Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(53,80)] 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("80.53"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.t (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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 77.43
Character \(\chi\) \(=\) 80.77
Dual form 80.11.t.a.53.43

$q$-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+(-13.4770 - 29.0236i) q^{2} +161.597 q^{3} +(-660.740 + 782.303i) q^{4} +(3090.65 - 462.093i) q^{5} +(-2177.85 - 4690.14i) q^{6} +(13285.8 - 13285.8i) q^{7} +(31610.1 + 8633.96i) q^{8} -32935.3 q^{9} +(-55064.3 - 83474.1i) q^{10} +(145925. - 145925. i) q^{11} +(-106774. + 126418. i) q^{12} +93073.8 q^{13} +(-564653. - 206548. i) q^{14} +(499440. - 74673.0i) q^{15} +(-175421. - 1.03380e6i) q^{16} +(481157. + 481157. i) q^{17} +(443869. + 955901. i) q^{18} +(-1.62060e6 + 1.62060e6i) q^{19} +(-1.68062e6 + 2.72315e6i) q^{20} +(2.14694e6 - 2.14694e6i) q^{21} +(-6.20190e6 - 2.26864e6i) q^{22} +(496004. + 496004. i) q^{23} +(5.10810e6 + 1.39522e6i) q^{24} +(9.33857e6 - 2.85633e6i) q^{25} +(-1.25436e6 - 2.70134e6i) q^{26} -1.48644e7 q^{27} +(1.61506e6 + 1.91719e7i) q^{28} +(1.67257e7 - 1.67257e7i) q^{29} +(-8.89824e6 - 1.34892e7i) q^{30} +2.35535e7 q^{31} +(-2.76404e7 + 1.90239e7i) q^{32} +(2.35811e7 - 2.35811e7i) q^{33} +(7.48036e6 - 2.04495e7i) q^{34} +(3.49223e7 - 4.72009e7i) q^{35} +(2.17617e7 - 2.57654e7i) q^{36} +8.53991e7 q^{37} +(6.88766e7 + 2.51948e7i) q^{38} +1.50405e7 q^{39} +(1.01685e8 + 1.20777e7i) q^{40} -1.28877e7i q^{41} +(-9.12465e7 - 3.33777e7i) q^{42} +9.53857e7i q^{43} +(1.77391e7 + 2.10576e8i) q^{44} +(-1.01791e8 + 1.52192e7i) q^{45} +(7.71118e6 - 2.10805e7i) q^{46} +(-1.87754e8 - 1.87754e8i) q^{47} +(-2.83475e7 - 1.67059e8i) q^{48} -7.05478e7i q^{49} +(-2.08757e8 - 2.32544e8i) q^{50} +(7.77538e7 + 7.77538e7i) q^{51} +(-6.14976e7 + 7.28120e7i) q^{52} -6.79121e6i q^{53} +(2.00328e8 + 4.31419e8i) q^{54} +(3.83571e8 - 5.18433e8i) q^{55} +(5.34673e8 - 3.05255e8i) q^{56} +(-2.61885e8 + 2.61885e8i) q^{57} +(-7.10852e8 - 2.60027e8i) q^{58} +(1.99395e7 + 1.99395e7i) q^{59} +(-2.71583e8 + 4.40053e8i) q^{60} +(1.02112e8 + 1.02112e8i) q^{61} +(-3.17431e8 - 6.83608e8i) q^{62} +(-4.37571e8 + 4.37571e8i) q^{63} +(9.24651e8 + 5.45840e8i) q^{64} +(2.87658e8 - 4.30087e7i) q^{65} +(-1.00221e9 - 3.66605e8i) q^{66} -1.06419e9i q^{67} +(-6.94331e8 + 5.84909e7i) q^{68} +(8.01529e7 + 8.01529e7i) q^{69} +(-1.84059e9 - 3.77446e8i) q^{70} -2.56049e9i q^{71} +(-1.04109e9 - 2.84362e8i) q^{72} +(-2.17841e9 - 2.17841e9i) q^{73} +(-1.15092e9 - 2.47859e9i) q^{74} +(1.50909e9 - 4.61575e8i) q^{75} +(-1.97005e8 - 2.33860e9i) q^{76} -3.87745e9i q^{77} +(-2.02701e8 - 4.36529e8i) q^{78} -1.19428e8i q^{79} +(-1.01987e9 - 3.11404e9i) q^{80} -4.57255e8 q^{81} +(-3.74047e8 + 1.73688e8i) q^{82} -4.47837e9 q^{83} +(2.60989e8 + 3.09813e9i) q^{84} +(1.70943e9 + 1.26475e9i) q^{85} +(2.76844e9 - 1.28551e9i) q^{86} +(2.70283e9 - 2.70283e9i) q^{87} +(5.87260e9 - 3.35279e9i) q^{88} -8.10767e9 q^{89} +(1.81356e9 + 2.74924e9i) q^{90} +(1.23656e9 - 1.23656e9i) q^{91} +(-7.15755e8 + 6.02957e7i) q^{92} +3.80619e9 q^{93} +(-2.91893e9 + 7.97965e9i) q^{94} +(-4.25984e9 + 5.75757e9i) q^{95} +(-4.46662e9 + 3.07420e9i) q^{96} +(3.22990e9 + 3.22990e9i) q^{97} +(-2.04755e9 + 9.50774e8i) q^{98} +(-4.80608e9 + 4.80608e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.4770 − 29.0236i −0.421157 − 0.906988i
\(3\) 161.597 0.665010 0.332505 − 0.943102i \(-0.392106\pi\)
0.332505 + 0.943102i \(0.392106\pi\)
\(4\) −660.740 + 782.303i −0.645254 + 0.763968i
\(5\) 3090.65 − 462.093i 0.989007 − 0.147870i
\(6\) −2177.85 − 4690.14i −0.280073 − 0.603156i
\(7\) 13285.8 − 13285.8i 0.790490 − 0.790490i −0.191084 − 0.981574i \(-0.561200\pi\)
0.981574 + 0.191084i \(0.0612002\pi\)
\(8\) 31610.1 + 8633.96i 0.964663 + 0.263488i
\(9\) −32935.3 −0.557762
\(10\) −55064.3 − 83474.1i −0.550643 − 0.834741i
\(11\) 145925. − 145925.i 0.906078 − 0.906078i −0.0898746 − 0.995953i \(-0.528647\pi\)
0.995953 + 0.0898746i \(0.0286466\pi\)
\(12\) −106774. + 126418.i −0.429100 + 0.508046i
\(13\) 93073.8 0.250675 0.125337 − 0.992114i \(-0.459999\pi\)
0.125337 + 0.992114i \(0.459999\pi\)
\(14\) −564653. − 206548.i −1.04988 − 0.384045i
\(15\) 499440. − 74673.0i 0.657699 − 0.0983348i
\(16\) −175421. − 1.03380e6i −0.167294 − 0.985907i
\(17\) 481157. + 481157.i 0.338877 + 0.338877i 0.855945 − 0.517067i \(-0.172976\pi\)
−0.517067 + 0.855945i \(0.672976\pi\)
\(18\) 443869. + 955901.i 0.234905 + 0.505883i
\(19\) −1.62060e6 + 1.62060e6i −0.654498 + 0.654498i −0.954073 − 0.299575i \(-0.903155\pi\)
0.299575 + 0.954073i \(0.403155\pi\)
\(20\) −1.68062e6 + 2.72315e6i −0.525193 + 0.850983i
\(21\) 2.14694e6 − 2.14694e6i 0.525683 − 0.525683i
\(22\) −6.20190e6 − 2.26864e6i −1.20340 − 0.440201i
\(23\) 496004. + 496004.i 0.0770630 + 0.0770630i 0.744588 − 0.667525i \(-0.232646\pi\)
−0.667525 + 0.744588i \(0.732646\pi\)
\(24\) 5.10810e6 + 1.39522e6i 0.641510 + 0.175222i
\(25\) 9.33857e6 − 2.85633e6i 0.956269 − 0.292488i
\(26\) −1.25436e6 − 2.70134e6i −0.105573 − 0.227359i
\(27\) −1.48644e7 −1.03593
\(28\) 1.61506e6 + 1.91719e7i 0.0938421 + 1.11398i
\(29\) 1.67257e7 − 1.67257e7i 0.815443 − 0.815443i −0.170001 − 0.985444i \(-0.554377\pi\)
0.985444 + 0.170001i \(0.0543770\pi\)
\(30\) −8.89824e6 − 1.34892e7i −0.366183 − 0.555111i
\(31\) 2.35535e7 0.822711 0.411356 − 0.911475i \(-0.365055\pi\)
0.411356 + 0.911475i \(0.365055\pi\)
\(32\) −2.76404e7 + 1.90239e7i −0.823749 + 0.566955i
\(33\) 2.35811e7 − 2.35811e7i 0.602551 − 0.602551i
\(34\) 7.48036e6 − 2.04495e7i 0.164637 − 0.450078i
\(35\) 3.49223e7 − 4.72009e7i 0.664910 − 0.898689i
\(36\) 2.17617e7 − 2.57654e7i 0.359898 − 0.426112i
\(37\) 8.53991e7 1.23153 0.615764 − 0.787930i \(-0.288847\pi\)
0.615764 + 0.787930i \(0.288847\pi\)
\(38\) 6.88766e7 + 2.51948e7i 0.869268 + 0.317975i
\(39\) 1.50405e7 0.166701
\(40\) 1.01685e8 + 1.20777e7i 0.993020 + 0.117947i
\(41\) − 1.28877e7i − 0.111239i −0.998452 − 0.0556193i \(-0.982287\pi\)
0.998452 − 0.0556193i \(-0.0177133\pi\)
\(42\) −9.12465e7 − 3.33777e7i −0.698184 − 0.255393i
\(43\) 9.53857e7i 0.648845i 0.945912 + 0.324422i \(0.105170\pi\)
−0.945912 + 0.324422i \(0.894830\pi\)
\(44\) 1.77391e7 + 2.10576e8i 0.107564 + 1.27687i
\(45\) −1.01791e8 + 1.52192e7i −0.551631 + 0.0824761i
\(46\) 7.71118e6 − 2.10805e7i 0.0374396 − 0.102351i
\(47\) −1.87754e8 − 1.87754e8i −0.818652 − 0.818652i 0.167261 − 0.985913i \(-0.446508\pi\)
−0.985913 + 0.167261i \(0.946508\pi\)
\(48\) −2.83475e7 − 1.67059e8i −0.111252 − 0.655638i
\(49\) − 7.05478e7i − 0.249749i
\(50\) −2.08757e8 − 2.32544e8i −0.668022 − 0.744141i
\(51\) 7.77538e7 + 7.77538e7i 0.225357 + 0.225357i
\(52\) −6.14976e7 + 7.28120e7i −0.161749 + 0.191508i
\(53\) − 6.79121e6i − 0.0162393i −0.999967 − 0.00811967i \(-0.997415\pi\)
0.999967 − 0.00811967i \(-0.00258460\pi\)
\(54\) 2.00328e8 + 4.31419e8i 0.436288 + 0.939573i
\(55\) 3.83571e8 − 5.18433e8i 0.762136 − 1.03010i
\(56\) 5.34673e8 − 3.05255e8i 0.970840 − 0.554272i
\(57\) −2.61885e8 + 2.61885e8i −0.435247 + 0.435247i
\(58\) −7.10852e8 − 2.60027e8i −1.08303 − 0.396168i
\(59\) 1.99395e7 + 1.99395e7i 0.0278904 + 0.0278904i 0.720914 − 0.693024i \(-0.243722\pi\)
−0.693024 + 0.720914i \(0.743722\pi\)
\(60\) −2.71583e8 + 4.40053e8i −0.349258 + 0.565912i
\(61\) 1.02112e8 + 1.02112e8i 0.120900 + 0.120900i 0.764968 − 0.644068i \(-0.222755\pi\)
−0.644068 + 0.764968i \(0.722755\pi\)
\(62\) −3.17431e8 − 6.83608e8i −0.346490 − 0.746189i
\(63\) −4.37571e8 + 4.37571e8i −0.440905 + 0.440905i
\(64\) 9.24651e8 + 5.45840e8i 0.861149 + 0.508353i
\(65\) 2.87658e8 − 4.30087e7i 0.247919 − 0.0370672i
\(66\) −1.00221e9 − 3.66605e8i −0.800275 − 0.292738i
\(67\) − 1.06419e9i − 0.788213i −0.919065 − 0.394107i \(-0.871054\pi\)
0.919065 − 0.394107i \(-0.128946\pi\)
\(68\) −6.94331e8 + 5.84909e7i −0.477554 + 0.0402294i
\(69\) 8.01529e7 + 8.01529e7i 0.0512477 + 0.0512477i
\(70\) −1.84059e9 − 3.77446e8i −1.09513 − 0.224577i
\(71\) − 2.56049e9i − 1.41916i −0.704624 − 0.709581i \(-0.748884\pi\)
0.704624 − 0.709581i \(-0.251116\pi\)
\(72\) −1.04109e9 − 2.84362e8i −0.538052 − 0.146963i
\(73\) −2.17841e9 − 2.17841e9i −1.05081 − 1.05081i −0.998638 − 0.0521741i \(-0.983385\pi\)
−0.0521741 − 0.998638i \(-0.516615\pi\)
\(74\) −1.15092e9 − 2.47859e9i −0.518667 − 1.11698i
\(75\) 1.50909e9 − 4.61575e8i 0.635928 − 0.194508i
\(76\) −1.97005e8 − 2.33860e9i −0.0776979 − 0.922333i
\(77\) − 3.87745e9i − 1.43249i
\(78\) −2.02701e8 − 4.36529e8i −0.0702073 − 0.151196i
\(79\) − 1.19428e8i − 0.0388123i −0.999812 − 0.0194062i \(-0.993822\pi\)
0.999812 − 0.0194062i \(-0.00617756\pi\)
\(80\) −1.01987e9 − 3.11404e9i −0.311241 − 0.950331i
\(81\) −4.57255e8 −0.131139
\(82\) −3.74047e8 + 1.73688e8i −0.100892 + 0.0468489i
\(83\) −4.47837e9 −1.13692 −0.568460 − 0.822711i \(-0.692461\pi\)
−0.568460 + 0.822711i \(0.692461\pi\)
\(84\) 2.60989e8 + 3.09813e9i 0.0624059 + 0.740805i
\(85\) 1.70943e9 + 1.26475e9i 0.385262 + 0.285042i
\(86\) 2.76844e9 − 1.28551e9i 0.588494 − 0.273265i
\(87\) 2.70283e9 − 2.70283e9i 0.542278 − 0.542278i
\(88\) 5.87260e9 − 3.35279e9i 1.11280 − 0.635320i
\(89\) −8.10767e9 −1.45193 −0.725965 − 0.687731i \(-0.758607\pi\)
−0.725965 + 0.687731i \(0.758607\pi\)
\(90\) 1.81356e9 + 2.74924e9i 0.307128 + 0.465587i
\(91\) 1.23656e9 − 1.23656e9i 0.198156 − 0.198156i
\(92\) −7.15755e8 + 6.02957e7i −0.108599 + 0.00914845i
\(93\) 3.80619e9 0.547111
\(94\) −2.91893e9 + 7.97965e9i −0.397727 + 1.08729i
\(95\) −4.25984e9 + 5.75757e9i −0.550522 + 0.744083i
\(96\) −4.46662e9 + 3.07420e9i −0.547801 + 0.377031i
\(97\) 3.22990e9 + 3.22990e9i 0.376123 + 0.376123i 0.869701 − 0.493578i \(-0.164311\pi\)
−0.493578 + 0.869701i \(0.664311\pi\)
\(98\) −2.04755e9 + 9.50774e8i −0.226519 + 0.105183i
\(99\) −4.80608e9 + 4.80608e9i −0.505376 + 0.505376i
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 80.11.t.a.77.43 yes 236
5.3 odd 4 80.11.i.a.13.17 ✓ 236
16.5 even 4 80.11.i.a.37.17 yes 236
80.53 odd 4 inner 80.11.t.a.53.43 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.17 ✓ 236 5.3 odd 4
80.11.i.a.37.17 yes 236 16.5 even 4
80.11.t.a.53.43 yes 236 80.53 odd 4 inner
80.11.t.a.77.43 yes 236 1.1 even 1 trivial