 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{48} + ( -5 - 2 \beta_{6} + 4 \beta_{7} ) q^{49} + 3 \beta_{5} q^{50} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} ) q^{52} + ( \beta_{1} + 4 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{54} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{56} + ( -5 - \beta_{3} - \beta_{5} ) q^{57} + ( 4 - 4 \beta_{2} ) q^{58} + 7 \beta_{6} q^{59} + ( -1 - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{60} + ( 2 \beta_{4} + \beta_{6} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{62} + ( 2 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{63} + ( 7 + 3 \beta_{2} ) q^{64} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{65} + ( 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{66} + ( 2 + 4 \beta_{2} ) q^{67} + ( -2 \beta_{4} + 4 \beta_{6} ) q^{69} + ( -2 + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{70} + ( 4 \beta_{3} + 4 \beta_{5} ) q^{71} + ( -1 + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{72} + ( 6 \beta_{6} - 12 \beta_{7} ) q^{73} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{75} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} ) q^{76} + ( 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} ) q^{77} + ( 3 + \beta_{2} - 5 \beta_{5} ) q^{78} -10 q^{79} + ( -2 \beta_{1} - \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{80} + ( -1 + 4 \beta_{3} + 4 \beta_{5} ) q^{81} + ( -6 \beta_{4} + 2 \beta_{6} - 10 \beta_{7} ) q^{82} -5 \beta_{6} q^{83} + ( 7 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{84} + ( 8 \beta_{3} - 2 \beta_{5} ) q^{86} + ( 8 \beta_{1} + 4 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} ) q^{87} + ( 10 + 2 \beta_{2} ) q^{88} + ( 2 \beta_{1} - \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{90} + ( -2 - 4 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{91} + ( -4 \beta_{3} + 6 \beta_{5} ) q^{92} + ( -2 - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{93} + ( 6 \beta_{4} - 2 \beta_{6} + 10 \beta_{7} ) q^{94} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{95} + ( -\beta_{1} - 3 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{96} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{97} + ( 4 \beta_{1} + 2 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{98} + ( 2 + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} + 8q^{7} + 16q^{9} + O(q^{10})$$ $$8q - 4q^{4} + 8q^{7} + 16q^{9} + 8q^{15} - 28q^{16} - 20q^{18} - 24q^{22} + 24q^{25} - 4q^{28} + 20q^{30} - 8q^{36} - 40q^{39} + 12q^{42} + 40q^{46} - 40q^{49} - 40q^{57} + 48q^{58} - 4q^{60} + 16q^{63} + 44q^{64} - 24q^{70} - 20q^{72} + 20q^{78} - 80q^{79} - 8q^{81} + 60q^{84} + 72q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 13 x^{4} + 12 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 4 \nu^{2} - 3$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 6 \nu^{4} - 17 \nu^{2} + 18$$$$)/24$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 14 \nu^{5} - 97 \nu^{3} + 138 \nu$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 6 \nu^{4} - 5 \nu^{2} - 6$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 14 \nu^{5} - 25 \nu^{3} - 78 \nu$$$$)/72$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} - 22 \nu^{5} + 5 \nu^{3} + 30 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2 \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{5} - 8 \beta_{3} + 3 \beta_{2} + 11$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{7} + 10 \beta_{6} - 5 \beta_{4} + 15 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{5} - 38 \beta_{3} + 18 \beta_{2} + 50$$ $$\nu^{7}$$ $$=$$ $$42 \beta_{7} + 43 \beta_{6} - 20 \beta_{4} + 57 \beta_{1}$$

## Character values

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

 $$n$$ $$73$$ $$85$$ $$113$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 0.578737 + 0.965926i −0.578737 − 0.965926i 0.578737 − 0.965926i −0.578737 + 0.965926i −2.15988 − 0.258819i 2.15988 + 0.258819i −2.15988 + 0.258819i 2.15988 − 0.258819i
−0.866025 1.11803i −1.58114 + 0.707107i −0.500000 + 1.93649i 1.41421i 2.15988 + 1.15539i 1.00000 + 2.44949i 2.59808 1.11803i 2.00000 2.23607i −1.58114 + 1.22474i
125.2 −0.866025 1.11803i 1.58114 0.707107i −0.500000 + 1.93649i 1.41421i −2.15988 1.15539i 1.00000 2.44949i 2.59808 1.11803i 2.00000 2.23607i 1.58114 1.22474i
125.3 −0.866025 + 1.11803i −1.58114 0.707107i −0.500000 1.93649i 1.41421i 2.15988 1.15539i 1.00000 2.44949i 2.59808 + 1.11803i 2.00000 + 2.23607i −1.58114 1.22474i
125.4 −0.866025 + 1.11803i 1.58114 + 0.707107i −0.500000 1.93649i 1.41421i −2.15988 + 1.15539i 1.00000 + 2.44949i 2.59808 + 1.11803i 2.00000 + 2.23607i 1.58114 + 1.22474i
125.5 0.866025 1.11803i −1.58114 + 0.707107i −0.500000 1.93649i 1.41421i −0.578737 + 2.38014i 1.00000 2.44949i −2.59808 1.11803i 2.00000 2.23607i −1.58114 1.22474i
125.6 0.866025 1.11803i 1.58114 0.707107i −0.500000 1.93649i 1.41421i 0.578737 2.38014i 1.00000 + 2.44949i −2.59808 1.11803i 2.00000 2.23607i 1.58114 + 1.22474i
125.7 0.866025 + 1.11803i −1.58114 0.707107i −0.500000 + 1.93649i 1.41421i −0.578737 2.38014i 1.00000 + 2.44949i −2.59808 + 1.11803i 2.00000 + 2.23607i −1.58114 + 1.22474i
125.8 0.866025 + 1.11803i 1.58114 + 0.707107i −0.500000 + 1.93649i 1.41421i 0.578737 + 2.38014i 1.00000 2.44949i −2.59808 + 1.11803i 2.00000 + 2.23607i 1.58114 1.22474i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.i.e 8
3.b odd 2 1 inner 168.2.i.e 8
4.b odd 2 1 672.2.i.d 8
7.b odd 2 1 inner 168.2.i.e 8
8.b even 2 1 inner 168.2.i.e 8
8.d odd 2 1 672.2.i.d 8
12.b even 2 1 672.2.i.d 8
21.c even 2 1 inner 168.2.i.e 8
24.f even 2 1 672.2.i.d 8
24.h odd 2 1 inner 168.2.i.e 8
28.d even 2 1 672.2.i.d 8
56.e even 2 1 672.2.i.d 8
56.h odd 2 1 inner 168.2.i.e 8
84.h odd 2 1 672.2.i.d 8
168.e odd 2 1 672.2.i.d 8
168.i even 2 1 inner 168.2.i.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.e 8 1.a even 1 1 trivial
168.2.i.e 8 3.b odd 2 1 inner
168.2.i.e 8 7.b odd 2 1 inner
168.2.i.e 8 8.b even 2 1 inner
168.2.i.e 8 21.c even 2 1 inner
168.2.i.e 8 24.h odd 2 1 inner
168.2.i.e 8 56.h odd 2 1 inner
168.2.i.e 8 168.i even 2 1 inner
672.2.i.d 8 4.b odd 2 1
672.2.i.d 8 8.d odd 2 1
672.2.i.d 8 12.b even 2 1
672.2.i.d 8 24.f even 2 1
672.2.i.d 8 28.d even 2 1
672.2.i.d 8 56.e even 2 1
672.2.i.d 8 84.h odd 2 1
672.2.i.d 8 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(168, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{11}^{2} - 12$$ $$T_{13}^{2} - 10$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )^{4}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{4}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 16 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 17 T^{2} )^{8}$$
$19$ $$( 1 + 28 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 26 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 + 10 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{8}$$
$41$ $$( 1 - 38 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 26 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 26 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{8}$$
$59$ $$( 1 - 20 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 112 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 74 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 62 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 70 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{8}$$
$83$ $$( 1 - 116 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{4}$$