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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 39.5
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.6

$q$-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} +(-9.99932 - 0.116190i) 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} +(19.4269 + 4.75331i) 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} +(-20.5270 - 0.238520i) 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} +(-35.6934 - 18.0549i) 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} +(-48.9520 + 10.1834i) 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} +(39.8804 + 9.75778i) 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} +(58.9655 + 0.685169i) 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} +(61.3599 + 51.3319i) 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} +(47.8553 + 0.556069i) 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 \(-0.388848\pi\)
0.342140 + 0.939649i \(0.388848\pi\)
\(4\) 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 \(-0.638395\pi\)
−0.421211 + 0.906963i \(0.638395\pi\)
\(8\) −6.19521 − 5.06155i −0.774402 − 0.632694i
\(9\) −4.78585 −0.531761
\(10\) −9.99932 − 0.116190i −0.999932 − 0.0116190i
\(11\) 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 \(0.183797\pi\)
−0.837877 + 0.545860i \(0.816203\pi\)
\(14\) 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 \(0.128850\pi\)
−0.919184 + 0.393829i \(0.871150\pi\)
\(18\) 9.32325 + 2.16668i 0.517958 + 0.120371i
\(19\) 4.35890i 0.229416i
\(20\) 19.4269 + 4.75331i 0.971347 + 0.237665i
\(21\) −12.1055 −0.576452
\(22\) 8.29928 − 35.7119i 0.377240 − 1.62327i
\(23\) −10.7531 −0.467526 −0.233763 − 0.972294i \(-0.575104\pi\)
−0.233763 + 0.972294i \(0.575104\pi\)
\(24\) −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 \(0.560209\pi\)
\(30\) −20.5270 − 0.238520i −0.684234 − 0.00795067i
\(31\) 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 \(-0.882415\pi\)
0.932543 − 0.361060i \(-0.117585\pi\)
\(38\) 1.97339 − 8.49151i 0.0519313 − 0.223461i
\(39\) 29.1346i 0.747042i
\(40\) −35.6934 − 18.0549i −0.892335 − 0.451374i
\(41\) 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 \(-0.0920099\pi\)
0.958513 + 0.285049i \(0.0920099\pi\)
\(44\) −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 \(-0.498380\pi\)
0.00509048 + 0.999987i \(0.498380\pi\)
\(48\) 20.0713 + 25.9994i 0.418151 + 0.541654i
\(49\) −14.2260 −0.290326
\(50\) −48.9520 + 10.1834i −0.979040 + 0.203669i
\(51\) 27.4879i 0.538978i
\(52\) −25.0339 + 50.9516i −0.481421 + 0.979839i
\(53\) 55.9368i 1.05541i 0.849427 + 0.527706i \(0.176948\pi\)
−0.849427 + 0.527706i \(0.823052\pi\)
\(54\) 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 \(-0.305473\pi\)
\(60\) 39.8804 + 9.75778i 0.664673 + 0.162630i
\(61\) −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 \(-0.556722\pi\)
−0.177254 + 0.984165i \(0.556722\pi\)
\(68\) −23.6189 + 48.0717i −0.347337 + 0.706937i
\(69\) −22.0744 −0.319919
\(70\) 58.9655 + 0.685169i 0.842365 + 0.00978812i
\(71\) − 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 \(0.367920\pi\)
−0.403136 + 0.915140i \(0.632080\pi\)
\(74\) −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 \(-0.0166039\pi\)
\(80\) 61.3599 + 51.3319i 0.766998 + 0.641649i
\(81\) −15.0230 −0.185469
\(82\) −120.876 − 28.0911i −1.47410 − 0.342574i
\(83\) −43.0532 −0.518714 −0.259357 − 0.965782i \(-0.583511\pi\)
−0.259357 + 0.965782i \(0.583511\pi\)
\(84\) −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 \(0.674587\pi\)
\(90\) 47.8553 + 0.556069i 0.531725 + 0.00617855i
\(91\) − 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 \(0.357704\pi\)
−0.432293 + 0.901733i \(0.642296\pi\)
\(98\) 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.39.5 ✓ 108
4.3 odd 2 inner 380.3.h.a.39.103 yes 108
5.4 even 2 inner 380.3.h.a.39.104 yes 108
20.19 odd 2 inner 380.3.h.a.39.6 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.5 ✓ 108 1.1 even 1 trivial
380.3.h.a.39.6 yes 108 20.19 odd 2 inner
380.3.h.a.39.103 yes 108 4.3 odd 2 inner
380.3.h.a.39.104 yes 108 5.4 even 2 inner