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Properties

Label 380.3.p.a.159.1
Level $380$
Weight $3$
Character 380.159
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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 159.1
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.1

$q$-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+(-1.99940 + 0.0491004i) q^{2} +(2.13447 + 3.69701i) q^{3} +(3.99518 - 0.196342i) q^{4} +(3.83161 - 3.21229i) q^{5} +(-4.44918 - 7.28700i) q^{6} -2.09290 q^{7} +(-7.97831 + 0.588731i) q^{8} +(-4.61194 + 7.98812i) q^{9} +(-7.50318 + 6.61077i) q^{10} -17.2373i q^{11} +(9.25348 + 14.3511i) q^{12} +(19.4015 + 11.2014i) q^{13} +(4.18454 - 0.102762i) q^{14} +(20.0543 + 7.30896i) q^{15} +(15.9229 - 1.56885i) q^{16} +(21.5372 - 12.4345i) q^{17} +(8.82888 - 16.1979i) q^{18} +(-8.71862 + 16.8815i) q^{19} +(14.6772 - 13.5860i) q^{20} +(-4.46724 - 7.73748i) q^{21} +(0.846360 + 34.4643i) q^{22} +(-3.67215 + 6.36036i) q^{23} +(-19.2060 - 28.2393i) q^{24} +(4.36240 - 24.6164i) q^{25} +(-39.3412 - 21.4435i) q^{26} -0.955736 q^{27} +(-8.36151 + 0.410925i) q^{28} +(2.05878 - 3.56591i) q^{29} +(-40.4554 - 13.6288i) q^{30} -21.0887i q^{31} +(-31.7592 + 3.91856i) q^{32} +(63.7267 - 36.7926i) q^{33} +(-42.4508 + 25.9190i) q^{34} +(-8.01917 + 6.72300i) q^{35} +(-16.8571 + 32.8195i) q^{36} -2.17705i q^{37} +(16.6031 - 34.1809i) q^{38} +95.6366i q^{39} +(-28.6786 + 27.8844i) q^{40} +(21.9937 + 38.0943i) q^{41} +(9.31169 + 15.2510i) q^{42} +(10.4957 + 18.1790i) q^{43} +(-3.38442 - 68.8662i) q^{44} +(7.98900 + 45.4222i) q^{45} +(7.02980 - 12.8972i) q^{46} +(-15.7965 + 27.3603i) q^{47} +(39.7870 + 55.5185i) q^{48} -44.6198 q^{49} +(-7.51349 + 49.4323i) q^{50} +(91.9410 + 53.0821i) q^{51} +(79.7116 + 40.9424i) q^{52} +(24.2276 + 13.9878i) q^{53} +(1.91090 - 0.0469270i) q^{54} +(-55.3713 - 66.0467i) q^{55} +(16.6978 - 1.23215i) q^{56} +(-81.0209 + 3.80026i) q^{57} +(-3.94123 + 7.23076i) q^{58} +(64.6792 - 37.3425i) q^{59} +(81.5557 + 25.2631i) q^{60} +(38.3821 - 66.4798i) q^{61} +(1.03546 + 42.1647i) q^{62} +(9.65233 - 16.7183i) q^{63} +(63.3068 - 9.39415i) q^{64} +(110.321 - 19.4036i) q^{65} +(-125.608 + 76.6920i) q^{66} +(-47.4943 + 82.2625i) q^{67} +(83.6034 - 53.9067i) q^{68} -31.3524 q^{69} +(15.7034 - 13.8357i) q^{70} +(-79.2164 + 45.7356i) q^{71} +(32.0926 - 66.4468i) q^{72} +(-74.1900 + 42.8336i) q^{73} +(0.106894 + 4.35279i) q^{74} +(100.319 - 36.4153i) q^{75} +(-31.5179 + 69.1565i) q^{76} +36.0760i q^{77} +(-4.69579 - 191.216i) q^{78} +(39.9297 - 23.0534i) q^{79} +(55.9707 - 57.1602i) q^{80} +(39.4675 + 68.3597i) q^{81} +(-45.8447 - 75.0857i) q^{82} +34.6765 q^{83} +(-19.3666 - 30.0355i) q^{84} +(42.5788 - 116.828i) q^{85} +(-21.8776 - 35.8317i) q^{86} +17.5776 q^{87} +(10.1482 + 137.525i) q^{88} +(8.02461 - 13.8990i) q^{89} +(-18.2034 - 90.4247i) q^{90} +(-40.6053 - 23.4435i) q^{91} +(-13.4221 + 26.1318i) q^{92} +(77.9653 - 45.0133i) q^{93} +(30.2400 - 55.4797i) q^{94} +(20.8220 + 92.6900i) q^{95} +(-82.2760 - 109.050i) q^{96} +(-132.859 + 76.7062i) q^{97} +(89.2126 - 2.19085i) q^{98} +(137.694 + 79.4976i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99940 + 0.0491004i −0.999699 + 0.0245502i
\(3\) 2.13447 + 3.69701i 0.711491 + 1.23234i 0.964298 + 0.264821i \(0.0853129\pi\)
−0.252807 + 0.967517i \(0.581354\pi\)
\(4\) 3.99518 − 0.196342i 0.998795 − 0.0490856i
\(5\) 3.83161 − 3.21229i 0.766321 − 0.642458i
\(6\) −4.44918 − 7.28700i −0.741530 − 1.21450i
\(7\) −2.09290 −0.298986 −0.149493 − 0.988763i \(-0.547764\pi\)
−0.149493 + 0.988763i \(0.547764\pi\)
\(8\) −7.97831 + 0.588731i −0.997288 + 0.0735914i
\(9\) −4.61194 + 7.98812i −0.512438 + 0.887568i
\(10\) −7.50318 + 6.61077i −0.750318 + 0.661077i
\(11\) − 17.2373i − 1.56703i −0.621373 − 0.783515i \(-0.713425\pi\)
0.621373 − 0.783515i \(-0.286575\pi\)
\(12\) 9.25348 + 14.3511i 0.771123 + 1.19593i
\(13\) 19.4015 + 11.2014i 1.49242 + 0.861649i 0.999962 − 0.00868805i \(-0.00276553\pi\)
0.492457 + 0.870337i \(0.336099\pi\)
\(14\) 4.18454 − 0.102762i 0.298896 − 0.00734015i
\(15\) 20.0543 + 7.30896i 1.33696 + 0.487264i
\(16\) 15.9229 − 1.56885i 0.995181 − 0.0980528i
\(17\) 21.5372 − 12.4345i 1.26689 − 0.731441i 0.292494 − 0.956268i \(-0.405515\pi\)
0.974399 + 0.224827i \(0.0721817\pi\)
\(18\) 8.82888 − 16.1979i 0.490493 − 0.899881i
\(19\) −8.71862 + 16.8815i −0.458875 + 0.888501i
\(20\) 14.6772 − 13.5860i 0.733862 − 0.679299i
\(21\) −4.46724 − 7.73748i −0.212726 − 0.368451i
\(22\) 0.846360 + 34.4643i 0.0384709 + 1.56656i
\(23\) −3.67215 + 6.36036i −0.159659 + 0.276537i −0.934746 − 0.355318i \(-0.884373\pi\)
0.775087 + 0.631855i \(0.217706\pi\)
\(24\) −19.2060 − 28.2393i −0.800251 − 1.17664i
\(25\) 4.36240 − 24.6164i 0.174496 − 0.984658i
\(26\) −39.3412 − 21.4435i −1.51312 − 0.824750i
\(27\) −0.955736 −0.0353976
\(28\) −8.36151 + 0.410925i −0.298625 + 0.0146759i
\(29\) 2.05878 − 3.56591i 0.0709924 − 0.122962i −0.828344 − 0.560220i \(-0.810717\pi\)
0.899337 + 0.437257i \(0.144050\pi\)
\(30\) −40.4554 − 13.6288i −1.34851 − 0.454294i
\(31\) − 21.0887i − 0.680281i −0.940375 − 0.340141i \(-0.889525\pi\)
0.940375 − 0.340141i \(-0.110475\pi\)
\(32\) −31.7592 + 3.91856i −0.992474 + 0.122455i
\(33\) 63.7267 − 36.7926i 1.93111 − 1.11493i
\(34\) −42.4508 + 25.9190i −1.24855 + 0.762323i
\(35\) −8.01917 + 6.72300i −0.229119 + 0.192086i
\(36\) −16.8571 + 32.8195i −0.468253 + 0.911652i
\(37\) − 2.17705i − 0.0588392i −0.999567 − 0.0294196i \(-0.990634\pi\)
0.999567 − 0.0294196i \(-0.00936590\pi\)
\(38\) 16.6031 − 34.1809i 0.436924 − 0.899499i
\(39\) 95.6366i 2.45222i
\(40\) −28.6786 + 27.8844i −0.716964 + 0.697110i
\(41\) 21.9937 + 38.0943i 0.536433 + 0.929128i 0.999093 + 0.0425926i \(0.0135618\pi\)
−0.462660 + 0.886536i \(0.653105\pi\)
\(42\) 9.31169 + 15.2510i 0.221707 + 0.363118i
\(43\) 10.4957 + 18.1790i 0.244085 + 0.422768i 0.961874 − 0.273493i \(-0.0881791\pi\)
−0.717789 + 0.696261i \(0.754846\pi\)
\(44\) −3.38442 − 68.8662i −0.0769186 − 1.56514i
\(45\) 7.98900 + 45.4222i 0.177533 + 1.00938i
\(46\) 7.02980 − 12.8972i 0.152822 − 0.280373i
\(47\) −15.7965 + 27.3603i −0.336095 + 0.582134i −0.983695 − 0.179847i \(-0.942440\pi\)
0.647600 + 0.761981i \(0.275773\pi\)
\(48\) 39.7870 + 55.5185i 0.828896 + 1.15664i
\(49\) −44.6198 −0.910608
\(50\) −7.51349 + 49.4323i −0.150270 + 0.988645i
\(51\) 91.9410 + 53.0821i 1.80276 + 1.04083i
\(52\) 79.7116 + 40.9424i 1.53291 + 0.787354i
\(53\) 24.2276 + 13.9878i 0.457125 + 0.263921i 0.710834 − 0.703359i \(-0.248318\pi\)
−0.253710 + 0.967280i \(0.581651\pi\)
\(54\) 1.91090 − 0.0469270i 0.0353870 − 0.000869019i
\(55\) −55.3713 − 66.0467i −1.00675 − 1.20085i
\(56\) 16.6978 − 1.23215i 0.298175 − 0.0220028i
\(57\) −81.0209 + 3.80026i −1.42142 + 0.0666712i
\(58\) −3.94123 + 7.23076i −0.0679523 + 0.124668i
\(59\) 64.6792 − 37.3425i 1.09626 − 0.632924i 0.161022 − 0.986951i \(-0.448521\pi\)
0.935235 + 0.354027i \(0.115188\pi\)
\(60\) 81.5557 + 25.2631i 1.35926 + 0.421051i
\(61\) 38.3821 − 66.4798i 0.629215 − 1.08983i −0.358494 − 0.933532i \(-0.616710\pi\)
0.987710 − 0.156301i \(-0.0499569\pi\)
\(62\) 1.03546 + 42.1647i 0.0167010 + 0.680076i
\(63\) 9.65233 − 16.7183i 0.153212 − 0.265370i
\(64\) 63.3068 − 9.39415i 0.989169 − 0.146784i
\(65\) 110.321 − 19.4036i 1.69725 − 0.298517i
\(66\) −125.608 + 76.6920i −1.90316 + 1.16200i
\(67\) −47.4943 + 82.2625i −0.708869 + 1.22780i 0.256407 + 0.966569i \(0.417461\pi\)
−0.965277 + 0.261229i \(0.915872\pi\)
\(68\) 83.6034 − 53.9067i 1.22946 − 0.792745i
\(69\) −31.3524 −0.454383
\(70\) 15.7034 − 13.8357i 0.224334 − 0.197653i
\(71\) −79.2164 + 45.7356i −1.11572 + 0.644164i −0.940306 − 0.340330i \(-0.889461\pi\)
−0.175418 + 0.984494i \(0.556128\pi\)
\(72\) 32.0926 − 66.4468i 0.445731 − 0.922873i
\(73\) −74.1900 + 42.8336i −1.01630 + 0.586762i −0.913031 − 0.407891i \(-0.866264\pi\)
−0.103271 + 0.994653i \(0.532931\pi\)
\(74\) 0.106894 + 4.35279i 0.00144451 + 0.0588215i
\(75\) 100.319 − 36.4153i 1.33758 − 0.485537i
\(76\) −31.5179 + 69.1565i −0.414709 + 0.909954i
\(77\) 36.0760i 0.468520i
\(78\) −4.69579 − 191.216i −0.0602025 − 2.45148i
\(79\) 39.9297 − 23.0534i 0.505439 − 0.291815i −0.225518 − 0.974239i \(-0.572407\pi\)
0.730957 + 0.682424i \(0.239074\pi\)
\(80\) 55.9707 − 57.1602i 0.699634 − 0.714502i
\(81\) 39.4675 + 68.3597i 0.487253 + 0.843947i
\(82\) −45.8447 − 75.0857i −0.559081 − 0.915679i
\(83\) 34.6765 0.417790 0.208895 − 0.977938i \(-0.433013\pi\)
0.208895 + 0.977938i \(0.433013\pi\)
\(84\) −19.3666 − 30.0355i −0.230555 − 0.357565i
\(85\) 42.5788 − 116.828i 0.500926 − 1.37444i
\(86\) −21.8776 − 35.8317i −0.254390 − 0.416648i
\(87\) 17.5776 0.202042
\(88\) 10.1482 + 137.525i 0.115320 + 1.56278i
\(89\) 8.02461 − 13.8990i 0.0901641 − 0.156169i −0.817416 − 0.576048i \(-0.804594\pi\)
0.907580 + 0.419879i \(0.137928\pi\)
\(90\) −18.2034 − 90.4247i −0.202260 − 1.00472i
\(91\) −40.6053 − 23.4435i −0.446212 − 0.257621i
\(92\) −13.4221 + 26.1318i −0.145892 + 0.284041i
\(93\) 77.9653 − 45.0133i 0.838336 − 0.484014i
\(94\) 30.2400 − 55.4797i 0.321702 − 0.590209i
\(95\) 20.8220 + 92.6900i 0.219179 + 0.975685i
\(96\) −82.2760 − 109.050i −0.857042 − 1.13594i
\(97\) −132.859 + 76.7062i −1.36968 + 0.790786i −0.990887 − 0.134696i \(-0.956994\pi\)
−0.378794 + 0.925481i \(0.623661\pi\)
\(98\) 89.2126 − 2.19085i 0.910333 − 0.0223556i
\(99\) 137.694 + 79.4976i 1.39085 + 0.803006i
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.159.1 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.38 yes 232
5.4 even 2 inner 380.3.p.a.159.116 yes 232
19.11 even 3 inner 380.3.p.a.239.79 yes 232
20.19 odd 2 inner 380.3.p.a.159.79 yes 232
76.11 odd 6 inner 380.3.p.a.239.116 yes 232
95.49 even 6 inner 380.3.p.a.239.38 yes 232
380.239 odd 6 inner 380.3.p.a.239.1 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.1 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.38 yes 232 4.3 odd 2 inner
380.3.p.a.159.79 yes 232 20.19 odd 2 inner
380.3.p.a.159.116 yes 232 5.4 even 2 inner
380.3.p.a.239.1 yes 232 380.239 odd 6 inner
380.3.p.a.239.38 yes 232 95.49 even 6 inner
380.3.p.a.239.79 yes 232 19.11 even 3 inner
380.3.p.a.239.116 yes 232 76.11 odd 6 inner