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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 8.11
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.99728 - 0.352174i) q^{2} +(-236.696 - 86.1504i) q^{4} +(-67.5893 - 80.5497i) q^{5} +(3385.55 - 1232.24i) q^{7} +(892.041 + 515.020i) q^{8} +(106.627 + 184.683i) q^{10} +(-5656.92 + 6741.66i) q^{11} +(4915.16 + 27875.3i) q^{13} +(-7195.84 + 1268.82i) q^{14} +(47796.6 + 40106.1i) q^{16} +(-115312. + 66575.1i) q^{17} +(56477.6 - 97822.1i) q^{19} +(9058.73 + 24888.7i) q^{20} +(13672.7 - 11472.7i) q^{22} +(141378. - 388433. i) q^{23} +(65911.4 - 373802. i) q^{25} -57405.7i q^{26} -907504. q^{28} +(970128. + 171060. i) q^{29} +(-23816.5 - 8668.51i) q^{31} +(-250836. - 298934. i) q^{32} +(253755. - 92359.3i) q^{34} +(-328083. - 189419. i) q^{35} +(-54478.7 - 94360.0i) q^{37} +(-147252. + 175488. i) q^{38} +(-18807.6 - 106663. i) q^{40} +(4.20215e6 - 740952. i) q^{41} +(-3.87489e6 - 3.25142e6i) q^{43} +(1.91977e6 - 1.10838e6i) q^{44} +(-419167. + 726019. i) q^{46} +(-917770. - 2.52155e6i) q^{47} +(5.52741e6 - 4.63805e6i) q^{49} +(-263287. + 723374. i) q^{50} +(1.23806e6 - 7.02141e6i) q^{52} -4.76381e6i q^{53} +925386. q^{55} +(3.65467e6 + 644417. i) q^{56} +(-1.87737e6 - 683308. i) q^{58} +(-1.05864e7 - 1.26163e7i) q^{59} +(-1.05057e6 + 382376. i) q^{61} +(44515.4 + 25701.0i) q^{62} +(-7.59072e6 - 1.31475e7i) q^{64} +(1.91313e6 - 2.27998e6i) q^{65} +(-1.51350e6 - 8.58350e6i) q^{67} +(3.30293e7 - 5.82395e6i) q^{68} +(588565. + 493864. i) q^{70} +(9.87592e6 - 5.70186e6i) q^{71} +(1.01458e7 - 1.75731e7i) q^{73} +(75578.1 + 207649. i) q^{74} +(-2.17954e7 + 1.82886e7i) q^{76} +(-1.08444e7 + 2.97949e7i) q^{77} +(-3.73057e6 + 2.11571e7i) q^{79} -6.56074e6i q^{80} -8.65380e6 q^{82} +(-8.64355e6 - 1.52409e6i) q^{83} +(1.31564e7 + 4.78855e6i) q^{85} +(6.59417e6 + 7.85863e6i) q^{86} +(-8.51829e6 + 3.10040e6i) q^{88} +(1.19561e7 + 6.90285e6i) q^{89} +(5.09895e7 + 8.83163e7i) q^{91} +(-6.69273e7 + 7.97608e7i) q^{92} +(945017. + 5.35946e6i) q^{94} +(-1.16968e7 + 2.06247e6i) q^{95} +(-5.38212e7 - 4.51614e7i) q^{97} +(-1.26732e7 + 7.31687e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99728 − 0.352174i −0.124830 − 0.0220109i 0.110884 − 0.993833i \(-0.464632\pi\)
−0.235714 + 0.971822i \(0.575743\pi\)
\(3\) 0 0
\(4\) −236.696 − 86.1504i −0.924595 − 0.336525i
\(5\) −67.5893 − 80.5497i −0.108143 − 0.128880i 0.709258 − 0.704949i \(-0.249030\pi\)
−0.817401 + 0.576070i \(0.804586\pi\)
\(6\) 0 0
\(7\) 3385.55 − 1232.24i 1.41006 − 0.513219i 0.478910 − 0.877864i \(-0.341032\pi\)
0.931146 + 0.364645i \(0.118810\pi\)
\(8\) 892.041 + 515.020i 0.217783 + 0.125737i
\(9\) 0 0
\(10\) 106.627 + 184.683i 0.0106627 + 0.0184683i
\(11\) −5656.92 + 6741.66i −0.386375 + 0.460464i −0.923816 − 0.382838i \(-0.874947\pi\)
0.537440 + 0.843302i \(0.319391\pi\)
\(12\) 0 0
\(13\) 4915.16 + 27875.3i 0.172093 + 0.975991i 0.941445 + 0.337165i \(0.109468\pi\)
−0.769352 + 0.638825i \(0.779421\pi\)
\(14\) −7195.84 + 1268.82i −0.187314 + 0.0330284i
\(15\) 0 0
\(16\) 47796.6 + 40106.1i 0.729318 + 0.611971i
\(17\) −115312. + 66575.1i −1.38063 + 0.797106i −0.992234 − 0.124386i \(-0.960304\pi\)
−0.388395 + 0.921493i \(0.626970\pi\)
\(18\) 0 0
\(19\) 56477.6 − 97822.1i 0.433373 − 0.750624i −0.563788 − 0.825919i \(-0.690657\pi\)
0.997161 + 0.0752952i \(0.0239899\pi\)
\(20\) 9058.73 + 24888.7i 0.0566171 + 0.155554i
\(21\) 0 0
\(22\) 13672.7 − 11472.7i 0.0583664 − 0.0489752i
\(23\) 141378. − 388433.i 0.505208 − 1.38805i −0.380920 − 0.924608i \(-0.624393\pi\)
0.886128 − 0.463440i \(-0.153385\pi\)
\(24\) 0 0
\(25\) 65911.4 − 373802.i 0.168733 − 0.956933i
\(26\) − 57405.7i − 0.125621i
\(27\) 0 0
\(28\) −907504. −1.47644
\(29\) 970128. + 171060.i 1.37163 + 0.241855i 0.810435 − 0.585828i \(-0.199231\pi\)
0.561195 + 0.827684i \(0.310342\pi\)
\(30\) 0 0
\(31\) −23816.5 − 8668.51i −0.0257888 − 0.00938637i 0.329094 − 0.944297i \(-0.393257\pi\)
−0.354882 + 0.934911i \(0.615479\pi\)
\(32\) −250836. − 298934.i −0.239215 − 0.285086i
\(33\) 0 0
\(34\) 253755. − 92359.3i 0.189889 − 0.0691139i
\(35\) −328083. − 189419.i −0.218631 − 0.126227i
\(36\) 0 0
\(37\) −54478.7 − 94360.0i −0.0290683 − 0.0503478i 0.851125 − 0.524963i \(-0.175921\pi\)
−0.880194 + 0.474615i \(0.842587\pi\)
\(38\) −147252. + 175488.i −0.0706198 + 0.0841614i
\(39\) 0 0
\(40\) −18807.6 − 106663.i −0.00734674 − 0.0416654i
\(41\) 4.20215e6 − 740952.i 1.48708 − 0.262213i 0.629679 − 0.776855i \(-0.283186\pi\)
0.857405 + 0.514642i \(0.172075\pi\)
\(42\) 0 0
\(43\) −3.87489e6 − 3.25142e6i −1.13341 − 0.951041i −0.134204 − 0.990954i \(-0.542848\pi\)
−0.999203 + 0.0399123i \(0.987292\pi\)
\(44\) 1.91977e6 − 1.10838e6i 0.512198 − 0.295718i
\(45\) 0 0
\(46\) −419167. + 726019.i −0.0936172 + 0.162150i
\(47\) −917770. − 2.52155e6i −0.188080 − 0.516745i 0.809434 − 0.587210i \(-0.199774\pi\)
−0.997514 + 0.0704651i \(0.977552\pi\)
\(48\) 0 0
\(49\) 5.52741e6 − 4.63805e6i 0.958821 − 0.804547i
\(50\) −263287. + 723374.i −0.0421259 + 0.115740i
\(51\) 0 0
\(52\) 1.23806e6 − 7.02141e6i 0.169328 − 0.960309i
\(53\) − 4.76381e6i − 0.603742i −0.953349 − 0.301871i \(-0.902389\pi\)
0.953349 − 0.301871i \(-0.0976111\pi\)
\(54\) 0 0
\(55\) 925386. 0.101128
\(56\) 3.65467e6 + 644417.i 0.371618 + 0.0655262i
\(57\) 0 0
\(58\) −1.87737e6 − 683308.i −0.165897 − 0.0603816i
\(59\) −1.05864e7 − 1.26163e7i −0.873652 − 1.04118i −0.998797 − 0.0490398i \(-0.984384\pi\)
0.125144 − 0.992139i \(-0.460061\pi\)
\(60\) 0 0
\(61\) −1.05057e6 + 382376.i −0.0758761 + 0.0276166i −0.379679 − 0.925118i \(-0.623966\pi\)
0.303803 + 0.952735i \(0.401743\pi\)
\(62\) 44515.4 + 25701.0i 0.00301262 + 0.00173933i
\(63\) 0 0
\(64\) −7.59072e6 − 1.31475e7i −0.452442 − 0.783653i
\(65\) 1.91313e6 − 2.27998e6i 0.107175 − 0.127726i
\(66\) 0 0
\(67\) −1.51350e6 − 8.58350e6i −0.0751076 − 0.425956i −0.999056 − 0.0434394i \(-0.986168\pi\)
0.923948 − 0.382517i \(-0.124943\pi\)
\(68\) 3.30293e7 − 5.82395e6i 1.54477 − 0.272384i
\(69\) 0 0
\(70\) 588565. + 493864.i 0.0245133 + 0.0205691i
\(71\) 9.87592e6 − 5.70186e6i 0.388637 − 0.224380i −0.292932 − 0.956133i \(-0.594631\pi\)
0.681569 + 0.731754i \(0.261298\pi\)
\(72\) 0 0
\(73\) 1.01458e7 − 1.75731e7i 0.357270 − 0.618810i −0.630234 − 0.776406i \(-0.717041\pi\)
0.987504 + 0.157595i \(0.0503742\pi\)
\(74\) 75578.1 + 207649.i 0.00252040 + 0.00692474i
\(75\) 0 0
\(76\) −2.17954e7 + 1.82886e7i −0.653298 + 0.548182i
\(77\) −1.08444e7 + 2.97949e7i −0.308492 + 0.847575i
\(78\) 0 0
\(79\) −3.73057e6 + 2.11571e7i −0.0957782 + 0.543185i 0.898728 + 0.438507i \(0.144492\pi\)
−0.994506 + 0.104678i \(0.966619\pi\)
\(80\) − 6.56074e6i − 0.160174i
\(81\) 0 0
\(82\) −8.65380e6 −0.191404
\(83\) −8.64355e6 − 1.52409e6i −0.182129 − 0.0321143i 0.0818395 − 0.996646i \(-0.473921\pi\)
−0.263969 + 0.964531i \(0.585032\pi\)
\(84\) 0 0
\(85\) 1.31564e7 + 4.78855e6i 0.252036 + 0.0917335i
\(86\) 6.59417e6 + 7.85863e6i 0.120550 + 0.143666i
\(87\) 0 0
\(88\) −8.51829e6 + 3.10040e6i −0.142044 + 0.0516997i
\(89\) 1.19561e7 + 6.90285e6i 0.190559 + 0.110019i 0.592244 − 0.805759i \(-0.298242\pi\)
−0.401685 + 0.915778i \(0.631575\pi\)
\(90\) 0 0
\(91\) 5.09895e7 + 8.83163e7i 0.743558 + 1.28788i
\(92\) −6.69273e7 + 7.97608e7i −0.934225 + 1.11337i
\(93\) 0 0
\(94\) 945017. + 5.35946e6i 0.0121040 + 0.0686451i
\(95\) −1.16968e7 + 2.06247e6i −0.143606 + 0.0253217i
\(96\) 0 0
\(97\) −5.38212e7 − 4.51614e7i −0.607948 − 0.510129i 0.286041 − 0.958217i \(-0.407661\pi\)
−0.893989 + 0.448088i \(0.852105\pi\)
\(98\) −1.26732e7 + 7.31687e6i −0.137398 + 0.0793270i
\(99\) 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 81.9.f.a.8.11 138
3.2 odd 2 27.9.f.a.2.13 ✓ 138
27.13 even 9 27.9.f.a.14.13 yes 138
27.14 odd 18 inner 81.9.f.a.71.11 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.13 ✓ 138 3.2 odd 2
27.9.f.a.14.13 yes 138 27.13 even 9
81.9.f.a.8.11 138 1.1 even 1 trivial
81.9.f.a.71.11 138 27.14 odd 18 inner