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 $2$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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
0.500000 0.866025i
0.500000 + 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:

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 \(S_{12}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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