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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 13.8
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.8

$q$-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+(-29.2557 + 12.9654i) q^{2} +(267.142 - 267.142i) q^{3} +(687.797 - 758.625i) q^{4} +(-1085.61 + 2930.37i) q^{5} +(-4351.84 + 11279.1i) q^{6} +(-2054.40 + 2054.40i) q^{7} +(-10286.1 + 31111.7i) q^{8} -83681.0i q^{9} +(-6233.15 - 99805.6i) q^{10} -250798. i q^{11} +(-18921.0 - 386400. i) q^{12} +(-319028. + 319028. i) q^{13} +(33466.8 - 86738.9i) q^{14} +(492814. + 1.07284e6i) q^{15} +(-102446. - 1.04356e6i) q^{16} +(447609. - 447609. i) q^{17} +(1.08496e6 + 2.44815e6i) q^{18} -995350. q^{19} +(1.47637e6 + 2.83907e6i) q^{20} +1.09763e6i q^{21} +(3.25170e6 + 7.33729e6i) q^{22} +(-3.50101e6 - 3.50101e6i) q^{23} +(5.56338e6 + 1.10591e7i) q^{24} +(-7.40853e6 - 6.36247e6i) q^{25} +(5.19708e6 - 1.34697e7i) q^{26} +(-6.58026e6 - 6.58026e6i) q^{27} +(145508. + 2.97152e6i) q^{28} -2.30459e7 q^{29} +(-2.83274e7 - 2.49971e7i) q^{30} +1.69998e7 q^{31} +(1.65273e7 + 2.92019e7i) q^{32} +(-6.69989e7 - 6.69989e7i) q^{33} +(-7.29171e6 + 1.88986e7i) q^{34} +(-3.78988e6 - 8.25041e6i) q^{35} +(-6.34825e7 - 5.75556e7i) q^{36} +(-6.52598e7 - 6.52598e7i) q^{37} +(2.91197e7 - 1.29051e7i) q^{38} +1.70452e8i q^{39} +(-8.00021e7 - 6.39173e7i) q^{40} -1.80994e8 q^{41} +(-1.42312e7 - 3.21120e7i) q^{42} +(1.38569e8 - 1.38569e8i) q^{43} +(-1.90262e8 - 1.72498e8i) q^{44} +(2.45217e8 + 9.08448e7i) q^{45} +(1.47816e8 + 5.70326e7i) q^{46} +(-1.42862e8 + 1.42862e8i) q^{47} +(-3.06147e8 - 2.51411e8i) q^{48} +2.74034e8i q^{49} +(2.99234e8 + 9.00843e7i) q^{50} -2.39151e8i q^{51} +(2.25960e7 + 4.61449e8i) q^{52} +(-3.71541e8 + 3.71541e8i) q^{53} +(2.77826e8 + 1.07195e8i) q^{54} +(7.34933e8 + 2.72269e8i) q^{55} +(-4.27839e7 - 8.50476e7i) q^{56} +(-2.65900e8 + 2.65900e8i) q^{57} +(6.74225e8 - 2.98799e8i) q^{58} -8.34858e8 q^{59} +(1.15284e9 + 3.64034e8i) q^{60} -1.20935e9i q^{61} +(-4.97342e8 + 2.20409e8i) q^{62} +(1.71914e8 + 1.71914e8i) q^{63} +(-8.62132e8 - 6.40039e8i) q^{64} +(-5.88532e8 - 1.28121e9i) q^{65} +(2.82877e9 + 1.09144e9i) q^{66} +(7.96862e7 + 7.96862e7i) q^{67} +(-3.17030e7 - 6.47432e8i) q^{68} -1.87053e9 q^{69} +(2.17846e8 + 1.92235e8i) q^{70} -6.82979e8 q^{71} +(2.60346e9 + 8.60755e8i) q^{72} +(-1.56541e8 - 1.56541e8i) q^{73} +(2.75534e9 + 1.06311e9i) q^{74} +(-3.67882e9 + 2.79447e8i) q^{75} +(-6.84599e8 + 7.55097e8i) q^{76} +(5.15239e8 + 5.15239e8i) q^{77} +(-2.20998e9 - 4.98670e9i) q^{78} -5.02581e8i q^{79} +(3.16923e9 + 8.32691e8i) q^{80} +1.42555e9 q^{81} +(5.29512e9 - 2.34666e9i) q^{82} +(1.59334e9 - 1.59334e9i) q^{83} +(8.32691e8 + 7.54948e8i) q^{84} +(8.25733e8 + 1.79759e9i) q^{85} +(-2.25733e9 + 5.85053e9i) q^{86} +(-6.15654e9 + 6.15654e9i) q^{87} +(7.80276e9 + 2.57975e9i) q^{88} +1.04831e10i q^{89} +(-8.35183e9 + 5.21596e8i) q^{90} -1.31082e9i q^{91} +(-5.06393e9 + 2.47967e8i) q^{92} +(4.54137e9 - 4.54137e9i) q^{93} +(2.32727e9 - 6.03180e9i) q^{94} +(1.08056e9 - 2.91675e9i) q^{95} +(1.22162e10 + 3.38591e9i) q^{96} +(3.68900e9 - 3.68900e9i) q^{97} +(-3.55296e9 - 8.01707e9i) q^{98} -2.09871e10 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.2557 + 12.9654i −0.914242 + 0.405169i
\(3\) 267.142 − 267.142i 1.09935 − 1.09935i 0.104865 − 0.994487i \(-0.466559\pi\)
0.994487 − 0.104865i \(-0.0334409\pi\)
\(4\) 687.797 − 758.625i 0.671677 − 0.740844i
\(5\) −1085.61 + 2930.37i −0.347395 + 0.937719i
\(6\) −4351.84 + 11279.1i −0.559650 + 1.45050i
\(7\) −2054.40 + 2054.40i −0.122235 + 0.122235i −0.765578 − 0.643343i \(-0.777547\pi\)
0.643343 + 0.765578i \(0.277547\pi\)
\(8\) −10286.1 + 31111.7i −0.313908 + 0.949453i
\(9\) − 83681.0i − 1.41715i
\(10\) −6233.15 − 99805.6i −0.0623315 − 0.998056i
\(11\) − 250798.i − 1.55726i −0.627483 − 0.778631i \(-0.715915\pi\)
0.627483 − 0.778631i \(-0.284085\pi\)
\(12\) −18921.0 − 386400.i −0.0760393 − 1.55286i
\(13\) −319028. + 319028.i −0.859236 + 0.859236i −0.991248 − 0.132012i \(-0.957856\pi\)
0.132012 + 0.991248i \(0.457856\pi\)
\(14\) 33466.8 − 86738.9i 0.0622264 − 0.161278i
\(15\) 492814. + 1.07284e6i 0.648974 + 1.41279i
\(16\) −102446. − 1.04356e6i −0.0977005 − 0.995216i
\(17\) 447609. − 447609.i 0.315250 − 0.315250i −0.531690 − 0.846939i \(-0.678443\pi\)
0.846939 + 0.531690i \(0.178443\pi\)
\(18\) 1.08496e6 + 2.44815e6i 0.574183 + 1.29561i
\(19\) −995350. −0.401983 −0.200992 − 0.979593i \(-0.564416\pi\)
−0.200992 + 0.979593i \(0.564416\pi\)
\(20\) 1.47637e6 + 2.83907e6i 0.461367 + 0.887209i
\(21\) 1.09763e6i 0.268757i
\(22\) 3.25170e6 + 7.33729e6i 0.630953 + 1.42371i
\(23\) −3.50101e6 − 3.50101e6i −0.543943 − 0.543943i 0.380739 − 0.924682i \(-0.375670\pi\)
−0.924682 + 0.380739i \(0.875670\pi\)
\(24\) 5.56338e6 + 1.10591e7i 0.698687 + 1.38888i
\(25\) −7.40853e6 − 6.36247e6i −0.758634 − 0.651517i
\(26\) 5.19708e6 − 1.34697e7i 0.437414 − 1.13368i
\(27\) −6.58026e6 − 6.58026e6i −0.458589 − 0.458589i
\(28\) 145508. + 2.97152e6i 0.00845465 + 0.172659i
\(29\) −2.30459e7 −1.12358 −0.561790 − 0.827280i \(-0.689887\pi\)
−0.561790 + 0.827280i \(0.689887\pi\)
\(30\) −2.83274e7 − 2.49971e7i −1.16574 − 1.02869i
\(31\) 1.69998e7 0.593794 0.296897 − 0.954910i \(-0.404048\pi\)
0.296897 + 0.954910i \(0.404048\pi\)
\(32\) 1.65273e7 + 2.92019e7i 0.492552 + 0.870283i
\(33\) −6.69989e7 − 6.69989e7i −1.71198 − 1.71198i
\(34\) −7.29171e6 + 1.88986e7i −0.160485 + 0.415944i
\(35\) −3.78988e6 − 8.25041e6i −0.0721580 − 0.157085i
\(36\) −6.34825e7 − 5.75556e7i −1.04988 − 0.951864i
\(37\) −6.52598e7 − 6.52598e7i −0.941103 − 0.941103i 0.0572561 − 0.998360i \(-0.481765\pi\)
−0.998360 + 0.0572561i \(0.981765\pi\)
\(38\) 2.91197e7 − 1.29051e7i 0.367510 − 0.162871i
\(39\) 1.70452e8i 1.88920i
\(40\) −8.00021e7 − 6.39173e7i −0.781270 − 0.624193i
\(41\) −1.80994e8 −1.56223 −0.781116 − 0.624386i \(-0.785349\pi\)
−0.781116 + 0.624386i \(0.785349\pi\)
\(42\) −1.42312e7 − 3.21120e7i −0.108892 − 0.245709i
\(43\) 1.38569e8 − 1.38569e8i 0.942590 − 0.942590i −0.0558489 − 0.998439i \(-0.517787\pi\)
0.998439 + 0.0558489i \(0.0177865\pi\)
\(44\) −1.90262e8 − 1.72498e8i −1.15369 − 1.04598i
\(45\) 2.45217e8 + 9.08448e7i 1.32888 + 0.492309i
\(46\) 1.47816e8 + 5.70326e7i 0.717684 + 0.276907i
\(47\) −1.42862e8 + 1.42862e8i −0.622914 + 0.622914i −0.946275 − 0.323362i \(-0.895187\pi\)
0.323362 + 0.946275i \(0.395187\pi\)
\(48\) −3.06147e8 − 2.51411e8i −1.20150 − 0.986684i
\(49\) 2.74034e8i 0.970117i
\(50\) 2.99234e8 + 9.00843e7i 0.957549 + 0.288270i
\(51\) − 2.39151e8i − 0.693140i
\(52\) 2.25960e7 + 4.61449e8i 0.0594311 + 1.21369i
\(53\) −3.71541e8 + 3.71541e8i −0.888440 + 0.888440i −0.994373 − 0.105934i \(-0.966217\pi\)
0.105934 + 0.994373i \(0.466217\pi\)
\(54\) 2.77826e8 + 1.07195e8i 0.605068 + 0.233456i
\(55\) 7.34933e8 + 2.72269e8i 1.46027 + 0.540984i
\(56\) −4.27839e7 − 8.50476e7i −0.0776856 − 0.154426i
\(57\) −2.65900e8 + 2.65900e8i −0.441921 + 0.441921i
\(58\) 6.74225e8 − 2.98799e8i 1.02722 − 0.455239i
\(59\) −8.34858e8 −1.16776 −0.583879 − 0.811841i \(-0.698466\pi\)
−0.583879 + 0.811841i \(0.698466\pi\)
\(60\) 1.15284e9 + 3.64034e8i 1.48256 + 0.468151i
\(61\) − 1.20935e9i − 1.43187i −0.698169 − 0.715933i \(-0.746002\pi\)
0.698169 − 0.715933i \(-0.253998\pi\)
\(62\) −4.97342e8 + 2.20409e8i −0.542871 + 0.240587i
\(63\) 1.71914e8 + 1.71914e8i 0.173224 + 0.173224i
\(64\) −8.62132e8 − 6.40039e8i −0.802923 − 0.596083i
\(65\) −5.88532e8 − 1.28121e9i −0.507228 − 1.10422i
\(66\) 2.82877e9 + 1.09144e9i 2.25880 + 0.871522i
\(67\) 7.96862e7 + 7.96862e7i 0.0590214 + 0.0590214i 0.736001 − 0.676980i \(-0.236712\pi\)
−0.676980 + 0.736001i \(0.736712\pi\)
\(68\) −3.17030e7 − 6.47432e8i −0.0218050 − 0.445297i
\(69\) −1.87053e9 −1.19597
\(70\) 2.17846e8 + 1.92235e8i 0.129616 + 0.114378i
\(71\) −6.82979e8 −0.378543 −0.189272 − 0.981925i \(-0.560613\pi\)
−0.189272 + 0.981925i \(0.560613\pi\)
\(72\) 2.60346e9 + 8.60755e8i 1.34551 + 0.444854i
\(73\) −1.56541e8 − 1.56541e8i −0.0755114 − 0.0755114i 0.668342 − 0.743854i \(-0.267004\pi\)
−0.743854 + 0.668342i \(0.767004\pi\)
\(74\) 2.75534e9 + 1.06311e9i 1.24170 + 0.479091i
\(75\) −3.67882e9 + 2.79447e8i −1.55025 + 0.117759i
\(76\) −6.84599e8 + 7.55097e8i −0.270003 + 0.297807i
\(77\) 5.15239e8 + 5.15239e8i 0.190351 + 0.190351i
\(78\) −2.20998e9 − 4.98670e9i −0.765446 − 1.72719i
\(79\) − 5.02581e8i − 0.163332i −0.996660 − 0.0816659i \(-0.973976\pi\)
0.996660 − 0.0816659i \(-0.0260241\pi\)
\(80\) 3.16923e9 + 8.32691e8i 0.967173 + 0.254117i
\(81\) 1.42555e9 0.408844
\(82\) 5.29512e9 − 2.34666e9i 1.42826 − 0.632967i
\(83\) 1.59334e9 − 1.59334e9i 0.404500 − 0.404500i −0.475315 − 0.879816i \(-0.657666\pi\)
0.879816 + 0.475315i \(0.157666\pi\)
\(84\) 8.32691e8 + 7.54948e8i 0.199107 + 0.180518i
\(85\) 8.25733e8 + 1.79759e9i 0.186100 + 0.405132i
\(86\) −2.25733e9 + 5.85053e9i −0.479848 + 1.24366i
\(87\) −6.15654e9 + 6.15654e9i −1.23521 + 1.23521i
\(88\) 7.80276e9 + 2.57975e9i 1.47855 + 0.488837i
\(89\) 1.04831e10i 1.87733i 0.344825 + 0.938667i \(0.387938\pi\)
−0.344825 + 0.938667i \(0.612062\pi\)
\(90\) −8.35183e9 + 5.21596e8i −1.41439 + 0.0883328i
\(91\) − 1.31082e9i − 0.210057i
\(92\) −5.06393e9 + 2.47967e8i −0.768331 + 0.0376232i
\(93\) 4.54137e9 − 4.54137e9i 0.652788 − 0.652788i
\(94\) 2.32727e9 − 6.03180e9i 0.317109 − 0.821879i
\(95\) 1.08056e9 − 2.91675e9i 0.139647 − 0.376947i
\(96\) 1.22162e10 + 3.38591e9i 1.49823 + 0.415259i
\(97\) 3.68900e9 − 3.68900e9i 0.429586 − 0.429586i −0.458901 − 0.888487i \(-0.651757\pi\)
0.888487 + 0.458901i \(0.151757\pi\)
\(98\) −3.55296e9 − 8.01707e9i −0.393061 − 0.886922i
\(99\) −2.09871e10 −2.20687
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.13.8 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.21 yes 116
8.5 even 2 inner 40.11.i.a.13.21 yes 116
40.37 odd 4 inner 40.11.i.a.37.8 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.8 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.21 yes 116 8.5 even 2 inner
40.11.i.a.37.8 yes 116 40.37 odd 4 inner
40.11.i.a.37.21 yes 116 5.2 odd 4 inner