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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 11.9
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.9

$q$-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} +(-57.9788 - 235.982i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(-1533.68 + 5280.24i) q^{6} +(4212.63 - 7296.48i) q^{7} -11585.2i q^{8} +(-52325.9 + 27363.9i) q^{9} -31622.8 q^{10} +(148786. + 85901.5i) q^{11} +(89793.0 - 86119.4i) q^{12} +(-63226.6 - 109512. i) q^{13} +(-165101. + 95320.9i) q^{14} +(-235069. - 245097. i) q^{15} +(-131072. + 227023. i) q^{16} -528692. i q^{17} +(1.33496e6 + 55780.1i) q^{18} +966945. q^{19} +(619677. + 357771. i) q^{20} +(-1.96608e6 - 571063. i) q^{21} +(-1.94373e6 - 3.36664e6i) q^{22} +(2.33452e6 - 1.34783e6i) q^{23} +(-2.73391e6 + 671698. i) q^{24} +(976562. - 1.69146e6i) q^{25} +2.86131e6i q^{26} +(9.49117e6 + 1.07614e7i) q^{27} +4.31373e6 q^{28} +(3.34260e7 + 1.92985e7i) q^{29} +(1.83345e6 + 7.46240e6i) q^{30} +(8.73537e6 + 1.51301e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(1.16448e7 - 4.00912e7i) q^{33} +(-5.98147e6 + 1.03602e7i) q^{34} -1.17747e7i q^{35} +(-2.55287e7 - 1.61964e7i) q^{36} +1.28931e8 q^{37} +(-1.89482e7 - 1.09397e7i) q^{38} +(-2.21770e7 + 2.12697e7i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-2.62381e7 + 1.51486e7i) q^{41} +(3.20663e7 + 3.34342e7i) q^{42} +(-5.90315e6 + 1.02246e7i) q^{43} +8.79631e7i q^{44} +(-4.42094e7 + 6.96826e7i) q^{45} -6.09960e7 q^{46} +(-3.99322e7 - 2.30548e7i) q^{47} +(6.11728e7 + 1.77681e7i) q^{48} +(1.05745e8 + 1.83156e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-1.24762e8 + 3.06529e7i) q^{51} +(3.23720e7 - 5.60700e7i) q^{52} -3.79949e8i q^{53} +(-6.42363e7 - 3.18261e8i) q^{54} +2.40102e8 q^{55} +(-8.45315e7 - 4.88043e7i) q^{56} +(-5.60622e7 - 2.28181e8i) q^{57} +(-4.36675e8 - 7.56344e8i) q^{58} +(-2.20119e8 + 1.27086e8i) q^{59} +(4.84993e7 - 1.66976e8i) q^{60} +(8.41880e7 - 1.45818e8i) q^{61} -3.95318e8i q^{62} +(-2.07696e7 + 4.97069e8i) q^{63} -1.34218e8 q^{64} +(-1.53047e8 - 8.83619e7i) q^{65} +(-6.81770e8 + 6.53878e8i) q^{66} +(-1.17274e9 - 2.03125e9i) q^{67} +(2.34425e8 - 1.35345e8i) q^{68} +(-4.53417e8 - 4.72758e8i) q^{69} +(-1.33215e8 + 2.30735e8i) q^{70} +2.40656e8i q^{71} +(3.17017e8 + 6.06208e8i) q^{72} +2.33989e9 q^{73} +(-2.52651e9 - 1.45868e9i) q^{74} +(-4.55773e8 - 1.32383e8i) q^{75} +(2.47538e8 + 4.28748e8i) q^{76} +(1.25356e9 - 7.23742e8i) q^{77} +(6.75218e8 - 1.65895e8i) q^{78} +(8.92681e8 - 1.54617e9i) q^{79} +3.66357e8i q^{80} +(1.98922e9 - 2.86368e9i) q^{81} +6.85546e8 q^{82} +(-5.56659e9 - 3.21387e9i) q^{83} +(-2.50105e8 - 1.01796e9i) q^{84} +(-3.69435e8 - 6.39880e8i) q^{85} +(2.31355e8 - 1.33573e8i) q^{86} +(2.61610e9 - 9.00683e9i) q^{87} +(9.95189e8 - 1.72372e9i) q^{88} -4.75295e9i q^{89} +(1.65469e9 - 8.65322e8i) q^{90} -1.06540e9 q^{91} +(1.19527e9 + 6.90091e8i) q^{92} +(3.06396e9 - 2.93861e9i) q^{93} +(5.21672e8 + 9.03562e8i) q^{94} +(1.17030e9 - 6.75673e8i) q^{95} +(-9.97714e8 - 1.04027e9i) q^{96} +(-1.29798e9 + 2.24816e9i) q^{97} -4.78548e9i q^{98} +(-1.01359e10 - 4.23521e8i) 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
\(3\) −57.9788 − 235.982i −0.238596 − 0.971119i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) −1533.68 + 5280.24i −0.197233 + 0.679043i
\(7\) 4212.63 − 7296.48i 0.250647 − 0.434134i −0.713057 − 0.701106i \(-0.752690\pi\)
0.963704 + 0.266972i \(0.0860233\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −52325.9 + 27363.9i −0.886144 + 0.463410i
\(10\) −31622.8 −0.316228
\(11\) 148786. + 85901.5i 0.923842 + 0.533380i 0.884859 − 0.465859i \(-0.154255\pi\)
0.0389833 + 0.999240i \(0.487588\pi\)
\(12\) 89793.0 − 86119.4i 0.360858 − 0.346095i
\(13\) −63226.6 − 109512.i −0.170288 − 0.294947i 0.768233 − 0.640171i \(-0.221136\pi\)
−0.938520 + 0.345224i \(0.887803\pi\)
\(14\) −165101. + 95320.9i −0.306979 + 0.177234i
\(15\) −235069. − 245097.i −0.309557 − 0.322761i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 528692.i − 0.372356i −0.982516 − 0.186178i \(-0.940390\pi\)
0.982516 − 0.186178i \(-0.0596101\pi\)
\(18\) 1.33496e6 + 55780.1i 0.706490 + 0.0295200i
\(19\) 966945. 0.390511 0.195256 − 0.980752i \(-0.437446\pi\)
0.195256 + 0.980752i \(0.437446\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) −1.96608e6 − 571063.i −0.481399 − 0.139826i
\(22\) −1.94373e6 − 3.36664e6i −0.377157 − 0.653255i
\(23\) 2.33452e6 − 1.34783e6i 0.362709 − 0.209410i −0.307560 − 0.951529i \(-0.599512\pi\)
0.670268 + 0.742119i \(0.266179\pi\)
\(24\) −2.73391e6 + 671698.i −0.343342 + 0.0843563i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 2.86131e6i 0.240823i
\(27\) 9.49117e6 + 1.07614e7i 0.661456 + 0.749984i
\(28\) 4.31373e6 0.250647
\(29\) 3.34260e7 + 1.92985e7i 1.62965 + 0.940879i 0.984197 + 0.177079i \(0.0566649\pi\)
0.645453 + 0.763800i \(0.276668\pi\)
\(30\) 1.83345e6 + 7.46240e6i 0.0754506 + 0.307095i
\(31\) 8.73537e6 + 1.51301e7i 0.305121 + 0.528486i 0.977288 − 0.211914i \(-0.0679696\pi\)
−0.672167 + 0.740400i \(0.734636\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 1.16448e7 − 4.00912e7i 0.297551 − 1.02442i
\(34\) −5.98147e6 + 1.03602e7i −0.131648 + 0.228020i
\(35\) − 1.17747e7i − 0.224186i
\(36\) −2.55287e7 − 1.61964e7i −0.422198 − 0.267859i
\(37\) 1.28931e8 1.85929 0.929646 − 0.368454i \(-0.120113\pi\)
0.929646 + 0.368454i \(0.120113\pi\)
\(38\) −1.89482e7 − 1.09397e7i −0.239138 − 0.138067i
\(39\) −2.21770e7 + 2.12697e7i −0.245799 + 0.235743i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −2.62381e7 + 1.51486e7i −0.226471 + 0.130753i −0.608943 − 0.793214i \(-0.708406\pi\)
0.382472 + 0.923967i \(0.375073\pi\)
\(42\) 3.20663e7 + 3.34342e7i 0.245359 + 0.255826i
\(43\) −5.90315e6 + 1.02246e7i −0.0401552 + 0.0695509i −0.885405 − 0.464821i \(-0.846119\pi\)
0.845249 + 0.534372i \(0.179452\pi\)
\(44\) 8.79631e7i 0.533380i
\(45\) −4.42094e7 + 6.96826e7i −0.239581 + 0.377626i
\(46\) −6.09960e7 −0.296150
\(47\) −3.99322e7 − 2.30548e7i −0.174114 − 0.100525i 0.410410 − 0.911901i \(-0.365385\pi\)
−0.584524 + 0.811376i \(0.698719\pi\)
\(48\) 6.11728e7 + 1.77681e7i 0.240078 + 0.0697324i
\(49\) 1.05745e8 + 1.83156e8i 0.374352 + 0.648397i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −1.24762e8 + 3.06529e7i −0.361602 + 0.0888425i
\(52\) 3.23720e7 − 5.60700e7i 0.0851439 − 0.147474i
\(53\) − 3.79949e8i − 0.908543i −0.890863 − 0.454272i \(-0.849900\pi\)
0.890863 − 0.454272i \(-0.150100\pi\)
\(54\) −6.42363e7 − 3.18261e8i −0.139898 − 0.693130i
\(55\) 2.40102e8 0.477070
\(56\) −8.45315e7 − 4.88043e7i −0.153489 − 0.0886172i
\(57\) −5.60622e7 − 2.28181e8i −0.0931743 − 0.379233i
\(58\) −4.36675e8 − 7.56344e8i −0.665302 − 1.15234i
\(59\) −2.20119e8 + 1.27086e8i −0.307891 + 0.177761i −0.645982 − 0.763352i \(-0.723552\pi\)
0.338091 + 0.941113i \(0.390219\pi\)
\(60\) 4.84993e7 − 1.66976e8i 0.0623705 − 0.214732i
\(61\) 8.41880e7 − 1.45818e8i 0.0996783 − 0.172648i −0.811873 − 0.583834i \(-0.801552\pi\)
0.911552 + 0.411186i \(0.134885\pi\)
\(62\) − 3.95318e8i − 0.431507i
\(63\) −2.07696e7 + 4.97069e8i −0.0209278 + 0.500857i
\(64\) −1.34218e8 −0.125000
\(65\) −1.53047e8 − 8.83619e7i −0.131904 − 0.0761550i
\(66\) −6.81770e8 + 6.53878e8i −0.544400 + 0.522128i
\(67\) −1.17274e9 − 2.03125e9i −0.868619 − 1.50449i −0.863409 − 0.504505i \(-0.831675\pi\)
−0.00520989 − 0.999986i \(-0.501658\pi\)
\(68\) 2.34425e8 − 1.35345e8i 0.161235 − 0.0930889i
\(69\) −4.53417e8 − 4.72758e8i −0.289903 − 0.302269i
\(70\) −1.33215e8 + 2.30735e8i −0.0792616 + 0.137285i
\(71\) 2.40656e8i 0.133384i 0.997774 + 0.0666921i \(0.0212445\pi\)
−0.997774 + 0.0666921i \(0.978755\pi\)
\(72\) 3.17017e8 + 6.06208e8i 0.163840 + 0.313299i
\(73\) 2.33989e9 1.12871 0.564353 − 0.825533i \(-0.309126\pi\)
0.564353 + 0.825533i \(0.309126\pi\)
\(74\) −2.52651e9 − 1.45868e9i −1.13858 − 0.657359i
\(75\) −4.55773e8 − 1.32383e8i −0.192062 − 0.0557859i
\(76\) 2.47538e8 + 4.28748e8i 0.0976278 + 0.169096i
\(77\) 1.25356e9 − 7.23742e8i 0.463117 − 0.267381i
\(78\) 6.75218e8 − 1.65895e8i 0.233868 − 0.0574594i
\(79\) 8.92681e8 − 1.54617e9i 0.290109 − 0.502483i −0.683726 − 0.729738i \(-0.739642\pi\)
0.973835 + 0.227255i \(0.0729751\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 1.98922e9 − 2.86368e9i 0.570503 − 0.821295i
\(82\) 6.85546e8 0.184913
\(83\) −5.56659e9 − 3.21387e9i −1.41318 − 0.815902i −0.417497 − 0.908678i \(-0.637093\pi\)
−0.995687 + 0.0927760i \(0.970426\pi\)
\(84\) −2.50105e8 − 1.01796e9i −0.0598033 − 0.243408i
\(85\) −3.69435e8 − 6.39880e8i −0.0832613 − 0.144213i
\(86\) 2.31355e8 − 1.33573e8i 0.0491799 − 0.0283940i
\(87\) 2.61610e9 − 9.00683e9i 0.524878 − 1.80707i
\(88\) 9.95189e8 − 1.72372e9i 0.188578 − 0.326628i
\(89\) − 4.75295e9i − 0.851165i −0.904920 − 0.425582i \(-0.860069\pi\)
0.904920 − 0.425582i \(-0.139931\pi\)
\(90\) 1.65469e9 − 8.65322e8i 0.280223 − 0.146543i
\(91\) −1.06540e9 −0.170729
\(92\) 1.19527e9 + 6.90091e8i 0.181354 + 0.104705i
\(93\) 3.06396e9 − 2.93861e9i 0.440422 − 0.422404i
\(94\) 5.21672e8 + 9.03562e8i 0.0710817 + 0.123117i
\(95\) 1.17030e9 − 6.75673e8i 0.151244 − 0.0873210i
\(96\) −9.97714e8 − 1.04027e9i −0.122363 − 0.127583i
\(97\) −1.29798e9 + 2.24816e9i −0.151150 + 0.261799i −0.931650 − 0.363356i \(-0.881631\pi\)
0.780501 + 0.625155i \(0.214964\pi\)
\(98\) − 4.78548e9i − 0.529414i
\(99\) −1.01359e10 − 4.23521e8i −1.06583 − 0.0445347i
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.11.9 ✓ 80
3.2 odd 2 270.11.h.a.251.24 80
9.4 even 3 270.11.h.a.71.24 80
9.5 odd 6 inner 90.11.h.a.41.9 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.9 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.9 yes 80 9.5 odd 6 inner
270.11.h.a.71.24 80 9.4 even 3
270.11.h.a.251.24 80 3.2 odd 2