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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 239.110
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.110

$q$-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} +(-3.94179 + 3.07608i) q^{5} +(2.73357 - 7.41619i) q^{6} +2.56913 q^{7} +(6.97662 - 3.91494i) q^{8} +(-3.30904 - 5.73143i) q^{9} +(-6.72574 + 7.40030i) q^{10} -17.7707i q^{11} +(2.87189 - 15.5448i) q^{12} +(4.58570 - 2.64756i) q^{13} +(5.06379 - 0.871606i) q^{14} +(2.73897 + 19.5691i) 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} +(-10.7459 + 16.8679i) 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} +(6.07548 - 24.2505i) q^{25} +(8.14028 - 6.77413i) q^{26} +9.41328 q^{27} +(9.68510 - 3.43590i) q^{28} +(24.5292 + 42.4859i) q^{29} +(12.0376 + 37.6418i) q^{30} +13.0380i q^{31} +(21.0647 - 24.0890i) q^{32} +(-60.8203 - 35.1146i) q^{33} +(-4.11851 - 1.51806i) q^{34} +(-10.1270 + 7.90283i) q^{35} +(-20.1395 - 17.1809i) q^{36} +53.5256i q^{37} +(-34.0209 - 16.9287i) q^{38} -20.9261i q^{39} +(-15.4577 + 36.8925i) 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} +(3.74760 - 49.8594i) 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} +(54.6640 + 70.0484i) 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} +(36.4967 + 70.1086i) 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} +(-17.2793 + 19.0123i) 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} +(-17.9512 + 77.9600i) q^{80} +(48.3819 - 83.7999i) q^{81} +(17.5670 - 47.6593i) q^{82} +124.038 q^{83} +(7.37824 - 39.9366i) q^{84} +(10.8675 - 1.52106i) q^{85} +(-50.8989 + 138.089i) q^{86} +193.878 q^{87} +(-69.5712 - 123.979i) q^{88} +(50.2003 + 86.9495i) q^{89} +(64.6700 + 14.0602i) 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} +(94.9559 - 2.89530i) 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 \(-0.604456\pi\)
0.980962 − 0.194199i \(-0.0622106\pi\)
\(4\) 3.76980 − 1.33738i 0.942451 − 0.334345i
\(5\) −3.94179 + 3.07608i −0.788359 + 0.615216i
\(6\) 2.73357 − 7.41619i 0.455596 − 1.23603i
\(7\) 2.56913 0.367018 0.183509 − 0.983018i \(-0.441254\pi\)
0.183509 + 0.983018i \(0.441254\pi\)
\(8\) 6.97662 − 3.91494i 0.872077 − 0.489368i
\(9\) −3.30904 − 5.73143i −0.367671 − 0.636825i
\(10\) −6.72574 + 7.40030i −0.672574 + 0.740030i
\(11\) − 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 −0.313148 − 0.949704i \(-0.601384\pi\)
0.665894 + 0.746046i \(0.268050\pi\)
\(14\) 5.06379 − 0.871606i 0.361699 − 0.0622576i
\(15\) 2.73897 + 19.5691i 0.182598 + 1.30461i
\(16\) 12.4228 − 10.0833i 0.776427 − 0.630207i
\(17\) −1.90065 − 1.09734i −0.111803 − 0.0645496i 0.443056 − 0.896494i \(-0.353894\pi\)
−0.554859 + 0.831945i \(0.687228\pi\)
\(18\) −8.46662 − 10.1741i −0.470368 − 0.565228i
\(19\) −15.3281 − 11.2272i −0.806742 − 0.590903i
\(20\) −10.7459 + 16.8679i −0.537295 + 0.843394i
\(21\) 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 0.987499 − 0.157627i \(-0.0503844\pi\)
−0.630259 + 0.776385i \(0.717051\pi\)
\(24\) 0.386770 − 31.6134i 0.0161154 − 1.31723i
\(25\) 6.07548 − 24.2505i 0.243019 − 0.970021i
\(26\) 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 \(0.512436\pi\)
\(30\) 12.0376 + 37.6418i 0.401253 + 1.25473i
\(31\) 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
\(35\) −10.1270 + 7.90283i −0.289342 + 0.225795i
\(36\) −20.1395 − 17.1809i −0.559431 − 0.477247i
\(37\) 53.5256i 1.44664i 0.690515 + 0.723318i \(0.257384\pi\)
−0.690515 + 0.723318i \(0.742616\pi\)
\(38\) −34.0209 − 16.9287i −0.895286 − 0.445491i
\(39\) − 20.9261i − 0.536568i
\(40\) −15.4577 + 36.8925i −0.386443 + 0.922313i
\(41\) 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 \(0.493505\pi\)
−0.876046 + 0.482227i \(0.839828\pi\)
\(44\) −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 \(-0.963832\pi\)
0.398586 − 0.917131i \(-0.369501\pi\)
\(48\) −9.96288 − 62.4417i −0.207560 − 1.30087i
\(49\) −42.3996 −0.865298
\(50\) 3.74760 − 49.8594i 0.0749519 − 0.997187i
\(51\) −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 −0.817917 − 0.575336i \(-0.804871\pi\)
0.0892969 + 0.996005i \(0.471538\pi\)
\(54\) 18.5537 − 3.19356i 0.343587 − 0.0591401i
\(55\) 54.6640 + 70.0484i 0.993891 + 1.27361i
\(56\) 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 \(0.472724\pi\)
\(60\) 36.4967 + 70.1086i 0.608279 + 1.16848i
\(61\) −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 \(0.108970\pi\)
−0.180267 + 0.983618i \(0.557696\pi\)
\(68\) −8.63266 − 1.59487i −0.126951 − 0.0234540i
\(69\) 64.9429 0.941202
\(70\) −17.2793 + 19.0123i −0.246847 + 0.271604i
\(71\) 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 0.190838 − 0.981622i \(-0.438880\pi\)
−0.754690 + 0.656081i \(0.772213\pi\)
\(74\) 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 \(0.819181\pi\)
\(80\) −17.9512 + 77.9600i −0.224390 + 0.974499i
\(81\) 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 \(-0.231388\pi\)
0.747219 + 0.664577i \(0.231388\pi\)
\(84\) 7.37824 − 39.9366i 0.0878362 − 0.475436i
\(85\) 10.8675 − 1.52106i 0.127853 − 0.0178948i
\(86\) −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 \(0.642576\pi\)
\(90\) 64.6700 + 14.0602i 0.718556 + 0.156225i
\(91\) 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
\(95\) 94.9559 − 2.89530i 0.999535 − 0.0304768i
\(96\) −40.8211 − 119.694i −0.425220 − 1.24681i
\(97\) 53.4146 + 30.8390i 0.550666 + 0.317927i 0.749391 − 0.662128i \(-0.230346\pi\)
−0.198724 + 0.980055i \(0.563680\pi\)
\(98\) −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.239.110 yes 232
4.3 odd 2 inner 380.3.p.a.239.72 yes 232
5.4 even 2 inner 380.3.p.a.239.7 yes 232
19.7 even 3 inner 380.3.p.a.159.45 yes 232
20.19 odd 2 inner 380.3.p.a.239.45 yes 232
76.7 odd 6 inner 380.3.p.a.159.7 ✓ 232
95.64 even 6 inner 380.3.p.a.159.72 yes 232
380.159 odd 6 inner 380.3.p.a.159.110 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 76.7 odd 6 inner
380.3.p.a.159.45 yes 232 19.7 even 3 inner
380.3.p.a.159.72 yes 232 95.64 even 6 inner
380.3.p.a.159.110 yes 232 380.159 odd 6 inner
380.3.p.a.239.7 yes 232 5.4 even 2 inner
380.3.p.a.239.45 yes 232 20.19 odd 2 inner
380.3.p.a.239.72 yes 232 4.3 odd 2 inner
380.3.p.a.239.110 yes 232 1.1 even 1 trivial