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Properties

Label 380.3.p.a.159.7
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 159.7
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.7

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97102 − 0.339262i −0.985508 − 0.169631i
\(3\) −1.97599 − 3.42251i −0.658662 − 1.14084i −0.980962 − 0.194199i \(-0.937789\pi\)
0.322300 − 0.946637i \(-0.395544\pi\)
\(4\) 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 \(0.299319\pi\)
−0.589514 + 0.807758i \(0.700681\pi\)
\(12\) −2.87189 − 15.5448i −0.239324 − 1.29540i
\(13\) −4.58570 − 2.64756i −0.352746 − 0.203658i 0.313148 − 0.949704i \(-0.398616\pi\)
−0.665894 + 0.746046i \(0.731950\pi\)
\(14\) 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 −0.443056 − 0.896494i \(-0.646106\pi\)
0.554859 + 0.831945i \(0.312772\pi\)
\(18\) 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 −0.987499 − 0.157627i \(-0.949616\pi\)
0.630259 + 0.776385i \(0.282949\pi\)
\(24\) 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 −0.0390576 − 0.999237i \(-0.512436\pi\)
0.884893 − 0.465794i \(-0.154231\pi\)
\(30\) 34.8285 − 18.6758i 1.16095 − 0.622528i
\(31\) − 13.0380i − 0.420581i −0.977639 − 0.210291i \(-0.932559\pi\)
0.977639 − 0.210291i \(-0.0674411\pi\)
\(32\) −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 \(0.257384\pi\)
−0.690515 + 0.723318i \(0.742616\pi\)
\(38\) 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 0.978301 − 0.207190i \(-0.0664319\pi\)
−0.668582 + 0.743638i \(0.733099\pi\)
\(42\) −7.02290 − 19.0531i −0.167212 − 0.453646i
\(43\) 36.7927 + 63.7268i 0.855644 + 1.48202i 0.876046 + 0.482227i \(0.160172\pi\)
−0.0204018 + 0.999792i \(0.506495\pi\)
\(44\) −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 −0.398586 − 0.917131i \(-0.630499\pi\)
0.993552 − 0.113380i \(-0.0361677\pi\)
\(48\) 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 0.817917 − 0.575336i \(-0.195129\pi\)
−0.0892969 + 0.996005i \(0.528462\pi\)
\(54\) 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 0.0855850 − 0.996331i \(-0.472724\pi\)
0.905640 + 0.424047i \(0.139391\pi\)
\(60\) −74.9834 + 24.9944i −1.24972 + 0.416573i
\(61\) −2.07481 + 3.59367i −0.0340132 + 0.0589127i −0.882531 − 0.470254i \(-0.844162\pi\)
0.848518 + 0.529167i \(0.177496\pi\)
\(62\) −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 0.180267 + 0.983618i \(0.442304\pi\)
−0.941971 + 0.335693i \(0.891030\pi\)
\(68\) 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 −0.225161 − 0.974322i \(-0.572291\pi\)
0.731207 + 0.682156i \(0.238958\pi\)
\(72\) 45.5241 − 27.0313i 0.632280 − 0.375434i
\(73\) 41.1613 − 23.7645i 0.563853 − 0.325541i −0.190838 − 0.981622i \(-0.561120\pi\)
0.754690 + 0.656081i \(0.227787\pi\)
\(74\) 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 −0.842947 − 0.537997i \(-0.819181\pi\)
0.0444457 + 0.999012i \(0.485848\pi\)
\(80\) 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 −0.433089 − 0.901351i \(-0.642576\pi\)
0.997137 − 0.0756097i \(-0.0240903\pi\)
\(90\) −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 −0.749391 − 0.662128i \(-0.769654\pi\)
0.198724 + 0.980055i \(0.436320\pi\)
\(98\) 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.159.7 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.45 yes 232
5.4 even 2 inner 380.3.p.a.159.110 yes 232
19.11 even 3 inner 380.3.p.a.239.72 yes 232
20.19 odd 2 inner 380.3.p.a.159.72 yes 232
76.11 odd 6 inner 380.3.p.a.239.110 yes 232
95.49 even 6 inner 380.3.p.a.239.45 yes 232
380.239 odd 6 inner 380.3.p.a.239.7 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.45 yes 232 4.3 odd 2 inner
380.3.p.a.159.72 yes 232 20.19 odd 2 inner
380.3.p.a.159.110 yes 232 5.4 even 2 inner
380.3.p.a.239.7 yes 232 380.239 odd 6 inner
380.3.p.a.239.45 yes 232 95.49 even 6 inner
380.3.p.a.239.72 yes 232 19.11 even 3 inner
380.3.p.a.239.110 yes 232 76.11 odd 6 inner