Show commands: Magma / Pari/GP / SageMath

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 209.2
Character \(\chi\) \(=\) 368.209
Dual form 368.2.m.e.81.2

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

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

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 0.982242 − 0.187619i \(-0.0600770\pi\)
−0.995313 + 0.0967104i \(0.969168\pi\)
\(4\) 0 0
\(5\) 0.0359441 + 0.0105542i 0.0160747 + 0.00471996i 0.289760 − 0.957099i \(-0.406425\pi\)
−0.273685 + 0.961819i \(0.588243\pi\)
\(6\) 0 0
\(7\) −2.08148 + 2.40216i −0.786726 + 0.907930i −0.997576 − 0.0695887i \(-0.977831\pi\)
0.210850 + 0.977518i \(0.432377\pi\)
\(8\) 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 \(-0.563648\pi\)
0.808996 + 0.587814i \(0.200011\pi\)
\(12\) 0 0
\(13\) 4.23678 + 4.88950i 1.17507 + 1.35610i 0.921306 + 0.388838i \(0.127123\pi\)
0.253764 + 0.967266i \(0.418331\pi\)
\(14\) 0 0
\(15\) 0.000848097 − 0.00589864i 0.000218978 − 0.00152302i
\(16\) 0 0
\(17\) 0.586837 − 1.28499i 0.142329 − 0.311657i −0.825021 − 0.565102i \(-0.808837\pi\)
0.967350 + 0.253446i \(0.0815639\pi\)
\(18\) 0 0
\(19\) 1.57592 + 3.45077i 0.361540 + 0.791662i 0.999762 + 0.0218095i \(0.00694273\pi\)
−0.638222 + 0.769852i \(0.720330\pi\)
\(20\) 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 −0.954319 − 0.298791i \(-0.903417\pi\)
0.850757 + 0.525560i \(0.176144\pi\)
\(30\) 0 0
\(31\) 0.749549 − 5.21323i 0.134623 − 0.936323i −0.804796 − 0.593552i \(-0.797725\pi\)
0.939419 − 0.342772i \(-0.111366\pi\)
\(32\) 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 −0.499719 − 0.866188i \(-0.666563\pi\)
0.0479065 + 0.998852i \(0.484745\pi\)
\(38\) 0 0
\(39\) 0.673976 − 0.777810i 0.107923 − 0.124549i
\(40\) 0 0
\(41\) −1.46876 − 0.431268i −0.229382 − 0.0673527i 0.165021 − 0.986290i \(-0.447231\pi\)
−0.394404 + 0.918937i \(0.629049\pi\)
\(42\) 0 0
\(43\) 0.448327 + 3.11818i 0.0683692 + 0.475518i 0.995026 + 0.0996110i \(0.0317598\pi\)
−0.926657 + 0.375907i \(0.877331\pi\)
\(44\) 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 −0.229108 + 0.973401i \(0.573581\pi\)
−0.930888 + 0.365305i \(0.880965\pi\)
\(54\) 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 \(-0.246599\pi\)
−0.794093 + 0.607796i \(0.792054\pi\)
\(60\) 0 0
\(61\) 0.617313 − 4.29351i 0.0790389 − 0.549727i −0.911373 − 0.411581i \(-0.864977\pi\)
0.990412 − 0.138146i \(-0.0441143\pi\)
\(62\) 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 \(-0.541689\pi\)
−0.956093 + 0.293064i \(0.905325\pi\)
\(68\) 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 \(-0.589842\pi\)
−0.989339 + 0.145631i \(0.953479\pi\)
\(72\) 0 0
\(73\) −5.25452 − 11.5058i −0.614995 − 1.34665i −0.919104 − 0.394015i \(-0.871086\pi\)
0.304109 − 0.952637i \(-0.401641\pi\)
\(74\) 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 \(-0.366087\pi\)
−0.961634 + 0.274337i \(0.911542\pi\)
\(80\) 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 \(-0.349286\pi\)
0.864764 + 0.502178i \(0.167468\pi\)
\(84\) 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 −0.960246 − 0.279154i \(-0.909946\pi\)
0.842703 − 0.538379i \(-0.180963\pi\)
\(90\) 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 0.391355 − 0.920240i \(-0.372006\pi\)
−0.168290 + 0.985738i \(0.553825\pi\)
\(98\) 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.209.2 30
4.3 odd 2 184.2.i.b.25.2 ✓ 30
23.9 even 11 8464.2.a.cg.1.8 15
23.12 even 11 inner 368.2.m.e.81.2 30
23.14 odd 22 8464.2.a.ch.1.8 15
92.35 odd 22 184.2.i.b.81.2 yes 30
92.55 odd 22 4232.2.a.bb.1.8 15
92.83 even 22 4232.2.a.ba.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.2 ✓ 30 4.3 odd 2
184.2.i.b.81.2 yes 30 92.35 odd 22
368.2.m.e.81.2 30 23.12 even 11 inner
368.2.m.e.209.2 30 1.1 even 1 trivial
4232.2.a.ba.1.8 15 92.83 even 22
4232.2.a.bb.1.8 15 92.55 odd 22
8464.2.a.cg.1.8 15 23.9 even 11
8464.2.a.ch.1.8 15 23.14 odd 22