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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 239.72
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.72

$q$-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+(0.691699 + 1.87658i) q^{2} +(-1.97599 + 3.42251i) q^{3} +(-3.04311 + 2.59606i) q^{4} +(-3.94179 + 3.07608i) q^{5} +(-7.78940 - 1.34075i) q^{6} -2.56913 q^{7} +(-6.97662 - 3.91494i) q^{8} +(-3.30904 - 5.73143i) q^{9} +(-8.49904 - 5.26937i) q^{10} +17.7707i q^{11} +(-2.87189 - 15.5448i) q^{12} +(4.58570 - 2.64756i) q^{13} +(-1.77706 - 4.82117i) q^{14} +(-2.73897 - 19.5691i) q^{15} +(2.52099 - 15.8001i) 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} +(-33.3481 + 12.2920i) q^{22} +(-8.21652 - 14.2314i) q^{23} +(27.1846 - 16.1417i) q^{24} +(6.07548 - 24.2505i) q^{25} +(8.14028 + 6.77413i) q^{26} -9.41328 q^{27} +(7.81812 - 6.66960i) q^{28} +(24.5292 + 42.4859i) q^{29} +(34.8285 - 18.6758i) q^{30} -13.0380i q^{31} +(31.3940 - 6.19810i) q^{32} +(-60.8203 - 35.1146i) q^{33} +(0.744573 - 4.32576i) q^{34} +(10.1270 - 7.90283i) q^{35} +(24.9489 + 8.85089i) q^{36} +53.5256i q^{37} +(-10.4662 + 36.5302i) q^{38} +20.9261i q^{39} +(39.5431 - 6.02872i) q^{40} +(12.6985 - 21.9944i) q^{41} +(20.0119 + 3.44456i) q^{42} +(36.7927 - 63.7268i) q^{43} +(-46.1337 - 54.0781i) q^{44} +(30.6739 + 12.4132i) q^{45} +(21.0231 - 25.2628i) q^{46} +(27.9634 + 48.4340i) q^{47} +(49.0947 + 39.8490i) q^{48} -42.3996 q^{49} +(49.7105 - 5.37293i) q^{50} +(7.51133 - 4.33667i) q^{51} +(-7.08158 + 19.9615i) q^{52} +(-38.6169 + 22.2955i) q^{53} +(-6.51115 - 17.6648i) 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} +(59.1375 + 52.4404i) q^{60} +(-2.07481 - 3.59367i) q^{61} +(24.4669 - 9.01838i) q^{62} +(8.50134 + 14.7248i) q^{63} +(33.3464 + 54.6262i) q^{64} +(-9.93181 + 24.5421i) q^{65} +(23.8261 - 138.423i) q^{66} +(-51.0342 - 88.3938i) q^{67} +(8.63266 - 1.59487i) q^{68} +64.9429 q^{69} +(21.8351 + 13.5377i) q^{70} +(-35.9292 - 20.7437i) q^{71} +(0.647696 + 52.9407i) q^{72} +(-41.1613 - 23.7645i) q^{73} +(-100.445 + 37.0236i) q^{74} +(70.9926 + 68.7121i) q^{75} +(-75.7914 + 5.62717i) q^{76} -45.6551i q^{77} +(-39.2696 + 14.4746i) q^{78} +(63.0816 + 36.4202i) q^{79} +(38.6653 + 70.0357i) q^{80} +(48.3819 - 83.7999i) q^{81} +(50.0577 + 8.61619i) q^{82} -124.038 q^{83} +(7.37824 + 39.9366i) q^{84} +(10.8675 - 1.52106i) q^{85} +(145.038 + 24.9647i) q^{86} -193.878 q^{87} +(69.5712 - 123.979i) q^{88} +(50.2003 + 86.9495i) q^{89} +(-2.07736 + 66.1482i) q^{90} +(-11.7813 + 6.80191i) q^{91} +(61.9494 + 21.9772i) 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} +(-29.3277 - 79.5662i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.691699 + 1.87658i 0.345849 + 0.938290i
\(3\) −1.97599 + 3.42251i −0.658662 + 1.14084i 0.322300 + 0.946637i \(0.395544\pi\)
−0.980962 + 0.194199i \(0.937789\pi\)
\(4\) −3.04311 + 2.59606i −0.760777 + 0.649014i
\(5\) −3.94179 + 3.07608i −0.788359 + 0.615216i
\(6\) −7.78940 − 1.34075i −1.29823 − 0.223459i
\(7\) −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
\(10\) −8.49904 − 5.26937i −0.849904 − 0.526937i
\(11\) 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 \(-0.601384\pi\)
0.665894 + 0.746046i \(0.268050\pi\)
\(14\) −1.77706 − 4.82117i −0.126933 − 0.344369i
\(15\) −2.73897 − 19.5691i −0.182598 − 1.30461i
\(16\) 2.52099 − 15.8001i 0.157562 − 0.987509i
\(17\) −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
\(20\) 4.00963 − 19.5940i 0.200481 − 0.979698i
\(21\) 5.07656 − 8.79286i 0.241741 − 0.418707i
\(22\) −33.3481 + 12.2920i −1.51582 + 0.558725i
\(23\) −8.21652 − 14.2314i −0.357240 − 0.618758i 0.630259 − 0.776385i \(-0.282949\pi\)
−0.987499 + 0.157627i \(0.949616\pi\)
\(24\) 27.1846 − 16.1417i 1.13269 − 0.672569i
\(25\) 6.07548 − 24.2505i 0.243019 − 0.970021i
\(26\) 8.14028 + 6.77413i 0.313088 + 0.260543i
\(27\) −9.41328 −0.348640
\(28\) 7.81812 − 6.66960i 0.279219 − 0.238200i
\(29\) 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 \(-0.932559\pi\)
0.977639 − 0.210291i \(-0.0674411\pi\)
\(32\) 31.3940 − 6.19810i 0.981063 − 0.193691i
\(33\) −60.8203 − 35.1146i −1.84304 − 1.06408i
\(34\) 0.744573 − 4.32576i 0.0218992 − 0.127228i
\(35\) 10.1270 − 7.90283i 0.289342 − 0.225795i
\(36\) 24.9489 + 8.85089i 0.693024 + 0.245858i
\(37\) 53.5256i 1.44664i 0.690515 + 0.723318i \(0.257384\pi\)
−0.690515 + 0.723318i \(0.742616\pi\)
\(38\) −10.4662 + 36.5302i −0.275427 + 0.961322i
\(39\) 20.9261i 0.536568i
\(40\) 39.5431 − 6.02872i 0.988577 − 0.150718i
\(41\) 12.6985 − 21.9944i 0.309718 − 0.536448i −0.668582 − 0.743638i \(-0.733099\pi\)
0.978301 + 0.207190i \(0.0664319\pi\)
\(42\) 20.0119 + 3.44456i 0.476475 + 0.0820134i
\(43\) 36.7927 − 63.7268i 0.855644 − 1.48202i −0.0204018 − 0.999792i \(-0.506495\pi\)
0.876046 − 0.482227i \(-0.160172\pi\)
\(44\) −46.1337 − 54.0781i −1.04849 − 1.22905i
\(45\) 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 \(0.630499\pi\)
\(48\) 49.0947 + 39.8490i 1.02281 + 0.830187i
\(49\) −42.3996 −0.865298
\(50\) 49.7105 − 5.37293i 0.994210 − 0.107459i
\(51\) 7.51133 − 4.33667i 0.147281 − 0.0850327i
\(52\) −7.08158 + 19.9615i −0.136184 + 0.383876i
\(53\) −38.6169 + 22.2955i −0.728620 + 0.420669i −0.817917 − 0.575336i \(-0.804871\pi\)
0.0892969 + 0.996005i \(0.471538\pi\)
\(54\) −6.51115 − 17.6648i −0.120577 − 0.327125i
\(55\) −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 −0.0855850 − 0.996331i \(-0.527276\pi\)
−0.905640 + 0.424047i \(0.860609\pi\)
\(60\) 59.1375 + 52.4404i 0.985625 + 0.874006i
\(61\) −2.07481 − 3.59367i −0.0340132 − 0.0589127i 0.848518 − 0.529167i \(-0.177496\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(62\) 24.4669 − 9.01838i 0.394627 − 0.145458i
\(63\) 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
\(66\) 23.8261 − 138.423i 0.361001 − 2.09732i
\(67\) −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
\(70\) 21.8351 + 13.5377i 0.311930 + 0.193396i
\(71\) −35.9292 − 20.7437i −0.506045 − 0.292165i 0.225161 − 0.974322i \(-0.427709\pi\)
−0.731207 + 0.682156i \(0.761042\pi\)
\(72\) 0.647696 + 52.9407i 0.00899578 + 0.735287i
\(73\) −41.1613 − 23.7645i −0.563853 − 0.325541i 0.190838 − 0.981622i \(-0.438880\pi\)
−0.754690 + 0.656081i \(0.772213\pi\)
\(74\) −100.445 + 37.0236i −1.35736 + 0.500318i
\(75\) 70.9926 + 68.7121i 0.946568 + 0.916161i
\(76\) −75.7914 + 5.62717i −0.997255 + 0.0740417i
\(77\) − 45.6551i − 0.592924i
\(78\) −39.2696 + 14.4746i −0.503456 + 0.185572i
\(79\) 63.0816 + 36.4202i 0.798501 + 0.461015i 0.842947 − 0.537997i \(-0.180819\pi\)
−0.0444457 + 0.999012i \(0.514152\pi\)
\(80\) 38.6653 + 70.0357i 0.483316 + 0.875446i
\(81\) 48.3819 − 83.7999i 0.597307 − 1.03457i
\(82\) 50.0577 + 8.61619i 0.610460 + 0.105076i
\(83\) −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
\(85\) 10.8675 − 1.52106i 0.127853 − 0.0178948i
\(86\) 145.038 + 24.9647i 1.68649 + 0.290287i
\(87\) −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 \(0.642576\pi\)
\(90\) −2.07736 + 66.1482i −0.0230817 + 0.734980i
\(91\) −11.7813 + 6.80191i −0.129464 + 0.0747463i
\(92\) 61.9494 + 21.9772i 0.673363 + 0.238883i
\(93\) 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 \(0.563680\pi\)
\(98\) −29.3277 − 79.5662i −0.299263 − 0.811900i
\(99\) 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.239.72 yes 232
4.3 odd 2 inner 380.3.p.a.239.110 yes 232
5.4 even 2 inner 380.3.p.a.239.45 yes 232
19.7 even 3 inner 380.3.p.a.159.7 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.7 yes 232
76.7 odd 6 inner 380.3.p.a.159.45 yes 232
95.64 even 6 inner 380.3.p.a.159.110 yes 232
380.159 odd 6 inner 380.3.p.a.159.72 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.7 ✓ 232 19.7 even 3 inner
380.3.p.a.159.45 yes 232 76.7 odd 6 inner
380.3.p.a.159.72 yes 232 380.159 odd 6 inner
380.3.p.a.159.110 yes 232 95.64 even 6 inner
380.3.p.a.239.7 yes 232 20.19 odd 2 inner
380.3.p.a.239.45 yes 232 5.4 even 2 inner
380.3.p.a.239.72 yes 232 1.1 even 1 trivial
380.3.p.a.239.110 yes 232 4.3 odd 2 inner