# Properties

 Label 980.2.e.f Level $980$ Weight $2$ Character orbit 980.e Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -\beta_{3} q^{5} +O(q^{10})$$ $$q -\beta_{2} q^{3} -\beta_{3} q^{5} + ( -2 - \beta_{1} - \beta_{3} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{13} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{15} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{23} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{25} -3 \beta_{2} q^{27} + ( -1 - \beta_{1} - \beta_{3} ) q^{29} + ( \beta_{1} + \beta_{3} ) q^{31} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{33} + ( \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{37} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{39} + ( -7 + \beta_{1} + \beta_{3} ) q^{41} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{43} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{47} + ( 4 - \beta_{1} - \beta_{3} ) q^{51} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{53} + ( 7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{57} + ( \beta_{1} + \beta_{3} ) q^{59} + ( -7 - 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -4 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -4 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{67} + ( 7 + 2 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{71} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} + ( 7 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{75} + ( -4 - \beta_{1} - \beta_{3} ) q^{79} -9 q^{81} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -1 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{87} + 7 q^{89} + ( 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{95} -4 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5} + O(q^{10})$$ $$4 q + q^{5} - 6 q^{11} - 3 q^{15} + 2 q^{19} + 9 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{39} - 30 q^{41} + 18 q^{51} + 27 q^{55} - 2 q^{59} - 24 q^{61} - 16 q^{65} + 24 q^{69} + 12 q^{71} + 27 q^{75} - 14 q^{79} - 36 q^{81} - 5 q^{85} + 28 q^{89} + 29 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} + 16 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 4 \nu^{2} + 4 \nu + 15$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - 5 \beta_{2} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 7$$

## 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
 2.13746 − 0.656712i −1.63746 + 1.52274i 2.13746 + 0.656712i −1.63746 − 1.52274i
0 1.73205i 0 −1.63746 1.52274i 0 0 0 0 0
589.2 0 1.73205i 0 2.13746 + 0.656712i 0 0 0 0 0
589.3 0 1.73205i 0 −1.63746 + 1.52274i 0 0 0 0 0
589.4 0 1.73205i 0 2.13746 0.656712i 0 0 0 0 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 980.2.e.f 4
5.b even 2 1 inner 980.2.e.f 4
5.c odd 4 2 4900.2.a.be 4
7.b odd 2 1 980.2.e.c 4
7.c even 3 1 140.2.q.a 4
7.c even 3 1 140.2.q.b yes 4
7.d odd 6 1 980.2.q.b 4
7.d odd 6 1 980.2.q.g 4
21.h odd 6 1 1260.2.bm.a 4
21.h odd 6 1 1260.2.bm.b 4
28.g odd 6 1 560.2.bw.a 4
28.g odd 6 1 560.2.bw.e 4
35.c odd 2 1 980.2.e.c 4
35.f even 4 2 4900.2.a.bf 4
35.i odd 6 1 980.2.q.b 4
35.i odd 6 1 980.2.q.g 4
35.j even 6 1 140.2.q.a 4
35.j even 6 1 140.2.q.b yes 4
35.l odd 12 4 700.2.i.f 8
105.o odd 6 1 1260.2.bm.a 4
105.o odd 6 1 1260.2.bm.b 4
140.p odd 6 1 560.2.bw.a 4
140.p odd 6 1 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 7.c even 3 1
140.2.q.a 4 35.j even 6 1
140.2.q.b yes 4 7.c even 3 1
140.2.q.b yes 4 35.j even 6 1
560.2.bw.a 4 28.g odd 6 1
560.2.bw.a 4 140.p odd 6 1
560.2.bw.e 4 28.g odd 6 1
560.2.bw.e 4 140.p odd 6 1
700.2.i.f 8 35.l odd 12 4
980.2.e.c 4 7.b odd 2 1
980.2.e.c 4 35.c odd 2 1
980.2.e.f 4 1.a even 1 1 trivial
980.2.e.f 4 5.b even 2 1 inner
980.2.q.b 4 7.d odd 6 1
980.2.q.b 4 35.i odd 6 1
980.2.q.g 4 7.d odd 6 1
980.2.q.g 4 35.i odd 6 1
1260.2.bm.a 4 21.h odd 6 1
1260.2.bm.a 4 105.o odd 6 1
1260.2.bm.b 4 21.h odd 6 1
1260.2.bm.b 4 105.o odd 6 1
4900.2.a.be 4 5.c odd 4 2
4900.2.a.bf 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 $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} + 3$$ $$T_{11}^{2} + 3 T_{11} - 12$$ $$T_{19}^{2} - T_{19} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$25 - 5 T - 4 T^{2} - T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -12 + 3 T + T^{2} )^{2}$$
$13$ $$256 + 44 T^{2} + T^{4}$$
$17$ $$4 + 23 T^{2} + T^{4}$$
$19$ $$( -14 - T + T^{2} )^{2}$$
$23$ $$49 + 62 T^{2} + T^{4}$$
$29$ $$( -14 + T + T^{2} )^{2}$$
$31$ $$( -14 + T + T^{2} )^{2}$$
$37$ $$3136 + 131 T^{2} + T^{4}$$
$41$ $$( 42 + 15 T + T^{2} )^{2}$$
$43$ $$196 + 47 T^{2} + T^{4}$$
$47$ $$196 + 47 T^{2} + T^{4}$$
$53$ $$1764 + 87 T^{2} + T^{4}$$
$59$ $$( -14 + T + T^{2} )^{2}$$
$61$ $$( -21 + 12 T + T^{2} )^{2}$$
$67$ $$2401 + 206 T^{2} + T^{4}$$
$71$ $$( -48 - 6 T + T^{2} )^{2}$$
$73$ $$196 + 47 T^{2} + T^{4}$$
$79$ $$( -2 + 7 T + T^{2} )^{2}$$
$83$ $$1764 + 87 T^{2} + T^{4}$$
$89$ $$( -7 + T )^{4}$$
$97$ $$( 48 + T^{2} )^{2}$$