# Properties

 Label 810.3.j.e Level $810$ Weight $3$ Character orbit 810.j Analytic conductor $22.071$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 810.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.0709014132$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{5} + ( - 5 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8}+O(q^{10})$$ q + (-z^7 - z) * q^2 + (2*z^4 - 2) * q^4 + (5*z^7 - 5*z^3) * q^5 + (-5*z^6 + 5*z^2) * q^7 + (-2*z^5 + 2*z^3 + 2*z) * q^8 $$q + ( - \zeta_{24}^{7} - \zeta_{24}) q^{2} + (2 \zeta_{24}^{4} - 2) q^{4} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{5} + ( - 5 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{7} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{8} + (5 \zeta_{24}^{6} + 5) q^{10} + (\zeta_{24}^{7} - \zeta_{24}) q^{11} - 9 \zeta_{24}^{2} q^{13} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3}) q^{14} - 4 \zeta_{24}^{4} q^{16} + (8 \zeta_{24}^{5} - 8 \zeta_{24}^{3} - 8 \zeta_{24}) q^{17} + 21 q^{19} - 10 \zeta_{24}^{7} q^{20} + 2 \zeta_{24}^{2} q^{22} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{23} + ( - 25 \zeta_{24}^{6} + 25 \zeta_{24}^{2}) q^{25} + (9 \zeta_{24}^{5} + 9 \zeta_{24}^{3} - 9 \zeta_{24}) q^{26} + 10 \zeta_{24}^{6} q^{28} + (27 \zeta_{24}^{7} - 27 \zeta_{24}) q^{29} + (40 \zeta_{24}^{4} - 40) q^{31} + (4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3}) q^{32} + 16 \zeta_{24}^{4} q^{34} + (25 \zeta_{24}^{5} - 25 \zeta_{24}) q^{35} - 25 \zeta_{24}^{6} q^{37} + ( - 21 \zeta_{24}^{7} - 21 \zeta_{24}) q^{38} + (10 \zeta_{24}^{4} - 10 \zeta_{24}^{2} - 10) q^{40} + ( - 37 \zeta_{24}^{7} + 37 \zeta_{24}^{5} + 37 \zeta_{24}^{3}) q^{41} + ( - 64 \zeta_{24}^{6} + 64 \zeta_{24}^{2}) q^{43} + ( - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{44} - 2 q^{46} + (16 \zeta_{24}^{7} + 16 \zeta_{24}) q^{47} + (24 \zeta_{24}^{4} - 24) q^{49} + (25 \zeta_{24}^{7} - 25 \zeta_{24}^{5} - 25 \zeta_{24}^{3}) q^{50} + ( - 18 \zeta_{24}^{6} + 18 \zeta_{24}^{2}) q^{52} + (51 \zeta_{24}^{5} - 51 \zeta_{24}^{3} - 51 \zeta_{24}) q^{53} + ( - 5 \zeta_{24}^{6} + 5) q^{55} + ( - 10 \zeta_{24}^{7} + 10 \zeta_{24}) q^{56} + 54 \zeta_{24}^{2} q^{58} + ( - 64 \zeta_{24}^{7} + 64 \zeta_{24}^{5} + 64 \zeta_{24}^{3}) q^{59} + 97 \zeta_{24}^{4} q^{61} + ( - 40 \zeta_{24}^{5} + 40 \zeta_{24}^{3} + 40 \zeta_{24}) q^{62} + 8 q^{64} + 45 \zeta_{24} q^{65} + 131 \zeta_{24}^{2} q^{67} + ( - 16 \zeta_{24}^{7} - 16 \zeta_{24}^{5} + 16 \zeta_{24}^{3}) q^{68} + ( - 25 \zeta_{24}^{6} + 25 \zeta_{24}^{4} + 25 \zeta_{24}^{2}) q^{70} + (63 \zeta_{24}^{5} + 63 \zeta_{24}^{3} - 63 \zeta_{24}) q^{71} - 17 \zeta_{24}^{6} q^{73} + (25 \zeta_{24}^{7} - 25 \zeta_{24}) q^{74} + (42 \zeta_{24}^{4} - 42) q^{76} + (5 \zeta_{24}^{7} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3}) q^{77} + 117 \zeta_{24}^{4} q^{79} + 20 \zeta_{24}^{3} q^{80} - 74 \zeta_{24}^{6} q^{82} + ( - 41 \zeta_{24}^{7} - 41 \zeta_{24}) q^{83} + ( - 40 \zeta_{24}^{4} + 40 \zeta_{24}^{2} + 40) q^{85} + (64 \zeta_{24}^{7} - 64 \zeta_{24}^{5} - 64 \zeta_{24}^{3}) q^{86} + (4 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{88} + ( - 104 \zeta_{24}^{5} - 104 \zeta_{24}^{3} + 104 \zeta_{24}) q^{89} - 45 q^{91} + (2 \zeta_{24}^{7} + 2 \zeta_{24}) q^{92} + ( - 32 \zeta_{24}^{4} + 32) q^{94} + (105 \zeta_{24}^{7} - 105 \zeta_{24}^{3}) q^{95} + (41 \zeta_{24}^{6} - 41 \zeta_{24}^{2}) q^{97} + ( - 24 \zeta_{24}^{5} + 24 \zeta_{24}^{3} + 24 \zeta_{24}) q^{98} +O(q^{100})$$ q + (-z^7 - z) * q^2 + (2*z^4 - 2) * q^4 + (5*z^7 - 5*z^3) * q^5 + (-5*z^6 + 5*z^2) * q^7 + (-2*z^5 + 2*z^3 + 2*z) * q^8 + (5*z^6 + 5) * q^10 + (z^7 - z) * q^11 - 9*z^2 * q^13 + (5*z^7 - 5*z^5 - 5*z^3) * q^14 - 4*z^4 * q^16 + (8*z^5 - 8*z^3 - 8*z) * q^17 + 21 * q^19 - 10*z^7 * q^20 + 2*z^2 * q^22 + (-z^7 - z^5 + z^3) * q^23 + (-25*z^6 + 25*z^2) * q^25 + (9*z^5 + 9*z^3 - 9*z) * q^26 + 10*z^6 * q^28 + (27*z^7 - 27*z) * q^29 + (40*z^4 - 40) * q^31 + (4*z^7 + 4*z^5 - 4*z^3) * q^32 + 16*z^4 * q^34 + (25*z^5 - 25*z) * q^35 - 25*z^6 * q^37 + (-21*z^7 - 21*z) * q^38 + (10*z^4 - 10*z^2 - 10) * q^40 + (-37*z^7 + 37*z^5 + 37*z^3) * q^41 + (-64*z^6 + 64*z^2) * q^43 + (-2*z^5 - 2*z^3 + 2*z) * q^44 - 2 * q^46 + (16*z^7 + 16*z) * q^47 + (24*z^4 - 24) * q^49 + (25*z^7 - 25*z^5 - 25*z^3) * q^50 + (-18*z^6 + 18*z^2) * q^52 + (51*z^5 - 51*z^3 - 51*z) * q^53 + (-5*z^6 + 5) * q^55 + (-10*z^7 + 10*z) * q^56 + 54*z^2 * q^58 + (-64*z^7 + 64*z^5 + 64*z^3) * q^59 + 97*z^4 * q^61 + (-40*z^5 + 40*z^3 + 40*z) * q^62 + 8 * q^64 + 45*z * q^65 + 131*z^2 * q^67 + (-16*z^7 - 16*z^5 + 16*z^3) * q^68 + (-25*z^6 + 25*z^4 + 25*z^2) * q^70 + (63*z^5 + 63*z^3 - 63*z) * q^71 - 17*z^6 * q^73 + (25*z^7 - 25*z) * q^74 + (42*z^4 - 42) * q^76 + (5*z^7 + 5*z^5 - 5*z^3) * q^77 + 117*z^4 * q^79 + 20*z^3 * q^80 - 74*z^6 * q^82 + (-41*z^7 - 41*z) * q^83 + (-40*z^4 + 40*z^2 + 40) * q^85 + (64*z^7 - 64*z^5 - 64*z^3) * q^86 + (4*z^6 - 4*z^2) * q^88 + (-104*z^5 - 104*z^3 + 104*z) * q^89 - 45 * q^91 + (2*z^7 + 2*z) * q^92 + (-32*z^4 + 32) * q^94 + (105*z^7 - 105*z^3) * q^95 + (41*z^6 - 41*z^2) * q^97 + (-24*z^5 + 24*z^3 + 24*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} + 40 q^{10} - 16 q^{16} + 168 q^{19} - 160 q^{31} + 64 q^{34} - 40 q^{40} - 16 q^{46} - 96 q^{49} + 40 q^{55} + 388 q^{61} + 64 q^{64} + 100 q^{70} - 168 q^{76} + 468 q^{79} + 160 q^{85} - 360 q^{91} + 128 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 + 40 * q^10 - 16 * q^16 + 168 * q^19 - 160 * q^31 + 64 * q^34 - 40 * q^40 - 16 * q^46 - 96 * q^49 + 40 * q^55 + 388 * q^61 + 64 * q^64 + 100 * q^70 - 168 * q^76 + 468 * q^79 + 160 * q^85 - 360 * q^91 + 128 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$$.

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −4.82963 1.29410i 0 4.33013 + 2.50000i 2.82843 0 5.00000 5.00000i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.29410 4.82963i 0 −4.33013 2.50000i 2.82843 0 5.00000 + 5.00000i
269.3 0.707107 1.22474i 0 −1.00000 1.73205i −1.29410 + 4.82963i 0 −4.33013 2.50000i −2.82843 0 5.00000 + 5.00000i
269.4 0.707107 1.22474i 0 −1.00000 1.73205i 4.82963 + 1.29410i 0 4.33013 + 2.50000i −2.82843 0 5.00000 5.00000i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.82963 + 1.29410i 0 4.33013 2.50000i 2.82843 0 5.00000 + 5.00000i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.29410 + 4.82963i 0 −4.33013 + 2.50000i 2.82843 0 5.00000 5.00000i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.29410 4.82963i 0 −4.33013 + 2.50000i −2.82843 0 5.00000 5.00000i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 4.82963 1.29410i 0 4.33013 2.50000i −2.82843 0 5.00000 + 5.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 539.4 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
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.j.e 8
3.b odd 2 1 inner 810.3.j.e 8
5.b even 2 1 inner 810.3.j.e 8
9.c even 3 1 270.3.b.c 4
9.c even 3 1 inner 810.3.j.e 8
9.d odd 6 1 270.3.b.c 4
9.d odd 6 1 inner 810.3.j.e 8
15.d odd 2 1 inner 810.3.j.e 8
36.f odd 6 1 2160.3.c.i 4
36.h even 6 1 2160.3.c.i 4
45.h odd 6 1 270.3.b.c 4
45.h odd 6 1 inner 810.3.j.e 8
45.j even 6 1 270.3.b.c 4
45.j even 6 1 inner 810.3.j.e 8
45.k odd 12 1 1350.3.d.f 2
45.k odd 12 1 1350.3.d.g 2
45.l even 12 1 1350.3.d.f 2
45.l even 12 1 1350.3.d.g 2
180.n even 6 1 2160.3.c.i 4
180.p odd 6 1 2160.3.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 9.c even 3 1
270.3.b.c 4 9.d odd 6 1
270.3.b.c 4 45.h odd 6 1
270.3.b.c 4 45.j even 6 1
810.3.j.e 8 1.a even 1 1 trivial
810.3.j.e 8 3.b odd 2 1 inner
810.3.j.e 8 5.b even 2 1 inner
810.3.j.e 8 9.c even 3 1 inner
810.3.j.e 8 9.d odd 6 1 inner
810.3.j.e 8 15.d odd 2 1 inner
810.3.j.e 8 45.h odd 6 1 inner
810.3.j.e 8 45.j even 6 1 inner
1350.3.d.f 2 45.k odd 12 1
1350.3.d.f 2 45.l even 12 1
1350.3.d.g 2 45.k odd 12 1
1350.3.d.g 2 45.l even 12 1
2160.3.c.i 4 36.f odd 6 1
2160.3.c.i 4 36.h even 6 1
2160.3.c.i 4 180.n even 6 1
2160.3.c.i 4 180.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{4} - 25T_{7}^{2} + 625$$ T7^4 - 25*T7^2 + 625 $$T_{17}^{2} - 128$$ T17^2 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 625 T^{4} + 390625$$
$7$ $$(T^{4} - 25 T^{2} + 625)^{2}$$
$11$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$13$ $$(T^{4} - 81 T^{2} + 6561)^{2}$$
$17$ $$(T^{2} - 128)^{4}$$
$19$ $$(T - 21)^{8}$$
$23$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$29$ $$(T^{4} - 1458 T^{2} + 2125764)^{2}$$
$31$ $$(T^{2} + 40 T + 1600)^{4}$$
$37$ $$(T^{2} + 625)^{4}$$
$41$ $$(T^{4} - 2738 T^{2} + 7496644)^{2}$$
$43$ $$(T^{4} - 4096 T^{2} + 16777216)^{2}$$
$47$ $$(T^{4} + 512 T^{2} + 262144)^{2}$$
$53$ $$(T^{2} - 5202)^{4}$$
$59$ $$(T^{4} - 8192 T^{2} + 67108864)^{2}$$
$61$ $$(T^{2} - 97 T + 9409)^{4}$$
$67$ $$(T^{4} - 17161 T^{2} + \cdots + 294499921)^{2}$$
$71$ $$(T^{2} + 7938)^{4}$$
$73$ $$(T^{2} + 289)^{4}$$
$79$ $$(T^{2} - 117 T + 13689)^{4}$$
$83$ $$(T^{4} + 3362 T^{2} + 11303044)^{2}$$
$89$ $$(T^{2} + 21632)^{4}$$
$97$ $$(T^{4} - 1681 T^{2} + 2825761)^{2}$$