Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [117,2,Mod(25,117)] 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("117.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,2,12,0] 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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.4
Root \(1.65391 + 0.514376i\) of defining polynomial
Character \(\chi\) \(=\) 117.103
Dual form 117.2.t.c.25.4

$q$-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+(-0.929969 + 0.536918i) q^{2} +(-0.744247 + 1.56400i) q^{3} +(-0.423439 + 0.733417i) q^{4} +(-1.10543 - 0.638222i) q^{5} +(-0.147613 - 1.85407i) q^{6} +(-0.890926 + 0.514376i) q^{7} -3.05708i q^{8} +(-1.89219 - 2.32800i) q^{9} +1.37069 q^{10} +(-4.03796 + 2.33132i) q^{11} +(-0.831922 - 1.20810i) q^{12} +(2.29741 + 2.77883i) q^{13} +(0.552355 - 0.956708i) q^{14} +(1.82089 - 1.25390i) q^{15} +(0.794522 + 1.37615i) q^{16} -0.476187 q^{17} +(3.00963 + 1.14902i) q^{18} +6.69096i q^{19} +(0.936166 - 0.540496i) q^{20} +(-0.141416 - 1.77623i) q^{21} +(2.50345 - 4.33610i) q^{22} +(-0.479867 + 0.831155i) q^{23} +(4.78127 + 2.27522i) q^{24} +(-1.68535 - 2.91910i) q^{25} +(-3.62852 - 1.35071i) q^{26} +(5.04926 - 1.22678i) q^{27} -0.871227i q^{28} +(4.68880 + 8.12123i) q^{29} +(-1.02013 + 2.14376i) q^{30} +(1.66927 + 0.963754i) q^{31} +(3.81725 + 2.20389i) q^{32} +(-0.640940 - 8.05044i) q^{33} +(0.442839 - 0.255673i) q^{34} +1.31314 q^{35} +(2.50863 - 0.402000i) q^{36} -4.94666i q^{37} +(-3.59249 - 6.22238i) q^{38} +(-6.05593 + 1.52501i) q^{39} +(-1.95109 + 3.37939i) q^{40} +(1.31994 + 0.762068i) q^{41} +(1.08520 + 1.57591i) q^{42} +(-1.31426 - 2.27637i) q^{43} -3.94868i q^{44} +(0.605908 + 3.78109i) q^{45} -1.03060i q^{46} +(-5.92316 + 3.41974i) q^{47} +(-2.74362 + 0.218435i) q^{48} +(-2.97083 + 5.14564i) q^{49} +(3.13464 + 1.80978i) q^{50} +(0.354400 - 0.744756i) q^{51} +(-3.01086 + 0.508296i) q^{52} +0.582145 q^{53} +(-4.03697 + 3.85190i) q^{54} +5.95159 q^{55} +(1.57249 + 2.72363i) q^{56} +(-10.4647 - 4.97973i) q^{57} +(-8.72087 - 5.03499i) q^{58} +(3.64799 + 2.10617i) q^{59} +(0.148597 + 1.86643i) q^{60} +(-4.71645 - 8.16913i) q^{61} -2.06983 q^{62} +(2.88327 + 1.10078i) q^{63} -7.91132 q^{64} +(-0.766122 - 4.53807i) q^{65} +(4.91848 + 7.14253i) q^{66} +(2.01156 + 1.16138i) q^{67} +(0.201636 - 0.349243i) q^{68} +(-0.942786 - 1.36910i) q^{69} +(-1.22118 + 0.705051i) q^{70} -1.35071i q^{71} +(-7.11689 + 5.78458i) q^{72} -12.8687i q^{73} +(2.65595 + 4.60024i) q^{74} +(5.81979 - 0.463346i) q^{75} +(-4.90726 - 2.83321i) q^{76} +(2.39835 - 4.15406i) q^{77} +(4.81302 - 4.66975i) q^{78} +(6.45415 + 11.1789i) q^{79} -2.02833i q^{80} +(-1.83921 + 8.81007i) q^{81} -1.63667 q^{82} +(8.86189 - 5.11641i) q^{83} +(1.36260 + 0.648408i) q^{84} +(0.526392 + 0.303913i) q^{85} +(2.44445 + 1.41130i) q^{86} +(-16.1912 + 1.28907i) q^{87} +(7.12701 + 12.3444i) q^{88} +6.85985i q^{89} +(-2.59361 - 3.19097i) q^{90} +(-3.47619 - 1.29400i) q^{91} +(-0.406389 - 0.703886i) q^{92} +(-2.74966 + 1.89347i) q^{93} +(3.67224 - 6.36050i) q^{94} +(4.27032 - 7.39640i) q^{95} +(-6.28787 + 4.32994i) q^{96} +(14.9635 - 8.63918i) q^{97} -6.38037i q^{98} +(13.0679 + 4.98909i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9} - 16 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 4 q^{16} - 12 q^{17} - 10 q^{22} + 24 q^{23} - 12 q^{25} - 12 q^{26} - 22 q^{27} + 12 q^{29} - 54 q^{30} - 12 q^{35} + 50 q^{36}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.929969 + 0.536918i −0.657587 + 0.379658i −0.791357 − 0.611354i \(-0.790625\pi\)
0.133770 + 0.991012i \(0.457292\pi\)
\(3\) −0.744247 + 1.56400i −0.429691 + 0.902976i
\(4\) −0.423439 + 0.733417i −0.211719 + 0.366709i
\(5\) −1.10543 − 0.638222i −0.494365 − 0.285422i 0.232019 − 0.972711i \(-0.425467\pi\)
−0.726383 + 0.687290i \(0.758800\pi\)
\(6\) −0.147613 − 1.85407i −0.0602627 − 0.756921i
\(7\) −0.890926 + 0.514376i −0.336738 + 0.194416i −0.658829 − 0.752293i \(-0.728948\pi\)
0.322090 + 0.946709i \(0.395614\pi\)
\(8\) − 3.05708i − 1.08084i
\(9\) −1.89219 − 2.32800i −0.630731 − 0.776002i
\(10\) 1.37069 0.433450
\(11\) −4.03796 + 2.33132i −1.21749 + 0.702918i −0.964380 − 0.264519i \(-0.914787\pi\)
−0.253110 + 0.967438i \(0.581453\pi\)
\(12\) −0.831922 − 1.20810i −0.240155 − 0.348749i
\(13\) 2.29741 + 2.77883i 0.637187 + 0.770709i
\(14\) 0.552355 − 0.956708i 0.147623 − 0.255691i
\(15\) 1.82089 − 1.25390i 0.470153 − 0.323756i
\(16\) 0.794522 + 1.37615i 0.198631 + 0.344038i
\(17\) −0.476187 −0.115492 −0.0577461 − 0.998331i \(-0.518391\pi\)
−0.0577461 + 0.998331i \(0.518391\pi\)
\(18\) 3.00963 + 1.14902i 0.709376 + 0.270827i
\(19\) 6.69096i 1.53501i 0.641042 + 0.767505i \(0.278502\pi\)
−0.641042 + 0.767505i \(0.721498\pi\)
\(20\) 0.936166 − 0.540496i 0.209333 − 0.120859i
\(21\) −0.141416 − 1.77623i −0.0308594 − 0.387605i
\(22\) 2.50345 − 4.33610i 0.533737 − 0.924460i
\(23\) −0.479867 + 0.831155i −0.100059 + 0.173308i −0.911709 − 0.410837i \(-0.865237\pi\)
0.811650 + 0.584145i \(0.198570\pi\)
\(24\) 4.78127 + 2.27522i 0.975973 + 0.464428i
\(25\) −1.68535 − 2.91910i −0.337069 − 0.583821i
\(26\) −3.62852 − 1.35071i −0.711612 − 0.264895i
\(27\) 5.04926 − 1.22678i 0.971730 − 0.236094i
\(28\) − 0.871227i − 0.164646i
\(29\) 4.68880 + 8.12123i 0.870687 + 1.50807i 0.861287 + 0.508119i \(0.169659\pi\)
0.00940050 + 0.999956i \(0.497008\pi\)
\(30\) −1.02013 + 2.14376i −0.186250 + 0.391395i
\(31\) 1.66927 + 0.963754i 0.299810 + 0.173095i 0.642357 − 0.766405i \(-0.277956\pi\)
−0.342548 + 0.939500i \(0.611290\pi\)
\(32\) 3.81725 + 2.20389i 0.674801 + 0.389597i
\(33\) −0.640940 − 8.05044i −0.111573 − 1.40140i
\(34\) 0.442839 − 0.255673i 0.0759462 − 0.0438476i
\(35\) 1.31314 0.221962
\(36\) 2.50863 − 0.402000i 0.418104 − 0.0669999i
\(37\) − 4.94666i − 0.813226i −0.913601 − 0.406613i \(-0.866710\pi\)
0.913601 − 0.406613i \(-0.133290\pi\)
\(38\) −3.59249 − 6.22238i −0.582779 − 1.00940i
\(39\) −6.05593 + 1.52501i −0.969725 + 0.244198i
\(40\) −1.95109 + 3.37939i −0.308495 + 0.534329i
\(41\) 1.31994 + 0.762068i 0.206140 + 0.119015i 0.599516 − 0.800363i \(-0.295360\pi\)
−0.393376 + 0.919378i \(0.628693\pi\)
\(42\) 1.08520 + 1.57591i 0.167450 + 0.243168i
\(43\) −1.31426 − 2.27637i −0.200423 − 0.347143i 0.748242 − 0.663426i \(-0.230898\pi\)
−0.948665 + 0.316283i \(0.897565\pi\)
\(44\) − 3.94868i − 0.595285i
\(45\) 0.605908 + 3.78109i 0.0903235 + 0.563652i
\(46\) − 1.03060i − 0.151953i
\(47\) −5.92316 + 3.41974i −0.863982 + 0.498820i −0.865344 − 0.501179i \(-0.832900\pi\)
0.00136148 + 0.999999i \(0.499567\pi\)
\(48\) −2.74362 + 0.218435i −0.396008 + 0.0315284i
\(49\) −2.97083 + 5.14564i −0.424405 + 0.735091i
\(50\) 3.13464 + 1.80978i 0.443305 + 0.255942i
\(51\) 0.354400 − 0.744756i 0.0496260 − 0.104287i
\(52\) −3.01086 + 0.508296i −0.417531 + 0.0704880i
\(53\) 0.582145 0.0799637 0.0399819 − 0.999200i \(-0.487270\pi\)
0.0399819 + 0.999200i \(0.487270\pi\)
\(54\) −4.03697 + 3.85190i −0.549363 + 0.524178i
\(55\) 5.95159 0.802512
\(56\) 1.57249 + 2.72363i 0.210133 + 0.363960i
\(57\) −10.4647 − 4.97973i −1.38608 − 0.659581i
\(58\) −8.72087 − 5.03499i −1.14511 − 0.661127i
\(59\) 3.64799 + 2.10617i 0.474927 + 0.274199i 0.718300 − 0.695733i \(-0.244920\pi\)
−0.243373 + 0.969933i \(0.578254\pi\)
\(60\) 0.148597 + 1.86643i 0.0191837 + 0.240955i
\(61\) −4.71645 − 8.16913i −0.603880 − 1.04595i −0.992227 − 0.124437i \(-0.960287\pi\)
0.388348 − 0.921513i \(-0.373046\pi\)
\(62\) −2.06983 −0.262868
\(63\) 2.88327 + 1.10078i 0.363258 + 0.138685i
\(64\) −7.91132 −0.988915
\(65\) −0.766122 − 4.53807i −0.0950257 − 0.562878i
\(66\) 4.91848 + 7.14253i 0.605423 + 0.879184i
\(67\) 2.01156 + 1.16138i 0.245751 + 0.141885i 0.617817 − 0.786322i \(-0.288017\pi\)
−0.372066 + 0.928206i \(0.621350\pi\)
\(68\) 0.201636 − 0.349243i 0.0244519 − 0.0423520i
\(69\) −0.942786 − 1.36910i −0.113498 − 0.164820i
\(70\) −1.22118 + 0.705051i −0.145959 + 0.0842697i
\(71\) − 1.35071i − 0.160299i −0.996783 − 0.0801497i \(-0.974460\pi\)
0.996783 − 0.0801497i \(-0.0255398\pi\)
\(72\) −7.11689 + 5.78458i −0.838734 + 0.681719i
\(73\) − 12.8687i − 1.50617i −0.657923 − 0.753085i \(-0.728565\pi\)
0.657923 − 0.753085i \(-0.271435\pi\)
\(74\) 2.65595 + 4.60024i 0.308748 + 0.534767i
\(75\) 5.81979 − 0.463346i 0.672012 − 0.0535026i
\(76\) −4.90726 − 2.83321i −0.562902 − 0.324991i
\(77\) 2.39835 − 4.15406i 0.273317 − 0.473399i
\(78\) 4.81302 − 4.66975i 0.544968 − 0.528745i
\(79\) 6.45415 + 11.1789i 0.726149 + 1.25773i 0.958499 + 0.285094i \(0.0920250\pi\)
−0.232351 + 0.972632i \(0.574642\pi\)
\(80\) − 2.02833i − 0.226774i
\(81\) −1.83921 + 8.81007i −0.204357 + 0.978896i
\(82\) −1.63667 −0.180740
\(83\) 8.86189 − 5.11641i 0.972718 − 0.561599i 0.0726545 − 0.997357i \(-0.476853\pi\)
0.900064 + 0.435758i \(0.143520\pi\)
\(84\) 1.36260 + 0.648408i 0.148672 + 0.0707471i
\(85\) 0.526392 + 0.303913i 0.0570953 + 0.0329640i
\(86\) 2.44445 + 1.41130i 0.263591 + 0.152185i
\(87\) −16.1912 + 1.28907i −1.73588 + 0.138203i
\(88\) 7.12701 + 12.3444i 0.759742 + 1.31591i
\(89\) 6.85985i 0.727143i 0.931566 + 0.363572i \(0.118443\pi\)
−0.931566 + 0.363572i \(0.881557\pi\)
\(90\) −2.59361 − 3.19097i −0.273391 − 0.336358i
\(91\) −3.47619 − 1.29400i −0.364403 − 0.135648i
\(92\) −0.406389 − 0.703886i −0.0423690 − 0.0733852i
\(93\) −2.74966 + 1.89347i −0.285127 + 0.196344i
\(94\) 3.67224 − 6.36050i 0.378763 − 0.656036i
\(95\) 4.27032 − 7.39640i 0.438125 − 0.758855i
\(96\) −6.28787 + 4.32994i −0.641753 + 0.441923i
\(97\) 14.9635 − 8.63918i 1.51931 − 0.877175i 0.519571 − 0.854427i \(-0.326092\pi\)
0.999741 − 0.0227483i \(-0.00724164\pi\)
\(98\) − 6.38037i − 0.644515i
\(99\) 13.0679 + 4.98909i 1.31337 + 0.501422i
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 117.2.t.c.103.4 yes 20
3.2 odd 2 351.2.t.c.64.7 20
9.2 odd 6 351.2.t.c.181.4 20
9.4 even 3 1053.2.b.j.649.4 10
9.5 odd 6 1053.2.b.i.649.7 10
9.7 even 3 inner 117.2.t.c.25.7 yes 20
13.12 even 2 inner 117.2.t.c.103.7 yes 20
39.38 odd 2 351.2.t.c.64.4 20
117.25 even 6 inner 117.2.t.c.25.4 ✓ 20
117.38 odd 6 351.2.t.c.181.7 20
117.77 odd 6 1053.2.b.i.649.4 10
117.103 even 6 1053.2.b.j.649.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.4 ✓ 20 117.25 even 6 inner
117.2.t.c.25.7 yes 20 9.7 even 3 inner
117.2.t.c.103.4 yes 20 1.1 even 1 trivial
117.2.t.c.103.7 yes 20 13.12 even 2 inner
351.2.t.c.64.4 20 39.38 odd 2
351.2.t.c.64.7 20 3.2 odd 2
351.2.t.c.181.4 20 9.2 odd 6
351.2.t.c.181.7 20 117.38 odd 6
1053.2.b.i.649.4 10 117.77 odd 6
1053.2.b.i.649.7 10 9.5 odd 6
1053.2.b.j.649.4 10 9.4 even 3
1053.2.b.j.649.7 10 117.103 even 6