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Properties

Label 380.3.p.a.159.8
Level $380$
Weight $3$
Character 380.159
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(159,380)] 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("380.159"); S:= CuspForms(chi, 3); 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, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.8
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.8

$q$-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+(-1.97064 - 0.341417i) q^{2} +(-2.91038 - 5.04092i) q^{3} +(3.76687 + 1.34562i) q^{4} +(-4.21638 + 2.68741i) q^{5} +(4.01426 + 10.9275i) q^{6} +4.09062 q^{7} +(-6.96373 - 3.93782i) q^{8} +(-12.4406 + 21.5477i) q^{9} +(9.22650 - 3.85638i) q^{10} -10.2061i q^{11} +(-4.17983 - 22.9047i) q^{12} +(15.4931 + 8.94495i) q^{13} +(-8.06115 - 1.39661i) q^{14} +(25.8183 + 13.4330i) q^{15} +(12.3786 + 10.1376i) q^{16} +(5.78026 - 3.33723i) q^{17} +(31.8727 - 38.2154i) q^{18} +(17.3301 + 7.78887i) q^{19} +(-19.4988 + 4.44947i) q^{20} +(-11.9052 - 20.6205i) q^{21} +(-3.48455 + 20.1126i) q^{22} +(-16.5272 + 28.6259i) q^{23} +(0.416869 + 46.5641i) q^{24} +(10.5556 - 22.6623i) q^{25} +(-27.4774 - 22.9169i) q^{26} +92.4402 q^{27} +(15.4088 + 5.50443i) q^{28} +(11.7692 - 20.3848i) q^{29} +(-46.2923 - 35.2865i) q^{30} +11.0838i q^{31} +(-20.9327 - 24.2038i) q^{32} +(-51.4482 + 29.7037i) q^{33} +(-12.5302 + 4.60302i) q^{34} +(-17.2476 + 10.9932i) q^{35} +(-75.8571 + 64.4270i) q^{36} +19.6187i q^{37} +(-31.4923 - 21.2659i) q^{38} -104.133i q^{39} +(39.9443 - 2.11110i) q^{40} +(-10.2422 - 17.7400i) q^{41} +(16.4208 + 44.7002i) q^{42} +(-37.5420 - 65.0247i) q^{43} +(13.7336 - 38.4451i) q^{44} +(-5.45340 - 124.286i) q^{45} +(42.3426 - 50.7689i) q^{46} +(13.4434 - 23.2846i) q^{47} +(15.0763 - 91.9036i) q^{48} -32.2669 q^{49} +(-28.5387 + 41.0554i) q^{50} +(-33.6454 - 19.4252i) q^{51} +(46.3240 + 54.5423i) q^{52} +(64.6386 + 37.3191i) q^{53} +(-182.167 - 31.5607i) q^{54} +(27.4281 + 43.0329i) q^{55} +(-28.4860 - 16.1081i) q^{56} +(-11.1742 - 110.028i) q^{57} +(-30.1526 + 36.1530i) q^{58} +(10.8711 - 6.27643i) q^{59} +(79.1782 + 85.3421i) q^{60} +(-27.4663 + 47.5730i) q^{61} +(3.78419 - 21.8422i) q^{62} +(-50.8896 + 88.1434i) q^{63} +(32.9872 + 54.8438i) q^{64} +(-89.3635 + 3.92108i) q^{65} +(111.527 - 40.9700i) q^{66} +(35.4685 - 61.4333i) q^{67} +(26.2641 - 4.79287i) q^{68} +192.401 q^{69} +(37.7421 - 15.7750i) q^{70} +(30.5036 - 17.6112i) q^{71} +(171.484 - 101.064i) q^{72} +(-31.3102 + 18.0769i) q^{73} +(6.69816 - 38.6614i) q^{74} +(-144.960 + 12.7456i) q^{75} +(54.7995 + 52.6595i) q^{76} -41.7493i q^{77} +(-35.5527 + 205.208i) q^{78} +(68.8954 - 39.7768i) q^{79} +(-79.4366 - 9.47743i) q^{80} +(-157.071 - 272.054i) q^{81} +(14.1270 + 38.4560i) q^{82} +85.8313 q^{83} +(-17.0981 - 93.6945i) q^{84} +(-15.4032 + 29.6050i) q^{85} +(51.7814 + 140.958i) q^{86} -137.011 q^{87} +(-40.1899 + 71.0727i) q^{88} +(5.09445 - 8.82385i) q^{89} +(-31.6867 + 246.786i) q^{90} +(63.3764 + 36.5904i) q^{91} +(-100.776 + 85.5908i) q^{92} +(55.8724 - 32.2580i) q^{93} +(-34.4419 + 41.2959i) q^{94} +(-94.0022 + 13.7324i) q^{95} +(-61.0875 + 175.962i) q^{96} +(19.6841 - 11.3646i) q^{97} +(63.5865 + 11.0165i) q^{98} +(219.919 + 126.970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97064 − 0.341417i −0.985322 − 0.170709i
\(3\) −2.91038 − 5.04092i −0.970125 − 1.68031i −0.695164 − 0.718851i \(-0.744668\pi\)
−0.274961 − 0.961455i \(-0.588665\pi\)
\(4\) 3.76687 + 1.34562i 0.941717 + 0.336406i
\(5\) −4.21638 + 2.68741i −0.843275 + 0.537482i
\(6\) 4.01426 + 10.9275i 0.669043 + 1.82125i
\(7\) 4.09062 0.584374 0.292187 − 0.956361i \(-0.405617\pi\)
0.292187 + 0.956361i \(0.405617\pi\)
\(8\) −6.96373 − 3.93782i −0.870467 − 0.492227i
\(9\) −12.4406 + 21.5477i −1.38229 + 2.39419i
\(10\) 9.22650 − 3.85638i 0.922650 − 0.385638i
\(11\) − 10.2061i − 0.927830i −0.885880 − 0.463915i \(-0.846444\pi\)
0.885880 − 0.463915i \(-0.153556\pi\)
\(12\) −4.17983 − 22.9047i −0.348319 − 1.90873i
\(13\) 15.4931 + 8.94495i 1.19178 + 0.688073i 0.958709 − 0.284387i \(-0.0917902\pi\)
0.233068 + 0.972460i \(0.425124\pi\)
\(14\) −8.06115 − 1.39661i −0.575796 − 0.0997577i
\(15\) 25.8183 + 13.4330i 1.72122 + 0.895535i
\(16\) 12.3786 + 10.1376i 0.773662 + 0.633598i
\(17\) 5.78026 − 3.33723i 0.340015 − 0.196308i −0.320264 − 0.947328i \(-0.603771\pi\)
0.660279 + 0.751021i \(0.270438\pi\)
\(18\) 31.8727 − 38.2154i 1.77070 − 2.12308i
\(19\) 17.3301 + 7.78887i 0.912112 + 0.409940i
\(20\) −19.4988 + 4.44947i −0.974939 + 0.222474i
\(21\) −11.9052 − 20.6205i −0.566916 − 0.981927i
\(22\) −3.48455 + 20.1126i −0.158389 + 0.914210i
\(23\) −16.5272 + 28.6259i −0.718574 + 1.24461i 0.242991 + 0.970029i \(0.421871\pi\)
−0.961565 + 0.274578i \(0.911462\pi\)
\(24\) 0.416869 + 46.5641i 0.0173695 + 1.94017i
\(25\) 10.5556 − 22.6623i 0.422226 − 0.906491i
\(26\) −27.4774 − 22.9169i −1.05682 − 0.881420i
\(27\) 92.4402 3.42371
\(28\) 15.4088 + 5.50443i 0.550315 + 0.196587i
\(29\) 11.7692 − 20.3848i 0.405834 − 0.702925i −0.588584 − 0.808436i \(-0.700315\pi\)
0.994418 + 0.105511i \(0.0336479\pi\)
\(30\) −46.2923 − 35.2865i −1.54308 − 1.17622i
\(31\) 11.0838i 0.357541i 0.983891 + 0.178771i \(0.0572120\pi\)
−0.983891 + 0.178771i \(0.942788\pi\)
\(32\) −20.9327 − 24.2038i −0.654145 − 0.756369i
\(33\) −51.4482 + 29.7037i −1.55904 + 0.900111i
\(34\) −12.5302 + 4.60302i −0.368536 + 0.135383i
\(35\) −17.2476 + 10.9932i −0.492788 + 0.314091i
\(36\) −75.8571 + 64.4270i −2.10714 + 1.78964i
\(37\) 19.6187i 0.530235i 0.964216 + 0.265118i \(0.0854108\pi\)
−0.964216 + 0.265118i \(0.914589\pi\)
\(38\) −31.4923 − 21.2659i −0.828744 − 0.559629i
\(39\) − 104.133i − 2.67007i
\(40\) 39.9443 − 2.11110i 0.998606 − 0.0527775i
\(41\) −10.2422 − 17.7400i −0.249809 − 0.432683i 0.713663 − 0.700489i \(-0.247035\pi\)
−0.963473 + 0.267806i \(0.913701\pi\)
\(42\) 16.4208 + 44.7002i 0.390971 + 1.06429i
\(43\) −37.5420 − 65.0247i −0.873070 − 1.51220i −0.858804 − 0.512304i \(-0.828792\pi\)
−0.0142664 − 0.999898i \(-0.504541\pi\)
\(44\) 13.7336 − 38.4451i 0.312127 − 0.873753i
\(45\) −5.45340 − 124.286i −0.121187 − 2.76191i
\(46\) 42.3426 − 50.7689i 0.920491 − 1.10367i
\(47\) 13.4434 − 23.2846i 0.286029 − 0.495417i −0.686829 − 0.726819i \(-0.740998\pi\)
0.972858 + 0.231402i \(0.0743312\pi\)
\(48\) 15.0763 − 91.9036i 0.314090 − 1.91466i
\(49\) −32.2669 −0.658507
\(50\) −28.5387 + 41.0554i −0.570774 + 0.821107i
\(51\) −33.6454 − 19.4252i −0.659715 − 0.380886i
\(52\) 46.3240 + 54.5423i 0.890845 + 1.04889i
\(53\) 64.6386 + 37.3191i 1.21960 + 0.704135i 0.964832 − 0.262868i \(-0.0846685\pi\)
0.254765 + 0.967003i \(0.418002\pi\)
\(54\) −182.167 − 31.5607i −3.37346 − 0.584457i
\(55\) 27.4281 + 43.0329i 0.498692 + 0.782416i
\(56\) −28.4860 − 16.1081i −0.508678 − 0.287645i
\(57\) −11.1742 − 110.028i −0.196038 − 1.93032i
\(58\) −30.1526 + 36.1530i −0.519872 + 0.623327i
\(59\) 10.8711 − 6.27643i 0.184256 − 0.106380i −0.405035 − 0.914301i \(-0.632741\pi\)
0.589291 + 0.807921i \(0.299407\pi\)
\(60\) 79.1782 + 85.3421i 1.31964 + 1.42237i
\(61\) −27.4663 + 47.5730i −0.450267 + 0.779886i −0.998402 − 0.0565041i \(-0.982005\pi\)
0.548135 + 0.836390i \(0.315338\pi\)
\(62\) 3.78419 − 21.8422i 0.0610354 − 0.352293i
\(63\) −50.8896 + 88.1434i −0.807772 + 1.39910i
\(64\) 32.9872 + 54.8438i 0.515425 + 0.856935i
\(65\) −89.3635 + 3.92108i −1.37482 + 0.0603242i
\(66\) 111.527 − 40.9700i 1.68981 − 0.620757i
\(67\) 35.4685 − 61.4333i 0.529381 − 0.916915i −0.470032 − 0.882649i \(-0.655758\pi\)
0.999413 − 0.0342653i \(-0.0109091\pi\)
\(68\) 26.2641 − 4.79287i 0.386237 − 0.0704834i
\(69\) 192.401 2.78843
\(70\) 37.7421 − 15.7750i 0.539173 − 0.225357i
\(71\) 30.5036 − 17.6112i 0.429628 − 0.248046i −0.269560 − 0.962983i \(-0.586878\pi\)
0.699188 + 0.714938i \(0.253545\pi\)
\(72\) 171.484 − 101.064i 2.38172 − 1.40366i
\(73\) −31.3102 + 18.0769i −0.428907 + 0.247629i −0.698881 − 0.715238i \(-0.746318\pi\)
0.269974 + 0.962868i \(0.412985\pi\)
\(74\) 6.69816 − 38.6614i 0.0905157 − 0.522452i
\(75\) −144.960 + 12.7456i −1.93279 + 0.169941i
\(76\) 54.7995 + 52.6595i 0.721045 + 0.692888i
\(77\) − 41.7493i − 0.542199i
\(78\) −35.5527 + 205.208i −0.455804 + 2.63088i
\(79\) 68.8954 − 39.7768i 0.872094 − 0.503504i 0.00405024 − 0.999992i \(-0.498711\pi\)
0.868043 + 0.496488i \(0.165377\pi\)
\(80\) −79.4366 − 9.47743i −0.992958 − 0.118468i
\(81\) −157.071 − 272.054i −1.93914 − 3.35869i
\(82\) 14.1270 + 38.4560i 0.172280 + 0.468976i
\(83\) 85.8313 1.03411 0.517056 − 0.855951i \(-0.327028\pi\)
0.517056 + 0.855951i \(0.327028\pi\)
\(84\) −17.0981 − 93.6945i −0.203548 − 1.11541i
\(85\) −15.4032 + 29.6050i −0.181214 + 0.348294i
\(86\) 51.7814 + 140.958i 0.602109 + 1.63905i
\(87\) −137.011 −1.57484
\(88\) −40.1899 + 71.0727i −0.456703 + 0.807645i
\(89\) 5.09445 − 8.82385i 0.0572410 − 0.0991444i −0.835985 − 0.548752i \(-0.815103\pi\)
0.893226 + 0.449608i \(0.148436\pi\)
\(90\) −31.6867 + 246.786i −0.352075 + 2.74206i
\(91\) 63.3764 + 36.5904i 0.696444 + 0.402092i
\(92\) −100.776 + 85.5908i −1.09539 + 0.930335i
\(93\) 55.8724 − 32.2580i 0.600779 − 0.346860i
\(94\) −34.4419 + 41.2959i −0.366403 + 0.439318i
\(95\) −94.0022 + 13.7324i −0.989497 + 0.144552i
\(96\) −61.0875 + 175.962i −0.636328 + 1.83294i
\(97\) 19.6841 − 11.3646i 0.202929 − 0.117161i −0.395092 − 0.918641i \(-0.629287\pi\)
0.598021 + 0.801481i \(0.295954\pi\)
\(98\) 63.5865 + 11.0165i 0.648841 + 0.112413i
\(99\) 219.919 + 126.970i 2.22140 + 1.28253i
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.p.a.159.8 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.46 yes 232
5.4 even 2 inner 380.3.p.a.159.109 yes 232
19.11 even 3 inner 380.3.p.a.239.71 yes 232
20.19 odd 2 inner 380.3.p.a.159.71 yes 232
76.11 odd 6 inner 380.3.p.a.239.109 yes 232
95.49 even 6 inner 380.3.p.a.239.46 yes 232
380.239 odd 6 inner 380.3.p.a.239.8 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.8 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.46 yes 232 4.3 odd 2 inner
380.3.p.a.159.71 yes 232 20.19 odd 2 inner
380.3.p.a.159.109 yes 232 5.4 even 2 inner
380.3.p.a.239.8 yes 232 380.239 odd 6 inner
380.3.p.a.239.46 yes 232 95.49 even 6 inner
380.3.p.a.239.71 yes 232 19.11 even 3 inner
380.3.p.a.239.109 yes 232 76.11 odd 6 inner