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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 119.15
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.15

$q$-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} +(-2.23430 + 0.0890267i) q^{5} +(-0.174294 - 1.99866i) q^{6} +(-0.189994 - 0.109693i) q^{7} +3.07916 q^{8} +(-2.95472 + 0.519284i) q^{9} +(2.58799 - 0.103120i) 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} +(-0.489817 - 3.84188i) q^{15} -2.24993 q^{16} +(1.53989 + 0.889057i) q^{17} +(3.42246 - 0.601488i) q^{18} +(0.113062 - 0.195829i) q^{19} +(1.47091 - 0.0586092i) q^{20} +(0.160687 - 0.344341i) q^{21} +(3.74755 + 6.49095i) q^{22} -6.66207i q^{23} +(0.463331 + 5.31309i) 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} +(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} +(1.94518 - 0.341861i) 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} +(-0.186124 + 0.398852i) q^{42} +(4.15057 - 7.18899i) q^{43} +(2.12995 + 3.68919i) q^{44} +(6.55548 - 1.42328i) q^{45} +7.71670i q^{46} +9.68203 q^{47} +(-0.338555 - 3.88227i) q^{48} +(-3.47593 - 6.02050i) q^{49} +(-5.77316 + 0.460801i) 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} +(7.72768 + 12.2326i) q^{55} +(-0.585022 - 0.337762i) q^{56} +(0.354916 + 0.165621i) 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} +(-4.65313 - 7.36572i) 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} +(1.43643 + 8.54030i) 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} +(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} +(-7.59323 + 1.64859i) 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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 \(0.634305\pi\)
\(3\) 0.150473 + 1.72550i 0.0868758 + 0.996219i
\(4\) −0.658332 −0.329166
\(5\) −2.23430 + 0.0890267i −0.999207 + 0.0398140i
\(6\) −0.174294 − 1.99866i −0.0711551 − 0.815948i
\(7\) −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
\(9\) −2.95472 + 0.519284i −0.984905 + 0.173095i
\(10\) 2.58799 − 0.103120i 0.818395 − 0.0326094i
\(11\) −3.23538 − 5.60384i −0.975503 − 1.68962i −0.678264 − 0.734818i \(-0.737268\pi\)
−0.297239 − 0.954803i \(-0.596066\pi\)
\(12\) −0.0990614 − 1.13595i −0.0285966 − 0.327922i
\(13\) 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
\(15\) −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 \(0.597487\pi\)
\(18\) 3.42246 − 0.601488i 0.806681 − 0.141772i
\(19\) 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
\(21\) 0.160687 − 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
\(25\) 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 \(-0.390959\pi\)
0.335903 + 0.941897i \(0.390959\pi\)
\(30\) 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
\(36\) 1.94518 − 0.341861i 0.324197 − 0.0569769i
\(37\) −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 −0.451299 − 0.892373i \(-0.649039\pi\)
0.547168 + 0.837023i \(0.315706\pi\)
\(42\) −0.186124 + 0.398852i −0.0287196 + 0.0615442i
\(43\) 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
\(45\) 6.55548 − 1.42328i 0.977233 − 0.212170i
\(46\) 7.71670i 1.13777i
\(47\) 9.68203 1.41227 0.706135 − 0.708077i \(-0.250437\pi\)
0.706135 + 0.708077i \(0.250437\pi\)
\(48\) −0.338555 − 3.88227i −0.0488662 − 0.560357i
\(49\) −3.47593 − 6.02050i −0.496562 − 0.860071i
\(50\) −5.77316 + 0.460801i −0.816448 + 0.0651671i
\(51\) −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 \(-0.883769\pi\)
−0.157787 + 0.987473i \(0.550436\pi\)
\(54\) 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
\(57\) 0.354916 + 0.165621i 0.0470098 + 0.0219371i
\(58\) −4.19050 −0.550239
\(59\) −7.20486 − 4.15973i −0.937993 − 0.541550i −0.0486621 − 0.998815i \(-0.515496\pi\)
−0.889330 + 0.457265i \(0.848829\pi\)
\(60\) 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
\(65\) −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
\(69\) 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 −0.234090 − 0.972215i \(-0.575211\pi\)
0.724918 + 0.688835i \(0.241878\pi\)
\(72\) −9.09803 + 1.59896i −1.07221 + 0.188439i
\(73\) −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
\(75\) 1.43643 + 8.54030i 0.165864 + 0.986149i
\(76\) −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
\(81\) 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 \(-0.721117\pi\)
0.345282 + 0.938499i \(0.387783\pi\)
\(84\) −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 \(0.660491\pi\)
\(90\) −7.59323 + 1.64859i −0.800397 + 0.173777i
\(91\) − 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.119.15 ✓ 104
3.2 odd 2 inner 465.2.t.d.119.37 yes 104
5.4 even 2 inner 465.2.t.d.119.38 yes 104
15.14 odd 2 inner 465.2.t.d.119.16 yes 104
31.6 odd 6 inner 465.2.t.d.254.16 yes 104
93.68 even 6 inner 465.2.t.d.254.38 yes 104
155.99 odd 6 inner 465.2.t.d.254.37 yes 104
465.254 even 6 inner 465.2.t.d.254.15 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.15 ✓ 104 1.1 even 1 trivial
465.2.t.d.119.16 yes 104 15.14 odd 2 inner
465.2.t.d.119.37 yes 104 3.2 odd 2 inner
465.2.t.d.119.38 yes 104 5.4 even 2 inner
465.2.t.d.254.15 yes 104 465.254 even 6 inner
465.2.t.d.254.16 yes 104 31.6 odd 6 inner
465.2.t.d.254.37 yes 104 155.99 odd 6 inner
465.2.t.d.254.38 yes 104 93.68 even 6 inner