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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 239.107
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.107

$q$-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} +(-1.72841 - 4.69176i) q^{5} +(-1.33338 + 1.38227i) q^{6} +5.58137 q^{7} +(5.95398 + 5.34323i) q^{8} +(4.03893 + 6.99563i) q^{9} +(-1.08952 - 9.94047i) q^{10} -16.9515i q^{11} +(-3.25523 + 2.03907i) q^{12} +(-4.31012 + 2.48845i) q^{13} +(10.8327 + 2.69456i) q^{14} +(4.73169 + 0.815314i) 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} +(2.68441 - 19.8190i) 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} +(-19.0252 + 16.2186i) q^{25} +(-9.56669 + 2.74890i) q^{26} -16.3996 q^{27} +(19.7238 + 10.4595i) q^{28} +(15.8167 + 27.3954i) q^{29} +(8.78992 + 3.86676i) q^{30} -7.75014i q^{31} +(11.0273 + 30.0399i) q^{32} +(14.0974 + 8.13915i) q^{33} +(33.8842 + 32.6856i) q^{34} +(-9.64690 - 26.1865i) q^{35} +(1.16318 + 32.2905i) q^{36} -28.8709i q^{37} +(33.7701 - 17.4236i) q^{38} -4.77924i q^{39} +(14.7782 - 37.1699i) 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} +(-44.7551 + 22.2930i) 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} +(-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} +(15.1932 + 11.7484i) 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} +(-6.08104 - 55.4815i) 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} +(46.6272 - 65.0070i) q^{80} +(-28.4762 + 49.3222i) q^{81} +(80.8289 - 83.7929i) q^{82} -22.6687 q^{83} +(-18.1686 + 11.3808i) q^{84} +(19.9861 - 115.990i) q^{85} +(-54.7329 + 56.7400i) q^{86} -30.3771 q^{87} +(90.5760 - 100.929i) q^{88} +(-5.95738 - 10.3185i) q^{89} +(65.1393 - 47.7707i) 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} +(-83.4744 - 45.3545i) 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 −0.934885 − 0.354950i \(-0.884498\pi\)
0.774838 + 0.632160i \(0.217831\pi\)
\(4\) 3.53385 + 1.87400i 0.883464 + 0.468500i
\(5\) −1.72841 − 4.69176i −0.345682 − 0.938352i
\(6\) −1.33338 + 1.38227i −0.222230 + 0.230379i
\(7\) 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
\(10\) −1.08952 − 9.94047i −0.108952 − 0.994047i
\(11\) − 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 \(-0.727976\pi\)
0.324980 + 0.945721i \(0.394642\pi\)
\(14\) 10.8327 + 2.69456i 0.773761 + 0.192468i
\(15\) 4.73169 + 0.815314i 0.315446 + 0.0543543i
\(16\) 8.97625 + 13.2449i 0.561016 + 0.827805i
\(17\) 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
\(20\) 2.68441 − 19.8190i 0.134221 − 0.990951i
\(21\) −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 0.829694 − 0.558219i \(-0.188515\pi\)
−0.898278 + 0.439427i \(0.855182\pi\)
\(24\) −7.30235 + 2.38601i −0.304265 + 0.0994169i
\(25\) −19.0252 + 16.2186i −0.761008 + 0.648742i
\(26\) −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 \(0.649710\pi\)
\(30\) 8.78992 + 3.86676i 0.292997 + 0.128892i
\(31\) − 7.75014i − 0.250005i −0.992156 − 0.125002i \(-0.960106\pi\)
0.992156 − 0.125002i \(-0.0398938\pi\)
\(32\) 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
\(35\) −9.64690 − 26.1865i −0.275626 − 0.748185i
\(36\) 1.16318 + 32.2905i 0.0323105 + 0.896958i
\(37\) − 28.8709i − 0.780294i −0.920753 − 0.390147i \(-0.872424\pi\)
0.920753 − 0.390147i \(-0.127576\pi\)
\(38\) 33.7701 − 17.4236i 0.888687 − 0.458515i
\(39\) − 4.77924i − 0.122545i
\(40\) 14.7782 − 37.1699i 0.369455 − 0.929248i
\(41\) 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 −0.998874 − 0.0474446i \(-0.984892\pi\)
0.540525 + 0.841328i \(0.318226\pi\)
\(44\) 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 \(0.183000\pi\)
0.0512907 + 0.998684i \(0.483667\pi\)
\(48\) −15.3247 + 1.10550i −0.319265 + 0.0230312i
\(49\) −17.8483 −0.364250
\(50\) −44.7551 + 22.2930i −0.895103 + 0.445860i
\(51\) −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 \(0.739289\pi\)
−0.974087 + 0.226175i \(0.927378\pi\)
\(54\) −31.8293 − 7.91733i −0.589431 − 0.146617i
\(55\) −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 −0.341839 − 0.939758i \(-0.611050\pi\)
−0.984774 + 0.173838i \(0.944383\pi\)
\(60\) 15.1932 + 11.7484i 0.253220 + 0.195806i
\(61\) −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 \(-0.832692\pi\)
−0.00201423 − 0.999998i \(-0.500641\pi\)
\(68\) 49.9846 + 79.7967i 0.735068 + 1.17348i
\(69\) 3.02959 0.0439071
\(70\) −6.08104 − 55.4815i −0.0868721 − 0.792593i
\(71\) 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 0.587039 − 0.809559i \(-0.300294\pi\)
−0.407579 + 0.913170i \(0.633627\pi\)
\(74\) 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 \(0.963005\pi\)
\(80\) 46.6272 − 65.0070i 0.582840 − 0.812587i
\(81\) −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 \(-0.543604\pi\)
−0.136559 + 0.990632i \(0.543604\pi\)
\(84\) −18.1686 + 11.3808i −0.216293 + 0.135486i
\(85\) 19.9861 − 115.990i 0.235131 − 1.36459i
\(86\) −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 \(0.854656\pi\)
\(90\) 65.1393 − 47.7707i 0.723770 − 0.530786i
\(91\) −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
\(95\) −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 \(0.726843\pi\)
\(98\) −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.239.107 yes 232
4.3 odd 2 inner 380.3.p.a.239.86 yes 232
5.4 even 2 inner 380.3.p.a.239.10 yes 232
19.7 even 3 inner 380.3.p.a.159.31 yes 232
20.19 odd 2 inner 380.3.p.a.239.31 yes 232
76.7 odd 6 inner 380.3.p.a.159.10 ✓ 232
95.64 even 6 inner 380.3.p.a.159.86 yes 232
380.159 odd 6 inner 380.3.p.a.159.107 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.10 ✓ 232 76.7 odd 6 inner
380.3.p.a.159.31 yes 232 19.7 even 3 inner
380.3.p.a.159.86 yes 232 95.64 even 6 inner
380.3.p.a.159.107 yes 232 380.159 odd 6 inner
380.3.p.a.239.10 yes 232 5.4 even 2 inner
380.3.p.a.239.31 yes 232 20.19 odd 2 inner
380.3.p.a.239.86 yes 232 4.3 odd 2 inner
380.3.p.a.239.107 yes 232 1.1 even 1 trivial