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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 11.6
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.6

$q$-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} +(-156.662 - 185.758i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(968.329 + 5412.53i) q^{6} +(-10611.8 + 18380.2i) q^{7} -11585.2i q^{8} +(-9962.94 + 58202.4i) q^{9} -31622.8 q^{10} +(-232171. - 134044. i) q^{11} +(42260.4 - 117019. i) q^{12} +(-266298. - 461242. i) q^{13} +(415896. - 240118. i) q^{14} +(-319412. - 115353. i) q^{15} +(-131072. + 227023. i) q^{16} +1.68584e6i q^{17} +(853718. - 1.02781e6i) q^{18} +314597. q^{19} +(619677. + 357771. i) q^{20} +(5.07673e6 - 908253. i) q^{21} +(3.03307e6 + 5.25343e6i) q^{22} +(-7.09048e6 + 4.09369e6i) q^{23} +(-2.15205e6 + 1.81497e6i) q^{24} +(976562. - 1.69146e6i) q^{25} +1.20513e7i q^{26} +(1.23724e7 - 7.26742e6i) q^{27} -1.08665e7 q^{28} +(-3.12690e7 - 1.80532e7i) q^{29} +(4.95409e6 + 5.87418e6i) q^{30} +(2.27961e7 + 3.94841e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(1.14727e7 + 6.41272e7i) q^{33} +(1.90731e7 - 3.30356e7i) q^{34} +2.96609e7i q^{35} +(-2.83578e7 + 1.04822e7i) q^{36} +2.77674e7 q^{37} +(-6.16482e6 - 3.55926e6i) q^{38} +(-4.39604e7 + 1.21726e8i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-1.24774e8 + 7.20382e7i) q^{41} +(-1.09759e8 - 3.96386e7i) q^{42} +(7.18757e7 - 1.24492e8i) q^{43} -1.37261e8i q^{44} +(2.86120e7 + 7.74047e7i) q^{45} +1.85259e8 q^{46} +(2.95221e8 + 1.70446e8i) q^{47} +(6.27054e7 - 1.12183e7i) q^{48} +(-8.39830e7 - 1.45463e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(3.13158e8 - 2.64107e8i) q^{51} +(1.36345e8 - 2.36156e8i) q^{52} -1.14407e8i q^{53} +(-3.24670e8 + 2.43441e6i) q^{54} -3.74664e8 q^{55} +(2.12939e8 + 1.22940e8i) q^{56} +(-4.92854e7 - 5.84389e7i) q^{57} +(4.08496e8 + 7.07536e8i) q^{58} +(4.76249e7 - 2.74963e7i) q^{59} +(-3.06212e7 - 1.71159e8i) q^{60} +(2.78802e8 - 4.82900e8i) q^{61} -1.03164e9i q^{62} +(-9.64046e8 - 8.00753e8i) q^{63} -1.34218e8 q^{64} +(-6.44605e8 - 3.72163e8i) q^{65} +(5.00698e8 - 1.38643e9i) q^{66} +(8.59232e8 + 1.48823e9i) q^{67} +(-7.47509e8 + 4.31575e8i) q^{68} +(1.87124e9 + 6.75786e8i) q^{69} +(3.35575e8 - 5.81232e8i) q^{70} -2.29652e9i q^{71} +(6.74289e8 + 1.15423e8i) q^{72} -2.42924e9 q^{73} +(-5.44127e8 - 3.14152e8i) q^{74} +(-4.67192e8 + 8.35830e7i) q^{75} +(8.05368e7 + 1.39494e8i) q^{76} +(4.92750e9 - 2.84490e9i) q^{77} +(2.23862e9 - 1.88798e9i) q^{78} +(1.81753e9 - 3.14806e9i) q^{79} +3.66357e8i q^{80} +(-3.28826e9 - 1.15974e9i) q^{81} +3.26008e9 q^{82} +(4.51472e9 + 2.60657e9i) q^{83} +(1.70237e9 + 2.01853e9i) q^{84} +(1.17802e9 + 2.04038e9i) q^{85} +(-2.81694e9 + 1.62636e9i) q^{86} +(1.54515e9 + 8.63670e9i) q^{87} +(-1.55293e9 + 2.68976e9i) q^{88} -6.74728e9i q^{89} +(3.15056e8 - 1.84052e9i) q^{90} +1.13036e10 q^{91} +(-3.63033e9 - 2.09597e9i) q^{92} +(3.76318e9 - 1.04202e10i) q^{93} +(-3.85675e9 - 6.68009e9i) q^{94} +(3.80759e8 - 2.19831e8i) q^{95} +(-1.35569e9 - 4.89597e8i) q^{96} +(-1.24083e8 + 2.14918e8i) q^{97} +3.80064e9i q^{98} +(1.01148e10 - 1.21774e10i) 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
\(3\) −156.662 − 185.758i −0.644700 − 0.764436i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) 968.329 + 5412.53i 0.124528 + 0.696055i
\(7\) −10611.8 + 18380.2i −0.631392 + 1.09360i 0.355876 + 0.934533i \(0.384183\pi\)
−0.987267 + 0.159069i \(0.949151\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −9962.94 + 58202.4i −0.168723 + 0.985663i
\(10\) −31622.8 −0.316228
\(11\) −232171. − 134044.i −1.44160 − 0.832307i −0.443642 − 0.896204i \(-0.646314\pi\)
−0.997957 + 0.0638965i \(0.979647\pi\)
\(12\) 42260.4 − 117019.i 0.169835 − 0.470272i
\(13\) −266298. − 461242.i −0.717218 − 1.24226i −0.962098 − 0.272704i \(-0.912082\pi\)
0.244880 − 0.969553i \(-0.421251\pi\)
\(14\) 415896. − 240118.i 0.773294 − 0.446461i
\(15\) −319412. − 115353.i −0.420624 − 0.151905i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 1.68584e6i 1.18733i 0.804712 + 0.593665i \(0.202320\pi\)
−0.804712 + 0.593665i \(0.797680\pi\)
\(18\) 853718. − 1.02781e6i 0.451806 − 0.543940i
\(19\) 314597. 0.127053 0.0635267 − 0.997980i \(-0.479765\pi\)
0.0635267 + 0.997980i \(0.479765\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) 5.07673e6 − 908253.i 1.24305 − 0.222388i
\(22\) 3.03307e6 + 5.25343e6i 0.588530 + 1.01936i
\(23\) −7.09048e6 + 4.09369e6i −1.10163 + 0.636027i −0.936649 − 0.350269i \(-0.886090\pi\)
−0.164983 + 0.986296i \(0.552757\pi\)
\(24\) −2.15205e6 + 1.81497e6i −0.270269 + 0.227936i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 1.20513e7i 1.01430i
\(27\) 1.23724e7 − 7.26742e6i 0.862252 − 0.506479i
\(28\) −1.08665e7 −0.631392
\(29\) −3.12690e7 − 1.80532e7i −1.52449 − 0.880163i −0.999579 − 0.0290037i \(-0.990767\pi\)
−0.524908 − 0.851159i \(-0.675900\pi\)
\(30\) 4.95409e6 + 5.87418e6i 0.203872 + 0.241736i
\(31\) 2.27961e7 + 3.94841e7i 0.796256 + 1.37916i 0.922038 + 0.387098i \(0.126523\pi\)
−0.125782 + 0.992058i \(0.540144\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 1.14727e7 + 6.41272e7i 0.293154 + 1.63860i
\(34\) 1.90731e7 − 3.30356e7i 0.419784 − 0.727088i
\(35\) 2.96609e7i 0.564734i
\(36\) −2.83578e7 + 1.04822e7i −0.468986 + 0.173357i
\(37\) 2.77674e7 0.400430 0.200215 − 0.979752i \(-0.435836\pi\)
0.200215 + 0.979752i \(0.435836\pi\)
\(38\) −6.16482e6 − 3.55926e6i −0.0778041 − 0.0449202i
\(39\) −4.39604e7 + 1.21726e8i −0.487235 + 1.34915i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −1.24774e8 + 7.20382e7i −1.07697 + 0.621790i −0.930078 − 0.367363i \(-0.880261\pi\)
−0.146893 + 0.989152i \(0.546927\pi\)
\(42\) −1.09759e8 − 3.96386e7i −0.839834 − 0.303299i
\(43\) 7.18757e7 − 1.24492e8i 0.488922 − 0.846838i −0.510996 − 0.859583i \(-0.670723\pi\)
0.999919 + 0.0127444i \(0.00405679\pi\)
\(44\) − 1.37261e8i − 0.832307i
\(45\) 2.86120e7 + 7.74047e7i 0.155055 + 0.419473i
\(46\) 1.85259e8 0.899479
\(47\) 2.95221e8 + 1.70446e8i 1.28724 + 0.743186i 0.978160 − 0.207852i \(-0.0666472\pi\)
0.309075 + 0.951038i \(0.399980\pi\)
\(48\) 6.27054e7 − 1.12183e7i 0.246093 − 0.0440272i
\(49\) −8.39830e7 − 1.45463e8i −0.297311 − 0.514958i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 3.13158e8 − 2.64107e8i 0.907637 − 0.765472i
\(52\) 1.36345e8 − 2.36156e8i 0.358609 − 0.621129i
\(53\) − 1.14407e8i − 0.273574i −0.990600 − 0.136787i \(-0.956322\pi\)
0.990600 − 0.136787i \(-0.0436776\pi\)
\(54\) −3.24670e8 + 2.43441e6i −0.707087 + 0.00530183i
\(55\) −3.74664e8 −0.744438
\(56\) 2.12939e8 + 1.22940e8i 0.386647 + 0.223231i
\(57\) −4.92854e7 − 5.84389e7i −0.0819114 − 0.0971242i
\(58\) 4.08496e8 + 7.07536e8i 0.622369 + 1.07798i
\(59\) 4.76249e7 − 2.74963e7i 0.0666154 − 0.0384604i −0.466322 − 0.884615i \(-0.654421\pi\)
0.532938 + 0.846154i \(0.321088\pi\)
\(60\) −3.06212e7 − 1.71159e8i −0.0393792 − 0.220112i
\(61\) 2.78802e8 − 4.82900e8i 0.330101 − 0.571752i −0.652430 − 0.757849i \(-0.726250\pi\)
0.982531 + 0.186097i \(0.0595838\pi\)
\(62\) − 1.03164e9i − 1.12608i
\(63\) −9.64046e8 − 8.00753e8i −0.971393 − 0.806856i
\(64\) −1.34218e8 −0.125000
\(65\) −6.44605e8 − 3.72163e8i −0.555554 − 0.320750i
\(66\) 5.00698e8 − 1.38643e9i 0.399813 − 1.10708i
\(67\) 8.59232e8 + 1.48823e9i 0.636409 + 1.10229i 0.986215 + 0.165470i \(0.0529143\pi\)
−0.349806 + 0.936822i \(0.613752\pi\)
\(68\) −7.47509e8 + 4.31575e8i −0.514129 + 0.296832i
\(69\) 1.87124e9 + 6.75786e8i 1.19642 + 0.432079i
\(70\) 3.35575e8 − 5.81232e8i 0.199664 − 0.345827i
\(71\) − 2.29652e9i − 1.27285i −0.771338 − 0.636426i \(-0.780412\pi\)
0.771338 − 0.636426i \(-0.219588\pi\)
\(72\) 6.74289e8 + 1.15423e8i 0.348485 + 0.0596527i
\(73\) −2.42924e9 −1.17181 −0.585904 − 0.810380i \(-0.699260\pi\)
−0.585904 + 0.810380i \(0.699260\pi\)
\(74\) −5.44127e8 − 3.14152e8i −0.245212 − 0.141573i
\(75\) −4.67192e8 + 8.35830e7i −0.196874 + 0.0352218i
\(76\) 8.05368e7 + 1.39494e8i 0.0317634 + 0.0550158i
\(77\) 4.92750e9 − 2.84490e9i 1.82043 − 1.05102i
\(78\) 2.23862e9 − 1.88798e9i 0.775366 − 0.653919i
\(79\) 1.81753e9 − 3.14806e9i 0.590672 − 1.02307i −0.403470 − 0.914993i \(-0.632196\pi\)
0.994142 − 0.108081i \(-0.0344706\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −3.28826e9 − 1.15974e9i −0.943065 − 0.332609i
\(82\) 3.26008e9 0.879343
\(83\) 4.51472e9 + 2.60657e9i 1.14615 + 0.661728i 0.947945 − 0.318433i \(-0.103157\pi\)
0.198202 + 0.980161i \(0.436490\pi\)
\(84\) 1.70237e9 + 2.01853e9i 0.407058 + 0.482658i
\(85\) 1.17802e9 + 2.04038e9i 0.265495 + 0.459851i
\(86\) −2.81694e9 + 1.62636e9i −0.598805 + 0.345720i
\(87\) 1.54515e9 + 8.63670e9i 0.310009 + 1.73281i
\(88\) −1.55293e9 + 2.68976e9i −0.294265 + 0.509682i
\(89\) − 6.74728e9i − 1.20831i −0.796867 − 0.604155i \(-0.793511\pi\)
0.796867 − 0.604155i \(-0.206489\pi\)
\(90\) 3.15056e8 − 1.84052e9i 0.0533550 − 0.311694i
\(91\) 1.13036e10 1.81138
\(92\) −3.63033e9 − 2.09597e9i −0.550816 − 0.318014i
\(93\) 3.76318e9 − 1.04202e10i 0.540929 − 1.49783i
\(94\) −3.85675e9 − 6.68009e9i −0.525512 − 0.910213i
\(95\) 3.80759e8 − 2.19831e8i 0.0492076 − 0.0284100i
\(96\) −1.35569e9 − 4.89597e8i −0.166266 − 0.0600458i
\(97\) −1.24083e8 + 2.14918e8i −0.0144495 + 0.0250273i −0.873160 − 0.487434i \(-0.837933\pi\)
0.858710 + 0.512461i \(0.171266\pi\)
\(98\) 3.80064e9i 0.420461i
\(99\) 1.01148e10 − 1.21774e10i 1.06361 − 1.28050i
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.11.6 ✓ 80
3.2 odd 2 270.11.h.a.251.37 80
9.4 even 3 270.11.h.a.71.37 80
9.5 odd 6 inner 90.11.h.a.41.6 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.6 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.6 yes 80 9.5 odd 6 inner
270.11.h.a.71.37 80 9.4 even 3
270.11.h.a.251.37 80 3.2 odd 2