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 121.1
Root \(-0.415415 + 0.909632i\) of defining polynomial
Character \(\chi\) \(=\) 138.121
Dual form 138.2.e.c.73.1

$q$-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+(-0.415415 + 0.909632i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(2.43450 - 1.56456i) q^{5} +(0.654861 - 0.755750i) q^{6} +(0.394306 - 2.74246i) q^{7} +(0.959493 - 0.281733i) q^{8} +(0.841254 + 0.540641i) q^{9} +(0.411844 + 2.86444i) q^{10} +(2.26413 + 4.95774i) q^{11} +(0.415415 + 0.909632i) q^{12} +(-0.0520365 - 0.361922i) q^{13} +(2.33083 + 1.49793i) q^{14} +(-2.77667 + 0.815304i) q^{15} +(-0.142315 + 0.989821i) q^{16} +(4.12405 - 4.75941i) q^{17} +(-0.841254 + 0.540641i) q^{18} +(-4.01801 - 4.63704i) q^{19} +(-2.77667 - 0.815304i) q^{20} +(-1.15098 + 2.52028i) q^{21} -5.45027 q^{22} +(-0.965501 + 4.69764i) q^{23} -1.00000 q^{24} +(1.40187 - 3.06967i) q^{25} +(0.350833 + 0.103014i) q^{26} +(-0.654861 - 0.755750i) q^{27} +(-2.33083 + 1.49793i) q^{28} +(-2.23325 + 2.57731i) q^{29} +(0.411844 - 2.86444i) q^{30} +(-7.37071 + 2.16423i) q^{31} +(-0.841254 - 0.540641i) q^{32} +(-0.775655 - 5.39480i) q^{33} +(2.61612 + 5.72850i) q^{34} +(-3.33080 - 7.29343i) q^{35} +(-0.142315 - 0.989821i) q^{36} +(4.06209 + 2.61055i) q^{37} +(5.88714 - 1.72862i) q^{38} +(-0.0520365 + 0.361922i) q^{39} +(1.89510 - 2.18706i) q^{40} +(-3.56427 + 2.29062i) q^{41} +(-1.81440 - 2.09393i) q^{42} +(6.75145 + 1.98241i) q^{43} +(2.26413 - 4.95774i) q^{44} +2.89389 q^{45} +(-3.87204 - 2.82972i) q^{46} -6.68409 q^{47} +(0.415415 - 0.909632i) q^{48} +(-0.649167 - 0.190613i) q^{49} +(2.20991 + 2.55037i) q^{50} +(-5.29788 + 3.40474i) q^{51} +(-0.239446 + 0.276335i) q^{52} +(-1.01536 + 7.06200i) q^{53} +(0.959493 - 0.281733i) q^{54} +(13.2687 + 8.52726i) q^{55} +(-0.394306 - 2.74246i) q^{56} +(2.54885 + 5.58121i) q^{57} +(-1.41668 - 3.10209i) q^{58} +(-0.0626320 - 0.435615i) q^{59} +(2.43450 + 1.56456i) q^{60} +(-7.80450 + 2.29161i) q^{61} +(1.09325 - 7.60369i) q^{62} +(1.81440 - 2.09393i) q^{63} +(0.841254 - 0.540641i) q^{64} +(-0.692931 - 0.799685i) q^{65} +(5.22950 + 1.53552i) q^{66} +(1.84829 - 4.04720i) q^{67} -6.29760 q^{68} +(2.24987 - 4.23534i) q^{69} +8.01801 q^{70} +(-0.586952 + 1.28525i) q^{71} +(0.959493 + 0.281733i) q^{72} +(2.58092 + 2.97855i) q^{73} +(-4.06209 + 2.61055i) q^{74} +(-2.20991 + 2.55037i) q^{75} +(-0.873198 + 6.07322i) q^{76} +(14.4892 - 4.25441i) q^{77} +(-0.307599 - 0.197682i) q^{78} +(0.565652 + 3.93420i) q^{79} +(1.20217 + 2.63238i) q^{80} +(0.415415 + 0.909632i) q^{81} +(-0.602967 - 4.19373i) q^{82} +(13.5461 + 8.70557i) q^{83} +(2.65843 - 0.780586i) q^{84} +(2.59363 - 18.0391i) q^{85} +(-4.60791 + 5.31782i) q^{86} +(2.86890 - 1.84373i) q^{87} +(3.56917 + 4.11904i) q^{88} +(-13.2033 - 3.87683i) q^{89} +(-1.20217 + 2.63238i) q^{90} -1.01308 q^{91} +(4.18251 - 2.34662i) q^{92} +7.68188 q^{93} +(2.77667 - 6.08006i) q^{94} +(-17.0368 - 5.00244i) q^{95} +(0.654861 + 0.755750i) q^{96} +(-3.64397 + 2.34184i) q^{97} +(0.443061 - 0.511320i) q^{98} +(-0.775655 + 5.39480i) 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 \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.415415 + 0.909632i −0.293743 + 0.643207i
\(3\) −0.959493 − 0.281733i −0.553964 − 0.162658i
\(4\) −0.654861 − 0.755750i −0.327430 − 0.377875i
\(5\) 2.43450 − 1.56456i 1.08874 − 0.699691i 0.132181 − 0.991226i \(-0.457802\pi\)
0.956560 + 0.291534i \(0.0941658\pi\)
\(6\) 0.654861 − 0.755750i 0.267346 − 0.308533i
\(7\) 0.394306 − 2.74246i 0.149034 − 1.03655i −0.768772 − 0.639523i \(-0.779132\pi\)
0.917806 − 0.397030i \(-0.129959\pi\)
\(8\) 0.959493 − 0.281733i 0.339232 − 0.0996075i
\(9\) 0.841254 + 0.540641i 0.280418 + 0.180214i
\(10\) 0.411844 + 2.86444i 0.130237 + 0.905815i
\(11\) 2.26413 + 4.95774i 0.682659 + 1.49482i 0.859800 + 0.510630i \(0.170588\pi\)
−0.177141 + 0.984185i \(0.556685\pi\)
\(12\) 0.415415 + 0.909632i 0.119920 + 0.262588i
\(13\) −0.0520365 − 0.361922i −0.0144323 − 0.100379i 0.981332 − 0.192321i \(-0.0616016\pi\)
−0.995764 + 0.0919422i \(0.970692\pi\)
\(14\) 2.33083 + 1.49793i 0.622941 + 0.400340i
\(15\) −2.77667 + 0.815304i −0.716933 + 0.210511i
\(16\) −0.142315 + 0.989821i −0.0355787 + 0.247455i
\(17\) 4.12405 − 4.75941i 1.00023 − 1.15433i 0.0122207 − 0.999925i \(-0.496110\pi\)
0.988008 − 0.154401i \(-0.0493446\pi\)
\(18\) −0.841254 + 0.540641i −0.198285 + 0.127430i
\(19\) −4.01801 − 4.63704i −0.921796 − 1.06381i −0.997773 − 0.0666993i \(-0.978753\pi\)
0.0759775 − 0.997110i \(-0.475792\pi\)
\(20\) −2.77667 − 0.815304i −0.620883 − 0.182308i
\(21\) −1.15098 + 2.52028i −0.251163 + 0.549971i
\(22\) −5.45027 −1.16200
\(23\) −0.965501 + 4.69764i −0.201321 + 0.979525i
\(24\) −1.00000 −0.204124
\(25\) 1.40187 − 3.06967i 0.280374 − 0.613933i
\(26\) 0.350833 + 0.103014i 0.0688039 + 0.0202027i
\(27\) −0.654861 − 0.755750i −0.126028 − 0.145444i
\(28\) −2.33083 + 1.49793i −0.440485 + 0.283083i
\(29\) −2.23325 + 2.57731i −0.414704 + 0.478594i −0.924216 − 0.381869i \(-0.875280\pi\)
0.509512 + 0.860463i \(0.329826\pi\)
\(30\) 0.411844 − 2.86444i 0.0751921 − 0.522973i
\(31\) −7.37071 + 2.16423i −1.32382 + 0.388708i −0.865869 − 0.500270i \(-0.833234\pi\)
−0.457949 + 0.888979i \(0.651416\pi\)
\(32\) −0.841254 − 0.540641i −0.148714 − 0.0955727i
\(33\) −0.775655 − 5.39480i −0.135024 − 0.939114i
\(34\) 2.61612 + 5.72850i 0.448660 + 0.982429i
\(35\) −3.33080 − 7.29343i −0.563008 − 1.23282i
\(36\) −0.142315 − 0.989821i −0.0237191 − 0.164970i
\(37\) 4.06209 + 2.61055i 0.667804 + 0.429171i 0.830133 − 0.557565i \(-0.188264\pi\)
−0.162329 + 0.986737i \(0.551901\pi\)
\(38\) 5.88714 − 1.72862i 0.955020 − 0.280419i
\(39\) −0.0520365 + 0.361922i −0.00833251 + 0.0579539i
\(40\) 1.89510 − 2.18706i 0.299641 − 0.345804i
\(41\) −3.56427 + 2.29062i −0.556645 + 0.357734i −0.788518 − 0.615012i \(-0.789151\pi\)
0.231873 + 0.972746i \(0.425515\pi\)
\(42\) −1.81440 − 2.09393i −0.279968 − 0.323100i
\(43\) 6.75145 + 1.98241i 1.02959 + 0.302314i 0.752543 − 0.658543i \(-0.228827\pi\)
0.277044 + 0.960857i \(0.410645\pi\)
\(44\) 2.26413 − 4.95774i 0.341330 − 0.747408i
\(45\) 2.89389 0.431396
\(46\) −3.87204 − 2.82972i −0.570901 − 0.417219i
\(47\) −6.68409 −0.974975 −0.487487 − 0.873130i \(-0.662086\pi\)
−0.487487 + 0.873130i \(0.662086\pi\)
\(48\) 0.415415 − 0.909632i 0.0599600 − 0.131294i
\(49\) −0.649167 − 0.190613i −0.0927382 − 0.0272304i
\(50\) 2.20991 + 2.55037i 0.312528 + 0.360677i
\(51\) −5.29788 + 3.40474i −0.741851 + 0.476759i
\(52\) −0.239446 + 0.276335i −0.0332051 + 0.0383208i
\(53\) −1.01536 + 7.06200i −0.139471 + 0.970040i 0.793110 + 0.609078i \(0.208461\pi\)
−0.932581 + 0.360962i \(0.882449\pi\)
\(54\) 0.959493 − 0.281733i 0.130570 − 0.0383389i
\(55\) 13.2687 + 8.52726i 1.78915 + 1.14982i
\(56\) −0.394306 − 2.74246i −0.0526914 − 0.366477i
\(57\) 2.54885 + 5.58121i 0.337604 + 0.739249i
\(58\) −1.41668 − 3.10209i −0.186019 − 0.407324i
\(59\) −0.0626320 − 0.435615i −0.00815399 − 0.0567122i 0.985338 − 0.170613i \(-0.0545746\pi\)
−0.993492 + 0.113900i \(0.963666\pi\)
\(60\) 2.43450 + 1.56456i 0.314292 + 0.201983i
\(61\) −7.80450 + 2.29161i −0.999264 + 0.293411i −0.740154 − 0.672437i \(-0.765247\pi\)
−0.259110 + 0.965848i \(0.583429\pi\)
\(62\) 1.09325 − 7.60369i 0.138842 − 0.965669i
\(63\) 1.81440 − 2.09393i 0.228593 − 0.263810i
\(64\) 0.841254 − 0.540641i 0.105157 − 0.0675801i
\(65\) −0.692931 − 0.799685i −0.0859475 − 0.0991887i
\(66\) 5.22950 + 1.53552i 0.643707 + 0.189009i
\(67\) 1.84829 − 4.04720i 0.225805 − 0.494444i −0.762490 − 0.647000i \(-0.776023\pi\)
0.988295 + 0.152556i \(0.0487505\pi\)
\(68\) −6.29760 −0.763696
\(69\) 2.24987 − 4.23534i 0.270852 − 0.509875i
\(70\) 8.01801 0.958335
\(71\) −0.586952 + 1.28525i −0.0696584 + 0.152531i −0.941258 − 0.337687i \(-0.890355\pi\)
0.871600 + 0.490218i \(0.163083\pi\)
\(72\) 0.959493 + 0.281733i 0.113077 + 0.0332025i
\(73\) 2.58092 + 2.97855i 0.302074 + 0.348612i 0.886411 − 0.462899i \(-0.153191\pi\)
−0.584337 + 0.811511i \(0.698645\pi\)
\(74\) −4.06209 + 2.61055i −0.472209 + 0.303470i
\(75\) −2.20991 + 2.55037i −0.255178 + 0.294491i
\(76\) −0.873198 + 6.07322i −0.100163 + 0.696647i
\(77\) 14.4892 − 4.25441i 1.65120 − 0.484835i
\(78\) −0.307599 − 0.197682i −0.0348287 − 0.0223831i
\(79\) 0.565652 + 3.93420i 0.0636408 + 0.442632i 0.996582 + 0.0826040i \(0.0263237\pi\)
−0.932942 + 0.360028i \(0.882767\pi\)
\(80\) 1.20217 + 2.63238i 0.134406 + 0.294309i
\(81\) 0.415415 + 0.909632i 0.0461572 + 0.101070i
\(82\) −0.602967 − 4.19373i −0.0665866 − 0.463120i
\(83\) 13.5461 + 8.70557i 1.48688 + 0.955561i 0.996459 + 0.0840759i \(0.0267938\pi\)
0.490422 + 0.871485i \(0.336843\pi\)
\(84\) 2.65843 − 0.780586i 0.290059 − 0.0851689i
\(85\) 2.59363 − 18.0391i 0.281319 − 1.95661i
\(86\) −4.60791 + 5.31782i −0.496884 + 0.573435i
\(87\) 2.86890 − 1.84373i 0.307578 − 0.197669i
\(88\) 3.56917 + 4.11904i 0.380475 + 0.439091i
\(89\) −13.2033 − 3.87683i −1.39954 − 0.410943i −0.507017 − 0.861936i \(-0.669252\pi\)
−0.892528 + 0.450993i \(0.851070\pi\)
\(90\) −1.20217 + 2.63238i −0.126720 + 0.277477i
\(91\) −1.01308 −0.106199
\(92\) 4.18251 − 2.34662i 0.436057 − 0.244652i
\(93\) 7.68188 0.796574
\(94\) 2.77667 − 6.08006i 0.286392 − 0.627110i
\(95\) −17.0368 − 5.00244i −1.74793 − 0.513240i
\(96\) 0.654861 + 0.755750i 0.0668364 + 0.0771334i
\(97\) −3.64397 + 2.34184i −0.369989 + 0.237778i −0.712401 − 0.701772i \(-0.752392\pi\)
0.342412 + 0.939550i \(0.388756\pi\)
\(98\) 0.443061 − 0.511320i 0.0447560 − 0.0516511i
\(99\) −0.775655 + 5.39480i −0.0779562 + 0.542198i
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.121.1 yes 10
3.2 odd 2 414.2.i.b.397.1 10
23.2 even 11 3174.2.a.z.1.3 5
23.4 even 11 inner 138.2.e.c.73.1 ✓ 10
23.21 odd 22 3174.2.a.y.1.3 5
69.2 odd 22 9522.2.a.bv.1.3 5
69.44 even 22 9522.2.a.ca.1.3 5
69.50 odd 22 414.2.i.b.73.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.c.73.1 ✓ 10 23.4 even 11 inner
138.2.e.c.121.1 yes 10 1.1 even 1 trivial
414.2.i.b.73.1 10 69.50 odd 22
414.2.i.b.397.1 10 3.2 odd 2
3174.2.a.y.1.3 5 23.21 odd 22
3174.2.a.z.1.3 5 23.2 even 11
9522.2.a.bv.1.3 5 69.2 odd 22
9522.2.a.ca.1.3 5 69.44 even 22