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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 239.86
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.86

$q$-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} +(-1.72841 - 4.69176i) q^{5} +(1.86377 - 0.463602i) q^{6} -5.58137 q^{7} +(-5.95398 + 5.34323i) q^{8} +(4.03893 + 6.99563i) q^{9} +(4.35358 - 9.00257i) q^{10} +16.9515i q^{11} +(3.25523 + 2.03907i) q^{12} +(-4.31012 + 2.48845i) q^{13} +(-7.74988 - 8.03407i) q^{14} +(-4.73169 - 0.815314i) 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} +(19.0038 - 6.23356i) 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} +(-19.0252 + 16.2186i) q^{25} +(-9.56669 - 2.74890i) q^{26} +16.3996 q^{27} +(0.803694 - 22.3110i) q^{28} +(15.8167 + 27.3954i) q^{29} +(-5.39648 - 7.94309i) q^{30} +7.75014i q^{31} +(-20.5017 - 24.5699i) q^{32} +(14.0974 + 8.13915i) q^{33} +(11.3645 + 45.6874i) q^{34} +(9.64690 + 26.1865i) q^{35} +(-28.5460 + 15.1379i) q^{36} -28.8709i q^{37} +(-37.8683 - 3.16105i) q^{38} +4.77924i q^{39} +(35.3601 + 18.6994i) 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} +(-49.7627 - 4.86584i) 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} +(79.5326 - 29.2992i) 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} +(3.94048 - 18.7971i) 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} +(-24.2990 + 50.2467i) 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} +(22.1816 + 76.8634i) q^{80} +(-28.4762 + 49.3222i) q^{81} +(112.981 - 28.1034i) q^{82} +22.6687 q^{83} +(-18.1686 - 11.3808i) q^{84} +(19.9861 - 115.990i) q^{85} +(76.5047 - 19.0301i) q^{86} +30.3771 q^{87} +(-90.5760 - 100.929i) q^{88} +(-5.95738 - 10.3185i) q^{89} +(80.5624 - 5.90469i) 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} +(83.4744 + 45.3545i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.38853 + 1.43944i 0.694263 + 0.719722i
\(3\) 0.480142 − 0.831631i 0.160047 − 0.277210i −0.774838 − 0.632160i \(-0.782169\pi\)
0.934885 + 0.354950i \(0.115502\pi\)
\(4\) −0.143996 + 3.99741i −0.0359989 + 0.999352i
\(5\) −1.72841 − 4.69176i −0.345682 − 0.938352i
\(6\) 1.86377 − 0.463602i 0.310629 − 0.0772671i
\(7\) −5.58137 −0.797339 −0.398670 − 0.917095i \(-0.630528\pi\)
−0.398670 + 0.917095i \(0.630528\pi\)
\(8\) −5.95398 + 5.34323i −0.744248 + 0.667903i
\(9\) 4.03893 + 6.99563i 0.448770 + 0.777292i
\(10\) 4.35358 − 9.00257i 0.435358 − 0.900257i
\(11\) 16.9515i 1.54105i 0.637410 + 0.770525i \(0.280006\pi\)
−0.637410 + 0.770525i \(0.719994\pi\)
\(12\) 3.25523 + 2.03907i 0.271269 + 0.169923i
\(13\) −4.31012 + 2.48845i −0.331548 + 0.191419i −0.656528 − 0.754302i \(-0.727976\pi\)
0.324980 + 0.945721i \(0.394642\pi\)
\(14\) −7.74988 − 8.03407i −0.553563 − 0.573862i
\(15\) −4.73169 − 0.815314i −0.315446 − 0.0543543i
\(16\) −15.9585 − 1.15122i −0.997408 − 0.0719512i
\(17\) 20.3861 + 11.7699i 1.19918 + 0.692348i 0.960373 − 0.278719i \(-0.0899098\pi\)
0.238809 + 0.971067i \(0.423243\pi\)
\(18\) −4.46166 + 15.5274i −0.247870 + 0.862634i
\(19\) −14.2828 + 12.5300i −0.751727 + 0.659475i
\(20\) 19.0038 − 6.23356i 0.950188 − 0.311678i
\(21\) −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 \(-0.144818\pi\)
−0.829694 + 0.558219i \(0.811485\pi\)
\(24\) 1.58483 + 7.51702i 0.0660347 + 0.313209i
\(25\) −19.0252 + 16.2186i −0.761008 + 0.648742i
\(26\) −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 0.998581 + 0.0532473i \(0.0169572\pi\)
−0.453177 + 0.891420i \(0.649710\pi\)
\(30\) −5.39648 − 7.94309i −0.179883 − 0.264770i
\(31\) 7.75014i 0.250005i 0.992156 + 0.125002i \(0.0398938\pi\)
−0.992156 + 0.125002i \(0.960106\pi\)
\(32\) −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
\(35\) 9.64690 + 26.1865i 0.275626 + 0.748185i
\(36\) −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
\(40\) 35.3601 + 18.6994i 0.884001 + 0.467484i
\(41\) 29.1060 − 50.4131i 0.709903 − 1.22959i −0.254990 − 0.966944i \(-0.582072\pi\)
0.964893 − 0.262644i \(-0.0845945\pi\)
\(42\) −10.4024 + 2.58754i −0.247677 + 0.0616081i
\(43\) 19.7090 − 34.1370i 0.458349 − 0.793883i −0.540525 − 0.841328i \(-0.681774\pi\)
0.998874 + 0.0474446i \(0.0151078\pi\)
\(44\) −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 −0.839240 − 0.543761i \(-0.817000\pi\)
−0.0512907 − 0.998684i \(-0.516333\pi\)
\(48\) −8.61975 + 12.7189i −0.179578 + 0.264976i
\(49\) −17.8483 −0.364250
\(50\) −49.7627 − 4.86584i −0.995253 − 0.0973169i
\(51\) 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 −0.682916 + 0.730497i \(0.739289\pi\)
−0.974087 + 0.226175i \(0.927378\pi\)
\(54\) 22.7712 + 23.6063i 0.421690 + 0.437154i
\(55\) 79.5326 − 29.2992i 1.44605 − 0.532713i
\(56\) 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 0.984774 − 0.173838i \(-0.0556168\pi\)
0.341839 + 0.939758i \(0.388950\pi\)
\(60\) 3.94048 − 18.7971i 0.0656747 − 0.313285i
\(61\) −35.1456 − 60.8740i −0.576158 − 0.997935i −0.995915 − 0.0902983i \(-0.971218\pi\)
0.419757 − 0.907637i \(-0.362115\pi\)
\(62\) −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 0.865017 + 0.501743i \(0.167308\pi\)
0.00201423 + 0.999998i \(0.499359\pi\)
\(68\) −49.9846 + 79.7967i −0.735068 + 1.17348i
\(69\) 3.02959 0.0439071
\(70\) −24.2990 + 50.2467i −0.347128 + 0.717810i
\(71\) −74.5015 − 43.0135i −1.04932 − 0.605824i −0.126859 − 0.991921i \(-0.540490\pi\)
−0.922458 + 0.386097i \(0.873823\pi\)
\(72\) −61.4269 − 20.0709i −0.853152 − 0.278763i
\(73\) 13.1006 + 7.56362i 0.179460 + 0.103611i 0.587039 − 0.809559i \(-0.300294\pi\)
−0.407579 + 0.913170i \(0.633627\pi\)
\(74\) 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 0.993254 − 0.115962i \(-0.0369951\pi\)
0.396201 + 0.918164i \(0.370328\pi\)
\(80\) 22.1816 + 76.8634i 0.277270 + 0.960792i
\(81\) −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 \(-0.456396\pi\)
0.136559 + 0.990632i \(0.456396\pi\)
\(84\) −18.1686 − 11.3808i −0.216293 − 0.135486i
\(85\) 19.9861 − 115.990i 0.235131 − 1.36459i
\(86\) 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 0.830615 − 0.556848i \(-0.187989\pi\)
−0.897552 + 0.440910i \(0.854656\pi\)
\(90\) 80.5624 − 5.90469i 0.895138 − 0.0656076i
\(91\) 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
\(95\) 83.4744 + 45.3545i 0.878678 + 0.477416i
\(96\) −30.2768 + 5.25280i −0.315383 + 0.0547167i
\(97\) −31.5731 − 18.2287i −0.325496 − 0.187925i 0.328344 − 0.944558i \(-0.393510\pi\)
−0.653840 + 0.756633i \(0.726843\pi\)
\(98\) −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.239.86 yes 232
4.3 odd 2 inner 380.3.p.a.239.107 yes 232
5.4 even 2 inner 380.3.p.a.239.31 yes 232
19.7 even 3 inner 380.3.p.a.159.10 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.10 yes 232
76.7 odd 6 inner 380.3.p.a.159.31 yes 232
95.64 even 6 inner 380.3.p.a.159.107 yes 232
380.159 odd 6 inner 380.3.p.a.159.86 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.10 ✓ 232 19.7 even 3 inner
380.3.p.a.159.31 yes 232 76.7 odd 6 inner
380.3.p.a.159.86 yes 232 380.159 odd 6 inner
380.3.p.a.159.107 yes 232 95.64 even 6 inner
380.3.p.a.239.10 yes 232 20.19 odd 2 inner
380.3.p.a.239.31 yes 232 5.4 even 2 inner
380.3.p.a.239.86 yes 232 1.1 even 1 trivial
380.3.p.a.239.107 yes 232 4.3 odd 2 inner