# Properties

 Label 1620.3.o.e 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: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ x^8 - 3*x^6 + 8*x^4 - 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 540) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_1) q^{5} + (\beta_{6} + 5 \beta_{2} - 5) q^{7}+O(q^{10})$$ q + (b4 - b1) * q^5 + (b6 + 5*b2 - 5) * q^7 $$q + (\beta_{4} - \beta_1) q^{5} + (\beta_{6} + 5 \beta_{2} - 5) q^{7} + ( - \beta_{3} - 3 \beta_1) q^{11} + ( - \beta_{5} + 7 \beta_{2}) q^{13} + (5 \beta_{7} + 5 \beta_{3}) q^{17} + ( - 2 \beta_{6} + 2 \beta_{5} - 1) q^{19} + (7 \beta_{7} + 6 \beta_{4} - 6 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + (\beta_{3} - 3 \beta_1) q^{29} + ( - 2 \beta_{5} + 7 \beta_{2}) q^{31} + ( - 5 \beta_{7} - 5 \beta_{4} - 5 \beta_{3}) q^{35} + ( - 6 \beta_{6} + 6 \beta_{5} - 16) q^{37} + (3 \beta_{7} - 15 \beta_{4} + 15 \beta_1) q^{41} + (7 \beta_{6} + 29 \beta_{2} - 29) q^{43} + ( - 4 \beta_{3} - 6 \beta_1) q^{47} + ( - 10 \beta_{5} - 21 \beta_{2}) q^{49} + (\beta_{7} - 30 \beta_{4} + \beta_{3}) q^{53} + ( - \beta_{6} + \beta_{5} + 15) q^{55} + ( - 17 \beta_{7} - 9 \beta_{4} + 9 \beta_1) q^{59} + (12 \beta_{6} - \beta_{2} + 1) q^{61} + (5 \beta_{3} - 7 \beta_1) q^{65} - 14 \beta_{2} q^{67} + ( - 15 \beta_{7} - 21 \beta_{4} - 15 \beta_{3}) q^{71} + ( - 3 \beta_{6} + 3 \beta_{5} - 7) q^{73} + ( - 20 \beta_{7} - 24 \beta_{4} + 24 \beta_1) q^{77} + ( - 12 \beta_{6} - 67 \beta_{2} + 67) q^{79} + ( - 11 \beta_{3} - 48 \beta_1) q^{83} - 5 \beta_{5} q^{85} + (3 \beta_{7} - 51 \beta_{4} + 3 \beta_{3}) q^{89} + (2 \beta_{6} - 2 \beta_{5} + 10) q^{91} + (10 \beta_{7} - \beta_{4} + \beta_1) q^{95} + (2 \beta_{6} + 2 \beta_{2} - 2) q^{97}+O(q^{100})$$ q + (b4 - b1) * q^5 + (b6 + 5*b2 - 5) * q^7 + (-b3 - 3*b1) * q^11 + (-b5 + 7*b2) * q^13 + (5*b7 + 5*b3) * q^17 + (-2*b6 + 2*b5 - 1) * q^19 + (7*b7 + 6*b4 - 6*b1) * q^23 + (-5*b2 + 5) * q^25 + (b3 - 3*b1) * q^29 + (-2*b5 + 7*b2) * q^31 + (-5*b7 - 5*b4 - 5*b3) * q^35 + (-6*b6 + 6*b5 - 16) * q^37 + (3*b7 - 15*b4 + 15*b1) * q^41 + (7*b6 + 29*b2 - 29) * q^43 + (-4*b3 - 6*b1) * q^47 + (-10*b5 - 21*b2) * q^49 + (b7 - 30*b4 + b3) * q^53 + (-b6 + b5 + 15) * q^55 + (-17*b7 - 9*b4 + 9*b1) * q^59 + (12*b6 - b2 + 1) * q^61 + (5*b3 - 7*b1) * q^65 - 14*b2 * q^67 + (-15*b7 - 21*b4 - 15*b3) * q^71 + (-3*b6 + 3*b5 - 7) * q^73 + (-20*b7 - 24*b4 + 24*b1) * q^77 + (-12*b6 - 67*b2 + 67) * q^79 + (-11*b3 - 48*b1) * q^83 - 5*b5 * q^85 + (3*b7 - 51*b4 + 3*b3) * q^89 + (2*b6 - 2*b5 + 10) * q^91 + (10*b7 - b4 + b1) * q^95 + (2*b6 + 2*b2 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 20 q^{7}+O(q^{10})$$ 8 * q - 20 * q^7 $$8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100})$$ 8 * q - 20 * q^7 + 28 * q^13 - 8 * q^19 + 20 * q^25 + 28 * q^31 - 128 * q^37 - 116 * q^43 - 84 * q^49 + 120 * q^55 + 4 * q^61 - 56 * q^67 - 56 * q^73 + 268 * q^79 + 80 * q^91 - 8 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 29\nu ) / 8$$ (v^7 + 29*v) / 8 $$\beta_{2}$$ $$=$$ $$( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 1 ) / 8$$ (3*v^6 - 8*v^4 + 24*v^2 - 1) / 8 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} - 39\nu ) / 8$$ (-3*v^7 - 39*v) / 8 $$\beta_{4}$$ $$=$$ $$( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2$$ (3*v^7 - 8*v^5 + 22*v^3 - v) / 2 $$\beta_{5}$$ $$=$$ $$( -27\nu^{6} + 72\nu^{4} - 168\nu^{2} + 9 ) / 8$$ (-27*v^6 + 72*v^4 - 168*v^2 + 9) / 8 $$\beta_{6}$$ $$=$$ $$( -21\nu^{6} + 72\nu^{4} - 168\nu^{2} + 63 ) / 8$$ (-21*v^6 + 72*v^4 - 168*v^2 + 63) / 8 $$\beta_{7}$$ $$=$$ $$( -15\nu^{7} + 48\nu^{5} - 120\nu^{3} + 45\nu ) / 8$$ (-15*v^7 + 48*v^5 - 120*v^3 + 45*v) / 8
 $$\nu$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 6$$ (b3 + 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 9\beta_{2} ) / 6$$ (b5 + 9*b2) / 6 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + 3\beta_{4} + 2\beta_{3} ) / 3$$ (2*b7 + 3*b4 + 2*b3) / 3 $$\nu^{4}$$ $$=$$ $$( \beta_{6} + 7\beta_{2} - 7 ) / 2$$ (b6 + 7*b2 - 7) / 2 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} + 15\beta_{4} - 15\beta_1 ) / 6$$ (11*b7 + 15*b4 - 15*b1) / 6 $$\nu^{6}$$ $$=$$ $$( 4\beta_{6} - 4\beta_{5} - 27 ) / 3$$ (4*b6 - 4*b5 - 27) / 3 $$\nu^{7}$$ $$=$$ $$( -29\beta_{3} - 39\beta_1 ) / 6$$ (-29*b3 - 39*b1) / 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
 1.40126 − 0.809017i 0.535233 − 0.309017i −1.40126 + 0.809017i −0.535233 + 0.309017i 1.40126 + 0.809017i 0.535233 + 0.309017i −1.40126 − 0.809017i −0.535233 − 0.309017i
0 0 0 −1.93649 1.11803i 0 −5.85410 10.1396i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 0.854102 + 1.47935i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −5.85410 10.1396i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 0.854102 + 1.47935i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 0.854102 1.47935i 0 0 0
1241.3 0 0 0 1.93649 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.4 0 0 0 1.93649 1.11803i 0 0.854102 1.47935i 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.e 8
3.b odd 2 1 inner 1620.3.o.e 8
9.c even 3 1 540.3.g.d 4
9.c even 3 1 inner 1620.3.o.e 8
9.d odd 6 1 540.3.g.d 4
9.d odd 6 1 inner 1620.3.o.e 8
36.f odd 6 1 2160.3.l.e 4
36.h even 6 1 2160.3.l.e 4
45.h odd 6 1 2700.3.g.n 4
45.j even 6 1 2700.3.g.n 4
45.k odd 12 1 2700.3.b.g 4
45.k odd 12 1 2700.3.b.l 4
45.l even 12 1 2700.3.b.g 4
45.l even 12 1 2700.3.b.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.d 4 9.c even 3 1
540.3.g.d 4 9.d odd 6 1
1620.3.o.e 8 1.a even 1 1 trivial
1620.3.o.e 8 3.b odd 2 1 inner
1620.3.o.e 8 9.c even 3 1 inner
1620.3.o.e 8 9.d odd 6 1 inner
2160.3.l.e 4 36.f odd 6 1
2160.3.l.e 4 36.h even 6 1
2700.3.b.g 4 45.k odd 12 1
2700.3.b.g 4 45.l even 12 1
2700.3.b.l 4 45.k odd 12 1
2700.3.b.l 4 45.l even 12 1
2700.3.g.n 4 45.h odd 6 1
2700.3.g.n 4 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 10T_{7}^{3} + 120T_{7}^{2} - 200T_{7} + 400$$ acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 5 T^{2} + 25)^{2}$$
$7$ $$(T^{4} + 10 T^{3} + 120 T^{2} - 200 T + 400)^{2}$$
$11$ $$T^{8} - 108 T^{6} + 10368 T^{4} + \cdots + 1679616$$
$13$ $$(T^{4} - 14 T^{3} + 192 T^{2} - 56 T + 16)^{2}$$
$17$ $$(T^{2} + 225)^{4}$$
$19$ $$(T^{2} + 2 T - 179)^{4}$$
$23$ $$T^{8} - 1242 T^{6} + \cdots + 4640470641$$
$29$ $$T^{8} - 108 T^{6} + 10368 T^{4} + \cdots + 1679616$$
$31$ $$(T^{4} - 14 T^{3} + 327 T^{2} + \cdots + 17161)^{2}$$
$37$ $$(T^{2} + 32 T - 1364)^{4}$$
$41$ $$T^{8} - 2412 T^{6} + \cdots + 1187960484096$$
$43$ $$(T^{4} + 58 T^{3} + 4728 T^{2} + \cdots + 1860496)^{2}$$
$47$ $$T^{8} - 648 T^{6} + 418608 T^{4} + \cdots + 1679616$$
$53$ $$(T^{4} + 9018 T^{2} + 20169081)^{2}$$
$59$ $$T^{8} - 6012 T^{6} + \cdots + 23255696077056$$
$61$ $$(T^{4} - 2 T^{3} + 6483 T^{2} + \cdots + 41977441)^{2}$$
$67$ $$(T^{2} + 14 T + 196)^{4}$$
$71$ $$(T^{4} + 8460 T^{2} + 32400)^{2}$$
$73$ $$(T^{2} + 14 T - 356)^{4}$$
$79$ $$(T^{4} - 134 T^{3} + 19947 T^{2} + \cdots + 3964081)^{2}$$
$83$ $$T^{8} - 25218 T^{6} + \cdots + 11\!\cdots\!21$$
$89$ $$(T^{4} + 26172 T^{2} + \cdots + 167029776)^{2}$$
$97$ $$(T^{4} + 4 T^{3} + 192 T^{2} - 704 T + 30976)^{2}$$