# Properties

 Label 980.4.i.n Level $980$ Weight $4$ Character orbit 980.i Analytic conductor $57.822$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 980.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.8218718056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 4 \zeta_{6} + 4) q^{3} + 5 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9}+O(q^{10})$$ q + (-4*z + 4) * q^3 + 5*z * q^5 + 11*z * q^9 $$q + ( - 4 \zeta_{6} + 4) q^{3} + 5 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} + ( - 60 \zeta_{6} + 60) q^{11} - 86 q^{13} + 20 q^{15} + ( - 18 \zeta_{6} + 18) q^{17} + 44 \zeta_{6} q^{19} - 48 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 152 q^{27} - 186 q^{29} + ( - 176 \zeta_{6} + 176) q^{31} - 240 \zeta_{6} q^{33} - 254 \zeta_{6} q^{37} + (344 \zeta_{6} - 344) q^{39} - 186 q^{41} - 100 q^{43} + (55 \zeta_{6} - 55) q^{45} + 168 \zeta_{6} q^{47} - 72 \zeta_{6} q^{51} + ( - 498 \zeta_{6} + 498) q^{53} + 300 q^{55} + 176 q^{57} + (252 \zeta_{6} - 252) q^{59} - 58 \zeta_{6} q^{61} - 430 \zeta_{6} q^{65} + ( - 1036 \zeta_{6} + 1036) q^{67} - 192 q^{69} + 168 q^{71} + ( - 506 \zeta_{6} + 506) q^{73} + 100 \zeta_{6} q^{75} - 272 \zeta_{6} q^{79} + ( - 311 \zeta_{6} + 311) q^{81} - 948 q^{83} + 90 q^{85} + (744 \zeta_{6} - 744) q^{87} - 1014 \zeta_{6} q^{89} - 704 \zeta_{6} q^{93} + (220 \zeta_{6} - 220) q^{95} + 766 q^{97} + 660 q^{99} +O(q^{100})$$ q + (-4*z + 4) * q^3 + 5*z * q^5 + 11*z * q^9 + (-60*z + 60) * q^11 - 86 * q^13 + 20 * q^15 + (-18*z + 18) * q^17 + 44*z * q^19 - 48*z * q^23 + (25*z - 25) * q^25 + 152 * q^27 - 186 * q^29 + (-176*z + 176) * q^31 - 240*z * q^33 - 254*z * q^37 + (344*z - 344) * q^39 - 186 * q^41 - 100 * q^43 + (55*z - 55) * q^45 + 168*z * q^47 - 72*z * q^51 + (-498*z + 498) * q^53 + 300 * q^55 + 176 * q^57 + (252*z - 252) * q^59 - 58*z * q^61 - 430*z * q^65 + (-1036*z + 1036) * q^67 - 192 * q^69 + 168 * q^71 + (-506*z + 506) * q^73 + 100*z * q^75 - 272*z * q^79 + (-311*z + 311) * q^81 - 948 * q^83 + 90 * q^85 + (744*z - 744) * q^87 - 1014*z * q^89 - 704*z * q^93 + (220*z - 220) * q^95 + 766 * q^97 + 660 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 5 q^{5} + 11 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 5 * q^5 + 11 * q^9 $$2 q + 4 q^{3} + 5 q^{5} + 11 q^{9} + 60 q^{11} - 172 q^{13} + 40 q^{15} + 18 q^{17} + 44 q^{19} - 48 q^{23} - 25 q^{25} + 304 q^{27} - 372 q^{29} + 176 q^{31} - 240 q^{33} - 254 q^{37} - 344 q^{39} - 372 q^{41} - 200 q^{43} - 55 q^{45} + 168 q^{47} - 72 q^{51} + 498 q^{53} + 600 q^{55} + 352 q^{57} - 252 q^{59} - 58 q^{61} - 430 q^{65} + 1036 q^{67} - 384 q^{69} + 336 q^{71} + 506 q^{73} + 100 q^{75} - 272 q^{79} + 311 q^{81} - 1896 q^{83} + 180 q^{85} - 744 q^{87} - 1014 q^{89} - 704 q^{93} - 220 q^{95} + 1532 q^{97} + 1320 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 5 * q^5 + 11 * q^9 + 60 * q^11 - 172 * q^13 + 40 * q^15 + 18 * q^17 + 44 * q^19 - 48 * q^23 - 25 * q^25 + 304 * q^27 - 372 * q^29 + 176 * q^31 - 240 * q^33 - 254 * q^37 - 344 * q^39 - 372 * q^41 - 200 * q^43 - 55 * q^45 + 168 * q^47 - 72 * q^51 + 498 * q^53 + 600 * q^55 + 352 * q^57 - 252 * q^59 - 58 * q^61 - 430 * q^65 + 1036 * q^67 - 384 * q^69 + 336 * q^71 + 506 * q^73 + 100 * q^75 - 272 * q^79 + 311 * q^81 - 1896 * q^83 + 180 * q^85 - 744 * q^87 - 1014 * q^89 - 704 * q^93 - 220 * q^95 + 1532 * q^97 + 1320 * q^99

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 2.00000 3.46410i 0 2.50000 + 4.33013i 0 0 0 5.50000 + 9.52628i 0
961.1 0 2.00000 + 3.46410i 0 2.50000 4.33013i 0 0 0 5.50000 9.52628i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.4.i.n 2
7.b odd 2 1 980.4.i.e 2
7.c even 3 1 980.4.a.c 1
7.c even 3 1 inner 980.4.i.n 2
7.d odd 6 1 20.4.a.a 1
7.d odd 6 1 980.4.i.e 2
21.g even 6 1 180.4.a.a 1
28.f even 6 1 80.4.a.c 1
35.i odd 6 1 100.4.a.a 1
35.k even 12 2 100.4.c.a 2
56.j odd 6 1 320.4.a.d 1
56.m even 6 1 320.4.a.k 1
63.i even 6 1 1620.4.i.j 2
63.k odd 6 1 1620.4.i.d 2
63.s even 6 1 1620.4.i.j 2
63.t odd 6 1 1620.4.i.d 2
77.i even 6 1 2420.4.a.d 1
84.j odd 6 1 720.4.a.k 1
105.p even 6 1 900.4.a.m 1
105.w odd 12 2 900.4.d.k 2
112.v even 12 2 1280.4.d.c 2
112.x odd 12 2 1280.4.d.n 2
140.s even 6 1 400.4.a.o 1
140.x odd 12 2 400.4.c.j 2
280.ba even 6 1 1600.4.a.p 1
280.bk odd 6 1 1600.4.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 7.d odd 6 1
80.4.a.c 1 28.f even 6 1
100.4.a.a 1 35.i odd 6 1
100.4.c.a 2 35.k even 12 2
180.4.a.a 1 21.g even 6 1
320.4.a.d 1 56.j odd 6 1
320.4.a.k 1 56.m even 6 1
400.4.a.o 1 140.s even 6 1
400.4.c.j 2 140.x odd 12 2
720.4.a.k 1 84.j odd 6 1
900.4.a.m 1 105.p even 6 1
900.4.d.k 2 105.w odd 12 2
980.4.a.c 1 7.c even 3 1
980.4.i.e 2 7.b odd 2 1
980.4.i.e 2 7.d odd 6 1
980.4.i.n 2 1.a even 1 1 trivial
980.4.i.n 2 7.c even 3 1 inner
1280.4.d.c 2 112.v even 12 2
1280.4.d.n 2 112.x odd 12 2
1600.4.a.p 1 280.ba even 6 1
1600.4.a.bl 1 280.bk odd 6 1
1620.4.i.d 2 63.k odd 6 1
1620.4.i.d 2 63.t odd 6 1
1620.4.i.j 2 63.i even 6 1
1620.4.i.j 2 63.s even 6 1
2420.4.a.d 1 77.i even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} - 4T_{3} + 16$$ T3^2 - 4*T3 + 16 $$T_{11}^{2} - 60T_{11} + 3600$$ T11^2 - 60*T11 + 3600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 4T + 16$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 60T + 3600$$
$13$ $$(T + 86)^{2}$$
$17$ $$T^{2} - 18T + 324$$
$19$ $$T^{2} - 44T + 1936$$
$23$ $$T^{2} + 48T + 2304$$
$29$ $$(T + 186)^{2}$$
$31$ $$T^{2} - 176T + 30976$$
$37$ $$T^{2} + 254T + 64516$$
$41$ $$(T + 186)^{2}$$
$43$ $$(T + 100)^{2}$$
$47$ $$T^{2} - 168T + 28224$$
$53$ $$T^{2} - 498T + 248004$$
$59$ $$T^{2} + 252T + 63504$$
$61$ $$T^{2} + 58T + 3364$$
$67$ $$T^{2} - 1036 T + 1073296$$
$71$ $$(T - 168)^{2}$$
$73$ $$T^{2} - 506T + 256036$$
$79$ $$T^{2} + 272T + 73984$$
$83$ $$(T + 948)^{2}$$
$89$ $$T^{2} + 1014 T + 1028196$$
$97$ $$(T - 766)^{2}$$