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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.16
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.16

$q$-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-1.15830 q^{2} +(1.56957 - 0.732437i) q^{3} -0.658332 q^{4} +(1.19425 - 1.89044i) q^{5} +(-1.81803 + 0.848385i) q^{6} +(0.189994 + 0.109693i) q^{7} +3.07916 q^{8} +(1.92707 - 2.29922i) q^{9} +(-1.38330 + 2.18971i) q^{10} +(3.23538 + 5.60384i) q^{11} +(-1.03330 + 0.482187i) q^{12} +(-1.94815 - 3.37429i) q^{13} +(-0.220071 - 0.127058i) q^{14} +(0.489817 - 3.84188i) q^{15} -2.24993 q^{16} +(1.53989 + 0.889057i) q^{17} +(-2.23213 + 2.66319i) q^{18} +(0.113062 - 0.195829i) q^{19} +(-0.786211 + 1.24454i) q^{20} +(0.378552 + 0.0330118i) q^{21} +(-3.74755 - 6.49095i) q^{22} -6.66207i q^{23} +(4.83294 - 2.25529i) q^{24} +(-2.14755 - 4.51531i) q^{25} +(2.25654 + 3.90845i) q^{26} +(1.34063 - 5.02023i) q^{27} +(-0.125079 - 0.0722145i) q^{28} -3.61779 q^{29} +(-0.567357 + 4.45007i) q^{30} +(5.53055 + 0.642639i) q^{31} -3.55221 q^{32} +(9.18260 + 6.42588i) q^{33} +(-1.78366 - 1.02980i) q^{34} +(0.434269 - 0.228172i) q^{35} +(-1.26865 + 1.51365i) q^{36} +(1.21541 - 2.10515i) q^{37} +(-0.130960 + 0.226829i) q^{38} +(-5.52920 - 3.86927i) q^{39} +(3.67727 - 5.82097i) q^{40} +(-0.613862 + 0.354413i) q^{41} +(-0.438478 - 0.0382377i) q^{42} +(-4.15057 + 7.18899i) q^{43} +(-2.12995 - 3.68919i) q^{44} +(-2.04514 - 6.38885i) q^{45} +7.71670i q^{46} +9.68203 q^{47} +(-3.53142 + 1.64794i) q^{48} +(-3.47593 - 6.02050i) q^{49} +(2.48751 + 5.23010i) q^{50} +(3.06814 + 0.267559i) q^{51} +(1.28253 + 2.22140i) q^{52} +(-7.94885 + 4.58927i) q^{53} +(-1.55286 + 5.81495i) q^{54} +(14.4576 + 0.576070i) q^{55} +(0.585022 + 0.337762i) q^{56} +(0.0340256 - 0.390177i) q^{57} +4.19050 q^{58} +(7.20486 + 4.15973i) q^{59} +(-0.322463 + 2.52924i) q^{60} -10.0815i q^{61} +(-6.40606 - 0.744372i) q^{62} +(0.618341 - 0.225451i) q^{63} +8.61440 q^{64} +(-8.70546 - 0.346874i) q^{65} +(-10.6362 - 7.44312i) q^{66} +(0.362559 - 0.209324i) q^{67} +(-1.01376 - 0.585295i) q^{68} +(-4.87955 - 10.4566i) q^{69} +(-0.503015 + 0.264293i) q^{70} +(-4.13579 + 2.38780i) q^{71} +(5.93375 - 7.07965i) q^{72} +(4.75906 + 8.24293i) q^{73} +(-1.40781 + 2.43840i) q^{74} +(-6.67790 - 5.51413i) q^{75} +(-0.0744322 + 0.128920i) q^{76} +1.41960i q^{77} +(6.40449 + 4.48179i) q^{78} +(-7.00270 - 4.04301i) q^{79} +(-2.68698 + 4.25337i) q^{80} +(-1.57280 - 8.86151i) q^{81} +(0.711039 - 0.410518i) q^{82} +(-2.68613 + 1.55084i) q^{83} +(-0.249213 - 0.0217327i) q^{84} +(3.51972 - 1.84932i) q^{85} +(4.80762 - 8.32704i) q^{86} +(-5.67835 + 2.64980i) q^{87} +(9.96224 + 17.2551i) q^{88} +9.11518 q^{89} +(2.36889 + 7.40023i) q^{90} -0.854793i q^{91} +4.38586i q^{92} +(9.15126 - 3.04212i) q^{93} -11.2147 q^{94} +(-0.235179 - 0.447605i) q^{95} +(-5.57542 + 2.60177i) q^{96} +3.68647i q^{97} +(4.02619 + 6.97356i) q^{98} +(19.1192 + 3.36016i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15830 −0.819044 −0.409522 − 0.912300i \(-0.634305\pi\)
−0.409522 + 0.912300i \(0.634305\pi\)
\(3\) 1.56957 − 0.732437i 0.906189 − 0.422873i
\(4\) −0.658332 −0.329166
\(5\) 1.19425 − 1.89044i 0.534083 − 0.845432i
\(6\) −1.81803 + 0.848385i −0.742209 + 0.346352i
\(7\) 0.189994 + 0.109693i 0.0718110 + 0.0414601i 0.535476 − 0.844551i \(-0.320132\pi\)
−0.463665 + 0.886011i \(0.653466\pi\)
\(8\) 3.07916 1.08865
\(9\) 1.92707 − 2.29922i 0.642357 − 0.766406i
\(10\) −1.38330 + 2.18971i −0.437438 + 0.692446i
\(11\) 3.23538 + 5.60384i 0.975503 + 1.68962i 0.678264 + 0.734818i \(0.262732\pi\)
0.297239 + 0.954803i \(0.403934\pi\)
\(12\) −1.03330 + 0.482187i −0.298287 + 0.139195i
\(13\) −1.94815 − 3.37429i −0.540318 − 0.935859i −0.998886 − 0.0471991i \(-0.984970\pi\)
0.458567 − 0.888660i \(-0.348363\pi\)
\(14\) −0.220071 − 0.127058i −0.0588164 − 0.0339577i
\(15\) 0.489817 − 3.84188i 0.126470 − 0.991970i
\(16\) −2.24993 −0.562484
\(17\) 1.53989 + 0.889057i 0.373479 + 0.215628i 0.674977 − 0.737839i \(-0.264153\pi\)
−0.301498 + 0.953467i \(0.597487\pi\)
\(18\) −2.23213 + 2.66319i −0.526119 + 0.627720i
\(19\) 0.113062 − 0.195829i 0.0259382 − 0.0449262i −0.852765 − 0.522295i \(-0.825076\pi\)
0.878703 + 0.477369i \(0.158409\pi\)
\(20\) −0.786211 + 1.24454i −0.175802 + 0.278287i
\(21\) 0.378552 + 0.0330118i 0.0826067 + 0.00720376i
\(22\) −3.74755 − 6.49095i −0.798981 − 1.38387i
\(23\) − 6.66207i − 1.38914i −0.719426 − 0.694569i \(-0.755595\pi\)
0.719426 − 0.694569i \(-0.244405\pi\)
\(24\) 4.83294 − 2.25529i 0.986519 − 0.460359i
\(25\) −2.14755 − 4.51531i −0.429510 − 0.903062i
\(26\) 2.25654 + 3.90845i 0.442545 + 0.766510i
\(27\) 1.34063 − 5.02023i 0.258005 − 0.966144i
\(28\) −0.125079 − 0.0722145i −0.0236378 − 0.0136473i
\(29\) −3.61779 −0.671806 −0.335903 − 0.941897i \(-0.609041\pi\)
−0.335903 + 0.941897i \(0.609041\pi\)
\(30\) −0.567357 + 4.45007i −0.103585 + 0.812468i
\(31\) 5.53055 + 0.642639i 0.993317 + 0.115421i
\(32\) −3.55221 −0.627947
\(33\) 9.18260 + 6.42588i 1.59849 + 1.11860i
\(34\) −1.78366 − 1.02980i −0.305896 − 0.176609i
\(35\) 0.434269 − 0.228172i 0.0734048 − 0.0385682i
\(36\) −1.26865 + 1.51365i −0.211442 + 0.252275i
\(37\) 1.21541 − 2.10515i 0.199812 − 0.346085i −0.748655 − 0.662959i \(-0.769300\pi\)
0.948467 + 0.316875i \(0.102634\pi\)
\(38\) −0.130960 + 0.226829i −0.0212445 + 0.0367966i
\(39\) −5.52920 − 3.86927i −0.885380 − 0.619579i
\(40\) 3.67727 − 5.82097i 0.581428 − 0.920376i
\(41\) −0.613862 + 0.354413i −0.0958691 + 0.0553501i −0.547168 − 0.837023i \(-0.684294\pi\)
0.451299 + 0.892373i \(0.350961\pi\)
\(42\) −0.438478 − 0.0382377i −0.0676586 − 0.00590020i
\(43\) −4.15057 + 7.18899i −0.632955 + 1.09631i 0.353989 + 0.935250i \(0.384825\pi\)
−0.986944 + 0.161061i \(0.948508\pi\)
\(44\) −2.12995 − 3.68919i −0.321103 − 0.556166i
\(45\) −2.04514 − 6.38885i −0.304871 − 0.952394i
\(46\) 7.71670i 1.13777i
\(47\) 9.68203 1.41227 0.706135 − 0.708077i \(-0.250437\pi\)
0.706135 + 0.708077i \(0.250437\pi\)
\(48\) −3.53142 + 1.64794i −0.509716 + 0.237859i
\(49\) −3.47593 − 6.02050i −0.496562 − 0.860071i
\(50\) 2.48751 + 5.23010i 0.351788 + 0.739648i
\(51\) 3.06814 + 0.267559i 0.429626 + 0.0374657i
\(52\) 1.28253 + 2.22140i 0.177855 + 0.308053i
\(53\) −7.94885 + 4.58927i −1.09186 + 0.630384i −0.934071 − 0.357089i \(-0.883769\pi\)
−0.157787 + 0.987473i \(0.550436\pi\)
\(54\) −1.55286 + 5.81495i −0.211317 + 0.791315i
\(55\) 14.4576 + 0.576070i 1.94946 + 0.0776773i
\(56\) 0.585022 + 0.337762i 0.0781768 + 0.0451354i
\(57\) 0.0340256 − 0.390177i 0.00450680 − 0.0516802i
\(58\) 4.19050 0.550239
\(59\) 7.20486 + 4.15973i 0.937993 + 0.541550i 0.889330 − 0.457265i \(-0.151171\pi\)
0.0486621 + 0.998815i \(0.484504\pi\)
\(60\) −0.322463 + 2.52924i −0.0416297 + 0.326523i
\(61\) − 10.0815i − 1.29081i −0.763842 − 0.645404i \(-0.776689\pi\)
0.763842 − 0.645404i \(-0.223311\pi\)
\(62\) −6.40606 − 0.744372i −0.813570 − 0.0945353i
\(63\) 0.618341 − 0.225451i 0.0779036 − 0.0284042i
\(64\) 8.61440 1.07680
\(65\) −8.70546 − 0.346874i −1.07978 − 0.0430244i
\(66\) −10.6362 − 7.44312i −1.30923 − 0.916185i
\(67\) 0.362559 − 0.209324i 0.0442937 − 0.0255730i −0.477690 − 0.878529i \(-0.658526\pi\)
0.521983 + 0.852956i \(0.325192\pi\)
\(68\) −1.01376 − 0.585295i −0.122937 − 0.0709775i
\(69\) −4.87955 − 10.4566i −0.587429 − 1.25882i
\(70\) −0.503015 + 0.264293i −0.0601218 + 0.0315890i
\(71\) −4.13579 + 2.38780i −0.490828 + 0.283380i −0.724918 − 0.688835i \(-0.758122\pi\)
0.234090 + 0.972215i \(0.424789\pi\)
\(72\) 5.93375 − 7.07965i 0.699299 − 0.834345i
\(73\) 4.75906 + 8.24293i 0.557006 + 0.964762i 0.997744 + 0.0671266i \(0.0213831\pi\)
−0.440739 + 0.897635i \(0.645284\pi\)
\(74\) −1.40781 + 2.43840i −0.163655 + 0.283459i
\(75\) −6.67790 − 5.51413i −0.771098 − 0.636717i
\(76\) −0.0744322 + 0.128920i −0.00853796 + 0.0147882i
\(77\) 1.41960i 0.161778i
\(78\) 6.40449 + 4.48179i 0.725166 + 0.507463i
\(79\) −7.00270 − 4.04301i −0.787865 − 0.454874i 0.0513452 − 0.998681i \(-0.483649\pi\)
−0.839211 + 0.543807i \(0.816982\pi\)
\(80\) −2.68698 + 4.25337i −0.300413 + 0.475541i
\(81\) −1.57280 − 8.86151i −0.174755 − 0.984612i
\(82\) 0.711039 − 0.410518i 0.0785211 − 0.0453342i
\(83\) −2.68613 + 1.55084i −0.294841 + 0.170226i −0.640123 − 0.768273i \(-0.721117\pi\)
0.345282 + 0.938499i \(0.387783\pi\)
\(84\) −0.249213 − 0.0217327i −0.0271913 − 0.00237123i
\(85\) 3.51972 − 1.84932i 0.381768 − 0.200587i
\(86\) 4.80762 − 8.32704i 0.518419 − 0.897927i
\(87\) −5.67835 + 2.64980i −0.608783 + 0.284089i
\(88\) 9.96224 + 17.2551i 1.06198 + 1.83940i
\(89\) 9.11518 0.966207 0.483103 − 0.875563i \(-0.339509\pi\)
0.483103 + 0.875563i \(0.339509\pi\)
\(90\) 2.36889 + 7.40023i 0.249703 + 0.780053i
\(91\) − 0.854793i − 0.0896067i
\(92\) 4.38586i 0.457257i
\(93\) 9.15126 − 3.04212i 0.948941 − 0.315453i
\(94\) −11.2147 −1.15671
\(95\) −0.235179 − 0.447605i −0.0241289 − 0.0459233i
\(96\) −5.57542 + 2.60177i −0.569039 + 0.265542i
\(97\) 3.68647i 0.374304i 0.982331 + 0.187152i \(0.0599257\pi\)
−0.982331 + 0.187152i \(0.940074\pi\)
\(98\) 4.02619 + 6.97356i 0.406706 + 0.704436i
\(99\) 19.1192 + 3.36016i 1.92156 + 0.337709i
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 465.2.t.d.119.16 yes 104
3.2 odd 2 inner 465.2.t.d.119.38 yes 104
5.4 even 2 inner 465.2.t.d.119.37 yes 104
15.14 odd 2 inner 465.2.t.d.119.15 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.15 yes 104
93.68 even 6 inner 465.2.t.d.254.37 yes 104
155.99 odd 6 inner 465.2.t.d.254.38 yes 104
465.254 even 6 inner 465.2.t.d.254.16 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.16 yes 104 1.1 even 1 trivial
465.2.t.d.119.37 yes 104 5.4 even 2 inner
465.2.t.d.119.38 yes 104 3.2 odd 2 inner
465.2.t.d.254.15 yes 104 31.6 odd 6 inner
465.2.t.d.254.16 yes 104 465.254 even 6 inner
465.2.t.d.254.37 yes 104 93.68 even 6 inner
465.2.t.d.254.38 yes 104 155.99 odd 6 inner