Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [115,2,Mod(4,115)] 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("115.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.j (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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 29.6
Character \(\chi\) \(=\) 115.29
Dual form 115.2.j.a.4.6

$q$-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+(0.175347 + 0.0800783i) q^{2} +(-0.167074 + 0.569000i) q^{3} +(-1.28539 - 1.48342i) q^{4} +(2.18641 + 0.468631i) q^{5} +(-0.0748604 + 0.0863935i) q^{6} +(3.28793 + 0.472733i) q^{7} +(-0.215217 - 0.732961i) q^{8} +(2.22791 + 1.43179i) q^{9} +(0.345853 + 0.257257i) q^{10} +(-0.928361 - 2.03283i) q^{11} +(1.05882 - 0.483546i) q^{12} +(-6.70483 + 0.964009i) q^{13} +(0.538673 + 0.346184i) q^{14} +(-0.631942 + 1.16577i) q^{15} +(-0.537726 + 3.73997i) q^{16} +(-1.46759 - 1.27168i) q^{17} +(0.276002 + 0.429468i) q^{18} +(-1.11465 - 1.28637i) q^{19} +(-2.11521 - 3.84573i) q^{20} +(-0.818312 + 1.79185i) q^{21} -0.430792i q^{22} +(-1.75636 - 4.46264i) q^{23} +0.453012 q^{24} +(4.56077 + 2.04924i) q^{25} +(-1.25287 - 0.367875i) q^{26} +(-2.53144 + 2.19351i) q^{27} +(-3.52501 - 5.48502i) q^{28} +(-4.03098 + 4.65200i) q^{29} +(-0.204162 + 0.153810i) q^{30} +(5.15374 - 1.51327i) q^{31} +(-1.21977 + 1.89801i) q^{32} +(1.31178 - 0.188606i) q^{33} +(-0.155504 - 0.340507i) q^{34} +(6.96723 + 2.57442i) q^{35} +(-0.739787 - 5.14533i) q^{36} +(-1.72562 + 2.68512i) q^{37} +(-0.0924398 - 0.314821i) q^{38} +(0.571678 - 3.97611i) q^{39} +(-0.127064 - 1.70341i) q^{40} +(-5.62116 + 3.61250i) q^{41} +(-0.286977 + 0.248667i) q^{42} +(-2.05844 + 7.01041i) q^{43} +(-1.82223 + 3.99012i) q^{44} +(4.20015 + 4.17455i) q^{45} +(0.0493883 - 0.923158i) q^{46} -9.63634i q^{47} +(-2.03820 - 0.930816i) q^{48} +(3.87057 + 1.13650i) q^{49} +(0.635618 + 0.724546i) q^{50} +(0.968780 - 0.622597i) q^{51} +(10.0483 + 8.70693i) q^{52} +(-1.44130 - 0.207228i) q^{53} +(-0.619533 + 0.181911i) q^{54} +(-1.07713 - 4.87965i) q^{55} +(-0.361123 - 2.51167i) q^{56} +(0.918175 - 0.419317i) q^{57} +(-1.07935 + 0.492921i) q^{58} +(-1.33162 - 9.26159i) q^{59} +(2.54162 - 0.561035i) q^{60} +(10.2194 - 3.00068i) q^{61} +(1.02487 + 0.147355i) q^{62} +(6.64837 + 5.76085i) q^{63} +(5.99136 - 3.85041i) q^{64} +(-15.1113 - 1.03437i) q^{65} +(0.245121 + 0.0719739i) q^{66} +(4.44321 + 2.02915i) q^{67} +3.81165i q^{68} +(2.83269 - 0.253780i) q^{69} +(1.01553 + 1.00934i) q^{70} +(-0.0212260 + 0.0464785i) q^{71} +(0.569964 - 1.94112i) q^{72} +(4.98167 - 4.31664i) q^{73} +(-0.517603 + 0.332643i) q^{74} +(-1.92800 + 2.25271i) q^{75} +(-0.475472 + 3.30698i) q^{76} +(-2.09140 - 7.12267i) q^{77} +(0.418642 - 0.651420i) q^{78} +(1.74906 + 12.1650i) q^{79} +(-2.92835 + 7.92510i) q^{80} +(2.47529 + 5.42013i) q^{81} +(-1.27494 + 0.183308i) q^{82} +(3.94928 - 6.14520i) q^{83} +(3.70991 - 1.08933i) q^{84} +(-2.61281 - 3.46816i) q^{85} +(-0.922323 + 1.06442i) q^{86} +(-1.97352 - 3.07086i) q^{87} +(-1.29018 + 1.11795i) q^{88} +(-4.61729 - 1.35576i) q^{89} +(0.402192 + 1.06834i) q^{90} -22.5007 q^{91} +(-4.36235 + 8.34164i) q^{92} +3.18531i q^{93} +(0.771662 - 1.68970i) q^{94} +(-1.83424 - 3.33490i) q^{95} +(-0.876174 - 1.01116i) q^{96} +(-1.58420 - 2.46507i) q^{97} +(0.587684 + 0.509231i) q^{98} +(0.842280 - 5.85818i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 14 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9} - 13 q^{10} - 26 q^{11} - 26 q^{14} - 10 q^{15} - 18 q^{16} - 14 q^{19} + 49 q^{20} - 22 q^{21} - 68 q^{24} + 21 q^{25} - 42 q^{26} - 24 q^{29} + 19 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.175347 + 0.0800783i 0.123989 + 0.0566239i 0.476444 − 0.879205i \(-0.341926\pi\)
−0.352455 + 0.935829i \(0.614653\pi\)
\(3\) −0.167074 + 0.569000i −0.0964600 + 0.328512i −0.993559 − 0.113317i \(-0.963852\pi\)
0.897099 + 0.441830i \(0.145670\pi\)
\(4\) −1.28539 − 1.48342i −0.642694 − 0.741708i
\(5\) 2.18641 + 0.468631i 0.977792 + 0.209578i
\(6\) −0.0748604 + 0.0863935i −0.0305616 + 0.0352700i
\(7\) 3.28793 + 0.472733i 1.24272 + 0.178676i 0.732141 − 0.681153i \(-0.238521\pi\)
0.510581 + 0.859830i \(0.329430\pi\)
\(8\) −0.215217 − 0.732961i −0.0760906 − 0.259141i
\(9\) 2.22791 + 1.43179i 0.742638 + 0.477264i
\(10\) 0.345853 + 0.257257i 0.109368 + 0.0813518i
\(11\) −0.928361 − 2.03283i −0.279911 − 0.612921i 0.716498 − 0.697589i \(-0.245744\pi\)
−0.996409 + 0.0846689i \(0.973017\pi\)
\(12\) 1.05882 − 0.483546i 0.305655 − 0.139588i
\(13\) −6.70483 + 0.964009i −1.85959 + 0.267368i −0.978599 − 0.205775i \(-0.934029\pi\)
−0.880986 + 0.473143i \(0.843119\pi\)
\(14\) 0.538673 + 0.346184i 0.143966 + 0.0925216i
\(15\) −0.631942 + 1.16577i −0.163167 + 0.301001i
\(16\) −0.537726 + 3.73997i −0.134432 + 0.934992i
\(17\) −1.46759 − 1.27168i −0.355944 − 0.308427i 0.458472 − 0.888709i \(-0.348397\pi\)
−0.814416 + 0.580282i \(0.802942\pi\)
\(18\) 0.276002 + 0.429468i 0.0650544 + 0.101227i
\(19\) −1.11465 − 1.28637i −0.255718 − 0.295114i 0.613346 − 0.789815i \(-0.289823\pi\)
−0.869064 + 0.494700i \(0.835278\pi\)
\(20\) −2.11521 − 3.84573i −0.472975 − 0.859931i
\(21\) −0.818312 + 1.79185i −0.178570 + 0.391014i
\(22\) − 0.430792i − 0.0918451i
\(23\) −1.75636 − 4.46264i −0.366227 − 0.930526i
\(24\) 0.453012 0.0924707
\(25\) 4.56077 + 2.04924i 0.912154 + 0.409848i
\(26\) −1.25287 − 0.367875i −0.245708 − 0.0721463i
\(27\) −2.53144 + 2.19351i −0.487177 + 0.422141i
\(28\) −3.52501 − 5.48502i −0.666164 − 1.03657i
\(29\) −4.03098 + 4.65200i −0.748535 + 0.863856i −0.994425 − 0.105442i \(-0.966374\pi\)
0.245890 + 0.969298i \(0.420920\pi\)
\(30\) −0.204162 + 0.153810i −0.0372747 + 0.0280817i
\(31\) 5.15374 − 1.51327i 0.925639 − 0.271792i 0.216029 − 0.976387i \(-0.430689\pi\)
0.709610 + 0.704595i \(0.248871\pi\)
\(32\) −1.21977 + 1.89801i −0.215628 + 0.335523i
\(33\) 1.31178 − 0.188606i 0.228352 − 0.0328321i
\(34\) −0.155504 − 0.340507i −0.0266688 − 0.0583965i
\(35\) 6.96723 + 2.57442i 1.17768 + 0.435156i
\(36\) −0.739787 − 5.14533i −0.123298 − 0.857555i
\(37\) −1.72562 + 2.68512i −0.283691 + 0.441432i −0.953629 − 0.300986i \(-0.902684\pi\)
0.669938 + 0.742417i \(0.266321\pi\)
\(38\) −0.0924398 − 0.314821i −0.0149957 − 0.0510707i
\(39\) 0.571678 − 3.97611i 0.0915418 − 0.636687i
\(40\) −0.127064 − 1.70341i −0.0200905 − 0.269333i
\(41\) −5.62116 + 3.61250i −0.877878 + 0.564178i −0.900153 − 0.435574i \(-0.856545\pi\)
0.0222748 + 0.999752i \(0.492909\pi\)
\(42\) −0.286977 + 0.248667i −0.0442815 + 0.0383701i
\(43\) −2.05844 + 7.01041i −0.313909 + 1.06908i 0.639848 + 0.768501i \(0.278997\pi\)
−0.953758 + 0.300576i \(0.902821\pi\)
\(44\) −1.82223 + 3.99012i −0.274711 + 0.601533i
\(45\) 4.20015 + 4.17455i 0.626121 + 0.622306i
\(46\) 0.0493883 − 0.923158i 0.00728190 − 0.136112i
\(47\) − 9.63634i − 1.40560i −0.711385 − 0.702802i \(-0.751932\pi\)
0.711385 − 0.702802i \(-0.248068\pi\)
\(48\) −2.03820 − 0.930816i −0.294189 − 0.134352i
\(49\) 3.87057 + 1.13650i 0.552939 + 0.162357i
\(50\) 0.635618 + 0.724546i 0.0898899 + 0.102466i
\(51\) 0.968780 − 0.622597i 0.135656 − 0.0871811i
\(52\) 10.0483 + 8.70693i 1.39345 + 1.20743i
\(53\) −1.44130 − 0.207228i −0.197978 − 0.0284649i 0.0426124 − 0.999092i \(-0.486432\pi\)
−0.240590 + 0.970627i \(0.577341\pi\)
\(54\) −0.619533 + 0.181911i −0.0843078 + 0.0247550i
\(55\) −1.07713 − 4.87965i −0.145240 − 0.657972i
\(56\) −0.361123 − 2.51167i −0.0482571 − 0.335636i
\(57\) 0.918175 − 0.419317i 0.121615 − 0.0555399i
\(58\) −1.07935 + 0.492921i −0.141725 + 0.0647236i
\(59\) −1.33162 − 9.26159i −0.173362 − 1.20576i −0.871719 − 0.490007i \(-0.836994\pi\)
0.698357 − 0.715749i \(-0.253915\pi\)
\(60\) 2.54162 − 0.561035i 0.328121 − 0.0724292i
\(61\) 10.2194 − 3.00068i 1.30846 − 0.384198i 0.448144 − 0.893962i \(-0.352085\pi\)
0.860315 + 0.509764i \(0.170267\pi\)
\(62\) 1.02487 + 0.147355i 0.130159 + 0.0187140i
\(63\) 6.64837 + 5.76085i 0.837616 + 0.725798i
\(64\) 5.99136 − 3.85041i 0.748919 − 0.481301i
\(65\) −15.1113 − 1.03437i −1.87432 − 0.128298i
\(66\) 0.245121 + 0.0719739i 0.0301723 + 0.00885937i
\(67\) 4.44321 + 2.02915i 0.542824 + 0.247900i 0.667904 − 0.744247i \(-0.267192\pi\)
−0.125080 + 0.992147i \(0.539919\pi\)
\(68\) 3.81165i 0.462230i
\(69\) 2.83269 − 0.253780i 0.341015 − 0.0305515i
\(70\) 1.01553 + 1.00934i 0.121379 + 0.120639i
\(71\) −0.0212260 + 0.0464785i −0.00251907 + 0.00551599i −0.910887 − 0.412655i \(-0.864601\pi\)
0.908368 + 0.418171i \(0.137329\pi\)
\(72\) 0.569964 − 1.94112i 0.0671709 − 0.228763i
\(73\) 4.98167 − 4.31664i 0.583061 − 0.505225i −0.312646 − 0.949870i \(-0.601216\pi\)
0.895707 + 0.444645i \(0.146670\pi\)
\(74\) −0.517603 + 0.332643i −0.0601701 + 0.0386690i
\(75\) −1.92800 + 2.25271i −0.222626 + 0.260120i
\(76\) −0.475472 + 3.30698i −0.0545403 + 0.379336i
\(77\) −2.09140 − 7.12267i −0.238338 − 0.811703i
\(78\) 0.418642 − 0.651420i 0.0474019 − 0.0737588i
\(79\) 1.74906 + 12.1650i 0.196784 + 1.36867i 0.813539 + 0.581510i \(0.197538\pi\)
−0.616754 + 0.787156i \(0.711553\pi\)
\(80\) −2.92835 + 7.92510i −0.327400 + 0.886054i
\(81\) 2.47529 + 5.42013i 0.275033 + 0.602237i
\(82\) −1.27494 + 0.183308i −0.140793 + 0.0202430i
\(83\) 3.94928 − 6.14520i 0.433490 − 0.674523i −0.553944 − 0.832554i \(-0.686878\pi\)
0.987434 + 0.158030i \(0.0505144\pi\)
\(84\) 3.70991 − 1.08933i 0.404785 − 0.118855i
\(85\) −2.61281 − 3.46816i −0.283399 − 0.376175i
\(86\) −0.922323 + 1.06442i −0.0994566 + 0.114779i
\(87\) −1.97352 − 3.07086i −0.211584 − 0.329231i
\(88\) −1.29018 + 1.11795i −0.137534 + 0.119174i
\(89\) −4.61729 − 1.35576i −0.489432 − 0.143710i 0.0276993 − 0.999616i \(-0.491182\pi\)
−0.517131 + 0.855906i \(0.673000\pi\)
\(90\) 0.402192 + 1.06834i 0.0423947 + 0.112612i
\(91\) −22.5007 −2.35872
\(92\) −4.36235 + 8.34164i −0.454807 + 0.869676i
\(93\) 3.18531i 0.330301i
\(94\) 0.771662 − 1.68970i 0.0795908 − 0.174280i
\(95\) −1.83424 − 3.33490i −0.188190 − 0.342153i
\(96\) −0.876174 − 1.01116i −0.0894242 − 0.103201i
\(97\) −1.58420 − 2.46507i −0.160851 − 0.250290i 0.751467 − 0.659770i \(-0.229346\pi\)
−0.912319 + 0.409480i \(0.865710\pi\)
\(98\) 0.587684 + 0.509231i 0.0593650 + 0.0514401i
\(99\) 0.842280 − 5.85818i 0.0846523 − 0.588770i
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 115.2.j.a.29.6 yes 100
5.2 odd 4 575.2.k.g.351.5 100
5.3 odd 4 575.2.k.g.351.6 100
5.4 even 2 inner 115.2.j.a.29.5 yes 100
23.4 even 11 inner 115.2.j.a.4.5 ✓ 100
115.4 even 22 inner 115.2.j.a.4.6 yes 100
115.27 odd 44 575.2.k.g.326.5 100
115.73 odd 44 575.2.k.g.326.6 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.4.5 ✓ 100 23.4 even 11 inner
115.2.j.a.4.6 yes 100 115.4 even 22 inner
115.2.j.a.29.5 yes 100 5.4 even 2 inner
115.2.j.a.29.6 yes 100 1.1 even 1 trivial
575.2.k.g.326.5 100 115.27 odd 44
575.2.k.g.326.6 100 115.73 odd 44
575.2.k.g.351.5 100 5.2 odd 4
575.2.k.g.351.6 100 5.3 odd 4