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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 159.31
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.31

$q$-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+(-1.38853 + 1.43944i) q^{2} +(-0.480142 - 0.831631i) q^{3} +(-0.143996 - 3.99741i) q^{4} +(4.92739 + 0.849034i) q^{5} +(1.86377 + 0.463602i) q^{6} +5.58137 q^{7} +(5.95398 + 5.34323i) q^{8} +(4.03893 - 6.99563i) q^{9} +(-8.06394 + 5.91379i) q^{10} -16.9515i q^{11} +(-3.25523 + 2.03907i) q^{12} +(4.31012 + 2.48845i) q^{13} +(-7.74988 + 8.03407i) q^{14} +(-1.65976 - 4.50542i) q^{15} +(-15.9585 + 1.15122i) q^{16} +(-20.3861 + 11.7699i) q^{17} +(4.46166 + 15.5274i) q^{18} +(-14.2828 - 12.5300i) q^{19} +(2.68441 - 19.8190i) q^{20} +(-2.67985 - 4.64164i) q^{21} +(24.4008 + 23.5377i) q^{22} +(-1.57744 + 2.73221i) q^{23} +(1.58483 - 7.51702i) q^{24} +(23.5583 + 8.36704i) q^{25} +(-9.56669 + 2.74890i) q^{26} -16.3996 q^{27} +(-0.803694 - 22.3110i) q^{28} +(15.8167 - 27.3954i) q^{29} +(8.78992 + 3.86676i) q^{30} -7.75014i q^{31} +(20.5017 - 24.5699i) q^{32} +(-14.0974 + 8.13915i) q^{33} +(11.3645 - 45.6874i) q^{34} +(27.5016 + 4.73878i) q^{35} +(-28.5460 - 15.1379i) q^{36} -28.8709i q^{37} +(37.8683 - 3.16105i) q^{38} -4.77924i q^{39} +(24.8010 + 31.3833i) q^{40} +(29.1060 + 50.4131i) q^{41} +(10.4024 + 2.58754i) q^{42} +(-19.7090 - 34.1370i) q^{43} +(-67.7622 + 2.44095i) q^{44} +(25.8409 - 31.0410i) q^{45} +(-1.74255 - 6.06439i) q^{46} +(41.8549 - 72.4949i) q^{47} +(8.61975 + 12.7189i) q^{48} -17.8483 q^{49} +(-44.7551 + 22.2930i) q^{50} +(19.5764 + 11.3025i) q^{51} +(9.32671 - 17.5876i) q^{52} +(87.8212 + 50.7036i) q^{53} +(22.7712 - 23.6063i) q^{54} +(14.3924 - 83.5268i) q^{55} +(33.2314 + 29.8226i) q^{56} +(-3.56257 + 17.8942i) q^{57} +(17.4722 + 60.8064i) q^{58} +(78.2702 - 45.1893i) q^{59} +(-17.7710 + 7.28351i) q^{60} +(-35.1456 + 60.8740i) q^{61} +(11.1559 + 10.7613i) q^{62} +(22.5428 - 39.0452i) q^{63} +(6.89985 + 63.6270i) q^{64} +(19.1249 + 15.9210i) q^{65} +(7.85878 - 31.5939i) q^{66} +(-58.0911 + 100.617i) q^{67} +(49.9846 + 79.7967i) q^{68} +3.02959 q^{69} +(-45.0079 + 33.0071i) q^{70} +(-74.5015 + 43.0135i) q^{71} +(61.4269 - 20.0709i) q^{72} +(-13.1006 + 7.56362i) q^{73} +(41.5580 + 40.0879i) q^{74} +(-4.35304 - 23.6092i) q^{75} +(-48.0309 + 58.8985i) q^{76} -94.6129i q^{77} +(6.87944 + 6.63609i) q^{78} +(109.767 - 63.3739i) q^{79} +(-79.6113 - 7.87683i) q^{80} +(-28.4762 - 49.3222i) q^{81} +(-112.981 - 28.1034i) q^{82} -22.6687 q^{83} +(-18.1686 + 11.3808i) q^{84} +(-110.443 + 40.6864i) q^{85} +(76.5047 + 19.0301i) q^{86} -30.3771 q^{87} +(90.5760 - 100.929i) q^{88} +(-5.95738 + 10.3185i) q^{89} +(8.80102 + 80.2977i) q^{90} +(24.0564 + 13.8890i) q^{91} +(11.1489 + 5.91226i) q^{92} +(-6.44526 + 3.72117i) q^{93} +(46.2357 + 160.909i) q^{94} +(-59.7385 - 73.8668i) q^{95} +(-30.2768 - 5.25280i) q^{96} +(31.5731 - 18.2287i) q^{97} +(24.7828 - 25.6916i) q^{98} +(-118.587 - 68.4661i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38853 + 1.43944i −0.694263 + 0.719722i
\(3\) −0.480142 − 0.831631i −0.160047 − 0.277210i 0.774838 − 0.632160i \(-0.217831\pi\)
−0.934885 + 0.354950i \(0.884498\pi\)
\(4\) −0.143996 − 3.99741i −0.0359989 − 0.999352i
\(5\) 4.92739 + 0.849034i 0.985477 + 0.169807i
\(6\) 1.86377 + 0.463602i 0.310629 + 0.0772671i
\(7\) 5.58137 0.797339 0.398670 − 0.917095i \(-0.369472\pi\)
0.398670 + 0.917095i \(0.369472\pi\)
\(8\) 5.95398 + 5.34323i 0.744248 + 0.667903i
\(9\) 4.03893 − 6.99563i 0.448770 − 0.777292i
\(10\) −8.06394 + 5.91379i −0.806394 + 0.591379i
\(11\) − 16.9515i − 1.54105i −0.637410 − 0.770525i \(-0.719994\pi\)
0.637410 − 0.770525i \(-0.280006\pi\)
\(12\) −3.25523 + 2.03907i −0.271269 + 0.169923i
\(13\) 4.31012 + 2.48845i 0.331548 + 0.191419i 0.656528 − 0.754302i \(-0.272024\pi\)
−0.324980 + 0.945721i \(0.605358\pi\)
\(14\) −7.74988 + 8.03407i −0.553563 + 0.573862i
\(15\) −1.65976 − 4.50542i −0.110651 − 0.300361i
\(16\) −15.9585 + 1.15122i −0.997408 + 0.0719512i
\(17\) −20.3861 + 11.7699i −1.19918 + 0.692348i −0.960373 − 0.278719i \(-0.910090\pi\)
−0.238809 + 0.971067i \(0.576757\pi\)
\(18\) 4.46166 + 15.5274i 0.247870 + 0.862634i
\(19\) −14.2828 − 12.5300i −0.751727 − 0.659475i
\(20\) 2.68441 − 19.8190i 0.134221 − 0.990951i
\(21\) −2.67985 − 4.64164i −0.127612 − 0.221031i
\(22\) 24.4008 + 23.5377i 1.10913 + 1.06989i
\(23\) −1.57744 + 2.73221i −0.0685845 + 0.118792i −0.898278 − 0.439427i \(-0.855182\pi\)
0.829694 + 0.558219i \(0.188515\pi\)
\(24\) 1.58483 − 7.51702i 0.0660347 − 0.313209i
\(25\) 23.5583 + 8.36704i 0.942331 + 0.334681i
\(26\) −9.56669 + 2.74890i −0.367950 + 0.105727i
\(27\) −16.3996 −0.607392
\(28\) −0.803694 − 22.3110i −0.0287034 − 0.796822i
\(29\) 15.8167 − 27.3954i 0.545404 − 0.944668i −0.453177 − 0.891420i \(-0.649710\pi\)
0.998581 − 0.0532473i \(-0.0169572\pi\)
\(30\) 8.78992 + 3.86676i 0.292997 + 0.128892i
\(31\) − 7.75014i − 0.250005i −0.992156 − 0.125002i \(-0.960106\pi\)
0.992156 − 0.125002i \(-0.0398938\pi\)
\(32\) 20.5017 − 24.5699i 0.640678 − 0.767809i
\(33\) −14.0974 + 8.13915i −0.427195 + 0.246641i
\(34\) 11.3645 − 45.6874i 0.334249 − 1.34375i
\(35\) 27.5016 + 4.73878i 0.785760 + 0.135394i
\(36\) −28.5460 − 15.1379i −0.792943 − 0.420497i
\(37\) − 28.8709i − 0.780294i −0.920753 − 0.390147i \(-0.872424\pi\)
0.920753 − 0.390147i \(-0.127576\pi\)
\(38\) 37.8683 − 3.16105i 0.996534 − 0.0831854i
\(39\) − 4.77924i − 0.122545i
\(40\) 24.8010 + 31.3833i 0.620025 + 0.784582i
\(41\) 29.1060 + 50.4131i 0.709903 + 1.22959i 0.964893 + 0.262644i \(0.0845945\pi\)
−0.254990 + 0.966944i \(0.582072\pi\)
\(42\) 10.4024 + 2.58754i 0.247677 + 0.0616081i
\(43\) −19.7090 − 34.1370i −0.458349 − 0.793883i 0.540525 − 0.841328i \(-0.318226\pi\)
−0.998874 + 0.0474446i \(0.984892\pi\)
\(44\) −67.7622 + 2.44095i −1.54005 + 0.0554761i
\(45\) 25.8409 − 31.0410i 0.574242 − 0.689799i
\(46\) −1.74255 − 6.06439i −0.0378814 − 0.131835i
\(47\) 41.8549 − 72.4949i 0.890531 − 1.54244i 0.0512907 − 0.998684i \(-0.483667\pi\)
0.839240 − 0.543761i \(-0.183000\pi\)
\(48\) 8.61975 + 12.7189i 0.179578 + 0.264976i
\(49\) −17.8483 −0.364250
\(50\) −44.7551 + 22.2930i −0.895103 + 0.445860i
\(51\) 19.5764 + 11.3025i 0.383852 + 0.221617i
\(52\) 9.32671 − 17.5876i 0.179360 − 0.338224i
\(53\) 87.8212 + 50.7036i 1.65700 + 0.956671i 0.974087 + 0.226175i \(0.0726220\pi\)
0.682916 + 0.730497i \(0.260711\pi\)
\(54\) 22.7712 − 23.6063i 0.421690 − 0.437154i
\(55\) 14.3924 − 83.5268i 0.261681 − 1.51867i
\(56\) 33.2314 + 29.8226i 0.593418 + 0.532546i
\(57\) −3.56257 + 17.8942i −0.0625012 + 0.313933i
\(58\) 17.4722 + 60.8064i 0.301244 + 1.04839i
\(59\) 78.2702 − 45.1893i 1.32661 − 0.765921i 0.341839 − 0.939758i \(-0.388950\pi\)
0.984774 + 0.173838i \(0.0556168\pi\)
\(60\) −17.7710 + 7.28351i −0.296183 + 0.121392i
\(61\) −35.1456 + 60.8740i −0.576158 + 0.997935i 0.419757 + 0.907637i \(0.362115\pi\)
−0.995915 + 0.0902983i \(0.971218\pi\)
\(62\) 11.1559 + 10.7613i 0.179934 + 0.173569i
\(63\) 22.5428 − 39.0452i 0.357822 − 0.619765i
\(64\) 6.89985 + 63.6270i 0.107810 + 0.994171i
\(65\) 19.1249 + 15.9210i 0.294229 + 0.244938i
\(66\) 7.85878 − 31.5939i 0.119072 − 0.478695i
\(67\) −58.0911 + 100.617i −0.867031 + 1.50174i −0.00201423 + 0.999998i \(0.500641\pi\)
−0.865017 + 0.501743i \(0.832692\pi\)
\(68\) 49.9846 + 79.7967i 0.735068 + 1.17348i
\(69\) 3.02959 0.0439071
\(70\) −45.0079 + 33.0071i −0.642969 + 0.471530i
\(71\) −74.5015 + 43.0135i −1.04932 + 0.605824i −0.922458 − 0.386097i \(-0.873823\pi\)
−0.126859 + 0.991921i \(0.540490\pi\)
\(72\) 61.4269 − 20.0709i 0.853152 − 0.278763i
\(73\) −13.1006 + 7.56362i −0.179460 + 0.103611i −0.587039 − 0.809559i \(-0.699706\pi\)
0.407579 + 0.913170i \(0.366373\pi\)
\(74\) 41.5580 + 40.0879i 0.561595 + 0.541729i
\(75\) −4.35304 − 23.6092i −0.0580406 − 0.314789i
\(76\) −48.0309 + 58.8985i −0.631986 + 0.774980i
\(77\) − 94.6129i − 1.22874i
\(78\) 6.87944 + 6.63609i 0.0881980 + 0.0850781i
\(79\) 109.767 − 63.3739i 1.38945 − 0.802202i 0.396201 − 0.918164i \(-0.370328\pi\)
0.993254 + 0.115962i \(0.0369951\pi\)
\(80\) −79.6113 − 7.87683i −0.995141 − 0.0984604i
\(81\) −28.4762 − 49.3222i −0.351558 − 0.608917i
\(82\) −112.981 − 28.1034i −1.37782 − 0.342724i
\(83\) −22.6687 −0.273117 −0.136559 − 0.990632i \(-0.543604\pi\)
−0.136559 + 0.990632i \(0.543604\pi\)
\(84\) −18.1686 + 11.3808i −0.216293 + 0.135486i
\(85\) −110.443 + 40.6864i −1.29933 + 0.478664i
\(86\) 76.5047 + 19.0301i 0.889589 + 0.221280i
\(87\) −30.3771 −0.349162
\(88\) 90.5760 − 100.929i 1.02927 − 1.14692i
\(89\) −5.95738 + 10.3185i −0.0669369 + 0.115938i −0.897552 − 0.440910i \(-0.854656\pi\)
0.830615 + 0.556848i \(0.187989\pi\)
\(90\) 8.80102 + 80.2977i 0.0977891 + 0.892196i
\(91\) 24.0564 + 13.8890i 0.264356 + 0.152626i
\(92\) 11.1489 + 5.91226i 0.121184 + 0.0642637i
\(93\) −6.44526 + 3.72117i −0.0693038 + 0.0400126i
\(94\) 46.2357 + 160.909i 0.491869 + 1.71180i
\(95\) −59.7385 − 73.8668i −0.628826 − 0.777546i
\(96\) −30.2768 − 5.25280i −0.315383 − 0.0547167i
\(97\) 31.5731 − 18.2287i 0.325496 − 0.187925i −0.328344 − 0.944558i \(-0.606490\pi\)
0.653840 + 0.756633i \(0.273157\pi\)
\(98\) 24.7828 − 25.6916i 0.252885 − 0.262159i
\(99\) −118.587 − 68.4661i −1.19785 − 0.691576i
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.159.31 yes 232
4.3 odd 2 inner 380.3.p.a.159.10 ✓ 232
5.4 even 2 inner 380.3.p.a.159.86 yes 232
19.11 even 3 inner 380.3.p.a.239.107 yes 232
20.19 odd 2 inner 380.3.p.a.159.107 yes 232
76.11 odd 6 inner 380.3.p.a.239.86 yes 232
95.49 even 6 inner 380.3.p.a.239.10 yes 232
380.239 odd 6 inner 380.3.p.a.239.31 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.10 ✓ 232 4.3 odd 2 inner
380.3.p.a.159.31 yes 232 1.1 even 1 trivial
380.3.p.a.159.86 yes 232 5.4 even 2 inner
380.3.p.a.159.107 yes 232 20.19 odd 2 inner
380.3.p.a.239.10 yes 232 95.49 even 6 inner
380.3.p.a.239.31 yes 232 380.239 odd 6 inner
380.3.p.a.239.86 yes 232 76.11 odd 6 inner
380.3.p.a.239.107 yes 232 19.11 even 3 inner