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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 37.36
Character \(\chi\) \(=\) 40.37
Dual form 40.11.i.a.13.36

$q$-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+(9.47256 - 30.5658i) 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} +(-25699.8 + 20328.8i) q^{8} -55643.3i q^{9} +(18818.6 - 98213.4i) q^{10} -204384. i q^{11} +(10954.8 + 58746.7i) 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} +(-1.70078e6 - 527084. i) q^{18} +759019. q^{19} +(-2.82371e6 - 1.50554e6i) q^{20} -367703. i q^{21} +(-6.24717e6 - 1.93604e6i) 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} +(-1.18274e6 - 6.34262e6i) q^{28} -3.88076e7 q^{29} +(-4.82942e6 + 3.27630e6i) q^{30} +2.62956e7 q^{31} +(3.34764e7 - 2.28644e6i) 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} +(7.18985e6 - 2.32001e7i) q^{38} -7.51405e6i q^{39} +(-7.27658e7 + 7.20479e7i) q^{40} +6.75298e6 q^{41} +(-1.12392e7 - 3.48309e6i) 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} +(2.47669e7 - 5.59576e7i) q^{48} -2.42776e8i q^{49} +(2.43840e7 - 3.11547e8i) q^{50} +1.29257e8i q^{51} +(-2.41694e7 - 1.29612e8i) 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} +(-3.67607e8 + 1.18619e9i) q^{58} -6.90408e8 q^{59} +(5.43957e7 + 1.78650e8i) q^{60} -2.62171e7i q^{61} +(2.49087e8 - 8.03748e8i) 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} +(4.15761e8 + 2.22959e9i) q^{68} +3.55186e8 q^{69} +(5.21412e8 - 3.53727e8i) q^{70} +2.58433e9 q^{71} +(1.13116e9 + 1.43002e9i) 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} +(-2.29673e8 - 7.11772e7i) q^{78} -3.06028e9i q^{79} +(1.51293e9 + 2.90662e9i) q^{80} -2.89507e9 q^{81} +(6.39680e7 - 2.06411e8i) 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} +(4.15488e9 + 5.25263e9i) q^{88} -4.48915e9i q^{89} +(-5.46491e9 - 1.04713e9i) q^{90} +8.11259e8i q^{91} +(6.12670e9 - 1.14248e9i) 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} +(-7.42065e9 - 2.29971e9i) 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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.47256 − 30.5658i 0.296017 − 0.955183i
\(3\) −41.2659 − 41.2659i −0.169818 − 0.169818i 0.617081 − 0.786900i \(-0.288315\pi\)
−0.786900 + 0.617081i \(0.788315\pi\)
\(4\) −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 0.827116 − 0.562031i \(-0.189980\pi\)
−0.562031 + 0.827116i \(0.689980\pi\)
\(8\) −25699.8 + 20328.8i −0.784296 + 0.620386i
\(9\) − 55643.3i − 0.942323i
\(10\) 18818.6 − 98213.4i 0.188186 − 0.982134i
\(11\) − 204384.i − 1.26906i −0.772897 − 0.634532i \(-0.781193\pi\)
0.772897 − 0.634532i \(-0.218807\pi\)
\(12\) 10954.8 + 58746.7i 0.0440248 + 0.236090i
\(13\) 91044.3 + 91044.3i 0.245209 + 0.245209i 0.819001 − 0.573792i \(-0.194528\pi\)
−0.573792 + 0.819001i \(0.694528\pi\)
\(14\) 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 −0.994043 − 0.108986i \(-0.965240\pi\)
−0.108986 − 0.994043i \(-0.534760\pi\)
\(18\) −1.70078e6 − 527084.i −0.900091 − 0.278944i
\(19\) 759019. 0.306538 0.153269 − 0.988184i \(-0.451020\pi\)
0.153269 + 0.988184i \(0.451020\pi\)
\(20\) −2.82371e6 − 1.50554e6i −0.882411 − 0.470480i
\(21\) − 367703.i − 0.0900328i
\(22\) −6.24717e6 − 1.93604e6i −1.21219 − 0.375665i
\(23\) −4.30362e6 + 4.30362e6i −0.668644 + 0.668644i −0.957402 − 0.288758i \(-0.906758\pi\)
0.288758 + 0.957402i \(0.406758\pi\)
\(24\) 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
\(28\) −1.18274e6 − 6.34262e6i −0.0687225 − 0.368535i
\(29\) −3.88076e7 −1.89203 −0.946013 − 0.324128i \(-0.894929\pi\)
−0.946013 + 0.324128i \(0.894929\pi\)
\(30\) −4.82942e6 + 3.27630e6i −0.198742 + 0.134827i
\(31\) 2.62956e7 0.918491 0.459245 − 0.888309i \(-0.348120\pi\)
0.459245 + 0.888309i \(0.348120\pi\)
\(32\) 3.34764e7 − 2.28644e6i 0.997676 − 0.0681413i
\(33\) −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 \(-0.969128\pi\)
0.0968361 + 0.995300i \(0.469128\pi\)
\(38\) 7.18985e6 − 2.32001e7i 0.0907406 − 0.292800i
\(39\) − 7.51405e6i − 0.0832819i
\(40\) −7.27658e7 + 7.20479e7i −0.710603 + 0.703593i
\(41\) 6.75298e6 0.0582876 0.0291438 − 0.999575i \(-0.490722\pi\)
0.0291438 + 0.999575i \(0.490722\pi\)
\(42\) −1.12392e7 − 3.48309e6i −0.0859978 − 0.0266513i
\(43\) −1.50455e8 − 1.50455e8i −1.02345 − 1.02345i −0.999718 − 0.0237277i \(-0.992447\pi\)
−0.0237277 − 0.999718i \(-0.507553\pi\)
\(44\) −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 0.866951 − 0.498393i \(-0.166077\pi\)
−0.498393 + 0.866951i \(0.666077\pi\)
\(48\) 2.47669e7 − 5.59576e7i 0.0971997 − 0.219610i
\(49\) − 2.42776e8i − 0.859459i
\(50\) 2.43840e7 − 3.11547e8i 0.0780288 − 0.996951i
\(51\) 1.29257e8i 0.374629i
\(52\) −2.41694e7 − 1.29612e8i −0.0635695 − 0.340901i
\(53\) −2.25788e6 − 2.25788e6i −0.00539910 − 0.00539910i 0.704402 − 0.709801i \(-0.251215\pi\)
−0.709801 + 0.704402i \(0.751215\pi\)
\(54\) 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
\(58\) −3.67607e8 + 1.18619e9i −0.560073 + 1.80723i
\(59\) −6.90408e8 −0.965708 −0.482854 − 0.875701i \(-0.660400\pi\)
−0.482854 + 0.875701i \(0.660400\pi\)
\(60\) 5.43957e7 + 1.78650e8i 0.0699534 + 0.229746i
\(61\) − 2.62171e7i − 0.0310410i −0.999880 − 0.0155205i \(-0.995059\pi\)
0.999880 − 0.0155205i \(-0.00494053\pi\)
\(62\) 2.49087e8 − 8.03748e8i 0.271889 − 0.877327i
\(63\) 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 \(-0.560655\pi\)
0.981900 + 0.189402i \(0.0606551\pi\)
\(68\) 4.15761e8 + 2.22959e9i 0.285956 + 1.53349i
\(69\) 3.55186e8 0.227096
\(70\) 5.21412e8 − 3.53727e8i 0.310235 − 0.210464i
\(71\) 2.58433e9 1.43237 0.716186 − 0.697909i \(-0.245886\pi\)
0.716186 + 0.697909i \(0.245886\pi\)
\(72\) 1.13116e9 + 1.43002e9i 0.584604 + 0.739061i
\(73\) 2.04749e9 − 2.04749e9i 0.987662 − 0.987662i −0.0122629 − 0.999925i \(-0.503903\pi\)
0.999925 + 0.0122629i \(0.00390349\pi\)
\(74\) 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
\(78\) −2.29673e8 − 7.11772e7i −0.0795495 − 0.0246529i
\(79\) − 3.06028e9i − 0.994548i −0.867594 − 0.497274i \(-0.834334\pi\)
0.867594 − 0.497274i \(-0.165666\pi\)
\(80\) 1.51293e9 + 2.90662e9i 0.461709 + 0.887032i
\(81\) −2.89507e9 −0.830297
\(82\) 6.39680e7 − 2.06411e8i 0.0172542 − 0.0556753i
\(83\) 1.92521e9 + 1.92521e9i 0.488750 + 0.488750i 0.907912 − 0.419161i \(-0.137676\pi\)
−0.419161 + 0.907912i \(0.637676\pi\)
\(84\) −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
\(88\) 4.15488e9 + 5.25263e9i 0.787309 + 0.995322i
\(89\) − 4.48915e9i − 0.803923i −0.915657 − 0.401961i \(-0.868329\pi\)
0.915657 − 0.401961i \(-0.131671\pi\)
\(90\) −5.46491e9 − 1.04713e9i −0.925487 − 0.177332i
\(91\) 8.11259e8i 0.130003i
\(92\) 6.12670e9 − 1.14248e9i 0.929582 − 0.173344i
\(93\) −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 0.703304 − 0.710889i \(-0.251707\pi\)
−0.710889 + 0.703304i \(0.751707\pi\)
\(98\) −7.42065e9 − 2.29971e9i −0.820940 − 0.254415i
\(99\) −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.37.36 yes 116
5.3 odd 4 inner 40.11.i.a.13.6 ✓ 116
8.5 even 2 inner 40.11.i.a.37.6 yes 116
40.13 odd 4 inner 40.11.i.a.13.36 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.6 ✓ 116 5.3 odd 4 inner
40.11.i.a.13.36 yes 116 40.13 odd 4 inner
40.11.i.a.37.6 yes 116 8.5 even 2 inner
40.11.i.a.37.36 yes 116 1.1 even 1 trivial