Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [23,2,Mod(2,23)] 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("23.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(23, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 23.c (of order \(11\), degree \(10\), 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: \(0.183655924649\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 3.1
Root \(0.142315 + 0.989821i\) of defining polynomial
Character \(\chi\) \(=\) 23.3
Dual form 23.2.c.a.8.1

$q$-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+(-0.313607 - 2.18119i) q^{2} +(-1.04408 + 2.28621i) q^{3} +(-2.74024 + 0.804606i) q^{4} +(0.809721 - 0.934468i) q^{5} +(5.31408 + 1.56036i) q^{6} +(-1.99611 + 1.28282i) q^{7} +(0.783524 + 1.71568i) q^{8} +(-2.17208 - 2.50672i) q^{9} +(-2.29218 - 1.47310i) q^{10} +(0.272084 - 1.89238i) q^{11} +(1.02152 - 7.10483i) q^{12} +(0.165284 + 0.106222i) q^{13} +(3.42408 + 3.95159i) q^{14} +(1.29098 + 2.82685i) q^{15} +(-1.30862 + 0.840996i) q^{16} +(1.49672 + 0.439476i) q^{17} +(-4.78644 + 5.52384i) q^{18} +(7.66103 - 2.24948i) q^{19} +(-1.46695 + 3.21217i) q^{20} +(-0.848710 - 5.90291i) q^{21} -4.21297 q^{22} +(-4.66752 + 1.10192i) q^{23} -4.74046 q^{24} +(0.493992 + 3.43579i) q^{25} +(0.179855 - 0.393828i) q^{26} +(0.764125 - 0.224367i) q^{27} +(4.43766 - 5.12133i) q^{28} +(-4.77570 - 1.40227i) q^{29} +(5.76103 - 3.70239i) q^{30} +(0.740552 + 1.62158i) q^{31} +(4.71506 + 5.44146i) q^{32} +(4.04231 + 2.59784i) q^{33} +(0.489198 - 3.40244i) q^{34} +(-0.417537 + 2.90404i) q^{35} +(7.96894 + 5.12133i) q^{36} +(-2.54297 - 2.93475i) q^{37} +(-7.30909 - 16.0047i) q^{38} +(-0.415415 + 0.266971i) q^{39} +(2.23768 + 0.657043i) q^{40} +(-0.279295 + 0.322324i) q^{41} +(-12.6092 + 3.70239i) q^{42} +(1.84991 - 4.05075i) q^{43} +(0.777050 + 5.40450i) q^{44} -4.10123 q^{45} +(3.86725 + 9.83517i) q^{46} -2.58842 q^{47} +(-0.556399 - 3.86984i) q^{48} +(-0.569072 + 1.24609i) q^{49} +(7.33918 - 2.15498i) q^{50} +(-2.56743 + 2.96297i) q^{51} +(-0.538385 - 0.158084i) q^{52} +(8.26060 - 5.30876i) q^{53} +(-0.729022 - 1.59634i) q^{54} +(-1.54806 - 1.78656i) q^{55} +(-3.76492 - 2.41956i) q^{56} +(-2.85592 + 19.8634i) q^{57} +(-1.56092 + 10.8565i) q^{58} +(-6.07293 - 3.90283i) q^{59} +(-5.81210 - 6.70752i) q^{60} +(3.08639 + 6.75826i) q^{61} +(3.30473 - 2.12382i) q^{62} +(7.55141 + 2.21729i) q^{63} +(8.35283 - 9.63968i) q^{64} +(0.233095 - 0.0684429i) q^{65} +(4.39867 - 9.63174i) q^{66} +(-1.03413 - 7.19254i) q^{67} -4.45497 q^{68} +(2.35404 - 11.8214i) q^{69} +6.46519 q^{70} +(0.103930 + 0.722850i) q^{71} +(2.59884 - 5.69067i) q^{72} +(6.18330 - 1.81558i) q^{73} +(-5.60373 + 6.46705i) q^{74} +(-8.37071 - 2.45786i) q^{75} +(-19.1831 + 12.3282i) q^{76} +(1.88449 + 4.12645i) q^{77} +(0.712591 + 0.822373i) q^{78} +(-4.77671 - 3.06980i) q^{79} +(-0.273730 + 1.90383i) q^{80} +(1.13126 - 7.86810i) q^{81} +(0.790638 + 0.508112i) q^{82} +(8.44098 + 9.74141i) q^{83} +(7.07518 + 15.4925i) q^{84} +(1.62260 - 1.04278i) q^{85} +(-9.41558 - 2.76466i) q^{86} +(8.19209 - 9.45418i) q^{87} +(3.45991 - 1.01592i) q^{88} +(-5.77436 + 12.6441i) q^{89} +(1.28618 + 8.94555i) q^{90} -0.466190 q^{91} +(11.9035 - 6.77503i) q^{92} -4.48047 q^{93} +(0.811746 + 5.64582i) q^{94} +(4.10123 - 8.98044i) q^{95} +(-17.3632 + 5.09830i) q^{96} +(-2.83147 + 3.26769i) q^{97} +(2.89643 + 0.850468i) q^{98} +(-5.33466 + 3.42838i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9} + q^{10} + 7 q^{11} + 12 q^{12} - 3 q^{13} + 9 q^{14} + 12 q^{15} + q^{16} - 10 q^{17} - 14 q^{18} + 2 q^{19} - 9 q^{20}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.313607 − 2.18119i −0.221754 − 1.54233i −0.731401 − 0.681948i \(-0.761133\pi\)
0.509647 − 0.860384i \(-0.329776\pi\)
\(3\) −1.04408 + 2.28621i −0.602799 + 1.31995i 0.324593 + 0.945854i \(0.394773\pi\)
−0.927392 + 0.374091i \(0.877955\pi\)
\(4\) −2.74024 + 0.804606i −1.37012 + 0.402303i
\(5\) 0.809721 − 0.934468i 0.362118 − 0.417907i −0.545230 − 0.838287i \(-0.683558\pi\)
0.907348 + 0.420380i \(0.138103\pi\)
\(6\) 5.31408 + 1.56036i 2.16947 + 0.637013i
\(7\) −1.99611 + 1.28282i −0.754460 + 0.484862i −0.860469 − 0.509503i \(-0.829829\pi\)
0.106009 + 0.994365i \(0.466193\pi\)
\(8\) 0.783524 + 1.71568i 0.277017 + 0.606584i
\(9\) −2.17208 − 2.50672i −0.724028 − 0.835573i
\(10\) −2.29218 − 1.47310i −0.724852 − 0.465834i
\(11\) 0.272084 − 1.89238i 0.0820363 − 0.570575i −0.906799 − 0.421563i \(-0.861482\pi\)
0.988835 − 0.149012i \(-0.0476093\pi\)
\(12\) 1.02152 − 7.10483i 0.294888 − 2.05099i
\(13\) 0.165284 + 0.106222i 0.0458416 + 0.0294606i 0.563361 − 0.826211i \(-0.309508\pi\)
−0.517519 + 0.855672i \(0.673144\pi\)
\(14\) 3.42408 + 3.95159i 0.915123 + 1.05611i
\(15\) 1.29098 + 2.82685i 0.333330 + 0.729890i
\(16\) −1.30862 + 0.840996i −0.327154 + 0.210249i
\(17\) 1.49672 + 0.439476i 0.363008 + 0.106589i 0.458150 − 0.888875i \(-0.348512\pi\)
−0.0951421 + 0.995464i \(0.530331\pi\)
\(18\) −4.78644 + 5.52384i −1.12817 + 1.30198i
\(19\) 7.66103 − 2.24948i 1.75756 − 0.516066i 0.765678 − 0.643224i \(-0.222403\pi\)
0.991883 + 0.127157i \(0.0405853\pi\)
\(20\) −1.46695 + 3.21217i −0.328020 + 0.718263i
\(21\) −0.848710 − 5.90291i −0.185204 − 1.28812i
\(22\) −4.21297 −0.898208
\(23\) −4.66752 + 1.10192i −0.973246 + 0.229765i
\(24\) −4.74046 −0.967643
\(25\) 0.493992 + 3.43579i 0.0987984 + 0.687158i
\(26\) 0.179855 − 0.393828i 0.0352725 − 0.0772359i
\(27\) 0.764125 − 0.224367i 0.147056 − 0.0431795i
\(28\) 4.43766 − 5.12133i 0.838638 − 0.967840i
\(29\) −4.77570 − 1.40227i −0.886825 − 0.260395i −0.193569 − 0.981087i \(-0.562006\pi\)
−0.693256 + 0.720691i \(0.743825\pi\)
\(30\) 5.76103 − 3.70239i 1.05182 − 0.675961i
\(31\) 0.740552 + 1.62158i 0.133007 + 0.291245i 0.964404 − 0.264435i \(-0.0851854\pi\)
−0.831397 + 0.555680i \(0.812458\pi\)
\(32\) 4.71506 + 5.44146i 0.833512 + 0.961924i
\(33\) 4.04231 + 2.59784i 0.703677 + 0.452226i
\(34\) 0.489198 − 3.40244i 0.0838967 − 0.583514i
\(35\) −0.417537 + 2.90404i −0.0705767 + 0.490872i
\(36\) 7.96894 + 5.12133i 1.32816 + 0.853555i
\(37\) −2.54297 − 2.93475i −0.418062 − 0.482469i 0.507184 − 0.861838i \(-0.330687\pi\)
−0.925246 + 0.379369i \(0.876141\pi\)
\(38\) −7.30909 − 16.0047i −1.18569 − 2.59630i
\(39\) −0.415415 + 0.266971i −0.0665196 + 0.0427496i
\(40\) 2.23768 + 0.657043i 0.353809 + 0.103888i
\(41\) −0.279295 + 0.322324i −0.0436186 + 0.0503386i −0.777140 − 0.629328i \(-0.783330\pi\)
0.733521 + 0.679667i \(0.237876\pi\)
\(42\) −12.6092 + 3.70239i −1.94564 + 0.571291i
\(43\) 1.84991 − 4.05075i 0.282109 − 0.617733i −0.714534 − 0.699601i \(-0.753361\pi\)
0.996643 + 0.0818677i \(0.0260885\pi\)
\(44\) 0.777050 + 5.40450i 0.117145 + 0.814759i
\(45\) −4.10123 −0.611375
\(46\) 3.86725 + 9.83517i 0.570195 + 1.45012i
\(47\) −2.58842 −0.377559 −0.188780 − 0.982019i \(-0.560453\pi\)
−0.188780 + 0.982019i \(0.560453\pi\)
\(48\) −0.556399 − 3.86984i −0.0803092 − 0.558563i
\(49\) −0.569072 + 1.24609i −0.0812960 + 0.178013i
\(50\) 7.33918 − 2.15498i 1.03792 − 0.304760i
\(51\) −2.56743 + 2.96297i −0.359512 + 0.414898i
\(52\) −0.538385 − 0.158084i −0.0746605 − 0.0219223i
\(53\) 8.26060 − 5.30876i 1.13468 − 0.729215i 0.168149 − 0.985762i \(-0.446221\pi\)
0.966532 + 0.256547i \(0.0825848\pi\)
\(54\) −0.729022 − 1.59634i −0.0992074 − 0.217234i
\(55\) −1.54806 − 1.78656i −0.208741 − 0.240899i
\(56\) −3.76492 − 2.41956i −0.503108 − 0.323328i
\(57\) −2.85592 + 19.8634i −0.378276 + 2.63097i
\(58\) −1.56092 + 10.8565i −0.204959 + 1.42552i
\(59\) −6.07293 − 3.90283i −0.790628 − 0.508106i 0.0819173 − 0.996639i \(-0.473896\pi\)
−0.872545 + 0.488533i \(0.837532\pi\)
\(60\) −5.81210 − 6.70752i −0.750338 − 0.865936i
\(61\) 3.08639 + 6.75826i 0.395172 + 0.865306i 0.997737 + 0.0672363i \(0.0214181\pi\)
−0.602565 + 0.798070i \(0.705855\pi\)
\(62\) 3.30473 − 2.12382i 0.419701 − 0.269726i
\(63\) 7.55141 + 2.21729i 0.951388 + 0.279353i
\(64\) 8.35283 − 9.63968i 1.04410 − 1.20496i
\(65\) 0.233095 − 0.0684429i 0.0289119 − 0.00848929i
\(66\) 4.39867 − 9.63174i 0.541439 − 1.18559i
\(67\) −1.03413 − 7.19254i −0.126339 − 0.878708i −0.950139 − 0.311827i \(-0.899059\pi\)
0.823800 − 0.566881i \(-0.191850\pi\)
\(68\) −4.45497 −0.540244
\(69\) 2.35404 − 11.8214i 0.283394 − 1.42313i
\(70\) 6.46519 0.772738
\(71\) 0.103930 + 0.722850i 0.0123342 + 0.0857866i 0.995059 − 0.0992879i \(-0.0316565\pi\)
−0.982725 + 0.185074i \(0.940747\pi\)
\(72\) 2.59884 − 5.69067i 0.306276 − 0.670652i
\(73\) 6.18330 − 1.81558i 0.723700 − 0.212498i 0.100920 − 0.994895i \(-0.467821\pi\)
0.622780 + 0.782397i \(0.286003\pi\)
\(74\) −5.60373 + 6.46705i −0.651421 + 0.751780i
\(75\) −8.37071 − 2.45786i −0.966566 − 0.283809i
\(76\) −19.1831 + 12.3282i −2.20045 + 1.41414i
\(77\) 1.88449 + 4.12645i 0.214757 + 0.470253i
\(78\) 0.712591 + 0.822373i 0.0806850 + 0.0931154i
\(79\) −4.77671 − 3.06980i −0.537422 − 0.345380i 0.243608 − 0.969874i \(-0.421669\pi\)
−0.781030 + 0.624494i \(0.785305\pi\)
\(80\) −0.273730 + 1.90383i −0.0306039 + 0.212855i
\(81\) 1.13126 − 7.86810i 0.125696 − 0.874233i
\(82\) 0.790638 + 0.508112i 0.0873113 + 0.0561116i
\(83\) 8.44098 + 9.74141i 0.926518 + 1.06926i 0.997421 + 0.0717758i \(0.0228666\pi\)
−0.0709025 + 0.997483i \(0.522588\pi\)
\(84\) 7.07518 + 15.4925i 0.771966 + 1.69037i
\(85\) 1.62260 − 1.04278i 0.175996 − 0.113106i
\(86\) −9.41558 − 2.76466i −1.01531 − 0.298121i
\(87\) 8.19209 − 9.45418i 0.878284 − 1.01359i
\(88\) 3.45991 − 1.01592i 0.368827 − 0.108297i
\(89\) −5.77436 + 12.6441i −0.612081 + 1.34027i 0.309060 + 0.951043i \(0.399986\pi\)
−0.921141 + 0.389228i \(0.872742\pi\)
\(90\) 1.28618 + 8.94555i 0.135575 + 0.942944i
\(91\) −0.466190 −0.0488700
\(92\) 11.9035 − 6.77503i 1.24103 − 0.706346i
\(93\) −4.48047 −0.464603
\(94\) 0.811746 + 5.64582i 0.0837252 + 0.582322i
\(95\) 4.10123 − 8.98044i 0.420777 − 0.921374i
\(96\) −17.3632 + 5.09830i −1.77213 + 0.520343i
\(97\) −2.83147 + 3.26769i −0.287492 + 0.331783i −0.881064 − 0.472998i \(-0.843172\pi\)
0.593572 + 0.804781i \(0.297717\pi\)
\(98\) 2.89643 + 0.850468i 0.292583 + 0.0859103i
\(99\) −5.33466 + 3.42838i −0.536154 + 0.344565i
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 23.2.c.a.3.1 ✓ 10
3.2 odd 2 207.2.i.c.118.1 10
4.3 odd 2 368.2.m.c.49.1 10
5.2 odd 4 575.2.p.b.49.2 20
5.3 odd 4 575.2.p.b.49.1 20
5.4 even 2 575.2.k.b.26.1 10
23.2 even 11 529.2.c.i.334.1 10
23.3 even 11 529.2.c.g.487.1 10
23.4 even 11 529.2.c.i.255.1 10
23.5 odd 22 529.2.c.f.466.1 10
23.6 even 11 529.2.c.d.177.1 10
23.7 odd 22 529.2.c.c.170.1 10
23.8 even 11 inner 23.2.c.a.8.1 yes 10
23.9 even 11 529.2.c.d.266.1 10
23.10 odd 22 529.2.a.j.1.5 5
23.11 odd 22 529.2.c.c.501.1 10
23.12 even 11 529.2.c.b.501.1 10
23.13 even 11 529.2.a.i.1.5 5
23.14 odd 22 529.2.c.e.266.1 10
23.15 odd 22 529.2.c.a.399.1 10
23.16 even 11 529.2.c.b.170.1 10
23.17 odd 22 529.2.c.e.177.1 10
23.18 even 11 529.2.c.g.466.1 10
23.19 odd 22 529.2.c.h.255.1 10
23.20 odd 22 529.2.c.f.487.1 10
23.21 odd 22 529.2.c.h.334.1 10
23.22 odd 2 529.2.c.a.118.1 10
69.8 odd 22 207.2.i.c.100.1 10
69.56 even 22 4761.2.a.bn.1.1 5
69.59 odd 22 4761.2.a.bo.1.1 5
92.31 odd 22 368.2.m.c.353.1 10
92.59 odd 22 8464.2.a.bs.1.5 5
92.79 even 22 8464.2.a.bt.1.5 5
115.8 odd 44 575.2.p.b.399.2 20
115.54 even 22 575.2.k.b.376.1 10
115.77 odd 44 575.2.p.b.399.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.c.a.3.1 ✓ 10 1.1 even 1 trivial
23.2.c.a.8.1 yes 10 23.8 even 11 inner
207.2.i.c.100.1 10 69.8 odd 22
207.2.i.c.118.1 10 3.2 odd 2
368.2.m.c.49.1 10 4.3 odd 2
368.2.m.c.353.1 10 92.31 odd 22
529.2.a.i.1.5 5 23.13 even 11
529.2.a.j.1.5 5 23.10 odd 22
529.2.c.a.118.1 10 23.22 odd 2
529.2.c.a.399.1 10 23.15 odd 22
529.2.c.b.170.1 10 23.16 even 11
529.2.c.b.501.1 10 23.12 even 11
529.2.c.c.170.1 10 23.7 odd 22
529.2.c.c.501.1 10 23.11 odd 22
529.2.c.d.177.1 10 23.6 even 11
529.2.c.d.266.1 10 23.9 even 11
529.2.c.e.177.1 10 23.17 odd 22
529.2.c.e.266.1 10 23.14 odd 22
529.2.c.f.466.1 10 23.5 odd 22
529.2.c.f.487.1 10 23.20 odd 22
529.2.c.g.466.1 10 23.18 even 11
529.2.c.g.487.1 10 23.3 even 11
529.2.c.h.255.1 10 23.19 odd 22
529.2.c.h.334.1 10 23.21 odd 22
529.2.c.i.255.1 10 23.4 even 11
529.2.c.i.334.1 10 23.2 even 11
575.2.k.b.26.1 10 5.4 even 2
575.2.k.b.376.1 10 115.54 even 22
575.2.p.b.49.1 20 5.3 odd 4
575.2.p.b.49.2 20 5.2 odd 4
575.2.p.b.399.1 20 115.77 odd 44
575.2.p.b.399.2 20 115.8 odd 44
4761.2.a.bn.1.1 5 69.56 even 22
4761.2.a.bo.1.1 5 69.59 odd 22
8464.2.a.bs.1.5 5 92.59 odd 22
8464.2.a.bt.1.5 5 92.79 even 22