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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.20
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.20

$q$-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+(-0.626498 + 1.26787i) q^{2} +(2.02176 + 1.16726i) q^{3} +(-1.21500 - 1.58864i) q^{4} +(1.88679 - 1.20001i) q^{5} +(-2.74657 + 1.83205i) q^{6} +0.943490 q^{7} +(2.77539 - 0.545184i) q^{8} +(1.22501 + 2.12179i) q^{9} +(0.339390 + 3.14401i) q^{10} -0.173788i q^{11} +(-0.602077 - 4.63008i) q^{12} +(-0.200082 - 0.346553i) q^{13} +(-0.591095 + 1.19622i) q^{14} +(5.21537 - 0.223754i) q^{15} +(-1.04755 + 3.86039i) q^{16} +(-2.08538 - 1.20400i) q^{17} +(-3.45762 + 0.223866i) q^{18} +(4.32892 + 0.510313i) q^{19} +(-4.19883 - 1.53942i) q^{20} +(1.90751 + 1.10130i) q^{21} +(0.220340 + 0.108878i) q^{22} +(3.88561 + 6.73008i) q^{23} +(6.24755 + 2.13738i) q^{24} +(2.11995 - 4.52834i) q^{25} +(0.564736 - 0.0365641i) q^{26} -1.28393i q^{27} +(-1.14634 - 1.49887i) q^{28} +(-4.11985 + 2.37860i) q^{29} +(-2.98373 + 6.75260i) q^{30} -6.51703 q^{31} +(-4.23820 - 3.74669i) q^{32} +(0.202856 - 0.351357i) q^{33} +(2.83300 - 1.88970i) q^{34} +(1.78017 - 1.13220i) q^{35} +(1.88236 - 4.52408i) q^{36} -9.07290 q^{37} +(-3.35908 + 5.16881i) q^{38} -0.934196i q^{39} +(4.58234 - 4.35914i) q^{40} +(-3.80514 - 2.19690i) q^{41} +(-2.59136 + 1.72852i) q^{42} +(-0.622036 + 1.07740i) q^{43} +(-0.276086 + 0.211152i) q^{44} +(4.85751 + 2.53333i) q^{45} +(-10.9672 + 0.710078i) q^{46} +(3.10145 + 5.37186i) q^{47} +(-6.62400 + 6.58203i) q^{48} -6.10983 q^{49} +(4.41321 + 5.52482i) q^{50} +(-2.81076 - 4.86839i) q^{51} +(-0.307447 + 0.738920i) q^{52} +(1.49617 + 2.59144i) q^{53} +(1.62785 + 0.804378i) q^{54} +(-0.208547 - 0.327901i) q^{55} +(2.61855 - 0.514376i) q^{56} +(8.15638 + 6.08473i) q^{57} +(-0.434677 - 6.71363i) q^{58} +(4.78221 - 8.28302i) q^{59} +(-6.69214 - 8.01348i) q^{60} +(-3.58110 - 6.20264i) q^{61} +(4.08291 - 8.26276i) q^{62} +(1.15579 + 2.00188i) q^{63} +(7.40555 - 3.02620i) q^{64} +(-0.793380 - 0.413771i) q^{65} +(0.318387 + 0.477320i) q^{66} +(-9.56423 + 5.52191i) q^{67} +(0.621023 + 4.77577i) q^{68} +18.1422i q^{69} +(0.320211 + 2.96634i) q^{70} +(4.54516 - 7.87245i) q^{71} +(4.55665 + 5.22092i) q^{72} +(7.54872 + 4.35826i) q^{73} +(5.68416 - 11.5033i) q^{74} +(9.57180 - 6.68068i) q^{75} +(-4.44894 - 7.49713i) q^{76} -0.163967i q^{77} +(1.18444 + 0.585272i) q^{78} +(-4.94885 + 8.57166i) q^{79} +(2.65601 + 8.54082i) q^{80} +(5.17372 - 8.96115i) q^{81} +(5.16930 - 3.44808i) q^{82} +5.08393 q^{83} +(-0.568054 - 4.36843i) q^{84} +(-5.37948 + 0.230795i) q^{85} +(-0.976298 - 1.46365i) q^{86} -11.1058 q^{87} +(-0.0947463 - 0.482328i) q^{88} +(3.34144 - 1.92918i) q^{89} +(-6.25516 + 4.57157i) q^{90} +(-0.188776 - 0.326969i) q^{91} +(5.97065 - 14.3499i) q^{92} +(-13.1759 - 7.60709i) q^{93} +(-8.75389 + 0.566775i) q^{94} +(8.78015 - 4.23190i) q^{95} +(-4.19524 - 12.5220i) q^{96} +(4.28716 - 7.42557i) q^{97} +(3.82780 - 7.74648i) q^{98} +(0.368740 - 0.212892i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.626498 + 1.26787i −0.443001 + 0.896521i
\(3\) 2.02176 + 1.16726i 1.16726 + 0.673921i 0.953034 − 0.302863i \(-0.0979423\pi\)
0.214230 + 0.976783i \(0.431276\pi\)
\(4\) −1.21500 − 1.58864i −0.607500 − 0.794320i
\(5\) 1.88679 − 1.20001i 0.843798 − 0.536661i
\(6\) −2.74657 + 1.83205i −1.12128 + 0.747930i
\(7\) 0.943490 0.356606 0.178303 − 0.983976i \(-0.442939\pi\)
0.178303 + 0.983976i \(0.442939\pi\)
\(8\) 2.77539 − 0.545184i 0.981248 − 0.192752i
\(9\) 1.22501 + 2.12179i 0.408338 + 0.707262i
\(10\) 0.339390 + 3.14401i 0.107325 + 0.994224i
\(11\) − 0.173788i − 0.0523989i −0.999657 − 0.0261995i \(-0.991659\pi\)
0.999657 − 0.0261995i \(-0.00834050\pi\)
\(12\) −0.602077 − 4.63008i −0.173805 − 1.33659i
\(13\) −0.200082 − 0.346553i −0.0554929 − 0.0961165i 0.836944 − 0.547288i \(-0.184340\pi\)
−0.892437 + 0.451171i \(0.851006\pi\)
\(14\) −0.591095 + 1.19622i −0.157977 + 0.319704i
\(15\) 5.21537 − 0.223754i 1.34660 − 0.0577729i
\(16\) −1.04755 + 3.86039i −0.261888 + 0.965098i
\(17\) −2.08538 − 1.20400i −0.505779 − 0.292012i 0.225318 − 0.974285i \(-0.427658\pi\)
−0.731097 + 0.682273i \(0.760991\pi\)
\(18\) −3.45762 + 0.223866i −0.814970 + 0.0527656i
\(19\) 4.32892 + 0.510313i 0.993123 + 0.117074i
\(20\) −4.19883 − 1.53942i −0.938888 − 0.344224i
\(21\) 1.90751 + 1.10130i 0.416253 + 0.240324i
\(22\) 0.220340 + 0.108878i 0.0469767 + 0.0232128i
\(23\) 3.88561 + 6.73008i 0.810206 + 1.40332i 0.912719 + 0.408587i \(0.133978\pi\)
−0.102513 + 0.994732i \(0.532688\pi\)
\(24\) 6.24755 + 2.13738i 1.27528 + 0.436291i
\(25\) 2.11995 − 4.52834i 0.423990 − 0.905667i
\(26\) 0.564736 − 0.0365641i 0.110754 − 0.00717081i
\(27\) − 1.28393i − 0.247092i
\(28\) −1.14634 − 1.49887i −0.216638 − 0.283259i
\(29\) −4.11985 + 2.37860i −0.765037 + 0.441694i −0.831101 − 0.556121i \(-0.812289\pi\)
0.0660643 + 0.997815i \(0.478956\pi\)
\(30\) −2.98373 + 6.75260i −0.544752 + 1.23285i
\(31\) −6.51703 −1.17049 −0.585246 − 0.810856i \(-0.699002\pi\)
−0.585246 + 0.810856i \(0.699002\pi\)
\(32\) −4.23820 − 3.74669i −0.749214 − 0.662328i
\(33\) 0.202856 − 0.351357i 0.0353127 − 0.0611634i
\(34\) 2.83300 − 1.88970i 0.485856 − 0.324080i
\(35\) 1.78017 − 1.13220i 0.300903 − 0.191376i
\(36\) 1.88236 − 4.52408i 0.313727 − 0.754013i
\(37\) −9.07290 −1.49158 −0.745788 − 0.666183i \(-0.767927\pi\)
−0.745788 + 0.666183i \(0.767927\pi\)
\(38\) −3.35908 + 5.16881i −0.544914 + 0.838492i
\(39\) − 0.934196i − 0.149591i
\(40\) 4.58234 − 4.35914i 0.724532 − 0.689241i
\(41\) −3.80514 − 2.19690i −0.594263 − 0.343098i 0.172518 − 0.985006i \(-0.444810\pi\)
−0.766781 + 0.641908i \(0.778143\pi\)
\(42\) −2.59136 + 1.72852i −0.399856 + 0.266716i
\(43\) −0.622036 + 1.07740i −0.0948596 + 0.164302i −0.909550 − 0.415595i \(-0.863574\pi\)
0.814690 + 0.579896i \(0.196907\pi\)
\(44\) −0.276086 + 0.211152i −0.0416215 + 0.0318323i
\(45\) 4.85751 + 2.53333i 0.724115 + 0.377647i
\(46\) −10.9672 + 0.710078i −1.61703 + 0.104695i
\(47\) 3.10145 + 5.37186i 0.452392 + 0.783567i 0.998534 − 0.0541261i \(-0.0172373\pi\)
−0.546142 + 0.837693i \(0.683904\pi\)
\(48\) −6.62400 + 6.58203i −0.956092 + 0.950034i
\(49\) −6.10983 −0.872832
\(50\) 4.41321 + 5.52482i 0.624122 + 0.781327i
\(51\) −2.81076 − 4.86839i −0.393586 − 0.681710i
\(52\) −0.307447 + 0.738920i −0.0426353 + 0.102470i
\(53\) 1.49617 + 2.59144i 0.205514 + 0.355961i 0.950296 − 0.311347i \(-0.100780\pi\)
−0.744782 + 0.667308i \(0.767447\pi\)
\(54\) 1.62785 + 0.804378i 0.221523 + 0.109462i
\(55\) −0.208547 − 0.327901i −0.0281205 − 0.0442141i
\(56\) 2.61855 − 0.514376i 0.349918 − 0.0687364i
\(57\) 8.15638 + 6.08473i 1.08034 + 0.805942i
\(58\) −0.434677 − 6.71363i −0.0570759 − 0.881543i
\(59\) 4.78221 − 8.28302i 0.622590 − 1.07836i −0.366411 − 0.930453i \(-0.619414\pi\)
0.989002 − 0.147905i \(-0.0472530\pi\)
\(60\) −6.69214 − 8.01348i −0.863951 − 1.03454i
\(61\) −3.58110 − 6.20264i −0.458513 − 0.794167i 0.540370 − 0.841427i \(-0.318284\pi\)
−0.998883 + 0.0472605i \(0.984951\pi\)
\(62\) 4.08291 − 8.26276i 0.518530 − 1.04937i
\(63\) 1.15579 + 2.00188i 0.145616 + 0.252214i
\(64\) 7.40555 − 3.02620i 0.925693 − 0.378274i
\(65\) −0.793380 − 0.413771i −0.0984067 − 0.0513220i
\(66\) 0.318387 + 0.477320i 0.0391907 + 0.0587541i
\(67\) −9.56423 + 5.52191i −1.16846 + 0.674609i −0.953316 − 0.301974i \(-0.902354\pi\)
−0.215141 + 0.976583i \(0.569021\pi\)
\(68\) 0.621023 + 4.77577i 0.0753101 + 0.579148i
\(69\) 18.1422i 2.18406i
\(70\) 0.320211 + 2.96634i 0.0382726 + 0.354546i
\(71\) 4.54516 − 7.87245i 0.539411 − 0.934288i −0.459524 − 0.888165i \(-0.651980\pi\)
0.998936 − 0.0461227i \(-0.0146865\pi\)
\(72\) 4.55665 + 5.22092i 0.537007 + 0.615291i
\(73\) 7.54872 + 4.35826i 0.883511 + 0.510095i 0.871814 − 0.489836i \(-0.162943\pi\)
0.0116966 + 0.999932i \(0.496277\pi\)
\(74\) 5.68416 − 11.5033i 0.660770 − 1.33723i
\(75\) 9.57180 − 6.68068i 1.10526 − 0.771418i
\(76\) −4.44894 − 7.49713i −0.510328 − 0.859980i
\(77\) − 0.163967i − 0.0186858i
\(78\) 1.18444 + 0.585272i 0.134112 + 0.0662691i
\(79\) −4.94885 + 8.57166i −0.556789 + 0.964387i 0.440973 + 0.897520i \(0.354634\pi\)
−0.997762 + 0.0668665i \(0.978700\pi\)
\(80\) 2.65601 + 8.54082i 0.296950 + 0.954893i
\(81\) 5.17372 − 8.96115i 0.574858 − 0.995684i
\(82\) 5.16930 − 3.44808i 0.570854 − 0.380777i
\(83\) 5.08393 0.558034 0.279017 − 0.960286i \(-0.409991\pi\)
0.279017 + 0.960286i \(0.409991\pi\)
\(84\) −0.568054 − 4.36843i −0.0619798 − 0.476635i
\(85\) −5.37948 + 0.230795i −0.583487 + 0.0250332i
\(86\) −0.976298 − 1.46365i −0.105277 − 0.157829i
\(87\) −11.1058 −1.19067
\(88\) −0.0947463 − 0.482328i −0.0101000 − 0.0514163i
\(89\) 3.34144 − 1.92918i 0.354192 − 0.204493i −0.312338 − 0.949971i \(-0.601112\pi\)
0.666530 + 0.745478i \(0.267779\pi\)
\(90\) −6.25516 + 4.57157i −0.659352 + 0.481886i
\(91\) −0.188776 − 0.326969i −0.0197891 − 0.0342757i
\(92\) 5.97065 − 14.3499i 0.622483 − 1.49608i
\(93\) −13.1759 − 7.60709i −1.36627 − 0.788819i
\(94\) −8.75389 + 0.566775i −0.902895 + 0.0584584i
\(95\) 8.78015 − 4.23190i 0.900824 − 0.434184i
\(96\) −4.19524 − 12.5220i −0.428175 − 1.27802i
\(97\) 4.28716 − 7.42557i 0.435295 − 0.753953i −0.562025 − 0.827120i \(-0.689977\pi\)
0.997320 + 0.0731676i \(0.0233108\pi\)
\(98\) 3.82780 − 7.74648i 0.386666 − 0.782513i
\(99\) 0.368740 − 0.212892i 0.0370598 − 0.0213965i
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 380.2.s.a.179.20 yes 112
4.3 odd 2 inner 380.2.s.a.179.18 ✓ 112
5.4 even 2 inner 380.2.s.a.179.37 yes 112
19.12 odd 6 inner 380.2.s.a.259.39 yes 112
20.19 odd 2 inner 380.2.s.a.179.39 yes 112
76.31 even 6 inner 380.2.s.a.259.37 yes 112
95.69 odd 6 inner 380.2.s.a.259.18 yes 112
380.259 even 6 inner 380.2.s.a.259.20 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.18 ✓ 112 4.3 odd 2 inner
380.2.s.a.179.20 yes 112 1.1 even 1 trivial
380.2.s.a.179.37 yes 112 5.4 even 2 inner
380.2.s.a.179.39 yes 112 20.19 odd 2 inner
380.2.s.a.259.18 yes 112 95.69 odd 6 inner
380.2.s.a.259.20 yes 112 380.259 even 6 inner
380.2.s.a.259.37 yes 112 76.31 even 6 inner
380.2.s.a.259.39 yes 112 19.12 odd 6 inner