Show commands: Magma / Pari/GP / SageMath

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 13.6
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.6

$q$-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+(-30.5658 - 9.47256i) q^{2} +(41.2659 - 41.2659i) q^{3} +(844.541 + 579.073i) q^{4} +(-3105.70 - 346.803i) q^{5} +(-1652.22 + 870.433i) q^{6} +(4455.29 - 4455.29i) q^{7} +(-20328.8 - 25699.8i) q^{8} +55643.3i q^{9} +(91643.1 + 40019.2i) q^{10} -204384. i q^{11} +(58746.7 - 10954.8i) q^{12} +(-91044.3 + 91044.3i) q^{13} +(-178383. + 93976.8i) q^{14} +(-142470. + 113848. i) q^{15} +(377924. + 978103. i) q^{16} +(-1.56614e6 + 1.56614e6i) q^{17} +(527084. - 1.70078e6i) q^{18} -759019. q^{19} +(-2.42206e6 - 2.09132e6i) q^{20} -367703. i q^{21} +(-1.93604e6 + 6.24717e6i) q^{22} +(-4.30362e6 - 4.30362e6i) q^{23} +(-1.89941e6 - 221639. i) q^{24} +(9.52508e6 + 2.15413e6i) q^{25} +(3.64527e6 - 1.92042e6i) q^{26} +(4.73288e6 + 4.73288e6i) q^{27} +(6.34262e6 - 1.18274e6i) q^{28} +3.88076e7 q^{29} +(5.43316e6 - 2.13031e6i) q^{30} +2.62956e7 q^{31} +(-2.28644e6 - 3.34764e7i) q^{32} +(-8.43408e6 - 8.43408e6i) q^{33} +(6.27059e7 - 3.30351e7i) q^{34} +(-1.53819e7 + 1.22917e7i) q^{35} +(-3.22215e7 + 4.69930e7i) q^{36} +(6.23031e7 + 6.23031e7i) q^{37} +(2.32001e7 + 7.18985e6i) q^{38} +7.51405e6i q^{39} +(5.42223e7 + 8.68660e7i) q^{40} +6.75298e6 q^{41} +(-3.48309e6 + 1.12392e7i) q^{42} +(1.50455e8 - 1.50455e8i) q^{43} +(1.18353e8 - 1.72611e8i) q^{44} +(1.92973e7 - 1.72811e8i) q^{45} +(9.07776e7 + 1.72310e8i) q^{46} +(8.45268e7 - 8.45268e7i) q^{47} +(5.59576e7 + 2.47669e7i) q^{48} +2.42776e8i q^{49} +(-2.70737e8 - 1.56070e8i) q^{50} +1.29257e8i q^{51} +(-1.29612e8 + 2.41694e7i) q^{52} +(2.25788e6 - 2.25788e6i) q^{53} +(-9.98319e7 - 1.89497e8i) q^{54} +(-7.08810e7 + 6.34754e8i) q^{55} +(-2.05071e8 - 2.39294e7i) q^{56} +(-3.13216e7 + 3.13216e7i) q^{57} +(-1.18619e9 - 3.67607e8i) q^{58} +6.90408e8 q^{59} +(-1.86249e8 + 1.36487e7i) q^{60} -2.62171e7i q^{61} +(-8.03748e8 - 2.49087e8i) q^{62} +(2.47907e8 + 2.47907e8i) q^{63} +(-2.47220e8 + 1.04489e9i) q^{64} +(3.14331e8 - 2.51182e8i) q^{65} +(1.77902e8 + 3.37687e8i) q^{66} +(-1.06997e9 - 1.06997e9i) q^{67} +(-2.22959e9 + 4.15761e8i) q^{68} -3.55186e8 q^{69} +(5.86595e8 - 2.30000e8i) q^{70} +2.58433e9 q^{71} +(1.43002e9 - 1.13116e9i) q^{72} +(2.04749e9 + 2.04749e9i) q^{73} +(-1.31418e9 - 2.49451e9i) q^{74} +(4.81953e8 - 3.04169e8i) q^{75} +(-6.41023e8 - 4.39528e8i) q^{76} +(-9.10591e8 - 9.10591e8i) q^{77} +(7.11772e7 - 2.29673e8i) q^{78} +3.06028e9i q^{79} +(-8.34509e8 - 3.16876e9i) q^{80} -2.89507e9 q^{81} +(-2.06411e8 - 6.39680e7i) q^{82} +(-1.92521e9 + 1.92521e9i) q^{83} +(2.12927e8 - 3.10541e8i) q^{84} +(5.40711e9 - 4.32082e9i) q^{85} +(-6.02399e9 + 3.17360e9i) q^{86} +(1.60143e9 - 1.60143e9i) q^{87} +(-5.25263e9 + 4.15488e9i) q^{88} +4.48915e9i q^{89} +(-2.22680e9 + 5.09932e9i) q^{90} +8.11259e8i q^{91} +(-1.14248e9 - 6.12670e9i) q^{92} +(1.08511e9 - 1.08511e9i) q^{93} +(-3.38432e9 + 1.78295e9i) q^{94} +(2.35728e9 + 2.63230e8i) q^{95} +(-1.47579e9 - 1.28708e9i) q^{96} +(-6.51344e7 + 6.51344e7i) q^{97} +(2.29971e9 - 7.42065e9i) q^{98} +1.13726e10 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.5658 − 9.47256i −0.955183 − 0.296017i
\(3\) 41.2659 − 41.2659i 0.169818 − 0.169818i −0.617081 − 0.786900i \(-0.711685\pi\)
0.786900 + 0.617081i \(0.211685\pi\)
\(4\) 844.541 + 579.073i 0.824747 + 0.565501i
\(5\) −3105.70 − 346.803i −0.993823 − 0.110977i
\(6\) −1652.22 + 870.433i −0.212477 + 0.111938i
\(7\) 4455.29 − 4455.29i 0.265086 − 0.265086i −0.562031 − 0.827116i \(-0.689980\pi\)
0.827116 + 0.562031i \(0.189980\pi\)
\(8\) −20328.8 − 25699.8i −0.620386 − 0.784296i
\(9\) 55643.3i 0.942323i
\(10\) 91643.1 + 40019.2i 0.916431 + 0.400192i
\(11\) − 204384.i − 1.26906i −0.772897 − 0.634532i \(-0.781193\pi\)
0.772897 − 0.634532i \(-0.218807\pi\)
\(12\) 58746.7 − 10954.8i 0.236090 − 0.0440248i
\(13\) −91044.3 + 91044.3i −0.245209 + 0.245209i −0.819001 − 0.573792i \(-0.805472\pi\)
0.573792 + 0.819001i \(0.305472\pi\)
\(14\) −178383. + 93976.8i −0.331675 + 0.174735i
\(15\) −142470. + 113848.i −0.187615 + 0.149923i
\(16\) 377924. + 978103.i 0.360417 + 0.932791i
\(17\) −1.56614e6 + 1.56614e6i −1.10303 + 1.10303i −0.108986 + 0.994043i \(0.534760\pi\)
−0.994043 + 0.108986i \(0.965240\pi\)
\(18\) 527084. − 1.70078e6i 0.278944 − 0.900091i
\(19\) −759019. −0.306538 −0.153269 − 0.988184i \(-0.548980\pi\)
−0.153269 + 0.988184i \(0.548980\pi\)
\(20\) −2.42206e6 − 2.09132e6i −0.756895 − 0.653536i
\(21\) − 367703.i − 0.0900328i
\(22\) −1.93604e6 + 6.24717e6i −0.375665 + 1.21219i
\(23\) −4.30362e6 − 4.30362e6i −0.668644 − 0.668644i 0.288758 − 0.957402i \(-0.406758\pi\)
−0.957402 + 0.288758i \(0.906758\pi\)
\(24\) −1.89941e6 − 221639.i −0.238541 − 0.0278350i
\(25\) 9.52508e6 + 2.15413e6i 0.975368 + 0.220583i
\(26\) 3.64527e6 − 1.92042e6i 0.306805 − 0.161633i
\(27\) 4.73288e6 + 4.73288e6i 0.329842 + 0.329842i
\(28\) 6.34262e6 − 1.18274e6i 0.368535 − 0.0687225i
\(29\) 3.88076e7 1.89203 0.946013 − 0.324128i \(-0.105071\pi\)
0.946013 + 0.324128i \(0.105071\pi\)
\(30\) 5.43316e6 − 2.13031e6i 0.223587 − 0.0876669i
\(31\) 2.62956e7 0.918491 0.459245 − 0.888309i \(-0.348120\pi\)
0.459245 + 0.888309i \(0.348120\pi\)
\(32\) −2.28644e6 − 3.34764e7i −0.0681413 − 0.997676i
\(33\) −8.43408e6 − 8.43408e6i −0.215510 − 0.215510i
\(34\) 6.27059e7 − 3.30351e7i 1.38011 − 0.727079i
\(35\) −1.53819e7 + 1.22917e7i −0.292867 + 0.234030i
\(36\) −3.22215e7 + 4.69930e7i −0.532885 + 0.777179i
\(37\) 6.23031e7 + 6.23031e7i 0.898464 + 0.898464i 0.995300 − 0.0968361i \(-0.0308723\pi\)
−0.0968361 + 0.995300i \(0.530872\pi\)
\(38\) 2.32001e7 + 7.18985e6i 0.292800 + 0.0907406i
\(39\) 7.51405e6i 0.0832819i
\(40\) 5.42223e7 + 8.68660e7i 0.529515 + 0.848300i
\(41\) 6.75298e6 0.0582876 0.0291438 − 0.999575i \(-0.490722\pi\)
0.0291438 + 0.999575i \(0.490722\pi\)
\(42\) −3.48309e6 + 1.12392e7i −0.0266513 + 0.0859978i
\(43\) 1.50455e8 − 1.50455e8i 1.02345 − 1.02345i 0.0237277 − 0.999718i \(-0.492447\pi\)
0.999718 − 0.0237277i \(-0.00755348\pi\)
\(44\) 1.18353e8 − 1.72611e8i 0.717657 − 1.04666i
\(45\) 1.92973e7 − 1.72811e8i 0.104576 − 0.936503i
\(46\) 9.07776e7 + 1.72310e8i 0.440747 + 0.836608i
\(47\) 8.45268e7 − 8.45268e7i 0.368557 − 0.368557i −0.498393 − 0.866951i \(-0.666077\pi\)
0.866951 + 0.498393i \(0.166077\pi\)
\(48\) 5.59576e7 + 2.47669e7i 0.219610 + 0.0971997i
\(49\) 2.42776e8i 0.859459i
\(50\) −2.70737e8 − 1.56070e8i −0.866358 − 0.499423i
\(51\) 1.29257e8i 0.374629i
\(52\) −1.29612e8 + 2.41694e7i −0.340901 + 0.0635695i
\(53\) 2.25788e6 − 2.25788e6i 0.00539910 − 0.00539910i −0.704402 − 0.709801i \(-0.748785\pi\)
0.709801 + 0.704402i \(0.248785\pi\)
\(54\) −9.98319e7 − 1.89497e8i −0.217421 − 0.412699i
\(55\) −7.08810e7 + 6.34754e8i −0.140837 + 1.26122i
\(56\) −2.05071e8 − 2.39294e7i −0.372361 − 0.0434503i
\(57\) −3.13216e7 + 3.13216e7i −0.0520558 + 0.0520558i
\(58\) −1.18619e9 − 3.67607e8i −1.80723 − 0.560073i
\(59\) 6.90408e8 0.965708 0.482854 − 0.875701i \(-0.339600\pi\)
0.482854 + 0.875701i \(0.339600\pi\)
\(60\) −1.86249e8 + 1.36487e7i −0.239517 + 0.0175523i
\(61\) − 2.62171e7i − 0.0310410i −0.999880 − 0.0155205i \(-0.995059\pi\)
0.999880 − 0.0155205i \(-0.00494053\pi\)
\(62\) −8.03748e8 − 2.49087e8i −0.877327 − 0.271889i
\(63\) 2.47907e8 + 2.47907e8i 0.249796 + 0.249796i
\(64\) −2.47220e8 + 1.04489e9i −0.230242 + 0.973133i
\(65\) 3.14331e8 − 2.51182e8i 0.270907 − 0.216482i
\(66\) 1.77902e8 + 3.37687e8i 0.142057 + 0.269646i
\(67\) −1.06997e9 − 1.06997e9i −0.792497 − 0.792497i 0.189402 − 0.981900i \(-0.439345\pi\)
−0.981900 + 0.189402i \(0.939345\pi\)
\(68\) −2.22959e9 + 4.15761e8i −1.53349 + 0.285956i
\(69\) −3.55186e8 −0.227096
\(70\) 5.86595e8 − 2.30000e8i 0.349018 − 0.136848i
\(71\) 2.58433e9 1.43237 0.716186 − 0.697909i \(-0.245886\pi\)
0.716186 + 0.697909i \(0.245886\pi\)
\(72\) 1.43002e9 − 1.13116e9i 0.739061 − 0.584604i
\(73\) 2.04749e9 + 2.04749e9i 0.987662 + 0.987662i 0.999925 − 0.0122629i \(-0.00390349\pi\)
−0.0122629 + 0.999925i \(0.503903\pi\)
\(74\) −1.31418e9 − 2.49451e9i −0.592236 − 1.12416i
\(75\) 4.81953e8 − 3.04169e8i 0.203095 − 0.128176i
\(76\) −6.41023e8 − 4.39528e8i −0.252817 − 0.173348i
\(77\) −9.10591e8 − 9.10591e8i −0.336411 − 0.336411i
\(78\) 7.11772e7 − 2.29673e8i 0.0246529 − 0.0795495i
\(79\) 3.06028e9i 0.994548i 0.867594 + 0.497274i \(0.165666\pi\)
−0.867594 + 0.497274i \(0.834334\pi\)
\(80\) −8.34509e8 − 3.16876e9i −0.254672 − 0.967028i
\(81\) −2.89507e9 −0.830297
\(82\) −2.06411e8 − 6.39680e7i −0.0556753 − 0.0172542i
\(83\) −1.92521e9 + 1.92521e9i −0.488750 + 0.488750i −0.907912 − 0.419161i \(-0.862324\pi\)
0.419161 + 0.907912i \(0.362324\pi\)
\(84\) 2.12927e8 − 3.10541e8i 0.0509137 − 0.0742544i
\(85\) 5.40711e9 − 4.32082e9i 1.21863 − 0.973805i
\(86\) −6.02399e9 + 3.17360e9i −1.28054 + 0.674620i
\(87\) 1.60143e9 − 1.60143e9i 0.321301 − 0.321301i
\(88\) −5.25263e9 + 4.15488e9i −0.995322 + 0.787309i
\(89\) 4.48915e9i 0.803923i 0.915657 + 0.401961i \(0.131671\pi\)
−0.915657 + 0.401961i \(0.868329\pi\)
\(90\) −2.22680e9 + 5.09932e9i −0.377110 + 0.863575i
\(91\) 8.11259e8i 0.130003i
\(92\) −1.14248e9 − 6.12670e9i −0.173344 − 0.929582i
\(93\) 1.08511e9 − 1.08511e9i 0.155977 − 0.155977i
\(94\) −3.38432e9 + 1.78295e9i −0.461139 + 0.242940i
\(95\) 2.35728e9 + 2.63230e8i 0.304645 + 0.0340187i
\(96\) −1.47579e9 − 1.28708e9i −0.180995 − 0.157852i
\(97\) −6.51344e7 + 6.51344e7i −0.00758493 + 0.00758493i −0.710889 − 0.703304i \(-0.751707\pi\)
0.703304 + 0.710889i \(0.251707\pi\)
\(98\) 2.29971e9 − 7.42065e9i 0.254415 − 0.820940i
\(99\) 1.13726e10 1.19587
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.13.6 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.36 yes 116
8.5 even 2 inner 40.11.i.a.13.36 yes 116
40.37 odd 4 inner 40.11.i.a.37.6 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.6 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.36 yes 116 8.5 even 2 inner
40.11.i.a.37.6 yes 116 40.37 odd 4 inner
40.11.i.a.37.36 yes 116 5.2 odd 4 inner