# Properties

 Label 980.2.e.d Level $980$ Weight $2$ Character orbit 980.e Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM discriminant -35 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-5}, \sqrt{-21})$$ Defining polynomial: $$x^{4} + 13 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{9} + ( 1 + \beta_{3} ) q^{11} + ( \beta_{1} + 2 \beta_{2} ) q^{13} + ( 3 - \beta_{3} ) q^{15} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{17} -5 q^{25} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{27} + ( -5 + \beta_{3} ) q^{29} + ( -\beta_{1} + 4 \beta_{2} ) q^{33} + ( -1 - \beta_{3} ) q^{39} + ( 5 \beta_{1} - \beta_{2} ) q^{45} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 15 - \beta_{3} ) q^{51} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{55} + ( -7 - \beta_{3} ) q^{65} -12 q^{71} -6 \beta_{2} q^{73} -5 \beta_{1} q^{75} + ( 1 - 3 \beta_{3} ) q^{79} + ( 21 - 4 \beta_{3} ) q^{81} + 4 \beta_{2} q^{83} + ( 1 + 3 \beta_{3} ) q^{85} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -7 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 22 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{9} + O(q^{10})$$ $$4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 13 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 7$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 3.40932i − 1.17325i 1.17325i 3.40932i
0 3.40932i 0 2.23607i 0 0 0 −8.62348 0
589.2 0 1.17325i 0 2.23607i 0 0 0 1.62348 0
589.3 0 1.17325i 0 2.23607i 0 0 0 1.62348 0
589.4 0 3.40932i 0 2.23607i 0 0 0 −8.62348 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.e.d 4
5.b even 2 1 inner 980.2.e.d 4
5.c odd 4 2 4900.2.a.bj 4
7.b odd 2 1 inner 980.2.e.d 4
7.c even 3 2 980.2.q.i 8
7.d odd 6 2 980.2.q.i 8
35.c odd 2 1 CM 980.2.e.d 4
35.f even 4 2 4900.2.a.bj 4
35.i odd 6 2 980.2.q.i 8
35.j even 6 2 980.2.q.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.d 4 1.a even 1 1 trivial
980.2.e.d 4 5.b even 2 1 inner
980.2.e.d 4 7.b odd 2 1 inner
980.2.e.d 4 35.c odd 2 1 CM
980.2.q.i 8 7.c even 3 2
980.2.q.i 8 7.d odd 6 2
980.2.q.i 8 35.i odd 6 2
980.2.q.i 8 35.j even 6 2
4900.2.a.bj 4 5.c odd 4 2
4900.2.a.bj 4 35.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{4} + 13 T_{3}^{2} + 16$$ $$T_{11}^{2} - 3 T_{11} - 24$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 13 T^{2} + T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -24 - 3 T + T^{2} )^{2}$$
$13$ $$36 + 33 T^{2} + T^{4}$$
$17$ $$2116 + 97 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -6 + 9 T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$256 + 157 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$( 180 + T^{2} )^{2}$$
$79$ $$( -236 + T + T^{2} )^{2}$$
$83$ $$( 80 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$60516 + 537 T^{2} + T^{4}$$