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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 11.19
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.19

$q$-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} +(235.959 - 58.0713i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(-5280.84 - 1531.61i) q^{6} +(-2866.21 + 4964.43i) q^{7} -11585.2i q^{8} +(52304.5 - 27404.9i) q^{9} +31622.8 q^{10} +(-177189. - 102300. i) q^{11} +(86154.6 + 89759.2i) q^{12} +(344864. + 597322. i) q^{13} +(112332. - 64855.0i) q^{14} +(-245005. + 235166. i) q^{15} +(-131072. + 227023. i) q^{16} -1.47016e6i q^{17} +(-1.33500e6 - 54733.0i) q^{18} -3.02442e6 q^{19} +(-619677. - 357771. i) q^{20} +(-388019. + 1.33785e6i) q^{21} +(2.31479e6 + 4.00933e6i) q^{22} +(5.66729e6 - 3.27201e6i) q^{23} +(-672770. - 2.73364e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.56068e7i q^{26} +(1.07503e7 - 9.50383e6i) q^{27} -2.93500e6 q^{28} +(-1.91534e6 - 1.10582e6i) q^{29} +(7.46168e6 - 1.83638e6i) q^{30} +(-1.64078e7 - 2.84191e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-4.77501e7 - 1.38491e7i) q^{33} +(-1.66330e7 + 2.88092e7i) q^{34} -8.01131e6i q^{35} +(2.55414e7 + 1.61764e7i) q^{36} +9.61827e7 q^{37} +(5.92664e7 + 3.42174e7i) q^{38} +(1.16061e8 + 1.20917e8i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(-1.48417e8 + 8.56887e7i) q^{41} +(2.27396e7 - 2.18264e7i) q^{42} +(8.98823e7 - 1.55681e8i) q^{43} -1.04755e8i q^{44} +(-4.41547e7 + 6.97172e7i) q^{45} -1.48074e8 q^{46} +(2.73128e8 + 1.57691e8i) q^{47} +(-1.77441e7 + 6.11798e7i) q^{48} +(1.24807e8 + 2.16173e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-8.53743e7 - 3.46899e8i) q^{51} +(-1.76570e8 + 3.05829e8i) q^{52} -5.68952e8i q^{53} +(-3.18185e8 + 6.46107e7i) q^{54} +2.85938e8 q^{55} +(5.75141e7 + 3.32058e7i) q^{56} +(-7.13641e8 + 1.75632e8i) q^{57} +(2.50218e7 + 4.33391e7i) q^{58} +(8.85464e8 - 5.11223e8i) q^{59} +(-1.66995e8 - 4.84339e7i) q^{60} +(-1.63176e8 + 2.82629e8i) q^{61} +7.42532e8i q^{62} +(-1.38661e7 + 3.38210e8i) q^{63} -1.34218e8 q^{64} +(-8.34783e8 - 4.81962e8i) q^{65} +(7.79023e8 + 8.11616e8i) q^{66} +(-1.27798e9 - 2.21352e9i) q^{67} +(6.51878e8 - 3.76362e8i) q^{68} +(1.14724e9 - 1.10117e9i) q^{69} +(-9.06376e7 + 1.56989e8i) q^{70} -2.91902e9i q^{71} +(-3.17492e8 - 6.05959e8i) q^{72} -2.61741e9 q^{73} +(-1.88479e9 - 1.08818e9i) q^{74} +(1.32204e8 - 4.55825e8i) q^{75} +(-7.74253e8 - 1.34104e9i) q^{76} +(1.01572e9 - 5.86429e8i) q^{77} +(-9.06306e8 - 3.68256e9i) q^{78} +(9.65645e8 - 1.67255e9i) q^{79} -3.66357e8i q^{80} +(1.98473e9 - 2.86680e9i) q^{81} +3.87783e9 q^{82} +(-5.10238e8 - 2.94586e8i) q^{83} +(-6.92541e8 + 1.70439e8i) q^{84} +(1.02731e9 + 1.77935e9i) q^{85} +(-3.52265e9 + 2.03380e9i) q^{86} +(-5.16157e8 - 1.49702e8i) q^{87} +(-1.18517e9 + 2.05278e9i) q^{88} -5.85504e9i q^{89} +(1.65401e9 - 8.66619e8i) q^{90} -3.95382e9 q^{91} +(2.90165e9 + 1.67527e9i) q^{92} +(-5.52191e9 - 5.75294e9i) q^{93} +(-3.56813e9 - 6.18018e9i) q^{94} +(3.66048e9 - 2.11338e9i) q^{95} +(1.03988e9 - 9.98122e8i) q^{96} +(4.23785e9 - 7.34016e9i) q^{97} -5.64813e9i q^{98} +(-1.20713e10 - 4.94904e8i) 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
\(3\) 235.959 − 58.0713i 0.971025 − 0.238977i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) −5280.84 − 1531.61i −0.679120 − 0.196967i
\(7\) −2866.21 + 4964.43i −0.170537 + 0.295379i −0.938608 − 0.344986i \(-0.887884\pi\)
0.768071 + 0.640365i \(0.221217\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 52304.5 − 27404.9i 0.885780 − 0.464104i
\(10\) 31622.8 0.316228
\(11\) −177189. − 102300.i −1.10021 − 0.635204i −0.163930 − 0.986472i \(-0.552417\pi\)
−0.936275 + 0.351268i \(0.885751\pi\)
\(12\) 86154.6 + 89759.2i 0.346236 + 0.360722i
\(13\) 344864. + 597322.i 0.928820 + 1.60876i 0.785300 + 0.619115i \(0.212509\pi\)
0.143520 + 0.989647i \(0.454158\pi\)
\(14\) 112332. − 64855.0i 0.208864 − 0.120588i
\(15\) −245005. + 235166.i −0.322640 + 0.309683i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 1.47016e6i − 1.03543i −0.855553 − 0.517716i \(-0.826783\pi\)
0.855553 − 0.517716i \(-0.173217\pi\)
\(18\) −1.33500e6 − 54733.0i −0.706513 − 0.0289659i
\(19\) −3.02442e6 −1.22145 −0.610724 − 0.791844i \(-0.709121\pi\)
−0.610724 + 0.791844i \(0.709121\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) −388019. + 1.33785e6i −0.0950071 + 0.327574i
\(22\) 2.31479e6 + 4.00933e6i 0.449157 + 0.777963i
\(23\) 5.66729e6 − 3.27201e6i 0.880513 − 0.508365i 0.00968573 − 0.999953i \(-0.496917\pi\)
0.870828 + 0.491588i \(0.163584\pi\)
\(24\) −672770. − 2.73364e6i −0.0844910 − 0.343309i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.56068e7i − 1.31355i
\(27\) 1.07503e7 − 9.50383e6i 0.749205 − 0.662338i
\(28\) −2.93500e6 −0.170537
\(29\) −1.91534e6 − 1.10582e6i −0.0933802 − 0.0539131i 0.452583 − 0.891722i \(-0.350503\pi\)
−0.545963 + 0.837809i \(0.683836\pi\)
\(30\) 7.46168e6 − 1.83638e6i 0.307065 − 0.0755710i
\(31\) −1.64078e7 − 2.84191e7i −0.573115 − 0.992665i −0.996244 − 0.0865949i \(-0.972401\pi\)
0.423128 − 0.906070i \(-0.360932\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −4.77501e7 − 1.38491e7i −1.22013 − 0.353876i
\(34\) −1.66330e7 + 2.88092e7i −0.366080 + 0.634070i
\(35\) − 8.01131e6i − 0.152533i
\(36\) 2.55414e7 + 1.61764e7i 0.422408 + 0.267528i
\(37\) 9.61827e7 1.38704 0.693519 − 0.720438i \(-0.256059\pi\)
0.693519 + 0.720438i \(0.256059\pi\)
\(38\) 5.92664e7 + 3.42174e7i 0.747981 + 0.431847i
\(39\) 1.16061e8 + 1.20917e8i 1.28636 + 1.34018i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) −1.48417e8 + 8.56887e7i −1.28105 + 0.739613i −0.977040 − 0.213057i \(-0.931658\pi\)
−0.304007 + 0.952670i \(0.598325\pi\)
\(42\) 2.27396e7 − 2.18264e7i 0.173995 − 0.167007i
\(43\) 8.98823e7 − 1.55681e8i 0.611409 − 1.05899i −0.379594 − 0.925153i \(-0.623936\pi\)
0.991003 − 0.133838i \(-0.0427303\pi\)
\(44\) − 1.04755e8i − 0.635204i
\(45\) −4.41547e7 + 6.97172e7i −0.239284 + 0.377813i
\(46\) −1.48074e8 −0.718936
\(47\) 2.73128e8 + 1.57691e8i 1.19090 + 0.687569i 0.958512 − 0.285053i \(-0.0920112\pi\)
0.232393 + 0.972622i \(0.425344\pi\)
\(48\) −1.77441e7 + 6.11798e7i −0.0696382 + 0.240105i
\(49\) 1.24807e8 + 2.16173e8i 0.441834 + 0.765280i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −8.53743e7 − 3.46899e8i −0.247444 − 1.00543i
\(52\) −1.76570e8 + 3.05829e8i −0.464410 + 0.804381i
\(53\) − 5.68952e8i − 1.36049i −0.732983 − 0.680247i \(-0.761873\pi\)
0.732983 − 0.680247i \(-0.238127\pi\)
\(54\) −3.18185e8 + 6.46107e7i −0.692964 + 0.140713i
\(55\) 2.85938e8 0.568144
\(56\) 5.75141e7 + 3.32058e7i 0.104432 + 0.0602939i
\(57\) −7.13641e8 + 1.75632e8i −1.18606 + 0.291897i
\(58\) 2.50218e7 + 4.33391e7i 0.0381223 + 0.0660298i
\(59\) 8.85464e8 − 5.11223e8i 1.23854 − 0.715073i 0.269746 − 0.962932i \(-0.413060\pi\)
0.968796 + 0.247859i \(0.0797270\pi\)
\(60\) −1.66995e8 − 4.84339e7i −0.214757 − 0.0622863i
\(61\) −1.63176e8 + 2.82629e8i −0.193200 + 0.334633i −0.946309 − 0.323263i \(-0.895220\pi\)
0.753109 + 0.657896i \(0.228553\pi\)
\(62\) 7.42532e8i 0.810507i
\(63\) −1.38661e7 + 3.38210e8i −0.0139717 + 0.340788i
\(64\) −1.34218e8 −0.125000
\(65\) −8.34783e8 − 4.81962e8i −0.719461 − 0.415381i
\(66\) 7.79023e8 + 8.11616e8i 0.622058 + 0.648084i
\(67\) −1.27798e9 − 2.21352e9i −0.946562 − 1.63949i −0.752593 − 0.658486i \(-0.771197\pi\)
−0.193969 − 0.981008i \(-0.562136\pi\)
\(68\) 6.51878e8 − 3.76362e8i 0.448355 − 0.258858i
\(69\) 1.14724e9 − 1.10117e9i 0.733514 − 0.704057i
\(70\) −9.06376e7 + 1.56989e8i −0.0539285 + 0.0934069i
\(71\) − 2.91902e9i − 1.61788i −0.587894 − 0.808938i \(-0.700043\pi\)
0.587894 − 0.808938i \(-0.299957\pi\)
\(72\) −3.17492e8 − 6.05959e8i −0.164086 − 0.313171i
\(73\) −2.61741e9 −1.26258 −0.631288 − 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(74\) −1.88479e9 − 1.08818e9i −0.849384 − 0.490392i
\(75\) 1.32204e8 − 4.55825e8i 0.0557106 − 0.192084i
\(76\) −7.74253e8 − 1.34104e9i −0.305362 − 0.528902i
\(77\) 1.01572e9 − 5.86429e8i 0.375251 − 0.216651i
\(78\) −9.06306e8 − 3.68256e9i −0.313907 − 1.27549i
\(79\) 9.65645e8 − 1.67255e9i 0.313821 − 0.543554i −0.665365 − 0.746518i \(-0.731724\pi\)
0.979186 + 0.202964i \(0.0650574\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 1.98473e9 − 2.86680e9i 0.569214 − 0.822189i
\(82\) 3.87783e9 1.04597
\(83\) −5.10238e8 − 2.94586e8i −0.129534 − 0.0747863i 0.433833 − 0.900993i \(-0.357161\pi\)
−0.563367 + 0.826207i \(0.690494\pi\)
\(84\) −6.92541e8 + 1.70439e8i −0.165596 + 0.0407543i
\(85\) 1.02731e9 + 1.77935e9i 0.231529 + 0.401021i
\(86\) −3.52265e9 + 2.03380e9i −0.748820 + 0.432332i
\(87\) −5.16157e8 − 1.49702e8i −0.103559 − 0.0300353i
\(88\) −1.18517e9 + 2.05278e9i −0.224579 + 0.388981i
\(89\) − 5.85504e9i − 1.04853i −0.851556 − 0.524264i \(-0.824341\pi\)
0.851556 − 0.524264i \(-0.175659\pi\)
\(90\) 1.65401e9 − 8.66619e8i 0.280108 − 0.146763i
\(91\) −3.95382e9 −0.633592
\(92\) 2.90165e9 + 1.67527e9i 0.440257 + 0.254182i
\(93\) −5.52191e9 − 5.75294e9i −0.793733 − 0.826941i
\(94\) −3.56813e9 − 6.18018e9i −0.486185 − 0.842097i
\(95\) 3.66048e9 − 2.11338e9i 0.473064 − 0.273124i
\(96\) 1.03988e9 − 9.98122e8i 0.127535 − 0.122413i
\(97\) 4.23785e9 − 7.34016e9i 0.493499 − 0.854766i −0.506473 − 0.862256i \(-0.669051\pi\)
0.999972 + 0.00749027i \(0.00238425\pi\)
\(98\) − 5.64813e9i − 0.624848i
\(99\) −1.20713e10 − 4.94904e8i −1.26934 − 0.0520409i
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.11.19 ✓ 80
3.2 odd 2 270.11.h.a.251.25 80
9.4 even 3 270.11.h.a.71.25 80
9.5 odd 6 inner 90.11.h.a.41.19 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.19 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.19 yes 80 9.5 odd 6 inner
270.11.h.a.71.25 80 9.4 even 3
270.11.h.a.251.25 80 3.2 odd 2