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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(2,145)] 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("145.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([7, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.o (of order \(28\), degree \(12\), 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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(13\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 113.11
Character \(\chi\) \(=\) 145.113
Dual form 145.2.o.a.77.11

$q$-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+(0.468883 + 2.05431i) q^{2} +(-0.913718 + 1.89735i) q^{3} +(-2.19840 + 1.05869i) q^{4} +(-2.20832 - 0.351181i) q^{5} +(-4.32618 - 0.987422i) q^{6} +(2.85563 - 0.999230i) q^{7} +(-0.578113 - 0.724931i) q^{8} +(-0.894605 - 1.12180i) q^{9} +(-0.314009 - 4.70123i) q^{10} +(-2.41178 + 0.271742i) q^{11} -5.13849i q^{12} +(0.211974 + 1.88132i) q^{13} +(3.39168 + 5.39783i) q^{14} +(2.68409 - 3.86908i) q^{15} +(-1.82451 + 2.28787i) q^{16} +4.91504 q^{17} +(1.88506 - 2.36379i) q^{18} +(0.708154 - 2.02379i) q^{19} +(5.22656 - 1.56590i) q^{20} +(-0.713351 + 6.33116i) q^{21} +(-1.68908 - 4.82713i) q^{22} +(-1.66941 + 1.04896i) q^{23} +(1.90368 - 0.434503i) q^{24} +(4.75334 + 1.55104i) q^{25} +(-3.76542 + 1.31758i) q^{26} +(-3.21345 + 0.733448i) q^{27} +(-5.21994 + 5.21994i) q^{28} +(-1.64321 + 5.12834i) q^{29} +(9.20682 + 3.69981i) q^{30} +(1.32648 - 2.11108i) q^{31} +(-7.22626 - 3.47998i) q^{32} +(1.68809 - 4.82430i) q^{33} +(2.30458 + 10.0970i) q^{34} +(-6.65706 + 1.20377i) q^{35} +(3.15434 + 1.51905i) q^{36} +(8.89447 - 7.09310i) q^{37} +(4.48952 + 0.505848i) q^{38} +(-3.76322 - 1.31681i) q^{39} +(1.02208 + 1.80390i) q^{40} +(6.33017 - 6.33017i) q^{41} +(-13.3406 + 1.50313i) q^{42} +(10.0274 + 2.28869i) q^{43} +(5.01436 - 3.15073i) q^{44} +(1.58162 + 2.79146i) q^{45} +(-2.93764 - 2.93764i) q^{46} +(-0.322734 - 0.257372i) q^{47} +(-2.67380 - 5.55221i) q^{48} +(1.68337 - 1.34244i) q^{49} +(-0.957550 + 10.4921i) q^{50} +(-4.49096 + 9.32558i) q^{51} +(-2.45774 - 3.91148i) q^{52} +(-2.85978 + 4.55132i) q^{53} +(-3.01346 - 6.25751i) q^{54} +(5.42141 + 0.246876i) q^{55} +(-2.37525 - 1.49247i) q^{56} +(3.19279 + 3.19279i) q^{57} +(-11.3057 - 0.971065i) q^{58} -7.52848i q^{59} +(-1.80454 + 11.3474i) q^{60} +(-4.61629 - 13.1926i) q^{61} +(4.95876 + 1.73515i) q^{62} +(-3.67560 - 2.30953i) q^{63} +(2.45837 - 10.7708i) q^{64} +(0.192577 - 4.22900i) q^{65} +(10.7021 + 1.20584i) q^{66} +(-0.918911 + 8.15556i) q^{67} +(-10.8052 + 5.20352i) q^{68} +(-0.464878 - 4.12591i) q^{69} +(-5.59431 - 13.1112i) q^{70} +(3.33359 + 2.65845i) q^{71} +(-0.296044 + 1.29705i) q^{72} +(2.41470 - 10.5795i) q^{73} +(18.7419 + 14.9462i) q^{74} +(-7.28608 + 7.60157i) q^{75} +(0.585765 + 5.19881i) q^{76} +(-6.61563 + 3.18592i) q^{77} +(0.940620 - 8.34824i) q^{78} +(-15.3317 - 1.72747i) q^{79} +(4.83256 - 4.41160i) q^{80} +(2.50242 - 10.9638i) q^{81} +(15.9722 + 10.0360i) q^{82} +(-6.65450 - 2.32851i) q^{83} +(-5.13453 - 14.6736i) q^{84} +(-10.8540 - 1.72607i) q^{85} +21.6726i q^{86} +(-8.22885 - 7.80360i) q^{87} +(1.59128 + 1.59128i) q^{88} +(-1.64302 - 1.03238i) q^{89} +(-4.99293 + 4.55800i) q^{90} +(2.48519 + 5.16055i) q^{91} +(2.55950 - 4.07342i) q^{92} +(2.79343 + 4.44572i) q^{93} +(0.377397 - 0.783673i) q^{94} +(-2.27454 + 4.22048i) q^{95} +(13.2055 - 10.5311i) q^{96} +(5.65349 + 11.7396i) q^{97} +(3.54709 + 2.82871i) q^{98} +(2.46243 + 2.46243i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q - 8 q^{2} - 14 q^{3} - 22 q^{4} - 14 q^{5} - 28 q^{6} - 10 q^{7} + 4 q^{8} + 10 q^{9} - 8 q^{10} - 20 q^{11} + 4 q^{14} - 24 q^{15} - 34 q^{16} - 48 q^{17} - 94 q^{18} + 6 q^{20} - 16 q^{21} - 6 q^{22}+ \cdots + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.468883 + 2.05431i 0.331550 + 1.45262i 0.816130 + 0.577869i \(0.196115\pi\)
−0.484579 + 0.874747i \(0.661027\pi\)
\(3\) −0.913718 + 1.89735i −0.527535 + 1.09544i 0.451598 + 0.892221i \(0.350854\pi\)
−0.979134 + 0.203217i \(0.934860\pi\)
\(4\) −2.19840 + 1.05869i −1.09920 + 0.529346i
\(5\) −2.20832 − 0.351181i −0.987590 − 0.157053i
\(6\) −4.32618 − 0.987422i −1.76616 − 0.403114i
\(7\) 2.85563 − 0.999230i 1.07933 − 0.377673i 0.268694 − 0.963226i \(-0.413408\pi\)
0.810635 + 0.585552i \(0.199122\pi\)
\(8\) −0.578113 − 0.724931i −0.204394 − 0.256302i
\(9\) −0.894605 − 1.12180i −0.298202 − 0.373933i
\(10\) −0.314009 − 4.70123i −0.0992984 − 1.48666i
\(11\) −2.41178 + 0.271742i −0.727179 + 0.0819334i −0.467790 − 0.883840i \(-0.654950\pi\)
−0.259389 + 0.965773i \(0.583521\pi\)
\(12\) − 5.13849i − 1.48335i
\(13\) 0.211974 + 1.88132i 0.0587910 + 0.521784i 0.988249 + 0.152855i \(0.0488469\pi\)
−0.929458 + 0.368929i \(0.879725\pi\)
\(14\) 3.39168 + 5.39783i 0.906466 + 1.44263i
\(15\) 2.68409 − 3.86908i 0.693030 − 0.998993i
\(16\) −1.82451 + 2.28787i −0.456128 + 0.571966i
\(17\) 4.91504 1.19207 0.596036 − 0.802957i \(-0.296741\pi\)
0.596036 + 0.802957i \(0.296741\pi\)
\(18\) 1.88506 − 2.36379i 0.444313 − 0.557150i
\(19\) 0.708154 − 2.02379i 0.162462 − 0.464288i −0.833660 − 0.552278i \(-0.813759\pi\)
0.996121 + 0.0879898i \(0.0280443\pi\)
\(20\) 5.22656 − 1.56590i 1.16869 − 0.350145i
\(21\) −0.713351 + 6.33116i −0.155666 + 1.38157i
\(22\) −1.68908 − 4.82713i −0.360114 − 1.02915i
\(23\) −1.66941 + 1.04896i −0.348096 + 0.218723i −0.694719 − 0.719282i \(-0.744471\pi\)
0.346623 + 0.938004i \(0.387328\pi\)
\(24\) 1.90368 − 0.434503i 0.388588 − 0.0886927i
\(25\) 4.75334 + 1.55104i 0.950669 + 0.310207i
\(26\) −3.76542 + 1.31758i −0.738460 + 0.258398i
\(27\) −3.21345 + 0.733448i −0.618428 + 0.141152i
\(28\) −5.21994 + 5.21994i −0.986477 + 0.986477i
\(29\) −1.64321 + 5.12834i −0.305136 + 0.952309i
\(30\) 9.20682 + 3.69981i 1.68093 + 0.675491i
\(31\) 1.32648 − 2.11108i 0.238242 − 0.379160i −0.706107 − 0.708105i \(-0.749550\pi\)
0.944349 + 0.328945i \(0.106693\pi\)
\(32\) −7.22626 − 3.47998i −1.27743 − 0.615180i
\(33\) 1.68809 − 4.82430i 0.293860 − 0.839802i
\(34\) 2.30458 + 10.0970i 0.395232 + 1.73162i
\(35\) −6.65706 + 1.20377i −1.12525 + 0.203475i
\(36\) 3.15434 + 1.51905i 0.525723 + 0.253175i
\(37\) 8.89447 − 7.09310i 1.46224 − 1.16610i 0.510229 − 0.860039i \(-0.329561\pi\)
0.952012 − 0.306060i \(-0.0990107\pi\)
\(38\) 4.48952 + 0.505848i 0.728297 + 0.0820594i
\(39\) −3.76322 − 1.31681i −0.602597 − 0.210858i
\(40\) 1.02208 + 1.80390i 0.161605 + 0.285222i
\(41\) 6.33017 − 6.33017i 0.988606 − 0.988606i −0.0113296 − 0.999936i \(-0.503606\pi\)
0.999936 + 0.0113296i \(0.00360641\pi\)
\(42\) −13.3406 + 1.50313i −2.05851 + 0.231938i
\(43\) 10.0274 + 2.28869i 1.52917 + 0.349023i 0.902647 − 0.430382i \(-0.141621\pi\)
0.626521 + 0.779405i \(0.284478\pi\)
\(44\) 5.01436 − 3.15073i 0.755943 − 0.474991i
\(45\) 1.58162 + 2.79146i 0.235774 + 0.416126i
\(46\) −2.93764 − 2.93764i −0.433132 − 0.433132i
\(47\) −0.322734 − 0.257372i −0.0470756 − 0.0375415i 0.599669 − 0.800248i \(-0.295299\pi\)
−0.646745 + 0.762706i \(0.723870\pi\)
\(48\) −2.67380 − 5.55221i −0.385930 − 0.801393i
\(49\) 1.68337 − 1.34244i 0.240481 − 0.191777i
\(50\) −0.957550 + 10.4921i −0.135418 + 1.48381i
\(51\) −4.49096 + 9.32558i −0.628860 + 1.30584i
\(52\) −2.45774 − 3.91148i −0.340828 − 0.542424i
\(53\) −2.85978 + 4.55132i −0.392822 + 0.625172i −0.983324 − 0.181864i \(-0.941787\pi\)
0.590502 + 0.807036i \(0.298930\pi\)
\(54\) −3.01346 − 6.25751i −0.410080 − 0.851540i
\(55\) 5.42141 + 0.246876i 0.731023 + 0.0332888i
\(56\) −2.37525 − 1.49247i −0.317406 − 0.199440i
\(57\) 3.19279 + 3.19279i 0.422895 + 0.422895i
\(58\) −11.3057 − 0.971065i −1.48451 − 0.127507i
\(59\) − 7.52848i − 0.980124i −0.871688 − 0.490062i \(-0.836974\pi\)
0.871688 − 0.490062i \(-0.163026\pi\)
\(60\) −1.80454 + 11.3474i −0.232965 + 1.46495i
\(61\) −4.61629 − 13.1926i −0.591055 − 1.68914i −0.716976 − 0.697098i \(-0.754474\pi\)
0.125921 − 0.992040i \(-0.459811\pi\)
\(62\) 4.95876 + 1.73515i 0.629764 + 0.220364i
\(63\) −3.67560 − 2.30953i −0.463082 − 0.290974i
\(64\) 2.45837 − 10.7708i 0.307297 − 1.34635i
\(65\) 0.192577 − 4.22900i 0.0238863 − 0.524542i
\(66\) 10.7021 + 1.20584i 1.31734 + 0.148429i
\(67\) −0.918911 + 8.15556i −0.112263 + 0.996360i 0.803683 + 0.595058i \(0.202871\pi\)
−0.915946 + 0.401302i \(0.868558\pi\)
\(68\) −10.8052 + 5.20352i −1.31033 + 0.631019i
\(69\) −0.464878 − 4.12591i −0.0559648 − 0.496701i
\(70\) −5.59431 − 13.1112i −0.668647 − 1.56709i
\(71\) 3.33359 + 2.65845i 0.395624 + 0.315500i 0.801016 − 0.598643i \(-0.204293\pi\)
−0.405391 + 0.914143i \(0.632865\pi\)
\(72\) −0.296044 + 1.29705i −0.0348891 + 0.152859i
\(73\) 2.41470 − 10.5795i 0.282619 − 1.23824i −0.611802 − 0.791011i \(-0.709555\pi\)
0.894422 − 0.447225i \(-0.147588\pi\)
\(74\) 18.7419 + 14.9462i 2.17870 + 1.73745i
\(75\) −7.28608 + 7.60157i −0.841325 + 0.877754i
\(76\) 0.585765 + 5.19881i 0.0671918 + 0.596344i
\(77\) −6.61563 + 3.18592i −0.753921 + 0.363069i
\(78\) 0.940620 − 8.34824i 0.106504 − 0.945252i
\(79\) −15.3317 − 1.72747i −1.72496 − 0.194356i −0.806868 − 0.590732i \(-0.798839\pi\)
−0.918088 + 0.396376i \(0.870268\pi\)
\(80\) 4.83256 − 4.41160i 0.540296 − 0.493232i
\(81\) 2.50242 − 10.9638i 0.278046 − 1.21820i
\(82\) 15.9722 + 10.0360i 1.76384 + 1.10829i
\(83\) −6.65450 − 2.32851i −0.730426 − 0.255587i −0.0606592 − 0.998159i \(-0.519320\pi\)
−0.669767 + 0.742571i \(0.733606\pi\)
\(84\) −5.13453 − 14.6736i −0.560223 − 1.60103i
\(85\) −10.8540 − 1.72607i −1.17728 − 0.187218i
\(86\) 21.6726i 2.33701i
\(87\) −8.22885 − 7.80360i −0.882226 − 0.836634i
\(88\) 1.59128 + 1.59128i 0.169631 + 0.169631i
\(89\) −1.64302 − 1.03238i −0.174160 − 0.109432i 0.442131 − 0.896950i \(-0.354223\pi\)
−0.616291 + 0.787519i \(0.711365\pi\)
\(90\) −4.99293 + 4.55800i −0.526301 + 0.480456i
\(91\) 2.48519 + 5.16055i 0.260519 + 0.540973i
\(92\) 2.55950 − 4.07342i 0.266846 − 0.424683i
\(93\) 2.79343 + 4.44572i 0.289666 + 0.461000i
\(94\) 0.377397 − 0.783673i 0.0389255 − 0.0808297i
\(95\) −2.27454 + 4.22048i −0.233363 + 0.433012i
\(96\) 13.2055 − 10.5311i 1.34778 − 1.07482i
\(97\) 5.65349 + 11.7396i 0.574025 + 1.19198i 0.962693 + 0.270597i \(0.0872212\pi\)
−0.388668 + 0.921378i \(0.627065\pi\)
\(98\) 3.54709 + 2.82871i 0.358310 + 0.285742i
\(99\) 2.46243 + 2.46243i 0.247484 + 0.247484i
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 145.2.o.a.113.11 yes 156
5.2 odd 4 145.2.t.a.142.3 yes 156
5.3 odd 4 725.2.bd.b.432.11 156
5.4 even 2 725.2.y.b.693.3 156
29.19 odd 28 145.2.t.a.48.3 yes 156
145.19 odd 28 725.2.bd.b.193.11 156
145.48 even 28 725.2.y.b.657.3 156
145.77 even 28 inner 145.2.o.a.77.11 ✓ 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.o.a.77.11 ✓ 156 145.77 even 28 inner
145.2.o.a.113.11 yes 156 1.1 even 1 trivial
145.2.t.a.48.3 yes 156 29.19 odd 28
145.2.t.a.142.3 yes 156 5.2 odd 4
725.2.y.b.657.3 156 145.48 even 28
725.2.y.b.693.3 156 5.4 even 2
725.2.bd.b.193.11 156 145.19 odd 28
725.2.bd.b.432.11 156 5.3 odd 4