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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 13.43
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.43

$q$-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+(20.7117 + 24.3931i) 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} +(-28087.1 + 16877.6i) q^{8} -97714.9i q^{9} +(91125.7 + 41183.8i) q^{10} -125984. i q^{11} +(-236404. - 329381. 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} +(2.38357e6 - 2.02384e6i) q^{18} +127448. q^{19} +(882769. + 3.07583e6i) q^{20} -1.08727e7i q^{21} +(3.07315e6 - 2.60935e6i) 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} +(-1.63964e7 - 2.28450e7i) q^{28} -2.84823e6 q^{29} +(-3.70424e7 + 1.39821e7i) q^{30} +4.46273e7 q^{31} +(-1.23901e7 - 3.11831e7i) 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} +(2.63966e6 + 3.10885e6i) q^{38} -2.20217e8i q^{39} +(-5.67454e7 + 8.52392e7i) q^{40} -8.39367e7 q^{41} +(2.65220e8 - 2.25193e8i) 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} +(3.72077e8 - 1.84180e8i) q^{48} -4.71628e8i q^{49} +(3.12450e8 - 5.59255e6i) q^{50} +4.78162e8i q^{51} +(-3.32093e8 - 4.62704e8i) 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} +(-5.89917e7 - 6.94772e7i) q^{58} -6.62453e8 q^{59} +(-1.10828e9 - 6.13986e8i) q^{60} -2.66227e8i q^{61} +(9.24308e8 + 1.08860e9i) 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} +(7.21082e8 + 1.00468e9i) q^{68} -3.81190e9 q^{69} +(-2.56916e9 + 9.69764e8i) q^{70} -1.41507e9 q^{71} +(1.64920e9 + 2.74453e9i) 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} +(5.37177e9 - 4.56107e9i) q^{78} +1.53877e9i q^{79} +(-3.25455e9 + 3.81252e8i) q^{80} -2.91450e8 q^{81} +(-1.73847e9 - 2.04748e9i) 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} +(2.12632e9 + 3.53854e9i) q^{88} -4.22766e8i q^{89} +(4.02427e9 - 8.90434e9i) q^{90} -1.52736e10i q^{91} +(-8.00930e9 + 5.74845e9i) 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} +(1.15045e10 - 9.76822e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.7117 + 24.3931i 0.647241 + 0.762285i
\(3\) −279.968 + 279.968i −1.15213 + 1.15213i −0.166006 + 0.986125i \(0.553087\pi\)
−0.986125 + 0.166006i \(0.946913\pi\)
\(4\) −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 \(0.945664\pi\)
\(8\) −28087.1 + 16877.6i −0.857152 + 0.515064i
\(9\) − 97714.9i − 1.65481i
\(10\) 91125.7 + 41183.8i 0.911257 + 0.411838i
\(11\) − 125984.i − 0.782264i −0.920335 − 0.391132i \(-0.872084\pi\)
0.920335 − 0.391132i \(-0.127916\pi\)
\(12\) −236404. − 329381.i −0.950055 − 1.32371i
\(13\) −393289. + 393289.i −1.05924 + 1.05924i −0.0611115 + 0.998131i \(0.519465\pi\)
−0.998131 + 0.0611115i \(0.980535\pi\)
\(14\) −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 \(0.110175\pi\)
\(18\) 2.38357e6 − 2.02384e6i 1.26144 − 1.07106i
\(19\) 127448. 0.0514711 0.0257356 − 0.999669i \(-0.491807\pi\)
0.0257356 + 0.999669i \(0.491807\pi\)
\(20\) 882769. + 3.07583e6i 0.275865 + 0.961196i
\(21\) − 1.08727e7i − 2.66221i
\(22\) 3.07315e6 − 2.60935e6i 0.596309 − 0.506314i
\(23\) 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
\(28\) −1.63964e7 − 2.28450e7i −0.952702 − 1.32740i
\(29\) −2.84823e6 −0.138862 −0.0694312 − 0.997587i \(-0.522118\pi\)
−0.0694312 + 0.997587i \(0.522118\pi\)
\(30\) −3.70424e7 + 1.39821e7i −1.52438 + 0.575397i
\(31\) 4.46273e7 1.55881 0.779403 − 0.626522i \(-0.215522\pi\)
0.779403 + 0.626522i \(0.215522\pi\)
\(32\) −1.23901e7 − 3.11831e7i −0.369254 − 0.929329i
\(33\) 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 0.144520 − 0.989502i \(-0.453836\pi\)
−0.989502 + 0.144520i \(0.953836\pi\)
\(38\) 2.63966e6 + 3.10885e6i 0.0333142 + 0.0392357i
\(39\) − 2.20217e8i − 2.44077i
\(40\) −5.67454e7 + 8.52392e7i −0.554154 + 0.832414i
\(41\) −8.39367e7 −0.724491 −0.362245 − 0.932083i \(-0.617990\pi\)
−0.362245 + 0.932083i \(0.617990\pi\)
\(42\) 2.65220e8 − 2.25193e8i 2.02936 − 1.72309i
\(43\) −4.40769e7 + 4.40769e7i −0.299826 + 0.299826i −0.840945 − 0.541120i \(-0.818001\pi\)
0.541120 + 0.840945i \(0.318001\pi\)
\(44\) 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 \(0.489163\pi\)
\(48\) 3.72077e8 − 1.84180e8i 1.46025 − 0.722831i
\(49\) − 4.71628e8i − 1.66963i
\(50\) 3.12450e8 − 5.59255e6i 0.999840 − 0.0178961i
\(51\) 4.78162e8i 1.38588i
\(52\) −3.32093e8 − 4.62704e8i −0.873459 − 1.21699i
\(53\) 7.67580e7 − 7.67580e7i 0.183546 − 0.183546i −0.609353 − 0.792899i \(-0.708571\pi\)
0.792899 + 0.609353i \(0.208571\pi\)
\(54\) −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
\(58\) −5.89917e7 − 6.94772e7i −0.0898775 − 0.105853i
\(59\) −6.62453e8 −0.926605 −0.463303 − 0.886200i \(-0.653336\pi\)
−0.463303 + 0.886200i \(0.653336\pi\)
\(60\) −1.10828e9 − 6.13986e8i −1.42526 − 0.789591i
\(61\) − 2.66227e8i − 0.315212i −0.987502 − 0.157606i \(-0.949622\pi\)
0.987502 − 0.157606i \(-0.0503776\pi\)
\(62\) 9.24308e8 + 1.08860e9i 1.00892 + 1.18826i
\(63\) 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 0.154253 − 0.988031i \(-0.450703\pi\)
−0.988031 + 0.154253i \(0.950703\pi\)
\(68\) 7.21082e8 + 1.00468e9i 0.495952 + 0.691009i
\(69\) −3.81190e9 −2.43723
\(70\) −2.56916e9 + 9.69764e8i −1.52863 + 0.577000i
\(71\) −1.41507e9 −0.784307 −0.392153 − 0.919900i \(-0.628270\pi\)
−0.392153 + 0.919900i \(0.628270\pi\)
\(72\) 1.64920e9 + 2.74453e9i 0.852334 + 1.41842i
\(73\) 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
\(78\) 5.37177e9 − 4.56107e9i 1.86056 − 1.57977i
\(79\) 1.53877e9i 0.500077i 0.968236 + 0.250039i \(0.0804434\pi\)
−0.968236 + 0.250039i \(0.919557\pi\)
\(80\) −3.25455e9 + 3.81252e8i −0.993208 + 0.116349i
\(81\) −2.91450e8 −0.0835869
\(82\) −1.73847e9 − 2.04748e9i −0.468920 − 0.552269i
\(83\) −1.52334e9 + 1.52334e9i −0.386727 + 0.386727i −0.873518 − 0.486791i \(-0.838167\pi\)
0.486791 + 0.873518i \(0.338167\pi\)
\(84\) 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
\(88\) 2.12632e9 + 3.53854e9i 0.402916 + 0.670519i
\(89\) − 4.22766e8i − 0.0757095i −0.999283 − 0.0378547i \(-0.987948\pi\)
0.999283 − 0.0378547i \(-0.0120524\pi\)
\(90\) 4.02427e9 − 8.90434e9i 0.681513 − 1.50796i
\(91\) − 1.52736e10i − 2.44757i
\(92\) −8.00930e9 + 5.74845e9i −1.21522 + 0.872191i
\(93\) −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 \(0.322527\pi\)
\(98\) 1.15045e10 − 9.76822e9i 1.27273 − 1.08065i
\(99\) −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.13.43 yes 116
5.2 odd 4 inner 40.11.i.a.37.13 yes 116
8.5 even 2 inner 40.11.i.a.13.13 ✓ 116
40.37 odd 4 inner 40.11.i.a.37.43 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.13 ✓ 116 8.5 even 2 inner
40.11.i.a.13.43 yes 116 1.1 even 1 trivial
40.11.i.a.37.13 yes 116 5.2 odd 4 inner
40.11.i.a.37.43 yes 116 40.37 odd 4 inner