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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 13.17
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.17

$q$-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+(-21.0650 - 24.0887i) q^{2} +(107.979 - 107.979i) q^{3} +(-136.529 + 1014.86i) q^{4} +(-95.2624 - 3123.55i) q^{5} +(-4875.66 - 326.491i) q^{6} +(7115.44 - 7115.44i) q^{7} +(27322.6 - 18089.2i) q^{8} +35730.0i q^{9} +(-73235.4 + 68092.4i) q^{10} -127709. i q^{11} +(94841.2 + 124326. i) q^{12} +(135042. - 135042. i) q^{13} +(-321289. - 21514.6i) q^{14} +(-347564. - 326992. i) q^{15} +(-1.01130e6 - 277115. i) q^{16} +(1.78454e6 - 1.78454e6i) q^{17} +(860689. - 752654. i) q^{18} +1.32184e6 q^{19} +(3.18296e6 + 329776. i) q^{20} -1.53664e6i q^{21} +(-3.07634e6 + 2.69019e6i) q^{22} +(-2.73554e6 - 2.73554e6i) q^{23} +(997009. - 4.90352e6i) q^{24} +(-9.74748e6 + 595113. i) q^{25} +(-6.09765e6 - 408319. i) q^{26} +(1.02342e7 + 1.02342e7i) q^{27} +(6.24970e6 + 8.19263e6i) q^{28} -1.74333e7 q^{29} +(-555342. + 1.52605e7i) q^{30} -1.95367e7 q^{31} +(1.46277e7 + 3.01982e7i) q^{32} +(-1.37899e7 - 1.37899e7i) q^{33} +(-8.05785e7 - 5.39581e6i) q^{34} +(-2.29033e7 - 2.15476e7i) q^{35} +(-3.62609e7 - 4.87818e6i) q^{36} +(-2.89071e7 - 2.89071e7i) q^{37} +(-2.78445e7 - 3.18413e7i) q^{38} -2.91635e7i q^{39} +(-5.91053e7 - 8.36201e7i) q^{40} -1.01726e8 q^{41} +(-3.70156e7 + 3.23693e7i) q^{42} +(-1.38368e8 + 1.38368e8i) q^{43} +(1.29606e8 + 1.74360e7i) q^{44} +(1.11604e8 - 3.40373e6i) q^{45} +(-8.27130e6 + 1.23520e8i) q^{46} +(-3.63632e7 + 3.63632e7i) q^{47} +(-1.39121e8 + 7.92762e7i) q^{48} +1.81216e8i q^{49} +(2.19666e8 + 2.22268e8i) q^{50} -3.85386e8i q^{51} +(1.18611e8 + 1.55486e8i) q^{52} +(2.99779e8 - 2.99779e8i) q^{53} +(3.09445e7 - 4.62110e8i) q^{54} +(-3.98905e8 + 1.21659e7i) q^{55} +(6.56994e7 - 3.23125e8i) q^{56} +(1.42731e8 - 1.42731e8i) q^{57} +(3.67232e8 + 4.19944e8i) q^{58} -4.40109e8 q^{59} +(3.79302e8 - 3.08084e8i) q^{60} -7.99817e8i q^{61} +(4.11542e8 + 4.70614e8i) q^{62} +(2.54235e8 + 2.54235e8i) q^{63} +(4.19303e8 - 9.88487e8i) q^{64} +(-4.34675e8 - 4.08946e8i) q^{65} +(-4.16958e7 + 6.22665e8i) q^{66} +(-1.61378e9 - 1.61378e9i) q^{67} +(1.56741e9 + 2.05469e9i) q^{68} -5.90763e8 q^{69} +(-3.65951e7 + 1.00561e9i) q^{70} +2.38025e9 q^{71} +(6.46328e8 + 9.76236e8i) q^{72} +(1.94956e9 + 1.94956e9i) q^{73} +(-8.74049e7 + 1.30526e9i) q^{74} +(-9.88264e8 + 1.11678e9i) q^{75} +(-1.80469e8 + 1.34147e9i) q^{76} +(-9.08706e8 - 9.08706e8i) q^{77} +(-7.02509e8 + 6.14329e8i) q^{78} -3.22986e9i q^{79} +(-7.69242e8 + 3.18523e9i) q^{80} +1.00328e8 q^{81} +(2.14286e9 + 2.45044e9i) q^{82} +(4.57436e9 - 4.57436e9i) q^{83} +(1.55947e9 + 2.09795e8i) q^{84} +(-5.74409e9 - 5.40409e9i) q^{85} +(6.24783e9 + 4.18376e8i) q^{86} +(-1.88243e9 + 1.88243e9i) q^{87} +(-2.31016e9 - 3.48934e9i) q^{88} +1.57146e8i q^{89} +(-2.43294e9 - 2.61670e9i) q^{90} -1.92177e9i q^{91} +(3.14966e9 - 2.40270e9i) q^{92} +(-2.10956e9 + 2.10956e9i) q^{93} +(1.64194e9 + 1.09950e8i) q^{94} +(-1.25921e8 - 4.12882e9i) q^{95} +(4.84026e9 + 1.68129e9i) q^{96} +(-1.19158e10 + 1.19158e10i) q^{97} +(4.36526e9 - 3.81732e9i) q^{98} +4.56305e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.0650 − 24.0887i −0.658282 − 0.752771i
\(3\) 107.979 − 107.979i 0.444359 − 0.444359i −0.449115 − 0.893474i \(-0.648261\pi\)
0.893474 + 0.449115i \(0.148261\pi\)
\(4\) −136.529 + 1014.86i −0.133329 + 0.991072i
\(5\) −95.2624 − 3123.55i −0.0304840 − 0.999535i
\(6\) −4875.66 − 326.491i −0.627014 − 0.0419870i
\(7\) 7115.44 − 7115.44i 0.423362 − 0.423362i −0.462998 − 0.886360i \(-0.653226\pi\)
0.886360 + 0.462998i \(0.153226\pi\)
\(8\) 27322.6 − 18089.2i 0.833818 − 0.552039i
\(9\) 35730.0i 0.605091i
\(10\) −73235.4 + 68092.4i −0.732354 + 0.680924i
\(11\) − 127709.i − 0.792972i −0.918041 − 0.396486i \(-0.870229\pi\)
0.918041 − 0.396486i \(-0.129771\pi\)
\(12\) 94841.2 + 124326.i 0.381145 + 0.499637i
\(13\) 135042. − 135042.i 0.363708 − 0.363708i −0.501468 − 0.865176i \(-0.667207\pi\)
0.865176 + 0.501468i \(0.167207\pi\)
\(14\) −321289. − 21514.6i −0.597386 − 0.0400030i
\(15\) −347564. − 326992.i −0.457698 − 0.430606i
\(16\) −1.01130e6 − 277115.i −0.964447 − 0.264277i
\(17\) 1.78454e6 − 1.78454e6i 1.25684 − 1.25684i 0.304251 − 0.952592i \(-0.401594\pi\)
0.952592 − 0.304251i \(-0.0984062\pi\)
\(18\) 860689. − 752654.i 0.455495 − 0.398321i
\(19\) 1.32184e6 0.533838 0.266919 − 0.963719i \(-0.413994\pi\)
0.266919 + 0.963719i \(0.413994\pi\)
\(20\) 3.18296e6 + 329776.i 0.994676 + 0.103055i
\(21\) − 1.53664e6i − 0.376249i
\(22\) −3.07634e6 + 2.69019e6i −0.596927 + 0.522000i
\(23\) −2.73554e6 − 2.73554e6i −0.425015 − 0.425015i 0.461911 − 0.886926i \(-0.347164\pi\)
−0.886926 + 0.461911i \(0.847164\pi\)
\(24\) 997009. − 4.90352e6i 0.125211 − 0.615817i
\(25\) −9.74748e6 + 595113.i −0.998141 + 0.0609396i
\(26\) −6.09765e6 − 408319.i −0.513211 − 0.0343663i
\(27\) 1.02342e7 + 1.02342e7i 0.713236 + 0.713236i
\(28\) 6.24970e6 + 8.19263e6i 0.363136 + 0.476028i
\(29\) −1.74333e7 −0.849941 −0.424970 − 0.905207i \(-0.639716\pi\)
−0.424970 + 0.905207i \(0.639716\pi\)
\(30\) −555342. + 1.52605e7i −0.0228536 + 0.628002i
\(31\) −1.95367e7 −0.682407 −0.341204 − 0.939989i \(-0.610835\pi\)
−0.341204 + 0.939989i \(0.610835\pi\)
\(32\) 1.46277e7 + 3.01982e7i 0.435938 + 0.899977i
\(33\) −1.37899e7 − 1.37899e7i −0.352364 − 0.352364i
\(34\) −8.05785e7 − 5.39581e6i −1.77347 − 0.118758i
\(35\) −2.29033e7 − 2.15476e7i −0.436071 − 0.410259i
\(36\) −3.62609e7 − 4.87818e6i −0.599689 − 0.0806761i
\(37\) −2.89071e7 − 2.89071e7i −0.416866 − 0.416866i 0.467256 − 0.884122i \(-0.345243\pi\)
−0.884122 + 0.467256i \(0.845243\pi\)
\(38\) −2.78445e7 − 3.18413e7i −0.351416 − 0.401858i
\(39\) − 2.91635e7i − 0.323233i
\(40\) −5.91053e7 − 8.36201e7i −0.577200 − 0.816603i
\(41\) −1.01726e8 −0.878035 −0.439018 − 0.898478i \(-0.644673\pi\)
−0.439018 + 0.898478i \(0.644673\pi\)
\(42\) −3.70156e7 + 3.23693e7i −0.283229 + 0.247678i
\(43\) −1.38368e8 + 1.38368e8i −0.941225 + 0.941225i −0.998366 − 0.0571411i \(-0.981802\pi\)
0.0571411 + 0.998366i \(0.481802\pi\)
\(44\) 1.29606e8 + 1.74360e7i 0.785893 + 0.105726i
\(45\) 1.11604e8 − 3.40373e6i 0.604810 − 0.0184456i
\(46\) −8.27130e6 + 1.23520e8i −0.0401592 + 0.599719i
\(47\) −3.63632e7 + 3.63632e7i −0.158553 + 0.158553i −0.781925 − 0.623372i \(-0.785762\pi\)
0.623372 + 0.781925i \(0.285762\pi\)
\(48\) −1.39121e8 + 7.92762e7i −0.545994 + 0.311126i
\(49\) 1.81216e8i 0.641529i
\(50\) 2.19666e8 + 2.22268e8i 0.702932 + 0.711257i
\(51\) − 3.85386e8i − 1.11698i
\(52\) 1.18611e8 + 1.55486e8i 0.311968 + 0.408953i
\(53\) 2.99779e8 − 2.99779e8i 0.716839 − 0.716839i −0.251118 − 0.967957i \(-0.580798\pi\)
0.967957 + 0.251118i \(0.0807982\pi\)
\(54\) 3.09445e7 − 4.62110e8i 0.0673929 − 1.00641i
\(55\) −3.98905e8 + 1.21659e7i −0.792604 + 0.0241729i
\(56\) 6.56994e7 − 3.23125e8i 0.119295 − 0.586719i
\(57\) 1.42731e8 − 1.42731e8i 0.237215 − 0.237215i
\(58\) 3.67232e8 + 4.19944e8i 0.559501 + 0.639811i
\(59\) −4.40109e8 −0.615602 −0.307801 − 0.951451i \(-0.599593\pi\)
−0.307801 + 0.951451i \(0.599593\pi\)
\(60\) 3.79302e8 − 3.08084e8i 0.487786 − 0.396199i
\(61\) − 7.99817e8i − 0.946981i −0.880799 − 0.473491i \(-0.842994\pi\)
0.880799 − 0.473491i \(-0.157006\pi\)
\(62\) 4.11542e8 + 4.70614e8i 0.449217 + 0.513697i
\(63\) 2.54235e8 + 2.54235e8i 0.256172 + 0.256172i
\(64\) 4.19303e8 − 9.88487e8i 0.390506 − 0.920600i
\(65\) −4.34675e8 − 4.08946e8i −0.374626 − 0.352451i
\(66\) −4.16958e7 + 6.22665e8i −0.0332945 + 0.497205i
\(67\) −1.61378e9 − 1.61378e9i −1.19528 − 1.19528i −0.975564 − 0.219714i \(-0.929487\pi\)
−0.219714 − 0.975564i \(-0.570513\pi\)
\(68\) 1.56741e9 + 2.05469e9i 1.07805 + 1.41320i
\(69\) −5.90763e8 −0.377718
\(70\) −3.65951e7 + 1.00561e9i −0.0217737 + 0.598328i
\(71\) 2.38025e9 1.31926 0.659629 − 0.751591i \(-0.270713\pi\)
0.659629 + 0.751591i \(0.270713\pi\)
\(72\) 6.46328e8 + 9.76236e8i 0.334034 + 0.504536i
\(73\) 1.94956e9 + 1.94956e9i 0.940422 + 0.940422i 0.998322 − 0.0579001i \(-0.0184405\pi\)
−0.0579001 + 0.998322i \(0.518440\pi\)
\(74\) −8.74049e7 + 1.30526e9i −0.0393892 + 0.588220i
\(75\) −9.88264e8 + 1.11678e9i −0.416454 + 0.470612i
\(76\) −1.80469e8 + 1.34147e9i −0.0711760 + 0.529072i
\(77\) −9.08706e8 − 9.08706e8i −0.335714 − 0.335714i
\(78\) −7.02509e8 + 6.14329e8i −0.243321 + 0.212779i
\(79\) − 3.22986e9i − 1.04966i −0.851207 − 0.524830i \(-0.824129\pi\)
0.851207 − 0.524830i \(-0.175871\pi\)
\(80\) −7.69242e8 + 3.18523e9i −0.234754 + 0.972055i
\(81\) 1.00328e8 0.0287739
\(82\) 2.14286e9 + 2.45044e9i 0.577995 + 0.660960i
\(83\) 4.57436e9 − 4.57436e9i 1.16129 − 1.16129i 0.177094 − 0.984194i \(-0.443330\pi\)
0.984194 − 0.177094i \(-0.0566696\pi\)
\(84\) 1.55947e9 + 2.09795e8i 0.372890 + 0.0501649i
\(85\) −5.74409e9 − 5.40409e9i −1.29457 − 1.21795i
\(86\) 6.24783e9 + 4.18376e8i 1.32812 + 0.0889353i
\(87\) −1.88243e9 + 1.88243e9i −0.377678 + 0.377678i
\(88\) −2.31016e9 − 3.48934e9i −0.437752 − 0.661195i
\(89\) 1.57146e8i 0.0281419i 0.999901 + 0.0140709i \(0.00447907\pi\)
−0.999901 + 0.0140709i \(0.995521\pi\)
\(90\) −2.43294e9 − 2.61670e9i −0.412021 − 0.443141i
\(91\) − 1.92177e9i − 0.307960i
\(92\) 3.14966e9 − 2.40270e9i 0.477887 − 0.364553i
\(93\) −2.10956e9 + 2.10956e9i −0.303234 + 0.303234i
\(94\) 1.64194e9 + 1.09950e8i 0.223726 + 0.0149815i
\(95\) −1.25921e8 − 4.12882e9i −0.0162735 − 0.533590i
\(96\) 4.84026e9 + 1.68129e9i 0.593625 + 0.206199i
\(97\) −1.19158e10 + 1.19158e10i −1.38760 + 1.38760i −0.557278 + 0.830326i \(0.688154\pi\)
−0.830326 + 0.557278i \(0.811846\pi\)
\(98\) 4.36526e9 − 3.81732e9i 0.482925 − 0.422307i
\(99\) 4.56305e9 0.479821
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.13.17 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.45 yes 116
8.5 even 2 inner 40.11.i.a.13.45 yes 116
40.37 odd 4 inner 40.11.i.a.37.17 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.17 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.45 yes 116 8.5 even 2 inner
40.11.i.a.37.17 yes 116 40.37 odd 4 inner
40.11.i.a.37.45 yes 116 5.2 odd 4 inner