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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 11.2
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.2

$q$-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+(-1.41405 - 0.0214454i) q^{2} +(-0.417492 + 2.90372i) q^{3} +(1.99908 + 0.0606497i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(0.652627 - 4.09706i) q^{6} +(2.59694 + 2.99703i) q^{7} +(-2.82550 - 0.128633i) q^{8} +(-5.37883 - 1.57937i) q^{9} +(1.36281 - 0.377808i) q^{10} +(1.45984 - 2.27156i) q^{11} +(-1.01071 + 5.77946i) q^{12} +(1.80368 + 1.56290i) q^{13} +(-3.60793 - 4.29364i) q^{14} +(-0.417492 - 2.90372i) q^{15} +(3.99264 + 0.242487i) q^{16} +(-6.50839 + 2.97228i) q^{17} +(7.57208 + 2.34866i) q^{18} +(4.78524 + 2.18535i) q^{19} +(-1.93519 + 0.505013i) q^{20} +(-9.78674 + 6.28955i) q^{21} +(-2.11301 + 3.18079i) q^{22} +(-2.91434 + 3.80876i) q^{23} +(1.55314 - 8.15077i) q^{24} +(0.841254 - 0.540641i) q^{25} +(-2.51698 - 2.24870i) q^{26} +(3.17571 - 6.95383i) q^{27} +(5.00972 + 6.14880i) q^{28} +(-3.38405 + 1.54544i) q^{29} +(0.528084 + 4.11497i) q^{30} +(-8.17242 + 1.17502i) q^{31} +(-5.64060 - 0.428513i) q^{32} +(5.98650 + 5.18734i) q^{33} +(9.26694 - 4.06338i) q^{34} +(-3.33610 - 2.14398i) q^{35} +(-10.6569 - 3.48351i) q^{36} +(-2.31872 - 0.680838i) q^{37} +(-6.71971 - 3.19281i) q^{38} +(-5.29125 + 4.58489i) q^{39} +(2.74729 - 0.672613i) q^{40} +(7.76984 - 2.28143i) q^{41} +(13.9738 - 8.68387i) q^{42} +(4.51454 + 0.649093i) q^{43} +(3.05611 - 4.45249i) q^{44} +5.60591 q^{45} +(4.20270 - 5.32328i) q^{46} -2.82882i q^{47} +(-2.37101 + 11.4923i) q^{48} +(-1.24187 + 8.63743i) q^{49} +(-1.20117 + 0.746453i) q^{50} +(-5.91348 - 20.1395i) q^{51} +(3.51091 + 3.23375i) q^{52} +(6.73907 + 7.77731i) q^{53} +(-4.63974 + 9.76497i) q^{54} +(-0.760736 + 2.59083i) q^{55} +(-6.95213 - 8.80215i) q^{56} +(-8.34345 + 12.9827i) q^{57} +(4.81836 - 2.11276i) q^{58} +(0.524253 - 0.605020i) q^{59} +(-0.658491 - 5.83010i) q^{60} +(-0.543398 - 3.77942i) q^{61} +(11.5814 - 1.48627i) q^{62} +(-9.23509 - 20.2220i) q^{63} +(7.96691 + 0.726904i) q^{64} +(-2.17094 - 0.991434i) q^{65} +(-8.35398 - 7.46354i) q^{66} +(-8.78544 - 13.6704i) q^{67} +(-13.1911 + 5.54710i) q^{68} +(-9.84286 - 10.0526i) q^{69} +(4.67144 + 3.10325i) q^{70} +(5.47514 + 8.51948i) q^{71} +(14.9947 + 5.15440i) q^{72} +(1.80492 - 3.95221i) q^{73} +(3.26419 + 1.01247i) q^{74} +(1.21865 + 2.66848i) q^{75} +(9.43354 + 4.65891i) q^{76} +(10.5990 - 1.52391i) q^{77} +(7.58042 - 6.36980i) q^{78} +(-5.70803 + 6.58742i) q^{79} +(-3.89923 + 0.892193i) q^{80} +(4.71819 + 3.03220i) q^{81} +(-11.0359 + 3.05943i) q^{82} +(3.96423 - 13.5009i) q^{83} +(-19.9459 + 11.9798i) q^{84} +(5.40737 - 4.68551i) q^{85} +(-6.36987 - 1.01467i) q^{86} +(-3.07472 - 10.4716i) q^{87} +(-4.41698 + 6.23051i) q^{88} +(-17.0823 - 2.45607i) q^{89} +(-7.92705 - 0.120221i) q^{90} +9.46443i q^{91} +(-6.05700 + 7.43726i) q^{92} -24.2210i q^{93} +(-0.0606650 + 4.00009i) q^{94} +(-5.20709 - 0.748667i) q^{95} +(3.59919 - 16.1998i) q^{96} +(-2.42670 - 8.26459i) q^{97} +(1.94131 - 12.1871i) q^{98} +(-11.4399 + 9.91271i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41405 − 0.0214454i −0.999885 − 0.0151642i
\(3\) −0.417492 + 2.90372i −0.241039 + 1.67647i 0.405895 + 0.913920i \(0.366960\pi\)
−0.646934 + 0.762546i \(0.723949\pi\)
\(4\) 1.99908 + 0.0606497i 0.999540 + 0.0303249i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) 0.652627 − 4.09706i 0.266434 − 1.67262i
\(7\) 2.59694 + 2.99703i 0.981550 + 1.13277i 0.991141 + 0.132815i \(0.0424017\pi\)
−0.00959057 + 0.999954i \(0.503053\pi\)
\(8\) −2.82550 − 0.128633i −0.998965 − 0.0454786i
\(9\) −5.37883 − 1.57937i −1.79294 − 0.526456i
\(10\) 1.36281 − 0.377808i 0.430960 − 0.119473i
\(11\) 1.45984 − 2.27156i 0.440159 − 0.684901i −0.548318 − 0.836270i \(-0.684732\pi\)
0.988477 + 0.151369i \(0.0483682\pi\)
\(12\) −1.01071 + 5.77946i −0.291767 + 1.66839i
\(13\) 1.80368 + 1.56290i 0.500251 + 0.433470i 0.868081 − 0.496423i \(-0.165354\pi\)
−0.367830 + 0.929893i \(0.619899\pi\)
\(14\) −3.60793 − 4.29364i −0.964260 − 1.14752i
\(15\) −0.417492 − 2.90372i −0.107796 − 0.749738i
\(16\) 3.99264 + 0.242487i 0.998161 + 0.0606218i
\(17\) −6.50839 + 2.97228i −1.57852 + 0.720884i −0.995780 − 0.0917766i \(-0.970745\pi\)
−0.582737 + 0.812661i \(0.698018\pi\)
\(18\) 7.57208 + 2.34866i 1.78476 + 0.553584i
\(19\) 4.78524 + 2.18535i 1.09781 + 0.501353i 0.880162 − 0.474674i \(-0.157434\pi\)
0.217648 + 0.976027i \(0.430161\pi\)
\(20\) −1.93519 + 0.505013i −0.432722 + 0.112924i
\(21\) −9.78674 + 6.28955i −2.13564 + 1.37249i
\(22\) −2.11301 + 3.18079i −0.450494 + 0.678147i
\(23\) −2.91434 + 3.80876i −0.607682 + 0.794181i
\(24\) 1.55314 − 8.15077i 0.317033 − 1.66377i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) −2.51698 − 2.24870i −0.493620 − 0.441006i
\(27\) 3.17571 − 6.95383i 0.611165 − 1.33827i
\(28\) 5.00972 + 6.14880i 0.946748 + 1.16201i
\(29\) −3.38405 + 1.54544i −0.628402 + 0.286982i −0.704047 − 0.710153i \(-0.748626\pi\)
0.0756452 + 0.997135i \(0.475898\pi\)
\(30\) 0.528084 + 4.11497i 0.0964145 + 0.751287i
\(31\) −8.17242 + 1.17502i −1.46781 + 0.211039i −0.829399 − 0.558656i \(-0.811317\pi\)
−0.638411 + 0.769696i \(0.720408\pi\)
\(32\) −5.64060 − 0.428513i −0.997127 − 0.0757511i
\(33\) 5.98650 + 5.18734i 1.04212 + 0.902999i
\(34\) 9.26694 − 4.06338i 1.58927 − 0.696865i
\(35\) −3.33610 − 2.14398i −0.563904 − 0.362399i
\(36\) −10.6569 − 3.48351i −1.77616 − 0.580585i
\(37\) −2.31872 − 0.680838i −0.381195 − 0.111929i 0.0855203 − 0.996336i \(-0.472745\pi\)
−0.466716 + 0.884407i \(0.654563\pi\)
\(38\) −6.71971 − 3.19281i −1.09008 − 0.517943i
\(39\) −5.29125 + 4.58489i −0.847278 + 0.734170i
\(40\) 2.74729 − 0.672613i 0.434384 − 0.106349i
\(41\) 7.76984 − 2.28143i 1.21344 − 0.356299i 0.388466 − 0.921463i \(-0.373005\pi\)
0.824979 + 0.565164i \(0.191187\pi\)
\(42\) 13.9738 − 8.68387i 2.15621 − 1.33995i
\(43\) 4.51454 + 0.649093i 0.688461 + 0.0989858i 0.477662 − 0.878544i \(-0.341484\pi\)
0.210799 + 0.977529i \(0.432393\pi\)
\(44\) 3.05611 − 4.45249i 0.460726 − 0.671238i
\(45\) 5.60591 0.835680
\(46\) 4.20270 − 5.32328i 0.619655 − 0.784874i
\(47\) − 2.82882i − 0.412625i −0.978486 − 0.206313i \(-0.933854\pi\)
0.978486 − 0.206313i \(-0.0661464\pi\)
\(48\) −2.37101 + 11.4923i −0.342226 + 1.65877i
\(49\) −1.24187 + 8.63743i −0.177411 + 1.23392i
\(50\) −1.20117 + 0.746453i −0.169871 + 0.105564i
\(51\) −5.91348 − 20.1395i −0.828053 − 2.82009i
\(52\) 3.51091 + 3.23375i 0.486876 + 0.448441i
\(53\) 6.73907 + 7.77731i 0.925683 + 1.06830i 0.997485 + 0.0708763i \(0.0225796\pi\)
−0.0718021 + 0.997419i \(0.522875\pi\)
\(54\) −4.63974 + 9.76497i −0.631389 + 1.32884i
\(55\) −0.760736 + 2.59083i −0.102578 + 0.349347i
\(56\) −6.95213 − 8.80215i −0.929018 − 1.17624i
\(57\) −8.34345 + 12.9827i −1.10512 + 1.71960i
\(58\) 4.81836 − 2.11276i 0.632681 − 0.277419i
\(59\) 0.524253 − 0.605020i 0.0682519 − 0.0787669i −0.720597 − 0.693354i \(-0.756132\pi\)
0.788849 + 0.614587i \(0.210678\pi\)
\(60\) −0.658491 − 5.83010i −0.0850108 − 0.752662i
\(61\) −0.543398 − 3.77942i −0.0695750 − 0.483905i −0.994582 − 0.103955i \(-0.966850\pi\)
0.925007 − 0.379950i \(-0.124059\pi\)
\(62\) 11.5814 − 1.48627i 1.47084 − 0.188757i
\(63\) −9.23509 − 20.2220i −1.16351 − 2.54774i
\(64\) 7.96691 + 0.726904i 0.995863 + 0.0908630i
\(65\) −2.17094 − 0.991434i −0.269272 − 0.122972i
\(66\) −8.35398 − 7.46354i −1.02830 − 0.918698i
\(67\) −8.78544 − 13.6704i −1.07331 − 1.67011i −0.638607 − 0.769533i \(-0.720489\pi\)
−0.434705 − 0.900573i \(-0.643148\pi\)
\(68\) −13.1911 + 5.54710i −1.59965 + 0.672685i
\(69\) −9.84286 − 10.0526i −1.18494 − 1.21019i
\(70\) 4.67144 + 3.10325i 0.558344 + 0.370909i
\(71\) 5.47514 + 8.51948i 0.649779 + 1.01108i 0.997302 + 0.0734047i \(0.0233865\pi\)
−0.347523 + 0.937671i \(0.612977\pi\)
\(72\) 14.9947 + 5.15440i 1.76715 + 0.607452i
\(73\) 1.80492 − 3.95221i 0.211249 − 0.462572i −0.774112 − 0.633048i \(-0.781803\pi\)
0.985362 + 0.170477i \(0.0545308\pi\)
\(74\) 3.26419 + 1.01247i 0.379454 + 0.117697i
\(75\) 1.21865 + 2.66848i 0.140718 + 0.308130i
\(76\) 9.43354 + 4.65891i 1.08210 + 0.534413i
\(77\) 10.5990 − 1.52391i 1.20787 − 0.173666i
\(78\) 7.58042 − 6.36980i 0.858313 − 0.721238i
\(79\) −5.70803 + 6.58742i −0.642204 + 0.741143i −0.979763 − 0.200161i \(-0.935853\pi\)
0.337559 + 0.941304i \(0.390399\pi\)
\(80\) −3.89923 + 0.892193i −0.435947 + 0.0997502i
\(81\) 4.71819 + 3.03220i 0.524243 + 0.336911i
\(82\) −11.0359 + 3.05943i −1.21871 + 0.337858i
\(83\) 3.96423 − 13.5009i 0.435131 − 1.48192i −0.392029 − 0.919953i \(-0.628227\pi\)
0.827160 − 0.561967i \(-0.189955\pi\)
\(84\) −19.9459 + 11.9798i −2.17628 + 1.30710i
\(85\) 5.40737 − 4.68551i 0.586511 − 0.508215i
\(86\) −6.36987 − 1.01467i −0.686881 − 0.109414i
\(87\) −3.07472 − 10.4716i −0.329645 − 1.12267i
\(88\) −4.41698 + 6.23051i −0.470852 + 0.664174i
\(89\) −17.0823 − 2.45607i −1.81072 − 0.260343i −0.847838 − 0.530256i \(-0.822096\pi\)
−0.962885 + 0.269913i \(0.913005\pi\)
\(90\) −7.92705 − 0.120221i −0.835584 − 0.0126724i
\(91\) 9.46443i 0.992141i
\(92\) −6.05700 + 7.43726i −0.631486 + 0.775388i
\(93\) − 24.2210i − 2.51160i
\(94\) −0.0606650 + 4.00009i −0.00625712 + 0.412578i
\(95\) −5.20709 − 0.748667i −0.534236 − 0.0768116i
\(96\) 3.59919 − 16.1998i 0.367341 − 1.65339i
\(97\) −2.42670 − 8.26459i −0.246394 − 0.839142i −0.986092 − 0.166202i \(-0.946850\pi\)
0.739697 − 0.672940i \(-0.234969\pi\)
\(98\) 1.94131 − 12.1871i 0.196102 − 1.23109i
\(99\) −11.4399 + 9.91271i −1.14975 + 0.996265i
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.11.2 ✓ 480
8.3 odd 2 920.2.bb.b.11.20 yes 480
23.21 odd 22 920.2.bb.b.251.20 yes 480
184.67 even 22 inner 920.2.bb.a.251.2 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.2 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.2 yes 480 184.67 even 22 inner
920.2.bb.b.11.20 yes 480 8.3 odd 2
920.2.bb.b.251.20 yes 480 23.21 odd 22