# Properties

 Label 1680.2.t.h Level $1680$ Weight $2$ Character orbit 1680.t Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} -\beta_{1} q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} -\beta_{1} q^{7} - q^{9} + 2 q^{11} -2 \beta_{2} q^{13} + \beta_{2} q^{15} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{17} + 2 q^{19} - q^{21} + 4 \beta_{1} q^{23} + 5 q^{25} + \beta_{1} q^{27} + ( 4 - 2 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} -2 \beta_{1} q^{33} + \beta_{2} q^{35} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{37} -2 \beta_{3} q^{39} + ( -8 - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{43} + \beta_{3} q^{45} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} - q^{49} + ( 2 - 2 \beta_{3} ) q^{51} -2 \beta_{1} q^{53} -2 \beta_{3} q^{55} -2 \beta_{1} q^{57} + ( -8 - 2 \beta_{3} ) q^{61} + \beta_{1} q^{63} + 10 \beta_{1} q^{65} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{67} + 4 q^{69} + ( -8 + 2 \beta_{3} ) q^{71} + ( -12 \beta_{1} - 2 \beta_{2} ) q^{73} -5 \beta_{1} q^{75} -2 \beta_{1} q^{77} -4 \beta_{3} q^{79} + q^{81} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 4 - 6 \beta_{3} ) q^{89} -2 \beta_{3} q^{91} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{93} -2 \beta_{3} q^{95} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 8q^{11} + 8q^{19} - 4q^{21} + 20q^{25} + 16q^{29} - 16q^{31} - 32q^{41} - 4q^{49} + 8q^{51} - 32q^{61} + 16q^{69} - 32q^{71} + 4q^{81} + 16q^{89} - 8q^{99} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 0.618034i − 1.61803i − 0.618034i 1.61803i
0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
1009.2 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 0
1009.3 0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
1009.4 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.t.h 4
3.b odd 2 1 5040.2.t.u 4
4.b odd 2 1 840.2.t.c 4
5.b even 2 1 inner 1680.2.t.h 4
5.c odd 4 1 8400.2.a.cz 2
5.c odd 4 1 8400.2.a.db 2
12.b even 2 1 2520.2.t.f 4
15.d odd 2 1 5040.2.t.u 4
20.d odd 2 1 840.2.t.c 4
20.e even 4 1 4200.2.a.bj 2
20.e even 4 1 4200.2.a.bk 2
60.h even 2 1 2520.2.t.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 4.b odd 2 1
840.2.t.c 4 20.d odd 2 1
1680.2.t.h 4 1.a even 1 1 trivial
1680.2.t.h 4 5.b even 2 1 inner
2520.2.t.f 4 12.b even 2 1
2520.2.t.f 4 60.h even 2 1
4200.2.a.bj 2 20.e even 4 1
4200.2.a.bk 2 20.e even 4 1
5040.2.t.u 4 3.b odd 2 1
5040.2.t.u 4 15.d odd 2 1
8400.2.a.cz 2 5.c odd 4 1
8400.2.a.db 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11} - 2$$ $$T_{13}^{2} + 20$$ $$T_{19} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$( -2 + T )^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -4 - 8 T + T^{2} )^{2}$$
$31$ $$( -4 + 8 T + T^{2} )^{2}$$
$37$ $$256 + 48 T^{2} + T^{4}$$
$41$ $$( 44 + 16 T + T^{2} )^{2}$$
$43$ $$256 + 48 T^{2} + T^{4}$$
$47$ $$256 + 48 T^{2} + T^{4}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 44 + 16 T + T^{2} )^{2}$$
$67$ $$256 + 112 T^{2} + T^{4}$$
$71$ $$( 44 + 16 T + T^{2} )^{2}$$
$73$ $$15376 + 328 T^{2} + T^{4}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$4096 + 192 T^{2} + T^{4}$$
$89$ $$( -164 - 8 T + T^{2} )^{2}$$
$97$ $$1936 + 168 T^{2} + T^{4}$$