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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 239.10
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.10

$q$-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+(-1.94086 - 0.482776i) q^{2} +(0.480142 - 0.831631i) q^{3} +(3.53385 + 1.87400i) q^{4} +(4.92739 - 0.849034i) q^{5} +(-1.33338 + 1.38227i) q^{6} -5.58137 q^{7} +(-5.95398 - 5.34323i) q^{8} +(4.03893 + 6.99563i) q^{9} +(-9.97325 - 0.730972i) q^{10} -16.9515i q^{11} +(3.25523 - 2.03907i) q^{12} +(4.31012 - 2.48845i) q^{13} +(10.8327 + 2.69456i) q^{14} +(1.65976 - 4.50542i) q^{15} +(8.97625 + 13.2449i) 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} +(-8.18381 + 32.9005i) q^{22} +(1.57744 + 2.73221i) q^{23} +(-7.30235 + 2.38601i) q^{24} +(23.5583 - 8.36704i) q^{25} +(-9.56669 + 2.74890i) q^{26} +16.3996 q^{27} +(-19.7238 - 10.4595i) q^{28} +(15.8167 + 27.3954i) q^{29} +(-5.39648 + 7.94309i) q^{30} -7.75014i q^{31} +(-11.0273 - 30.0399i) q^{32} +(-14.0974 - 8.13915i) q^{33} +(33.8842 + 32.6856i) q^{34} +(-27.5016 + 4.73878i) q^{35} +(1.16318 + 32.2905i) q^{36} +28.8709i q^{37} +(-33.7701 + 17.4236i) q^{38} -4.77924i q^{39} +(-33.8742 - 21.2730i) q^{40} +(29.1060 - 50.4131i) q^{41} +(7.44209 - 7.71499i) q^{42} +(19.7090 - 34.1370i) q^{43} +(31.7672 - 59.9043i) q^{44} +(25.8409 + 31.0410i) q^{45} +(-1.74255 - 6.06439i) q^{46} +(-41.8549 - 72.4949i) q^{47} +(15.3247 - 1.10550i) q^{48} -17.8483 q^{49} +(-49.7627 + 4.86584i) q^{50} +(-19.5764 + 11.3025i) q^{51} +(19.8947 - 0.716652i) q^{52} +(87.8212 - 50.7036i) q^{53} +(-31.8293 - 7.91733i) 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} +(14.3085 - 12.8111i) q^{60} +(-35.1456 - 60.8740i) q^{61} +(-3.74159 + 15.0419i) q^{62} +(-22.5428 - 39.0452i) q^{63} +(6.89985 + 63.6270i) q^{64} +(19.1249 - 15.9210i) q^{65} +(23.4317 + 22.6028i) q^{66} +(58.0911 + 100.617i) q^{67} +(-49.9846 - 79.7967i) q^{68} +3.02959 q^{69} +(55.6644 + 4.07983i) q^{70} +(74.5015 + 43.0135i) q^{71} +(13.3315 - 63.2328i) q^{72} +(-13.1006 - 7.56362i) q^{73} +(13.9382 - 56.0343i) q^{74} +(4.35304 - 23.6092i) q^{75} +(73.9546 - 17.5133i) q^{76} +94.6129i q^{77} +(-2.30730 + 9.27582i) q^{78} +(-109.767 - 63.3739i) q^{79} +(55.4748 + 57.6415i) q^{80} +(-28.4762 + 49.3222i) q^{81} +(-80.8289 + 83.7929i) q^{82} +22.6687 q^{83} +(-18.1686 + 11.3808i) q^{84} +(-110.443 - 40.6864i) q^{85} +(-54.7329 + 56.7400i) q^{86} +30.3771 q^{87} +(-90.5760 + 100.929i) q^{88} +(-5.95738 - 10.3185i) q^{89} +(-35.1676 - 72.7215i) q^{90} +(-24.0564 + 13.8890i) q^{91} +(0.454290 + 12.6114i) 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} +(34.6409 + 8.61672i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.94086 − 0.482776i −0.970429 − 0.241388i
\(3\) 0.480142 − 0.831631i 0.160047 − 0.277210i −0.774838 − 0.632160i \(-0.782169\pi\)
0.934885 + 0.354950i \(0.115502\pi\)
\(4\) 3.53385 + 1.87400i 0.883464 + 0.468500i
\(5\) 4.92739 − 0.849034i 0.985477 − 0.169807i
\(6\) −1.33338 + 1.38227i −0.222230 + 0.230379i
\(7\) −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
\(10\) −9.97325 − 0.730972i −0.997325 − 0.0730972i
\(11\) − 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 −0.324980 − 0.945721i \(-0.605358\pi\)
0.656528 + 0.754302i \(0.272024\pi\)
\(14\) 10.8327 + 2.69456i 0.773761 + 0.192468i
\(15\) 1.65976 − 4.50542i 0.110651 − 0.300361i
\(16\) 8.97625 + 13.2449i 0.561016 + 0.827805i
\(17\) −20.3861 − 11.7699i −1.19918 − 0.692348i −0.238809 − 0.971067i \(-0.576757\pi\)
−0.960373 + 0.278719i \(0.910090\pi\)
\(18\) −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
\(22\) −8.18381 + 32.9005i −0.371991 + 1.49548i
\(23\) 1.57744 + 2.73221i 0.0685845 + 0.118792i 0.898278 − 0.439427i \(-0.144818\pi\)
−0.829694 + 0.558219i \(0.811485\pi\)
\(24\) −7.30235 + 2.38601i −0.304265 + 0.0994169i
\(25\) 23.5583 − 8.36704i 0.942331 − 0.334681i
\(26\) −9.56669 + 2.74890i −0.367950 + 0.105727i
\(27\) 16.3996 0.607392
\(28\) −19.7238 − 10.4595i −0.704420 − 0.373553i
\(29\) 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 \(-0.960106\pi\)
0.992156 − 0.125002i \(-0.0398938\pi\)
\(32\) −11.0273 − 30.0399i −0.344603 − 0.938748i
\(33\) −14.0974 − 8.13915i −0.427195 − 0.246641i
\(34\) 33.8842 + 32.6856i 0.996595 + 0.961342i
\(35\) −27.5016 + 4.73878i −0.785760 + 0.135394i
\(36\) 1.16318 + 32.2905i 0.0323105 + 0.896958i
\(37\) 28.8709i 0.780294i 0.920753 + 0.390147i \(0.127576\pi\)
−0.920753 + 0.390147i \(0.872424\pi\)
\(38\) −33.7701 + 17.4236i −0.888687 + 0.458515i
\(39\) − 4.77924i − 0.122545i
\(40\) −33.8742 − 21.2730i −0.846854 − 0.531825i
\(41\) 29.1060 − 50.4131i 0.709903 − 1.22959i −0.254990 − 0.966944i \(-0.582072\pi\)
0.964893 − 0.262644i \(-0.0845945\pi\)
\(42\) 7.44209 − 7.71499i 0.177193 − 0.183690i
\(43\) 19.7090 − 34.1370i 0.458349 − 0.793883i −0.540525 − 0.841328i \(-0.681774\pi\)
0.998874 + 0.0474446i \(0.0151078\pi\)
\(44\) 31.7672 − 59.9043i 0.721982 − 1.36146i
\(45\) 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 \(-0.516333\pi\)
\(48\) 15.3247 − 1.10550i 0.319265 − 0.0230312i
\(49\) −17.8483 −0.364250
\(50\) −49.7627 + 4.86584i −0.995253 + 0.0973169i
\(51\) −19.5764 + 11.3025i −0.383852 + 0.221617i
\(52\) 19.8947 − 0.716652i 0.382590 − 0.0137818i
\(53\) 87.8212 − 50.7036i 1.65700 − 0.956671i 0.682916 − 0.730497i \(-0.260711\pi\)
0.974087 − 0.226175i \(-0.0726220\pi\)
\(54\) −31.8293 − 7.91733i −0.589431 − 0.146617i
\(55\) −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 \(-0.611050\pi\)
−0.984774 + 0.173838i \(0.944383\pi\)
\(60\) 14.3085 − 12.8111i 0.238475 − 0.213518i
\(61\) −35.1456 − 60.8740i −0.576158 − 0.997935i −0.995915 − 0.0902983i \(-0.971218\pi\)
0.419757 − 0.907637i \(-0.362115\pi\)
\(62\) −3.74159 + 15.0419i −0.0603481 + 0.242612i
\(63\) −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
\(66\) 23.4317 + 22.6028i 0.355026 + 0.342467i
\(67\) 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
\(70\) 55.6644 + 4.07983i 0.795206 + 0.0582833i
\(71\) 74.5015 + 43.0135i 1.04932 + 0.605824i 0.922458 − 0.386097i \(-0.126177\pi\)
0.126859 + 0.991921i \(0.459510\pi\)
\(72\) 13.3315 − 63.2328i 0.185160 − 0.878233i
\(73\) −13.1006 − 7.56362i −0.179460 − 0.103611i 0.407579 − 0.913170i \(-0.366373\pi\)
−0.587039 + 0.809559i \(0.699706\pi\)
\(74\) 13.9382 − 56.0343i 0.188354 − 0.757220i
\(75\) 4.35304 − 23.6092i 0.0580406 − 0.314789i
\(76\) 73.9546 − 17.5133i 0.973087 − 0.230438i
\(77\) 94.6129i 1.22874i
\(78\) −2.30730 + 9.27582i −0.0295808 + 0.118921i
\(79\) −109.767 − 63.3739i −1.38945 − 0.802202i −0.396201 − 0.918164i \(-0.629672\pi\)
−0.993254 + 0.115962i \(0.963005\pi\)
\(80\) 55.4748 + 57.6415i 0.693435 + 0.720519i
\(81\) −28.4762 + 49.3222i −0.351558 + 0.608917i
\(82\) −80.8289 + 83.7929i −0.985718 + 1.02186i
\(83\) 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
\(85\) −110.443 − 40.6864i −1.29933 − 0.478664i
\(86\) −54.7329 + 56.7400i −0.636429 + 0.659767i
\(87\) 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 \(0.854656\pi\)
\(90\) −35.1676 − 72.7215i −0.390751 − 0.808016i
\(91\) −24.0564 + 13.8890i −0.264356 + 0.152626i
\(92\) 0.454290 + 12.6114i 0.00493794 + 0.137080i
\(93\) −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 0.653840 − 0.756633i \(-0.273157\pi\)
−0.328344 + 0.944558i \(0.606490\pi\)
\(98\) 34.6409 + 8.61672i 0.353479 + 0.0879257i
\(99\) 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.239.10 yes 232
4.3 odd 2 inner 380.3.p.a.239.31 yes 232
5.4 even 2 inner 380.3.p.a.239.107 yes 232
19.7 even 3 inner 380.3.p.a.159.86 yes 232
20.19 odd 2 inner 380.3.p.a.239.86 yes 232
76.7 odd 6 inner 380.3.p.a.159.107 yes 232
95.64 even 6 inner 380.3.p.a.159.31 yes 232
380.159 odd 6 inner 380.3.p.a.159.10 ✓ 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.10 ✓ 232 380.159 odd 6 inner
380.3.p.a.159.31 yes 232 95.64 even 6 inner
380.3.p.a.159.86 yes 232 19.7 even 3 inner
380.3.p.a.159.107 yes 232 76.7 odd 6 inner
380.3.p.a.239.10 yes 232 1.1 even 1 trivial
380.3.p.a.239.31 yes 232 4.3 odd 2 inner
380.3.p.a.239.86 yes 232 20.19 odd 2 inner
380.3.p.a.239.107 yes 232 5.4 even 2 inner