↑

Properties

Label 380.3.p.a.159.19
Level $380$
Weight $3$
Character 380.159
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

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 159.19
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.19

$q$-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} +(2.78233 - 4.15435i) q^{5} +(-3.64473 - 2.11396i) q^{6} +8.55585 q^{7} +(-0.0476908 + 7.99986i) q^{8} +(2.28089 - 3.95061i) q^{9} +(-0.684605 + 9.97654i) q^{10} +10.2120i q^{11} +(8.42677 + 0.0334903i) q^{12} +(-10.2155 - 5.89792i) q^{13} +(-14.8361 + 8.52638i) q^{14} +(10.5102 + 0.700248i) 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} +(-8.75505 - 17.9819i) 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} +(-9.51731 - 23.1175i) q^{25} +(23.5916 + 0.0468796i) q^{26} +28.5707 q^{27} +(17.2294 - 29.5701i) q^{28} +(-6.36256 + 11.0203i) q^{29} +(-18.9230 + 9.25980i) q^{30} -8.11719i q^{31} +(27.5525 + 16.2746i) q^{32} +(-18.6314 + 10.7569i) q^{33} +(-18.6374 + 32.1332i) q^{34} +(23.8052 - 35.5440i) q^{35} +(-9.06067 - 15.8386i) q^{36} +43.7274i q^{37} +(37.3962 + 6.74726i) q^{38} -24.8504i q^{39} +(33.1015 + 22.4563i) 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} +(39.5413 + 30.6021i) 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} +(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} +(23.5852 - 34.9146i) 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} +(-5.85737 + 85.3577i) 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} +(-79.7782 - 5.95253i) q^{80} +(9.56712 + 16.5707i) q^{81} +(66.3329 + 38.4733i) q^{82} +144.479 q^{83} +(72.0982 + 0.286538i) q^{84} +(6.17362 - 92.6619i) q^{85} +(-100.562 - 58.3265i) q^{86} -26.8081 q^{87} +(-81.6946 - 0.487019i) q^{88} +(-9.75929 + 16.9036i) q^{89} +(37.8519 + 25.4600i) 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} +(-91.2857 + 26.3043i) 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 0.986446 − 0.164088i \(-0.0524682\pi\)
−0.635327 + 0.772243i \(0.719135\pi\)
\(4\) 2.01375 − 3.45613i 0.503438 − 0.864031i
\(5\) 2.78233 − 4.15435i 0.556465 − 0.830871i
\(6\) −3.64473 − 2.11396i −0.607456 − 0.352326i
\(7\) 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
\(10\) −0.684605 + 9.97654i −0.0684605 + 0.997654i
\(11\) 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 \(-0.483225\pi\)
−0.838485 + 0.544925i \(0.816558\pi\)
\(14\) −14.8361 + 8.52638i −1.05972 + 0.609027i
\(15\) 10.5102 + 0.700248i 0.700683 + 0.0466832i
\(16\) −7.88961 − 13.9196i −0.493101 − 0.869972i
\(17\) 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
\(20\) −8.75505 − 17.9819i −0.437753 − 0.899095i
\(21\) 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 \(-0.541692\pi\)
0.923910 − 0.382609i \(-0.124974\pi\)
\(24\) −14.6457 + 8.33968i −0.610237 + 0.347487i
\(25\) −9.51731 − 23.1175i −0.380692 − 0.924702i
\(26\) 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 \(0.237076\pi\)
\(30\) −18.9230 + 9.25980i −0.630766 + 0.308660i
\(31\) − 8.11719i − 0.261845i −0.991393 − 0.130922i \(-0.958206\pi\)
0.991393 − 0.130922i \(-0.0417939\pi\)
\(32\) 27.5525 + 16.2746i 0.861015 + 0.508580i
\(33\) −18.6314 + 10.7569i −0.564589 + 0.325966i
\(34\) −18.6374 + 32.1332i −0.548158 + 0.945095i
\(35\) 23.8052 − 35.5440i 0.680148 − 1.01554i
\(36\) −9.06067 − 15.8386i −0.251685 − 0.439961i
\(37\) 43.7274i 1.18182i 0.806737 + 0.590911i \(0.201231\pi\)
−0.806737 + 0.590911i \(0.798769\pi\)
\(38\) 37.3962 + 6.74726i 0.984110 + 0.177560i
\(39\) − 24.8504i − 0.637190i
\(40\) 33.1015 + 22.4563i 0.827539 + 0.561409i
\(41\) −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 \(0.0695739\pi\)
−0.300318 + 0.953839i \(0.597093\pi\)
\(44\) 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 −0.998711 − 0.0507603i \(-0.983836\pi\)
0.543315 + 0.839529i \(0.317169\pi\)
\(48\) 17.0852 − 29.0565i 0.355941 − 0.605345i
\(49\) 24.2025 0.493929
\(50\) 39.5413 + 30.6021i 0.790825 + 0.612042i
\(51\) 33.8866 + 19.5645i 0.664444 + 0.383617i
\(52\) −40.9554 + 23.4291i −0.787604 + 0.450560i
\(53\) −1.54174 − 0.890123i −0.0290894 − 0.0167948i 0.485385 − 0.874301i \(-0.338680\pi\)
−0.514474 + 0.857506i \(0.672013\pi\)
\(54\) −49.5426 + 28.4723i −0.917456 + 0.527265i
\(55\) 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 0.213648 − 0.976911i \(-0.431465\pi\)
0.952854 + 0.303430i \(0.0981320\pi\)
\(60\) 23.5852 − 34.9146i 0.393086 − 0.581910i
\(61\) −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 −0.978190 − 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
\(70\) −5.85737 + 85.3577i −0.0836768 + 1.21940i
\(71\) 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 0.218671 − 0.975799i \(-0.429828\pi\)
0.954402 + 0.298524i \(0.0964944\pi\)
\(74\) −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 \(0.368403\pi\)
\(80\) −79.7782 − 5.95253i −0.997228 − 0.0744066i
\(81\) 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 \(-0.163890\pi\)
0.870355 + 0.492426i \(0.163890\pi\)
\(84\) 72.0982 + 0.286538i 0.858312 + 0.00341117i
\(85\) 6.17362 − 92.6619i 0.0726308 − 1.09014i
\(86\) −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 \(0.201641\pi\)
\(90\) 37.8519 + 25.4600i 0.420577 + 0.282889i
\(91\) −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
\(95\) −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 \(0.305859\pi\)
\(98\) −41.9680 + 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.159.19 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.20 yes 232
5.4 even 2 inner 380.3.p.a.159.98 yes 232
19.11 even 3 inner 380.3.p.a.239.97 yes 232
20.19 odd 2 inner 380.3.p.a.159.97 yes 232
76.11 odd 6 inner 380.3.p.a.239.98 yes 232
95.49 even 6 inner 380.3.p.a.239.20 yes 232
380.239 odd 6 inner 380.3.p.a.239.19 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.19 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.20 yes 232 4.3 odd 2 inner
380.3.p.a.159.97 yes 232 20.19 odd 2 inner
380.3.p.a.159.98 yes 232 5.4 even 2 inner
380.3.p.a.239.19 yes 232 380.239 odd 6 inner
380.3.p.a.239.20 yes 232 95.49 even 6 inner
380.3.p.a.239.97 yes 232 19.11 even 3 inner
380.3.p.a.239.98 yes 232 76.11 odd 6 inner