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Properties

Label 81.8.c.j.55.4
Level $81$
Weight $8$
Character 81.55
Analytic conductor $25.303$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,8,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.28"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,15,0,-229] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.3031870642\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 229x^{6} + 13863x^{4} + 85327x^{2} + 40804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 55.4
Root \(-10.5078i\) of defining polynomial
Character \(\chi\) \(=\) 81.55
Dual form 81.8.c.j.28.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(10.6001 - 18.3598i) q^{2} +(-160.722 - 278.379i) q^{4} +(176.112 + 305.034i) q^{5} +(-381.456 + 660.702i) q^{7} -4101.05 q^{8} +7467.17 q^{10} +(-3574.94 + 6191.98i) q^{11} +(3568.34 + 6180.55i) q^{13} +(8086.92 + 14007.0i) q^{14} +(-22898.9 + 39662.1i) q^{16} -10229.9 q^{17} -31729.0 q^{19} +(56610.2 - 98051.7i) q^{20} +(75789.2 + 131271. i) q^{22} +(24668.3 + 42726.8i) q^{23} +(-22968.1 + 39781.9i) q^{25} +151299. q^{26} +245234. q^{28} +(30290.1 - 52464.0i) q^{29} +(-78176.2 - 135405. i) q^{31} +(222992. + 386234. i) q^{32} +(-108437. + 187819. i) q^{34} -268716. q^{35} -112163. q^{37} +(-336330. + 582540. i) q^{38} +(-722243. - 1.25096e6i) q^{40} +(-293129. - 507713. i) q^{41} +(-80982.0 + 140265. i) q^{43} +2.29829e6 q^{44} +1.04594e6 q^{46} +(-65431.3 + 113330. i) q^{47} +(120754. + 209151. i) q^{49} +(486926. + 843380. i) q^{50} +(1.14703e6 - 1.98671e6i) q^{52} +1.63679e6 q^{53} -2.51836e6 q^{55} +(1.56437e6 - 2.70957e6i) q^{56} +(-642154. - 1.11224e6i) q^{58} +(-40212.7 - 69650.4i) q^{59} +(-862508. + 1.49391e6i) q^{61} -3.31469e6 q^{62} +3.59281e6 q^{64} +(-1.25685e6 + 2.17693e6i) q^{65} +(1.17043e6 + 2.02724e6i) q^{67} +(1.64417e6 + 2.84779e6i) q^{68} +(-2.84840e6 + 4.93357e6i) q^{70} -469125. q^{71} +3.76695e6 q^{73} +(-1.18893e6 + 2.05930e6i) q^{74} +(5.09957e6 + 8.83271e6i) q^{76} +(-2.72737e6 - 4.72394e6i) q^{77} +(-4.15717e6 + 7.20043e6i) q^{79} -1.61311e7 q^{80} -1.24287e7 q^{82} +(494479. - 856463. i) q^{83} +(-1.80160e6 - 3.12046e6i) q^{85} +(1.71683e6 + 2.97363e6i) q^{86} +(1.46610e7 - 2.53936e7i) q^{88} +9.33980e6 q^{89} -5.44467e6 q^{91} +(7.92951e6 - 1.37343e7i) q^{92} +(1.38715e6 + 2.40262e6i) q^{94} +(-5.58785e6 - 9.67844e6i) q^{95} +(-4.44244e6 + 7.69452e6i) q^{97} +5.11998e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 15 q^{2} - 229 q^{4} + 192 q^{5} - 800 q^{7} - 5010 q^{8} + 10938 q^{10} - 5016 q^{11} + 2200 q^{13} + 19452 q^{14} - 40849 q^{16} + 39240 q^{17} + 22480 q^{19} + 96951 q^{20} + 41280 q^{22} + 154560 q^{23}+ \cdots - 31542030 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.6001 − 18.3598i 0.936922 − 1.62280i 0.165751 − 0.986168i \(-0.446995\pi\)
0.771171 − 0.636628i \(-0.219671\pi\)
\(3\) 0 0
\(4\) −160.722 − 278.379i −1.25564 − 2.17484i
\(5\) 176.112 + 305.034i 0.630076 + 1.09132i 0.987536 + 0.157395i \(0.0503096\pi\)
−0.357460 + 0.933929i \(0.616357\pi\)
\(6\) 0 0
\(7\) −381.456 + 660.702i −0.420341 + 0.728052i −0.995973 − 0.0896571i \(-0.971423\pi\)
0.575632 + 0.817709i \(0.304756\pi\)
\(8\) −4101.05 −2.83192
\(9\) 0 0
\(10\) 7467.17 2.36133
\(11\) −3574.94 + 6191.98i −0.809832 + 1.40267i 0.103149 + 0.994666i \(0.467108\pi\)
−0.912980 + 0.408003i \(0.866225\pi\)
\(12\) 0 0
\(13\) 3568.34 + 6180.55i 0.450469 + 0.780236i 0.998415 − 0.0562782i \(-0.0179234\pi\)
−0.547946 + 0.836514i \(0.684590\pi\)
\(14\) 8086.92 + 14007.0i 0.787653 + 1.36426i
\(15\) 0 0
\(16\) −22898.9 + 39662.1i −1.39764 + 2.42078i
\(17\) −10229.9 −0.505009 −0.252505 − 0.967596i \(-0.581254\pi\)
−0.252505 + 0.967596i \(0.581254\pi\)
\(18\) 0 0
\(19\) −31729.0 −1.06125 −0.530627 − 0.847606i \(-0.678043\pi\)
−0.530627 + 0.847606i \(0.678043\pi\)
\(20\) 56610.2 − 98051.7i 1.58230 − 2.74063i
\(21\) 0 0
\(22\) 75789.2 + 131271.i 1.51750 + 2.62838i
\(23\) 24668.3 + 42726.8i 0.422758 + 0.732239i 0.996208 − 0.0870017i \(-0.0277286\pi\)
−0.573450 + 0.819241i \(0.694395\pi\)
\(24\) 0 0
\(25\) −22968.1 + 39781.9i −0.293991 + 0.509208i
\(26\) 151299. 1.68822
\(27\) 0 0
\(28\) 245234. 2.11119
\(29\) 30290.1 − 52464.0i 0.230626 − 0.399456i −0.727367 − 0.686249i \(-0.759256\pi\)
0.957992 + 0.286793i \(0.0925893\pi\)
\(30\) 0 0
\(31\) −78176.2 − 135405.i −0.471312 − 0.816337i 0.528149 − 0.849152i \(-0.322886\pi\)
−0.999461 + 0.0328148i \(0.989553\pi\)
\(32\) 222992. + 386234.i 1.20300 + 2.08366i
\(33\) 0 0
\(34\) −108437. + 187819.i −0.473154 + 0.819527i
\(35\) −268716. −1.05939
\(36\) 0 0
\(37\) −112163. −0.364036 −0.182018 − 0.983295i \(-0.558263\pi\)
−0.182018 + 0.983295i \(0.558263\pi\)
\(38\) −336330. + 582540.i −0.994312 + 1.72220i
\(39\) 0 0
\(40\) −722243. − 1.25096e6i −1.78432 − 3.09054i
\(41\) −293129. − 507713.i −0.664224 − 1.15047i −0.979495 − 0.201468i \(-0.935429\pi\)
0.315271 − 0.949002i \(-0.397905\pi\)
\(42\) 0 0
\(43\) −80982.0 + 140265.i −0.155328 + 0.269035i −0.933178 − 0.359414i \(-0.882977\pi\)
0.777851 + 0.628449i \(0.216310\pi\)
\(44\) 2.29829e6 4.06744
\(45\) 0 0
\(46\) 1.04594e6 1.58437
\(47\) −65431.3 + 113330.i −0.0919271 + 0.159222i −0.908322 − 0.418272i \(-0.862636\pi\)
0.816395 + 0.577494i \(0.195969\pi\)
\(48\) 0 0
\(49\) 120754. + 209151.i 0.146627 + 0.253965i
\(50\) 486926. + 843380.i 0.550893 + 0.954175i
\(51\) 0 0
\(52\) 1.14703e6 − 1.98671e6i 1.13126 − 1.95940i
\(53\) 1.63679e6 1.51018 0.755090 − 0.655621i \(-0.227593\pi\)
0.755090 + 0.655621i \(0.227593\pi\)
\(54\) 0 0
\(55\) −2.51836e6 −2.04102
\(56\) 1.56437e6 − 2.70957e6i 1.19037 − 2.06178i
\(57\) 0 0
\(58\) −642154. − 1.11224e6i −0.432157 − 0.748517i
\(59\) −40212.7 − 69650.4i −0.0254907 − 0.0441511i 0.852999 − 0.521913i \(-0.174781\pi\)
−0.878489 + 0.477762i \(0.841448\pi\)
\(60\) 0 0
\(61\) −862508. + 1.49391e6i −0.486529 + 0.842693i −0.999880 − 0.0154855i \(-0.995071\pi\)
0.513351 + 0.858179i \(0.328404\pi\)
\(62\) −3.31469e6 −1.76633
\(63\) 0 0
\(64\) 3.59281e6 1.71318
\(65\) −1.25685e6 + 2.17693e6i −0.567660 + 0.983215i
\(66\) 0 0
\(67\) 1.17043e6 + 2.02724e6i 0.475426 + 0.823462i 0.999604 − 0.0281472i \(-0.00896071\pi\)
−0.524178 + 0.851609i \(0.675627\pi\)
\(68\) 1.64417e6 + 2.84779e6i 0.634112 + 1.09831i
\(69\) 0 0
\(70\) −2.84840e6 + 4.93357e6i −0.992562 + 1.71917i
\(71\) −469125. −0.155555 −0.0777775 − 0.996971i \(-0.524782\pi\)
−0.0777775 + 0.996971i \(0.524782\pi\)
\(72\) 0 0
\(73\) 3.76695e6 1.13334 0.566669 − 0.823945i \(-0.308232\pi\)
0.566669 + 0.823945i \(0.308232\pi\)
\(74\) −1.18893e6 + 2.05930e6i −0.341073 + 0.590755i
\(75\) 0 0
\(76\) 5.09957e6 + 8.83271e6i 1.33256 + 2.30806i
\(77\) −2.72737e6 − 4.72394e6i −0.680811 − 1.17920i
\(78\) 0 0
\(79\) −4.15717e6 + 7.20043e6i −0.948643 + 1.64310i −0.200356 + 0.979723i \(0.564210\pi\)
−0.748287 + 0.663375i \(0.769124\pi\)
\(80\) −1.61311e7 −3.52248
\(81\) 0 0
\(82\) −1.24287e7 −2.48930
\(83\) 494479. − 856463.i 0.0949237 − 0.164413i −0.814653 − 0.579949i \(-0.803073\pi\)
0.909577 + 0.415536i \(0.136406\pi\)
\(84\) 0 0
\(85\) −1.80160e6 − 3.12046e6i −0.318194 − 0.551128i
\(86\) 1.71683e6 + 2.97363e6i 0.291060 + 0.504130i
\(87\) 0 0
\(88\) 1.46610e7 − 2.53936e7i 2.29338 − 3.97224i
\(89\) 9.33980e6 1.40434 0.702170 − 0.712009i \(-0.252214\pi\)
0.702170 + 0.712009i \(0.252214\pi\)
\(90\) 0 0
\(91\) −5.44467e6 −0.757403
\(92\) 7.92951e6 − 1.37343e7i 1.06167 − 1.83886i
\(93\) 0 0
\(94\) 1.38715e6 + 2.40262e6i 0.172257 + 0.298358i
\(95\) −5.58785e6 − 9.67844e6i −0.668670 − 1.15817i
\(96\) 0 0
\(97\) −4.44244e6 + 7.69452e6i −0.494220 + 0.856014i −0.999978 − 0.00666169i \(-0.997880\pi\)
0.505758 + 0.862675i \(0.331213\pi\)
\(98\) 5.11998e6 0.549511
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 81.8.c.j.55.4 8
3.2 odd 2 81.8.c.i.55.1 8
9.2 odd 6 81.8.a.b.1.4 yes 4
9.4 even 3 inner 81.8.c.j.28.4 8
9.5 odd 6 81.8.c.i.28.1 8
9.7 even 3 81.8.a.a.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.8.a.a.1.1 ✓ 4 9.7 even 3
81.8.a.b.1.4 yes 4 9.2 odd 6
81.8.c.i.28.1 8 9.5 odd 6
81.8.c.i.55.1 8 3.2 odd 2
81.8.c.j.28.4 8 9.4 even 3 inner
81.8.c.j.55.4 8 1.1 even 1 trivial