Show commands: Magma / Pari/GP / SageMath

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 8.19
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.19

$q$-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+(21.5826 + 3.80559i) q^{2} +(210.764 + 76.7119i) q^{4} +(-129.757 - 154.639i) q^{5} +(1479.28 - 538.412i) q^{7} +(-601.827 - 347.465i) q^{8} +(-2212.01 - 3831.31i) q^{10} +(5647.40 - 6730.30i) q^{11} +(-2496.75 - 14159.8i) q^{13} +(33975.6 - 5990.81i) q^{14} +(-55651.7 - 46697.3i) q^{16} +(48327.4 - 27901.8i) q^{17} +(24124.8 - 41785.3i) q^{19} +(-15485.6 - 42546.2i) q^{20} +(147498. - 123766. i) q^{22} +(57310.9 - 157460. i) q^{23} +(60755.1 - 344560. i) q^{25} -315106. i q^{26} +353081. q^{28} +(52796.6 + 9309.47i) q^{29} +(971027. + 353425. i) q^{31} +(-909043. - 1.08336e6i) q^{32} +(1.14921e6 - 418280. i) q^{34} +(-275206. - 158890. i) q^{35} +(-932654. - 1.61540e6i) q^{37} +(679693. - 810026. i) q^{38} +(24359.9 + 138152. i) q^{40} +(4.40599e6 - 776896. i) q^{41} +(-3.96290e6 - 3.32527e6i) q^{43} +(1.70656e6 - 985285. i) q^{44} +(1.83615e6 - 3.18030e6i) q^{46} +(1.82983e6 + 5.02741e6i) q^{47} +(-2.51772e6 + 2.11262e6i) q^{49} +(2.62251e6 - 7.20528e6i) q^{50} +(559998. - 3.17591e6i) q^{52} +1.03085e7i q^{53} -1.77356e6 q^{55} +(-1.07735e6 - 189966. i) q^{56} +(1.10406e6 + 401845. i) q^{58} +(555817. + 662397. i) q^{59} +(-1.53716e7 + 5.59480e6i) q^{61} +(1.96123e7 + 1.13232e7i) q^{62} +(-6.19773e6 - 1.07348e7i) q^{64} +(-1.86568e6 + 2.22343e6i) q^{65} +(1.38317e6 + 7.84434e6i) q^{67} +(1.23261e7 - 2.17342e6i) q^{68} +(-5.33499e6 - 4.47659e6i) q^{70} +(1.63960e7 - 9.46623e6i) q^{71} +(2.49290e7 - 4.31782e7i) q^{73} +(-1.39815e7 - 3.84139e7i) q^{74} +(8.29006e6 - 6.95619e6i) q^{76} +(4.73038e6 - 1.29966e7i) q^{77} +(-2.63044e6 + 1.49180e7i) q^{79} +1.46652e7i q^{80} +9.80493e7 q^{82} +(-8.67787e7 - 1.53014e7i) q^{83} +(-1.05855e7 - 3.85282e6i) q^{85} +(-7.28750e7 - 8.68490e7i) q^{86} +(-5.73730e6 + 2.08821e6i) q^{88} +(6.02765e7 + 3.48007e7i) q^{89} +(-1.13172e7 - 1.96019e7i) q^{91} +(2.41582e7 - 2.87906e7i) q^{92} +(2.03602e7 + 1.15468e8i) q^{94} +(-9.59199e6 + 1.69133e6i) q^{95} +(-2.85952e7 - 2.39942e7i) q^{97} +(-6.23788e7 + 3.60144e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.5826 + 3.80559i 1.34891 + 0.237850i 0.800989 − 0.598679i \(-0.204308\pi\)
0.547923 + 0.836529i \(0.315419\pi\)
\(3\) 0 0
\(4\) 210.764 + 76.7119i 0.823297 + 0.299656i
\(5\) −129.757 − 154.639i −0.207612 − 0.247422i 0.652183 − 0.758061i \(-0.273853\pi\)
−0.859795 + 0.510639i \(0.829409\pi\)
\(6\) 0 0
\(7\) 1479.28 − 538.412i 0.616108 − 0.224245i −0.0150656 − 0.999887i \(-0.504796\pi\)
0.631174 + 0.775641i \(0.282573\pi\)
\(8\) −601.827 − 347.465i −0.146930 − 0.0848304i
\(9\) 0 0
\(10\) −2212.01 − 3831.31i −0.221201 − 0.383131i
\(11\) 5647.40 − 6730.30i 0.385725 − 0.459689i −0.537888 − 0.843017i \(-0.680778\pi\)
0.923612 + 0.383328i \(0.125222\pi\)
\(12\) 0 0
\(13\) −2496.75 − 14159.8i −0.0874182 − 0.495773i −0.996809 − 0.0798295i \(-0.974562\pi\)
0.909390 − 0.415944i \(-0.136549\pi\)
\(14\) 33975.6 − 5990.81i 0.884412 − 0.155946i
\(15\) 0 0
\(16\) −55651.7 − 46697.3i −0.849178 − 0.712545i
\(17\) 48327.4 − 27901.8i 0.578626 − 0.334070i −0.181961 − 0.983306i \(-0.558245\pi\)
0.760587 + 0.649236i \(0.224911\pi\)
\(18\) 0 0
\(19\) 24124.8 − 41785.3i 0.185118 − 0.320634i −0.758498 − 0.651675i \(-0.774067\pi\)
0.943616 + 0.331041i \(0.107400\pi\)
\(20\) −15485.6 − 42546.2i −0.0967847 − 0.265914i
\(21\) 0 0
\(22\) 147498. − 123766.i 0.629645 − 0.528335i
\(23\) 57310.9 − 157460.i 0.204798 − 0.562678i −0.794189 − 0.607671i \(-0.792104\pi\)
0.998987 + 0.0449922i \(0.0143263\pi\)
\(24\) 0 0
\(25\) 60755.1 − 344560.i 0.155533 − 0.882072i
\(26\) − 315106.i − 0.689547i
\(27\) 0 0
\(28\) 353081. 0.574437
\(29\) 52796.6 + 9309.47i 0.0746473 + 0.0131623i 0.210847 − 0.977519i \(-0.432378\pi\)
−0.136200 + 0.990681i \(0.543489\pi\)
\(30\) 0 0
\(31\) 971027. + 353425.i 1.05144 + 0.382693i 0.809206 − 0.587525i \(-0.199898\pi\)
0.242234 + 0.970218i \(0.422120\pi\)
\(32\) −909043. − 1.08336e6i −0.866931 − 1.03317i
\(33\) 0 0
\(34\) 1.14921e6 − 418280.i 0.859974 − 0.313005i
\(35\) −275206. − 158890.i −0.183394 − 0.105883i
\(36\) 0 0
\(37\) −932654. − 1.61540e6i −0.497638 − 0.861935i 0.502358 − 0.864660i \(-0.332466\pi\)
−0.999996 + 0.00272492i \(0.999133\pi\)
\(38\) 679693. − 810026.i 0.325970 − 0.388476i
\(39\) 0 0
\(40\) 24359.9 + 138152.i 0.00951559 + 0.0539656i
\(41\) 4.40599e6 − 776896.i 1.55922 − 0.274933i 0.673515 − 0.739174i \(-0.264784\pi\)
0.885709 + 0.464241i \(0.153673\pi\)
\(42\) 0 0
\(43\) −3.96290e6 − 3.32527e6i −1.15915 − 0.972641i −0.159255 − 0.987238i \(-0.550909\pi\)
−0.999893 + 0.0145968i \(0.995354\pi\)
\(44\) 1.70656e6 − 985285.i 0.455315 − 0.262876i
\(45\) 0 0
\(46\) 1.83615e6 − 3.18030e6i 0.410087 − 0.710292i
\(47\) 1.82983e6 + 5.02741e6i 0.374990 + 1.03028i 0.973405 + 0.229090i \(0.0735749\pi\)
−0.598416 + 0.801186i \(0.704203\pi\)
\(48\) 0 0
\(49\) −2.51772e6 + 2.11262e6i −0.436741 + 0.366469i
\(50\) 2.62251e6 − 7.20528e6i 0.419601 − 1.15284i
\(51\) 0 0
\(52\) 559998. − 3.17591e6i 0.0765901 − 0.434364i
\(53\) 1.03085e7i 1.30645i 0.757163 + 0.653226i \(0.226585\pi\)
−0.757163 + 0.653226i \(0.773415\pi\)
\(54\) 0 0
\(55\) −1.77356e6 −0.193818
\(56\) −1.07735e6 − 189966.i −0.109548 − 0.0193162i
\(57\) 0 0
\(58\) 1.10406e6 + 401845.i 0.0975620 + 0.0355097i
\(59\) 555817. + 662397.i 0.0458695 + 0.0546651i 0.788491 − 0.615046i \(-0.210862\pi\)
−0.742622 + 0.669711i \(0.766418\pi\)
\(60\) 0 0
\(61\) −1.53716e7 + 5.59480e6i −1.11020 + 0.404078i −0.831065 − 0.556175i \(-0.812268\pi\)
−0.279130 + 0.960253i \(0.590046\pi\)
\(62\) 1.96123e7 + 1.13232e7i 1.32728 + 0.766303i
\(63\) 0 0
\(64\) −6.19773e6 − 1.07348e7i −0.369414 − 0.639843i
\(65\) −1.86568e6 + 2.22343e6i −0.104516 + 0.124558i
\(66\) 0 0
\(67\) 1.38317e6 + 7.84434e6i 0.0686398 + 0.389276i 0.999702 + 0.0244149i \(0.00777228\pi\)
−0.931062 + 0.364861i \(0.881117\pi\)
\(68\) 1.23261e7 − 2.17342e6i 0.576487 − 0.101650i
\(69\) 0 0
\(70\) −5.33499e6 − 4.47659e6i −0.222199 − 0.186447i
\(71\) 1.63960e7 − 9.46623e6i 0.645215 − 0.372515i −0.141406 − 0.989952i \(-0.545162\pi\)
0.786620 + 0.617437i \(0.211829\pi\)
\(72\) 0 0
\(73\) 2.49290e7 − 4.31782e7i 0.877835 − 1.52045i 0.0241223 − 0.999709i \(-0.492321\pi\)
0.853712 − 0.520745i \(-0.174346\pi\)
\(74\) −1.39815e7 − 3.84139e7i −0.466259 − 1.28104i
\(75\) 0 0
\(76\) 8.29006e6 − 6.95619e6i 0.248487 − 0.208505i
\(77\) 4.73038e6 − 1.29966e7i 0.134565 − 0.369715i
\(78\) 0 0
\(79\) −2.63044e6 + 1.49180e7i −0.0675336 + 0.383002i 0.932242 + 0.361834i \(0.117849\pi\)
−0.999776 + 0.0211675i \(0.993262\pi\)
\(80\) 1.46652e7i 0.358038i
\(81\) 0 0
\(82\) 9.80493e7 2.16865
\(83\) −8.67787e7 − 1.53014e7i −1.82852 − 0.322418i −0.849723 − 0.527229i \(-0.823231\pi\)
−0.978802 + 0.204811i \(0.934342\pi\)
\(84\) 0 0
\(85\) −1.05855e7 − 3.85282e6i −0.202786 − 0.0738080i
\(86\) −7.28750e7 − 8.68490e7i −1.33225 − 1.58771i
\(87\) 0 0
\(88\) −5.73730e6 + 2.08821e6i −0.0956703 + 0.0348211i
\(89\) 6.02765e7 + 3.48007e7i 0.960701 + 0.554661i 0.896389 − 0.443269i \(-0.146181\pi\)
0.0643121 + 0.997930i \(0.479515\pi\)
\(90\) 0 0
\(91\) −1.13172e7 − 1.96019e7i −0.165034 − 0.285847i
\(92\) 2.41582e7 − 2.87906e7i 0.337220 − 0.401883i
\(93\) 0 0
\(94\) 2.03602e7 + 1.15468e8i 0.260777 + 1.47894i
\(95\) −9.59199e6 + 1.69133e6i −0.117765 + 0.0207651i
\(96\) 0 0
\(97\) −2.85952e7 − 2.39942e7i −0.323003 − 0.271032i 0.466839 − 0.884343i \(-0.345393\pi\)
−0.789842 + 0.613311i \(0.789837\pi\)
\(98\) −6.23788e7 + 3.60144e7i −0.676289 + 0.390456i
\(99\) 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.8.19 138
3.2 odd 2 27.9.f.a.2.5 ✓ 138
27.13 even 9 27.9.f.a.14.5 yes 138
27.14 odd 18 inner 81.9.f.a.71.19 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.5 ✓ 138 3.2 odd 2
27.9.f.a.14.5 yes 138 27.13 even 9
81.9.f.a.8.19 138 1.1 even 1 trivial
81.9.f.a.71.19 138 27.14 odd 18 inner