# Properties

 Label 980.2.c.b Level $980$ Weight $2$ Character orbit 980.c Analytic conductor $7.825$ Analytic rank $0$ Dimension $8$ CM discriminant -20 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: $$8$$ Coefficient field: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2 q^{4} + \beta_{6} q^{5} + ( -\beta_{5} + \beta_{6} ) q^{6} -2 \beta_{1} q^{8} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2 q^{4} + \beta_{6} q^{5} + ( -\beta_{5} + \beta_{6} ) q^{6} -2 \beta_{1} q^{8} + ( -1 + \beta_{2} ) q^{9} + ( \beta_{3} + \beta_{7} ) q^{10} + 2 \beta_{3} q^{12} + ( 3 \beta_{1} - \beta_{4} ) q^{15} + 4 q^{16} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{18} + 2 \beta_{6} q^{20} + ( -2 \beta_{1} + 3 \beta_{4} ) q^{23} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{24} -5 q^{25} + ( -2 \beta_{3} + \beta_{7} ) q^{27} + ( -3 + 2 \beta_{2} ) q^{29} + ( -5 + \beta_{2} ) q^{30} -4 \beta_{1} q^{32} + ( -2 + 2 \beta_{2} ) q^{36} + ( 2 \beta_{3} + 2 \beta_{7} ) q^{40} + ( 6 \beta_{5} + \beta_{6} ) q^{41} + ( -4 \beta_{1} - \beta_{4} ) q^{43} + ( 5 \beta_{5} - \beta_{6} ) q^{45} + ( 1 - 3 \beta_{2} ) q^{46} + ( -3 \beta_{3} - 3 \beta_{7} ) q^{47} + 4 \beta_{3} q^{48} + 5 \beta_{1} q^{50} + ( 3 \beta_{5} - \beta_{6} ) q^{54} + ( 5 \beta_{1} - 4 \beta_{4} ) q^{58} + ( 6 \beta_{1} - 2 \beta_{4} ) q^{60} + ( -4 \beta_{5} - 3 \beta_{6} ) q^{61} + 8 q^{64} + ( -\beta_{1} + 5 \beta_{4} ) q^{67} + ( -8 \beta_{5} + 5 \beta_{6} ) q^{69} + ( 4 \beta_{1} - 4 \beta_{4} ) q^{72} -5 \beta_{3} q^{75} + 4 \beta_{6} q^{80} + ( 4 + \beta_{2} ) q^{81} + ( -5 \beta_{3} + 7 \beta_{7} ) q^{82} + ( 4 \beta_{3} - 7 \beta_{7} ) q^{83} + ( 9 + \beta_{2} ) q^{86} + ( -11 \beta_{3} + 2 \beta_{7} ) q^{87} + ( 3 \beta_{5} + 4 \beta_{6} ) q^{89} + ( -6 \beta_{3} + 4 \beta_{7} ) q^{90} + ( -4 \beta_{1} + 6 \beta_{4} ) q^{92} -6 \beta_{6} q^{94} + ( -4 \beta_{5} + 4 \beta_{6} ) q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} - 8q^{9} + O(q^{10})$$ $$8q + 16q^{4} - 8q^{9} + 32q^{16} - 40q^{25} - 24q^{29} - 40q^{30} - 16q^{36} + 8q^{46} + 64q^{64} + 32q^{81} + 72q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + 7 \nu^{3} - 9 \nu$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 7 \nu^{3} + 63 \nu$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225$$$$)/63$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 22$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} + 4 \nu^{3} - 24 \nu$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{5} + \beta_{2} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{3} + 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{6} + \beta_{5} + 4 \beta_{2} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{7} + 5 \beta_{4} - 12 \beta_{3} + 7 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$21 \beta_{7} + 29 \beta_{4} + 8 \beta_{3} - 21 \beta_{1}$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
979.1
 −1.01575 − 1.40294i 1.72286 − 0.178197i 1.72286 + 0.178197i −1.01575 + 1.40294i 1.01575 − 1.40294i −1.72286 − 0.178197i −1.72286 + 0.178197i 1.01575 + 1.40294i
−1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 −2.82843 −4.87298 3.16228i
979.2 −1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 −2.82843 2.87298 3.16228i
979.3 −1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 −2.82843 2.87298 3.16228i
979.4 −1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 −2.82843 −4.87298 3.16228i
979.5 1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 2.82843 −4.87298 3.16228i
979.6 1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 2.82843 2.87298 3.16228i
979.7 1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 2.82843 2.87298 3.16228i
979.8 1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 2.82843 −4.87298 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 979.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b 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.b 8
4.b odd 2 1 inner 980.2.c.b 8
5.b even 2 1 inner 980.2.c.b 8
7.b odd 2 1 inner 980.2.c.b 8
7.c even 3 1 140.2.s.a 8
7.c even 3 1 980.2.s.a 8
7.d odd 6 1 140.2.s.a 8
7.d odd 6 1 980.2.s.a 8
20.d odd 2 1 CM 980.2.c.b 8
28.d even 2 1 inner 980.2.c.b 8
28.f even 6 1 140.2.s.a 8
28.f even 6 1 980.2.s.a 8
28.g odd 6 1 140.2.s.a 8
28.g odd 6 1 980.2.s.a 8
35.c odd 2 1 inner 980.2.c.b 8
35.i odd 6 1 140.2.s.a 8
35.i odd 6 1 980.2.s.a 8
35.j even 6 1 140.2.s.a 8
35.j even 6 1 980.2.s.a 8
35.k even 12 2 700.2.p.b 8
35.l odd 12 2 700.2.p.b 8
140.c even 2 1 inner 980.2.c.b 8
140.p odd 6 1 140.2.s.a 8
140.p odd 6 1 980.2.s.a 8
140.s even 6 1 140.2.s.a 8
140.s even 6 1 980.2.s.a 8
140.w even 12 2 700.2.p.b 8
140.x odd 12 2 700.2.p.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.a 8 7.c even 3 1
140.2.s.a 8 7.d odd 6 1
140.2.s.a 8 28.f even 6 1
140.2.s.a 8 28.g odd 6 1
140.2.s.a 8 35.i odd 6 1
140.2.s.a 8 35.j even 6 1
140.2.s.a 8 140.p odd 6 1
140.2.s.a 8 140.s even 6 1
700.2.p.b 8 35.k even 12 2
700.2.p.b 8 35.l odd 12 2
700.2.p.b 8 140.w even 12 2
700.2.p.b 8 140.x odd 12 2
980.2.c.b 8 1.a even 1 1 trivial
980.2.c.b 8 4.b odd 2 1 inner
980.2.c.b 8 5.b even 2 1 inner
980.2.c.b 8 7.b odd 2 1 inner
980.2.c.b 8 20.d odd 2 1 CM
980.2.c.b 8 28.d even 2 1 inner
980.2.c.b 8 35.c odd 2 1 inner
980.2.c.b 8 140.c even 2 1 inner
980.2.s.a 8 7.c even 3 1
980.2.s.a 8 7.d odd 6 1
980.2.s.a 8 28.f even 6 1
980.2.s.a 8 28.g odd 6 1
980.2.s.a 8 35.i odd 6 1
980.2.s.a 8 35.j even 6 1
980.2.s.a 8 140.p odd 6 1
980.2.s.a 8 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 8 T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{4}$$
$3$ $$( 1 + 8 T^{2} + T^{4} )^{2}$$
$5$ $$( 5 + T^{2} )^{4}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 4489 - 136 T^{2} + T^{4} )^{2}$$
$29$ $$( -51 + 6 T + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( 10609 + 226 T^{2} + T^{4} )^{2}$$
$43$ $$( 1089 - 96 T^{2} + T^{4} )^{2}$$
$47$ $$( 90 + T^{2} )^{4}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 9 + 186 T^{2} + T^{4} )^{2}$$
$67$ $$( 33489 - 384 T^{2} + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$( 25281 + 408 T^{2} + T^{4} )^{2}$$
$89$ $$( 2809 + 214 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$