# Properties

 Label 81.12.c.d Level $81$ Weight $12$ Character orbit 81.c Analytic conductor $62.236$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 81.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.2357976253$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) 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.

 $$f(q)$$ $$=$$ $$q + ( - 24 \zeta_{6} + 24) q^{2} + 1472 \zeta_{6} q^{4} - 4830 \zeta_{6} q^{5} + ( - 16744 \zeta_{6} + 16744) q^{7} + 84480 q^{8} +O(q^{10})$$ q + (-24*z + 24) * q^2 + 1472*z * q^4 - 4830*z * q^5 + (-16744*z + 16744) * q^7 + 84480 * q^8 $$q + ( - 24 \zeta_{6} + 24) q^{2} + 1472 \zeta_{6} q^{4} - 4830 \zeta_{6} q^{5} + ( - 16744 \zeta_{6} + 16744) q^{7} + 84480 q^{8} - 115920 q^{10} + (534612 \zeta_{6} - 534612) q^{11} + 577738 \zeta_{6} q^{13} - 401856 \zeta_{6} q^{14} + (987136 \zeta_{6} - 987136) q^{16} - 6905934 q^{17} + 10661420 q^{19} + ( - 7109760 \zeta_{6} + 7109760) q^{20} + 12830688 \zeta_{6} q^{22} - 18643272 \zeta_{6} q^{23} + ( - 25499225 \zeta_{6} + 25499225) q^{25} + 13865712 q^{26} + 24647168 q^{28} + (128406630 \zeta_{6} - 128406630) q^{29} + 52843168 \zeta_{6} q^{31} + 196706304 \zeta_{6} q^{32} + (165742416 \zeta_{6} - 165742416) q^{34} - 80873520 q^{35} - 182213314 q^{37} + ( - 255874080 \zeta_{6} + 255874080) q^{38} - 408038400 \zeta_{6} q^{40} - 308120442 \zeta_{6} q^{41} + ( - 17125708 \zeta_{6} + 17125708) q^{43} - 786948864 q^{44} - 447438528 q^{46} + (2687348496 \zeta_{6} - 2687348496) q^{47} + 1696965207 \zeta_{6} q^{49} - 611981400 \zeta_{6} q^{50} + (850430336 \zeta_{6} - 850430336) q^{52} - 1596055698 q^{53} + 2582175960 q^{55} + ( - 1414533120 \zeta_{6} + 1414533120) q^{56} + 3081759120 \zeta_{6} q^{58} + 5189203740 \zeta_{6} q^{59} + (6956478662 \zeta_{6} - 6956478662) q^{61} + 1268236032 q^{62} + 2699296768 q^{64} + ( - 2790474540 \zeta_{6} + 2790474540) q^{65} + 15481826884 \zeta_{6} q^{67} - 10165534848 \zeta_{6} q^{68} + (1940964480 \zeta_{6} - 1940964480) q^{70} + 9791485272 q^{71} + 1463791322 q^{73} + (4373119536 \zeta_{6} - 4373119536) q^{74} + 15693610240 \zeta_{6} q^{76} + 8951543328 \zeta_{6} q^{77} + (38116845680 \zeta_{6} - 38116845680) q^{79} + 4767866880 q^{80} - 7394890608 q^{82} + ( - 29335099668 \zeta_{6} + 29335099668) q^{83} + 33355661220 \zeta_{6} q^{85} - 411016992 \zeta_{6} q^{86} + (45164021760 \zeta_{6} - 45164021760) q^{88} - 24992917110 q^{89} + 9673645072 q^{91} + ( - 27442896384 \zeta_{6} + 27442896384) q^{92} + 64496363904 \zeta_{6} q^{94} - 51494658600 \zeta_{6} q^{95} + (75013568546 \zeta_{6} - 75013568546) q^{97} + 40727164968 q^{98} +O(q^{100})$$ q + (-24*z + 24) * q^2 + 1472*z * q^4 - 4830*z * q^5 + (-16744*z + 16744) * q^7 + 84480 * q^8 - 115920 * q^10 + (534612*z - 534612) * q^11 + 577738*z * q^13 - 401856*z * q^14 + (987136*z - 987136) * q^16 - 6905934 * q^17 + 10661420 * q^19 + (-7109760*z + 7109760) * q^20 + 12830688*z * q^22 - 18643272*z * q^23 + (-25499225*z + 25499225) * q^25 + 13865712 * q^26 + 24647168 * q^28 + (128406630*z - 128406630) * q^29 + 52843168*z * q^31 + 196706304*z * q^32 + (165742416*z - 165742416) * q^34 - 80873520 * q^35 - 182213314 * q^37 + (-255874080*z + 255874080) * q^38 - 408038400*z * q^40 - 308120442*z * q^41 + (-17125708*z + 17125708) * q^43 - 786948864 * q^44 - 447438528 * q^46 + (2687348496*z - 2687348496) * q^47 + 1696965207*z * q^49 - 611981400*z * q^50 + (850430336*z - 850430336) * q^52 - 1596055698 * q^53 + 2582175960 * q^55 + (-1414533120*z + 1414533120) * q^56 + 3081759120*z * q^58 + 5189203740*z * q^59 + (6956478662*z - 6956478662) * q^61 + 1268236032 * q^62 + 2699296768 * q^64 + (-2790474540*z + 2790474540) * q^65 + 15481826884*z * q^67 - 10165534848*z * q^68 + (1940964480*z - 1940964480) * q^70 + 9791485272 * q^71 + 1463791322 * q^73 + (4373119536*z - 4373119536) * q^74 + 15693610240*z * q^76 + 8951543328*z * q^77 + (38116845680*z - 38116845680) * q^79 + 4767866880 * q^80 - 7394890608 * q^82 + (-29335099668*z + 29335099668) * q^83 + 33355661220*z * q^85 - 411016992*z * q^86 + (45164021760*z - 45164021760) * q^88 - 24992917110 * q^89 + 9673645072 * q^91 + (-27442896384*z + 27442896384) * q^92 + 64496363904*z * q^94 - 51494658600*z * q^95 + (75013568546*z - 75013568546) * q^97 + 40727164968 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 24 q^{2} + 1472 q^{4} - 4830 q^{5} + 16744 q^{7} + 168960 q^{8}+O(q^{10})$$ 2 * q + 24 * q^2 + 1472 * q^4 - 4830 * q^5 + 16744 * q^7 + 168960 * q^8 $$2 q + 24 q^{2} + 1472 q^{4} - 4830 q^{5} + 16744 q^{7} + 168960 q^{8} - 231840 q^{10} - 534612 q^{11} + 577738 q^{13} - 401856 q^{14} - 987136 q^{16} - 13811868 q^{17} + 21322840 q^{19} + 7109760 q^{20} + 12830688 q^{22} - 18643272 q^{23} + 25499225 q^{25} + 27731424 q^{26} + 49294336 q^{28} - 128406630 q^{29} + 52843168 q^{31} + 196706304 q^{32} - 165742416 q^{34} - 161747040 q^{35} - 364426628 q^{37} + 255874080 q^{38} - 408038400 q^{40} - 308120442 q^{41} + 17125708 q^{43} - 1573897728 q^{44} - 894877056 q^{46} - 2687348496 q^{47} + 1696965207 q^{49} - 611981400 q^{50} - 850430336 q^{52} - 3192111396 q^{53} + 5164351920 q^{55} + 1414533120 q^{56} + 3081759120 q^{58} + 5189203740 q^{59} - 6956478662 q^{61} + 2536472064 q^{62} + 5398593536 q^{64} + 2790474540 q^{65} + 15481826884 q^{67} - 10165534848 q^{68} - 1940964480 q^{70} + 19582970544 q^{71} + 2927582644 q^{73} - 4373119536 q^{74} + 15693610240 q^{76} + 8951543328 q^{77} - 38116845680 q^{79} + 9535733760 q^{80} - 14789781216 q^{82} + 29335099668 q^{83} + 33355661220 q^{85} - 411016992 q^{86} - 45164021760 q^{88} - 49985834220 q^{89} + 19347290144 q^{91} + 27442896384 q^{92} + 64496363904 q^{94} - 51494658600 q^{95} - 75013568546 q^{97} + 81454329936 q^{98}+O(q^{100})$$ 2 * q + 24 * q^2 + 1472 * q^4 - 4830 * q^5 + 16744 * q^7 + 168960 * q^8 - 231840 * q^10 - 534612 * q^11 + 577738 * q^13 - 401856 * q^14 - 987136 * q^16 - 13811868 * q^17 + 21322840 * q^19 + 7109760 * q^20 + 12830688 * q^22 - 18643272 * q^23 + 25499225 * q^25 + 27731424 * q^26 + 49294336 * q^28 - 128406630 * q^29 + 52843168 * q^31 + 196706304 * q^32 - 165742416 * q^34 - 161747040 * q^35 - 364426628 * q^37 + 255874080 * q^38 - 408038400 * q^40 - 308120442 * q^41 + 17125708 * q^43 - 1573897728 * q^44 - 894877056 * q^46 - 2687348496 * q^47 + 1696965207 * q^49 - 611981400 * q^50 - 850430336 * q^52 - 3192111396 * q^53 + 5164351920 * q^55 + 1414533120 * q^56 + 3081759120 * q^58 + 5189203740 * q^59 - 6956478662 * q^61 + 2536472064 * q^62 + 5398593536 * q^64 + 2790474540 * q^65 + 15481826884 * q^67 - 10165534848 * q^68 - 1940964480 * q^70 + 19582970544 * q^71 + 2927582644 * q^73 - 4373119536 * q^74 + 15693610240 * q^76 + 8951543328 * q^77 - 38116845680 * q^79 + 9535733760 * q^80 - 14789781216 * q^82 + 29335099668 * q^83 + 33355661220 * q^85 - 411016992 * q^86 - 45164021760 * q^88 - 49985834220 * q^89 + 19347290144 * q^91 + 27442896384 * q^92 + 64496363904 * q^94 - 51494658600 * q^95 - 75013568546 * q^97 + 81454329936 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.5 − 0.866025i 0.5 + 0.866025i
12.0000 + 20.7846i 0 736.000 1274.79i −2415.00 + 4182.90i 0 8372.00 + 14500.7i 84480.0 0 −115920.
55.1 12.0000 20.7846i 0 736.000 + 1274.79i −2415.00 4182.90i 0 8372.00 14500.7i 84480.0 0 −115920.
 $$n$$: 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 81.12.c.d 2
3.b odd 2 1 81.12.c.b 2
9.c even 3 1 1.12.a.a 1
9.c even 3 1 inner 81.12.c.d 2
9.d odd 6 1 9.12.a.b 1
9.d odd 6 1 81.12.c.b 2
36.f odd 6 1 16.12.a.a 1
36.h even 6 1 144.12.a.d 1
45.h odd 6 1 225.12.a.b 1
45.j even 6 1 25.12.a.b 1
45.k odd 12 2 25.12.b.b 2
45.l even 12 2 225.12.b.d 2
63.g even 3 1 49.12.c.b 2
63.h even 3 1 49.12.c.b 2
63.k odd 6 1 49.12.c.c 2
63.l odd 6 1 49.12.a.a 1
63.t odd 6 1 49.12.c.c 2
72.n even 6 1 64.12.a.b 1
72.p odd 6 1 64.12.a.f 1
99.h odd 6 1 121.12.a.b 1
117.t even 6 1 169.12.a.a 1
144.v odd 12 2 256.12.b.c 2
144.x even 12 2 256.12.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 9.c even 3 1
9.12.a.b 1 9.d odd 6 1
16.12.a.a 1 36.f odd 6 1
25.12.a.b 1 45.j even 6 1
25.12.b.b 2 45.k odd 12 2
49.12.a.a 1 63.l odd 6 1
49.12.c.b 2 63.g even 3 1
49.12.c.b 2 63.h even 3 1
49.12.c.c 2 63.k odd 6 1
49.12.c.c 2 63.t odd 6 1
64.12.a.b 1 72.n even 6 1
64.12.a.f 1 72.p odd 6 1
81.12.c.b 2 3.b odd 2 1
81.12.c.b 2 9.d odd 6 1
81.12.c.d 2 1.a even 1 1 trivial
81.12.c.d 2 9.c even 3 1 inner
121.12.a.b 1 99.h odd 6 1
144.12.a.d 1 36.h even 6 1
169.12.a.a 1 117.t even 6 1
225.12.a.b 1 45.h odd 6 1
225.12.b.d 2 45.l even 12 2
256.12.b.c 2 144.v odd 12 2
256.12.b.e 2 144.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 24T_{2} + 576$$ acting on $$S_{12}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 24T + 576$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4830 T + 23328900$$
$7$ $$T^{2} - 16744 T + 280361536$$
$11$ $$T^{2} + 534612 T + 285809990544$$
$13$ $$T^{2} - 577738 T + 333781196644$$
$17$ $$(T + 6905934)^{2}$$
$19$ $$(T - 10661420)^{2}$$
$23$ $$T^{2} + \cdots + 347571590865984$$
$29$ $$T^{2} + 128406630 T + 16\!\cdots\!00$$
$31$ $$T^{2} - 52843168 T + 27\!\cdots\!24$$
$37$ $$(T + 182213314)^{2}$$
$41$ $$T^{2} + 308120442 T + 94\!\cdots\!64$$
$43$ $$T^{2} + \cdots + 293289874501264$$
$47$ $$T^{2} + 2687348496 T + 72\!\cdots\!16$$
$53$ $$(T + 1596055698)^{2}$$
$59$ $$T^{2} - 5189203740 T + 26\!\cdots\!00$$
$61$ $$T^{2} + 6956478662 T + 48\!\cdots\!44$$
$67$ $$T^{2} - 15481826884 T + 23\!\cdots\!56$$
$71$ $$(T - 9791485272)^{2}$$
$73$ $$(T - 1463791322)^{2}$$
$79$ $$T^{2} + 38116845680 T + 14\!\cdots\!00$$
$83$ $$T^{2} - 29335099668 T + 86\!\cdots\!24$$
$89$ $$(T + 24992917110)^{2}$$
$97$ $$T^{2} + 75013568546 T + 56\!\cdots\!16$$