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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 119.37
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.37

$q$-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} +(2.23430 - 0.0890267i) q^{5} +(-1.81803 + 0.848385i) q^{6} +(-0.189994 - 0.109693i) q^{7} -3.07916 q^{8} +(1.92707 - 2.29922i) q^{9} +(2.58799 - 0.103120i) 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} +(-3.44167 + 1.77621i) q^{15} -2.24993 q^{16} +(-1.53989 - 0.889057i) q^{17} +(2.23213 - 2.66319i) q^{18} +(0.113062 - 0.195829i) q^{19} +(-1.47091 + 0.0586092i) q^{20} +(0.378552 + 0.0330118i) q^{21} +(3.74755 + 6.49095i) q^{22} +6.66207i q^{23} +(4.83294 - 2.25529i) q^{24} +(4.98415 - 0.397824i) q^{25} +(2.25654 + 3.90845i) q^{26} +(-1.34063 + 5.02023i) q^{27} +(0.125079 + 0.0722145i) q^{28} -3.61779 q^{29} +(-3.98649 + 2.05740i) 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} +(-6.87974 + 0.274127i) 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} +(4.10095 - 5.30869i) q^{45} +7.71670i q^{46} -9.68203 q^{47} +(3.53142 - 1.64794i) q^{48} +(-3.47593 - 6.02050i) q^{49} +(5.77316 - 0.460801i) 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} +(7.72768 + 12.2326i) q^{55} +(0.585022 + 0.337762i) q^{56} +(-0.0340256 + 0.390177i) q^{57} -4.19050 q^{58} +(7.20486 + 4.15973i) q^{59} +(2.26576 - 1.16934i) q^{60} -10.0815i q^{61} +(6.40606 + 0.744372i) q^{62} +(-0.618341 + 0.225451i) q^{63} +8.61440 q^{64} +(4.65313 + 7.36572i) 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} +(-7.53157 + 4.27499i) q^{75} +(-0.0744322 + 0.128920i) q^{76} -1.41960i q^{77} +(-6.40449 - 4.48179i) q^{78} +(-7.00270 - 4.04301i) q^{79} +(-5.02702 + 0.200304i) 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} +(4.75015 - 6.14908i) 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 \(-0.365695\pi\)
0.409522 + 0.912300i \(0.365695\pi\)
\(3\) −1.56957 + 0.732437i −0.906189 + 0.422873i
\(4\) −0.658332 −0.329166
\(5\) 2.23430 − 0.0890267i 0.999207 − 0.0398140i
\(6\) −1.81803 + 0.848385i −0.742209 + 0.346352i
\(7\) −0.189994 − 0.109693i −0.0718110 − 0.0414601i 0.463665 − 0.886011i \(-0.346534\pi\)
−0.535476 + 0.844551i \(0.679868\pi\)
\(8\) −3.07916 −1.08865
\(9\) 1.92707 − 2.29922i 0.642357 − 0.766406i
\(10\) 2.58799 − 0.103120i 0.818395 − 0.0326094i
\(11\) 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 \(0.0150295\pi\)
−0.458567 + 0.888660i \(0.651637\pi\)
\(14\) −0.220071 − 0.127058i −0.0588164 − 0.0339577i
\(15\) −3.44167 + 1.77621i −0.888634 + 0.458617i
\(16\) −2.24993 −0.562484
\(17\) −1.53989 − 0.889057i −0.373479 − 0.215628i 0.301498 − 0.953467i \(-0.402513\pi\)
−0.674977 + 0.737839i \(0.735847\pi\)
\(18\) 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 \(0.158409\pi\)
\(20\) −1.47091 + 0.0586092i −0.328905 + 0.0131054i
\(21\) 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
\(25\) 4.98415 − 0.397824i 0.996830 − 0.0795648i
\(26\) 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 \(0.609041\pi\)
\(30\) −3.98649 + 2.05740i −0.727831 + 0.375627i
\(31\) 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 \(-0.897366\pi\)
0.748655 + 0.662959i \(0.230700\pi\)
\(38\) 0.130960 − 0.226829i 0.0212445 − 0.0367966i
\(39\) −5.52920 − 3.86927i −0.885380 − 0.619579i
\(40\) −6.87974 + 0.274127i −1.08778 + 0.0433433i
\(41\) −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 \(-0.615175\pi\)
0.986944 − 0.161061i \(-0.0514917\pi\)
\(44\) −2.12995 − 3.68919i −0.321103 − 0.556166i
\(45\) 4.10095 − 5.30869i 0.611334 − 0.791373i
\(46\) 7.71670i 1.13777i
\(47\) −9.68203 −1.41227 −0.706135 − 0.708077i \(-0.749563\pi\)
−0.706135 + 0.708077i \(0.749563\pi\)
\(48\) 3.53142 − 1.64794i 0.509716 − 0.237859i
\(49\) −3.47593 − 6.02050i −0.496562 − 0.860071i
\(50\) 5.77316 − 0.460801i 0.816448 − 0.0651671i
\(51\) 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 \(-0.449564\pi\)
0.934071 + 0.357089i \(0.116231\pi\)
\(54\) −1.55286 + 5.81495i −0.211317 + 0.791315i
\(55\) 7.72768 + 12.2326i 1.04200 + 1.64944i
\(56\) 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 \(0.484504\pi\)
\(60\) 2.26576 − 1.16934i 0.292508 − 0.150961i
\(61\) − 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
\(65\) 4.65313 + 7.36572i 0.577150 + 0.913605i
\(66\) −10.6362 − 7.44312i −1.30923 − 0.916185i
\(67\) −0.362559 + 0.209324i −0.0442937 + 0.0255730i −0.521983 − 0.852956i \(-0.674808\pi\)
0.477690 + 0.878529i \(0.341474\pi\)
\(68\) 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 \(-0.978617\pi\)
0.440739 − 0.897635i \(-0.354716\pi\)
\(74\) −1.40781 + 2.43840i −0.163655 + 0.283459i
\(75\) −7.53157 + 4.27499i −0.869670 + 0.493633i
\(76\) −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 \(0.816982\pi\)
\(80\) −5.02702 + 0.200304i −0.562038 + 0.0223947i
\(81\) −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 \(-0.612217\pi\)
0.640123 + 0.768273i \(0.278883\pi\)
\(84\) −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 \(0.339509\pi\)
\(90\) 4.75015 − 6.14908i 0.500710 − 0.648169i
\(91\) − 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.119.37 yes 104
3.2 odd 2 inner 465.2.t.d.119.15 ✓ 104
5.4 even 2 inner 465.2.t.d.119.16 yes 104
15.14 odd 2 inner 465.2.t.d.119.38 yes 104
31.6 odd 6 inner 465.2.t.d.254.38 yes 104
93.68 even 6 inner 465.2.t.d.254.16 yes 104
155.99 odd 6 inner 465.2.t.d.254.15 yes 104
465.254 even 6 inner 465.2.t.d.254.37 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 3.2 odd 2 inner
465.2.t.d.119.16 yes 104 5.4 even 2 inner
465.2.t.d.119.37 yes 104 1.1 even 1 trivial
465.2.t.d.119.38 yes 104 15.14 odd 2 inner
465.2.t.d.254.15 yes 104 155.99 odd 6 inner
465.2.t.d.254.16 yes 104 93.68 even 6 inner
465.2.t.d.254.37 yes 104 465.254 even 6 inner
465.2.t.d.254.38 yes 104 31.6 odd 6 inner