# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} -2 \beta_{2} q^{17} + ( -4 + \beta_{1} - \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( 4 + 3 \beta_{2} - \beta_{3} ) q^{27} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{31} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{33} + ( \beta_{1} + \beta_{3} ) q^{35} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -4 + 3 \beta_{1} + \beta_{3} ) q^{45} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{47} - q^{49} + ( -2 + 2 \beta_{3} ) q^{51} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{57} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} -4 \beta_{2} q^{65} + ( -2 - 4 \beta_{1} + 4 \beta_{3} ) q^{67} + ( -4 - 4 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( 4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{75} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{81} + ( \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{85} -6 \beta_{2} q^{89} + ( 2 + \beta_{1} - \beta_{3} ) q^{91} + ( -12 - 4 \beta_{1} + 8 \beta_{2} ) q^{93} + ( 4 - 6 \beta_{1} + 6 \beta_{3} ) q^{95} + ( 10 + 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 4 + 4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 4q^{5} + O(q^{10})$$ $$4q + 2q^{3} - 4q^{5} - 12q^{15} - 20q^{19} + 2q^{21} + 16q^{23} + 4q^{25} + 14q^{27} - 20q^{33} - 12q^{39} - 16q^{43} - 20q^{45} - 8q^{47} - 4q^{49} - 4q^{51} + 24q^{53} - 20q^{57} - 8q^{63} + 8q^{67} - 12q^{69} + 8q^{71} + 16q^{73} + 22q^{75} + 4q^{81} + 4q^{91} - 40q^{93} + 40q^{95} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 0.618034i − 0.618034i 1.61803i − 1.61803i
0 −0.618034 1.61803i 0 1.23607 0 1.00000i 0 −2.23607 + 2.00000i 0
239.2 0 −0.618034 + 1.61803i 0 1.23607 0 1.00000i 0 −2.23607 2.00000i 0
239.3 0 1.61803 0.618034i 0 −3.23607 0 1.00000i 0 2.23607 2.00000i 0
239.4 0 1.61803 + 0.618034i 0 −3.23607 0 1.00000i 0 2.23607 + 2.00000i 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.f 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.j.b 4
3.b odd 2 1 672.2.j.c 4
4.b odd 2 1 168.2.j.c yes 4
8.b even 2 1 168.2.j.a 4
8.d odd 2 1 672.2.j.c 4
12.b even 2 1 168.2.j.a 4
24.f even 2 1 inner 672.2.j.b 4
24.h odd 2 1 168.2.j.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.j.a 4 8.b even 2 1
168.2.j.a 4 12.b even 2 1
168.2.j.c yes 4 4.b odd 2 1
168.2.j.c yes 4 24.h odd 2 1
672.2.j.b 4 1.a even 1 1 trivial
672.2.j.b 4 24.f even 2 1 inner
672.2.j.c 4 3.b odd 2 1
672.2.j.c 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} - 4$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 10 T + 58 T^{2} + 190 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 + 18 T^{2} + 961 T^{4} )^{2}$$
$37$ $$1 - 100 T^{2} + 4918 T^{4} - 136900 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 8 T + 82 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 12 T + 122 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 224 T^{2} + 19486 T^{4} - 779744 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 104 T^{2} + 5646 T^{4} - 386984 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 - 4 T + 58 T^{2} - 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 4 T + 66 T^{2} - 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8}$$
$83$ $$1 - 80 T^{2} + 12958 T^{4} - 551120 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 142 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2}$$