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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 239.2
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.2

$q$-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+(-1.99929 + 0.0533207i) q^{2} +(-2.05947 + 3.56711i) q^{3} +(3.99431 - 0.213207i) q^{4} +(4.96777 - 0.566763i) q^{5} +(3.92728 - 7.24149i) q^{6} -2.21466 q^{7} +(-7.97442 + 0.639242i) q^{8} +(-3.98284 - 6.89847i) q^{9} +(-9.90180 + 1.39801i) q^{10} +4.95438i q^{11} +(-7.46564 + 14.6872i) q^{12} +(-1.28145 + 0.739846i) q^{13} +(4.42774 - 0.118087i) q^{14} +(-8.20928 + 18.8878i) q^{15} +(15.9091 - 1.70323i) q^{16} +(3.80694 + 2.19794i) q^{17} +(8.33067 + 13.5797i) q^{18} +(-3.26278 + 18.7178i) q^{19} +(19.7220 - 3.32300i) q^{20} +(4.56103 - 7.89993i) q^{21} +(-0.264171 - 9.90523i) q^{22} +(10.2902 + 17.8232i) q^{23} +(14.1428 - 29.7621i) q^{24} +(24.3576 - 5.63110i) q^{25} +(2.52254 - 1.54749i) q^{26} -4.26034 q^{27} +(-8.84605 + 0.472181i) q^{28} +(-6.55720 - 11.3574i) q^{29} +(15.4056 - 38.1999i) q^{30} +22.1910i q^{31} +(-31.7160 + 4.25354i) q^{32} +(-17.6728 - 10.2034i) q^{33} +(-7.72837 - 4.19132i) q^{34} +(-11.0019 + 1.25519i) q^{35} +(-17.3795 - 26.7055i) q^{36} +10.1149i q^{37} +(5.52519 - 37.5962i) q^{38} -6.09476i q^{39} +(-39.2528 + 7.69522i) q^{40} +(-16.9139 + 29.2957i) q^{41} +(-8.69758 + 16.0374i) q^{42} +(-11.5693 + 20.0387i) q^{43} +(1.05631 + 19.7893i) q^{44} +(-23.6956 - 32.0127i) q^{45} +(-21.5235 - 35.0850i) q^{46} +(-37.6516 - 65.2146i) q^{47} +(-26.6887 + 60.2572i) q^{48} -44.0953 q^{49} +(-48.3975 + 12.5570i) q^{50} +(-15.6806 + 9.05318i) q^{51} +(-4.96078 + 3.22839i) q^{52} +(-74.1430 + 42.8065i) q^{53} +(8.51765 - 0.227164i) q^{54} +(2.80796 + 24.6122i) q^{55} +(17.6606 - 1.41570i) q^{56} +(-60.0486 - 50.1873i) q^{57} +(13.7153 + 22.3571i) q^{58} +(-20.2512 - 11.6920i) q^{59} +(-28.7634 + 77.1941i) q^{60} +(17.5036 + 30.3171i) q^{61} +(-1.18324 - 44.3663i) q^{62} +(8.82062 + 15.2778i) q^{63} +(63.1827 - 10.1952i) q^{64} +(-5.94664 + 4.40167i) q^{65} +(35.8771 + 19.4572i) q^{66} +(-37.3107 - 64.6240i) q^{67} +(15.6747 + 7.96759i) q^{68} -84.7696 q^{69} +(21.9291 - 3.09611i) q^{70} +(22.8730 + 13.2058i) q^{71} +(36.1706 + 52.4653i) q^{72} +(36.3466 + 20.9847i) q^{73} +(-0.539336 - 20.2227i) q^{74} +(-30.0769 + 98.4831i) q^{75} +(-9.04180 + 75.4602i) q^{76} -10.9723i q^{77} +(0.324977 + 12.1852i) q^{78} +(72.8739 + 42.0738i) q^{79} +(78.0674 - 17.4780i) q^{80} +(44.6196 - 77.2833i) q^{81} +(32.2537 - 59.4725i) q^{82} -49.5874 q^{83} +(16.5338 - 32.5272i) q^{84} +(20.1577 + 8.76123i) q^{85} +(22.0620 - 40.6800i) q^{86} +54.0174 q^{87} +(-3.16705 - 39.5083i) q^{88} +(-72.7639 - 126.031i) q^{89} +(49.0814 + 62.7392i) q^{90} +(2.83798 - 1.63851i) q^{91} +(44.9024 + 68.9974i) q^{92} +(-79.1578 - 45.7018i) q^{93} +(78.7538 + 128.375i) q^{94} +(-5.60020 + 94.8348i) q^{95} +(50.1454 - 121.895i) q^{96} +(120.014 + 69.2899i) q^{97} +(88.1592 - 2.35119i) q^{98} +(34.1776 - 19.7325i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99929 + 0.0533207i −0.999645 + 0.0266604i
\(3\) −2.05947 + 3.56711i −0.686490 + 1.18904i 0.286476 + 0.958087i \(0.407516\pi\)
−0.972966 + 0.230948i \(0.925817\pi\)
\(4\) 3.99431 − 0.213207i 0.998578 − 0.0533018i
\(5\) 4.96777 − 0.566763i 0.993555 − 0.113353i
\(6\) 3.92728 − 7.24149i 0.654546 − 1.20692i
\(7\) −2.21466 −0.316380 −0.158190 − 0.987409i \(-0.550566\pi\)
−0.158190 + 0.987409i \(0.550566\pi\)
\(8\) −7.97442 + 0.639242i −0.996802 + 0.0799053i
\(9\) −3.98284 − 6.89847i −0.442537 − 0.766497i
\(10\) −9.90180 + 1.39801i −0.990180 + 0.139801i
\(11\) 4.95438i 0.450398i 0.974313 + 0.225199i \(0.0723032\pi\)
−0.974313 + 0.225199i \(0.927697\pi\)
\(12\) −7.46564 + 14.6872i −0.622136 + 1.22394i
\(13\) −1.28145 + 0.739846i −0.0985731 + 0.0569112i −0.548476 − 0.836166i \(-0.684792\pi\)
0.449903 + 0.893077i \(0.351459\pi\)
\(14\) 4.42774 − 0.118087i 0.316267 − 0.00843481i
\(15\) −8.20928 + 18.8878i −0.547285 + 1.25919i
\(16\) 15.9091 − 1.70323i 0.994318 − 0.106452i
\(17\) 3.80694 + 2.19794i 0.223938 + 0.129290i 0.607772 − 0.794111i \(-0.292063\pi\)
−0.383835 + 0.923402i \(0.625397\pi\)
\(18\) 8.33067 + 13.5797i 0.462815 + 0.754426i
\(19\) −3.26278 + 18.7178i −0.171725 + 0.985145i
\(20\) 19.7220 − 3.32300i 0.986101 − 0.166150i
\(21\) 4.56103 − 7.89993i 0.217192 − 0.376187i
\(22\) −0.264171 − 9.90523i −0.0120078 − 0.450238i
\(23\) 10.2902 + 17.8232i 0.447401 + 0.774921i 0.998216 − 0.0597065i \(-0.0190165\pi\)
−0.550815 + 0.834627i \(0.685683\pi\)
\(24\) 14.1428 − 29.7621i 0.589285 − 1.24009i
\(25\) 24.3576 − 5.63110i 0.974302 − 0.225244i
\(26\) 2.52254 − 1.54749i 0.0970208 − 0.0595190i
\(27\) −4.26034 −0.157790
\(28\) −8.84605 + 0.472181i −0.315930 + 0.0168636i
\(29\) −6.55720 − 11.3574i −0.226110 − 0.391635i 0.730542 − 0.682868i \(-0.239268\pi\)
−0.956652 + 0.291234i \(0.905934\pi\)
\(30\) 15.4056 − 38.1999i 0.513520 − 1.27333i
\(31\) 22.1910i 0.715840i 0.933752 + 0.357920i \(0.116514\pi\)
−0.933752 + 0.357920i \(0.883486\pi\)
\(32\) −31.7160 + 4.25354i −0.991126 + 0.132923i
\(33\) −17.6728 − 10.2034i −0.535539 − 0.309194i
\(34\) −7.72837 − 4.19132i −0.227305 − 0.123274i
\(35\) −11.0019 + 1.25519i −0.314341 + 0.0358625i
\(36\) −17.3795 − 26.7055i −0.482764 − 0.741819i
\(37\) 10.1149i 0.273377i 0.990614 + 0.136688i \(0.0436459\pi\)
−0.990614 + 0.136688i \(0.956354\pi\)
\(38\) 5.52519 − 37.5962i 0.145400 − 0.989373i
\(39\) − 6.09476i − 0.156276i
\(40\) −39.2528 + 7.69522i −0.981320 + 0.192380i
\(41\) −16.9139 + 29.2957i −0.412534 + 0.714530i −0.995166 − 0.0982061i \(-0.968690\pi\)
0.582632 + 0.812736i \(0.302023\pi\)
\(42\) −8.69758 + 16.0374i −0.207085 + 0.381844i
\(43\) −11.5693 + 20.0387i −0.269054 + 0.466016i −0.968618 − 0.248554i \(-0.920045\pi\)
0.699563 + 0.714570i \(0.253378\pi\)
\(44\) 1.05631 + 19.7893i 0.0240070 + 0.449758i
\(45\) −23.6956 − 32.0127i −0.526570 − 0.711394i
\(46\) −21.5235 − 35.0850i −0.467901 − 0.762717i
\(47\) −37.6516 − 65.2146i −0.801099 − 1.38754i −0.918893 − 0.394506i \(-0.870916\pi\)
0.117795 − 0.993038i \(-0.462417\pi\)
\(48\) −26.6887 + 60.2572i −0.556014 + 1.25536i
\(49\) −44.0953 −0.899904
\(50\) −48.3975 + 12.5570i −0.967951 + 0.251139i
\(51\) −15.6806 + 9.05318i −0.307462 + 0.177513i
\(52\) −4.96078 + 3.22839i −0.0953996 + 0.0620845i
\(53\) −74.1430 + 42.8065i −1.39892 + 0.807669i −0.994280 − 0.106805i \(-0.965938\pi\)
−0.404644 + 0.914474i \(0.632605\pi\)
\(54\) 8.51765 − 0.227164i 0.157734 − 0.00420675i
\(55\) 2.80796 + 24.6122i 0.0510538 + 0.447495i
\(56\) 17.6606 − 1.41570i 0.315368 − 0.0252804i
\(57\) −60.0486 − 50.1873i −1.05348 − 0.880479i
\(58\) 13.7153 + 22.3571i 0.236471 + 0.385467i
\(59\) −20.2512 − 11.6920i −0.343241 − 0.198170i 0.318463 − 0.947935i \(-0.396833\pi\)
−0.661704 + 0.749765i \(0.730167\pi\)
\(60\) −28.7634 + 77.1941i −0.479390 + 1.28657i
\(61\) 17.5036 + 30.3171i 0.286944 + 0.497002i 0.973079 − 0.230472i \(-0.0740272\pi\)
−0.686134 + 0.727475i \(0.740694\pi\)
\(62\) −1.18324 − 44.3663i −0.0190846 − 0.715585i
\(63\) 8.82062 + 15.2778i 0.140010 + 0.242504i
\(64\) 63.1827 − 10.1952i 0.987230 − 0.159300i
\(65\) −5.94664 + 4.40167i −0.0914868 + 0.0677180i
\(66\) 35.8771 + 19.4572i 0.543592 + 0.294806i
\(67\) −37.3107 − 64.6240i −0.556876 − 0.964537i −0.997755 − 0.0669704i \(-0.978667\pi\)
0.440879 − 0.897566i \(-0.354667\pi\)
\(68\) 15.6747 + 7.96759i 0.230511 + 0.117170i
\(69\) −84.7696 −1.22854
\(70\) 21.9291 − 3.09611i 0.313273 − 0.0442302i
\(71\) 22.8730 + 13.2058i 0.322156 + 0.185997i 0.652353 − 0.757915i \(-0.273782\pi\)
−0.330197 + 0.943912i \(0.607115\pi\)
\(72\) 36.1706 + 52.4653i 0.502369 + 0.728685i
\(73\) 36.3466 + 20.9847i 0.497899 + 0.287462i 0.727845 − 0.685741i \(-0.240522\pi\)
−0.229947 + 0.973203i \(0.573855\pi\)
\(74\) −0.539336 − 20.2227i −0.00728833 − 0.273280i
\(75\) −30.0769 + 98.4831i −0.401026 + 1.31311i
\(76\) −9.04180 + 75.4602i −0.118971 + 0.992898i
\(77\) − 10.9723i − 0.142497i
\(78\) 0.324977 + 12.1852i 0.00416638 + 0.156220i
\(79\) 72.8739 + 42.0738i 0.922454 + 0.532579i 0.884417 − 0.466697i \(-0.154556\pi\)
0.0380369 + 0.999276i \(0.487890\pi\)
\(80\) 78.0674 − 17.4780i 0.975843 − 0.218474i
\(81\) 44.6196 − 77.2833i 0.550859 − 0.954115i
\(82\) 32.2537 − 59.4725i 0.393338 − 0.725274i
\(83\) −49.5874 −0.597438 −0.298719 − 0.954341i \(-0.596559\pi\)
−0.298719 + 0.954341i \(0.596559\pi\)
\(84\) 16.5338 − 32.5272i 0.196831 − 0.387229i
\(85\) 20.1577 + 8.76123i 0.237150 + 0.103073i
\(86\) 22.0620 − 40.6800i 0.256535 − 0.473023i
\(87\) 54.0174 0.620890
\(88\) −3.16705 − 39.5083i −0.0359892 − 0.448958i
\(89\) −72.7639 − 126.031i −0.817571 − 1.41608i −0.907467 − 0.420124i \(-0.861987\pi\)
0.0898952 − 0.995951i \(-0.471347\pi\)
\(90\) 49.0814 + 62.7392i 0.545348 + 0.697103i
\(91\) 2.83798 − 1.63851i 0.0311866 − 0.0180056i
\(92\) 44.9024 + 68.9974i 0.488069 + 0.749972i
\(93\) −79.1578 − 45.7018i −0.851159 − 0.491417i
\(94\) 78.7538 + 128.375i 0.837806 + 1.36569i
\(95\) −5.60020 + 94.8348i −0.0589495 + 0.998261i
\(96\) 50.1454 − 121.895i 0.522348 − 1.26973i
\(97\) 120.014 + 69.2899i 1.23725 + 0.714329i 0.968532 − 0.248889i \(-0.0800655\pi\)
0.268722 + 0.963218i \(0.413399\pi\)
\(98\) 88.1592 − 2.35119i 0.899584 − 0.0239918i
\(99\) 34.1776 − 19.7325i 0.345229 − 0.199318i
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.239.2 yes 232
4.3 odd 2 inner 380.3.p.a.239.39 yes 232
5.4 even 2 inner 380.3.p.a.239.115 yes 232
19.7 even 3 inner 380.3.p.a.159.78 yes 232
20.19 odd 2 inner 380.3.p.a.239.78 yes 232
76.7 odd 6 inner 380.3.p.a.159.115 yes 232
95.64 even 6 inner 380.3.p.a.159.39 yes 232
380.159 odd 6 inner 380.3.p.a.159.2 ✓ 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.2 ✓ 232 380.159 odd 6 inner
380.3.p.a.159.39 yes 232 95.64 even 6 inner
380.3.p.a.159.78 yes 232 19.7 even 3 inner
380.3.p.a.159.115 yes 232 76.7 odd 6 inner
380.3.p.a.239.2 yes 232 1.1 even 1 trivial
380.3.p.a.239.39 yes 232 4.3 odd 2 inner
380.3.p.a.239.78 yes 232 20.19 odd 2 inner
380.3.p.a.239.115 yes 232 5.4 even 2 inner