Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [138,2,Mod(13,138)] 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("138.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(138, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,1,-1] 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: \(1.10193554789\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 31.1
Root \(0.142315 - 0.989821i\) of defining polynomial
Character \(\chi\) \(=\) 138.31
Dual form 138.2.e.c.49.1

$q$-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+(0.142315 - 0.989821i) q^{2} +(0.415415 + 0.909632i) q^{3} +(-0.959493 - 0.281733i) q^{4} +(1.59283 + 1.83823i) q^{5} +(0.959493 - 0.281733i) q^{6} +(1.56130 + 1.00339i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(-0.654861 + 0.755750i) q^{9} +(2.04620 - 1.31501i) q^{10} +(-0.790323 - 5.49682i) q^{11} +(-0.142315 - 0.989821i) q^{12} +(0.966031 - 0.620830i) q^{13} +(1.21537 - 1.40261i) q^{14} +(-1.01042 + 2.21252i) q^{15} +(0.841254 + 0.540641i) q^{16} +(-7.20366 + 2.11519i) q^{17} +(0.654861 + 0.755750i) q^{18} +(4.29408 + 1.26086i) q^{19} +(-1.01042 - 2.21252i) q^{20} +(-0.264125 + 1.83703i) q^{21} -5.55334 q^{22} +(-2.45093 - 4.12225i) q^{23} -1.00000 q^{24} +(-0.130388 + 0.906870i) q^{25} +(-0.477031 - 1.04455i) q^{26} +(-0.959493 - 0.281733i) q^{27} +(-1.21537 - 1.40261i) q^{28} +(-2.81913 + 0.827771i) q^{29} +(2.04620 + 1.31501i) q^{30} +(-0.863019 + 1.88975i) q^{31} +(0.654861 - 0.755750i) q^{32} +(4.67177 - 3.00236i) q^{33} +(1.06847 + 7.43136i) q^{34} +(0.642438 + 4.46825i) q^{35} +(0.841254 - 0.540641i) q^{36} +(-5.06718 + 5.84784i) q^{37} +(1.85913 - 4.07093i) q^{38} +(0.966031 + 0.620830i) q^{39} +(-2.33380 + 0.685265i) q^{40} +(-0.450833 - 0.520289i) q^{41} +(1.78074 + 0.522874i) q^{42} +(-3.55862 - 7.79229i) q^{43} +(-0.790323 + 5.49682i) q^{44} -2.43232 q^{45} +(-4.42909 + 1.83933i) q^{46} +7.09992 q^{47} +(-0.142315 + 0.989821i) q^{48} +(-1.47703 - 3.23425i) q^{49} +(0.879083 + 0.258122i) q^{50} +(-4.91655 - 5.67400i) q^{51} +(-1.10181 + 0.323520i) q^{52} +(4.53311 + 2.91326i) q^{53} +(-0.415415 + 0.909632i) q^{54} +(8.84555 - 10.2083i) q^{55} +(-1.56130 + 1.00339i) q^{56} +(0.636911 + 4.42981i) q^{57} +(0.418141 + 2.90824i) q^{58} +(-3.39399 + 2.18119i) q^{59} +(1.59283 - 1.83823i) q^{60} +(-0.157543 + 0.344971i) q^{61} +(1.74769 + 1.12317i) q^{62} +(-1.78074 + 0.522874i) q^{63} +(-0.654861 - 0.755750i) q^{64} +(2.67995 + 0.786905i) q^{65} +(-2.30694 - 5.05150i) q^{66} +(-0.420243 + 2.92285i) q^{67} +7.50778 q^{68} +(2.73158 - 3.94189i) q^{69} +4.51420 q^{70} +(2.14473 - 14.9169i) q^{71} +(-0.415415 - 0.909632i) q^{72} +(11.0242 + 3.23701i) q^{73} +(5.06718 + 5.84784i) q^{74} +(-0.879083 + 0.258122i) q^{75} +(-3.76492 - 2.41956i) q^{76} +(4.28150 - 9.37518i) q^{77} +(0.751992 - 0.867845i) q^{78} +(8.86394 - 5.69651i) q^{79} +(0.346156 + 2.40757i) q^{80} +(-0.142315 - 0.989821i) q^{81} +(-0.579153 + 0.372199i) q^{82} +(-6.84228 + 7.89641i) q^{83} +(0.770978 - 1.68821i) q^{84} +(-15.3624 - 9.87283i) q^{85} +(-8.21942 + 2.41344i) q^{86} +(-1.92408 - 2.22050i) q^{87} +(5.32839 + 1.56456i) q^{88} +(4.47642 + 9.80199i) q^{89} +(-0.346156 + 2.40757i) q^{90} +2.13120 q^{91} +(1.19028 + 4.64578i) q^{92} -2.07749 q^{93} +(1.01042 - 7.02765i) q^{94} +(4.52201 + 9.90183i) q^{95} +(0.959493 + 0.281733i) q^{96} +(6.51094 + 7.51402i) q^{97} +(-3.41153 + 1.00172i) q^{98} +(4.67177 + 3.00236i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9} + 11 q^{10} + 11 q^{11} - q^{12} - 13 q^{13} + 13 q^{14} - 11 q^{15} - q^{16} + q^{18} - 2 q^{19} - 11 q^{20} + 9 q^{21} - 22 q^{22}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.142315 − 0.989821i 0.100632 − 0.699909i
\(3\) 0.415415 + 0.909632i 0.239840 + 0.525176i
\(4\) −0.959493 − 0.281733i −0.479746 − 0.140866i
\(5\) 1.59283 + 1.83823i 0.712337 + 0.822080i 0.990363 − 0.138493i \(-0.0442259\pi\)
−0.278027 + 0.960573i \(0.589680\pi\)
\(6\) 0.959493 − 0.281733i 0.391711 − 0.115017i
\(7\) 1.56130 + 1.00339i 0.590116 + 0.379245i 0.801354 − 0.598190i \(-0.204113\pi\)
−0.211238 + 0.977435i \(0.567750\pi\)
\(8\) −0.415415 + 0.909632i −0.146871 + 0.321603i
\(9\) −0.654861 + 0.755750i −0.218287 + 0.251917i
\(10\) 2.04620 − 1.31501i 0.647065 − 0.415844i
\(11\) −0.790323 − 5.49682i −0.238291 − 1.65735i −0.660479 − 0.750844i \(-0.729647\pi\)
0.422188 − 0.906508i \(-0.361262\pi\)
\(12\) −0.142315 − 0.989821i −0.0410828 − 0.285737i
\(13\) 0.966031 − 0.620830i 0.267929 − 0.172187i −0.399777 − 0.916612i \(-0.630913\pi\)
0.667706 + 0.744425i \(0.267276\pi\)
\(14\) 1.21537 − 1.40261i 0.324821 − 0.374864i
\(15\) −1.01042 + 2.21252i −0.260890 + 0.571270i
\(16\) 0.841254 + 0.540641i 0.210313 + 0.135160i
\(17\) −7.20366 + 2.11519i −1.74714 + 0.513008i −0.990101 − 0.140357i \(-0.955175\pi\)
−0.757043 + 0.653365i \(0.773357\pi\)
\(18\) 0.654861 + 0.755750i 0.154352 + 0.178132i
\(19\) 4.29408 + 1.26086i 0.985130 + 0.289260i 0.734340 − 0.678781i \(-0.237492\pi\)
0.250789 + 0.968042i \(0.419310\pi\)
\(20\) −1.01042 − 2.21252i −0.225938 − 0.494734i
\(21\) −0.264125 + 1.83703i −0.0576368 + 0.400873i
\(22\) −5.55334 −1.18398
\(23\) −2.45093 − 4.12225i −0.511054 − 0.859548i
\(24\) −1.00000 −0.204124
\(25\) −0.130388 + 0.906870i −0.0260776 + 0.181374i
\(26\) −0.477031 − 1.04455i −0.0935534 − 0.204853i
\(27\) −0.959493 − 0.281733i −0.184655 − 0.0542195i
\(28\) −1.21537 − 1.40261i −0.229683 − 0.265069i
\(29\) −2.81913 + 0.827771i −0.523499 + 0.153713i −0.532796 − 0.846244i \(-0.678859\pi\)
0.00929715 + 0.999957i \(0.497041\pi\)
\(30\) 2.04620 + 1.31501i 0.373583 + 0.240087i
\(31\) −0.863019 + 1.88975i −0.155003 + 0.339409i −0.971163 − 0.238417i \(-0.923371\pi\)
0.816160 + 0.577826i \(0.196099\pi\)
\(32\) 0.654861 − 0.755750i 0.115764 − 0.133599i
\(33\) 4.67177 − 3.00236i 0.813251 − 0.522644i
\(34\) 1.06847 + 7.43136i 0.183241 + 1.27447i
\(35\) 0.642438 + 4.46825i 0.108592 + 0.755273i
\(36\) 0.841254 − 0.540641i 0.140209 − 0.0901068i
\(37\) −5.06718 + 5.84784i −0.833039 + 0.961379i −0.999696 − 0.0246361i \(-0.992157\pi\)
0.166657 + 0.986015i \(0.446703\pi\)
\(38\) 1.85913 − 4.07093i 0.301591 − 0.660393i
\(39\) 0.966031 + 0.620830i 0.154689 + 0.0994124i
\(40\) −2.33380 + 0.685265i −0.369006 + 0.108350i
\(41\) −0.450833 − 0.520289i −0.0704083 − 0.0812555i 0.719453 − 0.694542i \(-0.244393\pi\)
−0.789861 + 0.613286i \(0.789847\pi\)
\(42\) 1.78074 + 0.522874i 0.274775 + 0.0806811i
\(43\) −3.55862 − 7.79229i −0.542684 − 1.18831i −0.960115 − 0.279604i \(-0.909797\pi\)
0.417431 − 0.908709i \(-0.362931\pi\)
\(44\) −0.790323 + 5.49682i −0.119146 + 0.828676i
\(45\) −2.43232 −0.362589
\(46\) −4.42909 + 1.83933i −0.653034 + 0.271194i
\(47\) 7.09992 1.03563 0.517815 − 0.855493i \(-0.326746\pi\)
0.517815 + 0.855493i \(0.326746\pi\)
\(48\) −0.142315 + 0.989821i −0.0205414 + 0.142868i
\(49\) −1.47703 − 3.23425i −0.211004 − 0.462035i
\(50\) 0.879083 + 0.258122i 0.124321 + 0.0365040i
\(51\) −4.91655 − 5.67400i −0.688455 − 0.794519i
\(52\) −1.10181 + 0.323520i −0.152793 + 0.0448641i
\(53\) 4.53311 + 2.91326i 0.622671 + 0.400166i 0.813590 − 0.581439i \(-0.197510\pi\)
−0.190919 + 0.981606i \(0.561147\pi\)
\(54\) −0.415415 + 0.909632i −0.0565308 + 0.123785i
\(55\) 8.84555 − 10.2083i 1.19273 − 1.37649i
\(56\) −1.56130 + 1.00339i −0.208638 + 0.134083i
\(57\) 0.636911 + 4.42981i 0.0843609 + 0.586743i
\(58\) 0.418141 + 2.90824i 0.0549047 + 0.381870i
\(59\) −3.39399 + 2.18119i −0.441860 + 0.283966i −0.742594 − 0.669742i \(-0.766405\pi\)
0.300734 + 0.953708i \(0.402768\pi\)
\(60\) 1.59283 − 1.83823i 0.205634 − 0.237314i
\(61\) −0.157543 + 0.344971i −0.0201713 + 0.0441690i −0.919450 − 0.393208i \(-0.871365\pi\)
0.899278 + 0.437377i \(0.144092\pi\)
\(62\) 1.74769 + 1.12317i 0.221957 + 0.142643i
\(63\) −1.78074 + 0.522874i −0.224353 + 0.0658759i
\(64\) −0.654861 − 0.755750i −0.0818576 − 0.0944687i
\(65\) 2.67995 + 0.786905i 0.332407 + 0.0976036i
\(66\) −2.30694 − 5.05150i −0.283965 − 0.621796i
\(67\) −0.420243 + 2.92285i −0.0513408 + 0.357083i 0.947915 + 0.318522i \(0.103187\pi\)
−0.999256 + 0.0385611i \(0.987723\pi\)
\(68\) 7.50778 0.910452
\(69\) 2.73158 − 3.94189i 0.328843 − 0.474548i
\(70\) 4.51420 0.539550
\(71\) 2.14473 − 14.9169i 0.254532 − 1.77031i −0.315731 − 0.948849i \(-0.602250\pi\)
0.570264 − 0.821462i \(-0.306841\pi\)
\(72\) −0.415415 − 0.909632i −0.0489571 − 0.107201i
\(73\) 11.0242 + 3.23701i 1.29029 + 0.378863i 0.853685 − 0.520790i \(-0.174363\pi\)
0.436605 + 0.899653i \(0.356181\pi\)
\(74\) 5.06718 + 5.84784i 0.589048 + 0.679797i
\(75\) −0.879083 + 0.258122i −0.101508 + 0.0298054i
\(76\) −3.76492 − 2.41956i −0.431866 − 0.277543i
\(77\) 4.28150 − 9.37518i 0.487923 − 1.06840i
\(78\) 0.751992 − 0.867845i 0.0851463 − 0.0982640i
\(79\) 8.86394 − 5.69651i 0.997272 − 0.640908i 0.0632029 − 0.998001i \(-0.479868\pi\)
0.934069 + 0.357093i \(0.116232\pi\)
\(80\) 0.346156 + 2.40757i 0.0387014 + 0.269174i
\(81\) −0.142315 − 0.989821i −0.0158128 − 0.109980i
\(82\) −0.579153 + 0.372199i −0.0639568 + 0.0411025i
\(83\) −6.84228 + 7.89641i −0.751037 + 0.866743i −0.994668 − 0.103125i \(-0.967116\pi\)
0.243631 + 0.969868i \(0.421661\pi\)
\(84\) 0.770978 − 1.68821i 0.0841206 − 0.184198i
\(85\) −15.3624 − 9.87283i −1.66629 − 1.07086i
\(86\) −8.21942 + 2.41344i −0.886323 + 0.260248i
\(87\) −1.92408 − 2.22050i −0.206283 − 0.238063i
\(88\) 5.32839 + 1.56456i 0.568009 + 0.166782i
\(89\) 4.47642 + 9.80199i 0.474499 + 1.03901i 0.983939 + 0.178502i \(0.0571252\pi\)
−0.509440 + 0.860506i \(0.670147\pi\)
\(90\) −0.346156 + 2.40757i −0.0364880 + 0.253780i
\(91\) 2.13120 0.223410
\(92\) 1.19028 + 4.64578i 0.124095 + 0.484356i
\(93\) −2.07749 −0.215425
\(94\) 1.01042 − 7.02765i 0.104217 − 0.724847i
\(95\) 4.52201 + 9.90183i 0.463949 + 1.01591i
\(96\) 0.959493 + 0.281733i 0.0979278 + 0.0287542i
\(97\) 6.51094 + 7.51402i 0.661085 + 0.762933i 0.982954 − 0.183851i \(-0.0588564\pi\)
−0.321869 + 0.946784i \(0.604311\pi\)
\(98\) −3.41153 + 1.00172i −0.344616 + 0.101189i
\(99\) 4.67177 + 3.00236i 0.469530 + 0.301749i
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 138.2.e.c.31.1 ✓ 10
3.2 odd 2 414.2.i.b.307.1 10
23.3 even 11 inner 138.2.e.c.49.1 yes 10
23.7 odd 22 3174.2.a.y.1.5 5
23.16 even 11 3174.2.a.z.1.1 5
69.26 odd 22 414.2.i.b.325.1 10
69.53 even 22 9522.2.a.ca.1.1 5
69.62 odd 22 9522.2.a.bv.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.c.31.1 ✓ 10 1.1 even 1 trivial
138.2.e.c.49.1 yes 10 23.3 even 11 inner
414.2.i.b.307.1 10 3.2 odd 2
414.2.i.b.325.1 10 69.26 odd 22
3174.2.a.y.1.5 5 23.7 odd 22
3174.2.a.z.1.1 5 23.16 even 11
9522.2.a.bv.1.5 5 69.62 odd 22
9522.2.a.ca.1.1 5 69.53 even 22