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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 239.45
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.45

$q$-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+(-0.691699 - 1.87658i) q^{2} +(1.97599 - 3.42251i) q^{3} +(-3.04311 + 2.59606i) q^{4} +(-0.693065 + 4.95173i) q^{5} +(-7.78940 - 1.34075i) q^{6} +2.56913 q^{7} +(6.97662 + 3.91494i) q^{8} +(-3.30904 - 5.73143i) q^{9} +(9.77172 - 2.12451i) q^{10} +17.7707i q^{11} +(2.87189 + 15.5448i) q^{12} +(-4.58570 + 2.64756i) q^{13} +(-1.77706 - 4.82117i) q^{14} +(15.5779 + 12.1566i) q^{15} +(2.52099 - 15.8001i) q^{16} +(1.90065 + 1.09734i) q^{17} +(-8.46662 + 10.1741i) q^{18} +(15.3281 + 11.2272i) q^{19} +(-10.7459 - 16.8679i) q^{20} +(5.07656 - 8.79286i) q^{21} +(33.3481 - 12.2920i) q^{22} +(8.21652 + 14.2314i) q^{23} +(27.1846 - 16.1417i) q^{24} +(-24.0393 - 6.86374i) q^{25} +(8.14028 + 6.77413i) q^{26} +9.41328 q^{27} +(-7.81812 + 6.66960i) q^{28} +(24.5292 + 42.4859i) q^{29} +(12.0376 - 37.6418i) q^{30} -13.0380i q^{31} +(-31.3940 + 6.19810i) q^{32} +(60.8203 + 35.1146i) q^{33} +(0.744573 - 4.32576i) q^{34} +(-1.78057 + 12.7216i) q^{35} +(24.9489 + 8.85089i) q^{36} -53.5256i q^{37} +(10.4662 - 36.5302i) q^{38} +20.9261i q^{39} +(-24.2210 + 31.8330i) q^{40} +(12.6985 - 21.9944i) q^{41} +(-20.0119 - 3.44456i) q^{42} +(-36.7927 + 63.7268i) q^{43} +(-46.1337 - 54.0781i) q^{44} +(30.6739 - 12.4132i) q^{45} +(21.0231 - 25.2628i) q^{46} +(-27.9634 - 48.4340i) q^{47} +(-49.0947 - 39.8490i) q^{48} -42.3996 q^{49} +(3.74760 + 49.8594i) q^{50} +(7.51133 - 4.33667i) q^{51} +(7.08158 - 19.9615i) q^{52} +(38.6169 - 22.2955i) q^{53} +(-6.51115 - 17.6648i) q^{54} +(-87.9957 - 12.3162i) q^{55} +(17.9238 + 10.0580i) q^{56} +(68.7132 - 30.2759i) q^{57} +(62.7613 - 75.4185i) q^{58} +(-58.4823 - 33.7648i) q^{59} +(-78.9642 + 3.44724i) q^{60} +(-2.07481 - 3.59367i) q^{61} +(-24.4669 + 9.01838i) q^{62} +(-8.50134 - 14.7248i) q^{63} +(33.3464 + 54.6262i) q^{64} +(-9.93181 - 24.5421i) q^{65} +(23.8261 - 138.423i) q^{66} +(51.0342 + 88.3938i) q^{67} +(-8.63266 + 1.59487i) q^{68} +64.9429 q^{69} +(25.1048 - 5.45815i) q^{70} +(-35.9292 - 20.7437i) q^{71} +(-0.647696 - 52.9407i) q^{72} +(41.1613 + 23.7645i) q^{73} +(-100.445 + 37.0236i) q^{74} +(-70.9926 + 68.7121i) q^{75} +(-75.7914 + 5.62717i) q^{76} +45.6551i q^{77} +(39.2696 - 14.4746i) q^{78} +(63.0816 + 36.4202i) q^{79} +(76.4909 + 23.4338i) q^{80} +(48.3819 - 83.7999i) q^{81} +(-50.0577 - 8.61619i) q^{82} +124.038 q^{83} +(7.37824 + 39.9366i) q^{84} +(-6.75103 + 8.65100i) q^{85} +(145.038 + 24.9647i) q^{86} +193.878 q^{87} +(-69.5712 + 123.979i) q^{88} +(50.2003 + 86.9495i) q^{89} +(-44.5115 - 48.9758i) q^{90} +(-11.7813 + 6.80191i) q^{91} +(-61.9494 - 21.9772i) q^{92} +(-44.6227 - 25.7630i) q^{93} +(-71.5481 + 85.9773i) q^{94} +(-66.2173 + 68.1195i) q^{95} +(-40.8211 + 119.694i) q^{96} +(-53.4146 - 30.8390i) q^{97} +(29.3277 + 79.5662i) q^{98} +(101.851 - 58.8039i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.691699 − 1.87658i −0.345849 − 0.938290i
\(3\) 1.97599 − 3.42251i 0.658662 − 1.14084i −0.322300 − 0.946637i \(-0.604456\pi\)
0.980962 − 0.194199i \(-0.0622106\pi\)
\(4\) −3.04311 + 2.59606i −0.760777 + 0.649014i
\(5\) −0.693065 + 4.95173i −0.138613 + 0.990347i
\(6\) −7.78940 − 1.34075i −1.29823 − 0.223459i
\(7\) 2.56913 0.367018 0.183509 − 0.983018i \(-0.441254\pi\)
0.183509 + 0.983018i \(0.441254\pi\)
\(8\) 6.97662 + 3.91494i 0.872077 + 0.489368i
\(9\) −3.30904 − 5.73143i −0.367671 − 0.636825i
\(10\) 9.77172 − 2.12451i 0.977172 − 0.212451i
\(11\) 17.7707i 1.61552i 0.589514 + 0.807758i \(0.299319\pi\)
−0.589514 + 0.807758i \(0.700681\pi\)
\(12\) 2.87189 + 15.5448i 0.239324 + 1.29540i
\(13\) −4.58570 + 2.64756i −0.352746 + 0.203658i −0.665894 − 0.746046i \(-0.731950\pi\)
0.313148 + 0.949704i \(0.398616\pi\)
\(14\) −1.77706 − 4.82117i −0.126933 − 0.344369i
\(15\) 15.5779 + 12.1566i 1.03852 + 0.810438i
\(16\) 2.52099 − 15.8001i 0.157562 − 0.987509i
\(17\) 1.90065 + 1.09734i 0.111803 + 0.0645496i 0.554859 − 0.831945i \(-0.312772\pi\)
−0.443056 + 0.896494i \(0.646106\pi\)
\(18\) −8.46662 + 10.1741i −0.470368 + 0.565228i
\(19\) 15.3281 + 11.2272i 0.806742 + 0.590903i
\(20\) −10.7459 − 16.8679i −0.537295 − 0.843394i
\(21\) 5.07656 − 8.79286i 0.241741 − 0.418707i
\(22\) 33.3481 − 12.2920i 1.51582 − 0.558725i
\(23\) 8.21652 + 14.2314i 0.357240 + 0.618758i 0.987499 − 0.157627i \(-0.0503844\pi\)
−0.630259 + 0.776385i \(0.717051\pi\)
\(24\) 27.1846 − 16.1417i 1.13269 − 0.672569i
\(25\) −24.0393 − 6.86374i −0.961573 − 0.274550i
\(26\) 8.14028 + 6.77413i 0.313088 + 0.260543i
\(27\) 9.41328 0.348640
\(28\) −7.81812 + 6.66960i −0.279219 + 0.238200i
\(29\) 24.5292 + 42.4859i 0.845836 + 1.46503i 0.884893 + 0.465794i \(0.154231\pi\)
−0.0390576 + 0.999237i \(0.512436\pi\)
\(30\) 12.0376 − 37.6418i 0.401253 − 1.25473i
\(31\) − 13.0380i − 0.420581i −0.977639 − 0.210291i \(-0.932559\pi\)
0.977639 − 0.210291i \(-0.0674411\pi\)
\(32\) −31.3940 + 6.19810i −0.981063 + 0.193691i
\(33\) 60.8203 + 35.1146i 1.84304 + 1.06408i
\(34\) 0.744573 − 4.32576i 0.0218992 − 0.127228i
\(35\) −1.78057 + 12.7216i −0.0508735 + 0.363475i
\(36\) 24.9489 + 8.85089i 0.693024 + 0.245858i
\(37\) − 53.5256i − 1.44664i −0.690515 − 0.723318i \(-0.742616\pi\)
0.690515 − 0.723318i \(-0.257384\pi\)
\(38\) 10.4662 − 36.5302i 0.275427 − 0.961322i
\(39\) 20.9261i 0.536568i
\(40\) −24.2210 + 31.8330i −0.605525 + 0.795826i
\(41\) 12.6985 − 21.9944i 0.309718 − 0.536448i −0.668582 − 0.743638i \(-0.733099\pi\)
0.978301 + 0.207190i \(0.0664319\pi\)
\(42\) −20.0119 − 3.44456i −0.476475 − 0.0820134i
\(43\) −36.7927 + 63.7268i −0.855644 + 1.48202i 0.0204018 + 0.999792i \(0.493505\pi\)
−0.876046 + 0.482227i \(0.839828\pi\)
\(44\) −46.1337 − 54.0781i −1.04849 − 1.22905i
\(45\) 30.6739 − 12.4132i 0.681642 − 0.275850i
\(46\) 21.0231 − 25.2628i 0.457023 − 0.549192i
\(47\) −27.9634 − 48.4340i −0.594966 − 1.03051i −0.993552 − 0.113380i \(-0.963832\pi\)
0.398586 − 0.917131i \(-0.369501\pi\)
\(48\) −49.0947 − 39.8490i −1.02281 − 0.830187i
\(49\) −42.3996 −0.865298
\(50\) 3.74760 + 49.8594i 0.0749519 + 0.997187i
\(51\) 7.51133 − 4.33667i 0.147281 − 0.0850327i
\(52\) 7.08158 − 19.9615i 0.136184 − 0.383876i
\(53\) 38.6169 − 22.2955i 0.728620 − 0.420669i −0.0892969 − 0.996005i \(-0.528462\pi\)
0.817917 + 0.575336i \(0.195129\pi\)
\(54\) −6.51115 − 17.6648i −0.120577 − 0.327125i
\(55\) −87.9957 − 12.3162i −1.59992 − 0.223932i
\(56\) 17.9238 + 10.0580i 0.320068 + 0.179607i
\(57\) 68.7132 − 30.2759i 1.20549 − 0.531155i
\(58\) 62.7613 − 75.4185i 1.08209 − 1.30032i
\(59\) −58.4823 − 33.7648i −0.991225 − 0.572284i −0.0855850 − 0.996331i \(-0.527276\pi\)
−0.905640 + 0.424047i \(0.860609\pi\)
\(60\) −78.9642 + 3.44724i −1.31607 + 0.0574540i
\(61\) −2.07481 − 3.59367i −0.0340132 − 0.0589127i 0.848518 − 0.529167i \(-0.177496\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(62\) −24.4669 + 9.01838i −0.394627 + 0.145458i
\(63\) −8.50134 − 14.7248i −0.134942 − 0.233726i
\(64\) 33.3464 + 54.6262i 0.521038 + 0.853534i
\(65\) −9.93181 − 24.5421i −0.152797 − 0.377571i
\(66\) 23.8261 − 138.423i 0.361001 − 2.09732i
\(67\) 51.0342 + 88.3938i 0.761704 + 1.31931i 0.941971 + 0.335693i \(0.108970\pi\)
−0.180267 + 0.983618i \(0.557696\pi\)
\(68\) −8.63266 + 1.59487i −0.126951 + 0.0234540i
\(69\) 64.9429 0.941202
\(70\) 25.1048 − 5.45815i 0.358640 − 0.0779735i
\(71\) −35.9292 − 20.7437i −0.506045 − 0.292165i 0.225161 − 0.974322i \(-0.427709\pi\)
−0.731207 + 0.682156i \(0.761042\pi\)
\(72\) −0.647696 − 52.9407i −0.00899578 − 0.735287i
\(73\) 41.1613 + 23.7645i 0.563853 + 0.325541i 0.754690 − 0.656081i \(-0.227787\pi\)
−0.190838 + 0.981622i \(0.561120\pi\)
\(74\) −100.445 + 37.0236i −1.35736 + 0.500318i
\(75\) −70.9926 + 68.7121i −0.946568 + 0.916161i
\(76\) −75.7914 + 5.62717i −0.997255 + 0.0740417i
\(77\) 45.6551i 0.592924i
\(78\) 39.2696 − 14.4746i 0.503456 − 0.185572i
\(79\) 63.0816 + 36.4202i 0.798501 + 0.461015i 0.842947 − 0.537997i \(-0.180819\pi\)
−0.0444457 + 0.999012i \(0.514152\pi\)
\(80\) 76.4909 + 23.4338i 0.956136 + 0.292922i
\(81\) 48.3819 − 83.7999i 0.597307 − 1.03457i
\(82\) −50.0577 − 8.61619i −0.610460 − 0.105076i
\(83\) 124.038 1.49444 0.747219 − 0.664577i \(-0.231388\pi\)
0.747219 + 0.664577i \(0.231388\pi\)
\(84\) 7.37824 + 39.9366i 0.0878362 + 0.475436i
\(85\) −6.75103 + 8.65100i −0.0794239 + 0.101776i
\(86\) 145.038 + 24.9647i 1.68649 + 0.290287i
\(87\) 193.878 2.22848
\(88\) −69.5712 + 123.979i −0.790582 + 1.40886i
\(89\) 50.2003 + 86.9495i 0.564049 + 0.976961i 0.997137 + 0.0756097i \(0.0240903\pi\)
−0.433089 + 0.901351i \(0.642576\pi\)
\(90\) −44.5115 − 48.9758i −0.494572 − 0.544175i
\(91\) −11.7813 + 6.80191i −0.129464 + 0.0747463i
\(92\) −61.9494 − 21.9772i −0.673363 − 0.238883i
\(93\) −44.6227 − 25.7630i −0.479814 − 0.277021i
\(94\) −71.5481 + 85.9773i −0.761150 + 0.914652i
\(95\) −66.2173 + 68.1195i −0.697024 + 0.717048i
\(96\) −40.8211 + 119.694i −0.425220 + 1.24681i
\(97\) −53.4146 − 30.8390i −0.550666 − 0.317927i 0.198724 − 0.980055i \(-0.436320\pi\)
−0.749391 + 0.662128i \(0.769654\pi\)
\(98\) 29.3277 + 79.5662i 0.299263 + 0.811900i
\(99\) 101.851 − 58.8039i 1.02880 − 0.593979i
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.239.45 yes 232
4.3 odd 2 inner 380.3.p.a.239.7 yes 232
5.4 even 2 inner 380.3.p.a.239.72 yes 232
19.7 even 3 inner 380.3.p.a.159.110 yes 232
20.19 odd 2 inner 380.3.p.a.239.110 yes 232
76.7 odd 6 inner 380.3.p.a.159.72 yes 232
95.64 even 6 inner 380.3.p.a.159.7 ✓ 232
380.159 odd 6 inner 380.3.p.a.159.45 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 95.64 even 6 inner
380.3.p.a.159.45 yes 232 380.159 odd 6 inner
380.3.p.a.159.72 yes 232 76.7 odd 6 inner
380.3.p.a.159.110 yes 232 19.7 even 3 inner
380.3.p.a.239.7 yes 232 4.3 odd 2 inner
380.3.p.a.239.45 yes 232 1.1 even 1 trivial
380.3.p.a.239.72 yes 232 5.4 even 2 inner
380.3.p.a.239.110 yes 232 20.19 odd 2 inner