# 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 $2$

# 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 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.

 $$f(q)$$ $$=$$ $$q + (78 \zeta_{6} - 78) q^{2} - 4036 \zeta_{6} q^{4} + 5370 \zeta_{6} q^{5} + ( - 27760 \zeta_{6} + 27760) q^{7} + 155064 q^{8} +O(q^{10})$$ q + (78*z - 78) * q^2 - 4036*z * q^4 + 5370*z * q^5 + (-27760*z + 27760) * q^7 + 155064 * q^8 $$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})$$ q + (78*z - 78) * q^2 - 4036*z * q^4 + 5370*z * q^5 + (-27760*z + 27760) * q^7 + 155064 * q^8 - 418860 * q^10 + (637836*z - 637836) * q^11 - 766214*z * q^13 + 2165280*z * q^14 + (3829264*z - 3829264) * q^16 + 3084354 * q^17 - 19511404 * q^19 + (-21673320*z + 21673320) * q^20 - 49751208*z * q^22 - 15312360*z * q^23 + (-19991225*z + 19991225) * q^25 + 59764692 * q^26 - 112039360 * q^28 + (10751262*z - 10751262) * q^29 + 50937400*z * q^31 + 18888480*z * q^32 + (240579612*z - 240579612) * q^34 + 149071200 * q^35 + 664740830 * q^37 + (-1521889512*z + 1521889512) * q^38 + 832693680*z * q^40 - 898833450*z * q^41 + (-957947188*z + 957947188) * q^43 + 2574306096 * q^44 + 1194364080 * q^46 + (-1555741344*z + 1555741344) * q^47 + 1206709143*z * q^49 + 1559315550*z * q^50 + (3092439704*z - 3092439704) * q^52 + 3792417030 * q^53 - 3425179320 * q^55 + (-4304576640*z + 4304576640) * q^56 - 838598436*z * q^58 - 555306924*z * q^59 + (4950420998*z - 4950420998) * q^61 - 3973117200 * q^62 - 9315634112 * q^64 + (-4114569180*z + 4114569180) * q^65 - 5292399284*z * q^67 - 12448452744*z * q^68 + (11627553600*z - 11627553600) * q^70 - 14831086248 * q^71 + 13971005210 * q^73 + (51849784740*z - 51849784740) * q^74 + 78748026544*z * q^76 + 17706327360*z * q^77 + (3720542360*z - 3720542360) * q^79 - 20563147680 * q^80 + 70109009100 * q^82 + (8768454036*z - 8768454036) * q^83 + 16562980980*z * q^85 + 74719880664*z * q^86 + (98905401504*z - 98905401504) * q^88 - 25472769174 * q^89 - 21270100640 * q^91 + (61800684960*z - 61800684960) * q^92 + 121347824832*z * q^94 - 104776239480*z * q^95 + (-39092494846*z + 39092494846) * q^97 - 94123313154 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 78 q^{2} - 4036 q^{4} + 5370 q^{5} + 27760 q^{7} + 310128 q^{8}+O(q^{10})$$ 2 * q - 78 * q^2 - 4036 * q^4 + 5370 * q^5 + 27760 * q^7 + 310128 * q^8 $$2 q - 78 q^{2} - 4036 q^{4} + 5370 q^{5} + 27760 q^{7} + 310128 q^{8} - 837720 q^{10} - 637836 q^{11} - 766214 q^{13} + 2165280 q^{14} - 3829264 q^{16} + 6168708 q^{17} - 39022808 q^{19} + 21673320 q^{20} - 49751208 q^{22} - 15312360 q^{23} + 19991225 q^{25} + 119529384 q^{26} - 224078720 q^{28} - 10751262 q^{29} + 50937400 q^{31} + 18888480 q^{32} - 240579612 q^{34} + 298142400 q^{35} + 1329481660 q^{37} + 1521889512 q^{38} + 832693680 q^{40} - 898833450 q^{41} + 957947188 q^{43} + 5148612192 q^{44} + 2388728160 q^{46} + 1555741344 q^{47} + 1206709143 q^{49} + 1559315550 q^{50} - 3092439704 q^{52} + 7584834060 q^{53} - 6850358640 q^{55} + 4304576640 q^{56} - 838598436 q^{58} - 555306924 q^{59} - 4950420998 q^{61} - 7946234400 q^{62} - 18631268224 q^{64} + 4114569180 q^{65} - 5292399284 q^{67} - 12448452744 q^{68} - 11627553600 q^{70} - 29662172496 q^{71} + 27942010420 q^{73} - 51849784740 q^{74} + 78748026544 q^{76} + 17706327360 q^{77} - 3720542360 q^{79} - 41126295360 q^{80} + 140218018200 q^{82} - 8768454036 q^{83} + 16562980980 q^{85} + 74719880664 q^{86} - 98905401504 q^{88} - 50945538348 q^{89} - 42540201280 q^{91} - 61800684960 q^{92} + 121347824832 q^{94} - 104776239480 q^{95} + 39092494846 q^{97} - 188246626308 q^{98}+O(q^{100})$$ 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

## 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
−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: 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.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 $$S_{12}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

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