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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [184,2,Mod(13,184)] 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("184.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(184, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 184.p (of order \(22\), 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: \(1.46924739719\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 141.16
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.16

$q$-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+(0.851476 - 1.12915i) q^{2} +(0.496871 + 0.226913i) q^{3} +(-0.549977 - 1.92290i) q^{4} +(1.16498 + 1.00946i) q^{5} +(0.679294 - 0.367833i) q^{6} +(0.981244 - 0.630607i) q^{7} +(-2.63954 - 1.01629i) q^{8} +(-1.76919 - 2.04176i) q^{9} +(2.13178 - 0.455908i) q^{10} +(4.28986 + 0.616789i) q^{11} +(0.163063 - 1.08023i) q^{12} +(-1.76371 + 2.74438i) q^{13} +(0.123453 - 1.64492i) q^{14} +(0.349784 + 0.765920i) q^{15} +(-3.39505 + 2.11510i) q^{16} +(-3.79307 - 1.11375i) q^{17} +(-3.81188 + 0.259183i) q^{18} +(0.469533 + 1.59908i) q^{19} +(1.30037 - 2.79531i) q^{20} +(0.630645 - 0.0906731i) q^{21} +(4.34917 - 4.31873i) q^{22} +(-1.22189 + 4.63756i) q^{23} +(-1.08090 - 1.10391i) q^{24} +(-0.373409 - 2.59712i) q^{25} +(1.59707 + 4.32827i) q^{26} +(-0.877433 - 2.98826i) q^{27} +(-1.75225 - 1.54001i) q^{28} +(-1.96335 + 6.68655i) q^{29} +(1.16267 + 0.257203i) q^{30} +(1.28600 + 2.81594i) q^{31} +(-0.502537 + 5.63449i) q^{32} +(1.99155 + 1.27989i) q^{33} +(-4.48730 + 3.33463i) q^{34} +(1.77970 + 0.255882i) q^{35} +(-2.95307 + 4.52489i) q^{36} +(-2.91860 + 2.52898i) q^{37} +(2.20541 + 0.831406i) q^{38} +(-1.49907 + 0.963395i) q^{39} +(-2.04910 - 3.84846i) q^{40} +(4.80593 - 5.54634i) q^{41} +(0.434595 - 0.789301i) q^{42} +(-0.605421 - 0.276487i) q^{43} +(-1.17330 - 8.58818i) q^{44} -4.16452i q^{45} +(4.19611 + 5.32848i) q^{46} +8.53098 q^{47} +(-2.16685 + 0.280547i) q^{48} +(-2.34273 + 5.12986i) q^{49} +(-3.25049 - 1.78975i) q^{50} +(-1.63194 - 1.41409i) q^{51} +(6.24715 + 1.88208i) q^{52} +(-5.72300 - 8.90516i) q^{53} +(-4.12132 - 1.55368i) q^{54} +(4.37497 + 5.04898i) q^{55} +(-3.23091 + 0.667281i) q^{56} +(-0.129556 + 0.901082i) q^{57} +(5.87840 + 7.91036i) q^{58} +(0.863541 - 1.34370i) q^{59} +(1.28041 - 1.09384i) q^{60} +(-1.81729 + 0.829930i) q^{61} +(4.27463 + 0.945619i) q^{62} +(-3.02355 - 0.887795i) q^{63} +(5.93431 + 5.36507i) q^{64} +(-4.82501 + 1.41675i) q^{65} +(3.14095 - 1.15897i) q^{66} +(-2.22718 + 0.320221i) q^{67} +(-0.0555162 + 7.90622i) q^{68} +(-1.65945 + 2.02701i) q^{69} +(1.80430 - 1.79168i) q^{70} +(-1.69240 - 11.7709i) q^{71} +(2.59483 + 7.18730i) q^{72} +(-6.66904 + 1.95821i) q^{73} +(0.370490 + 5.44891i) q^{74} +(0.403785 - 1.37516i) q^{75} +(2.81664 - 1.78232i) q^{76} +(4.59835 - 2.10000i) q^{77} +(-0.188603 + 2.51299i) q^{78} +(-4.03633 - 2.59399i) q^{79} +(-6.09026 - 0.963125i) q^{80} +(-0.911340 + 6.33851i) q^{81} +(-2.17053 - 10.1492i) q^{82} +(13.6936 - 11.8656i) q^{83} +(-0.521195 - 1.16280i) q^{84} +(-3.29456 - 5.12644i) q^{85} +(-0.827697 + 0.448192i) q^{86} +(-2.49280 + 2.87684i) q^{87} +(-10.6964 - 5.98779i) q^{88} +(7.74096 - 16.9503i) q^{89} +(-4.70239 - 3.54599i) q^{90} +3.80511i q^{91} +(9.58956 - 0.200981i) q^{92} +1.69097i q^{93} +(7.26392 - 9.63278i) q^{94} +(-1.06721 + 2.33687i) q^{95} +(-1.52824 + 2.68558i) q^{96} +(-0.978945 + 1.12976i) q^{97} +(3.79763 + 7.01326i) q^{98} +(-6.33025 - 9.85007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q - 11 q^{2} - 11 q^{4} - 14 q^{6} - 22 q^{7} - 14 q^{8} - 9 q^{10} - 12 q^{12} - 3 q^{14} - 22 q^{15} + 5 q^{16} - 18 q^{17} - 4 q^{18} - 27 q^{20} - 42 q^{22} - 16 q^{23} - 22 q^{24} - 4 q^{25} - 16 q^{26}+ \cdots + 109 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.851476 − 1.12915i 0.602085 − 0.798432i
\(3\) 0.496871 + 0.226913i 0.286869 + 0.131009i 0.553652 − 0.832748i \(-0.313234\pi\)
−0.266784 + 0.963756i \(0.585961\pi\)
\(4\) −0.549977 − 1.92290i −0.274988 − 0.961448i
\(5\) 1.16498 + 1.00946i 0.520993 + 0.451443i 0.875226 − 0.483714i \(-0.160712\pi\)
−0.354233 + 0.935157i \(0.615258\pi\)
\(6\) 0.679294 − 0.367833i 0.277321 − 0.150167i
\(7\) 0.981244 − 0.630607i 0.370875 − 0.238347i −0.341905 − 0.939735i \(-0.611072\pi\)
0.712780 + 0.701387i \(0.247436\pi\)
\(8\) −2.63954 − 1.01629i −0.933217 − 0.359313i
\(9\) −1.76919 − 2.04176i −0.589730 − 0.680585i
\(10\) 2.13178 − 0.455908i 0.674129 − 0.144171i
\(11\) 4.28986 + 0.616789i 1.29344 + 0.185969i 0.754459 − 0.656347i \(-0.227899\pi\)
0.538984 + 0.842316i \(0.318808\pi\)
\(12\) 0.163063 − 1.08023i 0.0470723 − 0.311835i
\(13\) −1.76371 + 2.74438i −0.489164 + 0.761154i −0.994828 − 0.101577i \(-0.967611\pi\)
0.505664 + 0.862731i \(0.331248\pi\)
\(14\) 0.123453 − 1.64492i 0.0329943 − 0.439624i
\(15\) 0.349784 + 0.765920i 0.0903138 + 0.197760i
\(16\) −3.39505 + 2.11510i −0.848763 + 0.528774i
\(17\) −3.79307 − 1.11375i −0.919955 − 0.270123i −0.212730 − 0.977111i \(-0.568235\pi\)
−0.707226 + 0.706988i \(0.750054\pi\)
\(18\) −3.81188 + 0.259183i −0.898469 + 0.0610900i
\(19\) 0.469533 + 1.59908i 0.107718 + 0.366855i 0.995656 − 0.0931091i \(-0.0296805\pi\)
−0.887938 + 0.459964i \(0.847862\pi\)
\(20\) 1.30037 − 2.79531i 0.290772 − 0.625050i
\(21\) 0.630645 − 0.0906731i 0.137618 − 0.0197865i
\(22\) 4.34917 − 4.31873i 0.927245 − 0.920757i
\(23\) −1.22189 + 4.63756i −0.254782 + 0.966998i
\(24\) −1.08090 − 1.10391i −0.220638 − 0.225335i
\(25\) −0.373409 − 2.59712i −0.0746818 − 0.519423i
\(26\) 1.59707 + 4.32827i 0.313212 + 0.848843i
\(27\) −0.877433 − 2.98826i −0.168862 − 0.575092i
\(28\) −1.75225 − 1.54001i −0.331145 − 0.291035i
\(29\) −1.96335 + 6.68655i −0.364584 + 1.24166i 0.549279 + 0.835639i \(0.314902\pi\)
−0.913863 + 0.406022i \(0.866916\pi\)
\(30\) 1.16267 + 0.257203i 0.212274 + 0.0469586i
\(31\) 1.28600 + 2.81594i 0.230972 + 0.505759i 0.989261 − 0.146160i \(-0.0466915\pi\)
−0.758289 + 0.651919i \(0.773964\pi\)
\(32\) −0.502537 + 5.63449i −0.0888369 + 0.996046i
\(33\) 1.99155 + 1.27989i 0.346685 + 0.222801i
\(34\) −4.48730 + 3.33463i −0.769566 + 0.571885i
\(35\) 1.77970 + 0.255882i 0.300824 + 0.0432519i
\(36\) −2.95307 + 4.52489i −0.492178 + 0.754148i
\(37\) −2.91860 + 2.52898i −0.479814 + 0.415762i −0.860897 − 0.508779i \(-0.830097\pi\)
0.381082 + 0.924541i \(0.375551\pi\)
\(38\) 2.20541 + 0.831406i 0.357764 + 0.134872i
\(39\) −1.49907 + 0.963395i −0.240044 + 0.154267i
\(40\) −2.04910 − 3.84846i −0.323990 − 0.608494i
\(41\) 4.80593 − 5.54634i 0.750560 − 0.866192i −0.244062 − 0.969760i \(-0.578480\pi\)
0.994622 + 0.103567i \(0.0330257\pi\)
\(42\) 0.434595 − 0.789301i 0.0670595 − 0.121792i
\(43\) −0.605421 − 0.276487i −0.0923258 − 0.0421638i 0.368717 − 0.929542i \(-0.379797\pi\)
−0.461043 + 0.887378i \(0.652524\pi\)
\(44\) −1.17330 − 8.58818i −0.176882 − 1.29472i
\(45\) − 4.16452i − 0.620810i
\(46\) 4.19611 + 5.32848i 0.618682 + 0.785641i
\(47\) 8.53098 1.24437 0.622185 − 0.782870i \(-0.286245\pi\)
0.622185 + 0.782870i \(0.286245\pi\)
\(48\) −2.16685 + 0.280547i −0.312757 + 0.0404935i
\(49\) −2.34273 + 5.12986i −0.334676 + 0.732838i
\(50\) −3.25049 − 1.78975i −0.459689 − 0.253108i
\(51\) −1.63194 − 1.41409i −0.228518 − 0.198012i
\(52\) 6.24715 + 1.88208i 0.866324 + 0.260997i
\(53\) −5.72300 − 8.90516i −0.786114 − 1.22322i −0.970675 − 0.240396i \(-0.922723\pi\)
0.184561 − 0.982821i \(-0.440914\pi\)
\(54\) −4.12132 − 1.55368i −0.560841 − 0.211429i
\(55\) 4.37497 + 5.04898i 0.589921 + 0.680805i
\(56\) −3.23091 + 0.667281i −0.431748 + 0.0891692i
\(57\) −0.129556 + 0.901082i −0.0171601 + 0.119351i
\(58\) 5.87840 + 7.91036i 0.771871 + 1.03868i
\(59\) 0.863541 − 1.34370i 0.112423 − 0.174934i −0.780482 − 0.625178i \(-0.785026\pi\)
0.892906 + 0.450243i \(0.148663\pi\)
\(60\) 1.28041 − 1.09384i 0.165300 − 0.141214i
\(61\) −1.81729 + 0.829930i −0.232681 + 0.106262i −0.528344 − 0.849031i \(-0.677187\pi\)
0.295663 + 0.955292i \(0.404459\pi\)
\(62\) 4.27463 + 0.945619i 0.542879 + 0.120094i
\(63\) −3.02355 − 0.887795i −0.380932 − 0.111852i
\(64\) 5.93431 + 5.36507i 0.741788 + 0.670634i
\(65\) −4.82501 + 1.41675i −0.598469 + 0.175726i
\(66\) 3.14095 − 1.15897i 0.386625 − 0.142659i
\(67\) −2.22718 + 0.320221i −0.272094 + 0.0391212i −0.277011 − 0.960867i \(-0.589344\pi\)
0.00491729 + 0.999988i \(0.498435\pi\)
\(68\) −0.0555162 + 7.90622i −0.00673232 + 0.958770i
\(69\) −1.65945 + 2.02701i −0.199774 + 0.244023i
\(70\) 1.80430 − 1.79168i 0.215655 − 0.214146i
\(71\) −1.69240 − 11.7709i −0.200851 − 1.39695i −0.801766 − 0.597638i \(-0.796106\pi\)
0.600915 − 0.799313i \(-0.294803\pi\)
\(72\) 2.59483 + 7.18730i 0.305803 + 0.847031i
\(73\) −6.66904 + 1.95821i −0.780552 + 0.229191i −0.647650 − 0.761938i \(-0.724248\pi\)
−0.132903 + 0.991129i \(0.542430\pi\)
\(74\) 0.370490 + 5.44891i 0.0430686 + 0.633423i
\(75\) 0.403785 − 1.37516i 0.0466250 − 0.158790i
\(76\) 2.81664 − 1.78232i 0.323090 − 0.204446i
\(77\) 4.59835 − 2.10000i 0.524031 − 0.239317i
\(78\) −0.188603 + 2.51299i −0.0213551 + 0.284540i
\(79\) −4.03633 − 2.59399i −0.454123 − 0.291847i 0.293514 − 0.955955i \(-0.405175\pi\)
−0.747637 + 0.664108i \(0.768812\pi\)
\(80\) −6.09026 − 0.963125i −0.680911 − 0.107681i
\(81\) −0.911340 + 6.33851i −0.101260 + 0.704279i
\(82\) −2.17053 − 10.1492i −0.239695 − 1.12079i
\(83\) 13.6936 − 11.8656i 1.50307 − 1.30242i 0.682462 − 0.730921i \(-0.260909\pi\)
0.820606 − 0.571495i \(-0.193636\pi\)
\(84\) −0.521195 − 1.16280i −0.0568670 − 0.126871i
\(85\) −3.29456 − 5.12644i −0.357345 − 0.556040i
\(86\) −0.827697 + 0.448192i −0.0892529 + 0.0483298i
\(87\) −2.49280 + 2.87684i −0.267256 + 0.308430i
\(88\) −10.6964 − 5.98779i −1.14024 − 0.638300i
\(89\) 7.74096 − 16.9503i 0.820540 − 1.79673i 0.267552 − 0.963543i \(-0.413785\pi\)
0.552988 − 0.833189i \(-0.313487\pi\)
\(90\) −4.70239 − 3.54599i −0.495675 − 0.373780i
\(91\) 3.80511i 0.398884i
\(92\) 9.58956 − 0.200981i 0.999780 − 0.0209537i
\(93\) 1.69097i 0.175346i
\(94\) 7.26392 − 9.63278i 0.749216 − 0.993546i
\(95\) −1.06721 + 2.33687i −0.109494 + 0.239758i
\(96\) −1.52824 + 2.68558i −0.155975 + 0.274096i
\(97\) −0.978945 + 1.12976i −0.0993968 + 0.114710i −0.803268 − 0.595617i \(-0.796908\pi\)
0.703872 + 0.710327i \(0.251453\pi\)
\(98\) 3.79763 + 7.01326i 0.383618 + 0.708446i
\(99\) −6.33025 − 9.85007i −0.636215 − 0.989969i
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 184.2.p.a.141.16 yes 220
4.3 odd 2 736.2.x.a.49.9 220
8.3 odd 2 736.2.x.a.49.14 220
8.5 even 2 inner 184.2.p.a.141.5 yes 220
23.8 even 11 inner 184.2.p.a.77.5 ✓ 220
92.31 odd 22 736.2.x.a.721.14 220
184.77 even 22 inner 184.2.p.a.77.16 yes 220
184.123 odd 22 736.2.x.a.721.9 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.5 ✓ 220 23.8 even 11 inner
184.2.p.a.77.16 yes 220 184.77 even 22 inner
184.2.p.a.141.5 yes 220 8.5 even 2 inner
184.2.p.a.141.16 yes 220 1.1 even 1 trivial
736.2.x.a.49.9 220 4.3 odd 2
736.2.x.a.49.14 220 8.3 odd 2
736.2.x.a.721.9 220 184.123 odd 22
736.2.x.a.721.14 220 92.31 odd 22