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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 39.104
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.103

$q$-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+(1.94809 + 0.452727i) q^{2} -2.05284 q^{3} +(3.59008 + 1.76390i) q^{4} +(4.88304 + 1.07515i) q^{5} +(-3.99911 - 0.929375i) q^{6} +5.89695 q^{7} +(6.19521 + 5.06155i) q^{8} -4.78585 q^{9} +(9.02582 + 4.30517i) q^{10} +18.3318i q^{11} +(-7.36985 - 3.62100i) q^{12} -14.1924i q^{13} +(11.4878 + 2.66971i) q^{14} +(-10.0241 - 2.20712i) q^{15} +(9.77731 + 12.6651i) q^{16} -13.3902i q^{17} +(-9.32325 - 2.16668i) q^{18} +4.35890i q^{19} +(15.6340 + 12.4731i) q^{20} -12.1055 q^{21} +(-8.29928 + 35.7119i) q^{22} +10.7531 q^{23} +(-12.7178 - 10.3906i) q^{24} +(22.6881 + 10.5000i) q^{25} +(6.42525 - 27.6479i) q^{26} +28.3001 q^{27} +(21.1705 + 10.4016i) q^{28} -10.9056 q^{29} +(-18.5286 - 8.83782i) q^{30} +27.5721i q^{31} +(13.3132 + 29.0991i) q^{32} -37.6322i q^{33} +(6.06209 - 26.0852i) q^{34} +(28.7950 + 6.34012i) q^{35} +(-17.1816 - 8.44176i) q^{36} +26.7185i q^{37} +(-1.97339 + 8.49151i) q^{38} +29.1346i q^{39} +(24.8095 + 31.3765i) q^{40} +62.0487 q^{41} +(-23.5825 - 5.48048i) q^{42} -82.4321 q^{43} +(-32.3354 + 65.8125i) q^{44} +(-23.3695 - 5.14552i) q^{45} +(20.9480 + 4.86822i) q^{46} -0.478505 q^{47} +(-20.0713 - 25.9994i) q^{48} -14.2260 q^{49} +(39.4447 + 30.7264i) q^{50} +27.4879i q^{51} +(25.0339 - 50.9516i) q^{52} -55.9368i q^{53} +(55.1311 + 12.8122i) q^{54} +(-19.7095 + 89.5147i) q^{55} +(36.5329 + 29.8477i) q^{56} -8.94812i q^{57} +(-21.2450 - 4.93725i) q^{58} -96.6424i q^{59} +(-32.0941 - 25.6052i) q^{60} -1.94165 q^{61} +(-12.4826 + 53.7128i) q^{62} -28.2219 q^{63} +(12.7614 + 62.7148i) q^{64} +(15.2589 - 69.3018i) q^{65} +(17.0371 - 73.3108i) q^{66} +23.7521 q^{67} +(23.6189 - 48.0717i) q^{68} -22.0744 q^{69} +(53.2248 + 25.3874i) q^{70} -61.2979i q^{71} +(-29.6494 - 24.2238i) q^{72} -133.610i q^{73} +(-12.0962 + 52.0498i) q^{74} +(-46.5750 - 21.5549i) q^{75} +(-7.68866 + 15.6488i) q^{76} +108.102i q^{77} +(-13.1900 + 56.7568i) q^{78} -8.23795i q^{79} +(34.1261 + 72.3561i) q^{80} -15.0230 q^{81} +(120.876 + 28.0911i) q^{82} +43.0532 q^{83} +(-43.4597 - 21.3529i) q^{84} +(14.3965 - 65.3847i) q^{85} +(-160.585 - 37.3192i) q^{86} +22.3874 q^{87} +(-92.7873 + 113.569i) q^{88} -92.8078 q^{89} +(-43.1962 - 20.6039i) q^{90} -83.6916i q^{91} +(38.6045 + 18.9674i) q^{92} -56.6010i q^{93} +(-0.932169 - 0.216632i) q^{94} +(-4.68648 + 21.2847i) q^{95} +(-27.3299 - 59.7358i) q^{96} -174.936i q^{97} +(-27.7134 - 6.44048i) q^{98} -87.7331i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94809 + 0.452727i 0.974043 + 0.226363i
\(3\) −2.05284 −0.684280 −0.342140 − 0.939649i \(-0.611152\pi\)
−0.342140 + 0.939649i \(0.611152\pi\)
\(4\) 3.59008 + 1.76390i 0.897519 + 0.440975i
\(5\) 4.88304 + 1.07515i 0.976607 + 0.215031i
\(6\) −3.99911 − 0.929375i −0.666518 − 0.154896i
\(7\) 5.89695 0.842421 0.421211 − 0.906963i \(-0.361605\pi\)
0.421211 + 0.906963i \(0.361605\pi\)
\(8\) 6.19521 + 5.06155i 0.774402 + 0.632694i
\(9\) −4.78585 −0.531761
\(10\) 9.02582 + 4.30517i 0.902582 + 0.430517i
\(11\) 18.3318i 1.66653i 0.552877 + 0.833263i \(0.313530\pi\)
−0.552877 + 0.833263i \(0.686470\pi\)
\(12\) −7.36985 − 3.62100i −0.614154 − 0.301750i
\(13\) − 14.1924i − 1.09172i −0.837877 − 0.545860i \(-0.816203\pi\)
0.837877 − 0.545860i \(-0.183797\pi\)
\(14\) 11.4878 + 2.66971i 0.820555 + 0.190693i
\(15\) −10.0241 − 2.20712i −0.668273 − 0.147141i
\(16\) 9.77731 + 12.6651i 0.611082 + 0.791567i
\(17\) − 13.3902i − 0.787657i −0.919184 − 0.393829i \(-0.871150\pi\)
0.919184 − 0.393829i \(-0.128850\pi\)
\(18\) −9.32325 − 2.16668i −0.517958 − 0.120371i
\(19\) 4.35890i 0.229416i
\(20\) 15.6340 + 12.4731i 0.781701 + 0.623654i
\(21\) −12.1055 −0.576452
\(22\) −8.29928 + 35.7119i −0.377240 + 1.62327i
\(23\) 10.7531 0.467526 0.233763 − 0.972294i \(-0.424896\pi\)
0.233763 + 0.972294i \(0.424896\pi\)
\(24\) −12.7178 − 10.3906i −0.529908 − 0.432940i
\(25\) 22.6881 + 10.5000i 0.907524 + 0.420001i
\(26\) 6.42525 − 27.6479i 0.247125 − 1.06338i
\(27\) 28.3001 1.04815
\(28\) 21.1705 + 10.4016i 0.756090 + 0.371487i
\(29\) −10.9056 −0.376055 −0.188027 − 0.982164i \(-0.560209\pi\)
−0.188027 + 0.982164i \(0.560209\pi\)
\(30\) −18.5286 − 8.83782i −0.617619 − 0.294594i
\(31\) 27.5721i 0.889422i 0.895674 + 0.444711i \(0.146694\pi\)
−0.895674 + 0.444711i \(0.853306\pi\)
\(32\) 13.3132 + 29.0991i 0.416038 + 0.909347i
\(33\) − 37.6322i − 1.14037i
\(34\) 6.06209 − 26.0852i 0.178297 − 0.767212i
\(35\) 28.7950 + 6.34012i 0.822715 + 0.181146i
\(36\) −17.1816 − 8.44176i −0.477266 − 0.234493i
\(37\) 26.7185i 0.722120i 0.932543 + 0.361060i \(0.117585\pi\)
−0.932543 + 0.361060i \(0.882415\pi\)
\(38\) −1.97339 + 8.49151i −0.0519313 + 0.223461i
\(39\) 29.1346i 0.747042i
\(40\) 24.8095 + 31.3765i 0.620238 + 0.784414i
\(41\) 62.0487 1.51338 0.756692 − 0.653772i \(-0.226814\pi\)
0.756692 + 0.653772i \(0.226814\pi\)
\(42\) −23.5825 − 5.48048i −0.561489 − 0.130488i
\(43\) −82.4321 −1.91703 −0.958513 − 0.285049i \(-0.907990\pi\)
−0.958513 + 0.285049i \(0.907990\pi\)
\(44\) −32.3354 + 65.8125i −0.734896 + 1.49574i
\(45\) −23.3695 − 5.14552i −0.519322 − 0.114345i
\(46\) 20.9480 + 4.86822i 0.455391 + 0.105831i
\(47\) −0.478505 −0.0101810 −0.00509048 − 0.999987i \(-0.501620\pi\)
−0.00509048 + 0.999987i \(0.501620\pi\)
\(48\) −20.0713 − 25.9994i −0.418151 − 0.541654i
\(49\) −14.2260 −0.290326
\(50\) 39.4447 + 30.7264i 0.788894 + 0.614529i
\(51\) 27.4879i 0.538978i
\(52\) 25.0339 − 50.9516i 0.481421 − 0.979839i
\(53\) − 55.9368i − 1.05541i −0.849427 − 0.527706i \(-0.823052\pi\)
0.849427 − 0.527706i \(-0.176948\pi\)
\(54\) 55.1311 + 12.8122i 1.02095 + 0.237263i
\(55\) −19.7095 + 89.5147i −0.358354 + 1.62754i
\(56\) 36.5329 + 29.8477i 0.652373 + 0.532995i
\(57\) − 8.94812i − 0.156985i
\(58\) −21.2450 − 4.93725i −0.366294 − 0.0851250i
\(59\) − 96.6424i − 1.63801i −0.573789 − 0.819003i \(-0.694527\pi\)
0.573789 − 0.819003i \(-0.305473\pi\)
\(60\) −32.0941 − 25.6052i −0.534902 − 0.426754i
\(61\) −1.94165 −0.0318303 −0.0159151 − 0.999873i \(-0.505066\pi\)
−0.0159151 + 0.999873i \(0.505066\pi\)
\(62\) −12.4826 + 53.7128i −0.201332 + 0.866335i
\(63\) −28.2219 −0.447967
\(64\) 12.7614 + 62.7148i 0.199396 + 0.979919i
\(65\) 15.2589 − 69.3018i 0.234753 − 1.06618i
\(66\) 17.0371 − 73.3108i 0.258138 − 1.11077i
\(67\) 23.7521 0.354509 0.177254 − 0.984165i \(-0.443278\pi\)
0.177254 + 0.984165i \(0.443278\pi\)
\(68\) 23.6189 − 48.0717i 0.347337 − 0.706937i
\(69\) −22.0744 −0.319919
\(70\) 53.2248 + 25.3874i 0.760355 + 0.362677i
\(71\) − 61.2979i − 0.863351i −0.902029 − 0.431676i \(-0.857923\pi\)
0.902029 − 0.431676i \(-0.142077\pi\)
\(72\) −29.6494 − 24.2238i −0.411797 − 0.336442i
\(73\) − 133.610i − 1.83028i −0.403136 − 0.915140i \(-0.632080\pi\)
0.403136 − 0.915140i \(-0.367920\pi\)
\(74\) −12.0962 + 52.0498i −0.163462 + 0.703376i
\(75\) −46.5750 − 21.5549i −0.621000 − 0.287398i
\(76\) −7.68866 + 15.6488i −0.101167 + 0.205905i
\(77\) 108.102i 1.40392i
\(78\) −13.1900 + 56.7568i −0.169103 + 0.727651i
\(79\) − 8.23795i − 0.104278i −0.998640 − 0.0521390i \(-0.983396\pi\)
0.998640 − 0.0521390i \(-0.0166039\pi\)
\(80\) 34.1261 + 72.3561i 0.426576 + 0.904452i
\(81\) −15.0230 −0.185469
\(82\) 120.876 + 28.0911i 1.47410 + 0.342574i
\(83\) 43.0532 0.518714 0.259357 − 0.965782i \(-0.416489\pi\)
0.259357 + 0.965782i \(0.416489\pi\)
\(84\) −43.4597 − 21.3529i −0.517377 − 0.254201i
\(85\) 14.3965 − 65.3847i 0.169370 − 0.769232i
\(86\) −160.585 − 37.3192i −1.86727 − 0.433944i
\(87\) 22.3874 0.257327
\(88\) −92.7873 + 113.569i −1.05440 + 1.29056i
\(89\) −92.8078 −1.04278 −0.521392 − 0.853317i \(-0.674587\pi\)
−0.521392 + 0.853317i \(0.674587\pi\)
\(90\) −43.1962 − 20.6039i −0.479958 − 0.228932i
\(91\) − 83.6916i − 0.919688i
\(92\) 38.6045 + 18.9674i 0.419614 + 0.206167i
\(93\) − 56.6010i − 0.608613i
\(94\) −0.932169 − 0.216632i −0.00991669 − 0.00230460i
\(95\) −4.68648 + 21.2847i −0.0493314 + 0.224049i
\(96\) −27.3299 − 59.7358i −0.284687 − 0.622248i
\(97\) − 174.936i − 1.80347i −0.432293 − 0.901733i \(-0.642296\pi\)
0.432293 − 0.901733i \(-0.357704\pi\)
\(98\) −27.7134 − 6.44048i −0.282790 − 0.0657192i
\(99\) − 87.7331i − 0.886193i
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.39.104 yes 108
4.3 odd 2 inner 380.3.h.a.39.6 yes 108
5.4 even 2 inner 380.3.h.a.39.5 ✓ 108
20.19 odd 2 inner 380.3.h.a.39.103 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.5 ✓ 108 5.4 even 2 inner
380.3.h.a.39.6 yes 108 4.3 odd 2 inner
380.3.h.a.39.103 yes 108 20.19 odd 2 inner
380.3.h.a.39.104 yes 108 1.1 even 1 trivial