Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [114,2,Mod(25,114)] 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("114.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 61.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 114.61
Dual form 114.2.i.a.43.1

$q$-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+(0.939693 + 0.342020i) q^{2} +(0.173648 + 0.984808i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-0.907604 + 0.761570i) q^{5} +(-0.173648 + 0.984808i) q^{6} +(0.733956 - 1.27125i) q^{7} +(0.500000 + 0.866025i) q^{8} +(-0.939693 + 0.342020i) q^{9} +(-1.11334 + 0.405223i) q^{10} +(-0.592396 - 1.02606i) q^{11} +(-0.500000 + 0.866025i) q^{12} +(0.446967 - 2.53487i) q^{13} +(1.12449 - 0.943555i) q^{14} +(-0.907604 - 0.761570i) q^{15} +(0.173648 + 0.984808i) q^{16} +(-2.09240 - 0.761570i) q^{17} -1.00000 q^{18} +(0.819078 - 4.28125i) q^{19} -1.18479 q^{20} +(1.37939 + 0.502055i) q^{21} +(-0.205737 - 1.16679i) q^{22} +(0.907604 + 0.761570i) q^{23} +(-0.766044 + 0.642788i) q^{24} +(-0.624485 + 3.54163i) q^{25} +(1.28699 - 2.22913i) q^{26} +(-0.500000 - 0.866025i) q^{27} +(1.37939 - 0.502055i) q^{28} +(-8.84389 + 3.21891i) q^{29} +(-0.592396 - 1.02606i) q^{30} +(3.96451 - 6.86673i) q^{31} +(-0.173648 + 0.984808i) q^{32} +(0.907604 - 0.761570i) q^{33} +(-1.70574 - 1.43128i) q^{34} +(0.302004 + 1.71275i) q^{35} +(-0.939693 - 0.342020i) q^{36} -0.0641778 q^{37} +(2.23396 - 3.74292i) q^{38} +2.57398 q^{39} +(-1.11334 - 0.405223i) q^{40} +(1.68092 + 9.53298i) q^{41} +(1.12449 + 0.943555i) q^{42} +(-5.55303 + 4.65955i) q^{43} +(0.205737 - 1.16679i) q^{44} +(0.592396 - 1.02606i) q^{45} +(0.592396 + 1.02606i) q^{46} +(4.57145 - 1.66387i) q^{47} +(-0.939693 + 0.342020i) q^{48} +(2.42262 + 4.19610i) q^{49} +(-1.79813 + 3.11446i) q^{50} +(0.386659 - 2.19285i) q^{51} +(1.97178 - 1.65452i) q^{52} +(-1.11334 - 0.934204i) q^{53} +(-0.173648 - 0.984808i) q^{54} +(1.31908 + 0.480105i) q^{55} +1.46791 q^{56} +(4.35844 + 0.0632028i) q^{57} -9.41147 q^{58} +(-1.31908 - 0.480105i) q^{59} +(-0.205737 - 1.16679i) q^{60} +(5.97565 + 5.01417i) q^{61} +(6.07398 - 5.09667i) q^{62} +(-0.254900 + 1.44561i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(1.52481 + 2.64106i) q^{65} +(1.11334 - 0.405223i) q^{66} +(-8.19119 + 2.98135i) q^{67} +(-1.11334 - 1.92836i) q^{68} +(-0.592396 + 1.02606i) q^{69} +(-0.302004 + 1.71275i) q^{70} +(11.2135 - 9.40923i) q^{71} +(-0.766044 - 0.642788i) q^{72} +(1.13563 + 6.44047i) q^{73} +(-0.0603074 - 0.0219501i) q^{74} -3.59627 q^{75} +(3.37939 - 2.75314i) q^{76} -1.73917 q^{77} +(2.41875 + 0.880352i) q^{78} +(2.24035 + 12.7057i) q^{79} +(-0.907604 - 0.761570i) q^{80} +(0.766044 - 0.642788i) q^{81} +(-1.68092 + 9.53298i) q^{82} +(-1.94949 + 3.37662i) q^{83} +(0.733956 + 1.27125i) q^{84} +(2.47906 - 0.902302i) q^{85} +(-6.81180 + 2.47929i) q^{86} +(-4.70574 - 8.15058i) q^{87} +(0.592396 - 1.02606i) q^{88} +(2.15523 - 12.2229i) q^{89} +(0.907604 - 0.761570i) q^{90} +(-2.89440 - 2.42869i) q^{91} +(0.205737 + 1.16679i) q^{92} +(7.45084 + 2.71188i) q^{93} +4.86484 q^{94} +(2.51707 + 4.50946i) q^{95} -1.00000 q^{96} +(-12.1356 - 4.41701i) q^{97} +(0.841367 + 4.77163i) q^{98} +(0.907604 + 0.761570i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{5} + 9 q^{7} + 3 q^{8} - 3 q^{12} + 15 q^{13} - 6 q^{14} - 9 q^{15} - 9 q^{17} - 6 q^{18} - 12 q^{19} - 3 q^{21} + 9 q^{22} + 9 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{28} - 9 q^{29} - 9 q^{31} + 9 q^{33}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 + 0.342020i 0.664463 + 0.241845i
\(3\) 0.173648 + 0.984808i 0.100256 + 0.568579i
\(4\) 0.766044 + 0.642788i 0.383022 + 0.321394i
\(5\) −0.907604 + 0.761570i −0.405893 + 0.340584i −0.822766 − 0.568380i \(-0.807570\pi\)
0.416873 + 0.908965i \(0.363126\pi\)
\(6\) −0.173648 + 0.984808i −0.0708916 + 0.402046i
\(7\) 0.733956 − 1.27125i 0.277409 − 0.480487i −0.693331 − 0.720619i \(-0.743858\pi\)
0.970740 + 0.240133i \(0.0771909\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) −0.939693 + 0.342020i −0.313231 + 0.114007i
\(10\) −1.11334 + 0.405223i −0.352069 + 0.128143i
\(11\) −0.592396 − 1.02606i −0.178614 − 0.309369i 0.762792 − 0.646644i \(-0.223828\pi\)
−0.941406 + 0.337275i \(0.890495\pi\)
\(12\) −0.500000 + 0.866025i −0.144338 + 0.250000i
\(13\) 0.446967 − 2.53487i 0.123966 − 0.703047i −0.857950 − 0.513733i \(-0.828262\pi\)
0.981916 − 0.189315i \(-0.0606266\pi\)
\(14\) 1.12449 − 0.943555i 0.300531 − 0.252176i
\(15\) −0.907604 − 0.761570i −0.234342 − 0.196637i
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) −2.09240 − 0.761570i −0.507481 − 0.184708i 0.0755749 − 0.997140i \(-0.475921\pi\)
−0.583056 + 0.812432i \(0.698143\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.819078 − 4.28125i 0.187909 − 0.982186i
\(20\) −1.18479 −0.264928
\(21\) 1.37939 + 0.502055i 0.301007 + 0.109557i
\(22\) −0.205737 − 1.16679i −0.0438633 − 0.248761i
\(23\) 0.907604 + 0.761570i 0.189248 + 0.158798i 0.732489 − 0.680779i \(-0.238358\pi\)
−0.543241 + 0.839577i \(0.682803\pi\)
\(24\) −0.766044 + 0.642788i −0.156368 + 0.131208i
\(25\) −0.624485 + 3.54163i −0.124897 + 0.708326i
\(26\) 1.28699 − 2.22913i 0.252399 − 0.437168i
\(27\) −0.500000 − 0.866025i −0.0962250 − 0.166667i
\(28\) 1.37939 − 0.502055i 0.260679 − 0.0948795i
\(29\) −8.84389 + 3.21891i −1.64227 + 0.597737i −0.987434 − 0.158034i \(-0.949485\pi\)
−0.654836 + 0.755771i \(0.727262\pi\)
\(30\) −0.592396 − 1.02606i −0.108156 − 0.187332i
\(31\) 3.96451 − 6.86673i 0.712047 − 1.23330i −0.252041 − 0.967717i \(-0.581102\pi\)
0.964088 − 0.265584i \(-0.0855649\pi\)
\(32\) −0.173648 + 0.984808i −0.0306970 + 0.174091i
\(33\) 0.907604 − 0.761570i 0.157994 − 0.132572i
\(34\) −1.70574 − 1.43128i −0.292531 − 0.245463i
\(35\) 0.302004 + 1.71275i 0.0510479 + 0.289507i
\(36\) −0.939693 − 0.342020i −0.156615 − 0.0570034i
\(37\) −0.0641778 −0.0105508 −0.00527538 − 0.999986i \(-0.501679\pi\)
−0.00527538 + 0.999986i \(0.501679\pi\)
\(38\) 2.23396 − 3.74292i 0.362395 − 0.607182i
\(39\) 2.57398 0.412166
\(40\) −1.11334 − 0.405223i −0.176035 − 0.0640714i
\(41\) 1.68092 + 9.53298i 0.262516 + 1.48880i 0.776017 + 0.630713i \(0.217237\pi\)
−0.513501 + 0.858089i \(0.671652\pi\)
\(42\) 1.12449 + 0.943555i 0.173512 + 0.145594i
\(43\) −5.55303 + 4.65955i −0.846830 + 0.710574i −0.959089 − 0.283104i \(-0.908636\pi\)
0.112259 + 0.993679i \(0.464191\pi\)
\(44\) 0.205737 − 1.16679i 0.0310160 − 0.175901i
\(45\) 0.592396 − 1.02606i 0.0883092 − 0.152956i
\(46\) 0.592396 + 1.02606i 0.0873441 + 0.151284i
\(47\) 4.57145 − 1.66387i 0.666815 − 0.242701i 0.0136389 − 0.999907i \(-0.495658\pi\)
0.653176 + 0.757206i \(0.273436\pi\)
\(48\) −0.939693 + 0.342020i −0.135633 + 0.0493664i
\(49\) 2.42262 + 4.19610i 0.346088 + 0.599443i
\(50\) −1.79813 + 3.11446i −0.254294 + 0.440451i
\(51\) 0.386659 − 2.19285i 0.0541431 − 0.307061i
\(52\) 1.97178 − 1.65452i 0.273437 − 0.229441i
\(53\) −1.11334 − 0.934204i −0.152929 − 0.128323i 0.563113 − 0.826380i \(-0.309604\pi\)
−0.716042 + 0.698057i \(0.754048\pi\)
\(54\) −0.173648 − 0.984808i −0.0236305 − 0.134015i
\(55\) 1.31908 + 0.480105i 0.177864 + 0.0647374i
\(56\) 1.46791 0.196158
\(57\) 4.35844 + 0.0632028i 0.577290 + 0.00837141i
\(58\) −9.41147 −1.23579
\(59\) −1.31908 − 0.480105i −0.171729 − 0.0625044i 0.254725 − 0.967014i \(-0.418015\pi\)
−0.426454 + 0.904509i \(0.640237\pi\)
\(60\) −0.205737 − 1.16679i −0.0265605 − 0.150632i
\(61\) 5.97565 + 5.01417i 0.765104 + 0.641998i 0.939450 − 0.342686i \(-0.111337\pi\)
−0.174346 + 0.984684i \(0.555781\pi\)
\(62\) 6.07398 − 5.09667i 0.771396 − 0.647278i
\(63\) −0.254900 + 1.44561i −0.0321144 + 0.182130i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 1.52481 + 2.64106i 0.189130 + 0.327583i
\(66\) 1.11334 − 0.405223i 0.137043 − 0.0498795i
\(67\) −8.19119 + 2.98135i −1.00071 + 0.364230i −0.789859 − 0.613288i \(-0.789847\pi\)
−0.210854 + 0.977518i \(0.567624\pi\)
\(68\) −1.11334 − 1.92836i −0.135012 − 0.233848i
\(69\) −0.592396 + 1.02606i −0.0713161 + 0.123523i
\(70\) −0.302004 + 1.71275i −0.0360963 + 0.204713i
\(71\) 11.2135 − 9.40923i 1.33079 − 1.11667i 0.346904 − 0.937901i \(-0.387233\pi\)
0.983891 − 0.178769i \(-0.0572114\pi\)
\(72\) −0.766044 − 0.642788i −0.0902792 − 0.0757532i
\(73\) 1.13563 + 6.44047i 0.132915 + 0.753801i 0.976289 + 0.216473i \(0.0694552\pi\)
−0.843373 + 0.537328i \(0.819434\pi\)
\(74\) −0.0603074 − 0.0219501i −0.00701059 − 0.00255165i
\(75\) −3.59627 −0.415261
\(76\) 3.37939 − 2.75314i 0.387642 − 0.315806i
\(77\) −1.73917 −0.198197
\(78\) 2.41875 + 0.880352i 0.273869 + 0.0996803i
\(79\) 2.24035 + 12.7057i 0.252059 + 1.42950i 0.803510 + 0.595292i \(0.202963\pi\)
−0.551450 + 0.834208i \(0.685925\pi\)
\(80\) −0.907604 − 0.761570i −0.101473 − 0.0851461i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) −1.68092 + 9.53298i −0.185627 + 1.05274i
\(83\) −1.94949 + 3.37662i −0.213985 + 0.370632i −0.952958 − 0.303102i \(-0.901978\pi\)
0.738973 + 0.673735i \(0.235311\pi\)
\(84\) 0.733956 + 1.27125i 0.0800811 + 0.138705i
\(85\) 2.47906 − 0.902302i 0.268891 − 0.0978684i
\(86\) −6.81180 + 2.47929i −0.734536 + 0.267349i
\(87\) −4.70574 − 8.15058i −0.504508 − 0.873833i
\(88\) 0.592396 − 1.02606i 0.0631497 − 0.109378i
\(89\) 2.15523 − 12.2229i 0.228454 − 1.29563i −0.627517 − 0.778603i \(-0.715929\pi\)
0.855971 − 0.517024i \(-0.172960\pi\)
\(90\) 0.907604 − 0.761570i 0.0956698 − 0.0802765i
\(91\) −2.89440 − 2.42869i −0.303416 − 0.254596i
\(92\) 0.205737 + 1.16679i 0.0214496 + 0.121647i
\(93\) 7.45084 + 2.71188i 0.772616 + 0.281209i
\(94\) 4.86484 0.501770
\(95\) 2.51707 + 4.50946i 0.258246 + 0.462661i
\(96\) −1.00000 −0.102062
\(97\) −12.1356 − 4.41701i −1.23219 − 0.448479i −0.357841 − 0.933783i \(-0.616487\pi\)
−0.874346 + 0.485303i \(0.838709\pi\)
\(98\) 0.841367 + 4.77163i 0.0849909 + 0.482007i
\(99\) 0.907604 + 0.761570i 0.0912176 + 0.0765407i
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 114.2.i.a.61.1 yes 6
3.2 odd 2 342.2.u.e.289.1 6
4.3 odd 2 912.2.bo.a.289.1 6
19.5 even 9 inner 114.2.i.a.43.1 ✓ 6
19.9 even 9 2166.2.a.q.1.2 3
19.10 odd 18 2166.2.a.s.1.2 3
57.5 odd 18 342.2.u.e.271.1 6
57.29 even 18 6498.2.a.bm.1.2 3
57.47 odd 18 6498.2.a.br.1.2 3
76.43 odd 18 912.2.bo.a.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.a.43.1 ✓ 6 19.5 even 9 inner
114.2.i.a.61.1 yes 6 1.1 even 1 trivial
342.2.u.e.271.1 6 57.5 odd 18
342.2.u.e.289.1 6 3.2 odd 2
912.2.bo.a.289.1 6 4.3 odd 2
912.2.bo.a.385.1 6 76.43 odd 18
2166.2.a.q.1.2 3 19.9 even 9
2166.2.a.s.1.2 3 19.10 odd 18
6498.2.a.bm.1.2 3 57.29 even 18
6498.2.a.br.1.2 3 57.47 odd 18