# Properties

 Label 168.2.i.b Level 168 Weight 2 Character orbit 168.i Analytic conductor 1.341 Analytic rank 0 Dimension 4 CM discriminant -24 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + 2 \beta_{2} q^{5} + \beta_{3} q^{6} + ( 1 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + 2 \beta_{2} q^{5} + \beta_{3} q^{6} + ( 1 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} -2 \beta_{3} q^{10} -4 \beta_{1} q^{11} -2 \beta_{2} q^{12} + ( \beta_{1} - 2 \beta_{2} ) q^{14} + 6 q^{15} + 4 q^{16} -3 \beta_{1} q^{18} + 4 \beta_{2} q^{20} + ( -3 \beta_{1} - \beta_{2} ) q^{21} -8 q^{22} + 2 \beta_{3} q^{24} -7 q^{25} + 3 \beta_{2} q^{27} + ( 2 + 2 \beta_{3} ) q^{28} -2 \beta_{1} q^{29} + 6 \beta_{1} q^{30} -2 \beta_{3} q^{31} + 4 \beta_{1} q^{32} -4 \beta_{3} q^{33} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{35} -6 q^{36} -4 \beta_{3} q^{40} + ( -6 + \beta_{3} ) q^{42} -8 \beta_{1} q^{44} -6 \beta_{2} q^{45} -4 \beta_{2} q^{48} + ( -5 + 2 \beta_{3} ) q^{49} -7 \beta_{1} q^{50} + 10 \beta_{1} q^{53} -3 \beta_{3} q^{54} + 8 \beta_{3} q^{55} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{56} -4 q^{58} -6 \beta_{2} q^{59} + 12 q^{60} + 4 \beta_{2} q^{62} + ( -3 - 3 \beta_{3} ) q^{63} + 8 q^{64} + 8 \beta_{2} q^{66} + ( 12 - 2 \beta_{3} ) q^{70} -6 \beta_{1} q^{72} + 4 \beta_{3} q^{73} + 7 \beta_{2} q^{75} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{77} + 10 q^{79} + 8 \beta_{2} q^{80} + 9 q^{81} -10 \beta_{2} q^{83} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{84} -2 \beta_{3} q^{87} -16 q^{88} + 6 \beta_{3} q^{90} + 6 \beta_{1} q^{93} + 4 \beta_{3} q^{96} -8 \beta_{3} q^{97} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{98} + 12 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + 4q^{7} - 12q^{9} + O(q^{10})$$ $$4q + 8q^{4} + 4q^{7} - 12q^{9} + 24q^{15} + 16q^{16} - 32q^{22} - 28q^{25} + 8q^{28} - 24q^{36} - 24q^{42} - 20q^{49} - 16q^{58} + 48q^{60} - 12q^{63} + 32q^{64} + 48q^{70} + 40q^{79} + 36q^{81} - 64q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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
 0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i −0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 + 2.44949i −2.82843 −3.00000 4.89898i
125.2 −1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 2.44949i −2.82843 −3.00000 4.89898i
125.3 1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 2.44949i 2.82843 −3.00000 4.89898i
125.4 1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 + 2.44949i 2.82843 −3.00000 4.89898i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 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.b 4
3.b odd 2 1 inner 168.2.i.b 4
4.b odd 2 1 672.2.i.b 4
7.b odd 2 1 inner 168.2.i.b 4
8.b even 2 1 inner 168.2.i.b 4
8.d odd 2 1 672.2.i.b 4
12.b even 2 1 672.2.i.b 4
21.c even 2 1 inner 168.2.i.b 4
24.f even 2 1 672.2.i.b 4
24.h odd 2 1 CM 168.2.i.b 4
28.d even 2 1 672.2.i.b 4
56.e even 2 1 672.2.i.b 4
56.h odd 2 1 inner 168.2.i.b 4
84.h odd 2 1 672.2.i.b 4
168.e odd 2 1 672.2.i.b 4
168.i even 2 1 inner 168.2.i.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.b 4 1.a even 1 1 trivial
168.2.i.b 4 3.b odd 2 1 inner
168.2.i.b 4 7.b odd 2 1 inner
168.2.i.b 4 8.b even 2 1 inner
168.2.i.b 4 21.c even 2 1 inner
168.2.i.b 4 24.h odd 2 1 CM
168.2.i.b 4 56.h odd 2 1 inner
168.2.i.b 4 168.i even 2 1 inner
672.2.i.b 4 4.b odd 2 1
672.2.i.b 4 8.d odd 2 1
672.2.i.b 4 12.b even 2 1
672.2.i.b 4 24.f even 2 1
672.2.i.b 4 28.d even 2 1
672.2.i.b 4 56.e even 2 1
672.2.i.b 4 84.h odd 2 1
672.2.i.b 4 168.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 $$S_{2}^{\mathrm{new}}(168, [\chi])$$:

 $$T_{5}^{2} + 12$$ $$T_{11}^{2} - 32$$ $$T_{13}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ $$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 10 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 - 23 T^{2} )^{4}$$
$29$ $$( 1 + 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 10 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{4}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 134 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2}$$