# Properties

 Label 1620.4.i.r 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: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2x + 4$$ x^4 - x^3 - x^2 - 2*x + 4 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.

 $$f(q)$$ $$=$$ $$q - 5 \beta_1 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7}+O(q^{10})$$ q - 5*b1 * q^5 + (b3 + b1 + 1) * q^7 $$q - 5 \beta_1 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} - 21 \beta_1 - 21) q^{11} + (5 \beta_{3} - 5 \beta_{2} - 7 \beta_1) q^{13} + (7 \beta_{2} + 12) q^{17} + (\beta_{2} - 70) q^{19} + ( - 8 \beta_{3} + 8 \beta_{2} + 63 \beta_1) q^{23} + ( - 25 \beta_1 - 25) q^{25} + (11 \beta_{3} + 63 \beta_1 + 63) q^{29} + ( - 17 \beta_{3} + 17 \beta_{2} - 28 \beta_1) q^{31} + (5 \beta_{2} + 5) q^{35} - 286 q^{37} + (21 \beta_{3} - 21 \beta_{2} - 33 \beta_1) q^{41} + (7 \beta_{3} + 265 \beta_1 + 265) q^{43} + ( - 14 \beta_{3} + 198 \beta_1 + 198) q^{47} + (2 \beta_{3} - 2 \beta_{2} - 153 \beta_1) q^{49} + ( - 25 \beta_{2} - 252) q^{53} + (5 \beta_{2} - 105) q^{55} + (7 \beta_{3} - 7 \beta_{2} + 219 \beta_1) q^{59} + ( - 42 \beta_{3} + 301 \beta_1 + 301) q^{61} + (25 \beta_{3} - 35 \beta_1 - 35) q^{65} + (42 \beta_{3} - 42 \beta_{2} - 460 \beta_1) q^{67} + ( - 15 \beta_{2} + 399) q^{71} + ( - 27 \beta_{2} - 385) q^{73} + ( - 20 \beta_{3} + 20 \beta_{2} + 168 \beta_1) q^{77} + (21 \beta_{3} + 862 \beta_1 + 862) q^{79} + ( - 28 \beta_{3} + 243 \beta_1 + 243) q^{83} + ( - 35 \beta_{3} + 35 \beta_{2} - 60 \beta_1) q^{85} + (63 \beta_{2} - 147) q^{89} + (2 \beta_{2} - 938) q^{91} + ( - 5 \beta_{3} + 5 \beta_{2} + 350 \beta_1) q^{95} + (80 \beta_{3} - 56 \beta_1 - 56) q^{97}+O(q^{100})$$ q - 5*b1 * q^5 + (b3 + b1 + 1) * q^7 + (b3 - 21*b1 - 21) * q^11 + (5*b3 - 5*b2 - 7*b1) * q^13 + (7*b2 + 12) * q^17 + (b2 - 70) * q^19 + (-8*b3 + 8*b2 + 63*b1) * q^23 + (-25*b1 - 25) * q^25 + (11*b3 + 63*b1 + 63) * q^29 + (-17*b3 + 17*b2 - 28*b1) * q^31 + (5*b2 + 5) * q^35 - 286 * q^37 + (21*b3 - 21*b2 - 33*b1) * q^41 + (7*b3 + 265*b1 + 265) * q^43 + (-14*b3 + 198*b1 + 198) * q^47 + (2*b3 - 2*b2 - 153*b1) * q^49 + (-25*b2 - 252) * q^53 + (5*b2 - 105) * q^55 + (7*b3 - 7*b2 + 219*b1) * q^59 + (-42*b3 + 301*b1 + 301) * q^61 + (25*b3 - 35*b1 - 35) * q^65 + (42*b3 - 42*b2 - 460*b1) * q^67 + (-15*b2 + 399) * q^71 + (-27*b2 - 385) * q^73 + (-20*b3 + 20*b2 + 168*b1) * q^77 + (21*b3 + 862*b1 + 862) * q^79 + (-28*b3 + 243*b1 + 243) * q^83 + (-35*b3 + 35*b2 - 60*b1) * q^85 + (63*b2 - 147) * q^89 + (2*b2 - 938) * q^91 + (-5*b3 + 5*b2 + 350*b1) * q^95 + (80*b3 - 56*b1 - 56) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{5} + 2 q^{7}+O(q^{10})$$ 4 * q + 10 * q^5 + 2 * q^7 $$4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100})$$ 4 * q + 10 * q^5 + 2 * q^7 - 42 * q^11 + 14 * q^13 + 48 * q^17 - 280 * q^19 - 126 * q^23 - 50 * q^25 + 126 * q^29 + 56 * q^31 + 20 * q^35 - 1144 * q^37 + 66 * q^41 + 530 * q^43 + 396 * q^47 + 306 * q^49 - 1008 * q^53 - 420 * q^55 - 438 * q^59 + 602 * q^61 - 70 * q^65 + 920 * q^67 + 1596 * q^71 - 1540 * q^73 - 336 * q^77 + 1724 * q^79 + 486 * q^83 + 120 * q^85 - 588 * q^89 - 3752 * q^91 - 700 * q^95 - 112 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} - \nu - 4 ) / 2$$ (v^3 + v^2 - v - 4) / 2 $$\beta_{2}$$ $$=$$ $$-3\nu^{3} + 3\nu^{2} + 9\nu + 3$$ -3*v^3 + 3*v^2 + 9*v + 3 $$\beta_{3}$$ $$=$$ $$( 15\nu^{3} + 3\nu^{2} + 9\nu - 42 ) / 2$$ (15*v^3 + 3*v^2 + 9*v - 42) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 9\beta_1 ) / 18$$ (b3 + b2 - 9*b1) / 18 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 27\beta _1 + 27 ) / 18$$ (-b3 + 2*b2 + 27*b1 + 27) / 18 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} - \beta_{2} + 45 ) / 18$$ (2*b3 - b2 + 45) / 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
 −0.895644 + 1.09445i 1.39564 − 0.228425i −0.895644 − 1.09445i 1.39564 + 0.228425i
0 0 0 2.50000 + 4.33013i 0 −6.37386 + 11.0399i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 7.37386 12.7719i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −6.37386 11.0399i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 7.37386 + 12.7719i 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.r 4
3.b odd 2 1 1620.4.i.o 4
9.c even 3 1 540.4.a.e 2
9.c even 3 1 inner 1620.4.i.r 4
9.d odd 6 1 540.4.a.h yes 2
9.d odd 6 1 1620.4.i.o 4
36.f odd 6 1 2160.4.a.x 2
36.h even 6 1 2160.4.a.bc 2

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

 $$T_{7}^{4} - 2T_{7}^{3} + 192T_{7}^{2} + 376T_{7} + 35344$$ T7^4 - 2*T7^3 + 192*T7^2 + 376*T7 + 35344 $$T_{11}^{4} + 42T_{11}^{3} + 1512T_{11}^{2} + 10584T_{11} + 63504$$ T11^4 + 42*T11^3 + 1512*T11^2 + 10584*T11 + 63504

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4} - 2 T^{3} + 192 T^{2} + \cdots + 35344$$
$11$ $$T^{4} + 42 T^{3} + 1512 T^{2} + \cdots + 63504$$
$13$ $$T^{4} - 14 T^{3} + 4872 T^{2} + \cdots + 21864976$$
$17$ $$(T^{2} - 24 T - 9117)^{2}$$
$19$ $$(T^{2} + 140 T + 4711)^{2}$$
$23$ $$T^{4} + 126 T^{3} + \cdots + 66048129$$
$29$ $$T^{4} - 126 T^{3} + \cdots + 357210000$$
$31$ $$T^{4} - 56 T^{3} + \cdots + 2898422569$$
$37$ $$(T + 286)^{4}$$
$41$ $$T^{4} - 66 T^{3} + \cdots + 6766707600$$
$43$ $$T^{4} - 530 T^{3} + \cdots + 3716609296$$
$47$ $$T^{4} - 396 T^{3} + 154656 T^{2} + \cdots + 4665600$$
$53$ $$(T^{2} + 504 T - 54621)^{2}$$
$59$ $$T^{4} + 438 T^{3} + \cdots + 1497690000$$
$61$ $$T^{4} - 602 T^{3} + \cdots + 58949412025$$
$67$ $$T^{4} - 920 T^{3} + \cdots + 14834265616$$
$71$ $$(T^{2} - 798 T + 116676)^{2}$$
$73$ $$(T^{2} + 770 T + 10444)^{2}$$
$79$ $$T^{4} - 1724 T^{3} + \cdots + 435197493025$$
$83$ $$T^{4} - 486 T^{3} + \cdots + 7943622129$$
$89$ $$(T^{2} + 294 T - 728532)^{2}$$
$97$ $$T^{4} + 112 T^{3} + \cdots + 1455555383296$$