# Properties

 Label 1620.4.i.q Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-23})$$ Defining polynomial: $$x^{4} - x^{3} - 5x^{2} - 6x + 36$$ x^4 - x^3 - 5*x^2 - 6*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 \beta_1 q^{5} + ( - \beta_{3} - 5 \beta_1 - 5) q^{7}+O(q^{10})$$ q - 5*b1 * q^5 + (-b3 - 5*b1 - 5) * q^7 $$q - 5 \beta_1 q^{5} + ( - \beta_{3} - 5 \beta_1 - 5) q^{7} + ( - \beta_{3} + 9 \beta_1 + 9) q^{11} + (\beta_{3} - \beta_{2} + 17 \beta_1) q^{13} + ( - \beta_{2} - 12) q^{17} + ( - \beta_{2} + 14) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2} - 15 \beta_1) q^{23} + ( - 25 \beta_1 - 25) q^{25} + ( - 5 \beta_{3} - 21 \beta_1 - 21) q^{29} + ( - 7 \beta_{3} + 7 \beta_{2} + 128 \beta_1) q^{31} + ( - 5 \beta_{2} - 25) q^{35} + 74 q^{37} + (9 \beta_{3} - 9 \beta_{2} + 111 \beta_1) q^{41} + (5 \beta_{3} - 209 \beta_1 - 209) q^{43} + ( - 10 \beta_{3} - 6 \beta_1 - 6) q^{47} + (10 \beta_{3} - 10 \beta_{2} + 303 \beta_1) q^{49} + ( - 17 \beta_{2} + 24) q^{53} + ( - 5 \beta_{2} + 45) q^{55} + ( - 7 \beta_{3} + 7 \beta_{2} + 177 \beta_1) q^{59} + ( - 6 \beta_{3} - 419 \beta_1 - 419) q^{61} + (5 \beta_{3} + 85 \beta_1 + 85) q^{65} + (6 \beta_{3} - 6 \beta_{2} + 620 \beta_1) q^{67} + (27 \beta_{2} - 231) q^{71} + (21 \beta_{2} + 443) q^{73} + ( - 4 \beta_{3} + 4 \beta_{2} + 576 \beta_1) q^{77} + (3 \beta_{3} - 578 \beta_1 - 578) q^{79} + (28 \beta_{3} + 573 \beta_1 + 573) q^{83} + (5 \beta_{3} - 5 \beta_{2} + 60 \beta_1) q^{85} + (3 \beta_{2} - 75) q^{89} + (22 \beta_{2} + 706) q^{91} + (5 \beta_{3} - 5 \beta_{2} - 70 \beta_1) q^{95} + (16 \beta_{3} - 392 \beta_1 - 392) q^{97}+O(q^{100})$$ q - 5*b1 * q^5 + (-b3 - 5*b1 - 5) * q^7 + (-b3 + 9*b1 + 9) * q^11 + (b3 - b2 + 17*b1) * q^13 + (-b2 - 12) * q^17 + (-b2 + 14) * q^19 + (-4*b3 + 4*b2 - 15*b1) * q^23 + (-25*b1 - 25) * q^25 + (-5*b3 - 21*b1 - 21) * q^29 + (-7*b3 + 7*b2 + 128*b1) * q^31 + (-5*b2 - 25) * q^35 + 74 * q^37 + (9*b3 - 9*b2 + 111*b1) * q^41 + (5*b3 - 209*b1 - 209) * q^43 + (-10*b3 - 6*b1 - 6) * q^47 + (10*b3 - 10*b2 + 303*b1) * q^49 + (-17*b2 + 24) * q^53 + (-5*b2 + 45) * q^55 + (-7*b3 + 7*b2 + 177*b1) * q^59 + (-6*b3 - 419*b1 - 419) * q^61 + (5*b3 + 85*b1 + 85) * q^65 + (6*b3 - 6*b2 + 620*b1) * q^67 + (27*b2 - 231) * q^71 + (21*b2 + 443) * q^73 + (-4*b3 + 4*b2 + 576*b1) * q^77 + (3*b3 - 578*b1 - 578) * q^79 + (28*b3 + 573*b1 + 573) * q^83 + (5*b3 - 5*b2 + 60*b1) * q^85 + (3*b2 - 75) * q^89 + (22*b2 + 706) * q^91 + (5*b3 - 5*b2 - 70*b1) * q^95 + (16*b3 - 392*b1 - 392) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{5} - 10 q^{7}+O(q^{10})$$ 4 * q + 10 * q^5 - 10 * q^7 $$4 q + 10 q^{5} - 10 q^{7} + 18 q^{11} - 34 q^{13} - 48 q^{17} + 56 q^{19} + 30 q^{23} - 50 q^{25} - 42 q^{29} - 256 q^{31} - 100 q^{35} + 296 q^{37} - 222 q^{41} - 418 q^{43} - 12 q^{47} - 606 q^{49} + 96 q^{53} + 180 q^{55} - 354 q^{59} - 838 q^{61} + 170 q^{65} - 1240 q^{67} - 924 q^{71} + 1772 q^{73} - 1152 q^{77} - 1156 q^{79} + 1146 q^{83} - 120 q^{85} - 300 q^{89} + 2824 q^{91} + 140 q^{95} - 784 q^{97}+O(q^{100})$$ 4 * q + 10 * q^5 - 10 * q^7 + 18 * q^11 - 34 * q^13 - 48 * q^17 + 56 * q^19 + 30 * q^23 - 50 * q^25 - 42 * q^29 - 256 * q^31 - 100 * q^35 + 296 * q^37 - 222 * q^41 - 418 * q^43 - 12 * q^47 - 606 * q^49 + 96 * q^53 + 180 * q^55 - 354 * q^59 - 838 * q^61 + 170 * q^65 - 1240 * q^67 - 924 * q^71 + 1772 * q^73 - 1152 * q^77 - 1156 * q^79 + 1146 * q^83 - 120 * q^85 - 300 * q^89 + 2824 * q^91 + 140 * q^95 - 784 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5x^{2} - 6x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 5\nu^{2} - 5\nu - 36 ) / 30$$ (v^3 + 5*v^2 - 5*v - 36) / 30 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 11\nu + 3$$ -v^3 + v^2 + 11*v + 3 $$\beta_{3}$$ $$=$$ $$( 13\nu^{3} + 5\nu^{2} + 55\nu - 138 ) / 10$$ (13*v^3 + 5*v^2 + 55*v - 138) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 9\beta_1 ) / 18$$ (b3 + b2 - 9*b1) / 18 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 99\beta _1 + 99 ) / 18$$ (-b3 + 2*b2 + 99*b1 + 99) / 18 $$\nu^{3}$$ $$=$$ $$( 10\beta_{3} - 5\beta_{2} + 153 ) / 18$$ (10*b3 - 5*b2 + 153) / 18

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 2.32666 − 0.765945i −1.82666 + 1.63197i 2.32666 + 0.765945i −1.82666 − 1.63197i
0 0 0 2.50000 + 4.33013i 0 −14.9599 + 25.9114i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 9.95994 17.2511i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −14.9599 25.9114i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 9.95994 + 17.2511i 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.q 4
3.b odd 2 1 1620.4.i.n 4
9.c even 3 1 540.4.a.f 2
9.c even 3 1 inner 1620.4.i.q 4
9.d odd 6 1 540.4.a.i yes 2
9.d odd 6 1 1620.4.i.n 4
36.f odd 6 1 2160.4.a.v 2
36.h even 6 1 2160.4.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.f 2 9.c even 3 1
540.4.a.i yes 2 9.d odd 6 1
1620.4.i.n 4 3.b odd 2 1
1620.4.i.n 4 9.d odd 6 1
1620.4.i.q 4 1.a even 1 1 trivial
1620.4.i.q 4 9.c even 3 1 inner
2160.4.a.v 2 36.f odd 6 1
2160.4.a.ba 2 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 $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{4} + 10T_{7}^{3} + 696T_{7}^{2} - 5960T_{7} + 355216$$ T7^4 + 10*T7^3 + 696*T7^2 - 5960*T7 + 355216 $$T_{11}^{4} - 18T_{11}^{3} + 864T_{11}^{2} + 9720T_{11} + 291600$$ T11^4 - 18*T11^3 + 864*T11^2 + 9720*T11 + 291600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4} + 10 T^{3} + 696 T^{2} + \cdots + 355216$$
$11$ $$T^{4} - 18 T^{3} + 864 T^{2} + \cdots + 291600$$
$13$ $$T^{4} + 34 T^{3} + 1488 T^{2} + \cdots + 110224$$
$17$ $$(T^{2} + 24 T - 477)^{2}$$
$19$ $$(T^{2} - 28 T - 425)^{2}$$
$23$ $$T^{4} - 30 T^{3} + 10611 T^{2} + \cdots + 94303521$$
$29$ $$T^{4} + 42 T^{3} + \cdots + 227527056$$
$31$ $$T^{4} + 256 T^{3} + \cdots + 197262025$$
$37$ $$(T - 74)^{4}$$
$41$ $$T^{4} + 222 T^{3} + \cdots + 1442480400$$
$43$ $$T^{4} + 418 T^{3} + \cdots + 792760336$$
$47$ $$T^{4} + 12 T^{3} + \cdots + 3851940096$$
$53$ $$(T^{2} - 48 T - 178893)^{2}$$
$59$ $$T^{4} + 354 T^{3} + 124416 T^{2} + \cdots + 810000$$
$61$ $$T^{4} + 838 T^{3} + \cdots + 23471772025$$
$67$ $$T^{4} + 1240 T^{3} + \cdots + 131075857936$$
$71$ $$(T^{2} + 462 T - 399348)^{2}$$
$73$ $$(T^{2} - 886 T - 77612)^{2}$$
$79$ $$T^{4} + 1156 T^{3} + \cdots + 107908965025$$
$83$ $$T^{4} - 1146 T^{3} + \cdots + 25133346225$$
$89$ $$(T^{2} + 150 T + 36)^{2}$$
$97$ $$T^{4} + 784 T^{3} + \cdots + 28217344$$