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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 11.17
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.17

$q$-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} +(214.506 + 114.176i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(-2911.68 - 4664.24i) q^{6} +(15545.1 - 26924.9i) q^{7} -11585.2i q^{8} +(32976.6 + 48982.9i) q^{9} -31622.8 q^{10} +(234772. + 135546. i) q^{11} +(4287.23 + 124342. i) q^{12} +(276994. + 479767. i) q^{13} +(-609241. + 351745. i) q^{14} +(339401. - 11702.3i) q^{15} +(-131072. + 227023. i) q^{16} -534391. i q^{17} +(-92028.1 - 1.33295e6i) q^{18} +167539. q^{19} +(619677. + 357771. i) q^{20} +(6.40870e6 - 4.00067e6i) q^{21} +(-3.06705e6 - 5.31228e6i) q^{22} +(-747130. + 431356. i) q^{23} +(1.32276e6 - 2.48510e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.25353e7i q^{26} +(1.48099e6 + 1.42723e7i) q^{27} +1.59182e7 q^{28} +(-2.04189e7 - 1.17889e7i) q^{29} +(-6.78327e6 - 3.61057e6i) q^{30} +(-2.24824e6 - 3.89407e6i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(3.48839e7 + 5.58807e7i) q^{33} +(-6.04595e6 + 1.04719e7i) q^{34} -4.34499e7i q^{35} +(-1.32773e7 + 2.71616e7i) q^{36} +5.34610e7 q^{37} +(-3.28307e6 - 1.89548e6i) q^{38} +(4.63882e6 + 1.34539e8i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-5.45882e7 + 3.15165e7i) q^{41} +(-1.70847e8 + 5.89068e6i) q^{42} +(2.64717e7 - 4.58503e7i) q^{43} +1.38799e8i q^{44} +(7.41397e7 + 3.62413e7i) q^{45} +1.95209e7 q^{46} +(-3.24688e7 - 1.87459e7i) q^{47} +(-5.40364e7 + 3.37326e7i) q^{48} +(-3.42062e8 - 5.92469e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(6.10147e7 - 1.14630e8i) q^{51} +(-1.41821e8 + 2.45641e8i) q^{52} +7.23937e8i q^{53} +(1.32451e8 - 2.96434e8i) q^{54} +3.78862e8 q^{55} +(-3.11931e8 - 1.80094e8i) q^{56} +(3.59380e7 + 1.91289e7i) q^{57} +(2.66751e8 + 4.62027e8i) q^{58} +(9.77489e8 - 5.64354e8i) q^{59} +(9.20756e7 + 1.47496e8i) q^{60} +(-6.92108e8 + 1.19877e9i) q^{61} +1.01744e8i q^{62} +(1.83148e9 - 1.26447e8i) q^{63} -1.34218e8 q^{64} +(6.70495e8 + 3.87111e8i) q^{65} +(-5.13639e7 - 1.48970e9i) q^{66} +(-5.79876e8 - 1.00437e9i) q^{67} +(2.36952e8 - 1.36804e8i) q^{68} +(-2.09514e8 + 7.22391e6i) q^{69} +(-4.91579e8 + 8.51440e8i) q^{70} +9.14733e8i q^{71} +(5.67479e8 - 3.82042e8i) q^{72} +4.31063e8 q^{73} +(-1.04762e9 - 6.04842e8i) q^{74} +(4.02602e8 - 2.51327e8i) q^{75} +(4.28899e7 + 7.42875e7i) q^{76} +(7.29910e9 - 4.21414e9i) q^{77} +(1.43123e9 - 2.68890e9i) q^{78} +(2.08726e9 - 3.61524e9i) q^{79} +3.66357e8i q^{80} +(-1.31187e9 + 3.23058e9i) q^{81} +1.42628e9 q^{82} +(5.62364e9 + 3.24681e9i) q^{83} +(3.41454e9 + 1.81748e9i) q^{84} +(-3.73417e8 - 6.46777e8i) q^{85} +(-1.03747e9 + 5.98986e8i) q^{86} +(-3.03397e9 - 4.86013e9i) q^{87} +(1.57033e9 - 2.71989e9i) q^{88} +4.48046e9i q^{89} +(-1.04281e9 - 1.54898e9i) q^{90} +1.72236e10 q^{91} +(-3.82530e8 - 2.20854e8i) q^{92} +(-3.76513e7 - 1.09200e9i) q^{93} +(4.24171e8 + 7.34685e8i) q^{94} +(2.02773e8 - 1.17071e8i) q^{95} +(1.44053e9 - 4.96686e7i) q^{96} +(4.77216e9 - 8.26562e9i) q^{97} +1.54800e10i q^{98} +(1.10256e9 + 1.59697e10i) 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
\(3\) 214.506 + 114.176i 0.882741 + 0.469861i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) −2911.68 − 4664.24i −0.374445 − 0.599826i
\(7\) 15545.1 − 26924.9i 0.924918 − 1.60200i 0.133224 − 0.991086i \(-0.457467\pi\)
0.791693 − 0.610919i \(-0.209200\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 32976.6 + 48982.9i 0.558462 + 0.829530i
\(10\) −31622.8 −0.316228
\(11\) 234772. + 135546.i 1.45775 + 0.841632i 0.998900 − 0.0468829i \(-0.0149288\pi\)
0.458848 + 0.888515i \(0.348262\pi\)
\(12\) 4287.23 + 124342.i 0.0172294 + 0.499703i
\(13\) 276994. + 479767.i 0.746025 + 1.29215i 0.949714 + 0.313117i \(0.101373\pi\)
−0.203690 + 0.979035i \(0.565293\pi\)
\(14\) −609241. + 351745.i −1.13279 + 0.654016i
\(15\) 339401. − 11702.3i 0.446948 − 0.0154105i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 534391.i − 0.376370i −0.982134 − 0.188185i \(-0.939740\pi\)
0.982134 − 0.188185i \(-0.0602604\pi\)
\(18\) −92028.1 − 1.33295e6i −0.0487033 − 0.705428i
\(19\) 167539. 0.0676624 0.0338312 − 0.999428i \(-0.489229\pi\)
0.0338312 + 0.999428i \(0.489229\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) 6.40870e6 − 4.00067e6i 1.56918 − 0.979572i
\(22\) −3.06705e6 − 5.31228e6i −0.595123 − 1.03078i
\(23\) −747130. + 431356.i −0.116080 + 0.0670187i −0.556916 − 0.830569i \(-0.688015\pi\)
0.440836 + 0.897588i \(0.354682\pi\)
\(24\) 1.32276e6 − 2.48510e6i 0.166121 − 0.312096i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.25353e7i − 1.05504i
\(27\) 1.48099e6 + 1.42723e7i 0.103213 + 0.994659i
\(28\) 1.59182e7 0.924918
\(29\) −2.04189e7 − 1.17889e7i −0.995503 − 0.574754i −0.0885882 − 0.996068i \(-0.528235\pi\)
−0.906915 + 0.421315i \(0.861569\pi\)
\(30\) −6.78327e6 − 3.61057e6i −0.279147 − 0.148583i
\(31\) −2.24824e6 − 3.89407e6i −0.0785297 − 0.136018i 0.824086 − 0.566465i \(-0.191689\pi\)
−0.902616 + 0.430447i \(0.858356\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 3.48839e7 + 5.58807e7i 0.891364 + 1.42788i
\(34\) −6.04595e6 + 1.04719e7i −0.133067 + 0.230478i
\(35\) − 4.34499e7i − 0.827272i
\(36\) −1.32773e7 + 2.71616e7i −0.219582 + 0.449204i
\(37\) 5.34610e7 0.770954 0.385477 − 0.922718i \(-0.374037\pi\)
0.385477 + 0.922718i \(0.374037\pi\)
\(38\) −3.28307e6 − 1.89548e6i −0.0414346 − 0.0239223i
\(39\) 4.63882e6 + 1.34539e8i 0.0514143 + 1.49116i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −5.45882e7 + 3.15165e7i −0.471172 + 0.272031i −0.716730 − 0.697350i \(-0.754362\pi\)
0.245558 + 0.969382i \(0.421029\pi\)
\(42\) −1.70847e8 + 5.89068e6i −1.30725 + 0.0450733i
\(43\) 2.64717e7 − 4.58503e7i 0.180069 − 0.311889i −0.761835 − 0.647771i \(-0.775701\pi\)
0.941904 + 0.335883i \(0.109035\pi\)
\(44\) 1.38799e8i 0.841632i
\(45\) 7.41397e7 + 3.62413e7i 0.401780 + 0.196400i
\(46\) 1.95209e7 0.0947788
\(47\) −3.24688e7 − 1.87459e7i −0.141572 − 0.0817366i 0.427541 − 0.903996i \(-0.359380\pi\)
−0.569113 + 0.822259i \(0.692713\pi\)
\(48\) −5.40364e7 + 3.37326e7i −0.212070 + 0.132386i
\(49\) −3.42062e8 − 5.92469e8i −1.21095 − 2.09742i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 6.10147e7 − 1.14630e8i 0.176841 − 0.332237i
\(52\) −1.41821e8 + 2.45641e8i −0.373012 + 0.646076i
\(53\) 7.23937e8i 1.73110i 0.500825 + 0.865549i \(0.333030\pi\)
−0.500825 + 0.865549i \(0.666970\pi\)
\(54\) 1.32451e8 − 2.96434e8i 0.288460 − 0.645593i
\(55\) 3.78862e8 0.752778
\(56\) −3.11931e8 − 1.80094e8i −0.566394 − 0.327008i
\(57\) 3.59380e7 + 1.91289e7i 0.0597283 + 0.0317919i
\(58\) 2.66751e8 + 4.62027e8i 0.406412 + 0.703927i
\(59\) 9.77489e8 − 5.64354e8i 1.36726 − 0.789390i 0.376685 − 0.926341i \(-0.377064\pi\)
0.990578 + 0.136952i \(0.0437305\pi\)
\(60\) 9.20756e7 + 1.47496e8i 0.118410 + 0.189682i
\(61\) −6.92108e8 + 1.19877e9i −0.819454 + 1.41934i 0.0866315 + 0.996240i \(0.472390\pi\)
−0.906085 + 0.423095i \(0.860944\pi\)
\(62\) 1.01744e8i 0.111058i
\(63\) 1.83148e9 − 1.26447e8i 1.84544 − 0.127411i
\(64\) −1.34218e8 −0.125000
\(65\) 6.70495e8 + 3.87111e8i 0.577868 + 0.333632i
\(66\) −5.13639e7 − 1.48970e9i −0.0410146 − 1.18954i
\(67\) −5.79876e8 − 1.00437e9i −0.429498 − 0.743912i 0.567331 − 0.823490i \(-0.307976\pi\)
−0.996829 + 0.0795780i \(0.974643\pi\)
\(68\) 2.36952e8 − 1.36804e8i 0.162973 − 0.0940924i
\(69\) −2.09514e8 + 7.22391e6i −0.133958 + 0.00461878i
\(70\) −4.91579e8 + 8.51440e8i −0.292485 + 0.506598i
\(71\) 9.14733e8i 0.506994i 0.967336 + 0.253497i \(0.0815808\pi\)
−0.967336 + 0.253497i \(0.918419\pi\)
\(72\) 5.67479e8 − 3.82042e8i 0.293283 − 0.197446i
\(73\) 4.31063e8 0.207934 0.103967 − 0.994581i \(-0.466846\pi\)
0.103967 + 0.994581i \(0.466846\pi\)
\(74\) −1.04762e9 − 6.04842e8i −0.472111 − 0.272573i
\(75\) 4.02602e8 − 2.51327e8i 0.169656 − 0.105909i
\(76\) 4.28899e7 + 7.42875e7i 0.0169156 + 0.0292987i
\(77\) 7.29910e9 − 4.21414e9i 2.69660 − 1.55688i
\(78\) 1.43123e9 − 2.68890e9i 0.495721 − 0.931325i
\(79\) 2.08726e9 − 3.61524e9i 0.678331 − 1.17490i −0.297153 − 0.954830i \(-0.596037\pi\)
0.975483 − 0.220073i \(-0.0706296\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −1.31187e9 + 3.23058e9i −0.376241 + 0.926522i
\(82\) 1.42628e9 0.384711
\(83\) 5.62364e9 + 3.24681e9i 1.42767 + 0.824264i 0.996936 − 0.0782172i \(-0.0249228\pi\)
0.430730 + 0.902481i \(0.358256\pi\)
\(84\) 3.41454e9 + 1.81748e9i 0.816462 + 0.434583i
\(85\) −3.73417e8 − 6.46777e8i −0.0841588 − 0.145767i
\(86\) −1.03747e9 + 5.98986e8i −0.220539 + 0.127328i
\(87\) −3.03397e9 − 4.86013e9i −0.608716 − 0.975106i
\(88\) 1.57033e9 − 2.71989e9i 0.297562 − 0.515392i
\(89\) 4.48046e9i 0.802365i 0.915998 + 0.401183i \(0.131401\pi\)
−0.915998 + 0.401183i \(0.868599\pi\)
\(90\) −1.04281e9 − 1.54898e9i −0.176601 − 0.262321i
\(91\) 1.72236e10 2.76005
\(92\) −3.82530e8 − 2.20854e8i −0.0580399 − 0.0335094i
\(93\) −3.76513e7 − 1.09200e9i −0.00541209 − 0.156966i
\(94\) 4.24171e8 + 7.34685e8i 0.0577965 + 0.100106i
\(95\) 2.02773e8 − 1.17071e8i 0.0262055 − 0.0151298i
\(96\) 1.44053e9 − 4.96686e7i 0.176672 − 0.00609152i
\(97\) 4.77216e9 − 8.26562e9i 0.555720 − 0.962536i −0.442127 − 0.896953i \(-0.645776\pi\)
0.997847 − 0.0655833i \(-0.0208908\pi\)
\(98\) 1.54800e10i 1.71254i
\(99\) 1.10256e9 + 1.59697e10i 0.115938 + 1.67927i
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.11.17 ✓ 80
3.2 odd 2 270.11.h.a.251.21 80
9.4 even 3 270.11.h.a.71.21 80
9.5 odd 6 inner 90.11.h.a.41.17 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.17 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.17 yes 80 9.5 odd 6 inner
270.11.h.a.71.21 80 9.4 even 3
270.11.h.a.251.21 80 3.2 odd 2