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Properties

Label 380.3.h.a.39.2
Level $380$
Weight $3$
Character 380.39
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.2
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.1

$q$-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+(-1.99911 + 0.0595768i) q^{2} -2.57497 q^{3} +(3.99290 - 0.238201i) q^{4} +(2.22773 + 4.47629i) q^{5} +(5.14765 - 0.153408i) q^{6} -1.63196 q^{7} +(-7.96807 + 0.714075i) q^{8} -2.36954 q^{9} +(-4.72017 - 8.81589i) q^{10} +10.0708i q^{11} +(-10.2816 + 0.613361i) q^{12} -2.27729i q^{13} +(3.26246 - 0.0972267i) q^{14} +(-5.73634 - 11.5263i) q^{15} +(15.8865 - 1.90223i) q^{16} +5.69199i q^{17} +(4.73697 - 0.141169i) q^{18} -4.35890i q^{19} +(9.96138 + 17.3428i) q^{20} +4.20224 q^{21} +(-0.599987 - 20.1327i) q^{22} +7.62452 q^{23} +(20.5175 - 1.83872i) q^{24} +(-15.0744 + 19.9440i) q^{25} +(0.135674 + 4.55256i) q^{26} +29.2762 q^{27} +(-6.51624 + 0.388734i) q^{28} -11.2210 q^{29} +(12.1543 + 22.7006i) q^{30} -4.16767i q^{31} +(-31.6456 + 4.74924i) q^{32} -25.9321i q^{33} +(-0.339111 - 11.3789i) q^{34} +(-3.63556 - 7.30512i) q^{35} +(-9.46133 + 0.564427i) q^{36} -53.1159i q^{37} +(0.259689 + 8.71393i) q^{38} +5.86395i q^{39} +(-20.9471 - 34.0766i) q^{40} -29.2518 q^{41} +(-8.40074 + 0.250356i) q^{42} -52.6182 q^{43} +(2.39888 + 40.2118i) q^{44} +(-5.27869 - 10.6067i) q^{45} +(-15.2423 + 0.454244i) q^{46} -44.4213 q^{47} +(-40.9073 + 4.89818i) q^{48} -46.3367 q^{49} +(28.9473 - 40.7683i) q^{50} -14.6567i q^{51} +(-0.542453 - 9.09299i) q^{52} -47.8562i q^{53} +(-58.5264 + 1.74418i) q^{54} +(-45.0800 + 22.4351i) q^{55} +(13.0035 - 1.16534i) q^{56} +11.2240i q^{57} +(22.4321 - 0.668512i) q^{58} -32.2036i q^{59} +(-25.6502 - 44.6570i) q^{60} -81.8613 q^{61} +(0.248296 + 8.33165i) q^{62} +3.86698 q^{63} +(62.9802 - 11.3796i) q^{64} +(10.1938 - 5.07319i) q^{65} +(1.54495 + 51.8411i) q^{66} -27.8201 q^{67} +(1.35584 + 22.7276i) q^{68} -19.6329 q^{69} +(7.70311 + 14.3872i) q^{70} +66.6953i q^{71} +(18.8806 - 1.69203i) q^{72} -79.9780i q^{73} +(3.16448 + 106.185i) q^{74} +(38.8161 - 51.3551i) q^{75} +(-1.03830 - 17.4047i) q^{76} -16.4352i q^{77} +(-0.349355 - 11.7227i) q^{78} -87.0019i q^{79} +(43.9059 + 66.8751i) q^{80} -54.0595 q^{81} +(58.4776 - 1.74273i) q^{82} +55.1557 q^{83} +(16.7791 - 1.00098i) q^{84} +(-25.4790 + 12.6802i) q^{85} +(105.190 - 3.13482i) q^{86} +28.8938 q^{87} +(-7.19133 - 80.2451i) q^{88} +23.7889 q^{89} +(11.1846 + 20.8896i) q^{90} +3.71644i q^{91} +(30.4440 - 1.81617i) q^{92} +10.7316i q^{93} +(88.8032 - 2.64648i) q^{94} +(19.5117 - 9.71046i) q^{95} +(81.4865 - 12.2291i) q^{96} +173.823i q^{97} +(92.6323 - 2.76059i) q^{98} -23.8632i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99911 + 0.0595768i −0.999556 + 0.0297884i
\(3\) −2.57497 −0.858323 −0.429161 − 0.903228i \(-0.641191\pi\)
−0.429161 + 0.903228i \(0.641191\pi\)
\(4\) 3.99290 − 0.238201i 0.998225 − 0.0595503i
\(5\) 2.22773 + 4.47629i 0.445547 + 0.895259i
\(6\) 5.14765 − 0.153408i 0.857942 − 0.0255680i
\(7\) −1.63196 −0.233137 −0.116568 − 0.993183i \(-0.537189\pi\)
−0.116568 + 0.993183i \(0.537189\pi\)
\(8\) −7.96807 + 0.714075i −0.996008 + 0.0892594i
\(9\) −2.36954 −0.263282
\(10\) −4.72017 − 8.81589i −0.472017 − 0.881589i
\(11\) 10.0708i 0.915530i 0.889073 + 0.457765i \(0.151350\pi\)
−0.889073 + 0.457765i \(0.848650\pi\)
\(12\) −10.2816 + 0.613361i −0.856800 + 0.0511134i
\(13\) − 2.27729i − 0.175176i −0.996157 − 0.0875881i \(-0.972084\pi\)
0.996157 − 0.0875881i \(-0.0279159\pi\)
\(14\) 3.26246 − 0.0972267i 0.233033 − 0.00694476i
\(15\) −5.73634 − 11.5263i −0.382423 − 0.768421i
\(16\) 15.8865 − 1.90223i 0.992908 − 0.118889i
\(17\) 5.69199i 0.334823i 0.985887 + 0.167412i \(0.0535409\pi\)
−0.985887 + 0.167412i \(0.946459\pi\)
\(18\) 4.73697 − 0.141169i 0.263165 − 0.00784274i
\(19\) − 4.35890i − 0.229416i
\(20\) 9.96138 + 17.3428i 0.498069 + 0.867138i
\(21\) 4.20224 0.200106
\(22\) −0.599987 − 20.1327i −0.0272722 − 0.915124i
\(23\) 7.62452 0.331501 0.165751 − 0.986168i \(-0.446995\pi\)
0.165751 + 0.986168i \(0.446995\pi\)
\(24\) 20.5175 − 1.83872i 0.854897 − 0.0766134i
\(25\) −15.0744 + 19.9440i −0.602977 + 0.797759i
\(26\) 0.135674 + 4.55256i 0.00521821 + 0.175098i
\(27\) 29.2762 1.08430
\(28\) −6.51624 + 0.388734i −0.232723 + 0.0138834i
\(29\) −11.2210 −0.386932 −0.193466 − 0.981107i \(-0.561973\pi\)
−0.193466 + 0.981107i \(0.561973\pi\)
\(30\) 12.1543 + 22.7006i 0.405143 + 0.756688i
\(31\) − 4.16767i − 0.134441i −0.997738 − 0.0672205i \(-0.978587\pi\)
0.997738 − 0.0672205i \(-0.0214131\pi\)
\(32\) −31.6456 + 4.74924i −0.988925 + 0.148414i
\(33\) − 25.9321i − 0.785820i
\(34\) −0.339111 − 11.3789i −0.00997384 − 0.334675i
\(35\) −3.63556 − 7.30512i −0.103873 − 0.208718i
\(36\) −9.46133 + 0.564427i −0.262815 + 0.0156785i
\(37\) − 53.1159i − 1.43557i −0.696267 − 0.717783i \(-0.745157\pi\)
0.696267 − 0.717783i \(-0.254843\pi\)
\(38\) 0.259689 + 8.71393i 0.00683392 + 0.229314i
\(39\) 5.86395i 0.150358i
\(40\) −20.9471 − 34.0766i −0.523678 − 0.851916i
\(41\) −29.2518 −0.713458 −0.356729 − 0.934208i \(-0.616108\pi\)
−0.356729 + 0.934208i \(0.616108\pi\)
\(42\) −8.40074 + 0.250356i −0.200018 + 0.00596085i
\(43\) −52.6182 −1.22368 −0.611840 − 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(44\) 2.39888 + 40.2118i 0.0545201 + 0.913905i
\(45\) −5.27869 − 10.6067i −0.117304 − 0.235705i
\(46\) −15.2423 + 0.454244i −0.331354 + 0.00987488i
\(47\) −44.4213 −0.945134 −0.472567 − 0.881295i \(-0.656673\pi\)
−0.472567 + 0.881295i \(0.656673\pi\)
\(48\) −40.9073 + 4.89818i −0.852235 + 0.102045i
\(49\) −46.3367 −0.945647
\(50\) 28.9473 − 40.7683i 0.578945 − 0.815367i
\(51\) − 14.6567i − 0.287386i
\(52\) −0.542453 − 9.09299i −0.0104318 − 0.174865i
\(53\) − 47.8562i − 0.902947i −0.892285 − 0.451473i \(-0.850899\pi\)
0.892285 − 0.451473i \(-0.149101\pi\)
\(54\) −58.5264 + 1.74418i −1.08382 + 0.0322996i
\(55\) −45.0800 + 22.4351i −0.819636 + 0.407911i
\(56\) 13.0035 − 1.16534i 0.232206 − 0.0208096i
\(57\) 11.2240i 0.196913i
\(58\) 22.4321 − 0.668512i 0.386760 − 0.0115261i
\(59\) − 32.2036i − 0.545824i −0.962039 − 0.272912i \(-0.912013\pi\)
0.962039 − 0.272912i \(-0.0879867\pi\)
\(60\) −25.6502 − 44.6570i −0.427504 − 0.744284i
\(61\) −81.8613 −1.34199 −0.670994 − 0.741463i \(-0.734132\pi\)
−0.670994 + 0.741463i \(0.734132\pi\)
\(62\) 0.248296 + 8.33165i 0.00400478 + 0.134381i
\(63\) 3.86698 0.0613806
\(64\) 62.9802 − 11.3796i 0.984066 − 0.177806i
\(65\) 10.1938 − 5.07319i 0.156828 − 0.0780491i
\(66\) 1.54495 + 51.8411i 0.0234083 + 0.785472i
\(67\) −27.8201 −0.415226 −0.207613 − 0.978211i \(-0.566569\pi\)
−0.207613 + 0.978211i \(0.566569\pi\)
\(68\) 1.35584 + 22.7276i 0.0199388 + 0.334229i
\(69\) −19.6329 −0.284535
\(70\) 7.70311 + 14.3872i 0.110044 + 0.205531i
\(71\) 66.6953i 0.939370i 0.882834 + 0.469685i \(0.155633\pi\)
−0.882834 + 0.469685i \(0.844367\pi\)
\(72\) 18.8806 − 1.69203i 0.262231 − 0.0235004i
\(73\) − 79.9780i − 1.09559i −0.836613 − 0.547794i \(-0.815468\pi\)
0.836613 − 0.547794i \(-0.184532\pi\)
\(74\) 3.16448 + 106.185i 0.0427632 + 1.43493i
\(75\) 38.8161 − 51.3551i 0.517549 − 0.684735i
\(76\) −1.03830 − 17.4047i −0.0136618 − 0.229009i
\(77\) − 16.4352i − 0.213444i
\(78\) −0.349355 − 11.7227i −0.00447891 − 0.150291i
\(79\) − 87.0019i − 1.10129i −0.834740 − 0.550645i \(-0.814382\pi\)
0.834740 − 0.550645i \(-0.185618\pi\)
\(80\) 43.9059 + 66.8751i 0.548823 + 0.835938i
\(81\) −54.0595 −0.667401
\(82\) 58.4776 − 1.74273i 0.713142 − 0.0212528i
\(83\) 55.1557 0.664527 0.332264 − 0.943187i \(-0.392188\pi\)
0.332264 + 0.943187i \(0.392188\pi\)
\(84\) 16.7791 − 1.00098i 0.199751 − 0.0119164i
\(85\) −25.4790 + 12.6802i −0.299753 + 0.149179i
\(86\) 105.190 − 3.13482i 1.22314 − 0.0364514i
\(87\) 28.8938 0.332112
\(88\) −7.19133 − 80.2451i −0.0817197 − 0.911876i
\(89\) 23.7889 0.267291 0.133645 − 0.991029i \(-0.457332\pi\)
0.133645 + 0.991029i \(0.457332\pi\)
\(90\) 11.1846 + 20.8896i 0.124274 + 0.232107i
\(91\) 3.71644i 0.0408400i
\(92\) 30.4440 − 1.81617i 0.330913 − 0.0197410i
\(93\) 10.7316i 0.115394i
\(94\) 88.8032 − 2.64648i 0.944715 − 0.0281540i
\(95\) 19.5117 − 9.71046i 0.205386 − 0.102215i
\(96\) 81.4865 − 12.2291i 0.848817 − 0.127387i
\(97\) 173.823i 1.79199i 0.444061 + 0.895996i \(0.353537\pi\)
−0.444061 + 0.895996i \(0.646463\pi\)
\(98\) 92.6323 − 2.76059i 0.945228 − 0.0281693i
\(99\) − 23.8632i − 0.241042i
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.h.a.39.2 yes 108
4.3 odd 2 inner 380.3.h.a.39.108 yes 108
5.4 even 2 inner 380.3.h.a.39.107 yes 108
20.19 odd 2 inner 380.3.h.a.39.1 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.1 ✓ 108 20.19 odd 2 inner
380.3.h.a.39.2 yes 108 1.1 even 1 trivial
380.3.h.a.39.107 yes 108 5.4 even 2 inner
380.3.h.a.39.108 yes 108 4.3 odd 2 inner