Properties

 Label 81.12.c.b 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: $$x^{2} - x + 1$$ 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.

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

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.b 2
3.b odd 2 1 81.12.c.d 2
9.c even 3 1 9.12.a.b 1
9.c even 3 1 inner 81.12.c.b 2
9.d odd 6 1 1.12.a.a 1
9.d odd 6 1 81.12.c.d 2
36.f odd 6 1 144.12.a.d 1
36.h even 6 1 16.12.a.a 1
45.h odd 6 1 25.12.a.b 1
45.j even 6 1 225.12.a.b 1
45.k odd 12 2 225.12.b.d 2
45.l even 12 2 25.12.b.b 2
63.i even 6 1 49.12.c.c 2
63.j odd 6 1 49.12.c.b 2
63.n odd 6 1 49.12.c.b 2
63.o even 6 1 49.12.a.a 1
63.s even 6 1 49.12.c.c 2
72.j odd 6 1 64.12.a.b 1
72.l even 6 1 64.12.a.f 1
99.g even 6 1 121.12.a.b 1
117.n odd 6 1 169.12.a.a 1
144.u even 12 2 256.12.b.c 2
144.w odd 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.d odd 6 1
9.12.a.b 1 9.c even 3 1
16.12.a.a 1 36.h even 6 1
25.12.a.b 1 45.h odd 6 1
25.12.b.b 2 45.l even 12 2
49.12.a.a 1 63.o even 6 1
49.12.c.b 2 63.j odd 6 1
49.12.c.b 2 63.n odd 6 1
49.12.c.c 2 63.i even 6 1
49.12.c.c 2 63.s even 6 1
64.12.a.b 1 72.j odd 6 1
64.12.a.f 1 72.l even 6 1
81.12.c.b 2 1.a even 1 1 trivial
81.12.c.b 2 9.c even 3 1 inner
81.12.c.d 2 3.b odd 2 1
81.12.c.d 2 9.d odd 6 1
121.12.a.b 1 99.g even 6 1
144.12.a.d 1 36.f odd 6 1
169.12.a.a 1 117.n odd 6 1
225.12.a.b 1 45.j even 6 1
225.12.b.d 2 45.k odd 12 2
256.12.b.c 2 144.u even 12 2
256.12.b.e 2 144.w odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$576 + 24 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$23328900 - 4830 T + T^{2}$$
$7$ $$280361536 - 16744 T + T^{2}$$
$11$ $$285809990544 - 534612 T + T^{2}$$
$13$ $$333781196644 - 577738 T + T^{2}$$
$17$ $$( -6905934 + T )^{2}$$
$19$ $$( -10661420 + T )^{2}$$
$23$ $$347571590865984 - 18643272 T + T^{2}$$
$29$ $$16488262627956900 - 128406630 T + T^{2}$$
$31$ $$2792400404276224 - 52843168 T + T^{2}$$
$37$ $$( 182213314 + T )^{2}$$
$41$ $$94938206778275364 - 308120442 T + T^{2}$$
$43$ $$293289874501264 - 17125708 T + T^{2}$$
$47$ $$7221841938953462016 - 2687348496 T + T^{2}$$
$53$ $$( -1596055698 + T )^{2}$$
$59$ $$26927835455229987600 + 5189203740 T + T^{2}$$
$61$ $$48392595374861310244 + 6956478662 T + T^{2}$$
$67$ $$23\!\cdots\!56$$$$- 15481826884 T + T^{2}$$
$71$ $$( 9791485272 + T )^{2}$$
$73$ $$( -1463791322 + T )^{2}$$
$79$ $$14\!\cdots\!00$$$$+ 38116845680 T + T^{2}$$
$83$ $$86\!\cdots\!24$$$$+ 29335099668 T + T^{2}$$
$89$ $$( -24992917110 + T )^{2}$$
$97$ $$56\!\cdots\!16$$$$+ 75013568546 T + T^{2}$$