# Properties

 Label 672.2.i.b Level 672 Weight 2 Character orbit 672.i Analytic conductor 5.366 Analytic rank 0 Dimension 4 CM discriminant -24 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.i (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -3 q^{9} -4 \beta_{1} q^{11} -6 q^{15} + ( 3 \beta_{1} - \beta_{2} ) q^{21} -7 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{1} q^{29} -2 \beta_{3} q^{31} + 4 \beta_{3} q^{33} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{35} -6 \beta_{2} q^{45} + ( -5 - 2 \beta_{3} ) q^{49} -10 \beta_{1} q^{53} + 8 \beta_{3} q^{55} + 6 \beta_{2} q^{59} + ( 3 - 3 \beta_{3} ) q^{63} -4 \beta_{3} q^{73} -7 \beta_{2} q^{75} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{77} -10 q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} -2 \beta_{3} q^{87} -6 \beta_{1} q^{93} + 8 \beta_{3} q^{97} + 12 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{7} - 12q^{9} - 24q^{15} - 28q^{25} - 20q^{49} + 12q^{63} - 40q^{79} + 36q^{81} + O(q^{100})$$

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 $$\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.707107 − 1.22474i −0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.2 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 0
209.3 0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.4 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 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
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 672.2.i.b 4
3.b odd 2 1 inner 672.2.i.b 4
4.b odd 2 1 168.2.i.b 4
7.b odd 2 1 inner 672.2.i.b 4
8.b even 2 1 inner 672.2.i.b 4
8.d odd 2 1 168.2.i.b 4
12.b even 2 1 168.2.i.b 4
21.c even 2 1 inner 672.2.i.b 4
24.f even 2 1 168.2.i.b 4
24.h odd 2 1 CM 672.2.i.b 4
28.d even 2 1 168.2.i.b 4
56.e even 2 1 168.2.i.b 4
56.h odd 2 1 inner 672.2.i.b 4
84.h odd 2 1 168.2.i.b 4
168.e odd 2 1 168.2.i.b 4
168.i even 2 1 inner 672.2.i.b 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$:

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

## Hecke characteristic polynomials

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