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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 41.15
Character \(\chi\) \(=\) 90.41
Dual form 90.11.h.a.11.15

$q$-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} +(169.897 + 173.735i) q^{3} +(256.000 - 443.405i) q^{4} +(1210.31 + 698.771i) q^{5} +(-5294.88 - 1482.33i) q^{6} +(-13757.7 - 23829.0i) q^{7} +11585.2i q^{8} +(-1318.83 + 59034.3i) q^{9} -31622.8 q^{10} +(241492. - 139425. i) q^{11} +(120529. - 30857.1i) q^{12} +(-54722.5 + 94782.1i) q^{13} +(539189. + 311301. i) q^{14} +(84226.8 + 328992. i) q^{15} +(-131072. - 227023. i) q^{16} -253377. i q^{17} +(-642053. - 1.17175e6i) q^{18} -4.61535e6 q^{19} +(619677. - 357771. i) q^{20} +(1.80255e6 - 6.43868e6i) q^{21} +(-3.15484e6 + 5.46434e6i) q^{22} +(-1.38985e6 - 802432. i) q^{23} +(-2.01276e6 + 1.96830e6i) q^{24} +(976562. + 1.69146e6i) q^{25} -2.47646e6i q^{26} +(-1.04804e7 + 9.80063e6i) q^{27} -1.40879e7 q^{28} +(-1.52112e7 + 8.78220e6i) q^{29} +(-5.37262e6 - 5.49399e6i) q^{30} +(-1.21626e7 + 2.10662e7i) q^{31} +(5.13695e6 + 2.96582e6i) q^{32} +(6.52519e7 + 1.82676e7i) q^{33} +(2.86664e6 + 4.96516e6i) q^{34} -3.84539e7i q^{35} +(2.58385e7 + 1.56976e7i) q^{36} -2.74413e7 q^{37} +(9.04421e7 - 5.22168e7i) q^{38} +(-2.57642e7 + 6.59600e6i) q^{39} +(-8.09543e6 + 1.40217e7i) q^{40} +(-9.92296e7 - 5.72902e7i) q^{41} +(3.75228e7 + 1.46565e8i) q^{42} +(8.76234e7 + 1.51768e8i) q^{43} -1.42772e8i q^{44} +(-4.28476e7 + 7.05280e7i) q^{45} +3.63139e7 q^{46} +(-2.20818e8 + 1.27489e8i) q^{47} +(1.71732e7 - 6.13425e7i) q^{48} +(-2.37311e8 + 4.11034e8i) q^{49} +(-3.82733e7 - 2.20971e7i) q^{50} +(4.40205e7 - 4.30481e7i) q^{51} +(2.80179e7 + 4.85284e7i) q^{52} +2.33866e8i q^{53} +(9.44915e7 - 3.10625e8i) q^{54} +3.89706e8 q^{55} +(2.76065e8 - 1.59386e8i) q^{56} +(-7.84136e8 - 8.01849e8i) q^{57} +(1.98718e8 - 3.44190e8i) q^{58} +(3.36140e8 + 1.94070e8i) q^{59} +(1.67439e8 + 4.68755e7i) q^{60} +(-5.76153e8 - 9.97926e8i) q^{61} -5.50416e8i q^{62} +(1.42487e9 - 7.80749e8i) q^{63} -1.34218e8 q^{64} +(-1.32462e8 + 7.64770e7i) q^{65} +(-1.48535e9 + 3.80270e8i) q^{66} +(5.55398e8 - 9.61978e8i) q^{67} +(-1.12349e8 - 6.48645e7i) q^{68} +(-9.67215e7 - 3.77797e8i) q^{69} +(4.35057e8 + 7.53540e8i) q^{70} +2.28480e9i q^{71} +(-6.83926e8 - 1.52790e7i) q^{72} -1.52546e9 q^{73} +(5.37738e8 - 3.10463e8i) q^{74} +(-1.27950e8 + 4.57037e8i) q^{75} +(-1.18153e9 + 2.04647e9i) q^{76} +(-6.64474e9 - 3.83635e9i) q^{77} +(4.30248e8 - 4.20743e8i) q^{78} +(-2.77361e9 - 4.80404e9i) q^{79} -3.66357e8i q^{80} +(-3.48331e9 - 1.55713e8i) q^{81} +2.59266e9 q^{82} +(-3.73356e9 + 2.15557e9i) q^{83} +(-2.39349e9 - 2.44756e9i) q^{84} +(1.77053e8 - 3.06664e8i) q^{85} +(-3.43412e9 - 1.98269e9i) q^{86} +(-4.11012e9 - 1.15065e9i) q^{87} +(1.61528e9 + 2.79774e9i) q^{88} -6.44176e9i q^{89} +(4.17052e7 - 1.86683e9i) q^{90} +3.01142e9 q^{91} +(-7.11604e8 + 4.10845e8i) q^{92} +(-5.72634e9 + 1.46603e9i) q^{93} +(2.88476e9 - 4.99655e9i) q^{94} +(-5.58600e9 - 3.22508e9i) q^{95} +(3.57487e8 + 1.39635e9i) q^{96} +(-3.71029e9 - 6.42641e9i) q^{97} -1.07395e10i q^{98} +(7.91239e9 + 1.44402e10i) 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
\(3\) 169.897 + 173.735i 0.699166 + 0.714960i
\(4\) 256.000 − 443.405i 0.250000 − 0.433013i
\(5\) 1210.31 + 698.771i 0.387298 + 0.223607i
\(6\) −5294.88 − 1482.33i −0.680926 − 0.190629i
\(7\) −13757.7 − 23829.0i −0.818569 − 1.41780i −0.906736 − 0.421698i \(-0.861434\pi\)
0.0881668 − 0.996106i \(-0.471899\pi\)
\(8\) 11585.2i 0.353553i
\(9\) −1318.83 + 59034.3i −0.0223346 + 0.999751i
\(10\) −31622.8 −0.316228
\(11\) 241492. − 139425.i 1.49947 − 0.865722i 0.499475 − 0.866328i \(-0.333526\pi\)
1.00000 0.000606189i \(0.000192956\pi\)
\(12\) 120529. − 30857.1i 0.484378 − 0.124008i
\(13\) −54722.5 + 94782.1i −0.147384 + 0.255276i −0.930260 − 0.366902i \(-0.880418\pi\)
0.782876 + 0.622178i \(0.213752\pi\)
\(14\) 539189. + 311301.i 1.00254 + 0.578816i
\(15\) 84226.8 + 328992.i 0.110916 + 0.433241i
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) − 253377.i − 0.178453i −0.996011 − 0.0892263i \(-0.971561\pi\)
0.996011 − 0.0892263i \(-0.0284394\pi\)
\(18\) −642053. − 1.17175e6i −0.339788 − 0.620116i
\(19\) −4.61535e6 −1.86396 −0.931981 − 0.362507i \(-0.881921\pi\)
−0.931981 + 0.362507i \(0.881921\pi\)
\(20\) 619677. − 357771.i 0.193649 − 0.111803i
\(21\) 1.80255e6 − 6.43868e6i 0.441357 − 1.57652i
\(22\) −3.15484e6 + 5.46434e6i −0.612158 + 1.06029i
\(23\) −1.38985e6 − 802432.i −0.215938 − 0.124672i 0.388130 − 0.921605i \(-0.373121\pi\)
−0.604068 + 0.796933i \(0.706455\pi\)
\(24\) −2.01276e6 + 1.96830e6i −0.252776 + 0.247192i
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) − 2.47646e6i − 0.208432i
\(27\) −1.04804e7 + 9.80063e6i −0.730397 + 0.683023i
\(28\) −1.40879e7 −0.818569
\(29\) −1.52112e7 + 8.78220e6i −0.741607 + 0.428167i −0.822653 − 0.568543i \(-0.807507\pi\)
0.0810463 + 0.996710i \(0.474174\pi\)
\(30\) −5.37262e6 − 5.49399e6i −0.221096 − 0.226090i
\(31\) −1.21626e7 + 2.10662e7i −0.424833 + 0.735832i −0.996405 − 0.0847200i \(-0.973000\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(32\) 5.13695e6 + 2.96582e6i 0.153093 + 0.0883883i
\(33\) 6.52519e7 + 1.82676e7i 1.66734 + 0.466781i
\(34\) 2.86664e6 + 4.96516e6i 0.0630925 + 0.109279i
\(35\) − 3.84539e7i − 0.732151i
\(36\) 2.58385e7 + 1.56976e7i 0.427321 + 0.259609i
\(37\) −2.74413e7 −0.395728 −0.197864 − 0.980229i \(-0.563400\pi\)
−0.197864 + 0.980229i \(0.563400\pi\)
\(38\) 9.04421e7 − 5.22168e7i 1.14144 − 0.659010i
\(39\) −2.57642e7 + 6.59600e6i −0.285557 + 0.0731068i
\(40\) −8.09543e6 + 1.40217e7i −0.0790569 + 0.136931i
\(41\) −9.92296e7 − 5.72902e7i −0.856489 − 0.494494i 0.00634585 − 0.999980i \(-0.497980\pi\)
−0.862835 + 0.505486i \(0.831313\pi\)
\(42\) 3.75228e7 + 1.46565e8i 0.287111 + 1.12146i
\(43\) 8.76234e7 + 1.51768e8i 0.596043 + 1.03238i 0.993399 + 0.114713i \(0.0365947\pi\)
−0.397355 + 0.917665i \(0.630072\pi\)
\(44\) − 1.42772e8i − 0.865722i
\(45\) −4.28476e7 + 7.05280e7i −0.232201 + 0.382208i
\(46\) 3.63139e7 0.176313
\(47\) −2.20818e8 + 1.27489e8i −0.962821 + 0.555885i −0.897040 − 0.441949i \(-0.854287\pi\)
−0.0657811 + 0.997834i \(0.520954\pi\)
\(48\) 1.71732e7 − 6.13425e7i 0.0673976 − 0.240744i
\(49\) −2.37311e8 + 4.11034e8i −0.840112 + 1.45512i
\(50\) −3.82733e7 − 2.20971e7i −0.122474 − 0.0707107i
\(51\) 4.40205e7 − 4.30481e7i 0.127586 − 0.124768i
\(52\) 2.80179e7 + 4.85284e7i 0.0736918 + 0.127638i
\(53\) 2.33866e8i 0.559225i 0.960113 + 0.279613i \(0.0902060\pi\)
−0.960113 + 0.279613i \(0.909794\pi\)
\(54\) 9.44915e7 − 3.10625e8i 0.205790 − 0.676499i
\(55\) 3.89706e8 0.774325
\(56\) 2.76065e8 − 1.59386e8i 0.501269 − 0.289408i
\(57\) −7.84136e8 − 8.01849e8i −1.30322 − 1.33266i
\(58\) 1.98718e8 − 3.44190e8i 0.302760 − 0.524395i
\(59\) 3.36140e8 + 1.94070e8i 0.470176 + 0.271456i 0.716313 − 0.697779i \(-0.245828\pi\)
−0.246138 + 0.969235i \(0.579162\pi\)
\(60\) 1.67439e8 + 4.68755e7i 0.215328 + 0.0602822i
\(61\) −5.76153e8 − 9.97926e8i −0.682164 − 1.18154i −0.974319 − 0.225171i \(-0.927706\pi\)
0.292156 − 0.956371i \(-0.405627\pi\)
\(62\) − 5.50416e8i − 0.600804i
\(63\) 1.42487e9 − 7.80749e8i 1.43573 − 0.786699i
\(64\) −1.34218e8 −0.125000
\(65\) −1.32462e8 + 7.64770e7i −0.114163 + 0.0659119i
\(66\) −1.48535e9 + 3.80270e8i −1.18606 + 0.303649i
\(67\) 5.55398e8 − 9.61978e8i 0.411368 − 0.712510i −0.583672 − 0.811990i \(-0.698385\pi\)
0.995040 + 0.0994797i \(0.0317178\pi\)
\(68\) −1.12349e8 − 6.48645e7i −0.0772722 − 0.0446131i
\(69\) −9.67215e7 − 3.77797e8i −0.0618412 − 0.241553i
\(70\) 4.35057e8 + 7.53540e8i 0.258854 + 0.448349i
\(71\) 2.28480e9i 1.26636i 0.774005 + 0.633180i \(0.218251\pi\)
−0.774005 + 0.633180i \(0.781749\pi\)
\(72\) −6.83926e8 − 1.52790e7i −0.353465 − 0.00789646i
\(73\) −1.52546e9 −0.735846 −0.367923 − 0.929856i \(-0.619931\pi\)
−0.367923 + 0.929856i \(0.619931\pi\)
\(74\) 5.37738e8 − 3.10463e8i 0.242333 − 0.139911i
\(75\) −1.27950e8 + 4.57037e8i −0.0539181 + 0.192595i
\(76\) −1.18153e9 + 2.04647e9i −0.465990 + 0.807119i
\(77\) −6.64474e9 − 3.83635e9i −2.45485 − 1.41731i
\(78\) 4.30248e8 − 4.20743e8i 0.149020 − 0.145728i
\(79\) −2.77361e9 − 4.80404e9i −0.901385 − 1.56124i −0.825698 − 0.564113i \(-0.809218\pi\)
−0.0756875 − 0.997132i \(-0.524115\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) −3.48331e9 − 1.55713e8i −0.999002 − 0.0446580i
\(82\) 2.59266e9 0.699321
\(83\) −3.73356e9 + 2.15557e9i −0.947836 + 0.547233i −0.892408 − 0.451230i \(-0.850986\pi\)
−0.0554278 + 0.998463i \(0.517652\pi\)
\(84\) −2.39349e9 − 2.44756e9i −0.572316 − 0.585244i
\(85\) 1.77053e8 − 3.06664e8i 0.0399032 − 0.0691144i
\(86\) −3.43412e9 − 1.98269e9i −0.730001 − 0.421466i
\(87\) −4.11012e9 − 1.15065e9i −0.824628 − 0.230859i
\(88\) 1.61528e9 + 2.79774e9i 0.306079 + 0.530144i
\(89\) − 6.44176e9i − 1.15360i −0.816886 − 0.576799i \(-0.804302\pi\)
0.816886 − 0.576799i \(-0.195698\pi\)
\(90\) 4.17052e7 − 1.86683e9i 0.00706281 − 0.316149i
\(91\) 3.01142e9 0.482575
\(92\) −7.11604e8 + 4.10845e8i −0.107969 + 0.0623360i
\(93\) −5.72634e9 + 1.46603e9i −0.823119 + 0.210730i
\(94\) 2.88476e9 − 4.99655e9i 0.393070 − 0.680817i
\(95\) −5.58600e9 − 3.22508e9i −0.721909 − 0.416795i
\(96\) 3.57487e8 + 1.39635e9i 0.0438434 + 0.171253i
\(97\) −3.71029e9 − 6.42641e9i −0.432065 − 0.748358i 0.564986 − 0.825100i \(-0.308882\pi\)
−0.997051 + 0.0767421i \(0.975548\pi\)
\(98\) − 1.07395e10i − 1.18810i
\(99\) 7.91239e9 + 1.44402e10i 0.832016 + 1.51844i
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.41.15 yes 80
3.2 odd 2 270.11.h.a.71.40 80
9.2 odd 6 inner 90.11.h.a.11.15 ✓ 80
9.7 even 3 270.11.h.a.251.40 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.15 ✓ 80 9.2 odd 6 inner
90.11.h.a.41.15 yes 80 1.1 even 1 trivial
270.11.h.a.71.40 80 3.2 odd 2
270.11.h.a.251.40 80 9.7 even 3