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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 11.12
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.12

$q$-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} +(73.1613 + 231.725i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(1188.00 - 5368.59i) q^{6} +(9389.73 - 16263.5i) q^{7} -11585.2i q^{8} +(-48343.8 + 33906.6i) q^{9} +31622.8 q^{10} +(-140286. - 80994.4i) q^{11} +(-84018.7 + 91761.7i) q^{12} +(224244. + 388401. i) q^{13} +(-368001. + 212465. i) q^{14} +(-250470. - 229335. i) q^{15} +(-131072. + 227023. i) q^{16} -1.03997e6i q^{17} +(1.33095e6 - 117483. i) q^{18} +2.39774e6 q^{19} +(-619677. - 357771. i) q^{20} +(4.45562e6 + 985976. i) q^{21} +(1.83269e6 + 3.17432e6i) q^{22} +(1.94894e6 - 1.12522e6i) q^{23} +(2.68459e6 - 847591. i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.01481e7i q^{26} +(-1.13939e7 - 8.72182e6i) q^{27} +9.61509e6 q^{28} +(1.75835e7 + 1.01518e7i) q^{29} +(2.31356e6 + 7.32778e6i) q^{30} +(1.60901e7 + 2.78689e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(8.50487e6 - 3.84335e7i) q^{33} +(-1.17660e7 + 2.03792e7i) q^{34} +2.62451e7i q^{35} +(-2.74104e7 - 1.27558e7i) q^{36} -8.29575e7 q^{37} +(-4.69860e7 - 2.71274e7i) q^{38} +(-7.35963e7 + 8.03787e7i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(6.64008e7 - 3.83365e7i) q^{41} +(-7.61569e7 - 6.97307e7i) q^{42} +(3.65612e7 - 6.33258e7i) q^{43} -8.29382e7i q^{44} +(3.48179e7 - 7.48187e7i) q^{45} -5.09216e7 q^{46} +(-6.87571e6 - 3.96969e6i) q^{47} +(-6.21964e7 - 1.37633e7i) q^{48} +(-3.50965e7 - 6.07889e7i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(2.40988e8 - 7.60858e7i) q^{51} +(-1.14813e8 + 1.98861e8i) q^{52} -2.79670e8i q^{53} +(1.24598e8 + 2.99819e8i) q^{54} +2.26386e8 q^{55} +(-1.88416e8 - 1.08782e8i) q^{56} +(1.75422e8 + 5.55617e8i) q^{57} +(-2.29709e8 - 3.97868e8i) q^{58} +(-1.21549e9 + 7.01763e8i) q^{59} +(3.75680e7 - 1.69770e8i) q^{60} +(-2.24801e8 + 3.89366e8i) q^{61} -7.28155e8i q^{62} +(9.75041e7 + 1.10461e9i) q^{63} -1.34218e8 q^{64} +(-5.42807e8 - 3.13390e8i) q^{65} +(-6.01486e8 + 6.56918e8i) q^{66} +(1.26039e9 + 2.18305e9i) q^{67} +(4.61129e8 - 2.66233e8i) q^{68} +(4.03328e8 + 3.69294e8i) q^{69} +(2.96929e8 - 5.14297e8i) q^{70} +2.66915e9i q^{71} +(3.92816e8 + 5.60075e8i) q^{72} +2.06281e9 q^{73} +(1.62563e9 + 9.38557e8i) q^{74} +(4.63399e8 + 1.02545e8i) q^{75} +(6.13823e8 + 1.06317e9i) q^{76} +(-2.63450e9 + 1.52103e9i) q^{77} +(2.35157e9 - 7.42449e8i) q^{78} +(-2.42893e8 + 4.20704e8i) q^{79} -3.66357e8i q^{80} +(1.18747e9 - 3.27835e9i) q^{81} -1.73491e9 q^{82} +(3.39369e9 + 1.95935e9i) q^{83} +(7.03452e8 + 2.22805e9i) q^{84} +(7.26703e8 + 1.25869e9i) q^{85} +(-1.43290e9 + 8.27285e8i) q^{86} +(-1.06600e9 + 4.81725e9i) q^{87} +(-9.38339e8 + 1.62525e9i) q^{88} +4.62367e8i q^{89} +(-1.52877e9 + 1.07222e9i) q^{90} +8.42235e9 q^{91} +(9.97855e8 + 5.76112e8i) q^{92} +(-5.28074e9 + 5.76740e9i) q^{93} +(8.98238e7 + 1.55579e8i) q^{94} +(-2.90201e9 + 1.67547e9i) q^{95} +(1.06308e9 + 9.73376e8i) q^{96} +(-4.59144e9 + 7.95261e9i) q^{97} +1.58829e9i q^{98} +(9.52823e9 - 8.41055e8i) 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
\(3\) 73.1613 + 231.725i 0.301075 + 0.953600i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) 1188.00 − 5368.59i 0.152778 − 0.690405i
\(7\) 9389.73 − 16263.5i 0.558680 − 0.967662i −0.438927 − 0.898523i \(-0.644641\pi\)
0.997607 − 0.0691392i \(-0.0220253\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −48343.8 + 33906.6i −0.818707 + 0.574211i
\(10\) 31622.8 0.316228
\(11\) −140286. − 80994.4i −0.871068 − 0.502911i −0.00336500 − 0.999994i \(-0.501071\pi\)
−0.867703 + 0.497083i \(0.834404\pi\)
\(12\) −84018.7 + 91761.7i −0.337652 + 0.368770i
\(13\) 224244. + 388401.i 0.603953 + 1.04608i 0.992216 + 0.124529i \(0.0397419\pi\)
−0.388263 + 0.921549i \(0.626925\pi\)
\(14\) −368001. + 212465.i −0.684240 + 0.395046i
\(15\) −250470. − 229335.i −0.329838 − 0.302005i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 1.03997e6i − 0.732449i −0.930526 − 0.366225i \(-0.880650\pi\)
0.930526 − 0.366225i \(-0.119350\pi\)
\(18\) 1.33095e6 − 117483.i 0.704368 − 0.0621745i
\(19\) 2.39774e6 0.968356 0.484178 − 0.874970i \(-0.339119\pi\)
0.484178 + 0.874970i \(0.339119\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 4.45562e6 + 985976.i 1.09097 + 0.241418i
\(22\) 1.83269e6 + 3.17432e6i 0.355612 + 0.615938i
\(23\) 1.94894e6 − 1.12522e6i 0.302802 − 0.174823i −0.340899 − 0.940100i \(-0.610731\pi\)
0.643701 + 0.765277i \(0.277398\pi\)
\(24\) 2.68459e6 − 847591.i 0.337149 − 0.106446i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.01481e7i − 0.854118i
\(27\) −1.13939e7 − 8.72182e6i −0.794061 − 0.607838i
\(28\) 9.61509e6 0.558680
\(29\) 1.75835e7 + 1.01518e7i 0.857264 + 0.494941i 0.863095 − 0.505042i \(-0.168523\pi\)
−0.00583135 + 0.999983i \(0.501856\pi\)
\(30\) 2.31356e6 + 7.32778e6i 0.0952084 + 0.301555i
\(31\) 1.60901e7 + 2.78689e7i 0.562018 + 0.973444i 0.997320 + 0.0731597i \(0.0233083\pi\)
−0.435302 + 0.900285i \(0.643358\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 8.50487e6 − 3.84335e7i 0.217319 − 0.982065i
\(34\) −1.17660e7 + 2.03792e7i −0.258960 + 0.448532i
\(35\) 2.62451e7i 0.499698i
\(36\) −2.74104e7 − 1.27558e7i −0.453318 − 0.210958i
\(37\) −8.29575e7 −1.19632 −0.598159 − 0.801377i \(-0.704101\pi\)
−0.598159 + 0.801377i \(0.704101\pi\)
\(38\) −4.69860e7 − 2.71274e7i −0.592994 − 0.342365i
\(39\) −7.35963e7 + 8.03787e7i −0.815704 + 0.890878i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) 6.64008e7 − 3.83365e7i 0.573131 − 0.330897i −0.185268 − 0.982688i \(-0.559315\pi\)
0.758399 + 0.651791i \(0.225982\pi\)
\(42\) −7.61569e7 − 6.97307e7i −0.582724 − 0.533553i
\(43\) 3.65612e7 − 6.33258e7i 0.248701 − 0.430763i −0.714464 − 0.699672i \(-0.753330\pi\)
0.963166 + 0.268909i \(0.0866629\pi\)
\(44\) − 8.29382e7i − 0.502911i
\(45\) 3.48179e7 − 7.48187e7i 0.188686 − 0.405460i
\(46\) −5.09216e7 −0.247236
\(47\) −6.87571e6 − 3.96969e6i −0.0299798 − 0.0173088i 0.484935 − 0.874550i \(-0.338843\pi\)
−0.514915 + 0.857241i \(0.672177\pi\)
\(48\) −6.21964e7 − 1.37633e7i −0.244095 − 0.0540153i
\(49\) −3.50965e7 − 6.07889e7i −0.124246 − 0.215201i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 2.40988e8 − 7.60858e7i 0.698464 − 0.220523i
\(52\) −1.14813e8 + 1.98861e8i −0.301976 + 0.523039i
\(53\) − 2.79670e8i − 0.668754i −0.942439 − 0.334377i \(-0.891474\pi\)
0.942439 − 0.334377i \(-0.108526\pi\)
\(54\) 1.24598e8 + 2.99819e8i 0.271358 + 0.652966i
\(55\) 2.26386e8 0.449818
\(56\) −1.88416e8 − 1.08782e8i −0.342120 − 0.197523i
\(57\) 1.75422e8 + 5.55617e8i 0.291548 + 0.923424i
\(58\) −2.29709e8 − 3.97868e8i −0.349976 − 0.606177i
\(59\) −1.21549e9 + 7.01763e8i −1.70016 + 0.981590i −0.754583 + 0.656204i \(0.772161\pi\)
−0.945581 + 0.325386i \(0.894506\pi\)
\(60\) 3.75680e7 − 1.69770e8i 0.0483127 − 0.218325i
\(61\) −2.24801e8 + 3.89366e8i −0.266164 + 0.461009i −0.967868 − 0.251459i \(-0.919089\pi\)
0.701704 + 0.712468i \(0.252423\pi\)
\(62\) − 7.28155e8i − 0.794814i
\(63\) 9.75041e7 + 1.10461e9i 0.0982472 + 1.11303i
\(64\) −1.34218e8 −0.125000
\(65\) −5.42807e8 − 3.13390e8i −0.467820 − 0.270096i
\(66\) −6.01486e8 + 6.56918e8i −0.480293 + 0.524556i
\(67\) 1.26039e9 + 2.18305e9i 0.933534 + 1.61693i 0.777228 + 0.629219i \(0.216625\pi\)
0.156305 + 0.987709i \(0.450042\pi\)
\(68\) 4.61129e8 − 2.66233e8i 0.317160 − 0.183112i
\(69\) 4.03328e8 + 3.69294e8i 0.257877 + 0.236117i
\(70\) 2.96929e8 − 5.14297e8i 0.176670 − 0.306002i
\(71\) 2.66915e9i 1.47938i 0.672946 + 0.739692i \(0.265029\pi\)
−0.672946 + 0.739692i \(0.734971\pi\)
\(72\) 3.92816e8 + 5.60075e8i 0.203014 + 0.289457i
\(73\) 2.06281e9 0.995048 0.497524 − 0.867450i \(-0.334243\pi\)
0.497524 + 0.867450i \(0.334243\pi\)
\(74\) 1.62563e9 + 9.38557e8i 0.732593 + 0.422963i
\(75\) 4.63399e8 + 1.02545e8i 0.195276 + 0.0432122i
\(76\) 6.13823e8 + 1.06317e9i 0.242089 + 0.419310i
\(77\) −2.63450e9 + 1.52103e9i −0.973296 + 0.561933i
\(78\) 2.35157e9 − 7.42449e8i 0.814488 − 0.257154i
\(79\) −2.42893e8 + 4.20704e8i −0.0789369 + 0.136723i −0.902792 − 0.430078i \(-0.858486\pi\)
0.823855 + 0.566801i \(0.191819\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 1.18747e9 − 3.27835e9i 0.340563 − 0.940222i
\(82\) −1.73491e9 −0.467960
\(83\) 3.39369e9 + 1.95935e9i 0.861552 + 0.497418i 0.864532 − 0.502578i \(-0.167615\pi\)
−0.00297944 + 0.999996i \(0.500948\pi\)
\(84\) 7.03452e8 + 2.22805e9i 0.168205 + 0.532757i
\(85\) 7.26703e8 + 1.25869e9i 0.163781 + 0.283676i
\(86\) −1.43290e9 + 8.27285e8i −0.304596 + 0.175858i
\(87\) −1.06600e9 + 4.81725e9i −0.213875 + 0.966502i
\(88\) −9.38339e8 + 1.62525e9i −0.177806 + 0.307969i
\(89\) 4.62367e8i 0.0828013i 0.999143 + 0.0414007i \(0.0131820\pi\)
−0.999143 + 0.0414007i \(0.986818\pi\)
\(90\) −1.52877e9 + 1.07222e9i −0.258898 + 0.181582i
\(91\) 8.42235e9 1.34967
\(92\) 9.97855e8 + 5.76112e8i 0.151401 + 0.0874113i
\(93\) −5.28074e9 + 5.76740e9i −0.759067 + 0.829021i
\(94\) 8.98238e7 + 1.55579e8i 0.0122392 + 0.0211989i
\(95\) −2.90201e9 + 1.67547e9i −0.375043 + 0.216531i
\(96\) 1.06308e9 + 9.73376e8i 0.130380 + 0.119378i
\(97\) −4.59144e9 + 7.95261e9i −0.534676 + 0.926085i 0.464503 + 0.885571i \(0.346233\pi\)
−0.999179 + 0.0405141i \(0.987100\pi\)
\(98\) 1.58829e9i 0.175711i
\(99\) 9.52823e9 − 8.41055e8i 1.00193 − 0.0884400i
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.11.12 ✓ 80
3.2 odd 2 270.11.h.a.251.23 80
9.4 even 3 270.11.h.a.71.23 80
9.5 odd 6 inner 90.11.h.a.41.12 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.12 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.12 yes 80 9.5 odd 6 inner
270.11.h.a.71.23 80 9.4 even 3
270.11.h.a.251.23 80 3.2 odd 2