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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 159.97
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.97

$q$-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+(1.73006 - 1.00344i) q^{2} +(1.05335 + 1.82446i) q^{3} +(1.98622 - 3.47202i) q^{4} +(-4.98894 + 0.332389i) q^{5} +(3.65311 + 2.09945i) q^{6} +8.55585 q^{7} +(-0.0476908 - 7.99986i) q^{8} +(2.28089 - 3.95061i) q^{9} +(-8.29763 + 5.58115i) q^{10} -10.2120i q^{11} +(8.42677 - 0.0334903i) q^{12} +(10.2155 + 5.89792i) q^{13} +(14.8021 - 8.58528i) q^{14} +(-5.86156 - 8.75202i) q^{15} +(-8.10988 - 13.7924i) q^{16} +(-16.0851 + 9.28673i) q^{17} +(-0.0181296 - 9.12353i) q^{18} +(14.5306 + 12.2418i) q^{19} +(-8.75505 + 17.9819i) q^{20} +(9.01234 + 15.6098i) q^{21} +(-10.2471 - 17.6674i) q^{22} +(18.2460 - 31.6030i) q^{23} +(14.5452 - 8.51370i) q^{24} +(24.7790 - 3.31654i) q^{25} +(23.5916 - 0.0468796i) q^{26} +28.5707 q^{27} +(16.9938 - 29.7061i) q^{28} +(-6.36256 + 11.0203i) q^{29} +(-18.9230 - 9.25980i) q^{30} +8.11719i q^{31} +(-27.8704 - 15.7239i) q^{32} +(18.6314 - 10.7569i) q^{33} +(-18.5095 + 32.2070i) q^{34} +(-42.6846 + 2.84387i) q^{35} +(-9.18628 - 15.7661i) q^{36} -43.7274i q^{37} +(37.4227 + 6.59859i) q^{38} +24.8504i q^{39} +(2.89699 + 39.8950i) q^{40} +(-19.1707 - 33.2046i) q^{41} +(31.2554 + 17.9626i) q^{42} +(29.0633 + 50.3391i) q^{43} +(-35.4563 - 20.2833i) q^{44} +(-10.0661 + 20.4675i) q^{45} +(-0.145028 - 72.9839i) q^{46} +(-21.4036 + 37.0721i) q^{47} +(16.6211 - 29.3245i) q^{48} +24.2025 q^{49} +(39.5413 - 30.6021i) q^{50} +(-33.8866 - 19.5645i) q^{51} +(40.7679 - 23.7539i) q^{52} +(1.54174 + 0.890123i) q^{53} +(49.4291 - 28.6690i) 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} +(-42.0295 + 2.96805i) q^{60} +(-41.5021 + 71.8838i) q^{61} +(8.14511 + 14.0432i) q^{62} +(19.5149 - 33.8008i) q^{63} +(-63.9955 + 0.763039i) q^{64} +(-52.9249 - 26.0288i) q^{65} +(21.4396 - 37.3056i) q^{66} +(-20.7172 + 35.8833i) q^{67} +(0.295262 + 74.2933i) q^{68} +76.8780 q^{69} +(-70.9933 + 47.7515i) q^{70} +(-79.6311 + 45.9750i) q^{71} +(-31.7131 - 18.0584i) q^{72} +(-85.6344 + 49.4410i) q^{73} +(-43.8778 - 75.6510i) q^{74} +(32.1520 + 41.7150i) q^{75} +(71.3648 - 26.1355i) q^{76} -87.3724i q^{77} +(24.9359 + 42.9927i) q^{78} +(15.0449 - 8.68620i) q^{79} +(45.0442 + 66.1137i) q^{80} +(9.56712 + 16.5707i) q^{81} +(-66.4853 - 38.2094i) q^{82} +144.479 q^{83} +(72.0982 - 0.286538i) q^{84} +(77.1608 - 51.6775i) q^{85} +(100.793 + 57.9264i) q^{86} -26.8081 q^{87} +(-81.6946 + 0.487019i) q^{88} +(-9.75929 + 16.9036i) q^{89} +(3.12301 + 45.5107i) q^{90} +(87.4022 + 50.4617i) q^{91} +(-73.4858 - 126.121i) 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} +(41.8719 - 24.2858i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73006 − 1.00344i 0.865030 − 0.501720i
\(3\) 1.05335 + 1.82446i 0.351118 + 0.608155i 0.986446 − 0.164088i \(-0.0524682\pi\)
−0.635327 + 0.772243i \(0.719135\pi\)
\(4\) 1.98622 − 3.47202i 0.496554 − 0.868006i
\(5\) −4.98894 + 0.332389i −0.997788 + 0.0664779i
\(6\) 3.65311 + 2.09945i 0.608851 + 0.349909i
\(7\) 8.55585 1.22226 0.611132 − 0.791529i \(-0.290714\pi\)
0.611132 + 0.791529i \(0.290714\pi\)
\(8\) −0.0476908 − 7.99986i −0.00596135 − 0.999982i
\(9\) 2.28089 − 3.95061i 0.253432 − 0.438957i
\(10\) −8.29763 + 5.58115i −0.829763 + 0.558115i
\(11\) − 10.2120i − 0.928364i −0.885740 − 0.464182i \(-0.846348\pi\)
0.885740 − 0.464182i \(-0.153652\pi\)
\(12\) 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 \(0.516775\pi\)
\(14\) 14.8021 − 8.58528i 1.05730 − 0.613234i
\(15\) −5.86156 − 8.75202i −0.390770 − 0.583468i
\(16\) −8.10988 − 13.7924i −0.506868 − 0.862024i
\(17\) −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
\(20\) −8.75505 + 17.9819i −0.437753 + 0.899095i
\(21\) 9.01234 + 15.6098i 0.429159 + 0.743326i
\(22\) −10.2471 − 17.6674i −0.465779 − 0.803063i
\(23\) 18.2460 − 31.6030i 0.793304 − 1.37404i −0.130606 − 0.991434i \(-0.541692\pi\)
0.923910 − 0.382609i \(-0.124974\pi\)
\(24\) 14.5452 − 8.51370i 0.606051 − 0.354737i
\(25\) 24.7790 − 3.31654i 0.991161 − 0.132662i
\(26\) 23.5916 − 0.0468796i 0.907371 − 0.00180306i
\(27\) 28.5707 1.05817
\(28\) 16.9938 − 29.7061i 0.606920 − 1.06093i
\(29\) −6.36256 + 11.0203i −0.219398 + 0.380009i −0.954624 − 0.297813i \(-0.903743\pi\)
0.735226 + 0.677822i \(0.237076\pi\)
\(30\) −18.9230 − 9.25980i −0.630766 − 0.308660i
\(31\) 8.11719i 0.261845i 0.991393 + 0.130922i \(0.0417939\pi\)
−0.991393 + 0.130922i \(0.958206\pi\)
\(32\) −27.8704 − 15.7239i −0.870950 − 0.491371i
\(33\) 18.6314 − 10.7569i 0.564589 − 0.325966i
\(34\) −18.5095 + 32.2070i −0.544397 + 0.947266i
\(35\) −42.6846 + 2.84387i −1.21956 + 0.0812535i
\(36\) −9.18628 − 15.7661i −0.255174 − 0.437946i
\(37\) − 43.7274i − 1.18182i −0.806737 − 0.590911i \(-0.798769\pi\)
0.806737 − 0.590911i \(-0.201231\pi\)
\(38\) 37.4227 + 6.59859i 0.984808 + 0.173647i
\(39\) 24.8504i 0.637190i
\(40\) 2.89699 + 39.8950i 0.0724248 + 0.997374i
\(41\) −19.1707 − 33.2046i −0.467578 − 0.809869i 0.531736 − 0.846910i \(-0.321540\pi\)
−0.999314 + 0.0370415i \(0.988207\pi\)
\(42\) 31.2554 + 17.9626i 0.744177 + 0.427681i
\(43\) 29.0633 + 50.3391i 0.675890 + 1.17068i 0.976208 + 0.216837i \(0.0695739\pi\)
−0.300318 + 0.953839i \(0.597093\pi\)
\(44\) −35.4563 − 20.2833i −0.805826 − 0.460983i
\(45\) −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 −0.998711 − 0.0507603i \(-0.983836\pi\)
0.543315 + 0.839529i \(0.317169\pi\)
\(48\) 16.6211 − 29.3245i 0.346273 − 0.610926i
\(49\) 24.2025 0.493929
\(50\) 39.5413 − 30.6021i 0.790825 − 0.612042i
\(51\) −33.8866 − 19.5645i −0.664444 − 0.383617i
\(52\) 40.7679 − 23.7539i 0.783998 − 0.456806i
\(53\) 1.54174 + 0.890123i 0.0290894 + 0.0167948i 0.514474 − 0.857506i \(-0.327987\pi\)
−0.485385 + 0.874301i \(0.661320\pi\)
\(54\) 49.4291 − 28.6690i 0.915353 − 0.530907i
\(55\) 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 −0.952854 − 0.303430i \(-0.901868\pi\)
−0.213648 + 0.976911i \(0.568535\pi\)
\(60\) −42.0295 + 2.96805i −0.700492 + 0.0494675i
\(61\) −41.5021 + 71.8838i −0.680362 + 1.17842i 0.294508 + 0.955649i \(0.404844\pi\)
−0.974870 + 0.222773i \(0.928489\pi\)
\(62\) 8.14511 + 14.0432i 0.131373 + 0.226504i
\(63\) 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
\(66\) 21.4396 − 37.3056i 0.324843 − 0.565236i
\(67\) −20.7172 + 35.8833i −0.309212 + 0.535571i −0.978190 − 0.207711i \(-0.933399\pi\)
0.668978 + 0.743282i \(0.266732\pi\)
\(68\) 0.295262 + 74.2933i 0.00434209 + 1.09255i
\(69\) 76.8780 1.11417
\(70\) −70.9933 + 47.7515i −1.01419 + 0.682164i
\(71\) −79.6311 + 45.9750i −1.12156 + 0.647535i −0.941800 − 0.336174i \(-0.890867\pi\)
−0.179764 + 0.983710i \(0.557534\pi\)
\(72\) −31.7131 − 18.0584i −0.440460 − 0.250811i
\(73\) −85.6344 + 49.4410i −1.17307 + 0.677274i −0.954402 − 0.298524i \(-0.903506\pi\)
−0.218671 + 0.975799i \(0.570172\pi\)
\(74\) −43.8778 − 75.6510i −0.592943 − 1.02231i
\(75\) 32.1520 + 41.7150i 0.428694 + 0.556200i
\(76\) 71.3648 − 26.1355i 0.939011 − 0.343888i
\(77\) − 87.3724i − 1.13471i
\(78\) 24.9359 + 42.9927i 0.319691 + 0.551189i
\(79\) 15.0449 − 8.68620i 0.190442 − 0.109952i −0.401747 − 0.915751i \(-0.631597\pi\)
0.592190 + 0.805799i \(0.298264\pi\)
\(80\) 45.0442 + 66.1137i 0.563052 + 0.826421i
\(81\) 9.56712 + 16.5707i 0.118113 + 0.204577i
\(82\) −66.4853 − 38.2094i −0.810796 − 0.465968i
\(83\) 144.479 1.74071 0.870355 − 0.492426i \(-0.163890\pi\)
0.870355 + 0.492426i \(0.163890\pi\)
\(84\) 72.0982 − 0.286538i 0.858312 − 0.00341117i
\(85\) 77.1608 − 51.6775i 0.907774 − 0.607970i
\(86\) 100.793 + 57.9264i 1.17202 + 0.673562i
\(87\) −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 \(0.201641\pi\)
\(90\) 3.12301 + 45.5107i 0.0347001 + 0.505675i
\(91\) 87.4022 + 50.4617i 0.960464 + 0.554524i
\(92\) −73.4858 − 126.121i −0.798759 − 1.37088i
\(93\) −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 \(0.360808\pi\)
\(98\) 41.8719 − 24.2858i 0.427264 − 0.247814i
\(99\) −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.159.97 yes 232
4.3 odd 2 inner 380.3.p.a.159.98 yes 232
5.4 even 2 inner 380.3.p.a.159.20 yes 232
19.11 even 3 inner 380.3.p.a.239.19 yes 232
20.19 odd 2 inner 380.3.p.a.159.19 ✓ 232
76.11 odd 6 inner 380.3.p.a.239.20 yes 232
95.49 even 6 inner 380.3.p.a.239.98 yes 232
380.239 odd 6 inner 380.3.p.a.239.97 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.19 ✓ 232 20.19 odd 2 inner
380.3.p.a.159.20 yes 232 5.4 even 2 inner
380.3.p.a.159.97 yes 232 1.1 even 1 trivial
380.3.p.a.159.98 yes 232 4.3 odd 2 inner
380.3.p.a.239.19 yes 232 19.11 even 3 inner
380.3.p.a.239.20 yes 232 76.11 odd 6 inner
380.3.p.a.239.97 yes 232 380.239 odd 6 inner
380.3.p.a.239.98 yes 232 95.49 even 6 inner