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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [368,2,Mod(49,368)] 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("368.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.m (of order \(11\), degree \(10\), not 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 81.2
Character \(\chi\) \(=\) 368.81
Dual form 368.2.m.e.209.2

$q$-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 \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(e\left(\frac{10}{11}\right)\) \(1\)

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 0
\(3\) −0.0226391 + 0.157458i −0.0130707 + 0.0909086i −0.995313 − 0.0967104i \(-0.969168\pi\)
0.982242 + 0.187619i \(0.0600770\pi\)
\(4\) 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 0.210850 − 0.977518i \(-0.432377\pi\)
−0.997576 + 0.0695887i \(0.977831\pi\)
\(8\) 0 0
\(9\) 2.85420 + 0.838068i 0.951399 + 0.279356i
\(10\) 0 0
\(11\) 2.02437 + 1.30099i 0.610372 + 0.392262i 0.808996 − 0.587814i \(-0.200011\pi\)
−0.198625 + 0.980076i \(0.563648\pi\)
\(12\) 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 \(-0.720330\pi\)
0.999762 − 0.0218095i \(-0.00694273\pi\)
\(20\) 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 \(0.111366\pi\)
−0.804796 + 0.593552i \(0.797725\pi\)
\(32\) 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 \(-0.877331\pi\)
0.995026 − 0.0996110i \(-0.0317598\pi\)
\(44\) 0 0
\(45\) 0.111437 0.0166120
\(46\) 0 0
\(47\) 7.00603 1.02193 0.510967 − 0.859600i \(-0.329287\pi\)
0.510967 + 0.859600i \(0.329287\pi\)
\(48\) 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 −0.794093 − 0.607796i \(-0.792054\pi\)
0.714621 + 0.699512i \(0.246599\pi\)
\(60\) 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 −0.956093 − 0.293064i \(-0.905325\pi\)
−0.130595 + 0.991436i \(0.541689\pi\)
\(68\) 0 0
\(69\) 0.0174368 + 0.762710i 0.00209915 + 0.0918195i
\(70\) 0 0
\(71\) −10.6831 + 6.86563i −1.26785 + 0.814800i −0.989339 − 0.145631i \(-0.953479\pi\)
−0.278515 + 0.960432i \(0.589842\pi\)
\(72\) 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 −0.961634 − 0.274337i \(-0.911542\pi\)
0.408400 + 0.912803i \(0.366087\pi\)
\(80\) 0 0
\(81\) 7.38022 + 4.74298i 0.820025 + 0.526998i
\(82\) 0 0
\(83\) 12.0326 + 3.53310i 1.32075 + 0.387808i 0.864764 − 0.502178i \(-0.167468\pi\)
0.455988 + 0.889986i \(0.349286\pi\)
\(84\) 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 368.2.m.e.81.2 30
4.3 odd 2 184.2.i.b.81.2 yes 30
23.2 even 11 inner 368.2.m.e.209.2 30
23.5 odd 22 8464.2.a.ch.1.8 15
23.18 even 11 8464.2.a.cg.1.8 15
92.51 even 22 4232.2.a.ba.1.8 15
92.71 odd 22 184.2.i.b.25.2 ✓ 30
92.87 odd 22 4232.2.a.bb.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.2 ✓ 30 92.71 odd 22
184.2.i.b.81.2 yes 30 4.3 odd 2
368.2.m.e.81.2 30 1.1 even 1 trivial
368.2.m.e.209.2 30 23.2 even 11 inner
4232.2.a.ba.1.8 15 92.51 even 22
4232.2.a.bb.1.8 15 92.87 odd 22
8464.2.a.cg.1.8 15 23.18 even 11
8464.2.a.ch.1.8 15 23.5 odd 22