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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,Mod(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 41.11
Character \(\chi\) \(=\) 90.41
Dual form 90.11.h.a.11.11

$q$-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+(-19.5959 + 11.3137i) q^{2} +(60.9172 + 235.241i) q^{3} +(256.000 - 443.405i) q^{4} +(-1210.31 - 698.771i) q^{5} +(-3855.17 - 3920.55i) q^{6} +(-7265.28 - 12583.8i) q^{7} +11585.2i q^{8} +(-51627.2 + 28660.4i) q^{9} +31622.8 q^{10} +(-130195. + 75168.4i) q^{11} +(119902. + 33210.6i) q^{12} +(-235456. + 407821. i) q^{13} +(284740. + 164395. i) q^{14} +(90650.8 - 327280. i) q^{15} +(-131072. - 227023. i) q^{16} +1.85105e6i q^{17} +(687427. - 1.14572e6i) q^{18} -1.20948e6 q^{19} +(-619677. + 357771. i) q^{20} +(2.51765e6 - 2.47566e6i) q^{21} +(1.70087e6 - 2.94599e6i) q^{22} +(-2.05956e6 - 1.18909e6i) q^{23} +(-2.72532e6 + 705740. i) q^{24} +(976562. + 1.69146e6i) q^{25} -1.06555e7i q^{26} +(-9.88707e6 - 1.03989e7i) q^{27} -7.43965e6 q^{28} +(1.96374e7 - 1.13377e7i) q^{29} +(1.92637e6 + 7.43896e6i) q^{30} +(6.17161e6 - 1.06895e7i) q^{31} +(5.13695e6 + 2.96582e6i) q^{32} +(-2.56138e7 - 2.60482e7i) q^{33} +(-2.09423e7 - 3.62731e7i) q^{34} +2.03071e7i q^{35} +(-508404. + 3.02288e7i) q^{36} -3.05784e7 q^{37} +(2.37008e7 - 1.36837e7i) q^{38} +(-1.10279e8 - 3.05454e7i) q^{39} +(8.09543e6 - 1.40217e7i) q^{40} +(5.94254e7 + 3.43093e7i) q^{41} +(-2.13267e7 + 7.69968e7i) q^{42} +(-5.89535e7 - 1.02110e8i) q^{43} +7.69724e7i q^{44} +(8.25118e7 + 1.38773e6i) q^{45} +5.38121e7 q^{46} +(-2.07037e8 + 1.19533e8i) q^{47} +(4.54206e7 - 4.46631e7i) q^{48} +(3.56690e7 - 6.17805e7i) q^{49} +(-3.82733e7 - 2.20971e7i) q^{50} +(-4.35443e8 + 1.12761e8i) q^{51} +(1.20553e8 + 2.08804e8i) q^{52} -6.93864e8i q^{53} +(3.11396e8 + 9.19165e7i) q^{54} +2.10102e8 q^{55} +(1.45787e8 - 8.41700e7i) q^{56} +(-7.36780e7 - 2.84518e8i) q^{57} +(-2.56542e8 + 4.44344e8i) q^{58} +(1.06282e9 + 6.13619e8i) q^{59} +(-1.21911e8 - 1.23979e8i) q^{60} +(2.73855e8 + 4.74330e8i) q^{61} +2.79295e8i q^{62} +(7.35744e8 + 4.41442e8i) q^{63} -1.34218e8 q^{64} +(5.69947e8 - 3.29059e8i) q^{65} +(7.96628e8 + 2.20651e8i) q^{66} +(3.89775e7 - 6.75111e7i) q^{67} +(8.20767e8 + 4.73870e8i) q^{68} +(1.54259e8 - 5.56929e8i) q^{69} +(-2.29748e8 - 3.97936e8i) q^{70} -3.03040e9i q^{71} +(-3.32037e8 - 5.98113e8i) q^{72} +1.46051e9 q^{73} +(5.99211e8 - 3.45955e8i) q^{74} +(-3.38409e8 + 3.32766e8i) q^{75} +(-3.09626e8 + 5.36289e8i) q^{76} +(1.89181e9 + 1.09224e9i) q^{77} +(2.50661e9 - 6.49103e8i) q^{78} +(-1.79095e9 - 3.10202e9i) q^{79} +3.66357e8i q^{80} +(1.84395e9 - 2.95931e9i) q^{81} -1.55266e9 q^{82} +(3.68228e9 - 2.12597e9i) q^{83} +(-4.53203e8 - 1.75011e9i) q^{84} +(1.29346e9 - 2.24035e9i) q^{85} +(2.31050e9 + 1.33397e9i) q^{86} +(3.86333e9 + 3.92885e9i) q^{87} +(-8.70844e8 - 1.50835e9i) q^{88} -1.49064e9i q^{89} +(-1.63260e9 + 9.06321e8i) q^{90} +6.84260e9 q^{91} +(-1.05450e9 + 6.08814e8i) q^{92} +(2.89057e9 + 8.00636e8i) q^{93} +(2.70472e9 - 4.68471e9i) q^{94} +(1.46384e9 + 8.45148e8i) q^{95} +(-3.84752e8 + 1.38909e9i) q^{96} +(1.77425e9 + 3.07308e9i) q^{97} +1.61419e9i q^{98} +(4.56727e9 - 7.61219e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5959 + 11.3137i −0.612372 + 0.353553i
\(3\) 60.9172 + 235.241i 0.250688 + 0.968068i
\(4\) 256.000 − 443.405i 0.250000 − 0.433013i
\(5\) −1210.31 − 698.771i −0.387298 − 0.223607i
\(6\) −3855.17 − 3920.55i −0.495778 − 0.504187i
\(7\) −7265.28 − 12583.8i −0.432277 − 0.748726i 0.564792 − 0.825233i \(-0.308956\pi\)
−0.997069 + 0.0765074i \(0.975623\pi\)
\(8\) 11585.2i 0.353553i
\(9\) −51627.2 + 28660.4i −0.874311 + 0.485366i
\(10\) 31622.8 0.316228
\(11\) −130195. + 75168.4i −0.808412 + 0.466737i −0.846404 − 0.532541i \(-0.821237\pi\)
0.0379924 + 0.999278i \(0.487904\pi\)
\(12\) 119902. + 33210.6i 0.481858 + 0.133466i
\(13\) −235456. + 407821.i −0.634150 + 1.09838i 0.352544 + 0.935795i \(0.385317\pi\)
−0.986694 + 0.162585i \(0.948017\pi\)
\(14\) 284740. + 164395.i 0.529429 + 0.305666i
\(15\) 90650.8 − 327280.i 0.119376 − 0.430987i
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) 1.85105e6i 1.30369i 0.758352 + 0.651845i \(0.226005\pi\)
−0.758352 + 0.651845i \(0.773995\pi\)
\(18\) 687427. − 1.14572e6i 0.363801 − 0.606340i
\(19\) −1.20948e6 −0.488461 −0.244231 − 0.969717i \(-0.578535\pi\)
−0.244231 + 0.969717i \(0.578535\pi\)
\(20\) −619677. + 357771.i −0.193649 + 0.111803i
\(21\) 2.51765e6 − 2.47566e6i 0.616451 − 0.606170i
\(22\) 1.70087e6 − 2.94599e6i 0.330033 − 0.571633i
\(23\) −2.05956e6 − 1.18909e6i −0.319990 − 0.184746i 0.331398 − 0.943491i \(-0.392480\pi\)
−0.651388 + 0.758745i \(0.725813\pi\)
\(24\) −2.72532e6 + 705740.i −0.342264 + 0.0886316i
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) − 1.06555e7i − 0.896824i
\(27\) −9.88707e6 − 1.03989e7i −0.689047 − 0.724717i
\(28\) −7.43965e6 −0.432277
\(29\) 1.96374e7 − 1.13377e7i 0.957401 − 0.552756i 0.0620289 − 0.998074i \(-0.480243\pi\)
0.895372 + 0.445319i \(0.146910\pi\)
\(30\) 1.92637e6 + 7.43896e6i 0.0792745 + 0.306130i
\(31\) 6.17161e6 − 1.06895e7i 0.215571 − 0.373380i −0.737878 − 0.674934i \(-0.764172\pi\)
0.953449 + 0.301554i \(0.0975054\pi\)
\(32\) 5.13695e6 + 2.96582e6i 0.153093 + 0.0883883i
\(33\) −2.56138e7 − 2.60482e7i −0.654492 − 0.665592i
\(34\) −2.09423e7 − 3.62731e7i −0.460924 − 0.798344i
\(35\) 2.03071e7i 0.386640i
\(36\) −508404. + 3.02288e7i −0.00840807 + 0.499929i
\(37\) −3.05784e7 −0.440966 −0.220483 − 0.975391i \(-0.570763\pi\)
−0.220483 + 0.975391i \(0.570763\pi\)
\(38\) 2.37008e7 − 1.36837e7i 0.299120 − 0.172697i
\(39\) −1.10279e8 − 3.05454e7i −1.22228 − 0.338550i
\(40\) 8.09543e6 − 1.40217e7i 0.0790569 − 0.136931i
\(41\) 5.94254e7 + 3.43093e7i 0.512924 + 0.296137i 0.734035 − 0.679112i \(-0.237635\pi\)
−0.221111 + 0.975249i \(0.570968\pi\)
\(42\) −2.13267e7 + 7.69968e7i −0.163184 + 0.589150i
\(43\) −5.89535e7 − 1.02110e8i −0.401021 − 0.694589i 0.592828 − 0.805329i \(-0.298011\pi\)
−0.993849 + 0.110740i \(0.964678\pi\)
\(44\) 7.69724e7i 0.466737i
\(45\) 8.25118e7 + 1.38773e6i 0.447150 + 0.00752041i
\(46\) 5.38121e7 0.261271
\(47\) −2.07037e8 + 1.19533e8i −0.902731 + 0.521192i −0.878085 − 0.478504i \(-0.841179\pi\)
−0.0246460 + 0.999696i \(0.507846\pi\)
\(48\) 4.54206e7 − 4.46631e7i 0.178257 − 0.175284i
\(49\) 3.56690e7 − 6.17805e7i 0.126273 − 0.218711i
\(50\) −3.82733e7 − 2.20971e7i −0.122474 − 0.0707107i
\(51\) −4.35443e8 + 1.12761e8i −1.26206 + 0.326820i
\(52\) 1.20553e8 + 2.08804e8i 0.317075 + 0.549190i
\(53\) − 6.93864e8i − 1.65919i −0.558368 − 0.829593i \(-0.688572\pi\)
0.558368 − 0.829593i \(-0.311428\pi\)
\(54\) 3.11396e8 + 9.19165e7i 0.678179 + 0.200182i
\(55\) 2.10102e8 0.417462
\(56\) 1.45787e8 − 8.41700e7i 0.264715 − 0.152833i
\(57\) −7.36780e7 − 2.84518e8i −0.122451 − 0.472863i
\(58\) −2.56542e8 + 4.44344e8i −0.390857 + 0.676985i
\(59\) 1.06282e9 + 6.13619e8i 1.48662 + 0.858299i 0.999884 − 0.0152508i \(-0.00485468\pi\)
0.486734 + 0.873550i \(0.338188\pi\)
\(60\) −1.21911e8 − 1.23979e8i −0.156779 − 0.159438i
\(61\) 2.73855e8 + 4.74330e8i 0.324243 + 0.561606i 0.981359 − 0.192184i \(-0.0615570\pi\)
−0.657116 + 0.753790i \(0.728224\pi\)
\(62\) 2.79295e8i 0.304863i
\(63\) 7.35744e8 + 4.41442e8i 0.741351 + 0.444807i
\(64\) −1.34218e8 −0.125000
\(65\) 5.69947e8 − 3.29059e8i 0.491211 − 0.283601i
\(66\) 7.96628e8 + 2.20651e8i 0.636115 + 0.176192i
\(67\) 3.89775e7 − 6.75111e7i 0.0288696 − 0.0500036i −0.851230 − 0.524793i \(-0.824143\pi\)
0.880099 + 0.474790i \(0.157476\pi\)
\(68\) 8.20767e8 + 4.73870e8i 0.564515 + 0.325923i
\(69\) 1.54259e8 − 5.56929e8i 0.0986293 − 0.356086i
\(70\) −2.29748e8 − 3.97936e8i −0.136698 − 0.236768i
\(71\) − 3.03040e9i − 1.67961i −0.542887 − 0.839806i \(-0.682669\pi\)
0.542887 − 0.839806i \(-0.317331\pi\)
\(72\) −3.32037e8 − 5.98113e8i −0.171603 − 0.309116i
\(73\) 1.46051e9 0.704516 0.352258 − 0.935903i \(-0.385414\pi\)
0.352258 + 0.935903i \(0.385414\pi\)
\(74\) 5.99211e8 − 3.45955e8i 0.270036 − 0.155905i
\(75\) −3.38409e8 + 3.32766e8i −0.142605 + 0.140227i
\(76\) −3.09626e8 + 5.36289e8i −0.122115 + 0.211510i
\(77\) 1.89181e9 + 1.09224e9i 0.698916 + 0.403519i
\(78\) 2.50661e9 − 6.49103e8i 0.868187 − 0.224823i
\(79\) −1.79095e9 − 3.10202e9i −0.582034 − 1.00811i −0.995238 − 0.0974738i \(-0.968924\pi\)
0.413204 − 0.910638i \(-0.364410\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 1.84395e9 − 2.95931e9i 0.528840 − 0.848722i
\(82\) −1.55266e9 −0.418800
\(83\) 3.68228e9 − 2.12597e9i 0.934817 − 0.539717i 0.0464852 − 0.998919i \(-0.485198\pi\)
0.888332 + 0.459202i \(0.151865\pi\)
\(84\) −4.53203e8 − 1.75011e9i −0.108367 − 0.418474i
\(85\) 1.29346e9 − 2.24035e9i 0.291514 − 0.504917i
\(86\) 2.31050e9 + 1.33397e9i 0.491149 + 0.283565i
\(87\) 3.86333e9 + 3.92885e9i 0.775114 + 0.788260i
\(88\) −8.70844e8 − 1.50835e9i −0.165016 − 0.285817i
\(89\) − 1.49064e9i − 0.266946i −0.991052 − 0.133473i \(-0.957387\pi\)
0.991052 − 0.133473i \(-0.0426129\pi\)
\(90\) −1.63260e9 + 9.06321e8i −0.276481 + 0.153486i
\(91\) 6.84260e9 1.09651
\(92\) −1.05450e9 + 6.08814e8i −0.159995 + 0.0923731i
\(93\) 2.89057e9 + 8.00636e8i 0.415498 + 0.115085i
\(94\) 2.70472e9 − 4.68471e9i 0.368539 − 0.638327i
\(95\) 1.46384e9 + 8.45148e8i 0.189180 + 0.109223i
\(96\) −3.84752e8 + 1.38909e9i −0.0471873 + 0.170362i
\(97\) 1.77425e9 + 3.07308e9i 0.206612 + 0.357862i 0.950645 − 0.310280i \(-0.100423\pi\)
−0.744033 + 0.668143i \(0.767090\pi\)
\(98\) 1.61419e9i 0.178577i
\(99\) 4.56727e9 − 7.61219e9i 0.480265 − 0.800448i
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.h.a.41.11 yes 80
3.2 odd 2 270.11.h.a.71.32 80
9.2 odd 6 inner 90.11.h.a.11.11 ✓ 80
9.7 even 3 270.11.h.a.251.32 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.11 ✓ 80 9.2 odd 6 inner
90.11.h.a.41.11 yes 80 1.1 even 1 trivial
270.11.h.a.71.32 80 3.2 odd 2
270.11.h.a.251.32 80 9.7 even 3