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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.98
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.98

$q$-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+(1.73403 - 0.996556i) q^{2} +(-1.05335 - 1.82446i) q^{3} +(2.01375 - 3.45613i) q^{4} +(-4.98894 + 0.332389i) q^{5} +(-3.64473 - 2.11396i) q^{6} -8.55585 q^{7} +(0.0476908 - 7.99986i) q^{8} +(2.28089 - 3.95061i) q^{9} +(-8.31975 + 5.54813i) q^{10} +10.2120i q^{11} +(-8.42677 - 0.0334903i) q^{12} +(10.2155 + 5.89792i) q^{13} +(-14.8361 + 8.52638i) q^{14} +(5.86156 + 8.75202i) q^{15} +(-7.88961 - 13.9196i) q^{16} +(-16.0851 + 9.28673i) q^{17} +(0.0181296 - 9.12353i) q^{18} +(-14.5306 - 12.2418i) q^{19} +(-8.89770 + 17.9118i) q^{20} +(9.01234 + 15.6098i) q^{21} +(10.1768 + 17.7080i) q^{22} +(-18.2460 + 31.6030i) q^{23} +(-14.6457 + 8.33968i) q^{24} +(24.7790 - 3.31654i) q^{25} +(23.5916 + 0.0468796i) q^{26} -28.5707 q^{27} +(-17.2294 + 29.5701i) q^{28} +(-6.36256 + 11.0203i) q^{29} +(18.8860 + 9.33493i) q^{30} -8.11719i q^{31} +(-27.5525 - 16.2746i) q^{32} +(18.6314 - 10.7569i) q^{33} +(-18.6374 + 32.1332i) q^{34} +(42.6846 - 2.84387i) q^{35} +(-9.06067 - 15.8386i) q^{36} -43.7274i q^{37} +(-37.3962 - 6.74726i) q^{38} -24.8504i q^{39} +(2.42114 + 39.9267i) q^{40} +(-19.1707 - 33.2046i) q^{41} +(31.1838 + 18.0867i) q^{42} +(-29.0633 - 50.3391i) q^{43} +(35.2940 + 20.5644i) q^{44} +(-10.0661 + 20.4675i) q^{45} +(-0.145028 + 72.9839i) q^{46} +(21.4036 - 37.0721i) q^{47} +(-17.0852 + 29.0565i) q^{48} +24.2025 q^{49} +(39.6626 - 30.4447i) q^{50} +(33.8866 + 19.5645i) q^{51} +(40.9554 - 23.4291i) q^{52} +(1.54174 + 0.890123i) q^{53} +(-49.5426 + 28.4723i) q^{54} +(-3.39436 - 50.9471i) q^{55} +(-0.408035 + 68.4456i) q^{56} +(-7.02897 + 39.4055i) q^{57} +(-0.0505728 + 25.4502i) q^{58} +(68.8236 - 39.7353i) q^{59} +(42.0518 - 2.63389i) q^{60} +(-41.5021 + 71.8838i) q^{61} +(-8.08924 - 14.0755i) q^{62} +(-19.5149 + 33.8008i) q^{63} +(-63.9955 - 0.763039i) q^{64} +(-52.9249 - 26.0288i) q^{65} +(21.5877 - 37.2201i) q^{66} +(20.7172 - 35.8833i) q^{67} +(-0.295262 + 74.2933i) q^{68} +76.8780 q^{69} +(71.1825 - 47.4690i) q^{70} +(79.6311 - 45.9750i) q^{71} +(-31.4956 - 18.4352i) q^{72} +(-85.6344 + 49.4410i) q^{73} +(-43.5768 - 75.8248i) q^{74} +(-32.1520 - 41.7150i) q^{75} +(-71.5703 + 25.5674i) q^{76} -87.3724i q^{77} +(-24.7648 - 43.0915i) q^{78} +(-15.0449 + 8.68620i) q^{79} +(43.9875 + 66.8214i) q^{80} +(9.56712 + 16.5707i) q^{81} +(-66.3329 - 38.4733i) q^{82} -144.479 q^{83} +(72.0982 + 0.286538i) q^{84} +(77.1608 - 51.6775i) q^{85} +(-100.562 - 58.3265i) q^{86} +26.8081 q^{87} +(81.6946 + 0.487019i) q^{88} +(-9.75929 + 16.9036i) q^{89} +(2.94212 + 45.5228i) q^{90} +(-87.4022 - 50.4617i) q^{91} +(72.4810 + 126.701i) q^{92} +(-14.8095 + 8.55028i) q^{93} +(0.170127 - 85.6142i) q^{94} +(76.5611 + 56.2440i) q^{95} +(-0.669797 + 67.4114i) q^{96} +(-14.4836 + 8.36208i) q^{97} +(41.9680 - 24.1192i) q^{98} +(40.3437 + 23.2924i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73403 − 0.996556i 0.867017 − 0.498278i
\(3\) −1.05335 − 1.82446i −0.351118 − 0.608155i 0.635327 − 0.772243i \(-0.280865\pi\)
−0.986446 + 0.164088i \(0.947532\pi\)
\(4\) 2.01375 − 3.45613i 0.503438 − 0.864031i
\(5\) −4.98894 + 0.332389i −0.997788 + 0.0664779i
\(6\) −3.64473 − 2.11396i −0.607456 − 0.352326i
\(7\) −8.55585 −1.22226 −0.611132 − 0.791529i \(-0.709286\pi\)
−0.611132 + 0.791529i \(0.709286\pi\)
\(8\) 0.0476908 − 7.99986i 0.00596135 − 0.999982i
\(9\) 2.28089 − 3.95061i 0.253432 − 0.438957i
\(10\) −8.31975 + 5.54813i −0.831975 + 0.554813i
\(11\) 10.2120i 0.928364i 0.885740 + 0.464182i \(0.153652\pi\)
−0.885740 + 0.464182i \(0.846348\pi\)
\(12\) −8.42677 − 0.0334903i −0.702231 − 0.00279086i
\(13\) 10.2155 + 5.89792i 0.785808 + 0.453686i 0.838485 − 0.544925i \(-0.183442\pi\)
−0.0526770 + 0.998612i \(0.516775\pi\)
\(14\) −14.8361 + 8.52638i −1.05972 + 0.609027i
\(15\) 5.86156 + 8.75202i 0.390770 + 0.583468i
\(16\) −7.88961 − 13.9196i −0.493101 − 0.869972i
\(17\) −16.0851 + 9.28673i −0.946182 + 0.546278i −0.891893 − 0.452247i \(-0.850623\pi\)
−0.0542892 + 0.998525i \(0.517289\pi\)
\(18\) 0.0181296 − 9.12353i 0.00100720 − 0.506863i
\(19\) −14.5306 − 12.2418i −0.764766 − 0.644308i
\(20\) −8.89770 + 17.9118i −0.444885 + 0.895588i
\(21\) 9.01234 + 15.6098i 0.429159 + 0.743326i
\(22\) 10.1768 + 17.7080i 0.462584 + 0.804908i
\(23\) −18.2460 + 31.6030i −0.793304 + 1.37404i 0.130606 + 0.991434i \(0.458308\pi\)
−0.923910 + 0.382609i \(0.875026\pi\)
\(24\) −14.6457 + 8.33968i −0.610237 + 0.347487i
\(25\) 24.7790 − 3.31654i 0.991161 − 0.132662i
\(26\) 23.5916 + 0.0468796i 0.907371 + 0.00180306i
\(27\) −28.5707 −1.05817
\(28\) −17.2294 + 29.5701i −0.615334 + 1.05607i
\(29\) −6.36256 + 11.0203i −0.219398 + 0.380009i −0.954624 − 0.297813i \(-0.903743\pi\)
0.735226 + 0.677822i \(0.237076\pi\)
\(30\) 18.8860 + 9.33493i 0.629534 + 0.311164i
\(31\) − 8.11719i − 0.261845i −0.991393 − 0.130922i \(-0.958206\pi\)
0.991393 − 0.130922i \(-0.0417939\pi\)
\(32\) −27.5525 − 16.2746i −0.861015 − 0.508580i
\(33\) 18.6314 − 10.7569i 0.564589 − 0.325966i
\(34\) −18.6374 + 32.1332i −0.548158 + 0.945095i
\(35\) 42.6846 − 2.84387i 1.21956 − 0.0812535i
\(36\) −9.06067 − 15.8386i −0.251685 − 0.439961i
\(37\) − 43.7274i − 1.18182i −0.806737 − 0.590911i \(-0.798769\pi\)
0.806737 − 0.590911i \(-0.201231\pi\)
\(38\) −37.3962 − 6.74726i −0.984110 − 0.177560i
\(39\) − 24.8504i − 0.637190i
\(40\) 2.42114 + 39.9267i 0.0605285 + 0.998166i
\(41\) −19.1707 − 33.2046i −0.467578 − 0.809869i 0.531736 − 0.846910i \(-0.321540\pi\)
−0.999314 + 0.0370415i \(0.988207\pi\)
\(42\) 31.1838 + 18.0867i 0.742471 + 0.430635i
\(43\) −29.0633 − 50.3391i −0.675890 − 1.17068i −0.976208 − 0.216837i \(-0.930426\pi\)
0.300318 − 0.953839i \(-0.402907\pi\)
\(44\) 35.2940 + 20.5644i 0.802136 + 0.467374i
\(45\) −10.0661 + 20.4675i −0.223690 + 0.454834i
\(46\) −0.145028 + 72.9839i −0.00315279 + 1.58661i
\(47\) 21.4036 − 37.0721i 0.455396 − 0.788769i −0.543315 − 0.839529i \(-0.682831\pi\)
0.998711 + 0.0507603i \(0.0161644\pi\)
\(48\) −17.0852 + 29.0565i −0.355941 + 0.605345i
\(49\) 24.2025 0.493929
\(50\) 39.6626 − 30.4447i 0.793252 − 0.608894i
\(51\) 33.8866 + 19.5645i 0.664444 + 0.383617i
\(52\) 40.9554 − 23.4291i 0.787604 − 0.450560i
\(53\) 1.54174 + 0.890123i 0.0290894 + 0.0167948i 0.514474 − 0.857506i \(-0.327987\pi\)
−0.485385 + 0.874301i \(0.661320\pi\)
\(54\) −49.5426 + 28.4723i −0.917456 + 0.527265i
\(55\) −3.39436 − 50.9471i −0.0617157 − 0.926311i
\(56\) −0.408035 + 68.4456i −0.00728634 + 1.22224i
\(57\) −7.02897 + 39.4055i −0.123315 + 0.691324i
\(58\) −0.0505728 + 25.4502i −0.000871945 + 0.438796i
\(59\) 68.8236 − 39.7353i 1.16650 − 0.673480i 0.213648 − 0.976911i \(-0.431465\pi\)
0.952854 + 0.303430i \(0.0981320\pi\)
\(60\) 42.0518 − 2.63389i 0.700863 − 0.0438981i
\(61\) −41.5021 + 71.8838i −0.680362 + 1.17842i 0.294508 + 0.955649i \(0.404844\pi\)
−0.974870 + 0.222773i \(0.928489\pi\)
\(62\) −8.08924 − 14.0755i −0.130472 − 0.227024i
\(63\) −19.5149 + 33.8008i −0.309761 + 0.536521i
\(64\) −63.9955 − 0.763039i −0.999929 − 0.0119225i
\(65\) −52.9249 − 26.0288i −0.814229 − 0.400444i
\(66\) 21.5877 − 37.2201i 0.327087 − 0.563940i
\(67\) 20.7172 − 35.8833i 0.309212 − 0.535571i −0.668978 − 0.743282i \(-0.733268\pi\)
0.978190 + 0.207711i \(0.0666014\pi\)
\(68\) −0.295262 + 74.2933i −0.00434209 + 1.09255i
\(69\) 76.8780 1.11417
\(70\) 71.1825 − 47.4690i 1.01689 − 0.678128i
\(71\) 79.6311 − 45.9750i 1.12156 − 0.647535i 0.179764 − 0.983710i \(-0.442466\pi\)
0.941800 + 0.336174i \(0.109133\pi\)
\(72\) −31.4956 − 18.4352i −0.437438 − 0.256044i
\(73\) −85.6344 + 49.4410i −1.17307 + 0.677274i −0.954402 − 0.298524i \(-0.903506\pi\)
−0.218671 + 0.975799i \(0.570172\pi\)
\(74\) −43.5768 − 75.8248i −0.588876 − 1.02466i
\(75\) −32.1520 − 41.7150i −0.428694 − 0.556200i
\(76\) −71.5703 + 25.5674i −0.941714 + 0.336413i
\(77\) − 87.3724i − 1.13471i
\(78\) −24.7648 − 43.0915i −0.317498 − 0.552455i
\(79\) −15.0449 + 8.68620i −0.190442 + 0.109952i −0.592190 − 0.805799i \(-0.701736\pi\)
0.401747 + 0.915751i \(0.368403\pi\)
\(80\) 43.9875 + 66.8214i 0.549844 + 0.835268i
\(81\) 9.56712 + 16.5707i 0.118113 + 0.204577i
\(82\) −66.3329 − 38.4733i −0.808938 − 0.469186i
\(83\) −144.479 −1.74071 −0.870355 − 0.492426i \(-0.836110\pi\)
−0.870355 + 0.492426i \(0.836110\pi\)
\(84\) 72.0982 + 0.286538i 0.858312 + 0.00341117i
\(85\) 77.1608 − 51.6775i 0.907774 − 0.607970i
\(86\) −100.562 − 58.3265i −1.16933 − 0.678215i
\(87\) 26.8081 0.308139
\(88\) 81.6946 + 0.487019i 0.928348 + 0.00553430i
\(89\) −9.75929 + 16.9036i −0.109655 + 0.189928i −0.915630 − 0.402021i \(-0.868308\pi\)
0.805976 + 0.591949i \(0.201641\pi\)
\(90\) 2.94212 + 45.5228i 0.0326902 + 0.505809i
\(91\) −87.4022 − 50.4617i −0.960464 − 0.554524i
\(92\) 72.4810 + 126.701i 0.787837 + 1.37719i
\(93\) −14.8095 + 8.55028i −0.159242 + 0.0919385i
\(94\) 0.170127 − 85.6142i 0.00180986 − 0.910790i
\(95\) 76.5611 + 56.2440i 0.805907 + 0.592042i
\(96\) −0.669797 + 67.4114i −0.00697705 + 0.702202i
\(97\) −14.4836 + 8.36208i −0.149315 + 0.0862071i −0.572796 − 0.819698i \(-0.694141\pi\)
0.423481 + 0.905905i \(0.360808\pi\)
\(98\) 41.9680 − 24.1192i 0.428245 − 0.246114i
\(99\) 40.3437 + 23.2924i 0.407512 + 0.235277i
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 380.3.p.a.159.98 yes 232
4.3 odd 2 inner 380.3.p.a.159.97 yes 232
5.4 even 2 inner 380.3.p.a.159.19 ✓ 232
19.11 even 3 inner 380.3.p.a.239.20 yes 232
20.19 odd 2 inner 380.3.p.a.159.20 yes 232
76.11 odd 6 inner 380.3.p.a.239.19 yes 232
95.49 even 6 inner 380.3.p.a.239.97 yes 232
380.239 odd 6 inner 380.3.p.a.239.98 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.19 ✓ 232 5.4 even 2 inner
380.3.p.a.159.20 yes 232 20.19 odd 2 inner
380.3.p.a.159.97 yes 232 4.3 odd 2 inner
380.3.p.a.159.98 yes 232 1.1 even 1 trivial
380.3.p.a.239.19 yes 232 76.11 odd 6 inner
380.3.p.a.239.20 yes 232 19.11 even 3 inner
380.3.p.a.239.97 yes 232 95.49 even 6 inner
380.3.p.a.239.98 yes 232 380.239 odd 6 inner