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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 159.107
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.107

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94086 − 0.482776i 0.970429 − 0.241388i
\(3\) −0.480142 − 0.831631i −0.160047 − 0.277210i 0.774838 − 0.632160i \(-0.217831\pi\)
−0.934885 + 0.354950i \(0.884498\pi\)
\(4\) 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 \(0.280006\pi\)
−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 0.324980 − 0.945721i \(-0.394642\pi\)
−0.656528 + 0.754302i \(0.727976\pi\)
\(14\) 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 0.238809 − 0.971067i \(-0.423243\pi\)
0.960373 + 0.278719i \(0.0899098\pi\)
\(18\) 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 −0.898278 − 0.439427i \(-0.855182\pi\)
0.829694 + 0.558219i \(0.188515\pi\)
\(24\) −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 −0.453177 − 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 \(0.0398938\pi\)
−0.992156 + 0.125002i \(0.960106\pi\)
\(32\) 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 \(0.127576\pi\)
−0.920753 + 0.390147i \(0.872424\pi\)
\(38\) 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 0.964893 + 0.262644i \(0.0845945\pi\)
−0.254990 + 0.966944i \(0.582072\pi\)
\(42\) −7.44209 − 7.71499i −0.177193 − 0.183690i
\(43\) −19.7090 − 34.1370i −0.458349 − 0.793883i 0.540525 − 0.841328i \(-0.318226\pi\)
−0.998874 + 0.0474446i \(0.984892\pi\)
\(44\) 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 0.0512907 − 0.998684i \(-0.483667\pi\)
0.839240 − 0.543761i \(-0.183000\pi\)
\(48\) −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 −0.974087 − 0.226175i \(-0.927378\pi\)
−0.682916 − 0.730497i \(-0.739289\pi\)
\(54\) −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 −0.984774 − 0.173838i \(-0.944383\pi\)
−0.341839 + 0.939758i \(0.611050\pi\)
\(60\) 15.1932 − 11.7484i 0.253220 − 0.195806i
\(61\) −35.1456 + 60.8740i −0.576158 + 0.997935i 0.419757 + 0.907637i \(0.362115\pi\)
−0.995915 + 0.0902983i \(0.971218\pi\)
\(62\) 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 −0.00201423 + 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
\(70\) −6.08104 + 55.4815i −0.0868721 + 0.792593i
\(71\) 74.5015 − 43.0135i 1.04932 − 0.605824i 0.126859 − 0.991921i \(-0.459510\pi\)
0.922458 + 0.386097i \(0.126177\pi\)
\(72\) −13.3315 − 63.2328i −0.185160 − 0.878233i
\(73\) 13.1006 − 7.56362i 0.179460 − 0.103611i −0.407579 − 0.913170i \(-0.633627\pi\)
0.587039 + 0.809559i \(0.300294\pi\)
\(74\) 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 −0.993254 − 0.115962i \(-0.963005\pi\)
−0.396201 + 0.918164i \(0.629672\pi\)
\(80\) 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 −0.897552 − 0.440910i \(-0.854656\pi\)
0.830615 + 0.556848i \(0.187989\pi\)
\(90\) 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 −0.653840 − 0.756633i \(-0.726843\pi\)
0.328344 + 0.944558i \(0.393510\pi\)
\(98\) −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.159.107 yes 232
4.3 odd 2 inner 380.3.p.a.159.86 yes 232
5.4 even 2 inner 380.3.p.a.159.10 ✓ 232
19.11 even 3 inner 380.3.p.a.239.31 yes 232
20.19 odd 2 inner 380.3.p.a.159.31 yes 232
76.11 odd 6 inner 380.3.p.a.239.10 yes 232
95.49 even 6 inner 380.3.p.a.239.86 yes 232
380.239 odd 6 inner 380.3.p.a.239.107 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.10 ✓ 232 5.4 even 2 inner
380.3.p.a.159.31 yes 232 20.19 odd 2 inner
380.3.p.a.159.86 yes 232 4.3 odd 2 inner
380.3.p.a.159.107 yes 232 1.1 even 1 trivial
380.3.p.a.239.10 yes 232 76.11 odd 6 inner
380.3.p.a.239.31 yes 232 19.11 even 3 inner
380.3.p.a.239.86 yes 232 95.49 even 6 inner
380.3.p.a.239.107 yes 232 380.239 odd 6 inner