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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 159.12
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.12

$q$-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+(-1.87977 - 0.682971i) q^{2} +(1.12368 + 1.94628i) q^{3} +(3.06710 + 2.56766i) q^{4} +(2.49657 - 4.33210i) q^{5} +(-0.783020 - 4.42601i) q^{6} -2.72652 q^{7} +(-4.01181 - 6.92137i) q^{8} +(1.97467 - 3.42023i) q^{9} +(-7.65169 + 6.43829i) q^{10} -5.08353i q^{11} +(-1.55093 + 8.85467i) q^{12} +(-18.9487 - 10.9401i) q^{13} +(5.12524 + 1.86214i) q^{14} +(11.2368 - 0.00889531i) q^{15} +(2.81421 + 15.7506i) q^{16} +(-22.3835 + 12.9231i) q^{17} +(-6.04785 + 5.08061i) q^{18} +(-2.39820 - 18.8480i) q^{19} +(18.7806 - 6.87664i) q^{20} +(-3.06375 - 5.30656i) q^{21} +(-3.47191 + 9.55590i) q^{22} +(-14.2734 + 24.7223i) q^{23} +(8.96289 - 15.5855i) q^{24} +(-12.5343 - 21.6308i) q^{25} +(28.1476 + 33.5063i) q^{26} +29.1019 q^{27} +(-8.36251 - 7.00079i) q^{28} +(-4.54767 + 7.87680i) q^{29} +(-21.1288 - 7.65771i) q^{30} +5.65581i q^{31} +(5.46711 - 31.5295i) q^{32} +(9.89397 - 5.71229i) q^{33} +(50.9020 - 9.00524i) q^{34} +(-6.80695 + 11.8116i) q^{35} +(14.8385 - 5.41990i) q^{36} -52.6528i q^{37} +(-8.36460 + 37.0680i) q^{38} -49.1726i q^{39} +(-39.9999 + 0.0999120i) q^{40} +(-5.98199 - 10.3611i) q^{41} +(2.13492 + 12.0676i) q^{42} +(9.26038 + 16.0395i) q^{43} +(13.0528 - 15.5917i) q^{44} +(-9.88688 - 17.0933i) q^{45} +(43.7155 - 36.7240i) q^{46} +(-4.16671 + 7.21695i) q^{47} +(-27.4927 + 23.1759i) q^{48} -41.5661 q^{49} +(8.78835 + 49.2216i) q^{50} +(-50.3039 - 29.0430i) q^{51} +(-30.0273 - 82.2082i) q^{52} +(-25.1168 - 14.5012i) q^{53} +(-54.7050 - 19.8758i) q^{54} +(-22.0224 - 12.6914i) q^{55} +(10.9383 + 18.8712i) q^{56} +(33.9887 - 25.8468i) q^{57} +(13.9282 - 11.7007i) q^{58} +(39.6526 - 22.8935i) q^{59} +(34.4873 + 28.8251i) q^{60} +(7.37474 - 12.7734i) q^{61} +(3.86276 - 10.6316i) q^{62} +(-5.38398 + 9.32532i) q^{63} +(-31.8107 + 55.5345i) q^{64} +(-94.7003 + 54.7752i) q^{65} +(-22.4998 + 3.98051i) q^{66} +(53.2715 - 92.2689i) q^{67} +(-101.834 - 17.8368i) q^{68} -64.1554 q^{69} +(20.8625 - 17.5541i) q^{70} +(-36.3132 + 20.9654i) q^{71} +(-31.5947 + 0.0539063i) q^{72} +(-9.58915 + 5.53630i) q^{73} +(-35.9604 + 98.9754i) q^{74} +(28.0150 - 48.7014i) q^{75} +(41.0399 - 63.9666i) q^{76} +13.8604i q^{77} +(-33.5835 + 92.4335i) q^{78} +(-2.54597 + 1.46992i) q^{79} +(75.2590 + 27.1310i) q^{80} +(14.9293 + 25.8584i) q^{81} +(4.16845 + 23.5621i) q^{82} -95.1847 q^{83} +(4.22865 - 24.1424i) q^{84} +(0.102302 + 129.231i) q^{85} +(-6.45294 - 36.4751i) q^{86} -20.4406 q^{87} +(-35.1850 + 20.3942i) q^{88} +(58.7862 - 101.821i) q^{89} +(6.91086 + 38.8840i) q^{90} +(51.6641 + 29.8283i) q^{91} +(-107.257 + 39.1765i) q^{92} +(-11.0078 + 6.35534i) q^{93} +(12.7614 - 10.7205i) q^{94} +(-87.6390 - 36.6662i) q^{95} +(67.5085 - 24.7887i) q^{96} +(118.856 - 68.6217i) q^{97} +(78.1349 + 28.3884i) q^{98} +(-17.3868 - 10.0383i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87977 − 0.682971i −0.939887 − 0.341486i
\(3\) 1.12368 + 1.94628i 0.374561 + 0.648759i 0.990261 − 0.139222i \(-0.0444600\pi\)
−0.615700 + 0.787981i \(0.711127\pi\)
\(4\) 3.06710 + 2.56766i 0.766775 + 0.641916i
\(5\) 2.49657 − 4.33210i 0.499314 − 0.866421i
\(6\) −0.783020 − 4.42601i −0.130503 − 0.737668i
\(7\) −2.72652 −0.389503 −0.194751 − 0.980853i \(-0.562390\pi\)
−0.194751 + 0.980853i \(0.562390\pi\)
\(8\) −4.01181 − 6.92137i −0.501477 − 0.865171i
\(9\) 1.97467 − 3.42023i 0.219408 − 0.380025i
\(10\) −7.65169 + 6.43829i −0.765169 + 0.643829i
\(11\) − 5.08353i − 0.462140i −0.972937 − 0.231070i \(-0.925777\pi\)
0.972937 − 0.231070i \(-0.0742226\pi\)
\(12\) −1.55093 + 8.85467i −0.129245 + 0.737889i
\(13\) −18.9487 − 10.9401i −1.45759 − 0.841543i −0.458702 − 0.888590i \(-0.651685\pi\)
−0.998893 + 0.0470477i \(0.985019\pi\)
\(14\) 5.12524 + 1.86214i 0.366089 + 0.133010i
\(15\) 11.2368 − 0.00889531i 0.749122 − 0.000593021i
\(16\) 2.81421 + 15.7506i 0.175888 + 0.984410i
\(17\) −22.3835 + 12.9231i −1.31667 + 0.760182i −0.983192 − 0.182574i \(-0.941557\pi\)
−0.333482 + 0.942756i \(0.608224\pi\)
\(18\) −6.04785 + 5.08061i −0.335992 + 0.282256i
\(19\) −2.39820 − 18.8480i −0.126221 − 0.992002i
\(20\) 18.7806 − 6.87664i 0.939031 − 0.343832i
\(21\) −3.06375 − 5.30656i −0.145893 − 0.252694i
\(22\) −3.47191 + 9.55590i −0.157814 + 0.434359i
\(23\) −14.2734 + 24.7223i −0.620585 + 1.07488i 0.368792 + 0.929512i \(0.379771\pi\)
−0.989377 + 0.145372i \(0.953562\pi\)
\(24\) 8.96289 − 15.5855i 0.373454 − 0.649397i
\(25\) −12.5343 − 21.6308i −0.501370 − 0.865233i
\(26\) 28.1476 + 33.5063i 1.08260 + 1.28870i
\(27\) 29.1019 1.07785
\(28\) −8.36251 − 7.00079i −0.298661 − 0.250028i
\(29\) −4.54767 + 7.87680i −0.156816 + 0.271614i −0.933719 − 0.358007i \(-0.883456\pi\)
0.776903 + 0.629621i \(0.216790\pi\)
\(30\) −21.1288 − 7.65771i −0.704293 − 0.255257i
\(31\) 5.65581i 0.182445i 0.995831 + 0.0912227i \(0.0290775\pi\)
−0.995831 + 0.0912227i \(0.970922\pi\)
\(32\) 5.46711 − 31.5295i 0.170847 − 0.985298i
\(33\) 9.89397 − 5.71229i 0.299817 − 0.173100i
\(34\) 50.9020 − 9.00524i 1.49712 − 0.264860i
\(35\) −6.80695 + 11.8116i −0.194484 + 0.337473i
\(36\) 14.8385 − 5.41990i 0.412181 − 0.150553i
\(37\) − 52.6528i − 1.42305i −0.702661 − 0.711525i \(-0.748005\pi\)
0.702661 − 0.711525i \(-0.251995\pi\)
\(38\) −8.36460 + 37.0680i −0.220121 + 0.975473i
\(39\) − 49.1726i − 1.26084i
\(40\) −39.9999 + 0.0999120i −0.999997 + 0.00249780i
\(41\) −5.98199 − 10.3611i −0.145902 − 0.252710i 0.783807 − 0.621005i \(-0.213275\pi\)
−0.929709 + 0.368294i \(0.879942\pi\)
\(42\) 2.13492 + 12.0676i 0.0508314 + 0.287324i
\(43\) 9.26038 + 16.0395i 0.215358 + 0.373010i 0.953383 − 0.301762i \(-0.0975749\pi\)
−0.738025 + 0.674773i \(0.764242\pi\)
\(44\) 13.0528 − 15.5917i 0.296655 − 0.354357i
\(45\) −9.88688 − 17.0933i −0.219708 − 0.379852i
\(46\) 43.7155 − 36.7240i 0.950337 − 0.798349i
\(47\) −4.16671 + 7.21695i −0.0886534 + 0.153552i −0.906942 − 0.421255i \(-0.861590\pi\)
0.818289 + 0.574807i \(0.194923\pi\)
\(48\) −27.4927 + 23.1759i −0.572764 + 0.482831i
\(49\) −41.5661 −0.848288
\(50\) 8.78835 + 49.2216i 0.175767 + 0.984432i
\(51\) −50.3039 − 29.0430i −0.986350 − 0.569470i
\(52\) −30.0273 − 82.2082i −0.577448 − 1.58093i
\(53\) −25.1168 − 14.5012i −0.473901 − 0.273607i 0.243970 − 0.969783i \(-0.421550\pi\)
−0.717871 + 0.696176i \(0.754883\pi\)
\(54\) −54.7050 − 19.8758i −1.01306 − 0.368070i
\(55\) −22.0224 − 12.6914i −0.400407 − 0.230753i
\(56\) 10.9383 + 18.8712i 0.195327 + 0.336987i
\(57\) 33.9887 − 25.8468i 0.596293 − 0.453453i
\(58\) 13.9282 − 11.7007i 0.240142 − 0.201736i
\(59\) 39.6526 − 22.8935i 0.672079 − 0.388025i −0.124785 − 0.992184i \(-0.539824\pi\)
0.796864 + 0.604159i \(0.206491\pi\)
\(60\) 34.4873 + 28.8251i 0.574789 + 0.480419i
\(61\) 7.37474 − 12.7734i 0.120897 − 0.209400i −0.799224 − 0.601033i \(-0.794756\pi\)
0.920122 + 0.391632i \(0.128089\pi\)
\(62\) 3.86276 − 10.6316i 0.0623025 − 0.171478i
\(63\) −5.38398 + 9.32532i −0.0854599 + 0.148021i
\(64\) −31.8107 + 55.5345i −0.497042 + 0.867727i
\(65\) −94.7003 + 54.7752i −1.45693 + 0.842696i
\(66\) −22.4998 + 3.98051i −0.340905 + 0.0603107i
\(67\) 53.2715 − 92.2689i 0.795096 − 1.37715i −0.127681 − 0.991815i \(-0.540754\pi\)
0.922778 − 0.385332i \(-0.125913\pi\)
\(68\) −101.834 − 17.8368i −1.49757 − 0.262305i
\(69\) −64.1554 −0.929788
\(70\) 20.8625 − 17.5541i 0.298036 − 0.250773i
\(71\) −36.3132 + 20.9654i −0.511453 + 0.295288i −0.733431 − 0.679764i \(-0.762082\pi\)
0.221977 + 0.975052i \(0.428749\pi\)
\(72\) −31.5947 + 0.0539063i −0.438815 + 0.000748699i
\(73\) −9.58915 + 5.53630i −0.131358 + 0.0758397i −0.564239 − 0.825611i \(-0.690830\pi\)
0.432881 + 0.901451i \(0.357497\pi\)
\(74\) −35.9604 + 98.9754i −0.485951 + 1.33751i
\(75\) 28.0150 − 48.7014i 0.373534 − 0.649351i
\(76\) 41.0399 − 63.9666i 0.539999 − 0.841666i
\(77\) 13.8604i 0.180005i
\(78\) −33.5835 + 92.4335i −0.430558 + 1.18504i
\(79\) −2.54597 + 1.46992i −0.0322275 + 0.0186065i −0.516027 − 0.856572i \(-0.672590\pi\)
0.483800 + 0.875179i \(0.339256\pi\)
\(80\) 75.2590 + 27.1310i 0.940737 + 0.339137i
\(81\) 14.9293 + 25.8584i 0.184313 + 0.319239i
\(82\) 4.16845 + 23.5621i 0.0508347 + 0.287342i
\(83\) −95.1847 −1.14680 −0.573402 − 0.819274i \(-0.694377\pi\)
−0.573402 + 0.819274i \(0.694377\pi\)
\(84\) 4.22865 − 24.1424i 0.0503411 − 0.287410i
\(85\) 0.102302 + 129.231i 0.00120355 + 1.52036i
\(86\) −6.45294 − 36.4751i −0.0750342 − 0.424129i
\(87\) −20.4406 −0.234949
\(88\) −35.1850 + 20.3942i −0.399830 + 0.231752i
\(89\) 58.7862 − 101.821i 0.660519 − 1.14405i −0.319960 − 0.947431i \(-0.603670\pi\)
0.980479 − 0.196622i \(-0.0629971\pi\)
\(90\) 6.91086 + 38.8840i 0.0767873 + 0.432045i
\(91\) 51.6641 + 29.8283i 0.567737 + 0.327783i
\(92\) −107.257 + 39.1765i −1.16583 + 0.425831i
\(93\) −11.0078 + 6.35534i −0.118363 + 0.0683370i
\(94\) 12.7614 − 10.7205i 0.135760 − 0.114048i
\(95\) −87.6390 − 36.6662i −0.922515 − 0.385960i
\(96\) 67.5085 − 24.7887i 0.703213 − 0.258216i
\(97\) 118.856 − 68.6217i 1.22532 − 0.707441i 0.259275 − 0.965804i \(-0.416516\pi\)
0.966048 + 0.258363i \(0.0831830\pi\)
\(98\) 78.1349 + 28.3884i 0.797294 + 0.289678i
\(99\) −17.3868 − 10.0383i −0.175625 − 0.101397i
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.159.12 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.53 yes 232
5.4 even 2 inner 380.3.p.a.159.105 yes 232
19.11 even 3 inner 380.3.p.a.239.64 yes 232
20.19 odd 2 inner 380.3.p.a.159.64 yes 232
76.11 odd 6 inner 380.3.p.a.239.105 yes 232
95.49 even 6 inner 380.3.p.a.239.53 yes 232
380.239 odd 6 inner 380.3.p.a.239.12 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.12 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.53 yes 232 4.3 odd 2 inner
380.3.p.a.159.64 yes 232 20.19 odd 2 inner
380.3.p.a.159.105 yes 232 5.4 even 2 inner
380.3.p.a.239.12 yes 232 380.239 odd 6 inner
380.3.p.a.239.53 yes 232 95.49 even 6 inner
380.3.p.a.239.64 yes 232 19.11 even 3 inner
380.3.p.a.239.105 yes 232 76.11 odd 6 inner