# Properties

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

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## 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 210) 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} -2 q^{11} + 6 i q^{13} + ( 2 - i ) q^{15} -4 i q^{17} -6 q^{19} - q^{21} + 8 i q^{23} + ( -3 + 4 i ) q^{25} + i q^{27} -6 q^{29} + 2 q^{31} + 2 i q^{33} + ( 2 - i ) q^{35} -4 i q^{37} + 6 q^{39} + 2 q^{41} -4 i q^{43} + ( -1 - 2 i ) q^{45} + 8 i q^{47} - q^{49} -4 q^{51} + 6 i q^{53} + ( -2 - 4 i ) q^{55} + 6 i q^{57} -8 q^{59} -10 q^{61} + i q^{63} + ( -12 + 6 i ) q^{65} + 8 i q^{67} + 8 q^{69} + 6 q^{71} -14 i q^{73} + ( 4 + 3 i ) q^{75} + 2 i q^{77} -12 q^{79} + q^{81} + 8 i q^{83} + ( 8 - 4 i ) q^{85} + 6 i q^{87} + 10 q^{89} + 6 q^{91} -2 i q^{93} + ( -6 - 12 i ) q^{95} + 10 i q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{5} - 2q^{9} - 4q^{11} + 4q^{15} - 12q^{19} - 2q^{21} - 6q^{25} - 12q^{29} + 4q^{31} + 4q^{35} + 12q^{39} + 4q^{41} - 2q^{45} - 2q^{49} - 8q^{51} - 4q^{55} - 16q^{59} - 20q^{61} - 24q^{65} + 16q^{69} + 12q^{71} + 8q^{75} - 24q^{79} + 2q^{81} + 16q^{85} + 20q^{89} + 12q^{91} - 12q^{95} + 4q^{99} + O(q^{100})$$

## 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.e 2
3.b odd 2 1 5040.2.t.h 2
4.b odd 2 1 210.2.g.b 2
5.b even 2 1 inner 1680.2.t.e 2
5.c odd 4 1 8400.2.a.w 1
5.c odd 4 1 8400.2.a.bp 1
12.b even 2 1 630.2.g.c 2
15.d odd 2 1 5040.2.t.h 2
20.d odd 2 1 210.2.g.b 2
20.e even 4 1 1050.2.a.d 1
20.e even 4 1 1050.2.a.p 1
28.d even 2 1 1470.2.g.b 2
28.f even 6 2 1470.2.n.f 4
28.g odd 6 2 1470.2.n.b 4
60.h even 2 1 630.2.g.c 2
60.l odd 4 1 3150.2.a.d 1
60.l odd 4 1 3150.2.a.bk 1
140.c even 2 1 1470.2.g.b 2
140.j odd 4 1 7350.2.a.bk 1
140.j odd 4 1 7350.2.a.bz 1
140.p odd 6 2 1470.2.n.b 4
140.s even 6 2 1470.2.n.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 4.b odd 2 1
210.2.g.b 2 20.d odd 2 1
630.2.g.c 2 12.b even 2 1
630.2.g.c 2 60.h even 2 1
1050.2.a.d 1 20.e even 4 1
1050.2.a.p 1 20.e even 4 1
1470.2.g.b 2 28.d even 2 1
1470.2.g.b 2 140.c even 2 1
1470.2.n.b 4 28.g odd 6 2
1470.2.n.b 4 140.p odd 6 2
1470.2.n.f 4 28.f even 6 2
1470.2.n.f 4 140.s even 6 2
1680.2.t.e 2 1.a even 1 1 trivial
1680.2.t.e 2 5.b even 2 1 inner
3150.2.a.d 1 60.l odd 4 1
3150.2.a.bk 1 60.l odd 4 1
5040.2.t.h 2 3.b odd 2 1
5040.2.t.h 2 15.d odd 2 1
7350.2.a.bk 1 140.j odd 4 1
7350.2.a.bz 1 140.j odd 4 1
8400.2.a.w 1 5.c odd 4 1
8400.2.a.bp 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 $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11} + 2$$ $$T_{13}^{2} + 36$$ $$T_{19} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$100 + T^{2}$$
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