# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + (b3 + b2 - b1) * q^5 - 3*b2 * q^9 $$q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} - 3 \beta_{2} q^{9} + 2 \beta_{3} q^{13} + 2 \beta_1 q^{17} - 4 \beta_{2} q^{19} + (4 \beta_{3} - 4 \beta_1) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{25} - 2 q^{29} + (8 \beta_{2} - 8) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{37} - 6 q^{41} - 4 \beta_{3} q^{43} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{45} + (4 \beta_{3} - 4 \beta_1) q^{47} + ( - 4 \beta_{2} + 4) q^{59} - 6 \beta_{2} q^{61} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{65} - 4 \beta_1 q^{67} + 12 q^{71} + 2 \beta_1 q^{73} - 4 \beta_{2} q^{79} + (9 \beta_{2} - 9) q^{81} + (2 \beta_{3} - 8) q^{85} + 10 \beta_{2} q^{89} + ( - 4 \beta_{2} + 4 \beta_1 + 4) q^{95} + 6 \beta_{3} q^{97}+O(q^{100})$$ q + (b3 + b2 - b1) * q^5 - 3*b2 * q^9 + 2*b3 * q^13 + 2*b1 * q^17 - 4*b2 * q^19 + (4*b3 - 4*b1) * q^23 + (-3*b2 - 2*b1 + 3) * q^25 - 2 * q^29 + (8*b2 - 8) * q^31 + (4*b3 - 4*b1) * q^37 - 6 * q^41 - 4*b3 * q^43 + (-3*b2 + 3*b1 + 3) * q^45 + (4*b3 - 4*b1) * q^47 + (-4*b2 + 4) * q^59 - 6*b2 * q^61 + (2*b3 - 8*b2 - 2*b1) * q^65 - 4*b1 * q^67 + 12 * q^71 + 2*b1 * q^73 - 4*b2 * q^79 + (9*b2 - 9) * q^81 + (2*b3 - 8) * q^85 + 10*b2 * q^89 + (-4*b2 + 4*b1 + 4) * q^95 + 6*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} - 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^5 - 6 * q^9 $$4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100})$$ 4 * q + 2 * q^5 - 6 * q^9 - 8 * q^19 + 6 * q^25 - 8 * q^29 - 16 * q^31 - 24 * q^41 + 6 * q^45 + 8 * q^59 - 12 * q^61 - 16 * q^65 + 48 * q^71 - 8 * q^79 - 18 * q^81 - 32 * q^85 + 20 * q^89 + 8 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −1.23205 1.86603i 0 0 0 −1.50000 + 2.59808i 0
569.2 0 0 0 2.23205 + 0.133975i 0 0 0 −1.50000 + 2.59808i 0
949.1 0 0 0 −1.23205 + 1.86603i 0 0 0 −1.50000 2.59808i 0
949.2 0 0 0 2.23205 0.133975i 0 0 0 −1.50000 2.59808i 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.q.e 4
5.b even 2 1 inner 980.2.q.e 4
7.b odd 2 1 980.2.q.d 4
7.c even 3 1 980.2.e.a 2
7.c even 3 1 inner 980.2.q.e 4
7.d odd 6 1 140.2.e.b 2
7.d odd 6 1 980.2.q.d 4
21.g even 6 1 1260.2.k.b 2
28.f even 6 1 560.2.g.c 2
35.c odd 2 1 980.2.q.d 4
35.i odd 6 1 140.2.e.b 2
35.i odd 6 1 980.2.q.d 4
35.j even 6 1 980.2.e.a 2
35.j even 6 1 inner 980.2.q.e 4
35.k even 12 1 700.2.a.f 1
35.k even 12 1 700.2.a.h 1
35.l odd 12 1 4900.2.a.l 1
35.l odd 12 1 4900.2.a.m 1
56.j odd 6 1 2240.2.g.d 2
56.m even 6 1 2240.2.g.c 2
84.j odd 6 1 5040.2.t.g 2
105.p even 6 1 1260.2.k.b 2
105.w odd 12 1 6300.2.a.g 1
105.w odd 12 1 6300.2.a.y 1
140.s even 6 1 560.2.g.c 2
140.x odd 12 1 2800.2.a.o 1
140.x odd 12 1 2800.2.a.s 1
280.ba even 6 1 2240.2.g.c 2
280.bk odd 6 1 2240.2.g.d 2
420.be odd 6 1 5040.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 7.d odd 6 1
140.2.e.b 2 35.i odd 6 1
560.2.g.c 2 28.f even 6 1
560.2.g.c 2 140.s even 6 1
700.2.a.f 1 35.k even 12 1
700.2.a.h 1 35.k even 12 1
980.2.e.a 2 7.c even 3 1
980.2.e.a 2 35.j even 6 1
980.2.q.d 4 7.b odd 2 1
980.2.q.d 4 7.d odd 6 1
980.2.q.d 4 35.c odd 2 1
980.2.q.d 4 35.i odd 6 1
980.2.q.e 4 1.a even 1 1 trivial
980.2.q.e 4 5.b even 2 1 inner
980.2.q.e 4 7.c even 3 1 inner
980.2.q.e 4 35.j even 6 1 inner
1260.2.k.b 2 21.g even 6 1
1260.2.k.b 2 105.p even 6 1
2240.2.g.c 2 56.m even 6 1
2240.2.g.c 2 280.ba even 6 1
2240.2.g.d 2 56.j odd 6 1
2240.2.g.d 2 280.bk odd 6 1
2800.2.a.o 1 140.x odd 12 1
2800.2.a.s 1 140.x odd 12 1
4900.2.a.l 1 35.l odd 12 1
4900.2.a.m 1 35.l odd 12 1
5040.2.t.g 2 84.j odd 6 1
5040.2.t.g 2 420.be odd 6 1
6300.2.a.g 1 105.w odd 12 1
6300.2.a.y 1 105.w odd 12 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11}$$ T11 $$T_{19}^{2} + 4T_{19} + 16$$ T19^2 + 4*T19 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2 T^{3} - T^{2} - 10 T + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$T^{4} - 16T^{2} + 256$$
$19$ $$(T^{2} + 4 T + 16)^{2}$$
$23$ $$T^{4} - 64T^{2} + 4096$$
$29$ $$(T + 2)^{4}$$
$31$ $$(T^{2} + 8 T + 64)^{2}$$
$37$ $$T^{4} - 64T^{2} + 4096$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 64)^{2}$$
$47$ $$T^{4} - 64T^{2} + 4096$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 4 T + 16)^{2}$$
$61$ $$(T^{2} + 6 T + 36)^{2}$$
$67$ $$T^{4} - 64T^{2} + 4096$$
$71$ $$(T - 12)^{4}$$
$73$ $$T^{4} - 16T^{2} + 256$$
$79$ $$(T^{2} + 4 T + 16)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 10 T + 100)^{2}$$
$97$ $$(T^{2} + 144)^{2}$$