# Properties

 Label 1620.4.i.l Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1620.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$95.5830942093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{6} q^{5} + ( - 28 \zeta_{6} + 28) q^{7}+O(q^{10})$$ q + 5*z * q^5 + (-28*z + 28) * q^7 $$q + 5 \zeta_{6} q^{5} + ( - 28 \zeta_{6} + 28) q^{7} + ( - 24 \zeta_{6} + 24) q^{11} + 70 \zeta_{6} q^{13} + 102 q^{17} + 20 q^{19} + 72 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (306 \zeta_{6} - 306) q^{29} + 136 \zeta_{6} q^{31} + 140 q^{35} - 214 q^{37} + 150 \zeta_{6} q^{41} + ( - 292 \zeta_{6} + 292) q^{43} + ( - 72 \zeta_{6} + 72) q^{47} - 441 \zeta_{6} q^{49} - 414 q^{53} + 120 q^{55} + 744 \zeta_{6} q^{59} + ( - 418 \zeta_{6} + 418) q^{61} + (350 \zeta_{6} - 350) q^{65} - 188 \zeta_{6} q^{67} + 480 q^{71} + 434 q^{73} - 672 \zeta_{6} q^{77} + (1352 \zeta_{6} - 1352) q^{79} + ( - 612 \zeta_{6} + 612) q^{83} + 510 \zeta_{6} q^{85} - 30 q^{89} + 1960 q^{91} + 100 \zeta_{6} q^{95} + ( - 286 \zeta_{6} + 286) q^{97} +O(q^{100})$$ q + 5*z * q^5 + (-28*z + 28) * q^7 + (-24*z + 24) * q^11 + 70*z * q^13 + 102 * q^17 + 20 * q^19 + 72*z * q^23 + (25*z - 25) * q^25 + (306*z - 306) * q^29 + 136*z * q^31 + 140 * q^35 - 214 * q^37 + 150*z * q^41 + (-292*z + 292) * q^43 + (-72*z + 72) * q^47 - 441*z * q^49 - 414 * q^53 + 120 * q^55 + 744*z * q^59 + (-418*z + 418) * q^61 + (350*z - 350) * q^65 - 188*z * q^67 + 480 * q^71 + 434 * q^73 - 672*z * q^77 + (1352*z - 1352) * q^79 + (-612*z + 612) * q^83 + 510*z * q^85 - 30 * q^89 + 1960 * q^91 + 100*z * q^95 + (-286*z + 286) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{5} + 28 q^{7}+O(q^{10})$$ 2 * q + 5 * q^5 + 28 * q^7 $$2 q + 5 q^{5} + 28 q^{7} + 24 q^{11} + 70 q^{13} + 204 q^{17} + 40 q^{19} + 72 q^{23} - 25 q^{25} - 306 q^{29} + 136 q^{31} + 280 q^{35} - 428 q^{37} + 150 q^{41} + 292 q^{43} + 72 q^{47} - 441 q^{49} - 828 q^{53} + 240 q^{55} + 744 q^{59} + 418 q^{61} - 350 q^{65} - 188 q^{67} + 960 q^{71} + 868 q^{73} - 672 q^{77} - 1352 q^{79} + 612 q^{83} + 510 q^{85} - 60 q^{89} + 3920 q^{91} + 100 q^{95} + 286 q^{97}+O(q^{100})$$ 2 * q + 5 * q^5 + 28 * q^7 + 24 * q^11 + 70 * q^13 + 204 * q^17 + 40 * q^19 + 72 * q^23 - 25 * q^25 - 306 * q^29 + 136 * q^31 + 280 * q^35 - 428 * q^37 + 150 * q^41 + 292 * q^43 + 72 * q^47 - 441 * q^49 - 828 * q^53 + 240 * q^55 + 744 * q^59 + 418 * q^61 - 350 * q^65 - 188 * q^67 + 960 * q^71 + 868 * q^73 - 672 * q^77 - 1352 * q^79 + 612 * q^83 + 510 * q^85 - 60 * q^89 + 3920 * q^91 + 100 * q^95 + 286 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 2.50000 + 4.33013i 0 14.0000 24.2487i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 14.0000 + 24.2487i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.l 2
3.b odd 2 1 1620.4.i.f 2
9.c even 3 1 60.4.a.a 1
9.c even 3 1 inner 1620.4.i.l 2
9.d odd 6 1 180.4.a.d 1
9.d odd 6 1 1620.4.i.f 2
36.f odd 6 1 240.4.a.i 1
36.h even 6 1 720.4.a.bb 1
45.h odd 6 1 900.4.a.q 1
45.j even 6 1 300.4.a.i 1
45.k odd 12 2 300.4.d.b 2
45.l even 12 2 900.4.d.h 2
72.n even 6 1 960.4.a.bc 1
72.p odd 6 1 960.4.a.r 1
180.p odd 6 1 1200.4.a.a 1
180.x even 12 2 1200.4.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.a 1 9.c even 3 1
180.4.a.d 1 9.d odd 6 1
240.4.a.i 1 36.f odd 6 1
300.4.a.i 1 45.j even 6 1
300.4.d.b 2 45.k odd 12 2
720.4.a.bb 1 36.h even 6 1
900.4.a.q 1 45.h odd 6 1
900.4.d.h 2 45.l even 12 2
960.4.a.r 1 72.p odd 6 1
960.4.a.bc 1 72.n even 6 1
1200.4.a.a 1 180.p odd 6 1
1200.4.f.n 2 180.x even 12 2
1620.4.i.f 2 3.b odd 2 1
1620.4.i.f 2 9.d odd 6 1
1620.4.i.l 2 1.a even 1 1 trivial
1620.4.i.l 2 9.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{2} - 28T_{7} + 784$$ T7^2 - 28*T7 + 784 $$T_{11}^{2} - 24T_{11} + 576$$ T11^2 - 24*T11 + 576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} - 28T + 784$$
$11$ $$T^{2} - 24T + 576$$
$13$ $$T^{2} - 70T + 4900$$
$17$ $$(T - 102)^{2}$$
$19$ $$(T - 20)^{2}$$
$23$ $$T^{2} - 72T + 5184$$
$29$ $$T^{2} + 306T + 93636$$
$31$ $$T^{2} - 136T + 18496$$
$37$ $$(T + 214)^{2}$$
$41$ $$T^{2} - 150T + 22500$$
$43$ $$T^{2} - 292T + 85264$$
$47$ $$T^{2} - 72T + 5184$$
$53$ $$(T + 414)^{2}$$
$59$ $$T^{2} - 744T + 553536$$
$61$ $$T^{2} - 418T + 174724$$
$67$ $$T^{2} + 188T + 35344$$
$71$ $$(T - 480)^{2}$$
$73$ $$(T - 434)^{2}$$
$79$ $$T^{2} + 1352 T + 1827904$$
$83$ $$T^{2} - 612T + 374544$$
$89$ $$(T + 30)^{2}$$
$97$ $$T^{2} - 286T + 81796$$