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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 254.37
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.37

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

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

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 \(-0.365695\pi\)
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 −0.535476 − 0.844551i \(-0.679868\pi\)
0.463665 + 0.886011i \(0.346534\pi\)
\(8\) −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 0.297239 − 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 \(-0.651637\pi\)
0.998886 − 0.0471991i \(-0.0150295\pi\)
\(14\) −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 −0.674977 − 0.737839i \(-0.735847\pi\)
0.301498 + 0.953467i \(0.402513\pi\)
\(18\) 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 \(0.825076\pi\)
\(20\) −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 \(-0.755595\pi\)
0.719426 − 0.694569i \(-0.244405\pi\)
\(24\) 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 0.748655 − 0.662959i \(-0.230700\pi\)
−0.948467 + 0.316875i \(0.897366\pi\)
\(38\) 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 0.451299 − 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 \(0.0514917\pi\)
−0.353989 + 0.935250i \(0.615175\pi\)
\(44\) −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 0.934071 − 0.357089i \(-0.116231\pi\)
0.157787 + 0.987473i \(0.449564\pi\)
\(54\) −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 0.0486621 − 0.998815i \(-0.484504\pi\)
0.889330 + 0.457265i \(0.151171\pi\)
\(60\) 2.26576 + 1.16934i 0.292508 + 0.150961i
\(61\) 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
\(65\) 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 0.477690 − 0.878529i \(-0.341474\pi\)
−0.521983 + 0.852956i \(0.674808\pi\)
\(68\) 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 \(0.354716\pi\)
−0.997744 + 0.0671266i \(0.978617\pi\)
\(74\) −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 −0.839211 − 0.543807i \(-0.816982\pi\)
0.0513452 + 0.998681i \(0.483649\pi\)
\(80\) −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 0.640123 − 0.768273i \(-0.278883\pi\)
−0.345282 + 0.938499i \(0.612217\pi\)
\(84\) −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 \(0.0599257\pi\)
−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.254.37 yes 104
3.2 odd 2 inner 465.2.t.d.254.15 yes 104
5.4 even 2 inner 465.2.t.d.254.16 yes 104
15.14 odd 2 inner 465.2.t.d.254.38 yes 104
31.26 odd 6 inner 465.2.t.d.119.38 yes 104
93.26 even 6 inner 465.2.t.d.119.16 yes 104
155.119 odd 6 inner 465.2.t.d.119.15 ✓ 104
465.119 even 6 inner 465.2.t.d.119.37 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 155.119 odd 6 inner
465.2.t.d.119.16 yes 104 93.26 even 6 inner
465.2.t.d.119.37 yes 104 465.119 even 6 inner
465.2.t.d.119.38 yes 104 31.26 odd 6 inner
465.2.t.d.254.15 yes 104 3.2 odd 2 inner
465.2.t.d.254.16 yes 104 5.4 even 2 inner
465.2.t.d.254.37 yes 104 1.1 even 1 trivial
465.2.t.d.254.38 yes 104 15.14 odd 2 inner