# Properties

 Label 1620.4.i.k 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 540) 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.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{6} q^{5} + ( - 22 \zeta_{6} + 22) q^{7}+O(q^{10})$$ q + 5*z * q^5 + (-22*z + 22) * q^7 $$q + 5 \zeta_{6} q^{5} + ( - 22 \zeta_{6} + 22) q^{7} + ( - 9 \zeta_{6} + 9) q^{11} - 17 \zeta_{6} q^{13} - 75 q^{17} - 4 q^{19} - 183 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (129 \zeta_{6} - 129) q^{29} + 187 \zeta_{6} q^{31} + 110 q^{35} - 34 q^{37} - 264 \zeta_{6} q^{41} + (443 \zeta_{6} - 443) q^{43} + (609 \zeta_{6} - 609) q^{47} - 141 \zeta_{6} q^{49} - 228 q^{53} + 45 q^{55} - 60 \zeta_{6} q^{59} + ( - 454 \zeta_{6} + 454) q^{61} + ( - 85 \zeta_{6} + 85) q^{65} + 244 \zeta_{6} q^{67} + 444 q^{71} + 398 q^{73} - 198 \zeta_{6} q^{77} + ( - 349 \zeta_{6} + 349) q^{79} + (1038 \zeta_{6} - 1038) q^{83} - 375 \zeta_{6} q^{85} + 852 q^{89} - 374 q^{91} - 20 \zeta_{6} q^{95} + (914 \zeta_{6} - 914) q^{97} +O(q^{100})$$ q + 5*z * q^5 + (-22*z + 22) * q^7 + (-9*z + 9) * q^11 - 17*z * q^13 - 75 * q^17 - 4 * q^19 - 183*z * q^23 + (25*z - 25) * q^25 + (129*z - 129) * q^29 + 187*z * q^31 + 110 * q^35 - 34 * q^37 - 264*z * q^41 + (443*z - 443) * q^43 + (609*z - 609) * q^47 - 141*z * q^49 - 228 * q^53 + 45 * q^55 - 60*z * q^59 + (-454*z + 454) * q^61 + (-85*z + 85) * q^65 + 244*z * q^67 + 444 * q^71 + 398 * q^73 - 198*z * q^77 + (-349*z + 349) * q^79 + (1038*z - 1038) * q^83 - 375*z * q^85 + 852 * q^89 - 374 * q^91 - 20*z * q^95 + (914*z - 914) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{5} + 22 q^{7}+O(q^{10})$$ 2 * q + 5 * q^5 + 22 * q^7 $$2 q + 5 q^{5} + 22 q^{7} + 9 q^{11} - 17 q^{13} - 150 q^{17} - 8 q^{19} - 183 q^{23} - 25 q^{25} - 129 q^{29} + 187 q^{31} + 220 q^{35} - 68 q^{37} - 264 q^{41} - 443 q^{43} - 609 q^{47} - 141 q^{49} - 456 q^{53} + 90 q^{55} - 60 q^{59} + 454 q^{61} + 85 q^{65} + 244 q^{67} + 888 q^{71} + 796 q^{73} - 198 q^{77} + 349 q^{79} - 1038 q^{83} - 375 q^{85} + 1704 q^{89} - 748 q^{91} - 20 q^{95} - 914 q^{97}+O(q^{100})$$ 2 * q + 5 * q^5 + 22 * q^7 + 9 * q^11 - 17 * q^13 - 150 * q^17 - 8 * q^19 - 183 * q^23 - 25 * q^25 - 129 * q^29 + 187 * q^31 + 220 * q^35 - 68 * q^37 - 264 * q^41 - 443 * q^43 - 609 * q^47 - 141 * q^49 - 456 * q^53 + 90 * q^55 - 60 * q^59 + 454 * q^61 + 85 * q^65 + 244 * q^67 + 888 * q^71 + 796 * q^73 - 198 * q^77 + 349 * q^79 - 1038 * q^83 - 375 * q^85 + 1704 * q^89 - 748 * q^91 - 20 * q^95 - 914 * 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 11.0000 19.0526i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 11.0000 + 19.0526i 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.k 2
3.b odd 2 1 1620.4.i.e 2
9.c even 3 1 540.4.a.a 1
9.c even 3 1 inner 1620.4.i.k 2
9.d odd 6 1 540.4.a.c yes 1
9.d odd 6 1 1620.4.i.e 2
36.f odd 6 1 2160.4.a.h 1
36.h even 6 1 2160.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.a 1 9.c even 3 1
540.4.a.c yes 1 9.d odd 6 1
1620.4.i.e 2 3.b odd 2 1
1620.4.i.e 2 9.d odd 6 1
1620.4.i.k 2 1.a even 1 1 trivial
1620.4.i.k 2 9.c even 3 1 inner
2160.4.a.h 1 36.f odd 6 1
2160.4.a.s 1 36.h even 6 1

## Hecke kernels

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

 $$T_{7}^{2} - 22T_{7} + 484$$ T7^2 - 22*T7 + 484 $$T_{11}^{2} - 9T_{11} + 81$$ T11^2 - 9*T11 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} - 22T + 484$$
$11$ $$T^{2} - 9T + 81$$
$13$ $$T^{2} + 17T + 289$$
$17$ $$(T + 75)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 183T + 33489$$
$29$ $$T^{2} + 129T + 16641$$
$31$ $$T^{2} - 187T + 34969$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} + 264T + 69696$$
$43$ $$T^{2} + 443T + 196249$$
$47$ $$T^{2} + 609T + 370881$$
$53$ $$(T + 228)^{2}$$
$59$ $$T^{2} + 60T + 3600$$
$61$ $$T^{2} - 454T + 206116$$
$67$ $$T^{2} - 244T + 59536$$
$71$ $$(T - 444)^{2}$$
$73$ $$(T - 398)^{2}$$
$79$ $$T^{2} - 349T + 121801$$
$83$ $$T^{2} + 1038 T + 1077444$$
$89$ $$(T - 852)^{2}$$
$97$ $$T^{2} + 914T + 835396$$