# Properties

 Label 980.2.c.e Level $980$ Weight $2$ Character orbit 980.c Analytic conductor $7.825$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 16q^{4} - 64q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 16q^{4} - 64q^{9} + 16q^{16} - 16q^{25} - 48q^{29} - 8q^{30} + 176q^{36} - 48q^{44} - 32q^{46} + 32q^{50} + 24q^{60} - 80q^{64} - 16q^{65} - 112q^{74} - 48q^{81} - 64q^{85} - 112q^{86} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
979.1 −1.38298 0.295571i 1.66243i 1.82528 + 0.817539i 2.22517 0.220514i −0.491366 + 2.29911i 0 −2.28268 1.67014i 0.236337 −3.14254 0.352730i
979.2 −1.38298 0.295571i 1.66243i 1.82528 + 0.817539i −2.22517 + 0.220514i 0.491366 2.29911i 0 −2.28268 1.67014i 0.236337 3.14254 + 0.352730i
979.3 −1.38298 + 0.295571i 1.66243i 1.82528 0.817539i −2.22517 0.220514i 0.491366 + 2.29911i 0 −2.28268 + 1.67014i 0.236337 3.14254 0.352730i
979.4 −1.38298 + 0.295571i 1.66243i 1.82528 0.817539i 2.22517 + 0.220514i −0.491366 2.29911i 0 −2.28268 + 1.67014i 0.236337 −3.14254 + 0.352730i
979.5 −1.36075 0.385185i 0.423388i 1.70326 + 1.04828i 1.74517 1.39799i −0.163083 + 0.576124i 0 −1.91393 2.08252i 2.82074 −2.91323 + 1.23009i
979.6 −1.36075 0.385185i 0.423388i 1.70326 + 1.04828i −1.74517 + 1.39799i 0.163083 0.576124i 0 −1.91393 2.08252i 2.82074 2.91323 1.23009i
979.7 −1.36075 + 0.385185i 0.423388i 1.70326 1.04828i −1.74517 1.39799i 0.163083 + 0.576124i 0 −1.91393 + 2.08252i 2.82074 2.91323 + 1.23009i
979.8 −1.36075 + 0.385185i 0.423388i 1.70326 1.04828i 1.74517 + 1.39799i −0.163083 0.576124i 0 −1.91393 + 2.08252i 2.82074 −2.91323 1.23009i
979.9 −1.23364 0.691471i 2.08008i 1.04374 + 1.70605i 1.52262 + 1.63757i −1.43832 + 2.56607i 0 −0.107908 2.82637i −1.32674 −0.746026 3.07302i
979.10 −1.23364 0.691471i 2.08008i 1.04374 + 1.70605i −1.52262 1.63757i 1.43832 2.56607i 0 −0.107908 2.82637i −1.32674 0.746026 + 3.07302i
979.11 −1.23364 + 0.691471i 2.08008i 1.04374 1.70605i −1.52262 + 1.63757i 1.43832 + 2.56607i 0 −0.107908 + 2.82637i −1.32674 0.746026 3.07302i
979.12 −1.23364 + 0.691471i 2.08008i 1.04374 1.70605i 1.52262 1.63757i −1.43832 2.56607i 0 −0.107908 + 2.82637i −1.32674 −0.746026 + 3.07302i
979.13 −1.14730 0.826865i 1.52335i 0.632590 + 1.89732i 0.0967363 2.23397i −1.25961 + 1.74774i 0 0.843059 2.69986i 0.679393 −1.95818 + 2.48305i
979.14 −1.14730 0.826865i 1.52335i 0.632590 + 1.89732i −0.0967363 + 2.23397i 1.25961 1.74774i 0 0.843059 2.69986i 0.679393 1.95818 2.48305i
979.15 −1.14730 + 0.826865i 1.52335i 0.632590 1.89732i −0.0967363 2.23397i 1.25961 + 1.74774i 0 0.843059 + 2.69986i 0.679393 1.95818 + 2.48305i
979.16 −1.14730 + 0.826865i 1.52335i 0.632590 1.89732i 0.0967363 + 2.23397i −1.25961 1.74774i 0 0.843059 + 2.69986i 0.679393 −1.95818 2.48305i
979.17 −0.576258 1.29148i 2.50150i −1.33585 + 1.48845i −0.639901 + 2.14255i −3.23064 + 1.44151i 0 2.69211 + 0.867500i −3.25749 3.13582 0.408241i
979.18 −0.576258 1.29148i 2.50150i −1.33585 + 1.48845i 0.639901 2.14255i 3.23064 1.44151i 0 2.69211 + 0.867500i −3.25749 −3.13582 + 0.408241i
979.19 −0.576258 + 1.29148i 2.50150i −1.33585 1.48845i 0.639901 + 2.14255i 3.23064 + 1.44151i 0 2.69211 0.867500i −3.25749 −3.13582 0.408241i
979.20 −0.576258 + 1.29148i 2.50150i −1.33585 1.48845i −0.639901 2.14255i −3.23064 1.44151i 0 2.69211 0.867500i −3.25749 3.13582 + 0.408241i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 979.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.c.e 48
4.b odd 2 1 inner 980.2.c.e 48
5.b even 2 1 inner 980.2.c.e 48
7.b odd 2 1 inner 980.2.c.e 48
7.c even 3 2 980.2.s.g 96
7.d odd 6 2 980.2.s.g 96
20.d odd 2 1 inner 980.2.c.e 48
28.d even 2 1 inner 980.2.c.e 48
28.f even 6 2 980.2.s.g 96
28.g odd 6 2 980.2.s.g 96
35.c odd 2 1 inner 980.2.c.e 48
35.i odd 6 2 980.2.s.g 96
35.j even 6 2 980.2.s.g 96
140.c even 2 1 inner 980.2.c.e 48
140.p odd 6 2 980.2.s.g 96
140.s even 6 2 980.2.s.g 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.c.e 48 1.a even 1 1 trivial
980.2.c.e 48 4.b odd 2 1 inner
980.2.c.e 48 5.b even 2 1 inner
980.2.c.e 48 7.b odd 2 1 inner
980.2.c.e 48 20.d odd 2 1 inner
980.2.c.e 48 28.d even 2 1 inner
980.2.c.e 48 35.c odd 2 1 inner
980.2.c.e 48 140.c even 2 1 inner
980.2.s.g 96 7.c even 3 2
980.2.s.g 96 7.d odd 6 2
980.2.s.g 96 28.f even 6 2
980.2.s.g 96 28.g odd 6 2
980.2.s.g 96 35.i odd 6 2
980.2.s.g 96 35.j even 6 2
980.2.s.g 96 140.p odd 6 2
980.2.s.g 96 140.s even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 26 T_{3}^{10} + 251 T_{3}^{8} + 1136 T_{3}^{6} + 2456 T_{3}^{4} + 2168 T_{3}^{2} + 316$$ acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$.