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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [920,2,Mod(11,920)] 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("920.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 0, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [480,2,0,-4,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 11.18
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.18

$q$-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+(-0.476532 - 1.33151i) q^{2} +(0.0556596 - 0.387121i) q^{3} +(-1.54583 + 1.26901i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(-0.541979 + 0.110364i) q^{6} +(-2.16390 - 2.49727i) q^{7} +(2.42634 + 1.45357i) q^{8} +(2.73171 + 0.802104i) q^{9} +(0.832359 + 1.14332i) q^{10} +(2.24579 - 3.49451i) q^{11} +(0.405221 + 0.669058i) q^{12} +(1.07346 + 0.930155i) q^{13} +(-2.29398 + 4.07129i) q^{14} +(0.0556596 + 0.387121i) q^{15} +(0.779208 - 3.92337i) q^{16} +(-0.754891 + 0.344747i) q^{17} +(-0.233741 - 4.01953i) q^{18} +(-1.72989 - 0.790012i) q^{19} +(1.12569 - 1.65312i) q^{20} +(-1.08719 + 0.698694i) q^{21} +(-5.72317 - 1.32504i) q^{22} +(2.50326 - 4.09069i) q^{23} +(0.697756 - 0.858384i) q^{24} +(0.841254 - 0.540641i) q^{25} +(0.726974 - 1.87256i) q^{26} +(0.949966 - 2.08014i) q^{27} +(6.51411 + 1.11435i) q^{28} +(-4.26294 + 1.94682i) q^{29} +(0.488932 - 0.258587i) q^{30} +(-3.49022 + 0.501817i) q^{31} +(-5.59532 + 0.832089i) q^{32} +(-1.22780 - 1.06389i) q^{33} +(0.818764 + 0.840861i) q^{34} +(2.77981 + 1.78648i) q^{35} +(-5.24066 + 2.22666i) q^{36} +(-1.81652 - 0.533379i) q^{37} +(-0.227563 + 2.67982i) q^{38} +(0.419831 - 0.363785i) q^{39} +(-2.73758 - 0.711108i) q^{40} +(-4.95704 + 1.45552i) q^{41} +(1.44840 + 1.11465i) q^{42} +(-10.9976 - 1.58122i) q^{43} +(0.962970 + 8.25187i) q^{44} -2.84704 q^{45} +(-6.63967 - 1.38377i) q^{46} -7.64538i q^{47} +(-1.47545 - 0.520021i) q^{48} +(-0.557710 + 3.87896i) q^{49} +(-1.12075 - 0.862504i) q^{50} +(0.0914419 + 0.311423i) q^{51} +(-2.83976 - 0.0756351i) q^{52} +(-6.03616 - 6.96609i) q^{53} +(-3.22241 - 0.273638i) q^{54} +(-1.17030 + 3.98567i) q^{55} +(-1.62041 - 9.20462i) q^{56} +(-0.402115 + 0.625703i) q^{57} +(4.62363 + 4.74842i) q^{58} +(-1.19442 + 1.37844i) q^{59} +(-0.577302 - 0.527792i) q^{60} +(-0.522738 - 3.63572i) q^{61} +(2.33137 + 4.40812i) q^{62} +(-3.90809 - 8.55751i) q^{63} +(3.77428 + 7.05371i) q^{64} +(-1.29203 - 0.590049i) q^{65} +(-0.831500 + 2.14181i) q^{66} +(-0.686553 - 1.06830i) q^{67} +(0.729447 - 1.49089i) q^{68} +(-1.44426 - 1.19675i) q^{69} +(1.05404 - 4.55266i) q^{70} +(4.82480 + 7.50753i) q^{71} +(5.46216 + 5.91691i) q^{72} +(5.47173 - 11.9814i) q^{73} +(0.155432 + 2.67289i) q^{74} +(-0.162470 - 0.355759i) q^{75} +(3.67665 - 0.974020i) q^{76} +(-13.5864 + 1.95343i) q^{77} +(-0.684446 - 0.385653i) q^{78} +(1.69314 - 1.95399i) q^{79} +(0.357696 + 3.98397i) q^{80} +(6.43286 + 4.13415i) q^{81} +(4.30023 + 5.90674i) q^{82} +(0.649923 - 2.21344i) q^{83} +(0.793962 - 2.45972i) q^{84} +(0.627186 - 0.543460i) q^{85} +(3.13531 + 15.3969i) q^{86} +(0.516381 + 1.75863i) q^{87} +(10.5286 - 5.21449i) q^{88} +(0.927859 + 0.133406i) q^{89} +(1.35671 + 3.79086i) q^{90} -4.69348i q^{91} +(1.32152 + 9.50019i) q^{92} +1.37907i q^{93} +(-10.1799 + 3.64327i) q^{94} +(1.88238 + 0.270646i) q^{95} +(0.0106856 + 2.21238i) q^{96} +(-4.58080 - 15.6008i) q^{97} +(5.43064 - 1.10585i) q^{98} +(8.93781 - 7.74466i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 2 q^{2} - 4 q^{4} - 48 q^{5} + 5 q^{6} - q^{8} - 48 q^{9} + 2 q^{10} + 3 q^{12} + 32 q^{16} + 7 q^{18} - 4 q^{20} + 8 q^{21} + 4 q^{23} + 2 q^{24} - 48 q^{25} + 7 q^{26} + 12 q^{27} + 5 q^{30}+ \cdots - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{22}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.476532 − 1.33151i −0.336959 − 0.941519i
\(3\) 0.0556596 − 0.387121i 0.0321351 − 0.223504i −0.967425 − 0.253157i \(-0.918531\pi\)
0.999560 + 0.0296527i \(0.00944013\pi\)
\(4\) −1.54583 + 1.26901i −0.772917 + 0.634507i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) −0.541979 + 0.110364i −0.221262 + 0.0450560i
\(7\) −2.16390 − 2.49727i −0.817878 − 0.943881i 0.181340 − 0.983420i \(-0.441956\pi\)
−0.999218 + 0.0395393i \(0.987411\pi\)
\(8\) 2.42634 + 1.45357i 0.857842 + 0.513914i
\(9\) 2.73171 + 0.802104i 0.910571 + 0.267368i
\(10\) 0.832359 + 1.14332i 0.263215 + 0.361549i
\(11\) 2.24579 − 3.49451i 0.677130 − 1.05364i −0.317309 − 0.948322i \(-0.602779\pi\)
0.994439 − 0.105313i \(-0.0335845\pi\)
\(12\) 0.405221 + 0.669058i 0.116977 + 0.193140i
\(13\) 1.07346 + 0.930155i 0.297723 + 0.257979i 0.790893 − 0.611955i \(-0.209616\pi\)
−0.493170 + 0.869933i \(0.664162\pi\)
\(14\) −2.29398 + 4.07129i −0.613091 + 1.08810i
\(15\) 0.0556596 + 0.387121i 0.0143712 + 0.0999542i
\(16\) 0.779208 − 3.92337i 0.194802 − 0.980843i
\(17\) −0.754891 + 0.344747i −0.183088 + 0.0836135i −0.504850 − 0.863207i \(-0.668452\pi\)
0.321762 + 0.946821i \(0.395725\pi\)
\(18\) −0.233741 − 4.01953i −0.0550932 − 0.947413i
\(19\) −1.72989 − 0.790012i −0.396863 − 0.181241i 0.206979 − 0.978345i \(-0.433637\pi\)
−0.603842 + 0.797104i \(0.706364\pi\)
\(20\) 1.12569 − 1.65312i 0.251713 − 0.369649i
\(21\) −1.08719 + 0.698694i −0.237244 + 0.152468i
\(22\) −5.72317 − 1.32504i −1.22018 − 0.282499i
\(23\) 2.50326 − 4.09069i 0.521965 − 0.852967i
\(24\) 0.697756 − 0.858384i 0.142429 − 0.175217i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) 0.726974 − 1.87256i 0.142571 − 0.367240i
\(27\) 0.949966 − 2.08014i 0.182821 − 0.400322i
\(28\) 6.51411 + 1.11435i 1.23105 + 0.210593i
\(29\) −4.26294 + 1.94682i −0.791607 + 0.361515i −0.769831 − 0.638248i \(-0.779659\pi\)
−0.0217763 + 0.999763i \(0.506932\pi\)
\(30\) 0.488932 − 0.258587i 0.0892663 − 0.0472113i
\(31\) −3.49022 + 0.501817i −0.626861 + 0.0901291i −0.448422 − 0.893822i \(-0.648014\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(32\) −5.59532 + 0.832089i −0.989123 + 0.147094i
\(33\) −1.22780 − 1.06389i −0.213733 − 0.185200i
\(34\) 0.818764 + 0.840861i 0.140417 + 0.144207i
\(35\) 2.77981 + 1.78648i 0.469874 + 0.301970i
\(36\) −5.24066 + 2.22666i −0.873443 + 0.371111i
\(37\) −1.81652 − 0.533379i −0.298635 − 0.0876870i 0.128983 − 0.991647i \(-0.458829\pi\)
−0.427617 + 0.903960i \(0.640647\pi\)
\(38\) −0.227563 + 2.67982i −0.0369156 + 0.434725i
\(39\) 0.419831 − 0.363785i 0.0672267 − 0.0582523i
\(40\) −2.73758 − 0.711108i −0.432849 − 0.112436i
\(41\) −4.95704 + 1.45552i −0.774160 + 0.227314i −0.644869 − 0.764293i \(-0.723088\pi\)
−0.129290 + 0.991607i \(0.541270\pi\)
\(42\) 1.44840 + 1.11465i 0.223493 + 0.171995i
\(43\) −10.9976 − 1.58122i −1.67712 − 0.241134i −0.762948 − 0.646460i \(-0.776249\pi\)
−0.914172 + 0.405326i \(0.867158\pi\)
\(44\) 0.962970 + 8.25187i 0.145173 + 1.24402i
\(45\) −2.84704 −0.424412
\(46\) −6.63967 − 1.38377i −0.978966 − 0.204025i
\(47\) − 7.64538i − 1.11519i −0.830112 − 0.557597i \(-0.811723\pi\)
0.830112 − 0.557597i \(-0.188277\pi\)
\(48\) −1.47545 − 0.520021i −0.212963 − 0.0750586i
\(49\) −0.557710 + 3.87896i −0.0796729 + 0.554137i
\(50\) −1.12075 − 0.862504i −0.158498 − 0.121977i
\(51\) 0.0914419 + 0.311423i 0.0128044 + 0.0436079i
\(52\) −2.83976 − 0.0756351i −0.393804 − 0.0104887i
\(53\) −6.03616 − 6.96609i −0.829130 − 0.956867i 0.170464 − 0.985364i \(-0.445473\pi\)
−0.999594 + 0.0284972i \(0.990928\pi\)
\(54\) −3.22241 − 0.273638i −0.438514 − 0.0372373i
\(55\) −1.17030 + 3.98567i −0.157803 + 0.537428i
\(56\) −1.62041 − 9.20462i −0.216536 − 1.23002i
\(57\) −0.402115 + 0.625703i −0.0532614 + 0.0828764i
\(58\) 4.62363 + 4.74842i 0.607112 + 0.623498i
\(59\) −1.19442 + 1.37844i −0.155501 + 0.179457i −0.828154 − 0.560500i \(-0.810609\pi\)
0.672654 + 0.739957i \(0.265154\pi\)
\(60\) −0.577302 − 0.527792i −0.0745294 − 0.0681377i
\(61\) −0.522738 − 3.63572i −0.0669297 − 0.465506i −0.995532 − 0.0944288i \(-0.969898\pi\)
0.928602 − 0.371077i \(-0.121012\pi\)
\(62\) 2.33137 + 4.40812i 0.296085 + 0.559832i
\(63\) −3.90809 − 8.55751i −0.492372 − 1.07815i
\(64\) 3.77428 + 7.05371i 0.471786 + 0.881713i
\(65\) −1.29203 − 0.590049i −0.160256 − 0.0731866i
\(66\) −0.831500 + 2.14181i −0.102351 + 0.263638i
\(67\) −0.686553 − 1.06830i −0.0838758 − 0.130513i 0.796781 − 0.604268i \(-0.206535\pi\)
−0.880657 + 0.473755i \(0.842898\pi\)
\(68\) 0.729447 − 1.49089i 0.0884585 − 0.180797i
\(69\) −1.44426 − 1.19675i −0.173869 − 0.144072i
\(70\) 1.05404 − 4.55266i 0.125982 − 0.544147i
\(71\) 4.82480 + 7.50753i 0.572598 + 0.890980i 0.999914 − 0.0131227i \(-0.00417719\pi\)
−0.427316 + 0.904102i \(0.640541\pi\)
\(72\) 5.46216 + 5.91691i 0.643722 + 0.697314i
\(73\) 5.47173 − 11.9814i 0.640417 − 1.40232i −0.259279 − 0.965802i \(-0.583485\pi\)
0.899697 − 0.436516i \(-0.143788\pi\)
\(74\) 0.155432 + 2.67289i 0.0180686 + 0.310717i
\(75\) −0.162470 − 0.355759i −0.0187604 − 0.0410795i
\(76\) 3.67665 − 0.974020i 0.421741 − 0.111728i
\(77\) −13.5864 + 1.95343i −1.54832 + 0.222614i
\(78\) −0.684446 − 0.385653i −0.0774983 − 0.0436666i
\(79\) 1.69314 − 1.95399i 0.190493 − 0.219841i −0.652466 − 0.757818i \(-0.726266\pi\)
0.842960 + 0.537977i \(0.180811\pi\)
\(80\) 0.357696 + 3.98397i 0.0399917 + 0.445422i
\(81\) 6.43286 + 4.13415i 0.714762 + 0.459350i
\(82\) 4.30023 + 5.90674i 0.474880 + 0.652291i
\(83\) 0.649923 − 2.21344i 0.0713384 − 0.242956i −0.916103 − 0.400944i \(-0.868682\pi\)
0.987441 + 0.157987i \(0.0505005\pi\)
\(84\) 0.793962 − 2.45972i 0.0866284 − 0.268378i
\(85\) 0.627186 − 0.543460i 0.0680279 − 0.0589465i
\(86\) 3.13531 + 15.3969i 0.338089 + 1.66029i
\(87\) 0.516381 + 1.75863i 0.0553618 + 0.188545i
\(88\) 10.5286 − 5.21449i 1.12235 − 0.555866i
\(89\) 0.927859 + 0.133406i 0.0983529 + 0.0141410i 0.191316 − 0.981529i \(-0.438725\pi\)
−0.0929626 + 0.995670i \(0.529634\pi\)
\(90\) 1.35671 + 3.79086i 0.143009 + 0.399592i
\(91\) − 4.69348i − 0.492010i
\(92\) 1.32152 + 9.50019i 0.137778 + 0.990463i
\(93\) 1.37907i 0.143003i
\(94\) −10.1799 + 3.64327i −1.04998 + 0.375775i
\(95\) 1.88238 + 0.270646i 0.193129 + 0.0277677i
\(96\) 0.0106856 + 2.21238i 0.00109060 + 0.225800i
\(97\) −4.58080 − 15.6008i −0.465110 − 1.58402i −0.774171 − 0.632976i \(-0.781833\pi\)
0.309061 − 0.951042i \(-0.399985\pi\)
\(98\) 5.43064 − 1.10585i 0.548577 − 0.111708i
\(99\) 8.93781 − 7.74466i 0.898284 − 0.778367i
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 920.2.bb.a.11.18 ✓ 480
8.3 odd 2 920.2.bb.b.11.36 yes 480
23.21 odd 22 920.2.bb.b.251.36 yes 480
184.67 even 22 inner 920.2.bb.a.251.18 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.18 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.18 yes 480 184.67 even 22 inner
920.2.bb.b.11.36 yes 480 8.3 odd 2
920.2.bb.b.251.36 yes 480 23.21 odd 22