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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 159.45
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.45

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.691699 + 1.87658i −0.345849 + 0.938290i
\(3\) 1.97599 + 3.42251i 0.658662 + 1.14084i 0.980962 + 0.194199i \(0.0622106\pi\)
−0.322300 + 0.946637i \(0.604456\pi\)
\(4\) −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 \(-0.700681\pi\)
0.589514 − 0.807758i \(-0.299319\pi\)
\(12\) 2.87189 − 15.5448i 0.239324 − 1.29540i
\(13\) −4.58570 − 2.64756i −0.352746 − 0.203658i 0.313148 − 0.949704i \(-0.398616\pi\)
−0.665894 + 0.746046i \(0.731950\pi\)
\(14\) −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 −0.443056 − 0.896494i \(-0.646106\pi\)
0.554859 + 0.831945i \(0.312772\pi\)
\(18\) −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 −0.630259 − 0.776385i \(-0.717051\pi\)
0.987499 + 0.157627i \(0.0503844\pi\)
\(24\) 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 −0.0390576 − 0.999237i \(-0.512436\pi\)
0.884893 − 0.465794i \(-0.154231\pi\)
\(30\) 12.0376 + 37.6418i 0.401253 + 1.25473i
\(31\) 13.0380i 0.420581i 0.977639 + 0.210291i \(0.0674411\pi\)
−0.977639 + 0.210291i \(0.932559\pi\)
\(32\) −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 \(0.257384\pi\)
−0.690515 + 0.723318i \(0.742616\pi\)
\(38\) 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 0.978301 − 0.207190i \(-0.0664319\pi\)
−0.668582 + 0.743638i \(0.733099\pi\)
\(42\) −20.0119 + 3.44456i −0.476475 + 0.0820134i
\(43\) −36.7927 − 63.7268i −0.855644 − 1.48202i −0.876046 − 0.482227i \(-0.839828\pi\)
0.0204018 − 0.999792i \(-0.493505\pi\)
\(44\) −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 0.398586 + 0.917131i \(0.369501\pi\)
−0.993552 + 0.113380i \(0.963832\pi\)
\(48\) −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 0.817917 − 0.575336i \(-0.195129\pi\)
−0.0892969 + 0.996005i \(0.528462\pi\)
\(54\) −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 −0.905640 − 0.424047i \(-0.860609\pi\)
−0.0855850 + 0.996331i \(0.527276\pi\)
\(60\) −78.9642 − 3.44724i −1.31607 − 0.0574540i
\(61\) −2.07481 + 3.59367i −0.0340132 + 0.0589127i −0.882531 − 0.470254i \(-0.844162\pi\)
0.848518 + 0.529167i \(0.177496\pi\)
\(62\) −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 −0.180267 − 0.983618i \(-0.557696\pi\)
0.941971 − 0.335693i \(-0.108970\pi\)
\(68\) −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 −0.731207 − 0.682156i \(-0.761042\pi\)
0.225161 + 0.974322i \(0.427709\pi\)
\(72\) −0.647696 + 52.9407i −0.00899578 + 0.735287i
\(73\) 41.1613 − 23.7645i 0.563853 − 0.325541i −0.190838 − 0.981622i \(-0.561120\pi\)
0.754690 + 0.656081i \(0.227787\pi\)
\(74\) −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 −0.0444457 − 0.999012i \(-0.514152\pi\)
0.842947 + 0.537997i \(0.180819\pi\)
\(80\) 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 −0.433089 − 0.901351i \(-0.642576\pi\)
0.997137 − 0.0756097i \(-0.0240903\pi\)
\(90\) −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 −0.749391 − 0.662128i \(-0.769654\pi\)
0.198724 + 0.980055i \(0.436320\pi\)
\(98\) 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.159.45 yes 232
4.3 odd 2 inner 380.3.p.a.159.7 ✓ 232
5.4 even 2 inner 380.3.p.a.159.72 yes 232
19.11 even 3 inner 380.3.p.a.239.110 yes 232
20.19 odd 2 inner 380.3.p.a.159.110 yes 232
76.11 odd 6 inner 380.3.p.a.239.72 yes 232
95.49 even 6 inner 380.3.p.a.239.7 yes 232
380.239 odd 6 inner 380.3.p.a.239.45 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 4.3 odd 2 inner
380.3.p.a.159.45 yes 232 1.1 even 1 trivial
380.3.p.a.159.72 yes 232 5.4 even 2 inner
380.3.p.a.159.110 yes 232 20.19 odd 2 inner
380.3.p.a.239.7 yes 232 95.49 even 6 inner
380.3.p.a.239.45 yes 232 380.239 odd 6 inner
380.3.p.a.239.72 yes 232 76.11 odd 6 inner
380.3.p.a.239.110 yes 232 19.11 even 3 inner