# Properties

 Label 1620.3.o.f Level $1620$ Weight $3$ Character orbit 1620.o Analytic conductor $44.142$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1620.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.8 Defining polynomial: $$x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81$$ x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - \beta_{7} - 4 \beta_{2} + 4) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-b7 - 4*b2 + 4) * q^7 $$q + \beta_1 q^{5} + ( - \beta_{7} - 4 \beta_{2} + 4) q^{7} + (6 \beta_{5} - \beta_{4}) q^{11} + (\beta_{6} - 2 \beta_{2}) q^{13} - 4 \beta_{3} q^{17} + (2 \beta_{7} - 2 \beta_{6} + 8) q^{19} + (2 \beta_{4} - 2 \beta_{3} + 6 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + (6 \beta_{5} - 8 \beta_{4}) q^{29} + (2 \beta_{6} - 2 \beta_{2}) q^{31} + (4 \beta_{5} - 5 \beta_{3} + 4 \beta_1) q^{35} + ( - 3 \beta_{7} + 3 \beta_{6} - 34) q^{37} + ( - 3 \beta_{4} + 3 \beta_{3} - 24 \beta_1) q^{41} + (2 \beta_{7} + 20 \beta_{2} - 20) q^{43} + (12 \beta_{5} + 14 \beta_{4}) q^{47} + ( - 8 \beta_{6} - 57 \beta_{2}) q^{49} + (24 \beta_{5} + 10 \beta_{3} + 24 \beta_1) q^{53} + (\beta_{7} - \beta_{6} - 30) q^{55} + (17 \beta_{4} - 17 \beta_{3} - 18 \beta_1) q^{59} + (6 \beta_{7} - 10 \beta_{2} + 10) q^{61} + (2 \beta_{5} + 5 \beta_{4}) q^{65} + 76 \beta_{2} q^{67} + ( - 12 \beta_{5} + 12 \beta_{3} - 12 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{6} + 38) q^{73} + ( - 34 \beta_{4} + 34 \beta_{3} - 42 \beta_1) q^{77} + ( - 6 \beta_{7} + 50 \beta_{2} - 50) q^{79} + (24 \beta_{5} - 2 \beta_{4}) q^{83} - 4 \beta_{6} q^{85} + (12 \beta_{5} + 21 \beta_{3} + 12 \beta_1) q^{89} + ( - 2 \beta_{7} + 2 \beta_{6} + 82) q^{91} + ( - 10 \beta_{4} + 10 \beta_{3} + 8 \beta_1) q^{95} + ( - 2 \beta_{7} - 106 \beta_{2} + 106) q^{97}+O(q^{100})$$ q + b1 * q^5 + (-b7 - 4*b2 + 4) * q^7 + (6*b5 - b4) * q^11 + (b6 - 2*b2) * q^13 - 4*b3 * q^17 + (2*b7 - 2*b6 + 8) * q^19 + (2*b4 - 2*b3 + 6*b1) * q^23 + (-5*b2 + 5) * q^25 + (6*b5 - 8*b4) * q^29 + (2*b6 - 2*b2) * q^31 + (4*b5 - 5*b3 + 4*b1) * q^35 + (-3*b7 + 3*b6 - 34) * q^37 + (-3*b4 + 3*b3 - 24*b1) * q^41 + (2*b7 + 20*b2 - 20) * q^43 + (12*b5 + 14*b4) * q^47 + (-8*b6 - 57*b2) * q^49 + (24*b5 + 10*b3 + 24*b1) * q^53 + (b7 - b6 - 30) * q^55 + (17*b4 - 17*b3 - 18*b1) * q^59 + (6*b7 - 10*b2 + 10) * q^61 + (2*b5 + 5*b4) * q^65 + 76*b2 * q^67 + (-12*b5 + 12*b3 - 12*b1) * q^71 + (-6*b7 + 6*b6 + 38) * q^73 + (-34*b4 + 34*b3 - 42*b1) * q^77 + (-6*b7 + 50*b2 - 50) * q^79 + (24*b5 - 2*b4) * q^83 - 4*b6 * q^85 + (12*b5 + 21*b3 + 12*b1) * q^89 + (-2*b7 + 2*b6 + 82) * q^91 + (-10*b4 + 10*b3 + 8*b1) * q^95 + (-2*b7 - 106*b2 + 106) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{7}+O(q^{10})$$ 8 * q + 16 * q^7 $$8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100})$$ 8 * q + 16 * q^7 - 8 * q^13 + 64 * q^19 + 20 * q^25 - 8 * q^31 - 272 * q^37 - 80 * q^43 - 228 * q^49 - 240 * q^55 + 40 * q^61 + 304 * q^67 + 304 * q^73 - 200 * q^79 + 656 * q^91 + 424 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63$$ (-v^6 + 14*v^4 - 7*v^2 - 36) / 63 $$\beta_{2}$$ $$=$$ $$( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63$$ (4*v^6 + 7*v^4 + 28*v^2 + 144) / 63 $$\beta_{3}$$ $$=$$ $$( 4\nu^{7} + 7\nu^{5} - 35\nu^{3} + 81\nu ) / 63$$ (4*v^7 + 7*v^5 - 35*v^3 + 81*v) / 63 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 63$$ (-5*v^7 + 7*v^5 - 35*v^3 - 180*v) / 63 $$\beta_{5}$$ $$=$$ $$( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63$$ (-8*v^6 - 14*v^4 + 7*v^2 - 162) / 63 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 13\nu ) / 7$$ (-v^7 + 13*v) / 7 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} + 49\nu^{5} + 133\nu^{3} + 684\nu ) / 63$$ (19*v^7 + 49*v^5 + 133*v^3 + 684*v) / 63
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{4} + \beta_{3} ) / 6$$ (b6 - b4 + b3) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{2} - 2$$ b5 + 2*b2 - 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - \beta_{6} - 7\beta_{3} ) / 6$$ (b7 - b6 - 7*b3) / 6 $$\nu^{4}$$ $$=$$ $$\beta_{2} + 4\beta_1$$ b2 + 4*b1 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} + 19\beta_{4} ) / 6$$ (5*b7 + 19*b4) / 6 $$\nu^{6}$$ $$=$$ $$-7\beta_{5} - 7\beta _1 - 22$$ -7*b5 - 7*b1 - 22 $$\nu^{7}$$ $$=$$ $$( -29\beta_{6} - 13\beta_{4} + 13\beta_{3} ) / 6$$ (-29*b6 - 13*b4 + 13*b3) / 6

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 −1.40294 + 1.01575i 1.40294 − 1.01575i −0.178197 + 1.72286i 0.178197 − 1.72286i −1.40294 − 1.01575i 1.40294 + 1.01575i −0.178197 − 1.72286i 0.178197 + 1.72286i
0 0 0 −1.93649 1.11803i 0 −2.74342 4.75174i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 6.74342 + 11.6799i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −2.74342 4.75174i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 6.74342 + 11.6799i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 6.74342 11.6799i 0 0 0
1241.3 0 0 0 1.93649 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.4 0 0 0 1.93649 1.11803i 0 6.74342 11.6799i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1241.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.f 8
3.b odd 2 1 inner 1620.3.o.f 8
9.c even 3 1 180.3.g.a 4
9.c even 3 1 inner 1620.3.o.f 8
9.d odd 6 1 180.3.g.a 4
9.d odd 6 1 inner 1620.3.o.f 8
36.f odd 6 1 720.3.l.c 4
36.h even 6 1 720.3.l.c 4
45.h odd 6 1 900.3.g.d 4
45.j even 6 1 900.3.g.d 4
45.k odd 12 2 900.3.b.b 8
45.l even 12 2 900.3.b.b 8
72.j odd 6 1 2880.3.l.b 4
72.l even 6 1 2880.3.l.f 4
72.n even 6 1 2880.3.l.b 4
72.p odd 6 1 2880.3.l.f 4
180.n even 6 1 3600.3.l.n 4
180.p odd 6 1 3600.3.l.n 4
180.v odd 12 2 3600.3.c.k 8
180.x even 12 2 3600.3.c.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 9.c even 3 1
180.3.g.a 4 9.d odd 6 1
720.3.l.c 4 36.f odd 6 1
720.3.l.c 4 36.h even 6 1
900.3.b.b 8 45.k odd 12 2
900.3.b.b 8 45.l even 12 2
900.3.g.d 4 45.h odd 6 1
900.3.g.d 4 45.j even 6 1
1620.3.o.f 8 1.a even 1 1 trivial
1620.3.o.f 8 3.b odd 2 1 inner
1620.3.o.f 8 9.c even 3 1 inner
1620.3.o.f 8 9.d odd 6 1 inner
2880.3.l.b 4 72.j odd 6 1
2880.3.l.b 4 72.n even 6 1
2880.3.l.f 4 72.l even 6 1
2880.3.l.f 4 72.p odd 6 1
3600.3.c.k 8 180.v odd 12 2
3600.3.c.k 8 180.x even 12 2
3600.3.l.n 4 180.n even 6 1
3600.3.l.n 4 180.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 8T_{7}^{3} + 138T_{7}^{2} + 592T_{7} + 5476$$ acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 5 T^{2} + 25)^{2}$$
$7$ $$(T^{4} - 8 T^{3} + 138 T^{2} + 592 T + 5476)^{2}$$
$11$ $$T^{8} - 396 T^{6} + \cdots + 688747536$$
$13$ $$(T^{4} + 4 T^{3} + 102 T^{2} - 344 T + 7396)^{2}$$
$17$ $$(T^{2} + 288)^{4}$$
$19$ $$(T^{2} - 16 T - 296)^{4}$$
$23$ $$T^{8} - 504 T^{6} + \cdots + 136048896$$
$29$ $$T^{8} - 2664 T^{6} + \cdots + 892616806656$$
$31$ $$(T^{4} + 4 T^{3} + 372 T^{2} + \cdots + 126736)^{2}$$
$37$ $$(T^{2} + 68 T + 346)^{4}$$
$41$ $$T^{8} - 6084 T^{6} + \cdots + 54575510850576$$
$43$ $$(T^{4} + 40 T^{3} + 1560 T^{2} + \cdots + 1600)^{2}$$
$47$ $$T^{8} - 8496 T^{6} + \cdots + 62171080298496$$
$53$ $$(T^{4} + 9360 T^{2} + 1166400)^{2}$$
$59$ $$T^{8} + \cdots + 164627478364176$$
$61$ $$(T^{4} - 20 T^{3} + 3540 T^{2} + \cdots + 9859600)^{2}$$
$67$ $$(T^{2} - 76 T + 5776)^{4}$$
$71$ $$(T^{4} + 6624 T^{2} + 3504384)^{2}$$
$73$ $$(T^{2} - 76 T - 1796)^{4}$$
$79$ $$(T^{4} + 100 T^{3} + 10740 T^{2} + \cdots + 547600)^{2}$$
$83$ $$T^{8} - 5904 T^{6} + \cdots + 62171080298496$$
$89$ $$(T^{4} + 17316 T^{2} + 52099524)^{2}$$
$97$ $$(T^{4} - 212 T^{3} + 34068 T^{2} + \cdots + 118287376)^{2}$$