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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [184,2,Mod(9,184)] 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("184.9"); 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, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 184.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.46924739719\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 81.2
Character \(\chi\) \(=\) 184.81
Dual form 184.2.i.b.25.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0226391 - 0.157458i) q^{3} +(0.0359441 - 0.0105542i) q^{5} +(2.08148 + 2.40216i) q^{7} +(2.85420 + 0.838068i) q^{9} +(-2.02437 - 1.30099i) q^{11} +(4.23678 - 4.88950i) q^{13} +(-0.000848097 - 0.00589864i) q^{15} +(0.586837 + 1.28499i) q^{17} +(-1.57592 + 3.45077i) q^{19} +(0.425362 - 0.273364i) q^{21} +(-4.73018 + 0.790836i) q^{23} +(-4.20509 + 2.70245i) q^{25} +(0.394827 - 0.864550i) q^{27} +(-0.557699 - 1.22119i) q^{29} +(-0.749549 - 5.21323i) q^{31} +(-0.250681 + 0.289301i) q^{33} +(0.100170 + 0.0643752i) q^{35} +(-2.74827 - 0.806964i) q^{37} +(-0.673976 - 0.777810i) q^{39} +(-1.46876 + 0.431268i) q^{41} +(-0.448327 + 3.11818i) q^{43} +0.111437 q^{45} -7.00603 q^{47} +(-0.441591 + 3.07133i) q^{49} +(0.215618 - 0.0633113i) q^{51} +(-8.44489 - 9.74593i) q^{53} +(-0.0864952 - 0.0253973i) q^{55} +(0.507676 + 0.326263i) q^{57} +(0.610442 - 0.704487i) q^{59} +(0.617313 + 4.29351i) q^{61} +(3.92779 + 8.60065i) q^{63} +(0.100683 - 0.220465i) q^{65} +(8.89493 - 5.71642i) q^{67} +(0.0174368 + 0.762710i) q^{69} +(10.6831 - 6.86563i) q^{71} +(-5.25452 + 11.5058i) q^{73} +(0.330323 + 0.723307i) q^{75} +(-1.08852 - 7.57084i) q^{77} +(4.91725 - 5.67481i) q^{79} +(7.38022 + 4.74298i) q^{81} +(-12.0326 - 3.53310i) q^{83} +(0.0346554 + 0.0399944i) q^{85} +(-0.204912 + 0.0601677i) q^{87} +(-1.10890 + 7.71259i) q^{89} +20.5641 q^{91} -0.837836 q^{93} +(-0.0202249 + 0.140667i) q^{95} +(2.19694 - 0.645079i) q^{97} +(-4.68765 - 5.40983i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 2 q^{3} - 13 q^{7} + 21 q^{9} + 2 q^{11} - 2 q^{15} - 22 q^{17} + 3 q^{19} + 2 q^{21} + q^{23} + 13 q^{25} - 31 q^{27} + 7 q^{29} + 18 q^{31} - 8 q^{33} + 41 q^{35} - 62 q^{37} + 6 q^{39} - 15 q^{41}+ \cdots - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{10}{11}\right)\)

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 0
\(3\) 0.0226391 − 0.157458i 0.0130707 − 0.0909086i −0.982242 − 0.187619i \(-0.939923\pi\)
0.995313 + 0.0967104i \(0.0308321\pi\)
\(4\) 0 0
\(5\) 0.0359441 − 0.0105542i 0.0160747 − 0.00471996i −0.273685 − 0.961819i \(-0.588243\pi\)
0.289760 + 0.957099i \(0.406425\pi\)
\(6\) 0 0
\(7\) 2.08148 + 2.40216i 0.786726 + 0.907930i 0.997576 − 0.0695887i \(-0.0221687\pi\)
−0.210850 + 0.977518i \(0.567623\pi\)
\(8\) 0 0
\(9\) 2.85420 + 0.838068i 0.951399 + 0.279356i
\(10\) 0 0
\(11\) −2.02437 − 1.30099i −0.610372 − 0.392262i 0.198625 − 0.980076i \(-0.436352\pi\)
−0.808996 + 0.587814i \(0.799989\pi\)
\(12\) 0 0
\(13\) 4.23678 − 4.88950i 1.17507 − 1.35610i 0.253764 − 0.967266i \(-0.418331\pi\)
0.921306 − 0.388838i \(-0.127123\pi\)
\(14\) 0 0
\(15\) −0.000848097 − 0.00589864i −0.000218978 − 0.00152302i
\(16\) 0 0
\(17\) 0.586837 + 1.28499i 0.142329 + 0.311657i 0.967350 − 0.253446i \(-0.0815639\pi\)
−0.825021 + 0.565102i \(0.808837\pi\)
\(18\) 0 0
\(19\) −1.57592 + 3.45077i −0.361540 + 0.791662i 0.638222 + 0.769852i \(0.279670\pi\)
−0.999762 + 0.0218095i \(0.993057\pi\)
\(20\) 0 0
\(21\) 0.425362 − 0.273364i 0.0928217 − 0.0596529i
\(22\) 0 0
\(23\) −4.73018 + 0.790836i −0.986310 + 0.164901i
\(24\) 0 0
\(25\) −4.20509 + 2.70245i −0.841017 + 0.540489i
\(26\) 0 0
\(27\) 0.394827 − 0.864550i 0.0759845 − 0.166383i
\(28\) 0 0
\(29\) −0.557699 − 1.22119i −0.103562 − 0.226769i 0.850757 − 0.525560i \(-0.176144\pi\)
−0.954319 + 0.298791i \(0.903417\pi\)
\(30\) 0 0
\(31\) −0.749549 − 5.21323i −0.134623 − 0.936323i −0.939419 − 0.342772i \(-0.888634\pi\)
0.804796 − 0.593552i \(-0.202275\pi\)
\(32\) 0 0
\(33\) −0.250681 + 0.289301i −0.0436380 + 0.0503609i
\(34\) 0 0
\(35\) 0.100170 + 0.0643752i 0.0169318 + 0.0108814i
\(36\) 0 0
\(37\) −2.74827 − 0.806964i −0.451812 − 0.132664i 0.0479065 − 0.998852i \(-0.484745\pi\)
−0.499719 + 0.866188i \(0.666563\pi\)
\(38\) 0 0
\(39\) −0.673976 − 0.777810i −0.107923 − 0.124549i
\(40\) 0 0
\(41\) −1.46876 + 0.431268i −0.229382 + 0.0673527i −0.394404 − 0.918937i \(-0.629049\pi\)
0.165021 + 0.986290i \(0.447231\pi\)
\(42\) 0 0
\(43\) −0.448327 + 3.11818i −0.0683692 + 0.475518i 0.926657 + 0.375907i \(0.122669\pi\)
−0.995026 + 0.0996110i \(0.968240\pi\)
\(44\) 0 0
\(45\) 0.111437 0.0166120
\(46\) 0 0
\(47\) −7.00603 −1.02193 −0.510967 − 0.859600i \(-0.670713\pi\)
−0.510967 + 0.859600i \(0.670713\pi\)
\(48\) 0 0
\(49\) −0.441591 + 3.07133i −0.0630844 + 0.438762i
\(50\) 0 0
\(51\) 0.215618 − 0.0633113i 0.0301926 − 0.00886535i
\(52\) 0 0
\(53\) −8.44489 − 9.74593i −1.16000 − 1.33871i −0.930888 − 0.365305i \(-0.880965\pi\)
−0.229108 − 0.973401i \(-0.573581\pi\)
\(54\) 0 0
\(55\) −0.0864952 − 0.0253973i −0.0116630 − 0.00342457i
\(56\) 0 0
\(57\) 0.507676 + 0.326263i 0.0672433 + 0.0432146i
\(58\) 0 0
\(59\) 0.610442 − 0.704487i 0.0794727 − 0.0917164i −0.714621 − 0.699512i \(-0.753401\pi\)
0.794093 + 0.607796i \(0.207946\pi\)
\(60\) 0 0
\(61\) 0.617313 + 4.29351i 0.0790389 + 0.549727i 0.990412 + 0.138146i \(0.0441143\pi\)
−0.911373 + 0.411581i \(0.864977\pi\)
\(62\) 0 0
\(63\) 3.92779 + 8.60065i 0.494855 + 1.08358i
\(64\) 0 0
\(65\) 0.100683 − 0.220465i 0.0124882 − 0.0273453i
\(66\) 0 0
\(67\) 8.89493 − 5.71642i 1.08669 − 0.698372i 0.130595 − 0.991436i \(-0.458311\pi\)
0.956093 + 0.293064i \(0.0946748\pi\)
\(68\) 0 0
\(69\) 0.0174368 + 0.762710i 0.00209915 + 0.0918195i
\(70\) 0 0
\(71\) 10.6831 − 6.86563i 1.26785 − 0.814800i 0.278515 − 0.960432i \(-0.410158\pi\)
0.989339 + 0.145631i \(0.0465214\pi\)
\(72\) 0 0
\(73\) −5.25452 + 11.5058i −0.614995 + 1.34665i 0.304109 + 0.952637i \(0.401641\pi\)
−0.919104 + 0.394015i \(0.871086\pi\)
\(74\) 0 0
\(75\) 0.330323 + 0.723307i 0.0381425 + 0.0835203i
\(76\) 0 0
\(77\) −1.08852 − 7.57084i −0.124049 − 0.862777i
\(78\) 0 0
\(79\) 4.91725 − 5.67481i 0.553234 − 0.638466i −0.408400 − 0.912803i \(-0.633913\pi\)
0.961634 + 0.274337i \(0.0884585\pi\)
\(80\) 0 0
\(81\) 7.38022 + 4.74298i 0.820025 + 0.526998i
\(82\) 0 0
\(83\) −12.0326 − 3.53310i −1.32075 − 0.387808i −0.455988 − 0.889986i \(-0.650714\pi\)
−0.864764 + 0.502178i \(0.832532\pi\)
\(84\) 0 0
\(85\) 0.0346554 + 0.0399944i 0.00375890 + 0.00433800i
\(86\) 0 0
\(87\) −0.204912 + 0.0601677i −0.0219689 + 0.00645066i
\(88\) 0 0
\(89\) −1.10890 + 7.71259i −0.117543 + 0.817533i 0.842703 + 0.538379i \(0.180963\pi\)
−0.960246 + 0.279154i \(0.909946\pi\)
\(90\) 0 0
\(91\) 20.5641 2.15571
\(92\) 0 0
\(93\) −0.837836 −0.0868795
\(94\) 0 0
\(95\) −0.0202249 + 0.140667i −0.00207503 + 0.0144322i
\(96\) 0 0
\(97\) 2.19694 − 0.645079i 0.223065 − 0.0654979i −0.168290 − 0.985738i \(-0.553825\pi\)
0.391355 + 0.920240i \(0.372006\pi\)
\(98\) 0 0
\(99\) −4.68765 − 5.40983i −0.471126 − 0.543709i
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.i.b.81.2 yes 30
4.3 odd 2 368.2.m.e.81.2 30
23.2 even 11 inner 184.2.i.b.25.2 ✓ 30
23.5 odd 22 4232.2.a.ba.1.8 15
23.18 even 11 4232.2.a.bb.1.8 15
92.51 even 22 8464.2.a.ch.1.8 15
92.71 odd 22 368.2.m.e.209.2 30
92.87 odd 22 8464.2.a.cg.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.2 ✓ 30 23.2 even 11 inner
184.2.i.b.81.2 yes 30 1.1 even 1 trivial
368.2.m.e.81.2 30 4.3 odd 2
368.2.m.e.209.2 30 92.71 odd 22
4232.2.a.ba.1.8 15 23.5 odd 22
4232.2.a.bb.1.8 15 23.18 even 11
8464.2.a.cg.1.8 15 92.87 odd 22
8464.2.a.ch.1.8 15 92.51 even 22