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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 11.2
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.2

$q$-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} +(-242.497 - 15.6231i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(4575.20 + 3049.69i) q^{6} +(-2506.88 + 4342.04i) q^{7} -11585.2i q^{8} +(58560.8 + 7577.13i) q^{9} -31622.8 q^{10} +(60006.9 + 34645.0i) q^{11} +(-55151.9 - 111524. i) q^{12} +(224532. + 388901. i) q^{13} +(98249.1 - 56724.2i) q^{14} +(-304413. + 150541. i) q^{15} +(-131072. + 227023. i) q^{16} -1.64486e6i q^{17} +(-1.06183e6 - 811021. i) q^{18} +3.31204e6 q^{19} +(619677. + 357771. i) q^{20} +(675747. - 1.01377e6i) q^{21} +(-783927. - 1.35780e6i) q^{22} +(-9.48337e6 + 5.47523e6i) q^{23} +(-180998. + 2.80939e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.01612e7i q^{26} +(-1.40825e7 - 2.75234e6i) q^{27} -2.56704e6 q^{28} +(-7.37674e6 - 4.25896e6i) q^{29} +(7.66844e6 + 494046. i) q^{30} +(-2.15667e7 - 3.73546e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-1.40102e7 - 9.33881e6i) q^{33} +(-1.86094e7 + 3.22325e7i) q^{34} +7.00694e6i q^{35} +(1.16318e7 + 2.79059e7i) q^{36} +2.14552e7 q^{37} +(-6.49025e7 - 3.74715e7i) q^{38} +(-4.83726e7 - 9.78154e7i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-6.46878e7 + 3.73475e7i) q^{41} +(-2.47114e7 + 1.22205e7i) q^{42} +(-6.66001e7 + 1.15355e8i) q^{43} +3.54765e7i q^{44} +(7.61713e7 - 3.17500e7i) q^{45} +2.47780e8 q^{46} +(3.85643e7 + 2.22651e7i) q^{47} +(3.53314e7 - 5.30048e7i) q^{48} +(1.28669e8 + 2.22861e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-2.56978e7 + 3.98873e8i) q^{51} +(-1.14961e8 + 1.99117e8i) q^{52} -1.80813e8i q^{53} +(2.44820e8 + 2.13259e8i) q^{54} +9.68357e7 q^{55} +(5.03036e7 + 2.90428e7i) q^{56} +(-8.03161e8 - 5.17444e7i) q^{57} +(9.63693e7 + 1.66917e8i) q^{58} +(8.13925e8 - 4.69920e8i) q^{59} +(-1.44681e8 - 9.64397e7i) q^{60} +(4.32383e8 - 7.48909e8i) q^{61} +9.75996e8i q^{62} +(-1.79705e8 + 2.35279e8i) q^{63} -1.34218e8 q^{64} +(5.43506e8 + 3.13793e8i) q^{65} +(1.68887e8 + 3.41510e8i) q^{66} +(7.45279e8 + 1.29086e9i) q^{67} +(7.29338e8 - 4.21083e8i) q^{68} +(2.38523e9 - 1.17957e9i) q^{69} +(7.92744e7 - 1.37307e8i) q^{70} +2.18443e8i q^{71} +(8.77828e7 - 6.78441e8i) q^{72} +9.96051e8 q^{73} +(-4.20435e8 - 2.42738e8i) q^{74} +(-2.63240e8 + 3.94916e8i) q^{75} +(8.47882e8 + 1.46858e9i) q^{76} +(-3.00860e8 + 1.73702e8i) q^{77} +(-1.58749e8 + 2.46406e9i) q^{78} +(-2.23633e9 + 3.87344e9i) q^{79} +3.66357e8i q^{80} +(3.37196e9 + 8.87446e8i) q^{81} +1.69016e9 q^{82} +(4.67623e7 + 2.69983e7i) q^{83} +(6.22501e8 + 4.01052e7i) q^{84} +(-1.14938e9 - 1.99078e9i) q^{85} +(2.61018e9 - 1.50699e9i) q^{86} +(1.72230e9 + 1.14803e9i) q^{87} +(4.01371e8 - 6.95194e8i) q^{88} +2.21274e9i q^{89} +(-1.85186e9 - 2.39610e8i) q^{90} -2.25150e9 q^{91} +(-4.85548e9 - 2.80332e9i) q^{92} +(4.64627e9 + 9.39532e9i) q^{93} +(-5.03802e8 - 8.72611e8i) q^{94} +(4.00859e9 - 2.31436e9i) q^{95} +(-1.29203e9 + 6.38948e8i) q^{96} +(-3.09307e9 + 5.35735e9i) q^{97} -5.82288e9i q^{98} +(3.25154e9 + 2.48352e9i) 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
\(3\) −242.497 − 15.6231i −0.997931 − 0.0642927i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) 4575.20 + 3049.69i 0.588375 + 0.392193i
\(7\) −2506.88 + 4342.04i −0.149157 + 0.258347i −0.930916 − 0.365233i \(-0.880989\pi\)
0.781759 + 0.623580i \(0.214323\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 58560.8 + 7577.13i 0.991733 + 0.128319i
\(10\) −31622.8 −0.316228
\(11\) 60006.9 + 34645.0i 0.372596 + 0.215118i 0.674592 − 0.738191i \(-0.264320\pi\)
−0.301996 + 0.953309i \(0.597653\pi\)
\(12\) −55151.9 − 111524.i −0.221643 − 0.448190i
\(13\) 224532. + 388901.i 0.604731 + 1.04742i 0.992094 + 0.125497i \(0.0400526\pi\)
−0.387363 + 0.921927i \(0.626614\pi\)
\(14\) 98249.1 − 56724.2i 0.182679 − 0.105470i
\(15\) −304413. + 150541.i −0.400873 + 0.198244i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 1.64486e6i − 1.15847i −0.815162 − 0.579233i \(-0.803352\pi\)
0.815162 − 0.579233i \(-0.196648\pi\)
\(18\) −1.06183e6 − 811021.i −0.561942 − 0.429210i
\(19\) 3.31204e6 1.33760 0.668802 − 0.743440i \(-0.266807\pi\)
0.668802 + 0.743440i \(0.266807\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) 675747. − 1.01377e6i 0.165458 − 0.248223i
\(22\) −783927. − 1.35780e6i −0.152112 − 0.263465i
\(23\) −9.48337e6 + 5.47523e6i −1.47341 + 0.850673i −0.999552 − 0.0299283i \(-0.990472\pi\)
−0.473857 + 0.880602i \(0.657139\pi\)
\(24\) −180998. + 2.80939e6i −0.0227309 + 0.352822i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.01612e7i − 0.855218i
\(27\) −1.40825e7 − 2.75234e6i −0.981431 − 0.191815i
\(28\) −2.56704e6 −0.149157
\(29\) −7.37674e6 − 4.25896e6i −0.359645 − 0.207641i 0.309280 − 0.950971i \(-0.399912\pi\)
−0.668925 + 0.743330i \(0.733245\pi\)
\(30\) 7.66844e6 + 494046.i 0.315574 + 0.0203311i
\(31\) −2.15667e7 − 3.73546e7i −0.753312 − 1.30477i −0.946209 − 0.323555i \(-0.895122\pi\)
0.192897 − 0.981219i \(-0.438212\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −1.40102e7 − 9.33881e6i −0.357994 − 0.238628i
\(34\) −1.86094e7 + 3.22325e7i −0.409580 + 0.709413i
\(35\) 7.00694e6i 0.133410i
\(36\) 1.16318e7 + 2.79059e7i 0.192369 + 0.461513i
\(37\) 2.14552e7 0.309403 0.154702 − 0.987961i \(-0.450558\pi\)
0.154702 + 0.987961i \(0.450558\pi\)
\(38\) −6.49025e7 − 3.74715e7i −0.819112 − 0.472914i
\(39\) −4.83726e7 − 9.78154e7i −0.536138 − 1.08414i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −6.46878e7 + 3.73475e7i −0.558346 + 0.322361i −0.752481 − 0.658614i \(-0.771143\pi\)
0.194136 + 0.980975i \(0.437810\pi\)
\(42\) −2.47114e7 + 1.22205e7i −0.189082 + 0.0935066i
\(43\) −6.66001e7 + 1.15355e8i −0.453036 + 0.784681i −0.998573 − 0.0534054i \(-0.982992\pi\)
0.545537 + 0.838087i \(0.316326\pi\)
\(44\) 3.54765e7i 0.215118i
\(45\) 7.61713e7 − 3.17500e7i 0.412790 − 0.172060i
\(46\) 2.47780e8 1.20303
\(47\) 3.85643e7 + 2.22651e7i 0.168150 + 0.0970814i 0.581713 − 0.813394i \(-0.302383\pi\)
−0.413563 + 0.910475i \(0.635716\pi\)
\(48\) 3.53314e7 − 5.30048e7i 0.138661 − 0.208022i
\(49\) 1.28669e8 + 2.22861e8i 0.455505 + 0.788957i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −2.56978e7 + 3.98873e8i −0.0744809 + 1.15607i
\(52\) −1.14961e8 + 1.99117e8i −0.302365 + 0.523712i
\(53\) − 1.80813e8i − 0.432364i −0.976353 − 0.216182i \(-0.930639\pi\)
0.976353 − 0.216182i \(-0.0693605\pi\)
\(54\) 2.44820e8 + 2.13259e8i 0.533185 + 0.464450i
\(55\) 9.68357e7 0.192408
\(56\) 5.03036e7 + 2.90428e7i 0.0913395 + 0.0527349i
\(57\) −8.03161e8 − 5.17444e7i −1.33484 − 0.0859981i
\(58\) 9.63693e7 + 1.66917e8i 0.146825 + 0.254308i
\(59\) 8.13925e8 − 4.69920e8i 1.13848 − 0.657300i 0.192424 − 0.981312i \(-0.438365\pi\)
0.946053 + 0.324012i \(0.105032\pi\)
\(60\) −1.44681e8 − 9.64397e7i −0.186060 − 0.124022i
\(61\) 4.32383e8 − 7.48909e8i 0.511940 − 0.886706i −0.487964 − 0.872864i \(-0.662260\pi\)
0.999904 − 0.0138427i \(-0.00440641\pi\)
\(62\) 9.75996e8i 1.06534i
\(63\) −1.79705e8 + 2.35279e8i −0.181075 + 0.237072i
\(64\) −1.34218e8 −0.125000
\(65\) 5.43506e8 + 3.13793e8i 0.468422 + 0.270444i
\(66\) 1.68887e8 + 3.41510e8i 0.134858 + 0.272699i
\(67\) 7.45279e8 + 1.29086e9i 0.552008 + 0.956105i 0.998130 + 0.0611330i \(0.0194714\pi\)
−0.446122 + 0.894972i \(0.647195\pi\)
\(68\) 7.29338e8 − 4.21083e8i 0.501631 − 0.289617i
\(69\) 2.38523e9 − 1.17957e9i 1.52505 − 0.754184i
\(70\) 7.92744e7 − 1.37307e8i 0.0471675 − 0.0816965i
\(71\) 2.18443e8i 0.121073i 0.998166 + 0.0605364i \(0.0192811\pi\)
−0.998166 + 0.0605364i \(0.980719\pi\)
\(72\) 8.77828e7 − 6.78441e8i 0.0453677 − 0.350631i
\(73\) 9.96051e8 0.480471 0.240236 − 0.970715i \(-0.422775\pi\)
0.240236 + 0.970715i \(0.422775\pi\)
\(74\) −4.20435e8 − 2.42738e8i −0.189470 − 0.109391i
\(75\) −2.63240e8 + 3.94916e8i −0.110929 + 0.166417i
\(76\) 8.47882e8 + 1.46858e9i 0.334401 + 0.579200i
\(77\) −3.00860e8 + 1.73702e8i −0.111150 + 0.0641727i
\(78\) −1.58749e8 + 2.46406e9i −0.0549843 + 0.853449i
\(79\) −2.23633e9 + 3.87344e9i −0.726777 + 1.25881i 0.231462 + 0.972844i \(0.425649\pi\)
−0.958238 + 0.285970i \(0.907684\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 3.37196e9 + 8.87446e8i 0.967068 + 0.254517i
\(82\) 1.69016e9 0.455887
\(83\) 4.67623e7 + 2.69983e7i 0.0118715 + 0.00685402i 0.505924 − 0.862578i \(-0.331152\pi\)
−0.494053 + 0.869432i \(0.664485\pi\)
\(84\) 6.22501e8 + 4.01052e7i 0.148848 + 0.00958969i
\(85\) −1.14938e9 − 1.99078e9i −0.259041 − 0.448672i
\(86\) 2.61018e9 − 1.50699e9i 0.554854 − 0.320345i
\(87\) 1.72230e9 + 1.14803e9i 0.345551 + 0.230334i
\(88\) 4.01371e8 − 6.95194e8i 0.0760558 − 0.131732i
\(89\) 2.21274e9i 0.396261i 0.980176 + 0.198130i \(0.0634869\pi\)
−0.980176 + 0.198130i \(0.936513\pi\)
\(90\) −1.85186e9 − 2.39610e8i −0.313613 − 0.0405781i
\(91\) −2.25150e9 −0.360799
\(92\) −4.85548e9 − 2.80332e9i −0.736705 − 0.425337i
\(93\) 4.64627e9 + 9.39532e9i 0.667866 + 1.35051i
\(94\) −5.03802e8 − 8.72611e8i −0.0686469 − 0.118900i
\(95\) 4.00859e9 − 2.31436e9i 0.518052 − 0.299097i
\(96\) −1.29203e9 + 6.38948e8i −0.158459 + 0.0783627i
\(97\) −3.09307e9 + 5.35735e9i −0.360189 + 0.623866i −0.987992 − 0.154507i \(-0.950621\pi\)
0.627803 + 0.778373i \(0.283955\pi\)
\(98\) − 5.82288e9i − 0.644181i
\(99\) 3.25154e9 + 2.48352e9i 0.341912 + 0.261151i
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.11.2 ✓ 80
3.2 odd 2 270.11.h.a.251.26 80
9.4 even 3 270.11.h.a.71.26 80
9.5 odd 6 inner 90.11.h.a.41.2 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.2 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.2 yes 80 9.5 odd 6 inner
270.11.h.a.71.26 80 9.4 even 3
270.11.h.a.251.26 80 3.2 odd 2