↑

Properties

Label 90.11.h.a.11.11
Level $90$
Weight $11$
Character 90.11
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
Inner twists $2$

Related objects

Downloads

Learn more

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

$q$-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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

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

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 −0.997069 − 0.0765074i \(-0.975623\pi\)
0.564792 + 0.825233i \(0.308956\pi\)
\(8\) − 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 0.0379924 − 0.999278i \(-0.487904\pi\)
−0.846404 + 0.532541i \(0.821237\pi\)
\(12\) 119902. − 33210.6i 0.481858 − 0.133466i
\(13\) −235456. − 407821.i −0.634150 − 1.09838i −0.986694 − 0.162585i \(-0.948017\pi\)
0.352544 − 0.935795i \(-0.385317\pi\)
\(14\) 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 \(-0.773995\pi\)
0.758352 − 0.651845i \(-0.226005\pi\)
\(18\) 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 −0.651388 − 0.758745i \(-0.725813\pi\)
0.331398 + 0.943491i \(0.392480\pi\)
\(24\) −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 0.895372 − 0.445319i \(-0.146910\pi\)
0.0620289 + 0.998074i \(0.480243\pi\)
\(30\) 1.92637e6 − 7.43896e6i 0.0792745 − 0.306130i
\(31\) 6.17161e6 + 1.06895e7i 0.215571 + 0.373380i 0.953449 − 0.301554i \(-0.0975054\pi\)
−0.737878 + 0.674934i \(0.764172\pi\)
\(32\) 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 −0.221111 − 0.975249i \(-0.570968\pi\)
0.734035 + 0.679112i \(0.237635\pi\)
\(42\) −2.13267e7 − 7.69968e7i −0.163184 − 0.589150i
\(43\) −5.89535e7 + 1.02110e8i −0.401021 + 0.694589i −0.993849 − 0.110740i \(-0.964678\pi\)
0.592828 + 0.805329i \(0.298011\pi\)
\(44\) − 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 −0.0246460 − 0.999696i \(-0.507846\pi\)
−0.878085 + 0.478504i \(0.841179\pi\)
\(48\) 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 \(0.311428\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(54\) 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 0.486734 − 0.873550i \(-0.338188\pi\)
0.999884 + 0.0152508i \(0.00485468\pi\)
\(60\) −1.21911e8 + 1.23979e8i −0.156779 + 0.159438i
\(61\) 2.73855e8 − 4.74330e8i 0.324243 − 0.561606i −0.657116 − 0.753790i \(-0.728224\pi\)
0.981359 + 0.192184i \(0.0615570\pi\)
\(62\) − 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 0.880099 − 0.474790i \(-0.157476\pi\)
−0.851230 + 0.524793i \(0.824143\pi\)
\(68\) 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 \(0.317331\pi\)
−0.542887 + 0.839806i \(0.682669\pi\)
\(72\) −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 0.413204 + 0.910638i \(0.364410\pi\)
−0.995238 + 0.0974738i \(0.968924\pi\)
\(80\) − 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 0.888332 − 0.459202i \(-0.151865\pi\)
0.0464852 + 0.998919i \(0.485198\pi\)
\(84\) −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 \(0.0426129\pi\)
−0.991052 + 0.133473i \(0.957387\pi\)
\(90\) −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 −0.744033 − 0.668143i \(-0.767090\pi\)
0.950645 + 0.310280i \(0.100423\pi\)
\(98\) − 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.11.11 ✓ 80
3.2 odd 2 270.11.h.a.251.32 80
9.4 even 3 270.11.h.a.71.32 80
9.5 odd 6 inner 90.11.h.a.41.11 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.11 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.11 yes 80 9.5 odd 6 inner
270.11.h.a.71.32 80 9.4 even 3
270.11.h.a.251.32 80 3.2 odd 2