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Properties

Label 380.3.j.a.227.1
Level $380$
Weight $3$
Character 380.227
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), 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: \(232\)
Relative dimension: \(116\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.1
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.1

$q$-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+(-2.00000 + 0.00413025i) q^{2} +(-2.70754 + 2.70754i) q^{3} +(3.99997 - 0.0165209i) q^{4} +(-4.82977 - 1.29359i) q^{5} +(5.40389 - 5.42626i) q^{6} +(2.48709 - 2.48709i) q^{7} +(-7.99985 + 0.0495627i) q^{8} -5.66157i q^{9} +(9.66485 + 2.56722i) q^{10} -11.8674i q^{11} +(-10.7853 + 10.8748i) q^{12} +(12.4627 + 12.4627i) q^{13} +(-4.96391 + 4.98445i) q^{14} +(16.5792 - 9.57436i) q^{15} +(15.9995 - 0.132166i) q^{16} +(7.74445 + 7.74445i) q^{17} +(0.0233837 + 11.3231i) q^{18} +(-1.87967 - 18.9068i) q^{19} +(-19.3403 - 5.09451i) q^{20} +13.4678i q^{21} +(0.0490154 + 23.7348i) q^{22} +(7.81187 + 7.81187i) q^{23} +(21.5257 - 21.7941i) q^{24} +(21.6533 + 12.4954i) q^{25} +(-24.9769 - 24.8740i) q^{26} +(-9.03894 - 9.03894i) q^{27} +(9.90720 - 9.98938i) q^{28} -21.5360 q^{29} +(-33.1188 + 19.2171i) q^{30} +5.89571 q^{31} +(-31.9983 + 0.330414i) q^{32} +(32.1316 + 32.1316i) q^{33} +(-15.5208 - 15.4569i) q^{34} +(-15.2294 + 8.79481i) q^{35} +(-0.0935345 - 22.6461i) q^{36} +(-15.0638 + 15.0638i) q^{37} +(3.83743 + 37.8057i) q^{38} -67.4868 q^{39} +(38.7015 + 10.1091i) q^{40} +29.7784i q^{41} +(-0.0556254 - 26.9356i) q^{42} +(7.88534 + 7.88534i) q^{43} +(-0.196061 - 47.4693i) q^{44} +(-7.32373 + 27.3441i) q^{45} +(-15.6560 - 15.5914i) q^{46} +(-53.8523 + 53.8523i) q^{47} +(-42.9614 + 43.6770i) q^{48} +36.6287i q^{49} +(-43.3581 - 24.9014i) q^{50} -41.9368 q^{51} +(50.0564 + 49.6446i) q^{52} +(-6.60684 - 6.60684i) q^{53} +(18.1152 + 18.0405i) q^{54} +(-15.3515 + 57.3169i) q^{55} +(-19.7731 + 20.0196i) q^{56} +(56.2802 + 46.1016i) q^{57} +(43.0719 - 0.0889489i) q^{58} -6.08132i q^{59} +(66.1582 - 38.5710i) q^{60} -5.37083 q^{61} +(-11.7914 + 0.0243507i) q^{62} +(-14.0809 - 14.0809i) q^{63} +(63.9951 - 0.792987i) q^{64} +(-44.0705 - 76.3138i) q^{65} +(-64.3957 - 64.1303i) q^{66} +(29.7010 + 29.7010i) q^{67} +(31.1055 + 30.8496i) q^{68} -42.3019 q^{69} +(30.4223 - 17.6525i) q^{70} +55.3067 q^{71} +(0.280602 + 45.2917i) q^{72} +(-98.6691 + 98.6691i) q^{73} +(30.0653 - 30.1897i) q^{74} +(-92.4591 + 24.7952i) q^{75} +(-7.83098 - 75.5955i) q^{76} +(-29.5154 - 29.5154i) q^{77} +(134.973 - 0.278737i) q^{78} +87.2874i q^{79} +(-77.4446 - 20.0583i) q^{80} +99.9008 q^{81} +(-0.122992 - 59.5566i) q^{82} +(57.6920 + 57.6920i) q^{83} +(0.222501 + 53.8708i) q^{84} +(-27.3858 - 47.4220i) q^{85} +(-15.8032 - 15.7381i) q^{86} +(58.3096 - 58.3096i) q^{87} +(0.588181 + 94.9376i) q^{88} -107.902 q^{89} +(14.5345 - 54.7182i) q^{90} +61.9920 q^{91} +(31.3763 + 31.1181i) q^{92} +(-15.9629 + 15.9629i) q^{93} +(107.482 - 107.927i) q^{94} +(-15.3792 + 93.7469i) q^{95} +(85.7421 - 87.5313i) q^{96} +(8.54986 - 8.54986i) q^{97} +(-0.151286 - 73.2573i) q^{98} -67.1883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 + 0.00413025i −0.999998 + 0.00206512i
\(3\) −2.70754 + 2.70754i −0.902514 + 0.902514i −0.995653 − 0.0931391i \(-0.970310\pi\)
0.0931391 + 0.995653i \(0.470310\pi\)
\(4\) 3.99997 − 0.0165209i 0.999991 − 0.00413024i
\(5\) −4.82977 − 1.29359i −0.965953 − 0.258717i
\(6\) 5.40389 − 5.42626i 0.900648 − 0.904376i
\(7\) 2.48709 − 2.48709i 0.355299 − 0.355299i −0.506778 − 0.862077i \(-0.669163\pi\)
0.862077 + 0.506778i \(0.169163\pi\)
\(8\) −7.99985 + 0.0495627i −0.999981 + 0.00619533i
\(9\) − 5.66157i − 0.629063i
\(10\) 9.66485 + 2.56722i 0.966485 + 0.256722i
\(11\) − 11.8674i − 1.07886i −0.842031 − 0.539429i \(-0.818640\pi\)
0.842031 − 0.539429i \(-0.181360\pi\)
\(12\) −10.7853 + 10.8748i −0.898779 + 0.906234i
\(13\) 12.4627 + 12.4627i 0.958673 + 0.958673i 0.999179 − 0.0405068i \(-0.0128972\pi\)
−0.0405068 + 0.999179i \(0.512897\pi\)
\(14\) −4.96391 + 4.98445i −0.354565 + 0.356032i
\(15\) 16.5792 − 9.57436i 1.10528 − 0.638290i
\(16\) 15.9995 − 0.132166i 0.999966 − 0.00826040i
\(17\) 7.74445 + 7.74445i 0.455556 + 0.455556i 0.897193 − 0.441638i \(-0.145602\pi\)
−0.441638 + 0.897193i \(0.645602\pi\)
\(18\) 0.0233837 + 11.3231i 0.00129909 + 0.629062i
\(19\) −1.87967 − 18.9068i −0.0989301 − 0.995094i
\(20\) −19.3403 − 5.09451i −0.967013 − 0.254725i
\(21\) 13.4678i 0.641325i
\(22\) 0.0490154 + 23.7348i 0.00222797 + 1.07885i
\(23\) 7.81187 + 7.81187i 0.339646 + 0.339646i 0.856234 − 0.516588i \(-0.172798\pi\)
−0.516588 + 0.856234i \(0.672798\pi\)
\(24\) 21.5257 − 21.7941i 0.896905 − 0.908088i
\(25\) 21.6533 + 12.4954i 0.866131 + 0.499817i
\(26\) −24.9769 − 24.8740i −0.960650 − 0.956691i
\(27\) −9.03894 − 9.03894i −0.334776 − 0.334776i
\(28\) 9.90720 − 9.98938i 0.353829 − 0.356764i
\(29\) −21.5360 −0.742620 −0.371310 − 0.928509i \(-0.621091\pi\)
−0.371310 + 0.928509i \(0.621091\pi\)
\(30\) −33.1188 + 19.2171i −1.10396 + 0.640572i
\(31\) 5.89571 0.190184 0.0950922 − 0.995468i \(-0.469685\pi\)
0.0950922 + 0.995468i \(0.469685\pi\)
\(32\) −31.9983 + 0.330414i −0.999947 + 0.0103254i
\(33\) 32.1316 + 32.1316i 0.973684 + 0.973684i
\(34\) −15.5208 − 15.4569i −0.456496 − 0.454614i
\(35\) −15.2294 + 8.79481i −0.435124 + 0.251280i
\(36\) −0.0935345 − 22.6461i −0.00259818 − 0.629058i
\(37\) −15.0638 + 15.0638i −0.407129 + 0.407129i −0.880736 − 0.473607i \(-0.842952\pi\)
0.473607 + 0.880736i \(0.342952\pi\)
\(38\) 3.83743 + 37.8057i 0.100985 + 0.994888i
\(39\) −67.4868 −1.73043
\(40\) 38.7015 + 10.1091i 0.967537 + 0.252728i
\(41\) 29.7784i 0.726302i 0.931730 + 0.363151i \(0.118299\pi\)
−0.931730 + 0.363151i \(0.881701\pi\)
\(42\) −0.0556254 − 26.9356i −0.00132441 − 0.641324i
\(43\) 7.88534 + 7.88534i 0.183380 + 0.183380i 0.792827 − 0.609447i \(-0.208608\pi\)
−0.609447 + 0.792827i \(0.708608\pi\)
\(44\) −0.196061 − 47.4693i −0.00445594 − 1.07885i
\(45\) −7.32373 + 27.3441i −0.162749 + 0.607646i
\(46\) −15.6560 − 15.5914i −0.340347 − 0.338944i
\(47\) −53.8523 + 53.8523i −1.14579 + 1.14579i −0.158421 + 0.987372i \(0.550640\pi\)
−0.987372 + 0.158421i \(0.949360\pi\)
\(48\) −42.9614 + 43.6770i −0.895028 + 0.909938i
\(49\) 36.6287i 0.747525i
\(50\) −43.3581 − 24.9014i −0.867161 − 0.498028i
\(51\) −41.9368 −0.822291
\(52\) 50.0564 + 49.6446i 0.962624 + 0.954705i
\(53\) −6.60684 − 6.60684i −0.124657 − 0.124657i 0.642026 − 0.766683i \(-0.278094\pi\)
−0.766683 + 0.642026i \(0.778094\pi\)
\(54\) 18.1152 + 18.0405i 0.335466 + 0.334084i
\(55\) −15.3515 + 57.3169i −0.279119 + 1.04213i
\(56\) −19.7731 + 20.0196i −0.353091 + 0.357494i
\(57\) 56.2802 + 46.1016i 0.987373 + 0.808801i
\(58\) 43.0719 − 0.0889489i 0.742619 − 0.00153360i
\(59\) − 6.08132i − 0.103073i −0.998671 − 0.0515366i \(-0.983588\pi\)
0.998671 − 0.0515366i \(-0.0164119\pi\)
\(60\) 66.1582 − 38.5710i 1.10264 − 0.642850i
\(61\) −5.37083 −0.0880464 −0.0440232 − 0.999031i \(-0.514018\pi\)
−0.0440232 + 0.999031i \(0.514018\pi\)
\(62\) −11.7914 + 0.0243507i −0.190184 + 0.000392754i
\(63\) −14.0809 − 14.0809i −0.223506 − 0.223506i
\(64\) 63.9951 − 0.792987i 0.999923 − 0.0123904i
\(65\) −44.0705 − 76.3138i −0.678008 − 1.17406i
\(66\) −64.3957 − 64.1303i −0.975692 − 0.971671i
\(67\) 29.7010 + 29.7010i 0.443298 + 0.443298i 0.893119 − 0.449821i \(-0.148512\pi\)
−0.449821 + 0.893119i \(0.648512\pi\)
\(68\) 31.1055 + 30.8496i 0.457433 + 0.453670i
\(69\) −42.3019 −0.613071
\(70\) 30.4223 − 17.6525i 0.434605 − 0.252178i
\(71\) 55.3067 0.778967 0.389484 − 0.921033i \(-0.372653\pi\)
0.389484 + 0.921033i \(0.372653\pi\)
\(72\) 0.280602 + 45.2917i 0.00389726 + 0.629051i
\(73\) −98.6691 + 98.6691i −1.35163 + 1.35163i −0.467794 + 0.883838i \(0.654951\pi\)
−0.883838 + 0.467794i \(0.845049\pi\)
\(74\) 30.0653 − 30.1897i 0.406287 − 0.407969i
\(75\) −92.4591 + 24.7952i −1.23279 + 0.330603i
\(76\) −7.83098 − 75.5955i −0.103039 − 0.994677i
\(77\) −29.5154 − 29.5154i −0.383317 − 0.383317i
\(78\) 134.973 − 0.278737i 1.73043 − 0.00357355i
\(79\) 87.2874i 1.10490i 0.833545 + 0.552452i \(0.186308\pi\)
−0.833545 + 0.552452i \(0.813692\pi\)
\(80\) −77.4446 − 20.0583i −0.968057 − 0.250729i
\(81\) 99.9008 1.23334
\(82\) −0.122992 − 59.5566i −0.00149990 − 0.726300i
\(83\) 57.6920 + 57.6920i 0.695084 + 0.695084i 0.963346 − 0.268262i \(-0.0864493\pi\)
−0.268262 + 0.963346i \(0.586449\pi\)
\(84\) 0.222501 + 53.8708i 0.00264882 + 0.641320i
\(85\) −27.3858 − 47.4220i −0.322185 − 0.557906i
\(86\) −15.8032 − 15.7381i −0.183758 − 0.183001i
\(87\) 58.3096 − 58.3096i 0.670225 − 0.670225i
\(88\) 0.588181 + 94.9376i 0.00668388 + 1.07884i
\(89\) −107.902 −1.21238 −0.606192 − 0.795318i \(-0.707304\pi\)
−0.606192 + 0.795318i \(0.707304\pi\)
\(90\) 14.5345 − 54.7182i 0.161494 − 0.607980i
\(91\) 61.9920 0.681231
\(92\) 31.3763 + 31.1181i 0.341046 + 0.338241i
\(93\) −15.9629 + 15.9629i −0.171644 + 0.171644i
\(94\) 107.482 − 107.927i 1.14342 − 1.14816i
\(95\) −15.3792 + 93.7469i −0.161886 + 0.986809i
\(96\) 85.7421 − 87.5313i 0.893147 − 0.911785i
\(97\) 8.54986 − 8.54986i 0.0881429 − 0.0881429i −0.661661 − 0.749803i \(-0.730148\pi\)
0.749803 + 0.661661i \(0.230148\pi\)
\(98\) −0.151286 − 73.2573i −0.00154373 − 0.747523i
\(99\) −67.1883 −0.678669
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.j.a.227.1 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.58 yes 232
5.3 odd 4 inner 380.3.j.a.303.59 yes 232
19.18 odd 2 inner 380.3.j.a.227.116 yes 232
20.3 even 4 inner 380.3.j.a.303.116 yes 232
76.75 even 2 inner 380.3.j.a.227.59 yes 232
95.18 even 4 inner 380.3.j.a.303.58 yes 232
380.303 odd 4 inner 380.3.j.a.303.1 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.1 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.58 yes 232 4.3 odd 2 inner
380.3.j.a.227.59 yes 232 76.75 even 2 inner
380.3.j.a.227.116 yes 232 19.18 odd 2 inner
380.3.j.a.303.1 yes 232 380.303 odd 4 inner
380.3.j.a.303.58 yes 232 95.18 even 4 inner
380.3.j.a.303.59 yes 232 5.3 odd 4 inner
380.3.j.a.303.116 yes 232 20.3 even 4 inner