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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 311.61
Character \(\chi\) \(=\) 380.311
Dual form 380.3.q.a.11.61

$q$-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+(1.42975 - 1.39850i) q^{2} +(4.13473 - 2.38719i) q^{3} +(0.0883699 - 3.99902i) q^{4} +(1.11803 + 1.93649i) q^{5} +(2.57314 - 9.19551i) q^{6} -2.68655i q^{7} +(-5.46631 - 5.84119i) q^{8} +(6.89731 - 11.9465i) q^{9} +(4.30670 + 1.20512i) q^{10} +4.69031i q^{11} +(-9.18103 - 16.7458i) q^{12} +(-0.228760 + 0.396224i) q^{13} +(-3.75715 - 3.84109i) q^{14} +(9.24553 + 5.33791i) q^{15} +(-15.9844 - 0.706786i) q^{16} +(-6.39447 - 11.0756i) q^{17} +(-6.84579 - 26.7264i) q^{18} +(6.56692 + 17.8291i) q^{19} +(7.84288 - 4.29992i) q^{20} +(-6.41329 - 11.1081i) q^{21} +(6.55942 + 6.70597i) q^{22} +(14.4549 + 8.34553i) q^{23} +(-36.5457 - 11.1026i) q^{24} +(-2.50000 + 4.33013i) q^{25} +(0.227052 + 0.886424i) q^{26} -22.8913i q^{27} +(-10.7436 - 0.237410i) q^{28} +(7.50504 - 12.9991i) q^{29} +(20.6839 - 5.29804i) q^{30} -12.5190i q^{31} +(-23.8421 + 21.3437i) q^{32} +(11.1966 + 19.3932i) q^{33} +(-24.6317 - 6.89257i) q^{34} +(5.20248 - 3.00365i) q^{35} +(-47.1648 - 28.6382i) q^{36} +39.3914 q^{37} +(34.3231 + 16.3072i) q^{38} +2.18437i q^{39} +(5.19990 - 17.1161i) q^{40} +(18.7545 + 32.4838i) q^{41} +(-24.7042 - 6.91285i) q^{42} +(-64.2597 + 37.1004i) q^{43} +(18.7567 + 0.414482i) q^{44} +30.8457 q^{45} +(32.3381 - 8.28320i) q^{46} +(9.62255 + 5.55558i) q^{47} +(-67.7783 + 35.2353i) q^{48} +41.7825 q^{49} +(2.48133 + 9.68726i) q^{50} +(-52.8788 - 30.5296i) q^{51} +(1.56429 + 0.949832i) q^{52} +(-27.7604 + 48.0824i) q^{53} +(-32.0135 - 32.7288i) q^{54} +(-9.08275 + 5.24393i) q^{55} +(-15.6926 + 14.6855i) q^{56} +(69.7137 + 58.0418i) q^{57} +(-7.44898 - 29.0813i) q^{58} +(-40.2939 + 23.2637i) q^{59} +(22.1634 - 36.5014i) q^{60} +(10.4548 - 18.1083i) q^{61} +(-17.5079 - 17.8991i) q^{62} +(-32.0948 - 18.5300i) q^{63} +(-4.23899 + 63.8595i) q^{64} -1.02305 q^{65} +(43.1298 + 12.0688i) q^{66} +(-80.3487 - 46.3893i) q^{67} +(-44.8565 + 24.5929i) q^{68} +79.6893 q^{69} +(3.23762 - 11.5702i) q^{70} +(59.5305 - 34.3700i) q^{71} +(-107.484 + 25.0147i) q^{72} +(-18.6835 - 32.3608i) q^{73} +(56.3199 - 55.0891i) q^{74} +23.8719i q^{75} +(71.8792 - 24.6857i) q^{76} +12.6008 q^{77} +(3.05485 + 3.12310i) q^{78} +(39.1021 - 22.5756i) q^{79} +(-16.5024 - 31.7438i) q^{80} +(7.43008 + 12.8693i) q^{81} +(72.2430 + 20.2154i) q^{82} +96.9916i q^{83} +(-44.9885 + 24.6653i) q^{84} +(14.2985 - 24.7657i) q^{85} +(-39.9903 + 142.912i) q^{86} -71.6637i q^{87} +(27.3970 - 25.6387i) q^{88} +(-16.2918 + 28.2183i) q^{89} +(44.1016 - 43.1378i) q^{90} +(1.06448 + 0.614575i) q^{91} +(34.6513 - 57.0679i) q^{92} +(-29.8852 - 51.7627i) q^{93} +(21.5273 - 5.51409i) q^{94} +(-27.1838 + 32.6503i) q^{95} +(-47.6292 + 145.166i) q^{96} +(-38.7396 - 67.0990i) q^{97} +(59.7385 - 58.4330i) q^{98} +(56.0327 + 32.3505i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.42975 − 1.39850i 0.714875 − 0.699252i
\(3\) 4.13473 − 2.38719i 1.37824 − 0.795728i 0.386294 − 0.922376i \(-0.373755\pi\)
0.991948 + 0.126647i \(0.0404216\pi\)
\(4\) 0.0883699 − 3.99902i 0.0220925 − 0.999756i
\(5\) 1.11803 + 1.93649i 0.223607 + 0.387298i
\(6\) 2.57314 − 9.19551i 0.428856 − 1.53259i
\(7\) − 2.68655i − 0.383793i −0.981415 − 0.191896i \(-0.938536\pi\)
0.981415 − 0.191896i \(-0.0614637\pi\)
\(8\) −5.46631 − 5.84119i −0.683288 − 0.730149i
\(9\) 6.89731 − 11.9465i 0.766367 − 1.32739i
\(10\) 4.30670 + 1.20512i 0.430670 + 0.120512i
\(11\) 4.69031i 0.426392i 0.977009 + 0.213196i \(0.0683873\pi\)
−0.977009 + 0.213196i \(0.931613\pi\)
\(12\) −9.18103 − 16.7458i −0.765085 − 1.39549i
\(13\) −0.228760 + 0.396224i −0.0175969 + 0.0304788i −0.874690 − 0.484683i \(-0.838935\pi\)
0.857093 + 0.515162i \(0.172268\pi\)
\(14\) −3.75715 − 3.84109i −0.268368 − 0.274364i
\(15\) 9.24553 + 5.33791i 0.616369 + 0.355861i
\(16\) −15.9844 − 0.706786i −0.999024 − 0.0441742i
\(17\) −6.39447 − 11.0756i −0.376146 − 0.651503i 0.614352 − 0.789032i \(-0.289417\pi\)
−0.990498 + 0.137529i \(0.956084\pi\)
\(18\) −6.84579 − 26.7264i −0.380322 − 1.48480i
\(19\) 6.56692 + 17.8291i 0.345627 + 0.938372i
\(20\) 7.84288 − 4.29992i 0.392144 − 0.214996i
\(21\) −6.41329 − 11.1081i −0.305395 − 0.528959i
\(22\) 6.55942 + 6.70597i 0.298156 + 0.304817i
\(23\) 14.4549 + 8.34553i 0.628473 + 0.362849i 0.780161 − 0.625579i \(-0.215137\pi\)
−0.151687 + 0.988429i \(0.548471\pi\)
\(24\) −36.5457 − 11.1026i −1.52274 − 0.462610i
\(25\) −2.50000 + 4.33013i −0.100000 + 0.173205i
\(26\) 0.227052 + 0.886424i 0.00873275 + 0.0340932i
\(27\) − 22.8913i − 0.847825i
\(28\) −10.7436 − 0.237410i −0.383699 − 0.00847893i
\(29\) 7.50504 − 12.9991i 0.258794 − 0.448245i −0.707125 − 0.707089i \(-0.750008\pi\)
0.965919 + 0.258844i \(0.0833414\pi\)
\(30\) 20.6839 − 5.29804i 0.689463 − 0.176601i
\(31\) − 12.5190i − 0.403839i −0.979402 − 0.201920i \(-0.935282\pi\)
0.979402 − 0.201920i \(-0.0647179\pi\)
\(32\) −23.8421 + 21.3437i −0.745066 + 0.666991i
\(33\) 11.1966 + 19.3932i 0.339292 + 0.587671i
\(34\) −24.6317 − 6.89257i −0.724462 − 0.202723i
\(35\) 5.20248 − 3.00365i 0.148642 − 0.0858187i
\(36\) −47.1648 − 28.6382i −1.31013 − 0.795506i
\(37\) 39.3914 1.06463 0.532317 − 0.846545i \(-0.321322\pi\)
0.532317 + 0.846545i \(0.321322\pi\)
\(38\) 34.3231 + 16.3072i 0.903239 + 0.429138i
\(39\) 2.18437i 0.0560095i
\(40\) 5.19990 − 17.1161i 0.129997 − 0.427903i
\(41\) 18.7545 + 32.4838i 0.457427 + 0.792288i 0.998824 − 0.0484796i \(-0.0154376\pi\)
−0.541397 + 0.840767i \(0.682104\pi\)
\(42\) −24.7042 − 6.91285i −0.588195 − 0.164592i
\(43\) −64.2597 + 37.1004i −1.49441 + 0.862799i −0.999979 − 0.00641663i \(-0.997958\pi\)
−0.494433 + 0.869216i \(0.664624\pi\)
\(44\) 18.7567 + 0.414482i 0.426288 + 0.00942005i
\(45\) 30.8457 0.685460
\(46\) 32.3381 − 8.28320i 0.703003 − 0.180070i
\(47\) 9.62255 + 5.55558i 0.204735 + 0.118204i 0.598862 − 0.800852i \(-0.295620\pi\)
−0.394127 + 0.919056i \(0.628953\pi\)
\(48\) −67.7783 + 35.2353i −1.41205 + 0.734069i
\(49\) 41.7825 0.852703
\(50\) 2.48133 + 9.68726i 0.0496266 + 0.193745i
\(51\) −52.8788 − 30.5296i −1.03684 − 0.598619i
\(52\) 1.56429 + 0.949832i 0.0300826 + 0.0182660i
\(53\) −27.7604 + 48.0824i −0.523780 + 0.907214i 0.475836 + 0.879534i \(0.342145\pi\)
−0.999617 + 0.0276804i \(0.991188\pi\)
\(54\) −32.0135 − 32.7288i −0.592843 − 0.606089i
\(55\) −9.08275 + 5.24393i −0.165141 + 0.0953442i
\(56\) −15.6926 + 14.6855i −0.280226 + 0.262241i
\(57\) 69.7137 + 58.0418i 1.22305 + 1.01828i
\(58\) −7.44898 − 29.0813i −0.128431 − 0.501402i
\(59\) −40.2939 + 23.2637i −0.682947 + 0.394300i −0.800964 − 0.598712i \(-0.795679\pi\)
0.118017 + 0.993012i \(0.462346\pi\)
\(60\) 22.1634 − 36.5014i 0.369391 − 0.608356i
\(61\) 10.4548 − 18.1083i 0.171391 − 0.296858i −0.767515 − 0.641030i \(-0.778507\pi\)
0.938906 + 0.344173i \(0.111841\pi\)
\(62\) −17.5079 − 17.8991i −0.282385 − 0.288694i
\(63\) −32.0948 − 18.5300i −0.509442 − 0.294126i
\(64\) −4.23899 + 63.8595i −0.0662343 + 0.997804i
\(65\) −1.02305 −0.0157392
\(66\) 43.1298 + 12.0688i 0.653482 + 0.182861i
\(67\) −80.3487 − 46.3893i −1.19923 − 0.692378i −0.238849 − 0.971057i \(-0.576770\pi\)
−0.960384 + 0.278679i \(0.910104\pi\)
\(68\) −44.8565 + 24.5929i −0.659654 + 0.361660i
\(69\) 79.6893 1.15492
\(70\) 3.23762 − 11.5702i 0.0462517 − 0.165288i
\(71\) 59.5305 − 34.3700i 0.838458 − 0.484084i −0.0182818 − 0.999833i \(-0.505820\pi\)
0.856740 + 0.515749i \(0.172486\pi\)
\(72\) −107.484 + 25.0147i −1.49284 + 0.347426i
\(73\) −18.6835 − 32.3608i −0.255938 − 0.443298i 0.709212 − 0.704996i \(-0.249051\pi\)
−0.965150 + 0.261698i \(0.915718\pi\)
\(74\) 56.3199 − 55.0891i 0.761080 − 0.744447i
\(75\) 23.8719i 0.318291i
\(76\) 71.8792 − 24.6857i 0.945779 − 0.324812i
\(77\) 12.6008 0.163646
\(78\) 3.05485 + 3.12310i 0.0391648 + 0.0400398i
\(79\) 39.1021 − 22.5756i 0.494964 − 0.285768i −0.231668 − 0.972795i \(-0.574418\pi\)
0.726631 + 0.687027i \(0.241085\pi\)
\(80\) −16.5024 − 31.7438i −0.206280 − 0.396798i
\(81\) 7.43008 + 12.8693i 0.0917293 + 0.158880i
\(82\) 72.2430 + 20.2154i 0.881012 + 0.246529i
\(83\) 96.9916i 1.16857i 0.811547 + 0.584287i \(0.198626\pi\)
−0.811547 + 0.584287i \(0.801374\pi\)
\(84\) −44.9885 + 24.6653i −0.535577 + 0.293634i
\(85\) 14.2985 − 24.7657i 0.168217 − 0.291361i
\(86\) −39.9903 + 142.912i −0.465003 + 1.66176i
\(87\) − 71.6637i − 0.823720i
\(88\) 27.3970 − 25.6387i 0.311330 − 0.291349i
\(89\) −16.2918 + 28.2183i −0.183054 + 0.317060i −0.942919 − 0.333022i \(-0.891932\pi\)
0.759865 + 0.650081i \(0.225265\pi\)
\(90\) 44.1016 − 43.1378i 0.490018 − 0.479309i
\(91\) 1.06448 + 0.614575i 0.0116975 + 0.00675357i
\(92\) 34.6513 − 57.0679i 0.376645 − 0.620304i
\(93\) −29.8852 − 51.7627i −0.321346 − 0.556588i
\(94\) 21.5273 − 5.51409i 0.229014 − 0.0586605i
\(95\) −27.1838 + 32.6503i −0.286145 + 0.343687i
\(96\) −47.6292 + 145.166i −0.496138 + 1.51214i
\(97\) −38.7396 − 67.0990i −0.399378 − 0.691742i 0.594272 − 0.804264i \(-0.297440\pi\)
−0.993649 + 0.112522i \(0.964107\pi\)
\(98\) 59.7385 − 58.4330i 0.609576 − 0.596255i
\(99\) 56.0327 + 32.3505i 0.565987 + 0.326773i
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.311.61 yes 160
4.3 odd 2 inner 380.3.q.a.311.8 yes 160
19.11 even 3 inner 380.3.q.a.11.8 ✓ 160
76.11 odd 6 inner 380.3.q.a.11.61 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.8 ✓ 160 19.11 even 3 inner
380.3.q.a.11.61 yes 160 76.11 odd 6 inner
380.3.q.a.311.8 yes 160 4.3 odd 2 inner
380.3.q.a.311.61 yes 160 1.1 even 1 trivial