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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 41.6
Character \(\chi\) \(=\) 90.41
Dual form 90.11.h.a.11.6

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

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

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
\(3\) −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 −0.987267 − 0.159069i \(-0.949151\pi\)
0.355876 − 0.934533i \(-0.384183\pi\)
\(8\) 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 −0.997957 − 0.0638965i \(-0.979647\pi\)
−0.443642 + 0.896204i \(0.646314\pi\)
\(12\) 42260.4 + 117019.i 0.169835 + 0.470272i
\(13\) −266298. + 461242.i −0.717218 + 1.24226i 0.244880 + 0.969553i \(0.421251\pi\)
−0.962098 + 0.272704i \(0.912082\pi\)
\(14\) 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 \(-0.797680\pi\)
0.804712 − 0.593665i \(-0.202320\pi\)
\(18\) 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 −0.164983 − 0.986296i \(-0.552757\pi\)
−0.936649 + 0.350269i \(0.886090\pi\)
\(24\) −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 −0.524908 + 0.851159i \(0.675900\pi\)
−0.999579 + 0.0290037i \(0.990767\pi\)
\(30\) 4.95409e6 − 5.87418e6i 0.203872 − 0.241736i
\(31\) 2.27961e7 − 3.94841e7i 0.796256 − 1.37916i −0.125782 − 0.992058i \(-0.540144\pi\)
0.922038 − 0.387098i \(-0.126523\pi\)
\(32\) 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 −0.146893 − 0.989152i \(-0.546927\pi\)
−0.930078 + 0.367363i \(0.880261\pi\)
\(42\) −1.09759e8 + 3.96386e7i −0.839834 + 0.303299i
\(43\) 7.18757e7 + 1.24492e8i 0.488922 + 0.846838i 0.999919 − 0.0127444i \(-0.00405679\pi\)
−0.510996 + 0.859583i \(0.670723\pi\)
\(44\) 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 0.309075 − 0.951038i \(-0.399980\pi\)
0.978160 + 0.207852i \(0.0666472\pi\)
\(48\) 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 \(0.0436776\pi\)
−0.990600 + 0.136787i \(0.956322\pi\)
\(54\) −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 0.532938 − 0.846154i \(-0.321088\pi\)
−0.466322 + 0.884615i \(0.654421\pi\)
\(60\) −3.06212e7 + 1.71159e8i −0.0393792 + 0.220112i
\(61\) 2.78802e8 + 4.82900e8i 0.330101 + 0.571752i 0.982531 − 0.186097i \(-0.0595838\pi\)
−0.652430 + 0.757849i \(0.726250\pi\)
\(62\) 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 −0.349806 − 0.936822i \(-0.613752\pi\)
0.986215 − 0.165470i \(-0.0529143\pi\)
\(68\) −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 \(0.219588\pi\)
−0.771338 + 0.636426i \(0.780412\pi\)
\(72\) 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 0.994142 + 0.108081i \(0.0344706\pi\)
−0.403470 + 0.914993i \(0.632196\pi\)
\(80\) − 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 0.198202 − 0.980161i \(-0.436490\pi\)
0.947945 + 0.318433i \(0.103157\pi\)
\(84\) 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 \(0.206489\pi\)
−0.796867 + 0.604155i \(0.793511\pi\)
\(90\) 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 0.858710 − 0.512461i \(-0.171266\pi\)
−0.873160 + 0.487434i \(0.837933\pi\)
\(98\) − 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.41.6 yes 80
3.2 odd 2 270.11.h.a.71.37 80
9.2 odd 6 inner 90.11.h.a.11.6 ✓ 80
9.7 even 3 270.11.h.a.251.37 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.6 ✓ 80 9.2 odd 6 inner
90.11.h.a.41.6 yes 80 1.1 even 1 trivial
270.11.h.a.71.37 80 3.2 odd 2
270.11.h.a.251.37 80 9.7 even 3