Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(64,351)] 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("351.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,12,0] 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: \(2.80274911095\)
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: \( 3^{9} \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 181.7
Root \(-1.65391 + 0.514376i\) of defining polynomial
Character \(\chi\) \(=\) 351.181
Dual form 351.2.t.c.64.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.929969 + 0.536918i) q^{2} +(-0.423439 - 0.733417i) q^{4} +(1.10543 - 0.638222i) q^{5} +(-0.890926 - 0.514376i) q^{7} -3.05708i q^{8} +1.37069 q^{10} +(4.03796 + 2.33132i) q^{11} +(2.29741 - 2.77883i) q^{13} +(-0.552355 - 0.956708i) q^{14} +(0.794522 - 1.37615i) q^{16} +0.476187 q^{17} -6.69096i q^{19} +(-0.936166 - 0.540496i) q^{20} +(2.50345 + 4.33610i) q^{22} +(0.479867 + 0.831155i) q^{23} +(-1.68535 + 2.91910i) q^{25} +(3.62852 - 1.35071i) q^{26} +0.871227i q^{28} +(-4.68880 + 8.12123i) q^{29} +(1.66927 - 0.963754i) q^{31} +(-3.81725 + 2.20389i) q^{32} +(0.442839 + 0.255673i) q^{34} -1.31314 q^{35} +4.94666i q^{37} +(3.59249 - 6.22238i) q^{38} +(-1.95109 - 3.37939i) q^{40} +(-1.31994 + 0.762068i) q^{41} +(-1.31426 + 2.27637i) q^{43} -3.94868i q^{44} +1.03060i q^{46} +(5.92316 + 3.41974i) q^{47} +(-2.97083 - 5.14564i) q^{49} +(-3.13464 + 1.80978i) q^{50} +(-3.01086 - 0.508296i) q^{52} -0.582145 q^{53} +5.95159 q^{55} +(-1.57249 + 2.72363i) q^{56} +(-8.72087 + 5.03499i) q^{58} +(-3.64799 + 2.10617i) q^{59} +(-4.71645 + 8.16913i) q^{61} +2.06983 q^{62} -7.91132 q^{64} +(0.766122 - 4.53807i) q^{65} +(2.01156 - 1.16138i) q^{67} +(-0.201636 - 0.349243i) q^{68} +(-1.22118 - 0.705051i) q^{70} -1.35071i q^{71} +12.8687i q^{73} +(-2.65595 + 4.60024i) q^{74} +(-4.90726 + 2.83321i) q^{76} +(-2.39835 - 4.15406i) q^{77} +(6.45415 - 11.1789i) q^{79} -2.02833i q^{80} -1.63667 q^{82} +(-8.86189 - 5.11641i) q^{83} +(0.526392 - 0.303913i) q^{85} +(-2.44445 + 1.41130i) q^{86} +(7.12701 - 12.3444i) q^{88} +6.85985i q^{89} +(-3.47619 + 1.29400i) q^{91} +(0.406389 - 0.703886i) q^{92} +(3.67224 + 6.36050i) q^{94} +(-4.27032 - 7.39640i) q^{95} +(14.9635 + 8.63918i) q^{97} -6.38037i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{4} - 16 q^{10} - 4 q^{13} + 18 q^{14} + 4 q^{16} + 12 q^{17} - 10 q^{22} - 24 q^{23} - 12 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{35} - 12 q^{38} - 8 q^{40} + 4 q^{43} - 10 q^{49} - 108 q^{53}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 \(-0.209375\pi\)
−0.133770 + 0.991012i \(0.542708\pi\)
\(3\) 0 0
\(4\) −0.423439 − 0.733417i −0.211719 − 0.366709i
\(5\) 1.10543 − 0.638222i 0.494365 − 0.285422i −0.232019 − 0.972711i \(-0.574533\pi\)
0.726383 + 0.687290i \(0.241200\pi\)
\(6\) 0 0
\(7\) −0.890926 − 0.514376i −0.336738 − 0.194416i 0.322090 − 0.946709i \(-0.395614\pi\)
−0.658829 + 0.752293i \(0.728948\pi\)
\(8\) − 3.05708i − 1.08084i
\(9\) 0 0
\(10\) 1.37069 0.433450
\(11\) 4.03796 + 2.33132i 1.21749 + 0.702918i 0.964380 − 0.264519i \(-0.0852133\pi\)
0.253110 + 0.967438i \(0.418547\pi\)
\(12\) 0 0
\(13\) 2.29741 − 2.77883i 0.637187 − 0.770709i
\(14\) −0.552355 − 0.956708i −0.147623 − 0.255691i
\(15\) 0 0
\(16\) 0.794522 − 1.37615i 0.198631 − 0.344038i
\(17\) 0.476187 0.115492 0.0577461 − 0.998331i \(-0.481609\pi\)
0.0577461 + 0.998331i \(0.481609\pi\)
\(18\) 0 0
\(19\) − 6.69096i − 1.53501i −0.641042 − 0.767505i \(-0.721498\pi\)
0.641042 − 0.767505i \(-0.278502\pi\)
\(20\) −0.936166 − 0.540496i −0.209333 − 0.120859i
\(21\) 0 0
\(22\) 2.50345 + 4.33610i 0.533737 + 0.924460i
\(23\) 0.479867 + 0.831155i 0.100059 + 0.173308i 0.911709 − 0.410837i \(-0.134763\pi\)
−0.811650 + 0.584145i \(0.801430\pi\)
\(24\) 0 0
\(25\) −1.68535 + 2.91910i −0.337069 + 0.583821i
\(26\) 3.62852 − 1.35071i 0.711612 − 0.264895i
\(27\) 0 0
\(28\) 0.871227i 0.164646i
\(29\) −4.68880 + 8.12123i −0.870687 + 1.50807i −0.00940050 + 0.999956i \(0.502992\pi\)
−0.861287 + 0.508119i \(0.830341\pi\)
\(30\) 0 0
\(31\) 1.66927 − 0.963754i 0.299810 − 0.173095i −0.342548 − 0.939500i \(-0.611290\pi\)
0.642357 + 0.766405i \(0.277956\pi\)
\(32\) −3.81725 + 2.20389i −0.674801 + 0.389597i
\(33\) 0 0
\(34\) 0.442839 + 0.255673i 0.0759462 + 0.0438476i
\(35\) −1.31314 −0.221962
\(36\) 0 0
\(37\) 4.94666i 0.813226i 0.913601 + 0.406613i \(0.133290\pi\)
−0.913601 + 0.406613i \(0.866710\pi\)
\(38\) 3.59249 − 6.22238i 0.582779 − 1.00940i
\(39\) 0 0
\(40\) −1.95109 − 3.37939i −0.308495 − 0.534329i
\(41\) −1.31994 + 0.762068i −0.206140 + 0.119015i −0.599516 − 0.800363i \(-0.704640\pi\)
0.393376 + 0.919378i \(0.371307\pi\)
\(42\) 0 0
\(43\) −1.31426 + 2.27637i −0.200423 + 0.347143i −0.948665 − 0.316283i \(-0.897565\pi\)
0.748242 + 0.663426i \(0.230898\pi\)
\(44\) − 3.94868i − 0.595285i
\(45\) 0 0
\(46\) 1.03060i 0.151953i
\(47\) 5.92316 + 3.41974i 0.863982 + 0.498820i 0.865344 − 0.501179i \(-0.167100\pi\)
−0.00136148 + 0.999999i \(0.500433\pi\)
\(48\) 0 0
\(49\) −2.97083 − 5.14564i −0.424405 − 0.735091i
\(50\) −3.13464 + 1.80978i −0.443305 + 0.255942i
\(51\) 0 0
\(52\) −3.01086 − 0.508296i −0.417531 − 0.0704880i
\(53\) −0.582145 −0.0799637 −0.0399819 − 0.999200i \(-0.512730\pi\)
−0.0399819 + 0.999200i \(0.512730\pi\)
\(54\) 0 0
\(55\) 5.95159 0.802512
\(56\) −1.57249 + 2.72363i −0.210133 + 0.363960i
\(57\) 0 0
\(58\) −8.72087 + 5.03499i −1.14511 + 0.661127i
\(59\) −3.64799 + 2.10617i −0.474927 + 0.274199i −0.718300 − 0.695733i \(-0.755080\pi\)
0.243373 + 0.969933i \(0.421746\pi\)
\(60\) 0 0
\(61\) −4.71645 + 8.16913i −0.603880 + 1.04595i 0.388348 + 0.921513i \(0.373046\pi\)
−0.992227 + 0.124437i \(0.960287\pi\)
\(62\) 2.06983 0.262868
\(63\) 0 0
\(64\) −7.91132 −0.988915
\(65\) 0.766122 − 4.53807i 0.0950257 − 0.562878i
\(66\) 0 0
\(67\) 2.01156 − 1.16138i 0.245751 − 0.141885i −0.372066 − 0.928206i \(-0.621350\pi\)
0.617817 + 0.786322i \(0.288017\pi\)
\(68\) −0.201636 − 0.349243i −0.0244519 − 0.0423520i
\(69\) 0 0
\(70\) −1.22118 − 0.705051i −0.145959 − 0.0842697i
\(71\) − 1.35071i − 0.160299i −0.996783 − 0.0801497i \(-0.974460\pi\)
0.996783 − 0.0801497i \(-0.0255398\pi\)
\(72\) 0 0
\(73\) 12.8687i 1.50617i 0.657923 + 0.753085i \(0.271435\pi\)
−0.657923 + 0.753085i \(0.728565\pi\)
\(74\) −2.65595 + 4.60024i −0.308748 + 0.534767i
\(75\) 0 0
\(76\) −4.90726 + 2.83321i −0.562902 + 0.324991i
\(77\) −2.39835 − 4.15406i −0.273317 − 0.473399i
\(78\) 0 0
\(79\) 6.45415 − 11.1789i 0.726149 − 1.25773i −0.232351 − 0.972632i \(-0.574642\pi\)
0.958499 − 0.285094i \(-0.0920250\pi\)
\(80\) − 2.02833i − 0.226774i
\(81\) 0 0
\(82\) −1.63667 −0.180740
\(83\) −8.86189 − 5.11641i −0.972718 − 0.561599i −0.0726545 − 0.997357i \(-0.523147\pi\)
−0.900064 + 0.435758i \(0.856480\pi\)
\(84\) 0 0
\(85\) 0.526392 − 0.303913i 0.0570953 − 0.0329640i
\(86\) −2.44445 + 1.41130i −0.263591 + 0.152185i
\(87\) 0 0
\(88\) 7.12701 − 12.3444i 0.759742 − 1.31591i
\(89\) 6.85985i 0.727143i 0.931566 + 0.363572i \(0.118443\pi\)
−0.931566 + 0.363572i \(0.881557\pi\)
\(90\) 0 0
\(91\) −3.47619 + 1.29400i −0.364403 + 0.135648i
\(92\) 0.406389 − 0.703886i 0.0423690 − 0.0733852i
\(93\) 0 0
\(94\) 3.67224 + 6.36050i 0.378763 + 0.656036i
\(95\) −4.27032 − 7.39640i −0.438125 − 0.758855i
\(96\) 0 0
\(97\) 14.9635 + 8.63918i 1.51931 + 0.877175i 0.999741 + 0.0227483i \(0.00724164\pi\)
0.519571 + 0.854427i \(0.326092\pi\)
\(98\) − 6.38037i − 0.644515i
\(99\) 0 0
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 351.2.t.c.181.7 20
3.2 odd 2 117.2.t.c.25.4 ✓ 20
9.2 odd 6 1053.2.b.j.649.7 10
9.4 even 3 inner 351.2.t.c.64.4 20
9.5 odd 6 117.2.t.c.103.7 yes 20
9.7 even 3 1053.2.b.i.649.4 10
13.12 even 2 inner 351.2.t.c.181.4 20
39.38 odd 2 117.2.t.c.25.7 yes 20
117.25 even 6 1053.2.b.i.649.7 10
117.38 odd 6 1053.2.b.j.649.4 10
117.77 odd 6 117.2.t.c.103.4 yes 20
117.103 even 6 inner 351.2.t.c.64.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.4 ✓ 20 3.2 odd 2
117.2.t.c.25.7 yes 20 39.38 odd 2
117.2.t.c.103.4 yes 20 117.77 odd 6
117.2.t.c.103.7 yes 20 9.5 odd 6
351.2.t.c.64.4 20 9.4 even 3 inner
351.2.t.c.64.7 20 117.103 even 6 inner
351.2.t.c.181.4 20 13.12 even 2 inner
351.2.t.c.181.7 20 1.1 even 1 trivial
1053.2.b.i.649.4 10 9.7 even 3
1053.2.b.i.649.7 10 117.25 even 6
1053.2.b.j.649.4 10 117.38 odd 6
1053.2.b.j.649.7 10 9.2 odd 6