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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 239.78
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.78

$q$-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+(0.953467 + 1.75810i) q^{2} +(-2.05947 + 3.56711i) q^{3} +(-2.18180 + 3.35257i) q^{4} +(-1.99306 - 4.58560i) q^{5} +(-8.23495 - 0.219625i) q^{6} -2.21466 q^{7} +(-7.97442 - 0.639242i) q^{8} +(-3.98284 - 6.89847i) q^{9} +(6.16161 - 7.87620i) q^{10} -4.95438i q^{11} +(-7.46564 - 14.6872i) q^{12} +(1.28145 - 0.739846i) q^{13} +(-2.11161 - 3.89358i) q^{14} +(20.4620 + 2.33446i) q^{15} +(-6.47950 - 14.6293i) 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} +(8.71027 - 4.72384i) q^{22} +(10.2902 + 17.8232i) q^{23} +(18.7033 - 27.1291i) q^{24} +(-17.0555 + 18.2787i) q^{25} +(2.52254 + 1.54749i) q^{26} -4.26034 q^{27} +(4.83194 - 7.42481i) q^{28} +(-6.55720 - 11.3574i) q^{29} +(15.4056 + 38.1999i) q^{30} -22.1910i q^{31} +(19.5417 - 25.3401i) q^{32} +(17.6728 + 10.2034i) q^{33} +(0.234391 - 8.78863i) q^{34} +(4.41394 + 10.1555i) q^{35} +(31.8174 + 1.69834i) q^{36} -10.1149i q^{37} +(36.0185 - 12.1105i) q^{38} +6.09476i q^{39} +(12.9622 + 37.8415i) q^{40} +(-16.9139 + 29.2957i) q^{41} +(18.2376 + 0.486395i) q^{42} +(-11.5693 + 20.0387i) q^{43} +(16.6099 + 10.8095i) q^{44} +(-23.6956 + 32.0127i) q^{45} +(-21.5235 + 35.0850i) q^{46} +(-37.6516 - 65.2146i) q^{47} +(65.5286 + 7.01551i) q^{48} -44.0953 q^{49} +(-48.3975 - 12.5570i) q^{50} +(15.6806 - 9.05318i) q^{51} +(-0.315481 + 5.91035i) q^{52} +(74.1430 - 42.8065i) q^{53} +(-4.06210 - 7.49008i) q^{54} +(-22.7188 + 9.87435i) 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} +(-52.4704 + 63.5069i) q^{60} +(17.5036 + 30.3171i) q^{61} +(39.0140 - 21.1584i) q^{62} +(8.82062 + 15.2778i) q^{63} +(63.1827 + 10.1952i) q^{64} +(-5.94664 - 4.40167i) q^{65} +(-1.08810 + 40.7991i) q^{66} +(-37.3107 - 64.6240i) q^{67} +(15.6747 - 7.96759i) q^{68} -84.7696 q^{69} +(-13.6459 + 17.4431i) q^{70} +(-22.8730 - 13.2058i) q^{71} +(27.3510 + 57.5573i) q^{72} +(-36.3466 - 20.9847i) q^{73} +(17.7830 - 9.64427i) q^{74} +(-30.0769 - 98.4831i) q^{75} +(55.6339 + 51.7771i) q^{76} +10.9723i q^{77} +(-10.7152 + 5.81116i) q^{78} +(-72.8739 - 42.0738i) q^{79} +(-54.1701 + 58.8694i) q^{80} +(44.6196 - 77.2833i) q^{81} +(-67.6315 - 1.80372i) q^{82} -49.5874 q^{83} +(16.5338 + 32.5272i) q^{84} +(-2.49142 + 21.8377i) q^{85} +(-46.2609 - 1.23377i) q^{86} +54.0174 q^{87} +(-3.16705 + 39.5083i) q^{88} +(-72.7639 - 126.031i) q^{89} +(-78.8745 - 11.1361i) q^{90} +(-2.83798 + 1.63851i) q^{91} +(-82.2047 - 4.38789i) q^{92} +(79.1578 + 45.7018i) q^{93} +(78.7538 - 128.375i) q^{94} +(-92.3350 + 22.3437i) q^{95} +(50.1454 + 121.895i) q^{96} +(-120.014 - 69.2899i) q^{97} +(-42.0434 - 77.5237i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.953467 + 1.75810i 0.476734 + 0.879048i
\(3\) −2.05947 + 3.56711i −0.686490 + 1.18904i 0.286476 + 0.958087i \(0.407516\pi\)
−0.972966 + 0.230948i \(0.925817\pi\)
\(4\) −2.18180 + 3.35257i −0.545450 + 0.838143i
\(5\) −1.99306 − 4.58560i −0.398611 − 0.917120i
\(6\) −8.23495 − 0.219625i −1.37249 − 0.0366042i
\(7\) −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
\(10\) 6.16161 − 7.87620i 0.616161 − 0.787620i
\(11\) − 4.95438i − 0.450398i −0.974313 − 0.225199i \(-0.927697\pi\)
0.974313 − 0.225199i \(-0.0723032\pi\)
\(12\) −7.46564 − 14.6872i −0.622136 − 1.22394i
\(13\) 1.28145 − 0.739846i 0.0985731 − 0.0569112i −0.449903 − 0.893077i \(-0.648541\pi\)
0.548476 + 0.836166i \(0.315208\pi\)
\(14\) −2.11161 − 3.89358i −0.150829 − 0.278113i
\(15\) 20.4620 + 2.33446i 1.36413 + 0.155631i
\(16\) −6.47950 − 14.6293i −0.404969 − 0.914331i
\(17\) −3.80694 − 2.19794i −0.223938 − 0.129290i 0.383835 − 0.923402i \(-0.374603\pi\)
−0.607772 + 0.794111i \(0.707937\pi\)
\(18\) 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
\(22\) 8.71027 − 4.72384i 0.395921 − 0.214720i
\(23\) 10.2902 + 17.8232i 0.447401 + 0.774921i 0.998216 − 0.0597065i \(-0.0190165\pi\)
−0.550815 + 0.834627i \(0.685683\pi\)
\(24\) 18.7033 − 27.1291i 0.779305 − 1.13038i
\(25\) −17.0555 + 18.2787i −0.682218 + 0.731149i
\(26\) 2.52254 + 1.54749i 0.0970208 + 0.0595190i
\(27\) −4.26034 −0.157790
\(28\) 4.83194 − 7.42481i 0.172569 − 0.265172i
\(29\) −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 \(-0.883486\pi\)
0.933752 − 0.357920i \(-0.116514\pi\)
\(32\) 19.5417 − 25.3401i 0.610678 − 0.791879i
\(33\) 17.6728 + 10.2034i 0.535539 + 0.309194i
\(34\) 0.234391 − 8.78863i 0.00689386 − 0.258489i
\(35\) 4.41394 + 10.1555i 0.126113 + 0.290158i
\(36\) 31.8174 + 1.69834i 0.883816 + 0.0471761i
\(37\) − 10.1149i − 0.273377i −0.990614 − 0.136688i \(-0.956354\pi\)
0.990614 − 0.136688i \(-0.0436459\pi\)
\(38\) 36.0185 − 12.1105i 0.947857 − 0.318697i
\(39\) 6.09476i 0.156276i
\(40\) 12.9622 + 37.8415i 0.324054 + 0.946039i
\(41\) −16.9139 + 29.2957i −0.412534 + 0.714530i −0.995166 − 0.0982061i \(-0.968690\pi\)
0.582632 + 0.812736i \(0.302023\pi\)
\(42\) 18.2376 + 0.486395i 0.434229 + 0.0115808i
\(43\) −11.5693 + 20.0387i −0.269054 + 0.466016i −0.968618 − 0.248554i \(-0.920045\pi\)
0.699563 + 0.714570i \(0.253378\pi\)
\(44\) 16.6099 + 10.8095i 0.377498 + 0.245669i
\(45\) −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 \(-0.462417\pi\)
\(48\) 65.5286 + 7.01551i 1.36518 + 0.146157i
\(49\) −44.0953 −0.899904
\(50\) −48.3975 − 12.5570i −0.967951 − 0.251139i
\(51\) 15.6806 − 9.05318i 0.307462 − 0.177513i
\(52\) −0.315481 + 5.91035i −0.00606694 + 0.113661i
\(53\) 74.1430 − 42.8065i 1.39892 − 0.807669i 0.404644 − 0.914474i \(-0.367395\pi\)
0.994280 + 0.106805i \(0.0340620\pi\)
\(54\) −4.06210 − 7.49008i −0.0752240 − 0.138705i
\(55\) −22.7188 + 9.87435i −0.413069 + 0.179534i
\(56\) 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 0.661704 − 0.749765i \(-0.269833\pi\)
−0.318463 + 0.947935i \(0.603167\pi\)
\(60\) −52.4704 + 63.5069i −0.874506 + 1.05845i
\(61\) 17.5036 + 30.3171i 0.286944 + 0.497002i 0.973079 − 0.230472i \(-0.0740272\pi\)
−0.686134 + 0.727475i \(0.740694\pi\)
\(62\) 39.0140 − 21.1584i 0.629257 − 0.341265i
\(63\) 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
\(66\) −1.08810 + 40.7991i −0.0164864 + 0.618168i
\(67\) −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
\(70\) −13.6459 + 17.4431i −0.194941 + 0.249187i
\(71\) −22.8730 − 13.2058i −0.322156 − 0.185997i 0.330197 − 0.943912i \(-0.392885\pi\)
−0.652353 + 0.757915i \(0.726218\pi\)
\(72\) 27.3510 + 57.5573i 0.379875 + 0.799407i
\(73\) −36.3466 − 20.9847i −0.497899 − 0.287462i 0.229947 − 0.973203i \(-0.426145\pi\)
−0.727845 + 0.685741i \(0.759478\pi\)
\(74\) 17.7830 − 9.64427i 0.240311 − 0.130328i
\(75\) −30.0769 − 98.4831i −0.401026 − 1.31311i
\(76\) 55.6339 + 51.7771i 0.732025 + 0.681277i
\(77\) 10.9723i 0.142497i
\(78\) −10.7152 + 5.81116i −0.137374 + 0.0745020i
\(79\) −72.8739 − 42.0738i −0.922454 − 0.532579i −0.0380369 − 0.999276i \(-0.512110\pi\)
−0.884417 + 0.466697i \(0.845444\pi\)
\(80\) −54.1701 + 58.8694i −0.677126 + 0.735867i
\(81\) 44.6196 − 77.2833i 0.550859 − 0.954115i
\(82\) −67.6315 − 1.80372i −0.824775 − 0.0219966i
\(83\) −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
\(85\) −2.49142 + 21.8377i −0.0293108 + 0.256914i
\(86\) −46.2609 − 1.23377i −0.537918 − 0.0143462i
\(87\) 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 \(-0.471347\pi\)
\(90\) −78.8745 − 11.1361i −0.876383 − 0.123734i
\(91\) −2.83798 + 1.63851i −0.0311866 + 0.0180056i
\(92\) −82.2047 − 4.38789i −0.893529 − 0.0476945i
\(93\) 79.1578 + 45.7018i 0.851159 + 0.491417i
\(94\) 78.7538 − 128.375i 0.837806 − 1.36569i
\(95\) −92.3350 + 22.3437i −0.971948 + 0.235197i
\(96\) 50.1454 + 121.895i 0.522348 + 1.26973i
\(97\) −120.014 − 69.2899i −1.23725 − 0.714329i −0.268722 − 0.963218i \(-0.586601\pi\)
−0.968532 + 0.248889i \(0.919934\pi\)
\(98\) −42.0434 − 77.5237i −0.429014 − 0.791058i
\(99\) −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.239.78 yes 232
4.3 odd 2 inner 380.3.p.a.239.115 yes 232
5.4 even 2 inner 380.3.p.a.239.39 yes 232
19.7 even 3 inner 380.3.p.a.159.2 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.2 yes 232
76.7 odd 6 inner 380.3.p.a.159.39 yes 232
95.64 even 6 inner 380.3.p.a.159.115 yes 232
380.159 odd 6 inner 380.3.p.a.159.78 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.2 ✓ 232 19.7 even 3 inner
380.3.p.a.159.39 yes 232 76.7 odd 6 inner
380.3.p.a.159.78 yes 232 380.159 odd 6 inner
380.3.p.a.159.115 yes 232 95.64 even 6 inner
380.3.p.a.239.2 yes 232 20.19 odd 2 inner
380.3.p.a.239.39 yes 232 5.4 even 2 inner
380.3.p.a.239.78 yes 232 1.1 even 1 trivial
380.3.p.a.239.115 yes 232 4.3 odd 2 inner