Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 43.30
Character \(\chi\) \(=\) 90.43
Dual form 90.11.k.b.67.30

$q$-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+(-5.85641 - 21.8564i) q^{2} +(242.937 + 5.52060i) q^{3} +(-443.405 + 256.000i) q^{4} +(991.496 + 2963.54i) q^{5} +(-1302.08 - 5342.07i) q^{6} +(7864.30 + 29350.0i) q^{7} +(8192.00 + 8192.00i) q^{8} +(58988.0 + 2682.32i) q^{9} +(58965.7 - 39026.2i) q^{10} +(78077.6 - 135234. i) q^{11} +(-109133. + 59744.1i) q^{12} +(7383.23 - 27554.6i) q^{13} +(595429. - 343771. i) q^{14} +(224511. + 725428. i) q^{15} +(131072. - 227023. i) q^{16} +(372065. - 372065. i) q^{17} +(-286832. - 1.30498e6i) q^{18} -2.95019e6i q^{19} +(-1.19830e6 - 1.06022e6i) q^{20} +(1.74850e6 + 7.17362e6i) q^{21} +(-3.41299e6 - 914508. i) q^{22} +(-1.67330e6 + 6.24485e6i) q^{23} +(1.94492e6 + 2.03537e6i) q^{24} +(-7.79950e6 + 5.87667e6i) q^{25} -645483. q^{26} +(1.43156e7 + 977285. i) q^{27} +(-1.10007e7 - 1.10007e7i) q^{28} +(2.20435e7 + 1.27268e7i) q^{29} +(1.45404e7 - 9.15540e6i) q^{30} +(2.08765e7 + 3.61592e7i) q^{31} +(-5.72953e6 - 1.53522e6i) q^{32} +(1.97145e7 - 3.24224e7i) q^{33} +(-1.03110e7 - 5.95303e6i) q^{34} +(-7.91824e7 + 5.24066e7i) q^{35} +(-2.68423e7 + 1.39116e7i) q^{36} +(4.96124e6 - 4.96124e6i) q^{37} +(-6.44805e7 + 1.72775e7i) q^{38} +(1.94578e6 - 6.65327e6i) q^{39} +(-1.61550e7 + 3.23996e7i) q^{40} +(4.38786e7 + 7.59999e7i) q^{41} +(1.46550e8 - 8.02276e7i) q^{42} +(-2.35035e8 + 6.29775e7i) q^{43} +7.99514e7i q^{44} +(5.05373e7 + 1.77473e8i) q^{45} +1.46289e8 q^{46} +(-5.32740e7 - 1.98821e8i) q^{47} +(3.30956e7 - 5.44288e7i) q^{48} +(-5.54943e8 + 3.20397e8i) q^{49} +(1.74120e8 + 1.36053e8i) q^{50} +(9.24424e7 - 8.83344e7i) q^{51} +(3.78021e6 + 1.41079e7i) q^{52} +(-1.86524e8 - 1.86524e8i) q^{53} +(-6.24780e7 - 3.18611e8i) q^{54} +(4.78186e8 + 9.73016e7i) q^{55} +(-1.76011e8 + 3.04859e8i) q^{56} +(1.62868e7 - 7.16711e8i) q^{57} +(1.49067e8 - 5.56326e8i) q^{58} +(4.55548e8 - 2.63011e8i) q^{59} +(-2.85259e8 - 2.64183e8i) q^{60} +(-3.33505e8 + 5.77647e8i) q^{61} +(6.68049e8 - 6.68049e8i) q^{62} +(3.85174e8 + 1.75239e9i) q^{63} +1.34218e8i q^{64} +(8.89795e7 - 5.43978e6i) q^{65} +(-8.24094e8 - 2.41010e8i) q^{66} +(1.03774e9 + 2.78062e8i) q^{67} +(-6.97268e7 + 2.60224e8i) q^{68} +(-4.40983e8 + 1.50787e9i) q^{69} +(1.60914e9 + 1.42373e9i) q^{70} -8.07040e8 q^{71} +(4.61257e8 + 5.05204e8i) q^{72} +(-5.02972e8 - 5.02972e8i) q^{73} +(-1.37490e8 - 7.93798e7i) q^{74} +(-1.92723e9 + 1.38461e9i) q^{75} +(7.55248e8 + 1.30813e9i) q^{76} +(4.58315e9 + 1.22805e9i) q^{77} +(-1.56812e8 - 3.56346e6i) q^{78} +(1.87875e9 + 1.08470e9i) q^{79} +(8.02750e8 + 1.63344e8i) q^{80} +(3.47239e9 + 3.16450e8i) q^{81} +(1.40411e9 - 1.40411e9i) q^{82} +(5.77171e8 - 1.54652e8i) q^{83} +(-2.61174e9 - 2.73320e9i) q^{84} +(1.47153e9 + 7.33727e8i) q^{85} +(2.75292e9 + 4.76820e9i) q^{86} +(5.28494e9 + 3.21352e9i) q^{87} +(1.74745e9 - 4.68228e8i) q^{88} +7.54357e9i q^{89} +(3.58295e9 - 2.14392e9i) q^{90} +8.66790e8 q^{91} +(-8.56731e8 - 3.19736e9i) q^{92} +(4.87207e9 + 8.89968e9i) q^{93} +(-4.03353e9 + 2.32876e9i) q^{94} +(8.74300e9 - 2.92510e9i) q^{95} +(-1.38344e9 - 4.04593e8i) q^{96} +(-2.64995e9 - 9.88975e9i) q^{97} +(1.02527e10 + 1.02527e10i) q^{98} +(4.96839e9 - 7.76778e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.85641 − 21.8564i −0.183013 − 0.683013i
\(3\) 242.937 + 5.52060i 0.999742 + 0.0227185i
\(4\) −443.405 + 256.000i −0.433013 + 0.250000i
\(5\) 991.496 + 2963.54i 0.317279 + 0.948332i
\(6\) −1302.08 − 5342.07i −0.167448 − 0.686994i
\(7\) 7864.30 + 29350.0i 0.467918 + 1.74630i 0.647028 + 0.762466i \(0.276012\pi\)
−0.179110 + 0.983829i \(0.557322\pi\)
\(8\) 8192.00 + 8192.00i 0.250000 + 0.250000i
\(9\) 58988.0 + 2682.32i 0.998968 + 0.0454253i
\(10\) 58965.7 − 39026.2i 0.589657 − 0.390262i
\(11\) 78077.6 − 135234.i 0.484800 − 0.839699i −0.515047 − 0.857162i \(-0.672226\pi\)
0.999848 + 0.0174630i \(0.00555892\pi\)
\(12\) −109133. + 59744.1i −0.438581 + 0.240098i
\(13\) 7383.23 − 27554.6i 0.0198852 − 0.0742125i −0.955270 − 0.295734i \(-0.904436\pi\)
0.975155 + 0.221522i \(0.0711024\pi\)
\(14\) 595429. − 343771.i 1.10711 − 0.639188i
\(15\) 224511. + 725428.i 0.295652 + 0.955296i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 372065. − 372065.i 0.262044 − 0.262044i −0.563840 − 0.825884i \(-0.690677\pi\)
0.825884 + 0.563840i \(0.190677\pi\)
\(18\) −286832. − 1.30498e6i −0.151798 − 0.690621i
\(19\) − 2.95019e6i − 1.19147i −0.803182 − 0.595733i \(-0.796862\pi\)
0.803182 − 0.595733i \(-0.203138\pi\)
\(20\) −1.19830e6 − 1.06022e6i −0.374469 − 0.331320i
\(21\) 1.74850e6 + 7.17362e6i 0.428124 + 1.75647i
\(22\) −3.41299e6 − 914508.i −0.662250 − 0.177449i
\(23\) −1.67330e6 + 6.24485e6i −0.259977 + 0.970248i 0.705276 + 0.708933i \(0.250823\pi\)
−0.965253 + 0.261315i \(0.915844\pi\)
\(24\) 1.94492e6 + 2.03537e6i 0.244256 + 0.255615i
\(25\) −7.79950e6 + 5.87667e6i −0.798668 + 0.601771i
\(26\) −645483. −0.0543273
\(27\) 1.43156e7 + 977285.i 0.997678 + 0.0681087i
\(28\) −1.10007e7 − 1.10007e7i −0.639188 − 0.639188i
\(29\) 2.20435e7 + 1.27268e7i 1.07471 + 0.620484i 0.929464 − 0.368912i \(-0.120269\pi\)
0.145245 + 0.989396i \(0.453603\pi\)
\(30\) 1.45404e7 − 9.15540e6i 0.598371 − 0.376765i
\(31\) 2.08765e7 + 3.61592e7i 0.729206 + 1.26302i 0.957219 + 0.289363i \(0.0934436\pi\)
−0.228014 + 0.973658i \(0.573223\pi\)
\(32\) −5.72953e6 − 1.53522e6i −0.170753 − 0.0457532i
\(33\) 1.97145e7 − 3.24224e7i 0.503752 − 0.828468i
\(34\) −1.03110e7 − 5.95303e6i −0.226937 − 0.131022i
\(35\) −7.91824e7 + 5.24066e7i −1.50761 + 0.997805i
\(36\) −2.68423e7 + 1.39116e7i −0.443922 + 0.230072i
\(37\) 4.96124e6 − 4.96124e6i 0.0715454 − 0.0715454i −0.670429 − 0.741974i \(-0.733890\pi\)
0.741974 + 0.670429i \(0.233890\pi\)
\(38\) −6.44805e7 + 1.72775e7i −0.813787 + 0.218053i
\(39\) 1.94578e6 − 6.65327e6i 0.0215660 − 0.0737416i
\(40\) −1.61550e7 + 3.23996e7i −0.157763 + 0.316403i
\(41\) 4.38786e7 + 7.59999e7i 0.378733 + 0.655985i 0.990878 − 0.134761i \(-0.0430265\pi\)
−0.612145 + 0.790745i \(0.709693\pi\)
\(42\) 1.46550e8 − 8.02276e7i 1.12134 − 0.613872i
\(43\) −2.35035e8 + 6.29775e7i −1.59879 + 0.428394i −0.944676 − 0.328005i \(-0.893624\pi\)
−0.654111 + 0.756398i \(0.726957\pi\)
\(44\) 7.99514e7i 0.484800i
\(45\) 5.05373e7 + 1.77473e8i 0.273873 + 0.961766i
\(46\) 1.46289e8 0.710271
\(47\) −5.32740e7 − 1.98821e8i −0.232288 − 0.866909i −0.979353 − 0.202159i \(-0.935204\pi\)
0.747065 − 0.664751i \(-0.231462\pi\)
\(48\) 3.30956e7 − 5.44288e7i 0.129886 − 0.213611i
\(49\) −5.54943e8 + 3.20397e8i −1.96457 + 1.13425i
\(50\) 1.74120e8 + 1.36053e8i 0.557184 + 0.435369i
\(51\) 9.24424e7 − 8.83344e7i 0.267929 − 0.256023i
\(52\) 3.78021e6 + 1.41079e7i 0.00994259 + 0.0371062i
\(53\) −1.86524e8 − 1.86524e8i −0.446021 − 0.446021i 0.448008 − 0.894029i \(-0.352134\pi\)
−0.894029 + 0.448008i \(0.852134\pi\)
\(54\) −6.24780e7 − 3.18611e8i −0.136069 − 0.693891i
\(55\) 4.78186e8 + 9.73016e7i 0.950130 + 0.193333i
\(56\) −1.76011e8 + 3.04859e8i −0.319594 + 0.553553i
\(57\) 1.62868e7 − 7.16711e8i 0.0270684 − 1.19116i
\(58\) 1.49067e8 − 5.56326e8i 0.227113 − 0.847597i
\(59\) 4.55548e8 − 2.63011e8i 0.637198 − 0.367886i −0.146337 − 0.989235i \(-0.546748\pi\)
0.783534 + 0.621349i \(0.213415\pi\)
\(60\) −2.85259e8 − 2.64183e8i −0.366845 − 0.339742i
\(61\) −3.33505e8 + 5.77647e8i −0.394869 + 0.683933i −0.993084 − 0.117402i \(-0.962543\pi\)
0.598216 + 0.801335i \(0.295877\pi\)
\(62\) 6.68049e8 − 6.68049e8i 0.729206 − 0.729206i
\(63\) 3.85174e8 + 1.75239e9i 0.388109 + 1.76575i
\(64\) 1.34218e8i 0.125000i
\(65\) 8.89795e7 − 5.43978e6i 0.0766872 − 0.00468829i
\(66\) −8.24094e8 − 2.41010e8i −0.658047 − 0.192449i
\(67\) 1.03774e9 + 2.78062e8i 0.768626 + 0.205953i 0.621765 − 0.783204i \(-0.286416\pi\)
0.146862 + 0.989157i \(0.453083\pi\)
\(68\) −6.97268e7 + 2.60224e8i −0.0479573 + 0.178979i
\(69\) −4.40983e8 + 1.50787e9i −0.281953 + 0.964091i
\(70\) 1.60914e9 + 1.42373e9i 0.957424 + 0.847104i
\(71\) −8.07040e8 −0.447304 −0.223652 − 0.974669i \(-0.571798\pi\)
−0.223652 + 0.974669i \(0.571798\pi\)
\(72\) 4.61257e8 + 5.05204e8i 0.238386 + 0.261098i
\(73\) −5.02972e8 − 5.02972e8i −0.242622 − 0.242622i 0.575312 − 0.817934i \(-0.304881\pi\)
−0.817934 + 0.575312i \(0.804881\pi\)
\(74\) −1.37490e8 − 7.93798e7i −0.0619601 − 0.0357727i
\(75\) −1.92723e9 + 1.38461e9i −0.812134 + 0.583472i
\(76\) 7.55248e8 + 1.30813e9i 0.297867 + 0.515920i
\(77\) 4.58315e9 + 1.22805e9i 1.69321 + 0.453694i
\(78\) −1.56812e8 − 3.56346e6i −0.0543133 − 0.00123424i
\(79\) 1.87875e9 + 1.08470e9i 0.610568 + 0.352511i 0.773188 − 0.634177i \(-0.218661\pi\)
−0.162620 + 0.986689i \(0.551994\pi\)
\(80\) 8.02750e8 + 1.63344e8i 0.244980 + 0.0498487i
\(81\) 3.47239e9 + 3.16450e8i 0.995873 + 0.0907569i
\(82\) 1.40411e9 − 1.40411e9i 0.378733 − 0.378733i
\(83\) 5.77171e8 − 1.54652e8i 0.146526 − 0.0392614i −0.184811 − 0.982774i \(-0.559167\pi\)
0.331336 + 0.943513i \(0.392501\pi\)
\(84\) −2.61174e9 − 2.73320e9i −0.624502 − 0.653545i
\(85\) 1.47153e9 + 7.33727e8i 0.331645 + 0.165364i
\(86\) 2.75292e9 + 4.76820e9i 0.585197 + 1.01359i
\(87\) 5.28494e9 + 3.21352e9i 1.06034 + 0.644740i
\(88\) 1.74745e9 − 4.68228e8i 0.331125 − 0.0887246i
\(89\) 7.54357e9i 1.35091i 0.737401 + 0.675455i \(0.236053\pi\)
−0.737401 + 0.675455i \(0.763947\pi\)
\(90\) 3.58295e9 − 2.14392e9i 0.606776 − 0.363074i
\(91\) 8.66790e8 0.138902
\(92\) −8.56731e8 − 3.19736e9i −0.129989 − 0.485124i
\(93\) 4.87207e9 + 8.89968e9i 0.700324 + 1.27926i
\(94\) −4.03353e9 + 2.32876e9i −0.549599 + 0.317311i
\(95\) 8.74300e9 − 2.92510e9i 1.12991 − 0.378027i
\(96\) −1.38344e9 − 4.04593e8i −0.169670 − 0.0496206i
\(97\) −2.64995e9 − 9.88975e9i −0.308588 − 1.15167i −0.929812 − 0.368034i \(-0.880031\pi\)
0.621224 − 0.783633i \(-0.286636\pi\)
\(98\) 1.02527e10 + 1.02527e10i 1.13425 + 1.13425i
\(99\) 4.96839e9 − 7.76778e9i 0.522444 − 0.816810i
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 90.11.k.b.43.30 yes 120
5.2 odd 4 inner 90.11.k.b.7.16 ✓ 120
9.4 even 3 inner 90.11.k.b.13.16 yes 120
45.22 odd 12 inner 90.11.k.b.67.30 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.16 ✓ 120 5.2 odd 4 inner
90.11.k.b.13.16 yes 120 9.4 even 3 inner
90.11.k.b.43.30 yes 120 1.1 even 1 trivial
90.11.k.b.67.30 yes 120 45.22 odd 12 inner