# Properties

 Label 672.2.i.c Level 672 Weight 2 Character orbit 672.i Analytic conductor 5.366 Analytic rank 0 Dimension 4 CM no Inner twists 4

# 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{9} + \beta_{3} q^{11} + 2 q^{13} + ( 2 + \beta_{1} ) q^{15} -3 \beta_{3} q^{17} + 4 q^{19} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} -\beta_{1} q^{23} + 3 q^{25} + ( -5 - \beta_{1} ) q^{27} + 2 \beta_{3} q^{29} -4 \beta_{2} q^{31} + ( -2 \beta_{2} + \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{3} ) q^{35} + 6 \beta_{2} q^{37} + ( 2 - 2 \beta_{1} ) q^{39} + \beta_{3} q^{41} -2 \beta_{2} q^{43} + ( 4 - \beta_{1} ) q^{45} -2 \beta_{3} q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{51} + 2 \beta_{2} q^{55} + ( 4 - 4 \beta_{1} ) q^{57} -4 \beta_{1} q^{59} -10 q^{61} + ( -2 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{63} + 2 \beta_{1} q^{65} + 2 \beta_{2} q^{67} + ( -2 - \beta_{1} ) q^{69} -\beta_{1} q^{71} + ( 3 - 3 \beta_{1} ) q^{75} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{77} -8 q^{79} + ( -7 + 4 \beta_{1} ) q^{81} + 8 \beta_{1} q^{83} -6 \beta_{2} q^{85} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{87} + 3 \beta_{3} q^{89} + ( 4 - 2 \beta_{2} ) q^{91} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{93} + 4 \beta_{1} q^{95} + 8 \beta_{2} q^{97} + ( -4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 8q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 8q^{7} - 4q^{9} + 8q^{13} + 8q^{15} + 16q^{19} + 8q^{21} + 12q^{25} - 20q^{27} + 8q^{39} + 16q^{45} + 4q^{49} + 16q^{57} - 40q^{61} - 8q^{63} - 8q^{69} + 12q^{75} - 32q^{79} - 28q^{81} + 16q^{91} + O(q^{100})$$

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
 1.22474 + 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i
0 1.00000 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 2.82843i 0
209.2 0 1.00000 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 2.82843i 0
209.3 0 1.00000 + 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 + 2.82843i 0
209.4 0 1.00000 + 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 + 2.82843i 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
3.b 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 672.2.i.c 4
3.b odd 2 1 inner 672.2.i.c 4
4.b odd 2 1 168.2.i.a 4
7.b odd 2 1 672.2.i.a 4
8.b even 2 1 672.2.i.a 4
8.d odd 2 1 168.2.i.c yes 4
12.b even 2 1 168.2.i.a 4
21.c even 2 1 672.2.i.a 4
24.f even 2 1 168.2.i.c yes 4
24.h odd 2 1 672.2.i.a 4
28.d even 2 1 168.2.i.c yes 4
56.e even 2 1 168.2.i.a 4
56.h odd 2 1 inner 672.2.i.c 4
84.h odd 2 1 168.2.i.c yes 4
168.e odd 2 1 168.2.i.a 4
168.i even 2 1 inner 672.2.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.a 4 4.b odd 2 1
168.2.i.a 4 12.b even 2 1
168.2.i.a 4 56.e even 2 1
168.2.i.a 4 168.e odd 2 1
168.2.i.c yes 4 8.d odd 2 1
168.2.i.c yes 4 24.f even 2 1
168.2.i.c yes 4 28.d even 2 1
168.2.i.c yes 4 84.h odd 2 1
672.2.i.a 4 7.b odd 2 1
672.2.i.a 4 8.b even 2 1
672.2.i.a 4 21.c even 2 1
672.2.i.a 4 24.h odd 2 1
672.2.i.c 4 1.a even 1 1 trivial
672.2.i.c 4 3.b odd 2 1 inner
672.2.i.c 4 56.h odd 2 1 inner
672.2.i.c 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 $$S_{2}^{\mathrm{new}}(672, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{11}^{2} - 6$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 2 T + 3 T^{2} )^{2}$$
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 16 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{4}$$
$23$ $$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 34 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 14 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 34 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 76 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 - 86 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 38 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 124 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$