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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 159.20
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.20

$q$-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} +(2.78233 - 4.15435i) q^{5} +(3.65311 + 2.09945i) q^{6} -8.55585 q^{7} +(0.0476908 + 7.99986i) q^{8} +(2.28089 - 3.95061i) q^{9} +(-0.644950 + 9.97918i) q^{10} -10.2120i q^{11} +(-8.42677 + 0.0334903i) q^{12} +(-10.2155 - 5.89792i) q^{13} +(14.8021 - 8.58528i) q^{14} +(-10.5102 - 0.700248i) 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} +(-8.89770 - 17.9118i) 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} +(-9.51731 - 23.1175i) q^{25} +(23.5916 - 0.0468796i) q^{26} -28.5707 q^{27} +(-16.9938 + 29.7061i) q^{28} +(-6.36256 + 11.0203i) q^{29} +(18.8860 - 9.33493i) q^{30} +8.11719i q^{31} +(27.8704 + 15.7239i) q^{32} +(-18.6314 + 10.7569i) q^{33} +(-18.5095 + 32.2070i) q^{34} +(-23.8052 + 35.5440i) q^{35} +(-9.18628 - 15.7661i) q^{36} +43.7274i q^{37} +(-37.4227 - 6.59859i) q^{38} +24.8504i q^{39} +(33.3669 + 22.0601i) 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} +(39.6626 + 30.4447i) 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} +(-42.4243 - 28.4132i) 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} +(-23.3069 + 35.1010i) 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} +(5.51809 - 85.3804i) 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} +(-79.8628 - 4.68359i) q^{80} +(9.56712 + 16.5707i) q^{81} +(66.4853 + 38.2094i) q^{82} -144.479 q^{83} +(72.0982 - 0.286538i) q^{84} +(6.17362 - 92.6619i) q^{85} +(100.793 + 57.9264i) q^{86} +26.8081 q^{87} +(81.6946 - 0.487019i) q^{88} +(-9.75929 + 16.9036i) q^{89} +(37.9528 + 25.3093i) 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} +(91.2857 - 26.3043i) 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 0.635327 − 0.772243i \(-0.280865\pi\)
−0.986446 + 0.164088i \(0.947532\pi\)
\(4\) 1.98622 − 3.47202i 0.496554 − 0.868006i
\(5\) 2.78233 − 4.15435i 0.556465 − 0.830871i
\(6\) 3.65311 + 2.09945i 0.608851 + 0.349909i
\(7\) −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
\(10\) −0.644950 + 9.97918i −0.0644950 + 0.997918i
\(11\) − 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 0.0526770 − 0.998612i \(-0.483225\pi\)
−0.838485 + 0.544925i \(0.816558\pi\)
\(14\) 14.8021 − 8.58528i 1.05730 − 0.613234i
\(15\) −10.5102 − 0.700248i −0.700683 − 0.0466832i
\(16\) −8.10988 − 13.7924i −0.506868 − 0.862024i
\(17\) 16.0851 − 9.28673i 0.946182 − 0.546278i 0.0542892 − 0.998525i \(-0.482711\pi\)
0.891893 + 0.452247i \(0.149377\pi\)
\(18\) 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
\(22\) 10.2471 + 17.6674i 0.465779 + 0.803063i
\(23\) −18.2460 + 31.6030i −0.793304 + 1.37404i 0.130606 + 0.991434i \(0.458308\pi\)
−0.923910 + 0.382609i \(0.875026\pi\)
\(24\) 14.5452 − 8.51370i 0.606051 − 0.354737i
\(25\) −9.51731 − 23.1175i −0.380692 − 0.924702i
\(26\) 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 \(0.237076\pi\)
\(30\) 18.8860 − 9.33493i 0.629534 − 0.311164i
\(31\) 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
\(35\) −23.8052 + 35.5440i −0.680148 + 1.01554i
\(36\) −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
\(40\) 33.3669 + 22.0601i 0.834173 + 0.551502i
\(41\) −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 \(-0.930426\pi\)
0.300318 − 0.953839i \(-0.402907\pi\)
\(44\) −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 \(-0.682831\pi\)
0.998711 + 0.0507603i \(0.0161644\pi\)
\(48\) −16.6211 + 29.3245i −0.346273 + 0.610926i
\(49\) 24.2025 0.493929
\(50\) 39.6626 + 30.4447i 0.793252 + 0.608894i
\(51\) −33.8866 − 19.5645i −0.664444 − 0.383617i
\(52\) −40.7679 + 23.7539i −0.783998 + 0.456806i
\(53\) −1.54174 − 0.890123i −0.0290894 − 0.0167948i 0.485385 − 0.874301i \(-0.338680\pi\)
−0.514474 + 0.857506i \(0.672013\pi\)
\(54\) 49.4291 − 28.6690i 0.915353 − 0.530907i
\(55\) −42.4243 − 28.4132i −0.771351 − 0.516603i
\(56\) −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 \(0.568535\pi\)
\(60\) −23.3069 + 35.1010i −0.388448 + 0.585016i
\(61\) −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 −0.668978 − 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
\(70\) 5.51809 − 85.3804i 0.0788299 − 1.21972i
\(71\) −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 0.218671 − 0.975799i \(-0.429828\pi\)
0.954402 + 0.298524i \(0.0964944\pi\)
\(74\) −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 \(0.298264\pi\)
\(80\) −79.8628 − 4.68359i −0.998285 − 0.0585449i
\(81\) 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 \(-0.836110\pi\)
−0.870355 + 0.492426i \(0.836110\pi\)
\(84\) 72.0982 − 0.286538i 0.858312 − 0.00341117i
\(85\) 6.17362 − 92.6619i 0.0726308 − 1.09014i
\(86\) 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 \(0.201641\pi\)
\(90\) 37.9528 + 25.3093i 0.421698 + 0.281215i
\(91\) 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
\(95\) 91.2857 − 26.3043i 0.960902 − 0.276887i
\(96\) −0.669797 − 67.4114i −0.00697705 − 0.702202i
\(97\) 14.4836 − 8.36208i 0.149315 − 0.0862071i −0.423481 − 0.905905i \(-0.639192\pi\)
0.572796 + 0.819698i \(0.305859\pi\)
\(98\) −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.159.20 yes 232
4.3 odd 2 inner 380.3.p.a.159.19 ✓ 232
5.4 even 2 inner 380.3.p.a.159.97 yes 232
19.11 even 3 inner 380.3.p.a.239.98 yes 232
20.19 odd 2 inner 380.3.p.a.159.98 yes 232
76.11 odd 6 inner 380.3.p.a.239.97 yes 232
95.49 even 6 inner 380.3.p.a.239.19 yes 232
380.239 odd 6 inner 380.3.p.a.239.20 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.19 ✓ 232 4.3 odd 2 inner
380.3.p.a.159.20 yes 232 1.1 even 1 trivial
380.3.p.a.159.97 yes 232 5.4 even 2 inner
380.3.p.a.159.98 yes 232 20.19 odd 2 inner
380.3.p.a.239.19 yes 232 95.49 even 6 inner
380.3.p.a.239.20 yes 232 380.239 odd 6 inner
380.3.p.a.239.97 yes 232 76.11 odd 6 inner
380.3.p.a.239.98 yes 232 19.11 even 3 inner