Properties

Label 81.12.c.a
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 3)
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.

\(f(q)\) \(=\) \( q + (78 \zeta_{6} - 78) q^{2} - 4036 \zeta_{6} q^{4} + 5370 \zeta_{6} q^{5} + ( - 27760 \zeta_{6} + 27760) q^{7} + 155064 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (78 \zeta_{6} - 78) q^{2} - 4036 \zeta_{6} q^{4} + 5370 \zeta_{6} q^{5} + ( - 27760 \zeta_{6} + 27760) q^{7} + 155064 q^{8} - 418860 q^{10} + (637836 \zeta_{6} - 637836) q^{11} - 766214 \zeta_{6} q^{13} + 2165280 \zeta_{6} q^{14} + (3829264 \zeta_{6} - 3829264) q^{16} + 3084354 q^{17} - 19511404 q^{19} + ( - 21673320 \zeta_{6} + 21673320) q^{20} - 49751208 \zeta_{6} q^{22} - 15312360 \zeta_{6} q^{23} + ( - 19991225 \zeta_{6} + 19991225) q^{25} + 59764692 q^{26} - 112039360 q^{28} + (10751262 \zeta_{6} - 10751262) q^{29} + 50937400 \zeta_{6} q^{31} + 18888480 \zeta_{6} q^{32} + (240579612 \zeta_{6} - 240579612) q^{34} + 149071200 q^{35} + 664740830 q^{37} + ( - 1521889512 \zeta_{6} + 1521889512) q^{38} + 832693680 \zeta_{6} q^{40} - 898833450 \zeta_{6} q^{41} + ( - 957947188 \zeta_{6} + 957947188) q^{43} + 2574306096 q^{44} + 1194364080 q^{46} + ( - 1555741344 \zeta_{6} + 1555741344) q^{47} + 1206709143 \zeta_{6} q^{49} + 1559315550 \zeta_{6} q^{50} + (3092439704 \zeta_{6} - 3092439704) q^{52} + 3792417030 q^{53} - 3425179320 q^{55} + ( - 4304576640 \zeta_{6} + 4304576640) q^{56} - 838598436 \zeta_{6} q^{58} - 555306924 \zeta_{6} q^{59} + (4950420998 \zeta_{6} - 4950420998) q^{61} - 3973117200 q^{62} - 9315634112 q^{64} + ( - 4114569180 \zeta_{6} + 4114569180) q^{65} - 5292399284 \zeta_{6} q^{67} - 12448452744 \zeta_{6} q^{68} + (11627553600 \zeta_{6} - 11627553600) q^{70} - 14831086248 q^{71} + 13971005210 q^{73} + (51849784740 \zeta_{6} - 51849784740) q^{74} + 78748026544 \zeta_{6} q^{76} + 17706327360 \zeta_{6} q^{77} + (3720542360 \zeta_{6} - 3720542360) q^{79} - 20563147680 q^{80} + 70109009100 q^{82} + (8768454036 \zeta_{6} - 8768454036) q^{83} + 16562980980 \zeta_{6} q^{85} + 74719880664 \zeta_{6} q^{86} + (98905401504 \zeta_{6} - 98905401504) q^{88} - 25472769174 q^{89} - 21270100640 q^{91} + (61800684960 \zeta_{6} - 61800684960) q^{92} + 121347824832 \zeta_{6} q^{94} - 104776239480 \zeta_{6} q^{95} + ( - 39092494846 \zeta_{6} + 39092494846) q^{97} - 94123313154 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 78 q^{2} - 4036 q^{4} + 5370 q^{5} + 27760 q^{7} + 310128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 78 q^{2} - 4036 q^{4} + 5370 q^{5} + 27760 q^{7} + 310128 q^{8} - 837720 q^{10} - 637836 q^{11} - 766214 q^{13} + 2165280 q^{14} - 3829264 q^{16} + 6168708 q^{17} - 39022808 q^{19} + 21673320 q^{20} - 49751208 q^{22} - 15312360 q^{23} + 19991225 q^{25} + 119529384 q^{26} - 224078720 q^{28} - 10751262 q^{29} + 50937400 q^{31} + 18888480 q^{32} - 240579612 q^{34} + 298142400 q^{35} + 1329481660 q^{37} + 1521889512 q^{38} + 832693680 q^{40} - 898833450 q^{41} + 957947188 q^{43} + 5148612192 q^{44} + 2388728160 q^{46} + 1555741344 q^{47} + 1206709143 q^{49} + 1559315550 q^{50} - 3092439704 q^{52} + 7584834060 q^{53} - 6850358640 q^{55} + 4304576640 q^{56} - 838598436 q^{58} - 555306924 q^{59} - 4950420998 q^{61} - 7946234400 q^{62} - 18631268224 q^{64} + 4114569180 q^{65} - 5292399284 q^{67} - 12448452744 q^{68} - 11627553600 q^{70} - 29662172496 q^{71} + 27942010420 q^{73} - 51849784740 q^{74} + 78748026544 q^{76} + 17706327360 q^{77} - 3720542360 q^{79} - 41126295360 q^{80} + 140218018200 q^{82} - 8768454036 q^{83} + 16562980980 q^{85} + 74719880664 q^{86} - 98905401504 q^{88} - 50945538348 q^{89} - 42540201280 q^{91} - 61800684960 q^{92} + 121347824832 q^{94} - 104776239480 q^{95} + 39092494846 q^{97} - 188246626308 q^{98}+O(q^{100}) \) 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
−39.0000 67.5500i 0 −2018.00 + 3495.28i 2685.00 4650.56i 0 13880.0 + 24040.9i 155064. 0 −418860.
55.1 −39.0000 + 67.5500i 0 −2018.00 3495.28i 2685.00 + 4650.56i 0 13880.0 24040.9i 155064. 0 −418860.
\(n\): 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.a 2
3.b odd 2 1 81.12.c.e 2
9.c even 3 1 3.12.a.a 1
9.c even 3 1 inner 81.12.c.a 2
9.d odd 6 1 9.12.a.a 1
9.d odd 6 1 81.12.c.e 2
36.f odd 6 1 48.12.a.f 1
36.h even 6 1 144.12.a.l 1
45.h odd 6 1 225.12.a.f 1
45.j even 6 1 75.12.a.a 1
45.k odd 12 2 75.12.b.a 2
45.l even 12 2 225.12.b.a 2
63.l odd 6 1 147.12.a.c 1
72.n even 6 1 192.12.a.q 1
72.p odd 6 1 192.12.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 9.c even 3 1
9.12.a.a 1 9.d odd 6 1
48.12.a.f 1 36.f odd 6 1
75.12.a.a 1 45.j even 6 1
75.12.b.a 2 45.k odd 12 2
81.12.c.a 2 1.a even 1 1 trivial
81.12.c.a 2 9.c even 3 1 inner
81.12.c.e 2 3.b odd 2 1
81.12.c.e 2 9.d odd 6 1
144.12.a.l 1 36.h even 6 1
147.12.a.c 1 63.l odd 6 1
192.12.a.g 1 72.p odd 6 1
192.12.a.q 1 72.n even 6 1
225.12.a.f 1 45.h odd 6 1
225.12.b.a 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 78T_{2} + 6084 \) acting on \(S_{12}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 78T + 6084 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5370 T + 28836900 \) Copy content Toggle raw display
$7$ \( T^{2} - 27760 T + 770617600 \) Copy content Toggle raw display
$11$ \( T^{2} + 637836 T + 406834762896 \) Copy content Toggle raw display
$13$ \( T^{2} + 766214 T + 587083893796 \) Copy content Toggle raw display
$17$ \( (T - 3084354)^{2} \) Copy content Toggle raw display
$19$ \( (T + 19511404)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 234468368769600 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 115589634592644 \) Copy content Toggle raw display
$31$ \( T^{2} - 50937400 T + 25\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T - 664740830)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 898833450 T + 80\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} - 957947188 T + 91\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} - 1555741344 T + 24\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T - 3792417030)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 555306924 T + 30\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{2} + 4950420998 T + 24\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + 5292399284 T + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T + 14831086248)^{2} \) Copy content Toggle raw display
$73$ \( (T - 13971005210)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 3720542360 T + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 8768454036 T + 76\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T + 25472769174)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 39092494846 T + 15\!\cdots\!16 \) Copy content Toggle raw display
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