# Properties

 Label 1680.2.t.f Level $1680$ Weight $2$ Character orbit 1680.t Analytic conductor $13.415$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( 1 + 2 i ) q^{5} + i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + ( 1 + 2 i ) q^{5} + i q^{7} - q^{9} + 6 q^{11} + 2 i q^{13} + ( -2 + i ) q^{15} + 4 i q^{17} -6 q^{19} - q^{21} + ( -3 + 4 i ) q^{25} -i q^{27} + 2 q^{29} + 10 q^{31} + 6 i q^{33} + ( -2 + i ) q^{35} -4 i q^{37} -2 q^{39} + 2 q^{41} -4 i q^{43} + ( -1 - 2 i ) q^{45} - q^{49} -4 q^{51} -6 i q^{53} + ( 6 + 12 i ) q^{55} -6 i q^{57} -8 q^{59} -2 q^{61} -i q^{63} + ( -4 + 2 i ) q^{65} + 16 i q^{67} -10 q^{71} + 6 i q^{73} + ( -4 - 3 i ) q^{75} + 6 i q^{77} + 4 q^{79} + q^{81} + 8 i q^{83} + ( -8 + 4 i ) q^{85} + 2 i q^{87} -6 q^{89} -2 q^{91} + 10 i q^{93} + ( -6 - 12 i ) q^{95} -2 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{5} - 2q^{9} + 12q^{11} - 4q^{15} - 12q^{19} - 2q^{21} - 6q^{25} + 4q^{29} + 20q^{31} - 4q^{35} - 4q^{39} + 4q^{41} - 2q^{45} - 2q^{49} - 8q^{51} + 12q^{55} - 16q^{59} - 4q^{61} - 8q^{65} - 20q^{71} - 8q^{75} + 8q^{79} + 2q^{81} - 16q^{85} - 12q^{89} - 4q^{91} - 12q^{95} - 12q^{99} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 1.00000i 0 1.00000 2.00000i 0 1.00000i 0 −1.00000 0
1009.2 0 1.00000i 0 1.00000 + 2.00000i 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.f 2
3.b odd 2 1 5040.2.t.e 2
4.b odd 2 1 105.2.d.a 2
5.b even 2 1 inner 1680.2.t.f 2
5.c odd 4 1 8400.2.a.bj 1
5.c odd 4 1 8400.2.a.ch 1
12.b even 2 1 315.2.d.c 2
15.d odd 2 1 5040.2.t.e 2
20.d odd 2 1 105.2.d.a 2
20.e even 4 1 525.2.a.b 1
20.e even 4 1 525.2.a.c 1
28.d even 2 1 735.2.d.a 2
28.f even 6 2 735.2.q.b 4
28.g odd 6 2 735.2.q.a 4
60.h even 2 1 315.2.d.c 2
60.l odd 4 1 1575.2.a.e 1
60.l odd 4 1 1575.2.a.i 1
84.h odd 2 1 2205.2.d.f 2
140.c even 2 1 735.2.d.a 2
140.j odd 4 1 3675.2.a.d 1
140.j odd 4 1 3675.2.a.l 1
140.p odd 6 2 735.2.q.a 4
140.s even 6 2 735.2.q.b 4
420.o odd 2 1 2205.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 4.b odd 2 1
105.2.d.a 2 20.d odd 2 1
315.2.d.c 2 12.b even 2 1
315.2.d.c 2 60.h even 2 1
525.2.a.b 1 20.e even 4 1
525.2.a.c 1 20.e even 4 1
735.2.d.a 2 28.d even 2 1
735.2.d.a 2 140.c even 2 1
735.2.q.a 4 28.g odd 6 2
735.2.q.a 4 140.p odd 6 2
735.2.q.b 4 28.f even 6 2
735.2.q.b 4 140.s even 6 2
1575.2.a.e 1 60.l odd 4 1
1575.2.a.i 1 60.l odd 4 1
1680.2.t.f 2 1.a even 1 1 trivial
1680.2.t.f 2 5.b even 2 1 inner
2205.2.d.f 2 84.h odd 2 1
2205.2.d.f 2 420.o odd 2 1
3675.2.a.d 1 140.j odd 4 1
3675.2.a.l 1 140.j odd 4 1
5040.2.t.e 2 3.b odd 2 1
5040.2.t.e 2 15.d odd 2 1
8400.2.a.bj 1 5.c odd 4 1
8400.2.a.ch 1 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 $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11} - 6$$ $$T_{13}^{2} + 4$$ $$T_{19} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$4 + T^{2}$$