# Properties

 Label 1620.4.i.d 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 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.

 $$f(q)$$ $$=$$ $$q - 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} +O(q^{10})$$ q - 5*z * q^5 + (-16*z + 16) * q^7 $$q - 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} + ( - 60 \zeta_{6} + 60) q^{11} - 86 \zeta_{6} q^{13} + 18 q^{17} + 44 q^{19} - 48 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + ( - 186 \zeta_{6} + 186) q^{29} - 176 \zeta_{6} q^{31} - 80 q^{35} + 254 q^{37} - 186 \zeta_{6} q^{41} + ( - 100 \zeta_{6} + 100) q^{43} + (168 \zeta_{6} - 168) q^{47} + 87 \zeta_{6} q^{49} - 498 q^{53} - 300 q^{55} + 252 \zeta_{6} q^{59} + ( - 58 \zeta_{6} + 58) q^{61} + (430 \zeta_{6} - 430) q^{65} + 1036 \zeta_{6} q^{67} + 168 q^{71} + 506 q^{73} - 960 \zeta_{6} q^{77} + (272 \zeta_{6} - 272) q^{79} + (948 \zeta_{6} - 948) q^{83} - 90 \zeta_{6} q^{85} - 1014 q^{89} - 1376 q^{91} - 220 \zeta_{6} q^{95} + ( - 766 \zeta_{6} + 766) q^{97} +O(q^{100})$$ q - 5*z * q^5 + (-16*z + 16) * q^7 + (-60*z + 60) * q^11 - 86*z * q^13 + 18 * q^17 + 44 * q^19 - 48*z * q^23 + (25*z - 25) * q^25 + (-186*z + 186) * q^29 - 176*z * q^31 - 80 * q^35 + 254 * q^37 - 186*z * q^41 + (-100*z + 100) * q^43 + (168*z - 168) * q^47 + 87*z * q^49 - 498 * q^53 - 300 * q^55 + 252*z * q^59 + (-58*z + 58) * q^61 + (430*z - 430) * q^65 + 1036*z * q^67 + 168 * q^71 + 506 * q^73 - 960*z * q^77 + (272*z - 272) * q^79 + (948*z - 948) * q^83 - 90*z * q^85 - 1014 * q^89 - 1376 * q^91 - 220*z * q^95 + (-766*z + 766) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{5} + 16 q^{7}+O(q^{10})$$ 2 * q - 5 * q^5 + 16 * q^7 $$2 q - 5 q^{5} + 16 q^{7} + 60 q^{11} - 86 q^{13} + 36 q^{17} + 88 q^{19} - 48 q^{23} - 25 q^{25} + 186 q^{29} - 176 q^{31} - 160 q^{35} + 508 q^{37} - 186 q^{41} + 100 q^{43} - 168 q^{47} + 87 q^{49} - 996 q^{53} - 600 q^{55} + 252 q^{59} + 58 q^{61} - 430 q^{65} + 1036 q^{67} + 336 q^{71} + 1012 q^{73} - 960 q^{77} - 272 q^{79} - 948 q^{83} - 90 q^{85} - 2028 q^{89} - 2752 q^{91} - 220 q^{95} + 766 q^{97}+O(q^{100})$$ 2 * q - 5 * q^5 + 16 * q^7 + 60 * q^11 - 86 * q^13 + 36 * q^17 + 88 * q^19 - 48 * q^23 - 25 * q^25 + 186 * q^29 - 176 * q^31 - 160 * q^35 + 508 * q^37 - 186 * q^41 + 100 * q^43 - 168 * q^47 + 87 * q^49 - 996 * q^53 - 600 * q^55 + 252 * q^59 + 58 * q^61 - 430 * q^65 + 1036 * q^67 + 336 * q^71 + 1012 * q^73 - 960 * q^77 - 272 * q^79 - 948 * q^83 - 90 * q^85 - 2028 * q^89 - 2752 * q^91 - 220 * q^95 + 766 * 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 8.00000 13.8564i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 8.00000 + 13.8564i 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.d 2
3.b odd 2 1 1620.4.i.j 2
9.c even 3 1 20.4.a.a 1
9.c even 3 1 inner 1620.4.i.d 2
9.d odd 6 1 180.4.a.a 1
9.d odd 6 1 1620.4.i.j 2
36.f odd 6 1 80.4.a.c 1
36.h even 6 1 720.4.a.k 1
45.h odd 6 1 900.4.a.m 1
45.j even 6 1 100.4.a.a 1
45.k odd 12 2 100.4.c.a 2
45.l even 12 2 900.4.d.k 2
63.g even 3 1 980.4.i.e 2
63.h even 3 1 980.4.i.e 2
63.k odd 6 1 980.4.i.n 2
63.l odd 6 1 980.4.a.c 1
63.t odd 6 1 980.4.i.n 2
72.n even 6 1 320.4.a.d 1
72.p odd 6 1 320.4.a.k 1
99.h odd 6 1 2420.4.a.d 1
144.v odd 12 2 1280.4.d.c 2
144.x even 12 2 1280.4.d.n 2
180.p odd 6 1 400.4.a.o 1
180.x even 12 2 400.4.c.j 2
360.z odd 6 1 1600.4.a.p 1
360.bk even 6 1 1600.4.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 9.c even 3 1
80.4.a.c 1 36.f odd 6 1
100.4.a.a 1 45.j even 6 1
100.4.c.a 2 45.k odd 12 2
180.4.a.a 1 9.d odd 6 1
320.4.a.d 1 72.n even 6 1
320.4.a.k 1 72.p odd 6 1
400.4.a.o 1 180.p odd 6 1
400.4.c.j 2 180.x even 12 2
720.4.a.k 1 36.h even 6 1
900.4.a.m 1 45.h odd 6 1
900.4.d.k 2 45.l even 12 2
980.4.a.c 1 63.l odd 6 1
980.4.i.e 2 63.g even 3 1
980.4.i.e 2 63.h even 3 1
980.4.i.n 2 63.k odd 6 1
980.4.i.n 2 63.t odd 6 1
1280.4.d.c 2 144.v odd 12 2
1280.4.d.n 2 144.x even 12 2
1600.4.a.p 1 360.z odd 6 1
1600.4.a.bl 1 360.bk even 6 1
1620.4.i.d 2 1.a even 1 1 trivial
1620.4.i.d 2 9.c even 3 1 inner
1620.4.i.j 2 3.b odd 2 1
1620.4.i.j 2 9.d odd 6 1
2420.4.a.d 1 99.h odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} - 16T_{7} + 256$$ T7^2 - 16*T7 + 256 $$T_{11}^{2} - 60T_{11} + 3600$$ T11^2 - 60*T11 + 3600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 16T + 256$$
$11$ $$T^{2} - 60T + 3600$$
$13$ $$T^{2} + 86T + 7396$$
$17$ $$(T - 18)^{2}$$
$19$ $$(T - 44)^{2}$$
$23$ $$T^{2} + 48T + 2304$$
$29$ $$T^{2} - 186T + 34596$$
$31$ $$T^{2} + 176T + 30976$$
$37$ $$(T - 254)^{2}$$
$41$ $$T^{2} + 186T + 34596$$
$43$ $$T^{2} - 100T + 10000$$
$47$ $$T^{2} + 168T + 28224$$
$53$ $$(T + 498)^{2}$$
$59$ $$T^{2} - 252T + 63504$$
$61$ $$T^{2} - 58T + 3364$$
$67$ $$T^{2} - 1036 T + 1073296$$
$71$ $$(T - 168)^{2}$$
$73$ $$(T - 506)^{2}$$
$79$ $$T^{2} + 272T + 73984$$
$83$ $$T^{2} + 948T + 898704$$
$89$ $$(T + 1014)^{2}$$
$97$ $$T^{2} - 766T + 586756$$