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 13.1
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.1

$q$-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+(-31.8852 - 2.70791i) q^{2} +(-181.420 + 181.420i) q^{3} +(1009.33 + 172.685i) q^{4} +(1980.27 - 2417.47i) q^{5} +(6275.90 - 5293.36i) q^{6} +(16061.1 - 16061.1i) q^{7} +(-31715.2 - 8239.28i) q^{8} -6777.67i q^{9} +(-69687.7 + 71719.1i) q^{10} +9369.49i q^{11} +(-214442. + 151785. i) q^{12} +(-44722.8 + 44722.8i) q^{13} +(-555602. + 468618. i) q^{14} +(79316.0 + 797839. i) q^{15} +(988936. + 348593. i) q^{16} +(409687. - 409687. i) q^{17} +(-18353.4 + 216108. i) q^{18} -2.36738e6 q^{19} +(2.41622e6 - 2.09807e6i) q^{20} +5.82760e6i q^{21} +(25371.8 - 298748. i) q^{22} +(2.24613e6 + 2.24613e6i) q^{23} +(7.24856e6 - 4.25901e6i) q^{24} +(-1.92267e6 - 9.57449e6i) q^{25} +(1.54710e6 - 1.30489e6i) q^{26} +(-9.48308e6 - 9.48308e6i) q^{27} +(1.89845e7 - 1.34375e7i) q^{28} -2.96608e7 q^{29} +(-368530. - 2.56541e7i) q^{30} +3.83325e7 q^{31} +(-3.05885e7 - 1.37929e7i) q^{32} +(-1.69982e6 - 1.69982e6i) q^{33} +(-1.41724e7 + 1.19536e7i) q^{34} +(-7.02181e6 - 7.06323e7i) q^{35} +(1.17040e6 - 6.84094e6i) q^{36} +(-1.26503e7 - 1.26503e7i) q^{37} +(7.54843e7 + 6.41065e6i) q^{38} -1.62272e7i q^{39} +(-8.27230e7 + 6.03545e7i) q^{40} -7.38613e7 q^{41} +(1.57806e7 - 1.85814e8i) q^{42} +(1.96061e8 - 1.96061e8i) q^{43} +(-1.61797e6 + 9.45695e6i) q^{44} +(-1.63848e7 - 1.34216e7i) q^{45} +(-6.55359e7 - 7.77005e7i) q^{46} +(2.72087e8 - 2.72087e8i) q^{47} +(-2.42655e8 + 1.16171e8i) q^{48} -2.33439e8i q^{49} +(3.53778e7 + 3.10491e8i) q^{50} +1.48651e8i q^{51} +(-5.28632e7 + 3.74173e7i) q^{52} +(2.93142e7 - 2.93142e7i) q^{53} +(2.76691e8 + 3.28049e8i) q^{54} +(2.26504e7 + 1.85541e7i) q^{55} +(-6.41711e8 + 3.77048e8i) q^{56} +(4.29490e8 - 4.29490e8i) q^{57} +(9.45742e8 + 8.03190e7i) q^{58} -6.79472e8 q^{59} +(-5.77183e7 + 8.18983e8i) q^{60} +1.01534e9i q^{61} +(-1.22224e9 - 1.03801e8i) q^{62} +(-1.08857e8 - 1.08857e8i) q^{63} +(9.37970e8 + 5.22622e8i) q^{64} +(1.95526e7 + 1.96679e8i) q^{65} +(4.95960e7 + 5.88019e7i) q^{66} +(-3.41798e8 - 3.41798e8i) q^{67} +(4.84258e8 - 3.42765e8i) q^{68} -8.14986e8 q^{69} +(3.26257e7 + 2.27114e9i) q^{70} -6.24796e8 q^{71} +(-5.58432e7 + 2.14956e8i) q^{72} +(-2.00067e9 - 2.00067e9i) q^{73} +(3.69103e8 + 4.37615e8i) q^{74} +(2.08582e9 + 1.38820e9i) q^{75} +(-2.38947e9 - 4.08810e8i) q^{76} +(1.50484e8 + 1.50484e8i) q^{77} +(-4.39420e7 + 5.17409e8i) q^{78} -5.65597e9i q^{79} +(2.80108e9 - 1.70041e9i) q^{80} +3.84106e9 q^{81} +(2.35508e9 + 2.00010e8i) q^{82} +(1.18232e9 - 1.18232e9i) q^{83} +(-1.00634e9 + 5.88200e9i) q^{84} +(-1.79113e8 - 1.80170e9i) q^{85} +(-6.78237e9 + 5.72054e9i) q^{86} +(5.38108e9 - 5.38108e9i) q^{87} +(7.71979e7 - 2.97156e8i) q^{88} +1.86976e9i q^{89} +(4.86088e8 + 4.72321e8i) q^{90} +1.43659e9i q^{91} +(1.87922e9 + 2.65496e9i) q^{92} +(-6.95429e9 + 6.95429e9i) q^{93} +(-9.41235e9 + 7.93877e9i) q^{94} +(-4.68805e9 + 5.72305e9i) q^{95} +(8.05169e9 - 3.04705e9i) q^{96} +(4.01722e9 - 4.01722e9i) q^{97} +(-6.32134e8 + 7.44327e9i) q^{98} +6.35033e7 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.8852 − 2.70791i −0.996413 − 0.0846223i
\(3\) −181.420 + 181.420i −0.746586 + 0.746586i −0.973836 − 0.227251i \(-0.927026\pi\)
0.227251 + 0.973836i \(0.427026\pi\)
\(4\) 1009.33 + 172.685i 0.985678 + 0.168638i
\(5\) 1980.27 − 2417.47i 0.633687 − 0.773589i
\(6\) 6275.90 − 5293.36i 0.807086 − 0.680730i
\(7\) 16061.1 − 16061.1i 0.955617 − 0.955617i −0.0434393 − 0.999056i \(-0.513832\pi\)
0.999056 + 0.0434393i \(0.0138315\pi\)
\(8\) −31715.2 − 8239.28i −0.967872 − 0.251443i
\(9\) − 6777.67i − 0.114781i
\(10\) −69687.7 + 71719.1i −0.696877 + 0.717191i
\(11\) 9369.49i 0.0581771i 0.999577 + 0.0290886i \(0.00926049\pi\)
−0.999577 + 0.0290886i \(0.990740\pi\)
\(12\) −214442. + 151785.i −0.861796 + 0.609991i
\(13\) −44722.8 + 44722.8i −0.120451 + 0.120451i −0.764763 − 0.644312i \(-0.777144\pi\)
0.644312 + 0.764763i \(0.277144\pi\)
\(14\) −555602. + 468618.i −1.03306 + 0.871323i
\(15\) 79316.0 + 797839.i 0.104449 + 1.05065i
\(16\) 988936. + 348593.i 0.943123 + 0.332445i
\(17\) 409687. − 409687.i 0.288541 − 0.288541i −0.547962 − 0.836503i \(-0.684596\pi\)
0.836503 + 0.547962i \(0.184596\pi\)
\(18\) −18353.4 + 216108.i −0.00971299 + 0.114369i
\(19\) −2.36738e6 −0.956091 −0.478046 − 0.878335i \(-0.658655\pi\)
−0.478046 + 0.878335i \(0.658655\pi\)
\(20\) 2.41622e6 − 2.09807e6i 0.755068 − 0.655647i
\(21\) 5.82760e6i 1.42690i
\(22\) 25371.8 − 298748.i 0.00492308 − 0.0579685i
\(23\) 2.24613e6 + 2.24613e6i 0.348976 + 0.348976i 0.859728 − 0.510752i \(-0.170633\pi\)
−0.510752 + 0.859728i \(0.670633\pi\)
\(24\) 7.24856e6 − 4.25901e6i 0.910323 − 0.534876i
\(25\) −1.92267e6 − 9.57449e6i −0.196881 − 0.980427i
\(26\) 1.54710e6 − 1.30489e6i 0.130212 − 0.109827i
\(27\) −9.48308e6 − 9.48308e6i −0.660892 − 0.660892i
\(28\) 1.89845e7 − 1.34375e7i 1.10308 − 0.780778i
\(29\) −2.96608e7 −1.44608 −0.723042 − 0.690804i \(-0.757257\pi\)
−0.723042 + 0.690804i \(0.757257\pi\)
\(30\) −368530. − 2.56541e7i −0.0151658 − 1.05572i
\(31\) 3.83325e7 1.33893 0.669466 − 0.742843i \(-0.266523\pi\)
0.669466 + 0.742843i \(0.266523\pi\)
\(32\) −3.05885e7 − 1.37929e7i −0.911608 − 0.411061i
\(33\) −1.69982e6 − 1.69982e6i −0.0434342 − 0.0434342i
\(34\) −1.41724e7 + 1.19536e7i −0.311923 + 0.263089i
\(35\) −7.02181e6 − 7.06323e7i −0.133693 − 1.34482i
\(36\) 1.17040e6 − 6.84094e6i 0.0193563 − 0.113137i
\(37\) −1.26503e7 − 1.26503e7i −0.182429 − 0.182429i 0.609984 − 0.792413i \(-0.291176\pi\)
−0.792413 + 0.609984i \(0.791176\pi\)
\(38\) 7.54843e7 + 6.41065e6i 0.952662 + 0.0809066i
\(39\) − 1.62272e7i − 0.179855i
\(40\) −8.27230e7 + 6.03545e7i −0.807842 + 0.589400i
\(41\) −7.38613e7 −0.637525 −0.318763 − 0.947835i \(-0.603267\pi\)
−0.318763 + 0.947835i \(0.603267\pi\)
\(42\) 1.57806e7 − 1.85814e8i 0.120748 − 1.42178i
\(43\) 1.96061e8 − 1.96061e8i 1.33367 − 1.33367i 0.431614 − 0.902058i \(-0.357944\pi\)
0.902058 − 0.431614i \(-0.142056\pi\)
\(44\) −1.61797e6 + 9.45695e6i −0.00981085 + 0.0573439i
\(45\) −1.63848e7 − 1.34216e7i −0.0887930 − 0.0727349i
\(46\) −6.55359e7 − 7.77005e7i −0.318193 − 0.377255i
\(47\) 2.72087e8 − 2.72087e8i 1.18637 − 1.18637i 0.208302 − 0.978065i \(-0.433206\pi\)
0.978065 − 0.208302i \(-0.0667938\pi\)
\(48\) −2.42655e8 + 1.16171e8i −0.952320 + 0.455924i
\(49\) − 2.33439e8i − 0.826407i
\(50\) 3.53778e7 + 3.10491e8i 0.113209 + 0.993571i
\(51\) 1.48651e8i 0.430842i
\(52\) −5.28632e7 + 3.74173e7i −0.139039 + 0.0984137i
\(53\) 2.93142e7 − 2.93142e7i 0.0700969 − 0.0700969i −0.671189 − 0.741286i \(-0.734216\pi\)
0.741286 + 0.671189i \(0.234216\pi\)
\(54\) 2.76691e8 + 3.28049e8i 0.602595 + 0.714448i
\(55\) 2.26504e7 + 1.85541e7i 0.0450052 + 0.0368661i
\(56\) −6.41711e8 + 3.77048e8i −1.16520 + 0.684632i
\(57\) 4.29490e8 − 4.29490e8i 0.713804 − 0.713804i
\(58\) 9.45742e8 + 8.03190e7i 1.44090 + 0.122371i
\(59\) −6.79472e8 −0.950411 −0.475206 − 0.879875i \(-0.657626\pi\)
−0.475206 + 0.879875i \(0.657626\pi\)
\(60\) −5.77183e7 + 8.18983e8i −0.0742262 + 1.05322i
\(61\) 1.01534e9i 1.20217i 0.799187 + 0.601083i \(0.205264\pi\)
−0.799187 + 0.601083i \(0.794736\pi\)
\(62\) −1.22224e9 − 1.03801e8i −1.33413 − 0.113304i
\(63\) −1.08857e8 − 1.08857e8i −0.109686 − 0.109686i
\(64\) 9.37970e8 + 5.22622e8i 0.873553 + 0.486729i
\(65\) 1.95526e7 + 1.96679e8i 0.0168514 + 0.169508i
\(66\) 4.95960e7 + 5.88019e7i 0.0396029 + 0.0469539i
\(67\) −3.41798e8 − 3.41798e8i −0.253160 − 0.253160i 0.569105 − 0.822265i \(-0.307290\pi\)
−0.822265 + 0.569105i \(0.807290\pi\)
\(68\) 4.84258e8 − 3.42765e8i 0.333068 − 0.235750i
\(69\) −8.14986e8 −0.521080
\(70\) 3.26257e7 + 2.27114e9i 0.0194120 + 1.35131i
\(71\) −6.24796e8 −0.346295 −0.173148 − 0.984896i \(-0.555394\pi\)
−0.173148 + 0.984896i \(0.555394\pi\)
\(72\) −5.58432e7 + 2.14956e8i −0.0288608 + 0.111093i
\(73\) −2.00067e9 − 2.00067e9i −0.965075 − 0.965075i 0.0343352 − 0.999410i \(-0.489069\pi\)
−0.999410 + 0.0343352i \(0.989069\pi\)
\(74\) 3.69103e8 + 4.37615e8i 0.166337 + 0.197212i
\(75\) 2.08582e9 + 1.38820e9i 0.878962 + 0.584984i
\(76\) −2.38947e9 − 4.08810e8i −0.942398 − 0.161233i
\(77\) 1.50484e8 + 1.50484e8i 0.0555951 + 0.0555951i
\(78\) −4.39420e7 + 5.17409e8i −0.0152197 + 0.179210i
\(79\) − 5.65597e9i − 1.83811i −0.394130 − 0.919055i \(-0.628954\pi\)
0.394130 − 0.919055i \(-0.371046\pi\)
\(80\) 2.80108e9 − 1.70041e9i 0.854820 − 0.518924i
\(81\) 3.84106e9 1.10161
\(82\) 2.35508e9 + 2.00010e8i 0.635239 + 0.0539489i
\(83\) 1.18232e9 − 1.18232e9i 0.300153 − 0.300153i −0.540920 − 0.841074i \(-0.681924\pi\)
0.841074 + 0.540920i \(0.181924\pi\)
\(84\) −1.00634e9 + 5.88200e9i −0.240629 + 1.40646i
\(85\) −1.79113e8 − 1.80170e9i −0.0403676 − 0.406057i
\(86\) −6.78237e9 + 5.72054e9i −1.44175 + 1.21603i
\(87\) 5.38108e9 − 5.38108e9i 1.07963 − 1.07963i
\(88\) 7.71979e7 − 2.97156e8i 0.0146282 − 0.0563080i
\(89\) 1.86976e9i 0.334840i 0.985886 + 0.167420i \(0.0535435\pi\)
−0.985886 + 0.167420i \(0.946456\pi\)
\(90\) 4.86088e8 + 4.72321e8i 0.0823195 + 0.0799879i
\(91\) 1.43659e9i 0.230211i
\(92\) 1.87922e9 + 2.65496e9i 0.285127 + 0.402828i
\(93\) −6.95429e9 + 6.95429e9i −0.999628 + 0.999628i
\(94\) −9.41235e9 + 7.93877e9i −1.28250 + 1.08172i
\(95\) −4.68805e9 + 5.72305e9i −0.605863 + 0.739622i
\(96\) 8.05169e9 − 3.04705e9i 0.987486 − 0.373701i
\(97\) 4.01722e9 − 4.01722e9i 0.467808 − 0.467808i −0.433396 − 0.901204i \(-0.642685\pi\)
0.901204 + 0.433396i \(0.142685\pi\)
\(98\) −6.32134e8 + 7.44327e9i −0.0699324 + 0.823442i
\(99\) 6.35033e7 0.00667760
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.13.1 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.31 yes 116
8.5 even 2 inner 40.11.i.a.13.31 yes 116
40.37 odd 4 inner 40.11.i.a.37.1 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.1 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.31 yes 116 8.5 even 2 inner
40.11.i.a.37.1 yes 116 40.37 odd 4 inner
40.11.i.a.37.31 yes 116 5.2 odd 4 inner