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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 254.16
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.16

$q$-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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 −0.463665 − 0.886011i \(-0.653466\pi\)
0.535476 + 0.844551i \(0.320132\pi\)
\(8\) 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 0.297239 − 0.954803i \(-0.403934\pi\)
0.678264 − 0.734818i \(-0.262732\pi\)
\(12\) −1.03330 − 0.482187i −0.298287 − 0.139195i
\(13\) −1.94815 + 3.37429i −0.540318 + 0.935859i 0.458567 + 0.888660i \(0.348363\pi\)
−0.998886 + 0.0471991i \(0.984970\pi\)
\(14\) −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 −0.301498 − 0.953467i \(-0.597487\pi\)
0.674977 + 0.737839i \(0.264153\pi\)
\(18\) −2.23213 − 2.66319i −0.526119 − 0.627720i
\(19\) 0.113062 + 0.195829i 0.0259382 + 0.0449262i 0.878703 − 0.477369i \(-0.158409\pi\)
−0.852765 + 0.522295i \(0.825076\pi\)
\(20\) −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 \(0.244405\pi\)
−0.719426 + 0.694569i \(0.755595\pi\)
\(24\) 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 0.948467 − 0.316875i \(-0.102634\pi\)
−0.748655 + 0.662959i \(0.769300\pi\)
\(38\) −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 0.451299 − 0.892373i \(-0.350961\pi\)
−0.547168 + 0.837023i \(0.684294\pi\)
\(42\) −0.438478 + 0.0382377i −0.0676586 + 0.00590020i
\(43\) −4.15057 − 7.18899i −0.632955 − 1.09631i −0.986944 − 0.161061i \(-0.948508\pi\)
0.353989 − 0.935250i \(-0.384825\pi\)
\(44\) −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 −0.157787 − 0.987473i \(-0.550436\pi\)
−0.934071 + 0.357089i \(0.883769\pi\)
\(54\) −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 0.0486621 − 0.998815i \(-0.484504\pi\)
0.889330 + 0.457265i \(0.151171\pi\)
\(60\) −0.322463 − 2.52924i −0.0416297 − 0.326523i
\(61\) 10.0815i 1.29081i 0.763842 + 0.645404i \(0.223311\pi\)
−0.763842 + 0.645404i \(0.776689\pi\)
\(62\) −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 0.521983 − 0.852956i \(-0.325192\pi\)
−0.477690 + 0.878529i \(0.658526\pi\)
\(68\) −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 0.234090 − 0.972215i \(-0.424789\pi\)
−0.724918 + 0.688835i \(0.758122\pi\)
\(72\) 5.93375 + 7.07965i 0.699299 + 0.834345i
\(73\) 4.75906 − 8.24293i 0.557006 − 0.964762i −0.440739 − 0.897635i \(-0.645284\pi\)
0.997744 − 0.0671266i \(-0.0213831\pi\)
\(74\) −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 −0.839211 − 0.543807i \(-0.816982\pi\)
0.0513452 + 0.998681i \(0.483649\pi\)
\(80\) −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 0.345282 − 0.938499i \(-0.387783\pi\)
−0.640123 + 0.768273i \(0.721117\pi\)
\(84\) −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 \(-0.940074\pi\)
0.982331 − 0.187152i \(-0.0599257\pi\)
\(98\) 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.254.16 yes 104
3.2 odd 2 inner 465.2.t.d.254.38 yes 104
5.4 even 2 inner 465.2.t.d.254.37 yes 104
15.14 odd 2 inner 465.2.t.d.254.15 yes 104
31.26 odd 6 inner 465.2.t.d.119.15 ✓ 104
93.26 even 6 inner 465.2.t.d.119.37 yes 104
155.119 odd 6 inner 465.2.t.d.119.38 yes 104
465.119 even 6 inner 465.2.t.d.119.16 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 31.26 odd 6 inner
465.2.t.d.119.16 yes 104 465.119 even 6 inner
465.2.t.d.119.37 yes 104 93.26 even 6 inner
465.2.t.d.119.38 yes 104 155.119 odd 6 inner
465.2.t.d.254.15 yes 104 15.14 odd 2 inner
465.2.t.d.254.16 yes 104 1.1 even 1 trivial
465.2.t.d.254.37 yes 104 5.4 even 2 inner
465.2.t.d.254.38 yes 104 3.2 odd 2 inner