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 7.16
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.16

$q$-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+(21.8564 - 5.85641i) q^{2} +(5.52060 - 242.937i) q^{3} +(443.405 - 256.000i) q^{4} +(2070.75 + 2340.43i) q^{5} +(-1302.08 - 5342.07i) q^{6} +(-29350.0 + 7864.30i) 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} +(-59744.1 - 109133. i) q^{12} +(-27554.6 - 7383.23i) q^{13} +(-595429. + 343771. i) q^{14} +(580010. - 490142. i) q^{15} +(131072. - 227023. i) q^{16} +(372065. + 372065. i) q^{17} +(-1.30498e6 + 286832. i) q^{18} +2.95019e6i q^{19} +(1.51733e6 + 507646. i) q^{20} +(1.74850e6 + 7.17362e6i) q^{21} +(914508. - 3.41299e6i) q^{22} +(6.24485e6 + 1.67330e6i) q^{23} +(-1.94492e6 - 2.03537e6i) q^{24} +(-1.18960e6 + 9.69290e6i) q^{25} -645483. q^{26} +(-977285. + 1.43156e7i) q^{27} +(-1.10007e7 + 1.10007e7i) q^{28} +(-2.20435e7 - 1.27268e7i) q^{29} +(9.80645e6 - 1.41095e7i) q^{30} +(2.08765e7 + 3.61592e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-3.24224e7 - 1.97145e7i) 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} +(1.72775e7 + 6.44805e7i) q^{38} +(-1.94578e6 + 6.65327e6i) q^{39} +(3.61364e7 + 2.20921e6i) q^{40} +(4.38786e7 + 7.59999e7i) q^{41} +(8.02276e7 + 1.46550e8i) q^{42} +(6.29775e7 + 2.35035e8i) q^{43} -7.99514e7i q^{44} +(-1.15872e8 - 1.43612e8i) q^{45} +1.46289e8 q^{46} +(1.98821e8 - 5.32740e7i) q^{47} +(-5.44288e7 - 3.30956e7i) q^{48} +(5.54943e8 - 3.20397e8i) q^{49} +(3.07651e7 + 2.18819e8i) q^{50} +(9.24424e7 - 8.83344e7i) q^{51} +(-1.41079e7 + 3.78021e6i) 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} +(7.16711e8 + 1.62868e7i) q^{57} +(-5.56326e8 - 1.49067e8i) q^{58} +(-4.55548e8 + 2.63011e8i) q^{59} +(1.31703e8 - 3.65814e8i) q^{60} +(-3.33505e8 + 5.77647e8i) q^{61} +(6.68049e8 + 6.68049e8i) q^{62} +(1.75239e9 - 3.85174e8i) q^{63} -1.34218e8i q^{64} +(-3.97788e7 - 7.97784e7i) q^{65} +(-8.24094e8 - 2.41010e8i) q^{66} +(-2.78062e8 + 1.03774e9i) q^{67} +(2.60224e8 + 6.97268e7i) q^{68} +(4.40983e8 - 1.50787e9i) q^{69} +(-2.03756e9 - 6.81695e8i) q^{70} -8.07040e8 q^{71} +(-5.05204e8 + 4.61257e8i) q^{72} +(-5.02972e8 + 5.02972e8i) q^{73} +(1.37490e8 + 7.93798e7i) q^{74} +(2.34820e9 + 3.42509e8i) q^{75} +(7.55248e8 + 1.30813e9i) q^{76} +(-1.22805e9 + 4.58315e9i) q^{77} +(-3.56346e6 + 1.56812e8i) 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} +(-1.54652e8 - 5.77171e8i) q^{83} +(2.61174e9 + 2.73320e9i) q^{84} +(-1.00338e8 + 1.64124e9i) q^{85} +(2.75292e9 + 4.76820e9i) q^{86} +(-3.21352e9 + 5.28494e9i) q^{87} +(-4.68228e8 - 1.74745e9i) q^{88} -7.54357e9i q^{89} +(-3.37359e9 - 2.46025e9i) q^{90} +8.66790e8 q^{91} +(3.19736e9 - 8.56731e8i) q^{92} +(8.89968e9 - 4.87207e9i) q^{93} +(4.03353e9 - 2.32876e9i) q^{94} +(-6.90471e9 + 6.10911e9i) q^{95} +(-1.38344e9 - 4.04593e8i) q^{96} +(9.88975e9 - 2.64995e9i) 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) 5.52060 − 242.937i 0.0227185 − 0.999742i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 2070.75 + 2340.43i 0.662640 + 0.748938i
\(6\) −1302.08 − 5342.07i −0.167448 − 0.686994i
\(7\) −29350.0 + 7864.30i −1.74630 + 0.467918i −0.983829 − 0.179110i \(-0.942678\pi\)
−0.762466 + 0.647028i \(0.776012\pi\)
\(8\) 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 \(0.00555892\pi\)
\(12\) −59744.1 − 109133.i −0.240098 − 0.438581i
\(13\) −27554.6 − 7383.23i −0.0742125 − 0.0198852i 0.221522 − 0.975155i \(-0.428898\pi\)
−0.295734 + 0.955270i \(0.595564\pi\)
\(14\) −595429. + 343771.i −1.10711 + 0.639188i
\(15\) 580010. − 490142.i 0.763799 − 0.645455i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 372065. + 372065.i 0.262044 + 0.262044i 0.825884 − 0.563840i \(-0.190677\pi\)
−0.563840 + 0.825884i \(0.690677\pi\)
\(18\) −1.30498e6 + 286832.i −0.690621 + 0.151798i
\(19\) 2.95019e6i 1.19147i 0.803182 + 0.595733i \(0.203138\pi\)
−0.803182 + 0.595733i \(0.796862\pi\)
\(20\) 1.51733e6 + 507646.i 0.474166 + 0.158639i
\(21\) 1.74850e6 + 7.17362e6i 0.428124 + 1.75647i
\(22\) 914508. − 3.41299e6i 0.177449 − 0.662250i
\(23\) 6.24485e6 + 1.67330e6i 0.970248 + 0.259977i 0.708933 − 0.705276i \(-0.249177\pi\)
0.261315 + 0.965253i \(0.415844\pi\)
\(24\) −1.94492e6 − 2.03537e6i −0.244256 − 0.255615i
\(25\) −1.18960e6 + 9.69290e6i −0.121815 + 0.992553i
\(26\) −645483. −0.0543273
\(27\) −977285. + 1.43156e7i −0.0681087 + 0.997678i
\(28\) −1.10007e7 + 1.10007e7i −0.639188 + 0.639188i
\(29\) −2.20435e7 − 1.27268e7i −1.07471 − 0.620484i −0.145245 − 0.989396i \(-0.546397\pi\)
−0.929464 + 0.368912i \(0.879731\pi\)
\(30\) 9.80645e6 − 1.41095e7i 0.403558 − 0.580639i
\(31\) 2.08765e7 + 3.61592e7i 0.729206 + 1.26302i 0.957219 + 0.289363i \(0.0934436\pi\)
−0.228014 + 0.973658i \(0.573223\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −3.24224e7 − 1.97145e7i −0.828468 − 0.503752i
\(34\) 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 0.741974 − 0.670429i \(-0.233890\pi\)
−0.670429 + 0.741974i \(0.733890\pi\)
\(38\) 1.72775e7 + 6.44805e7i 0.218053 + 0.813787i
\(39\) −1.94578e6 + 6.65327e6i −0.0215660 + 0.0737416i
\(40\) 3.61364e7 + 2.20921e6i 0.352895 + 0.0215743i
\(41\) 4.38786e7 + 7.59999e7i 0.378733 + 0.655985i 0.990878 − 0.134761i \(-0.0430265\pi\)
−0.612145 + 0.790745i \(0.709693\pi\)
\(42\) 8.02276e7 + 1.46550e8i 0.613872 + 1.12134i
\(43\) 6.29775e7 + 2.35035e8i 0.428394 + 1.59879i 0.756398 + 0.654111i \(0.226957\pi\)
−0.328005 + 0.944676i \(0.606376\pi\)
\(44\) − 7.99514e7i − 0.484800i
\(45\) −1.15872e8 − 1.43612e8i −0.627936 − 0.778265i
\(46\) 1.46289e8 0.710271
\(47\) 1.98821e8 − 5.32740e7i 0.866909 − 0.232288i 0.202159 − 0.979353i \(-0.435204\pi\)
0.664751 + 0.747065i \(0.268538\pi\)
\(48\) −5.44288e7 − 3.30956e7i −0.213611 − 0.129886i
\(49\) 5.54943e8 − 3.20397e8i 1.96457 − 1.13425i
\(50\) 3.07651e7 + 2.18819e8i 0.0984484 + 0.700220i
\(51\) 9.24424e7 − 8.83344e7i 0.267929 − 0.256023i
\(52\) −1.41079e7 + 3.78021e6i −0.0371062 + 0.00994259i
\(53\) −1.86524e8 + 1.86524e8i −0.446021 + 0.446021i −0.894029 − 0.448008i \(-0.852134\pi\)
0.448008 + 0.894029i \(0.352134\pi\)
\(54\) 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
\(57\) 7.16711e8 + 1.62868e7i 1.19116 + 0.0270684i
\(58\) −5.56326e8 − 1.49067e8i −0.847597 − 0.227113i
\(59\) −4.55548e8 + 2.63011e8i −0.637198 + 0.367886i −0.783534 − 0.621349i \(-0.786585\pi\)
0.146337 + 0.989235i \(0.453252\pi\)
\(60\) 1.31703e8 − 3.65814e8i 0.169371 − 0.470440i
\(61\) −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
\(63\) 1.75239e9 − 3.85174e8i 1.76575 − 0.388109i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −3.97788e7 − 7.97784e7i −0.0342834 − 0.0687572i
\(66\) −8.24094e8 − 2.41010e8i −0.658047 − 0.192449i
\(67\) −2.78062e8 + 1.03774e9i −0.205953 + 0.768626i 0.783204 + 0.621765i \(0.213584\pi\)
−0.989157 + 0.146862i \(0.953083\pi\)
\(68\) 2.60224e8 + 6.97268e7i 0.178979 + 0.0479573i
\(69\) 4.40983e8 − 1.50787e9i 0.281953 − 0.964091i
\(70\) −2.03756e9 − 6.81695e8i −1.21233 − 0.405602i
\(71\) −8.07040e8 −0.447304 −0.223652 − 0.974669i \(-0.571798\pi\)
−0.223652 + 0.974669i \(0.571798\pi\)
\(72\) −5.05204e8 + 4.61257e8i −0.261098 + 0.238386i
\(73\) −5.02972e8 + 5.02972e8i −0.242622 + 0.242622i −0.817934 − 0.575312i \(-0.804881\pi\)
0.575312 + 0.817934i \(0.304881\pi\)
\(74\) 1.37490e8 + 7.93798e7i 0.0619601 + 0.0357727i
\(75\) 2.34820e9 + 3.42509e8i 0.989529 + 0.144333i
\(76\) 7.55248e8 + 1.30813e9i 0.297867 + 0.515920i
\(77\) −1.22805e9 + 4.58315e9i −0.453694 + 1.69321i
\(78\) −3.56346e6 + 1.56812e8i −0.00123424 + 0.0543133i
\(79\) −1.87875e9 − 1.08470e9i −0.610568 − 0.352511i 0.162620 − 0.986689i \(-0.448006\pi\)
−0.773188 + 0.634177i \(0.781339\pi\)
\(80\) 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
\(83\) −1.54652e8 − 5.77171e8i −0.0392614 − 0.146526i 0.943513 − 0.331336i \(-0.107499\pi\)
−0.982774 + 0.184811i \(0.940833\pi\)
\(84\) 2.61174e9 + 2.73320e9i 0.624502 + 0.653545i
\(85\) −1.00338e8 + 1.64124e9i −0.0226136 + 0.369895i
\(86\) 2.75292e9 + 4.76820e9i 0.585197 + 1.01359i
\(87\) −3.21352e9 + 5.28494e9i −0.644740 + 1.06034i
\(88\) −4.68228e8 − 1.74745e9i −0.0887246 − 0.331125i
\(89\) − 7.54357e9i − 1.35091i −0.737401 − 0.675455i \(-0.763947\pi\)
0.737401 − 0.675455i \(-0.236053\pi\)
\(90\) −3.37359e9 − 2.46025e9i −0.571320 − 0.416645i
\(91\) 8.66790e8 0.138902
\(92\) 3.19736e9 − 8.56731e8i 0.485124 − 0.129989i
\(93\) 8.89968e9 − 4.87207e9i 1.27926 − 0.700324i
\(94\) 4.03353e9 − 2.32876e9i 0.549599 − 0.317311i
\(95\) −6.90471e9 + 6.10911e9i −0.892334 + 0.789514i
\(96\) −1.38344e9 − 4.04593e8i −0.169670 − 0.0496206i
\(97\) 9.88975e9 − 2.64995e9i 1.15167 − 0.308588i 0.368034 − 0.929812i \(-0.380031\pi\)
0.783633 + 0.621224i \(0.213364\pi\)
\(98\) 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.7.16 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.30 yes 120
9.4 even 3 inner 90.11.k.b.67.30 yes 120
45.13 odd 12 inner 90.11.k.b.13.16 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.16 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.16 yes 120 45.13 odd 12 inner
90.11.k.b.43.30 yes 120 5.3 odd 4 inner
90.11.k.b.67.30 yes 120 9.4 even 3 inner