# Properties

 Label 810.4.e.a Level $810$ Weight $4$ Character orbit 810.e Analytic conductor $47.792$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$47.7915471046$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (14*z - 14) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} + 8 q^{8} + 10 q^{10} + ( - 6 \zeta_{6} + 6) q^{11} - 68 \zeta_{6} q^{13} - 28 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 78 q^{17} + 44 q^{19} + (20 \zeta_{6} - 20) q^{20} + 12 \zeta_{6} q^{22} + 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 136 q^{26} + 56 q^{28} + ( - 126 \zeta_{6} + 126) q^{29} + 244 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 156 \zeta_{6} + 156) q^{34} + 70 q^{35} - 304 q^{37} + (88 \zeta_{6} - 88) q^{38} - 40 \zeta_{6} q^{40} - 480 \zeta_{6} q^{41} + (104 \zeta_{6} - 104) q^{43} - 24 q^{44} - 240 q^{46} + ( - 600 \zeta_{6} + 600) q^{47} + 147 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (272 \zeta_{6} - 272) q^{52} + 258 q^{53} - 30 q^{55} + (112 \zeta_{6} - 112) q^{56} + 252 \zeta_{6} q^{58} + 534 \zeta_{6} q^{59} + (362 \zeta_{6} - 362) q^{61} - 488 q^{62} + 64 q^{64} + (340 \zeta_{6} - 340) q^{65} + 268 \zeta_{6} q^{67} + 312 \zeta_{6} q^{68} + (140 \zeta_{6} - 140) q^{70} + 972 q^{71} + 470 q^{73} + ( - 608 \zeta_{6} + 608) q^{74} - 176 \zeta_{6} q^{76} + 84 \zeta_{6} q^{77} + (1244 \zeta_{6} - 1244) q^{79} + 80 q^{80} + 960 q^{82} + ( - 396 \zeta_{6} + 396) q^{83} + 390 \zeta_{6} q^{85} - 208 \zeta_{6} q^{86} + ( - 48 \zeta_{6} + 48) q^{88} + 972 q^{89} + 952 q^{91} + ( - 480 \zeta_{6} + 480) q^{92} + 1200 \zeta_{6} q^{94} - 220 \zeta_{6} q^{95} + ( - 46 \zeta_{6} + 46) q^{97} - 294 q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (14*z - 14) * q^7 + 8 * q^8 + 10 * q^10 + (-6*z + 6) * q^11 - 68*z * q^13 - 28*z * q^14 + (16*z - 16) * q^16 - 78 * q^17 + 44 * q^19 + (20*z - 20) * q^20 + 12*z * q^22 + 120*z * q^23 + (25*z - 25) * q^25 + 136 * q^26 + 56 * q^28 + (-126*z + 126) * q^29 + 244*z * q^31 - 32*z * q^32 + (-156*z + 156) * q^34 + 70 * q^35 - 304 * q^37 + (88*z - 88) * q^38 - 40*z * q^40 - 480*z * q^41 + (104*z - 104) * q^43 - 24 * q^44 - 240 * q^46 + (-600*z + 600) * q^47 + 147*z * q^49 - 50*z * q^50 + (272*z - 272) * q^52 + 258 * q^53 - 30 * q^55 + (112*z - 112) * q^56 + 252*z * q^58 + 534*z * q^59 + (362*z - 362) * q^61 - 488 * q^62 + 64 * q^64 + (340*z - 340) * q^65 + 268*z * q^67 + 312*z * q^68 + (140*z - 140) * q^70 + 972 * q^71 + 470 * q^73 + (-608*z + 608) * q^74 - 176*z * q^76 + 84*z * q^77 + (1244*z - 1244) * q^79 + 80 * q^80 + 960 * q^82 + (-396*z + 396) * q^83 + 390*z * q^85 - 208*z * q^86 + (-48*z + 48) * q^88 + 972 * q^89 + 952 * q^91 + (-480*z + 480) * q^92 + 1200*z * q^94 - 220*z * q^95 + (-46*z + 46) * q^97 - 294 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8} + 20 q^{10} + 6 q^{11} - 68 q^{13} - 28 q^{14} - 16 q^{16} - 156 q^{17} + 88 q^{19} - 20 q^{20} + 12 q^{22} + 120 q^{23} - 25 q^{25} + 272 q^{26} + 112 q^{28} + 126 q^{29} + 244 q^{31} - 32 q^{32} + 156 q^{34} + 140 q^{35} - 608 q^{37} - 88 q^{38} - 40 q^{40} - 480 q^{41} - 104 q^{43} - 48 q^{44} - 480 q^{46} + 600 q^{47} + 147 q^{49} - 50 q^{50} - 272 q^{52} + 516 q^{53} - 60 q^{55} - 112 q^{56} + 252 q^{58} + 534 q^{59} - 362 q^{61} - 976 q^{62} + 128 q^{64} - 340 q^{65} + 268 q^{67} + 312 q^{68} - 140 q^{70} + 1944 q^{71} + 940 q^{73} + 608 q^{74} - 176 q^{76} + 84 q^{77} - 1244 q^{79} + 160 q^{80} + 1920 q^{82} + 396 q^{83} + 390 q^{85} - 208 q^{86} + 48 q^{88} + 1944 q^{89} + 1904 q^{91} + 480 q^{92} + 1200 q^{94} - 220 q^{95} + 46 q^{97} - 588 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8 + 20 * q^10 + 6 * q^11 - 68 * q^13 - 28 * q^14 - 16 * q^16 - 156 * q^17 + 88 * q^19 - 20 * q^20 + 12 * q^22 + 120 * q^23 - 25 * q^25 + 272 * q^26 + 112 * q^28 + 126 * q^29 + 244 * q^31 - 32 * q^32 + 156 * q^34 + 140 * q^35 - 608 * q^37 - 88 * q^38 - 40 * q^40 - 480 * q^41 - 104 * q^43 - 48 * q^44 - 480 * q^46 + 600 * q^47 + 147 * q^49 - 50 * q^50 - 272 * q^52 + 516 * q^53 - 60 * q^55 - 112 * q^56 + 252 * q^58 + 534 * q^59 - 362 * q^61 - 976 * q^62 + 128 * q^64 - 340 * q^65 + 268 * q^67 + 312 * q^68 - 140 * q^70 + 1944 * q^71 + 940 * q^73 + 608 * q^74 - 176 * q^76 + 84 * q^77 - 1244 * q^79 + 160 * q^80 + 1920 * q^82 + 396 * q^83 + 390 * q^85 - 208 * q^86 + 48 * q^88 + 1944 * q^89 + 1904 * q^91 + 480 * q^92 + 1200 * q^94 - 220 * q^95 + 46 * q^97 - 588 * q^98

## Character values

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

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 −7.00000 12.1244i 8.00000 0 10.0000
541.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −2.50000 4.33013i 0 −7.00000 + 12.1244i 8.00000 0 10.0000
 $$n$$: 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 810.4.e.a 2
3.b odd 2 1 810.4.e.u 2
9.c even 3 1 90.4.a.e yes 1
9.c even 3 1 inner 810.4.e.a 2
9.d odd 6 1 90.4.a.b 1
9.d odd 6 1 810.4.e.u 2
36.f odd 6 1 720.4.a.t 1
36.h even 6 1 720.4.a.e 1
45.h odd 6 1 450.4.a.m 1
45.j even 6 1 450.4.a.c 1
45.k odd 12 2 450.4.c.f 2
45.l even 12 2 450.4.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 9.d odd 6 1
90.4.a.e yes 1 9.c even 3 1
450.4.a.c 1 45.j even 6 1
450.4.a.m 1 45.h odd 6 1
450.4.c.f 2 45.k odd 12 2
450.4.c.g 2 45.l even 12 2
720.4.a.e 1 36.h even 6 1
720.4.a.t 1 36.f odd 6 1
810.4.e.a 2 1.a even 1 1 trivial
810.4.e.a 2 9.c even 3 1 inner
810.4.e.u 2 3.b odd 2 1
810.4.e.u 2 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{2} + 14T_{7} + 196$$ T7^2 + 14*T7 + 196 $$T_{11}^{2} - 6T_{11} + 36$$ T11^2 - 6*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} + 14T + 196$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} + 68T + 4624$$
$17$ $$(T + 78)^{2}$$
$19$ $$(T - 44)^{2}$$
$23$ $$T^{2} - 120T + 14400$$
$29$ $$T^{2} - 126T + 15876$$
$31$ $$T^{2} - 244T + 59536$$
$37$ $$(T + 304)^{2}$$
$41$ $$T^{2} + 480T + 230400$$
$43$ $$T^{2} + 104T + 10816$$
$47$ $$T^{2} - 600T + 360000$$
$53$ $$(T - 258)^{2}$$
$59$ $$T^{2} - 534T + 285156$$
$61$ $$T^{2} + 362T + 131044$$
$67$ $$T^{2} - 268T + 71824$$
$71$ $$(T - 972)^{2}$$
$73$ $$(T - 470)^{2}$$
$79$ $$T^{2} + 1244 T + 1547536$$
$83$ $$T^{2} - 396T + 156816$$
$89$ $$(T - 972)^{2}$$
$97$ $$T^{2} - 46T + 2116$$