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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,11,Mod(13,40)] 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("40.13"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.i (of order \(4\), 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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.13
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.13

$q$-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+(-24.3931 - 20.7117i) q^{2} +(279.968 - 279.968i) q^{3} +(166.050 + 1010.45i) q^{4} +(-2824.19 + 1337.75i) q^{5} +(-12627.9 + 1030.68i) q^{6} +(-19417.8 + 19417.8i) q^{7} +(16877.6 - 28087.1i) q^{8} -97714.9i q^{9} +(96597.9 + 25862.0i) q^{10} +125984. i q^{11} +(329381. + 236404. i) q^{12} +(393289. - 393289. i) q^{13} +(875838. - 71485.0i) q^{14} +(-416156. + 1.16521e6i) q^{15} +(-993431. + 335569. i) q^{16} +(853960. - 853960. i) q^{17} +(-2.02384e6 + 2.38357e6i) q^{18} -127448. q^{19} +(-1.82068e6 - 2.63156e6i) q^{20} +1.08727e7i q^{21} +(2.60935e6 - 3.07315e6i) q^{22} +(6.80775e6 + 6.80775e6i) q^{23} +(-3.13830e6 - 1.25887e7i) q^{24} +(6.18648e6 - 7.55612e6i) q^{25} +(-1.77393e7 + 1.44786e6i) q^{26} +(-1.08252e7 - 1.08252e7i) q^{27} +(-2.28450e7 - 1.63964e7i) q^{28} +2.84823e6 q^{29} +(3.42848e7 - 1.98038e7i) q^{30} +4.46273e7 q^{31} +(3.11831e7 + 1.23901e7i) q^{32} +(3.52716e7 + 3.52716e7i) q^{33} +(-3.85177e7 + 3.14378e6i) q^{34} +(2.88634e7 - 8.08158e7i) q^{35} +(9.87357e7 - 1.62255e7i) q^{36} +(5.85944e7 + 5.85944e7i) q^{37} +(3.10885e6 + 2.63966e6i) q^{38} -2.20217e8i q^{39} +(-1.00921e7 + 1.01901e8i) q^{40} -8.39367e7 q^{41} +(2.25193e8 - 2.65220e8i) q^{42} +(4.40769e7 - 4.40769e7i) q^{43} +(-1.27301e8 + 2.09197e7i) q^{44} +(1.30718e8 + 2.75965e8i) q^{45} +(-2.50621e7 - 3.07063e8i) q^{46} +(-2.21406e8 + 2.21406e8i) q^{47} +(-1.84180e8 + 3.72077e8i) q^{48} -4.71628e8i q^{49} +(-3.07408e8 + 5.61847e7i) q^{50} -4.78162e8i q^{51} +(4.62704e8 + 3.32093e8i) q^{52} +(-7.67580e7 + 7.67580e7i) q^{53} +(3.98520e7 + 4.88269e8i) q^{54} +(-1.68536e8 - 3.55804e8i) q^{55} +(2.17664e8 + 8.73118e8i) q^{56} +(-3.56812e7 + 3.56812e7i) q^{57} +(-6.94772e7 - 5.89917e7i) q^{58} +6.62453e8 q^{59} +(-1.24648e9 - 2.27021e8i) q^{60} +2.66227e8i q^{61} +(-1.08860e9 - 9.24308e8i) q^{62} +(1.89741e9 + 1.89741e9i) q^{63} +(-5.04033e8 - 9.48089e8i) q^{64} +(-5.84602e8 + 1.63685e9i) q^{65} +(-1.29849e8 - 1.59092e9i) q^{66} +(1.12570e9 + 1.12570e9i) q^{67} +(1.00468e9 + 7.21082e8i) q^{68} +3.81190e9 q^{69} +(-2.37790e9 + 1.37354e9i) q^{70} -1.41507e9 q^{71} +(-2.74453e9 - 1.64920e9i) q^{72} +(1.40586e8 + 1.40586e8i) q^{73} +(-2.15710e8 - 2.64289e9i) q^{74} +(-3.83454e8 - 3.84748e9i) q^{75} +(-2.11626e7 - 1.28779e8i) q^{76} +(-2.44634e9 - 2.44634e9i) q^{77} +(-4.56107e9 + 5.37177e9i) q^{78} +1.53877e9i q^{79} +(2.35673e9 - 2.27667e9i) q^{80} -2.91450e8 q^{81} +(2.04748e9 + 1.73847e9i) q^{82} +(1.52334e9 - 1.52334e9i) q^{83} +(-1.09863e10 + 1.80541e9i) q^{84} +(-1.26936e9 + 3.55413e9i) q^{85} +(-1.98808e9 + 1.62265e8i) q^{86} +(7.97412e8 - 7.97412e8i) q^{87} +(3.53854e9 + 2.12632e9i) q^{88} -4.22766e8i q^{89} +(2.52710e9 - 9.43906e9i) q^{90} +1.52736e10i q^{91} +(-5.74845e9 + 8.00930e9i) q^{92} +(1.24942e10 - 1.24942e10i) q^{93} +(9.98647e9 - 8.15085e8i) q^{94} +(3.59936e8 - 1.70493e8i) q^{95} +(1.21991e10 - 5.26143e9i) q^{96} +(-2.74322e9 + 2.74322e9i) q^{97} +(-9.76822e9 + 1.15045e10i) q^{98} +1.23106e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 2 q^{2} - 4864 q^{6} - 4 q^{7} + 69124 q^{8} + 256166 q^{10} + 502036 q^{12} - 4 q^{15} + 3502536 q^{16} - 905772 q^{17} - 5688470 q^{18} + 4385256 q^{20} + 9808012 q^{22} - 4 q^{23} - 1476988 q^{25}+ \cdots - 65612488734 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.3931 − 20.7117i −0.762285 − 0.647241i
\(3\) 279.968 − 279.968i 1.15213 − 1.15213i 0.166006 − 0.986125i \(-0.446913\pi\)
0.986125 − 0.166006i \(-0.0530871\pi\)
\(4\) 166.050 + 1010.45i 0.162158 + 0.986765i
\(5\) −2824.19 + 1337.75i −0.903741 + 0.428080i
\(6\) −12627.9 + 1030.68i −1.62396 + 0.132546i
\(7\) −19417.8 + 19417.8i −1.15534 + 1.15534i −0.169875 + 0.985466i \(0.554336\pi\)
−0.985466 + 0.169875i \(0.945664\pi\)
\(8\) 16877.6 − 28087.1i 0.515064 − 0.857152i
\(9\) − 97714.9i − 1.65481i
\(10\) 96597.9 + 25862.0i 0.965979 + 0.258620i
\(11\) 125984.i 0.782264i 0.920335 + 0.391132i \(0.127916\pi\)
−0.920335 + 0.391132i \(0.872084\pi\)
\(12\) 329381. + 236404.i 1.32371 + 0.950055i
\(13\) 393289. − 393289.i 1.05924 − 1.05924i 0.0611115 − 0.998131i \(-0.480535\pi\)
0.998131 − 0.0611115i \(-0.0194645\pi\)
\(14\) 875838. − 71485.0i 1.62848 − 0.132915i
\(15\) −416156. + 1.16521e6i −0.548024 + 1.53443i
\(16\) −993431. + 335569.i −0.947410 + 0.320023i
\(17\) 853960. − 853960.i 0.601441 − 0.601441i −0.339254 − 0.940695i \(-0.610175\pi\)
0.940695 + 0.339254i \(0.110175\pi\)
\(18\) −2.02384e6 + 2.38357e6i −1.07106 + 1.26144i
\(19\) −127448. −0.0514711 −0.0257356 − 0.999669i \(-0.508193\pi\)
−0.0257356 + 0.999669i \(0.508193\pi\)
\(20\) −1.82068e6 − 2.63156e6i −0.568963 − 0.822363i
\(21\) 1.08727e7i 2.66221i
\(22\) 2.60935e6 − 3.07315e6i 0.506314 − 0.596309i
\(23\) 6.80775e6 + 6.80775e6i 1.05771 + 1.05771i 0.998230 + 0.0594753i \(0.0189428\pi\)
0.0594753 + 0.998230i \(0.481057\pi\)
\(24\) −3.13830e6 − 1.25887e7i −0.394129 − 1.58097i
\(25\) 6.18648e6 − 7.55612e6i 0.633496 − 0.773746i
\(26\) −1.77393e7 + 1.44786e6i −1.49303 + 0.121860i
\(27\) −1.08252e7 − 1.08252e7i −0.754427 − 0.754427i
\(28\) −2.28450e7 − 1.63964e7i −1.32740 − 0.952702i
\(29\) 2.84823e6 0.138862 0.0694312 − 0.997587i \(-0.477882\pi\)
0.0694312 + 0.997587i \(0.477882\pi\)
\(30\) 3.42848e7 − 1.98038e7i 1.41090 − 0.814971i
\(31\) 4.46273e7 1.55881 0.779403 − 0.626522i \(-0.215522\pi\)
0.779403 + 0.626522i \(0.215522\pi\)
\(32\) 3.11831e7 + 1.23901e7i 0.929329 + 0.369254i
\(33\) 3.52716e7 + 3.52716e7i 0.901271 + 0.901271i
\(34\) −3.85177e7 + 3.14378e6i −0.847747 + 0.0691922i
\(35\) 2.88634e7 − 8.08158e7i 0.549551 − 1.53871i
\(36\) 9.87357e7 − 1.62255e7i 1.63291 − 0.268340i
\(37\) 5.85944e7 + 5.85944e7i 0.844982 + 0.844982i 0.989502 − 0.144520i \(-0.0461638\pi\)
−0.144520 + 0.989502i \(0.546164\pi\)
\(38\) 3.10885e6 + 2.63966e6i 0.0392357 + 0.0333142i
\(39\) − 2.20217e8i − 2.44077i
\(40\) −1.00921e7 + 1.01901e8i −0.0985556 + 0.995132i
\(41\) −8.39367e7 −0.724491 −0.362245 − 0.932083i \(-0.617990\pi\)
−0.362245 + 0.932083i \(0.617990\pi\)
\(42\) 2.25193e8 − 2.65220e8i 1.72309 − 2.02936i
\(43\) 4.40769e7 − 4.40769e7i 0.299826 − 0.299826i −0.541120 − 0.840945i \(-0.681999\pi\)
0.840945 + 0.541120i \(0.181999\pi\)
\(44\) −1.27301e8 + 2.09197e7i −0.771911 + 0.126850i
\(45\) 1.30718e8 + 2.75965e8i 0.708391 + 1.49552i
\(46\) −2.50621e7 − 3.07063e8i −0.121683 − 1.49086i
\(47\) −2.21406e8 + 2.21406e8i −0.965382 + 0.965382i −0.999421 − 0.0340381i \(-0.989163\pi\)
0.0340381 + 0.999421i \(0.489163\pi\)
\(48\) −1.84180e8 + 3.72077e8i −0.722831 + 1.46025i
\(49\) − 4.71628e8i − 1.66963i
\(50\) −3.07408e8 + 5.61847e7i −0.983705 + 0.179791i
\(51\) − 4.78162e8i − 1.38588i
\(52\) 4.62704e8 + 3.32093e8i 1.21699 + 0.873459i
\(53\) −7.67580e7 + 7.67580e7i −0.183546 + 0.183546i −0.792899 − 0.609353i \(-0.791429\pi\)
0.609353 + 0.792899i \(0.291429\pi\)
\(54\) 3.98520e7 + 4.88269e8i 0.0867924 + 1.06338i
\(55\) −1.68536e8 − 3.55804e8i −0.334871 − 0.706964i
\(56\) 2.17664e8 + 8.73118e8i 0.395227 + 1.58538i
\(57\) −3.56812e7 + 3.56812e7i −0.0593015 + 0.0593015i
\(58\) −6.94772e7 − 5.89917e7i −0.105853 − 0.0898775i
\(59\) 6.62453e8 0.926605 0.463303 − 0.886200i \(-0.346664\pi\)
0.463303 + 0.886200i \(0.346664\pi\)
\(60\) −1.24648e9 − 2.27021e8i −1.60299 − 0.291951i
\(61\) 2.66227e8i 0.315212i 0.987502 + 0.157606i \(0.0503776\pi\)
−0.987502 + 0.157606i \(0.949622\pi\)
\(62\) −1.08860e9 − 9.24308e8i −1.18826 − 1.00892i
\(63\) 1.89741e9 + 1.89741e9i 1.91187 + 1.91187i
\(64\) −5.04033e8 − 9.48089e8i −0.469417 − 0.882976i
\(65\) −5.84602e8 + 1.63685e9i −0.503841 + 1.41072i
\(66\) −1.29849e8 − 1.59092e9i −0.103686 − 1.27036i
\(67\) 1.12570e9 + 1.12570e9i 0.833778 + 0.833778i 0.988031 − 0.154253i \(-0.0492972\pi\)
−0.154253 + 0.988031i \(0.549297\pi\)
\(68\) 1.00468e9 + 7.21082e8i 0.691009 + 0.495952i
\(69\) 3.81190e9 2.43723
\(70\) −2.37790e9 + 1.37354e9i −1.41483 + 0.817242i
\(71\) −1.41507e9 −0.784307 −0.392153 − 0.919900i \(-0.628270\pi\)
−0.392153 + 0.919900i \(0.628270\pi\)
\(72\) −2.74453e9 − 1.64920e9i −1.41842 − 0.852334i
\(73\) 1.40586e8 + 1.40586e8i 0.0678153 + 0.0678153i 0.740201 − 0.672386i \(-0.234730\pi\)
−0.672386 + 0.740201i \(0.734730\pi\)
\(74\) −2.15710e8 − 2.64289e9i −0.0972102 − 1.19102i
\(75\) −3.83454e8 − 3.84748e9i −0.161587 − 1.62133i
\(76\) −2.11626e7 − 1.28779e8i −0.00834644 − 0.0507899i
\(77\) −2.44634e9 − 2.44634e9i −0.903782 − 0.903782i
\(78\) −4.56107e9 + 5.37177e9i −1.57977 + 1.86056i
\(79\) 1.53877e9i 0.500077i 0.968236 + 0.250039i \(0.0804434\pi\)
−0.968236 + 0.250039i \(0.919557\pi\)
\(80\) 2.35673e9 − 2.27667e9i 0.719218 − 0.694785i
\(81\) −2.91450e8 −0.0835869
\(82\) 2.04748e9 + 1.73847e9i 0.552269 + 0.468920i
\(83\) 1.52334e9 − 1.52334e9i 0.386727 − 0.386727i −0.486791 − 0.873518i \(-0.661833\pi\)
0.873518 + 0.486791i \(0.161833\pi\)
\(84\) −1.09863e10 + 1.80541e9i −2.62697 + 0.431698i
\(85\) −1.26936e9 + 3.55413e9i −0.286082 + 0.801011i
\(86\) −1.98808e9 + 1.62265e8i −0.422612 + 0.0344932i
\(87\) 7.97412e8 − 7.97412e8i 0.159988 − 0.159988i
\(88\) 3.53854e9 + 2.12632e9i 0.670519 + 0.402916i
\(89\) − 4.22766e8i − 0.0757095i −0.999283 − 0.0378547i \(-0.987948\pi\)
0.999283 − 0.0378547i \(-0.0120524\pi\)
\(90\) 2.52710e9 − 9.43906e9i 0.427966 − 1.59851i
\(91\) 1.52736e10i 2.44757i
\(92\) −5.74845e9 + 8.00930e9i −0.872191 + 1.21522i
\(93\) 1.24942e10 − 1.24942e10i 1.79595 − 1.79595i
\(94\) 9.98647e9 − 8.15085e8i 1.36073 − 0.111062i
\(95\) 3.59936e8 − 1.70493e8i 0.0465166 − 0.0220337i
\(96\) 1.21991e10 − 5.26143e9i 1.49614 − 0.645280i
\(97\) −2.74322e9 + 2.74322e9i −0.319450 + 0.319450i −0.848556 − 0.529106i \(-0.822527\pi\)
0.529106 + 0.848556i \(0.322527\pi\)
\(98\) −9.76822e9 + 1.15045e10i −1.08065 + 1.27273i
\(99\) 1.23106e10 1.29450
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 40.11.i.a.13.13 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.43 yes 116
8.5 even 2 inner 40.11.i.a.13.43 yes 116
40.37 odd 4 inner 40.11.i.a.37.13 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.13 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.43 yes 116 8.5 even 2 inner
40.11.i.a.37.13 yes 116 40.37 odd 4 inner
40.11.i.a.37.43 yes 116 5.2 odd 4 inner