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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 239.97
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.97

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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 −0.635327 − 0.772243i \(-0.719135\pi\)
0.986446 + 0.164088i \(0.0524682\pi\)
\(4\) 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 \(0.153652\pi\)
−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 −0.0526770 − 0.998612i \(-0.516775\pi\)
0.838485 + 0.544925i \(0.183442\pi\)
\(14\) 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 −0.0542892 − 0.998525i \(-0.517289\pi\)
−0.891893 + 0.452247i \(0.850623\pi\)
\(18\) −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 0.923910 + 0.382609i \(0.124974\pi\)
−0.130606 + 0.991434i \(0.541692\pi\)
\(24\) 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 0.735226 − 0.677822i \(-0.237076\pi\)
−0.954624 + 0.297813i \(0.903743\pi\)
\(30\) −18.9230 + 9.25980i −0.630766 + 0.308660i
\(31\) − 8.11719i − 0.261845i −0.991393 − 0.130922i \(-0.958206\pi\)
0.991393 − 0.130922i \(-0.0417939\pi\)
\(32\) −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 \(0.201231\pi\)
−0.806737 + 0.590911i \(0.798769\pi\)
\(38\) 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 −0.999314 − 0.0370415i \(-0.988207\pi\)
0.531736 + 0.846910i \(0.321540\pi\)
\(42\) 31.2554 − 17.9626i 0.744177 − 0.427681i
\(43\) 29.0633 − 50.3391i 0.675890 − 1.17068i −0.300318 − 0.953839i \(-0.597093\pi\)
0.976208 − 0.216837i \(-0.0695739\pi\)
\(44\) −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 0.543315 − 0.839529i \(-0.317169\pi\)
−0.998711 + 0.0507603i \(0.983836\pi\)
\(48\) 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 −0.485385 − 0.874301i \(-0.661320\pi\)
0.514474 + 0.857506i \(0.327987\pi\)
\(54\) 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 −0.213648 − 0.976911i \(-0.568535\pi\)
−0.952854 + 0.303430i \(0.901868\pi\)
\(60\) −42.0295 − 2.96805i −0.700492 − 0.0494675i
\(61\) −41.5021 − 71.8838i −0.680362 − 1.17842i −0.974870 − 0.222773i \(-0.928489\pi\)
0.294508 − 0.955649i \(-0.404844\pi\)
\(62\) 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 0.668978 − 0.743282i \(-0.266732\pi\)
−0.978190 + 0.207711i \(0.933399\pi\)
\(68\) 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 −0.179764 − 0.983710i \(-0.557534\pi\)
−0.941800 + 0.336174i \(0.890867\pi\)
\(72\) −31.7131 + 18.0584i −0.440460 + 0.250811i
\(73\) −85.6344 − 49.4410i −1.17307 − 0.677274i −0.218671 − 0.975799i \(-0.570172\pi\)
−0.954402 + 0.298524i \(0.903506\pi\)
\(74\) −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 0.592190 − 0.805799i \(-0.298264\pi\)
−0.401747 + 0.915751i \(0.631597\pi\)
\(80\) 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 0.805976 − 0.591949i \(-0.201641\pi\)
−0.915630 + 0.402021i \(0.868308\pi\)
\(90\) 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 0.423481 − 0.905905i \(-0.360808\pi\)
−0.572796 + 0.819698i \(0.694141\pi\)
\(98\) 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.239.97 yes 232
4.3 odd 2 inner 380.3.p.a.239.98 yes 232
5.4 even 2 inner 380.3.p.a.239.20 yes 232
19.7 even 3 inner 380.3.p.a.159.19 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.19 yes 232
76.7 odd 6 inner 380.3.p.a.159.20 yes 232
95.64 even 6 inner 380.3.p.a.159.98 yes 232
380.159 odd 6 inner 380.3.p.a.159.97 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.19 ✓ 232 19.7 even 3 inner
380.3.p.a.159.20 yes 232 76.7 odd 6 inner
380.3.p.a.159.97 yes 232 380.159 odd 6 inner
380.3.p.a.159.98 yes 232 95.64 even 6 inner
380.3.p.a.239.19 yes 232 20.19 odd 2 inner
380.3.p.a.239.20 yes 232 5.4 even 2 inner
380.3.p.a.239.97 yes 232 1.1 even 1 trivial
380.3.p.a.239.98 yes 232 4.3 odd 2 inner