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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); 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, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (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: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.13
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.13

$q$-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+(-1.72336 - 1.01491i) q^{2} +(4.96001 + 2.86366i) q^{3} +(1.93992 + 3.49810i) q^{4} +(1.11803 - 1.93649i) q^{5} +(-5.64152 - 9.96907i) q^{6} -1.47840i q^{7} +(0.207068 - 7.99732i) q^{8} +(11.9011 + 20.6134i) q^{9} +(-3.89213 + 2.20257i) q^{10} +19.1158i q^{11} +(-0.395339 + 22.9059i) q^{12} +(-5.39021 - 9.33611i) q^{13} +(-1.50044 + 2.54781i) q^{14} +(11.0909 - 6.40335i) q^{15} +(-8.47340 + 13.5721i) q^{16} +(-3.84445 + 6.65878i) q^{17} +(0.410764 - 47.6028i) q^{18} +(17.3653 + 7.71006i) q^{19} +(8.94294 + 0.154349i) q^{20} +(4.23363 - 7.33287i) q^{21} +(19.4007 - 32.9433i) q^{22} +(26.2364 - 15.1476i) q^{23} +(23.9287 - 39.0738i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-0.186041 + 21.5600i) q^{26} +84.7775i q^{27} +(5.17158 - 2.86798i) q^{28} +(-20.4613 - 35.4400i) q^{29} +(-25.6124 - 0.221010i) q^{30} +31.0982i q^{31} +(28.3771 - 14.7898i) q^{32} +(-54.7411 + 94.8144i) q^{33} +(13.3834 - 7.57369i) q^{34} +(-2.86291 - 1.65290i) q^{35} +(-49.0203 + 81.6197i) q^{36} -48.5758 q^{37} +(-22.1017 - 30.9114i) q^{38} -61.7430i q^{39} +(-15.2552 - 9.34226i) q^{40} +(13.1096 - 22.7065i) q^{41} +(-14.7383 + 8.34041i) q^{42} +(31.9066 + 18.4213i) q^{43} +(-66.8688 + 37.0831i) q^{44} +53.2235 q^{45} +(-60.5880 - 0.522814i) q^{46} +(-2.63389 + 1.52068i) q^{47} +(-80.8940 + 43.0527i) q^{48} +46.8143 q^{49} +(-0.0862868 + 9.99963i) q^{50} +(-38.1370 + 22.0184i) q^{51} +(22.2021 - 36.9668i) q^{52} +(-1.58536 - 2.74593i) q^{53} +(86.0413 - 146.102i) q^{54} +(37.0175 + 21.3721i) q^{55} +(-11.8232 - 0.306129i) q^{56} +(64.0532 + 87.9705i) q^{57} +(-0.706216 + 81.8422i) q^{58} +(-37.7019 - 21.7672i) q^{59} +(43.9151 + 26.3751i) q^{60} +(38.6988 + 67.0283i) q^{61} +(31.5619 - 53.5934i) q^{62} +(30.4748 - 17.5946i) q^{63} +(-63.9142 - 3.31197i) q^{64} -24.1057 q^{65} +(190.566 - 107.842i) q^{66} +(26.2608 - 15.1617i) q^{67} +(-30.7510 - 0.530740i) q^{68} +173.510 q^{69} +(3.25627 + 5.75412i) q^{70} +(-100.913 - 58.2620i) q^{71} +(167.316 - 90.9088i) q^{72} +(55.7742 - 96.6037i) q^{73} +(83.7135 + 49.3000i) q^{74} -28.6366i q^{75} +(6.71685 + 75.7026i) q^{76} +28.2607 q^{77} +(-62.6634 + 106.405i) q^{78} +(68.4437 + 39.5160i) q^{79} +(16.8087 + 31.5827i) q^{80} +(-135.664 + 234.977i) q^{81} +(-45.6375 + 25.8263i) q^{82} -73.2143i q^{83} +(33.8640 + 0.584469i) q^{84} +(8.59644 + 14.8895i) q^{85} +(-36.2906 - 64.1287i) q^{86} -234.377i q^{87} +(152.875 + 3.95826i) q^{88} +(-26.8569 - 46.5175i) q^{89} +(-91.7231 - 54.0170i) q^{90} +(-13.8025 + 7.96887i) q^{91} +(103.884 + 62.3923i) q^{92} +(-89.0549 + 154.248i) q^{93} +(6.08249 + 0.0524858i) q^{94} +(34.3455 - 25.0077i) q^{95} +(183.104 + 7.90473i) q^{96} +(-13.3790 + 23.1731i) q^{97} +(-80.6778 - 47.5123i) q^{98} +(-394.040 + 227.499i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.72336 − 1.01491i −0.861679 − 0.507454i
\(3\) 4.96001 + 2.86366i 1.65334 + 0.954554i 0.975687 + 0.219170i \(0.0703350\pi\)
0.677650 + 0.735384i \(0.262998\pi\)
\(4\) 1.93992 + 3.49810i 0.484981 + 0.874525i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) −5.64152 − 9.96907i −0.940253 − 1.66151i
\(7\) − 1.47840i − 0.211200i −0.994409 − 0.105600i \(-0.966324\pi\)
0.994409 − 0.105600i \(-0.0336763\pi\)
\(8\) 0.207068 − 7.99732i 0.0258835 − 0.999665i
\(9\) 11.9011 + 20.6134i 1.32235 + 2.29037i
\(10\) −3.89213 + 2.20257i −0.389213 + 0.220257i
\(11\) 19.1158i 1.73780i 0.494991 + 0.868898i \(0.335171\pi\)
−0.494991 + 0.868898i \(0.664829\pi\)
\(12\) −0.395339 + 22.9059i −0.0329449 + 1.90882i
\(13\) −5.39021 − 9.33611i −0.414631 − 0.718163i 0.580758 − 0.814076i \(-0.302756\pi\)
−0.995390 + 0.0959135i \(0.969423\pi\)
\(14\) −1.50044 + 2.54781i −0.107174 + 0.181986i
\(15\) 11.0909 − 6.40335i 0.739395 − 0.426890i
\(16\) −8.47340 + 13.5721i −0.529587 + 0.848255i
\(17\) −3.84445 + 6.65878i −0.226144 + 0.391693i −0.956662 − 0.291201i \(-0.905945\pi\)
0.730518 + 0.682893i \(0.239279\pi\)
\(18\) 0.410764 − 47.6028i 0.0228202 − 2.64460i
\(19\) 17.3653 + 7.71006i 0.913965 + 0.405793i
\(20\) 8.94294 + 0.154349i 0.447147 + 0.00771744i
\(21\) 4.23363 − 7.33287i 0.201602 − 0.349184i
\(22\) 19.4007 − 32.9433i 0.881852 − 1.49742i
\(23\) 26.2364 − 15.1476i 1.14071 − 0.658590i 0.194105 − 0.980981i \(-0.437820\pi\)
0.946607 + 0.322391i \(0.104487\pi\)
\(24\) 23.9287 − 39.0738i 0.997029 − 1.62808i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −0.186041 + 21.5600i −0.00715544 + 0.829232i
\(27\) 84.7775i 3.13991i
\(28\) 5.17158 − 2.86798i 0.184699 − 0.102428i
\(29\) −20.4613 − 35.4400i −0.705562 − 1.22207i −0.966488 − 0.256711i \(-0.917361\pi\)
0.260926 − 0.965359i \(-0.415972\pi\)
\(30\) −25.6124 − 0.221010i −0.853748 − 0.00736699i
\(31\) 31.0982i 1.00317i 0.865109 + 0.501585i \(0.167249\pi\)
−0.865109 + 0.501585i \(0.832751\pi\)
\(32\) 28.3771 − 14.7898i 0.886785 − 0.462183i
\(33\) −54.7411 + 94.8144i −1.65882 + 2.87316i
\(34\) 13.3834 − 7.57369i 0.393629 − 0.222756i
\(35\) −2.86291 − 1.65290i −0.0817973 − 0.0472257i
\(36\) −49.0203 + 81.6197i −1.36168 + 2.26721i
\(37\) −48.5758 −1.31286 −0.656430 − 0.754387i \(-0.727934\pi\)
−0.656430 + 0.754387i \(0.727934\pi\)
\(38\) −22.1017 − 30.9114i −0.581623 − 0.813458i
\(39\) − 61.7430i − 1.58315i
\(40\) −15.2552 − 9.34226i −0.381381 − 0.233557i
\(41\) 13.1096 − 22.7065i 0.319746 − 0.553816i −0.660689 − 0.750660i \(-0.729736\pi\)
0.980435 + 0.196843i \(0.0630691\pi\)
\(42\) −14.7383 + 8.34041i −0.350911 + 0.198581i
\(43\) 31.9066 + 18.4213i 0.742014 + 0.428402i 0.822801 − 0.568329i \(-0.192410\pi\)
−0.0807871 + 0.996731i \(0.525743\pi\)
\(44\) −66.8688 + 37.0831i −1.51975 + 0.842798i
\(45\) 53.2235 1.18274
\(46\) −60.5880 − 0.522814i −1.31713 − 0.0113655i
\(47\) −2.63389 + 1.52068i −0.0560403 + 0.0323549i −0.527758 − 0.849395i \(-0.676967\pi\)
0.471718 + 0.881749i \(0.343634\pi\)
\(48\) −80.8940 + 43.0527i −1.68529 + 0.896932i
\(49\) 46.8143 0.955395
\(50\) −0.0862868 + 9.99963i −0.00172574 + 0.199993i
\(51\) −38.1370 + 22.0184i −0.747784 + 0.431733i
\(52\) 22.2021 − 36.9668i 0.426963 − 0.710900i
\(53\) −1.58536 − 2.74593i −0.0299125 − 0.0518100i 0.850682 − 0.525681i \(-0.176190\pi\)
−0.880594 + 0.473871i \(0.842856\pi\)
\(54\) 86.0413 − 146.102i 1.59336 − 2.70559i
\(55\) 37.0175 + 21.3721i 0.673046 + 0.388583i
\(56\) −11.8232 − 0.306129i −0.211129 − 0.00546658i
\(57\) 64.0532 + 87.9705i 1.12374 + 1.54334i
\(58\) −0.706216 + 81.8422i −0.0121761 + 1.41107i
\(59\) −37.7019 − 21.7672i −0.639016 − 0.368936i 0.145219 − 0.989399i \(-0.453611\pi\)
−0.784235 + 0.620463i \(0.786945\pi\)
\(60\) 43.9151 + 26.3751i 0.731918 + 0.439586i
\(61\) 38.6988 + 67.0283i 0.634406 + 1.09882i 0.986641 + 0.162912i \(0.0520887\pi\)
−0.352234 + 0.935912i \(0.614578\pi\)
\(62\) 31.5619 − 53.5934i 0.509062 − 0.864410i
\(63\) 30.4748 − 17.5946i 0.483727 − 0.279280i
\(64\) −63.9142 − 3.31197i −0.998660 − 0.0517496i
\(65\) −24.1057 −0.370858
\(66\) 190.566 − 107.842i 2.88737 − 1.63397i
\(67\) 26.2608 − 15.1617i 0.391952 − 0.226294i −0.291054 − 0.956707i \(-0.594006\pi\)
0.683006 + 0.730413i \(0.260672\pi\)
\(68\) −30.7510 − 0.530740i −0.452220 − 0.00780500i
\(69\) 173.510 2.51464
\(70\) 3.25627 + 5.75412i 0.0465181 + 0.0822018i
\(71\) −100.913 − 58.2620i −1.42131 − 0.820591i −0.424895 − 0.905243i \(-0.639689\pi\)
−0.996411 + 0.0846516i \(0.973022\pi\)
\(72\) 167.316 − 90.9088i 2.32383 − 1.26262i
\(73\) 55.7742 − 96.6037i 0.764030 − 1.32334i −0.176729 − 0.984260i \(-0.556551\pi\)
0.940758 − 0.339078i \(-0.110115\pi\)
\(74\) 83.7135 + 49.3000i 1.13126 + 0.666216i
\(75\) − 28.6366i − 0.381822i
\(76\) 6.71685 + 75.7026i 0.0883796 + 0.996087i
\(77\) 28.2607 0.367022
\(78\) −62.6634 + 106.405i −0.803377 + 1.36417i
\(79\) 68.4437 + 39.5160i 0.866376 + 0.500202i 0.866142 − 0.499798i \(-0.166592\pi\)
0.000233442 1.00000i \(0.499926\pi\)
\(80\) 16.8087 + 31.5827i 0.210109 + 0.394784i
\(81\) −135.664 + 234.977i −1.67486 + 2.90095i
\(82\) −45.6375 + 25.8263i −0.556555 + 0.314955i
\(83\) − 73.2143i − 0.882100i −0.897483 − 0.441050i \(-0.854606\pi\)
0.897483 − 0.441050i \(-0.145394\pi\)
\(84\) 33.8640 + 0.584469i 0.403143 + 0.00695796i
\(85\) 8.59644 + 14.8895i 0.101135 + 0.175170i
\(86\) −36.2906 − 64.1287i −0.421983 − 0.745683i
\(87\) − 234.377i − 2.69399i
\(88\) 152.875 + 3.95826i 1.73721 + 0.0449802i
\(89\) −26.8569 − 46.5175i −0.301763 − 0.522668i 0.674773 − 0.738026i \(-0.264242\pi\)
−0.976535 + 0.215358i \(0.930908\pi\)
\(90\) −91.7231 − 54.0170i −1.01915 − 0.600188i
\(91\) −13.8025 + 7.96887i −0.151676 + 0.0875700i
\(92\) 103.884 + 62.3923i 1.12918 + 0.678177i
\(93\) −89.0549 + 154.248i −0.957580 + 1.65858i
\(94\) 6.08249 + 0.0524858i 0.0647073 + 0.000558359i
\(95\) 34.3455 − 25.0077i 0.361532 − 0.263239i
\(96\) 183.104 + 7.90473i 1.90733 + 0.0823409i
\(97\) −13.3790 + 23.1731i −0.137928 + 0.238898i −0.926712 − 0.375772i \(-0.877377\pi\)
0.788784 + 0.614670i \(0.210711\pi\)
\(98\) −80.6778 − 47.5123i −0.823243 − 0.484819i
\(99\) −394.040 + 227.499i −3.98021 + 2.29797i
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.q.a.11.13 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.67 yes 160
19.7 even 3 inner 380.3.q.a.311.67 yes 160
76.7 odd 6 inner 380.3.q.a.311.13 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.13 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.67 yes 160 4.3 odd 2 inner
380.3.q.a.311.13 yes 160 76.7 odd 6 inner
380.3.q.a.311.67 yes 160 19.7 even 3 inner