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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 254.38
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.38

$q$-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} +(-0.150473 + 1.72550i) q^{3} -0.658332 q^{4} +(-1.19425 - 1.89044i) q^{5} +(-0.174294 + 1.99866i) q^{6} +(0.189994 - 0.109693i) q^{7} -3.07916 q^{8} +(-2.95472 - 0.519284i) q^{9} +(-1.38330 - 2.18971i) q^{10} +(-3.23538 + 5.60384i) q^{11} +(0.0990614 - 1.13595i) 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} +(-3.42246 - 0.601488i) q^{18} +(0.113062 + 0.195829i) q^{19} +(0.786211 + 1.24454i) q^{20} +(0.160687 + 0.344341i) q^{21} +(-3.74755 + 6.49095i) q^{22} -6.66207i q^{23} +(0.463331 - 5.31309i) 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} +(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} +(1.94518 + 0.341861i) 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} +(0.186124 + 0.398852i) q^{42} +(-4.15057 - 7.18899i) q^{43} +(2.12995 - 3.68919i) q^{44} +(2.54698 + 6.20587i) q^{45} -7.71670i q^{46} -9.68203 q^{47} +(0.338555 - 3.88227i) q^{48} +(-3.47593 + 6.02050i) q^{49} +(-2.48751 + 5.23010i) q^{50} +(-1.30236 - 2.79087i) 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} +(-0.354916 + 0.165621i) 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} +(8.70546 - 0.346874i) q^{65} +(-10.6362 - 7.44312i) q^{66} +(0.362559 + 0.209324i) q^{67} +(1.01376 - 0.585295i) q^{68} +(11.4954 + 1.00246i) q^{69} +(-0.503015 - 0.264293i) q^{70} +(4.13579 + 2.38780i) q^{71} +(9.09803 + 1.59896i) q^{72} +(4.75906 - 8.24293i) q^{73} +(1.40781 + 2.43840i) q^{74} +(-7.46803 - 4.38503i) 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} +(8.46069 + 3.06867i) q^{81} +(0.711039 + 0.410518i) q^{82} +(2.68613 + 1.55084i) q^{83} +(-0.105785 - 0.226691i) q^{84} +(3.51972 + 1.84932i) q^{85} +(-4.80762 - 8.32704i) q^{86} +(-0.544380 + 6.24250i) q^{87} +(9.96224 - 17.2551i) q^{88} -9.11518 q^{89} +(2.95018 + 7.18829i) q^{90} +0.854793i q^{91} +4.38586i q^{92} +(0.276675 + 9.63968i) q^{93} -11.2147 q^{94} +(0.235179 - 0.447605i) q^{95} +(-0.534512 + 6.12934i) q^{96} -3.68647i q^{97} +(-4.02619 + 6.97356i) q^{98} +(12.4696 - 14.8777i) 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 \(0.365695\pi\)
\(3\) −0.150473 + 1.72550i −0.0868758 + 0.996219i
\(4\) −0.658332 −0.329166
\(5\) −1.19425 − 1.89044i −0.534083 − 0.845432i
\(6\) −0.174294 + 1.99866i −0.0711551 + 0.815948i
\(7\) 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
\(9\) −2.95472 − 0.519284i −0.984905 − 0.173095i
\(10\) −1.38330 − 2.18971i −0.437438 − 0.692446i
\(11\) −3.23538 + 5.60384i −0.975503 + 1.68962i −0.297239 + 0.954803i \(0.596066\pi\)
−0.678264 + 0.734818i \(0.737268\pi\)
\(12\) 0.0990614 − 1.13595i 0.0285966 − 0.327922i
\(13\) −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
\(15\) 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 \(0.402513\pi\)
\(18\) −3.42246 − 0.601488i −0.806681 − 0.141772i
\(19\) 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
\(21\) 0.160687 + 0.344341i 0.0350647 + 0.0751414i
\(22\) −3.74755 + 6.49095i −0.798981 + 1.38387i
\(23\) − 6.66207i − 1.38914i −0.719426 − 0.694569i \(-0.755595\pi\)
0.719426 − 0.694569i \(-0.244405\pi\)
\(24\) 0.463331 − 5.31309i 0.0945770 − 1.08453i
\(25\) −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 \(-0.390959\pi\)
0.335903 + 0.941897i \(0.390959\pi\)
\(30\) 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
\(36\) 1.94518 + 0.341861i 0.324197 + 0.0569769i
\(37\) 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 0.547168 − 0.837023i \(-0.315706\pi\)
−0.451299 + 0.892373i \(0.649039\pi\)
\(42\) 0.186124 + 0.398852i 0.0287196 + 0.0615442i
\(43\) −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
\(45\) 2.54698 + 6.20587i 0.379682 + 0.925117i
\(46\) − 7.71670i − 1.13777i
\(47\) −9.68203 −1.41227 −0.706135 − 0.708077i \(-0.749563\pi\)
−0.706135 + 0.708077i \(0.749563\pi\)
\(48\) 0.338555 − 3.88227i 0.0488662 − 0.560357i
\(49\) −3.47593 + 6.02050i −0.496562 + 0.860071i
\(50\) −2.48751 + 5.23010i −0.351788 + 0.739648i
\(51\) −1.30236 − 2.79087i −0.182367 − 0.390800i
\(52\) 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
\(55\) 14.4576 − 0.576070i 1.94946 − 0.0776773i
\(56\) −0.585022 + 0.337762i −0.0781768 + 0.0451354i
\(57\) −0.354916 + 0.165621i −0.0470098 + 0.0219371i
\(58\) 4.19050 0.550239
\(59\) −7.20486 + 4.15973i −0.937993 + 0.541550i −0.889330 − 0.457265i \(-0.848829\pi\)
−0.0486621 + 0.998815i \(0.515496\pi\)
\(60\) −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
\(65\) 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
\(69\) 11.4954 + 1.00246i 1.38389 + 0.120682i
\(70\) −0.503015 − 0.264293i −0.0601218 − 0.0315890i
\(71\) 4.13579 + 2.38780i 0.490828 + 0.283380i 0.724918 − 0.688835i \(-0.241878\pi\)
−0.234090 + 0.972215i \(0.575211\pi\)
\(72\) 9.09803 + 1.59896i 1.07221 + 0.188439i
\(73\) 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
\(75\) −7.46803 − 4.38503i −0.862334 − 0.506340i
\(76\) −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
\(81\) 8.46069 + 3.06867i 0.940077 + 0.340964i
\(82\) 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 \(0.612217\pi\)
\(84\) −0.105785 − 0.226691i −0.0115421 − 0.0247340i
\(85\) 3.51972 + 1.84932i 0.381768 + 0.200587i
\(86\) −4.80762 − 8.32704i −0.518419 − 0.897927i
\(87\) −0.544380 + 6.24250i −0.0583637 + 0.669266i
\(88\) 9.96224 − 17.2551i 1.06198 − 1.83940i
\(89\) −9.11518 −0.966207 −0.483103 − 0.875563i \(-0.660491\pi\)
−0.483103 + 0.875563i \(0.660491\pi\)
\(90\) 2.95018 + 7.18829i 0.310976 + 0.757712i
\(91\) 0.854793i 0.0896067i
\(92\) 4.38586i 0.457257i
\(93\) 0.276675 + 9.63968i 0.0286899 + 0.999588i
\(94\) −11.2147 −1.15671
\(95\) 0.235179 − 0.447605i 0.0241289 − 0.0459233i
\(96\) −0.534512 + 6.12934i −0.0545534 + 0.625573i
\(97\) − 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
\(99\) 12.4696 − 14.8777i 1.25324 − 1.49526i
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.254.38 yes 104
3.2 odd 2 inner 465.2.t.d.254.16 yes 104
5.4 even 2 inner 465.2.t.d.254.15 yes 104
15.14 odd 2 inner 465.2.t.d.254.37 yes 104
31.26 odd 6 inner 465.2.t.d.119.37 yes 104
93.26 even 6 inner 465.2.t.d.119.15 ✓ 104
155.119 odd 6 inner 465.2.t.d.119.16 yes 104
465.119 even 6 inner 465.2.t.d.119.38 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 93.26 even 6 inner
465.2.t.d.119.16 yes 104 155.119 odd 6 inner
465.2.t.d.119.37 yes 104 31.26 odd 6 inner
465.2.t.d.119.38 yes 104 465.119 even 6 inner
465.2.t.d.254.15 yes 104 5.4 even 2 inner
465.2.t.d.254.16 yes 104 3.2 odd 2 inner
465.2.t.d.254.37 yes 104 15.14 odd 2 inner
465.2.t.d.254.38 yes 104 1.1 even 1 trivial