# Properties

 Label 810.4.e.q 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 270) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - 8 q^{8} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (-4*z + 4) * q^7 - 8 * q^8 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - 8 q^{8} - 10 q^{10} + ( - 42 \zeta_{6} + 42) q^{11} - 20 \zeta_{6} q^{13} - 8 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 93 q^{17} + 59 q^{19} + (20 \zeta_{6} - 20) q^{20} - 84 \zeta_{6} q^{22} + 9 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 40 q^{26} - 16 q^{28} + ( - 120 \zeta_{6} + 120) q^{29} - 47 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + (186 \zeta_{6} - 186) q^{34} - 20 q^{35} - 262 q^{37} + ( - 118 \zeta_{6} + 118) q^{38} + 40 \zeta_{6} q^{40} + 126 \zeta_{6} q^{41} + ( - 178 \zeta_{6} + 178) q^{43} - 168 q^{44} + 18 q^{46} + ( - 144 \zeta_{6} + 144) q^{47} + 327 \zeta_{6} q^{49} + 50 \zeta_{6} q^{50} + (80 \zeta_{6} - 80) q^{52} - 741 q^{53} - 210 q^{55} + (32 \zeta_{6} - 32) q^{56} - 240 \zeta_{6} q^{58} - 444 \zeta_{6} q^{59} + (221 \zeta_{6} - 221) q^{61} - 94 q^{62} + 64 q^{64} + (100 \zeta_{6} - 100) q^{65} + 538 \zeta_{6} q^{67} + 372 \zeta_{6} q^{68} + (40 \zeta_{6} - 40) q^{70} - 690 q^{71} - 1126 q^{73} + (524 \zeta_{6} - 524) q^{74} - 236 \zeta_{6} q^{76} - 168 \zeta_{6} q^{77} + (665 \zeta_{6} - 665) q^{79} + 80 q^{80} + 252 q^{82} + ( - 75 \zeta_{6} + 75) q^{83} + 465 \zeta_{6} q^{85} - 356 \zeta_{6} q^{86} + (336 \zeta_{6} - 336) q^{88} + 1086 q^{89} - 80 q^{91} + ( - 36 \zeta_{6} + 36) q^{92} - 288 \zeta_{6} q^{94} - 295 \zeta_{6} q^{95} + (1544 \zeta_{6} - 1544) q^{97} + 654 q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (-4*z + 4) * q^7 - 8 * q^8 - 10 * q^10 + (-42*z + 42) * q^11 - 20*z * q^13 - 8*z * q^14 + (16*z - 16) * q^16 - 93 * q^17 + 59 * q^19 + (20*z - 20) * q^20 - 84*z * q^22 + 9*z * q^23 + (25*z - 25) * q^25 - 40 * q^26 - 16 * q^28 + (-120*z + 120) * q^29 - 47*z * q^31 + 32*z * q^32 + (186*z - 186) * q^34 - 20 * q^35 - 262 * q^37 + (-118*z + 118) * q^38 + 40*z * q^40 + 126*z * q^41 + (-178*z + 178) * q^43 - 168 * q^44 + 18 * q^46 + (-144*z + 144) * q^47 + 327*z * q^49 + 50*z * q^50 + (80*z - 80) * q^52 - 741 * q^53 - 210 * q^55 + (32*z - 32) * q^56 - 240*z * q^58 - 444*z * q^59 + (221*z - 221) * q^61 - 94 * q^62 + 64 * q^64 + (100*z - 100) * q^65 + 538*z * q^67 + 372*z * q^68 + (40*z - 40) * q^70 - 690 * q^71 - 1126 * q^73 + (524*z - 524) * q^74 - 236*z * q^76 - 168*z * q^77 + (665*z - 665) * q^79 + 80 * q^80 + 252 * q^82 + (-75*z + 75) * q^83 + 465*z * q^85 - 356*z * q^86 + (336*z - 336) * q^88 + 1086 * q^89 - 80 * q^91 + (-36*z + 36) * q^92 - 288*z * q^94 - 295*z * q^95 + (1544*z - 1544) * q^97 + 654 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} - 16 q^{8} - 20 q^{10} + 42 q^{11} - 20 q^{13} - 8 q^{14} - 16 q^{16} - 186 q^{17} + 118 q^{19} - 20 q^{20} - 84 q^{22} + 9 q^{23} - 25 q^{25} - 80 q^{26} - 32 q^{28} + 120 q^{29} - 47 q^{31} + 32 q^{32} - 186 q^{34} - 40 q^{35} - 524 q^{37} + 118 q^{38} + 40 q^{40} + 126 q^{41} + 178 q^{43} - 336 q^{44} + 36 q^{46} + 144 q^{47} + 327 q^{49} + 50 q^{50} - 80 q^{52} - 1482 q^{53} - 420 q^{55} - 32 q^{56} - 240 q^{58} - 444 q^{59} - 221 q^{61} - 188 q^{62} + 128 q^{64} - 100 q^{65} + 538 q^{67} + 372 q^{68} - 40 q^{70} - 1380 q^{71} - 2252 q^{73} - 524 q^{74} - 236 q^{76} - 168 q^{77} - 665 q^{79} + 160 q^{80} + 504 q^{82} + 75 q^{83} + 465 q^{85} - 356 q^{86} - 336 q^{88} + 2172 q^{89} - 160 q^{91} + 36 q^{92} - 288 q^{94} - 295 q^{95} - 1544 q^{97} + 1308 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 - 16 * q^8 - 20 * q^10 + 42 * q^11 - 20 * q^13 - 8 * q^14 - 16 * q^16 - 186 * q^17 + 118 * q^19 - 20 * q^20 - 84 * q^22 + 9 * q^23 - 25 * q^25 - 80 * q^26 - 32 * q^28 + 120 * q^29 - 47 * q^31 + 32 * q^32 - 186 * q^34 - 40 * q^35 - 524 * q^37 + 118 * q^38 + 40 * q^40 + 126 * q^41 + 178 * q^43 - 336 * q^44 + 36 * q^46 + 144 * q^47 + 327 * q^49 + 50 * q^50 - 80 * q^52 - 1482 * q^53 - 420 * q^55 - 32 * q^56 - 240 * q^58 - 444 * q^59 - 221 * q^61 - 188 * q^62 + 128 * q^64 - 100 * q^65 + 538 * q^67 + 372 * q^68 - 40 * q^70 - 1380 * q^71 - 2252 * q^73 - 524 * q^74 - 236 * q^76 - 168 * q^77 - 665 * q^79 + 160 * q^80 + 504 * q^82 + 75 * q^83 + 465 * q^85 - 356 * q^86 - 336 * q^88 + 2172 * q^89 - 160 * q^91 + 36 * q^92 - 288 * q^94 - 295 * q^95 - 1544 * q^97 + 1308 * 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 2.00000 + 3.46410i −8.00000 0 −10.0000
541.1 1.00000 1.73205i 0 −2.00000 3.46410i −2.50000 4.33013i 0 2.00000 3.46410i −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.q 2
3.b odd 2 1 810.4.e.h 2
9.c even 3 1 270.4.a.e 1
9.c even 3 1 inner 810.4.e.q 2
9.d odd 6 1 270.4.a.i yes 1
9.d odd 6 1 810.4.e.h 2
36.f odd 6 1 2160.4.a.o 1
36.h even 6 1 2160.4.a.e 1
45.h odd 6 1 1350.4.a.g 1
45.j even 6 1 1350.4.a.u 1
45.k odd 12 2 1350.4.c.d 2
45.l even 12 2 1350.4.c.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 9.c even 3 1
270.4.a.i yes 1 9.d odd 6 1
810.4.e.h 2 3.b odd 2 1
810.4.e.h 2 9.d odd 6 1
810.4.e.q 2 1.a even 1 1 trivial
810.4.e.q 2 9.c even 3 1 inner
1350.4.a.g 1 45.h odd 6 1
1350.4.a.u 1 45.j even 6 1
1350.4.c.d 2 45.k odd 12 2
1350.4.c.q 2 45.l even 12 2
2160.4.a.e 1 36.h even 6 1
2160.4.a.o 1 36.f odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16 $$T_{11}^{2} - 42T_{11} + 1764$$ T11^2 - 42*T11 + 1764

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} - 42T + 1764$$
$13$ $$T^{2} + 20T + 400$$
$17$ $$(T + 93)^{2}$$
$19$ $$(T - 59)^{2}$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} - 120T + 14400$$
$31$ $$T^{2} + 47T + 2209$$
$37$ $$(T + 262)^{2}$$
$41$ $$T^{2} - 126T + 15876$$
$43$ $$T^{2} - 178T + 31684$$
$47$ $$T^{2} - 144T + 20736$$
$53$ $$(T + 741)^{2}$$
$59$ $$T^{2} + 444T + 197136$$
$61$ $$T^{2} + 221T + 48841$$
$67$ $$T^{2} - 538T + 289444$$
$71$ $$(T + 690)^{2}$$
$73$ $$(T + 1126)^{2}$$
$79$ $$T^{2} + 665T + 442225$$
$83$ $$T^{2} - 75T + 5625$$
$89$ $$(T - 1086)^{2}$$
$97$ $$T^{2} + 1544 T + 2383936$$