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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 239.7
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.7

$q$-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+(-1.97102 + 0.339262i) q^{2} +(-1.97599 + 3.42251i) q^{3} +(3.76980 - 1.33738i) q^{4} +(-0.693065 + 4.95173i) q^{5} +(2.73357 - 7.41619i) q^{6} -2.56913 q^{7} +(-6.97662 + 3.91494i) q^{8} +(-3.30904 - 5.73143i) q^{9} +(-0.313891 - 9.99507i) q^{10} -17.7707i q^{11} +(-2.87189 + 15.5448i) q^{12} +(-4.58570 + 2.64756i) q^{13} +(5.06379 - 0.871606i) q^{14} +(-15.5779 - 12.1566i) q^{15} +(12.4228 - 10.0833i) q^{16} +(1.90065 + 1.09734i) q^{17} +(8.46662 + 10.1741i) q^{18} +(-15.3281 - 11.2272i) q^{19} +(4.00963 + 19.5940i) q^{20} +(5.07656 - 8.79286i) q^{21} +(6.02891 + 35.0263i) q^{22} +(-8.21652 - 14.2314i) q^{23} +(0.386770 - 31.6134i) q^{24} +(-24.0393 - 6.86374i) q^{25} +(8.14028 - 6.77413i) q^{26} -9.41328 q^{27} +(-9.68510 + 3.43590i) q^{28} +(24.5292 + 42.4859i) q^{29} +(34.8285 + 18.6758i) q^{30} +13.0380i q^{31} +(-21.0647 + 24.0890i) q^{32} +(60.8203 + 35.1146i) q^{33} +(-4.11851 - 1.51806i) q^{34} +(1.78057 - 12.7216i) q^{35} +(-20.1395 - 17.1809i) q^{36} -53.5256i q^{37} +(34.0209 + 16.9287i) q^{38} -20.9261i q^{39} +(-14.5505 - 37.2597i) q^{40} +(12.6985 - 21.9944i) q^{41} +(-7.02290 + 19.0531i) q^{42} +(36.7927 - 63.7268i) q^{43} +(-23.7661 - 66.9920i) q^{44} +(30.6739 - 12.4132i) q^{45} +(21.0231 + 25.2628i) q^{46} +(27.9634 + 48.4340i) q^{47} +(9.96288 + 62.4417i) q^{48} -42.3996 q^{49} +(49.7105 + 5.37293i) q^{50} +(-7.51133 + 4.33667i) q^{51} +(-13.7464 + 16.1136i) q^{52} +(38.6169 - 22.2955i) q^{53} +(18.5537 - 3.19356i) q^{54} +(87.9957 + 12.3162i) q^{55} +(17.9238 - 10.0580i) q^{56} +(68.7132 - 30.2759i) q^{57} +(-62.7613 - 75.4185i) q^{58} +(58.4823 + 33.7648i) q^{59} +(-74.9834 - 24.9944i) q^{60} +(-2.07481 - 3.59367i) q^{61} +(-4.42330 - 25.6981i) q^{62} +(8.50134 + 14.7248i) q^{63} +(33.3464 - 54.6262i) q^{64} +(-9.93181 - 24.5421i) q^{65} +(-131.791 - 48.5775i) q^{66} +(-51.0342 - 88.3938i) q^{67} +(8.63266 + 1.59487i) q^{68} +64.9429 q^{69} +(0.806426 + 25.6786i) q^{70} +(35.9292 + 20.7437i) q^{71} +(45.5241 + 27.0313i) q^{72} +(41.1613 + 23.7645i) q^{73} +(18.1592 + 105.500i) q^{74} +(70.9926 - 68.7121i) q^{75} +(-72.7989 - 21.8247i) q^{76} +45.6551i q^{77} +(7.09944 + 41.2457i) q^{78} +(-63.0816 - 36.4202i) q^{79} +(41.3201 + 68.5029i) q^{80} +(48.3819 - 83.7999i) q^{81} +(-17.5670 + 47.6593i) q^{82} -124.038 q^{83} +(7.37824 - 39.9366i) q^{84} +(-6.75103 + 8.65100i) q^{85} +(-50.8989 + 138.089i) q^{86} -193.878 q^{87} +(69.5712 + 123.979i) q^{88} +(50.2003 + 86.9495i) q^{89} +(-56.2473 + 34.8731i) q^{90} +(11.7813 - 6.80191i) q^{91} +(-50.0075 - 42.6611i) q^{92} +(-44.6227 - 25.7630i) q^{93} +(-71.5481 - 85.9773i) q^{94} +(66.2173 - 68.1195i) q^{95} +(-40.8211 - 119.694i) q^{96} +(-53.4146 - 30.8390i) q^{97} +(83.5702 - 14.3846i) q^{98} +(-101.851 + 58.8039i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97102 + 0.339262i −0.985508 + 0.169631i
\(3\) −1.97599 + 3.42251i −0.658662 + 1.14084i 0.322300 + 0.946637i \(0.395544\pi\)
−0.980962 + 0.194199i \(0.937789\pi\)
\(4\) 3.76980 − 1.33738i 0.942451 − 0.334345i
\(5\) −0.693065 + 4.95173i −0.138613 + 0.990347i
\(6\) 2.73357 − 7.41619i 0.455596 − 1.23603i
\(7\) −2.56913 −0.367018 −0.183509 − 0.983018i \(-0.558746\pi\)
−0.183509 + 0.983018i \(0.558746\pi\)
\(8\) −6.97662 + 3.91494i −0.872077 + 0.489368i
\(9\) −3.30904 − 5.73143i −0.367671 − 0.636825i
\(10\) −0.313891 − 9.99507i −0.0313891 − 0.999507i
\(11\) − 17.7707i − 1.61552i −0.589514 − 0.807758i \(-0.700681\pi\)
0.589514 − 0.807758i \(-0.299319\pi\)
\(12\) −2.87189 + 15.5448i −0.239324 + 1.29540i
\(13\) −4.58570 + 2.64756i −0.352746 + 0.203658i −0.665894 − 0.746046i \(-0.731950\pi\)
0.313148 + 0.949704i \(0.398616\pi\)
\(14\) 5.06379 − 0.871606i 0.361699 − 0.0622576i
\(15\) −15.5779 − 12.1566i −1.03852 − 0.810438i
\(16\) 12.4228 − 10.0833i 0.776427 − 0.630207i
\(17\) 1.90065 + 1.09734i 0.111803 + 0.0645496i 0.554859 − 0.831945i \(-0.312772\pi\)
−0.443056 + 0.896494i \(0.646106\pi\)
\(18\) 8.46662 + 10.1741i 0.470368 + 0.565228i
\(19\) −15.3281 − 11.2272i −0.806742 − 0.590903i
\(20\) 4.00963 + 19.5940i 0.200481 + 0.979698i
\(21\) 5.07656 − 8.79286i 0.241741 − 0.418707i
\(22\) 6.02891 + 35.0263i 0.274041 + 1.59210i
\(23\) −8.21652 − 14.2314i −0.357240 − 0.618758i 0.630259 − 0.776385i \(-0.282949\pi\)
−0.987499 + 0.157627i \(0.949616\pi\)
\(24\) 0.386770 − 31.6134i 0.0161154 − 1.31723i
\(25\) −24.0393 − 6.86374i −0.961573 − 0.274550i
\(26\) 8.14028 − 6.77413i 0.313088 − 0.260543i
\(27\) −9.41328 −0.348640
\(28\) −9.68510 + 3.43590i −0.345896 + 0.122711i
\(29\) 24.5292 + 42.4859i 0.845836 + 1.46503i 0.884893 + 0.465794i \(0.154231\pi\)
−0.0390576 + 0.999237i \(0.512436\pi\)
\(30\) 34.8285 + 18.6758i 1.16095 + 0.622528i
\(31\) 13.0380i 0.420581i 0.977639 + 0.210291i \(0.0674411\pi\)
−0.977639 + 0.210291i \(0.932559\pi\)
\(32\) −21.0647 + 24.0890i −0.658272 + 0.752780i
\(33\) 60.8203 + 35.1146i 1.84304 + 1.06408i
\(34\) −4.11851 − 1.51806i −0.121133 − 0.0446489i
\(35\) 1.78057 − 12.7216i 0.0508735 − 0.363475i
\(36\) −20.1395 − 17.1809i −0.559431 − 0.477247i
\(37\) − 53.5256i − 1.44664i −0.690515 − 0.723318i \(-0.742616\pi\)
0.690515 − 0.723318i \(-0.257384\pi\)
\(38\) 34.0209 + 16.9287i 0.895286 + 0.445491i
\(39\) − 20.9261i − 0.536568i
\(40\) −14.5505 − 37.2597i −0.363763 − 0.931492i
\(41\) 12.6985 − 21.9944i 0.309718 − 0.536448i −0.668582 − 0.743638i \(-0.733099\pi\)
0.978301 + 0.207190i \(0.0664319\pi\)
\(42\) −7.02290 + 19.0531i −0.167212 + 0.453646i
\(43\) 36.7927 − 63.7268i 0.855644 − 1.48202i −0.0204018 − 0.999792i \(-0.506495\pi\)
0.876046 − 0.482227i \(-0.160172\pi\)
\(44\) −23.7661 − 66.9920i −0.540140 − 1.52254i
\(45\) 30.6739 − 12.4132i 0.681642 − 0.275850i
\(46\) 21.0231 + 25.2628i 0.457023 + 0.549192i
\(47\) 27.9634 + 48.4340i 0.594966 + 1.03051i 0.993552 + 0.113380i \(0.0361677\pi\)
−0.398586 + 0.917131i \(0.630499\pi\)
\(48\) 9.96288 + 62.4417i 0.207560 + 1.30087i
\(49\) −42.3996 −0.865298
\(50\) 49.7105 + 5.37293i 0.994210 + 0.107459i
\(51\) −7.51133 + 4.33667i −0.147281 + 0.0850327i
\(52\) −13.7464 + 16.1136i −0.264354 + 0.309877i
\(53\) 38.6169 − 22.2955i 0.728620 − 0.420669i −0.0892969 − 0.996005i \(-0.528462\pi\)
0.817917 + 0.575336i \(0.195129\pi\)
\(54\) 18.5537 − 3.19356i 0.343587 − 0.0591401i
\(55\) 87.9957 + 12.3162i 1.59992 + 0.223932i
\(56\) 17.9238 − 10.0580i 0.320068 − 0.179607i
\(57\) 68.7132 − 30.2759i 1.20549 − 0.531155i
\(58\) −62.7613 − 75.4185i −1.08209 − 1.30032i
\(59\) 58.4823 + 33.7648i 0.991225 + 0.572284i 0.905640 − 0.424047i \(-0.139391\pi\)
0.0855850 + 0.996331i \(0.472724\pi\)
\(60\) −74.9834 − 24.9944i −1.24972 − 0.416573i
\(61\) −2.07481 − 3.59367i −0.0340132 − 0.0589127i 0.848518 − 0.529167i \(-0.177496\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(62\) −4.42330 − 25.6981i −0.0713436 − 0.414486i
\(63\) 8.50134 + 14.7248i 0.134942 + 0.233726i
\(64\) 33.3464 − 54.6262i 0.521038 − 0.853534i
\(65\) −9.93181 − 24.5421i −0.152797 − 0.377571i
\(66\) −131.791 − 48.5775i −1.99683 − 0.736022i
\(67\) −51.0342 − 88.3938i −0.761704 − 1.31931i −0.941971 − 0.335693i \(-0.891030\pi\)
0.180267 − 0.983618i \(-0.442304\pi\)
\(68\) 8.63266 + 1.59487i 0.126951 + 0.0234540i
\(69\) 64.9429 0.941202
\(70\) 0.806426 + 25.6786i 0.0115204 + 0.366837i
\(71\) 35.9292 + 20.7437i 0.506045 + 0.292165i 0.731207 − 0.682156i \(-0.238958\pi\)
−0.225161 + 0.974322i \(0.572291\pi\)
\(72\) 45.5241 + 27.0313i 0.632280 + 0.375434i
\(73\) 41.1613 + 23.7645i 0.563853 + 0.325541i 0.754690 − 0.656081i \(-0.227787\pi\)
−0.190838 + 0.981622i \(0.561120\pi\)
\(74\) 18.1592 + 105.500i 0.245394 + 1.42567i
\(75\) 70.9926 − 68.7121i 0.946568 − 0.916161i
\(76\) −72.7989 − 21.8247i −0.957881 − 0.287167i
\(77\) 45.6551i 0.592924i
\(78\) 7.09944 + 41.2457i 0.0910184 + 0.528792i
\(79\) −63.0816 − 36.4202i −0.798501 − 0.461015i 0.0444457 − 0.999012i \(-0.485848\pi\)
−0.842947 + 0.537997i \(0.819181\pi\)
\(80\) 41.3201 + 68.5029i 0.516501 + 0.856287i
\(81\) 48.3819 − 83.7999i 0.597307 − 1.03457i
\(82\) −17.5670 + 47.6593i −0.214232 + 0.581211i
\(83\) −124.038 −1.49444 −0.747219 − 0.664577i \(-0.768612\pi\)
−0.747219 + 0.664577i \(0.768612\pi\)
\(84\) 7.37824 − 39.9366i 0.0878362 − 0.475436i
\(85\) −6.75103 + 8.65100i −0.0794239 + 0.101776i
\(86\) −50.8989 + 138.089i −0.591848 + 1.60569i
\(87\) −193.878 −2.22848
\(88\) 69.5712 + 123.979i 0.790582 + 1.40886i
\(89\) 50.2003 + 86.9495i 0.564049 + 0.976961i 0.997137 + 0.0756097i \(0.0240903\pi\)
−0.433089 + 0.901351i \(0.642576\pi\)
\(90\) −56.2473 + 34.8731i −0.624970 + 0.387479i
\(91\) 11.7813 − 6.80191i 0.129464 − 0.0747463i
\(92\) −50.0075 − 42.6611i −0.543560 − 0.463708i
\(93\) −44.6227 − 25.7630i −0.479814 − 0.277021i
\(94\) −71.5481 − 85.9773i −0.761150 − 0.914652i
\(95\) 66.2173 − 68.1195i 0.697024 − 0.717048i
\(96\) −40.8211 − 119.694i −0.425220 − 1.24681i
\(97\) −53.4146 − 30.8390i −0.550666 − 0.317927i 0.198724 − 0.980055i \(-0.436320\pi\)
−0.749391 + 0.662128i \(0.769654\pi\)
\(98\) 83.5702 − 14.3846i 0.852758 − 0.146781i
\(99\) −101.851 + 58.8039i −1.02880 + 0.593979i
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.239.7 yes 232
4.3 odd 2 inner 380.3.p.a.239.45 yes 232
5.4 even 2 inner 380.3.p.a.239.110 yes 232
19.7 even 3 inner 380.3.p.a.159.72 yes 232
20.19 odd 2 inner 380.3.p.a.239.72 yes 232
76.7 odd 6 inner 380.3.p.a.159.110 yes 232
95.64 even 6 inner 380.3.p.a.159.45 yes 232
380.159 odd 6 inner 380.3.p.a.159.7 ✓ 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 380.159 odd 6 inner
380.3.p.a.159.45 yes 232 95.64 even 6 inner
380.3.p.a.159.72 yes 232 19.7 even 3 inner
380.3.p.a.159.110 yes 232 76.7 odd 6 inner
380.3.p.a.239.7 yes 232 1.1 even 1 trivial
380.3.p.a.239.45 yes 232 4.3 odd 2 inner
380.3.p.a.239.72 yes 232 20.19 odd 2 inner
380.3.p.a.239.110 yes 232 5.4 even 2 inner