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Properties

Label 80.11.i.a.13.17
Level $80$
Weight $11$
Character 80.13
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
Inner twists $2$

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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(13,80)] 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("80.13"); 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, 3, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(50.8285802139\)
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 13.17
Character \(\chi\) \(=\) 80.13
Dual form 80.11.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+(-29.0236 + 13.4770i) q^{2} +161.597i q^{3} +(660.740 - 782.303i) q^{4} +(462.093 - 3090.65i) q^{5} +(-2177.85 - 4690.14i) q^{6} +(-13285.8 - 13285.8i) q^{7} +(-8633.96 + 31610.1i) q^{8} +32935.3 q^{9} +(28241.1 + 95929.4i) q^{10} +(145925. - 145925. i) q^{11} +(126418. + 106774. i) q^{12} +93073.8i q^{13} +(564653. + 206548. i) q^{14} +(499440. + 74673.0i) q^{15} +(-175421. - 1.03380e6i) q^{16} +(481157. - 481157. i) q^{17} +(-955901. + 443869. i) q^{18} +(1.62060e6 - 1.62060e6i) q^{19} +(-2.11250e6 - 2.40361e6i) q^{20} +(2.14694e6 - 2.14694e6i) q^{21} +(-2.26864e6 + 6.20190e6i) 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} +1.48644e7i q^{27} +(-1.91719e7 + 1.61506e6i) q^{28} +(-1.67257e7 + 1.67257e7i) q^{29} +(-1.55019e7 + 4.56368e6i) q^{30} +2.35535e7 q^{31} +(1.90239e7 + 2.76404e7i) q^{32} +(2.35811e7 + 2.35811e7i) q^{33} +(-7.48036e6 + 2.04495e7i) q^{34} +(-4.72009e7 + 3.49223e7i) q^{35} +(2.17617e7 - 2.57654e7i) q^{36} -8.53991e7i q^{37} +(-2.51948e7 + 6.88766e7i) q^{38} -1.50405e7 q^{39} +(9.37059e7 + 4.12913e7i) q^{40} -1.28877e7i q^{41} +(-3.33777e7 + 9.12465e7i) q^{42} -9.53857e7 q^{43} +(-1.77391e7 - 2.10576e8i) q^{44} +(1.52192e7 - 1.01791e8i) q^{45} +(7.71118e6 - 2.10805e7i) q^{46} +(-1.87754e8 + 1.87754e8i) q^{47} +(1.67059e8 - 2.83475e7i) q^{48} +7.05478e7i q^{49} +(3.09534e8 - 4.29549e7i) q^{50} +(7.77538e7 + 7.77538e7i) q^{51} +(7.28120e7 + 6.14976e7i) q^{52} +6.79121e6 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} +(2.60027e8 - 7.10852e8i) q^{58} +(-1.99395e7 - 1.99395e7i) q^{59} +(3.88417e8 - 3.41374e8i) q^{60} +(1.02112e8 + 1.02112e8i) q^{61} +(-6.83608e8 + 3.17431e8i) 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} -1.06419e9 q^{67} +(-5.84909e7 - 6.94331e8i) q^{68} +(-8.01529e7 - 8.01529e7i) q^{69} +(8.99290e8 - 1.64970e9i) q^{70} -2.56049e9i q^{71} +(-2.84362e8 + 1.04109e9i) q^{72} +(2.17841e9 - 2.17841e9i) q^{73} +(1.15092e9 + 2.47859e9i) q^{74} +(4.61575e8 - 1.50909e9i) q^{75} +(-1.97005e8 - 2.33860e9i) q^{76} -3.87745e9 q^{77} +(4.36529e8 - 2.02701e8i) q^{78} +1.19428e8i q^{79} +(-3.27617e9 + 6.44525e7i) q^{80} -4.57255e8 q^{81} +(1.73688e8 + 3.74047e8i) q^{82} -4.47837e9i q^{83} +(-2.60989e8 - 3.09813e9i) q^{84} +(-1.26475e9 - 1.70943e9i) q^{85} +(2.76844e9 - 1.28551e9i) q^{86} +(-2.70283e9 - 2.70283e9i) q^{87} +(3.35279e9 + 5.87260e9i) q^{88} +8.10767e9 q^{89} +(9.30128e8 + 3.15946e9i) q^{90} +(1.23656e9 - 1.23656e9i) q^{91} +(6.02957e7 + 7.15755e8i) q^{92} +3.80619e9i 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} +(-9.50774e8 - 2.04755e9i) q^{98} +(4.80608e9 - 4.80608e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.0236 + 13.4770i −0.906988 + 0.421157i
\(3\) 161.597i 0.665010i 0.943102 + 0.332505i \(0.107894\pi\)
−0.943102 + 0.332505i \(0.892106\pi\)
\(4\) 660.740 − 782.303i 0.645254 − 0.763968i
\(5\) 462.093 − 3090.65i 0.147870 − 0.989007i
\(6\) −2177.85 − 4690.14i −0.280073 − 0.603156i
\(7\) −13285.8 − 13285.8i −0.790490 − 0.790490i 0.191084 − 0.981574i \(-0.438800\pi\)
−0.981574 + 0.191084i \(0.938800\pi\)
\(8\) −8633.96 + 31610.1i −0.263488 + 0.964663i
\(9\) 32935.3 0.557762
\(10\) 28241.1 + 95929.4i 0.282411 + 0.959294i
\(11\) 145925. − 145925.i 0.906078 − 0.906078i −0.0898746 − 0.995953i \(-0.528647\pi\)
0.995953 + 0.0898746i \(0.0286466\pi\)
\(12\) 126418. + 106774.i 0.508046 + 0.429100i
\(13\) 93073.8i 0.250675i 0.992114 + 0.125337i \(0.0400014\pi\)
−0.992114 + 0.125337i \(0.959999\pi\)
\(14\) 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 −0.517067 − 0.855945i \(-0.672976\pi\)
0.855945 + 0.517067i \(0.172976\pi\)
\(18\) −955901. + 443869.i −0.505883 + 0.234905i
\(19\) 1.62060e6 − 1.62060e6i 0.654498 − 0.654498i −0.299575 − 0.954073i \(-0.596845\pi\)
0.954073 + 0.299575i \(0.0968449\pi\)
\(20\) −2.11250e6 − 2.40361e6i −0.660156 − 0.751128i
\(21\) 2.14694e6 − 2.14694e6i 0.525683 − 0.525683i
\(22\) −2.26864e6 + 6.20190e6i −0.440201 + 1.20340i
\(23\) −496004. + 496004.i −0.0770630 + 0.0770630i −0.744588 − 0.667525i \(-0.767354\pi\)
0.667525 + 0.744588i \(0.267354\pi\)
\(24\) −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
\(27\) 1.48644e7i 1.03593i
\(28\) −1.91719e7 + 1.61506e6i −1.11398 + 0.0938421i
\(29\) −1.67257e7 + 1.67257e7i −0.815443 + 0.815443i −0.985444 − 0.170001i \(-0.945623\pi\)
0.170001 + 0.985444i \(0.445623\pi\)
\(30\) −1.55019e7 + 4.56368e6i −0.637940 + 0.187806i
\(31\) 2.35535e7 0.822711 0.411356 − 0.911475i \(-0.365055\pi\)
0.411356 + 0.911475i \(0.365055\pi\)
\(32\) 1.90239e7 + 2.76404e7i 0.566955 + 0.823749i
\(33\) 2.35811e7 + 2.35811e7i 0.602551 + 0.602551i
\(34\) −7.48036e6 + 2.04495e7i −0.164637 + 0.450078i
\(35\) −4.72009e7 + 3.49223e7i −0.898689 + 0.664910i
\(36\) 2.17617e7 − 2.57654e7i 0.359898 − 0.426112i
\(37\) − 8.53991e7i − 1.23153i −0.787930 − 0.615764i \(-0.788847\pi\)
0.787930 − 0.615764i \(-0.211153\pi\)
\(38\) −2.51948e7 + 6.88766e7i −0.317975 + 0.869268i
\(39\) −1.50405e7 −0.166701
\(40\) 9.37059e7 + 4.12913e7i 0.915096 + 0.403235i
\(41\) − 1.28877e7i − 0.111239i −0.998452 − 0.0556193i \(-0.982287\pi\)
0.998452 − 0.0556193i \(-0.0177133\pi\)
\(42\) −3.33777e7 + 9.12465e7i −0.255393 + 0.698184i
\(43\) −9.53857e7 −0.648845 −0.324422 − 0.945912i \(-0.605170\pi\)
−0.324422 + 0.945912i \(0.605170\pi\)
\(44\) −1.77391e7 − 2.10576e8i −0.107564 − 1.27687i
\(45\) 1.52192e7 − 1.01791e8i 0.0824761 − 0.551631i
\(46\) 7.71118e6 − 2.10805e7i 0.0374396 − 0.102351i
\(47\) −1.87754e8 + 1.87754e8i −0.818652 + 0.818652i −0.985913 − 0.167261i \(-0.946508\pi\)
0.167261 + 0.985913i \(0.446508\pi\)
\(48\) 1.67059e8 − 2.83475e7i 0.655638 − 0.111252i
\(49\) 7.05478e7i 0.249749i
\(50\) 3.09534e8 − 4.29549e7i 0.990508 − 0.137456i
\(51\) 7.77538e7 + 7.77538e7i 0.225357 + 0.225357i
\(52\) 7.28120e7 + 6.14976e7i 0.191508 + 0.161749i
\(53\) 6.79121e6 0.0162393 0.00811967 − 0.999967i \(-0.497415\pi\)
0.00811967 + 0.999967i \(0.497415\pi\)
\(54\) −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
\(58\) 2.60027e8 − 7.10852e8i 0.396168 − 1.08303i
\(59\) −1.99395e7 − 1.99395e7i −0.0278904 − 0.0278904i 0.693024 − 0.720914i \(-0.256278\pi\)
−0.720914 + 0.693024i \(0.756278\pi\)
\(60\) 3.88417e8 − 3.41374e8i 0.499508 − 0.439010i
\(61\) 1.02112e8 + 1.02112e8i 0.120900 + 0.120900i 0.764968 − 0.644068i \(-0.222755\pi\)
−0.644068 + 0.764968i \(0.722755\pi\)
\(62\) −6.83608e8 + 3.17431e8i −0.746189 + 0.346490i
\(63\) −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
\(67\) −1.06419e9 −0.788213 −0.394107 − 0.919065i \(-0.628946\pi\)
−0.394107 + 0.919065i \(0.628946\pi\)
\(68\) −5.84909e7 − 6.94331e8i −0.0402294 − 0.477554i
\(69\) −8.01529e7 − 8.01529e7i −0.0512477 − 0.0512477i
\(70\) 8.99290e8 − 1.64970e9i 0.535069 − 0.981555i
\(71\) − 2.56049e9i − 1.41916i −0.704624 − 0.709581i \(-0.748884\pi\)
0.704624 − 0.709581i \(-0.251116\pi\)
\(72\) −2.84362e8 + 1.04109e9i −0.146963 + 0.538052i
\(73\) 2.17841e9 − 2.17841e9i 1.05081 − 1.05081i 0.0521741 − 0.998638i \(-0.483385\pi\)
0.998638 − 0.0521741i \(-0.0166151\pi\)
\(74\) 1.15092e9 + 2.47859e9i 0.518667 + 1.11698i
\(75\) 4.61575e8 − 1.50909e9i 0.194508 − 0.635928i
\(76\) −1.97005e8 − 2.33860e9i −0.0776979 − 0.922333i
\(77\) −3.87745e9 −1.43249
\(78\) 4.36529e8 − 2.02701e8i 0.151196 − 0.0702073i
\(79\) 1.19428e8i 0.0388123i 0.999812 + 0.0194062i \(0.00617756\pi\)
−0.999812 + 0.0194062i \(0.993822\pi\)
\(80\) −3.27617e9 + 6.44525e7i −0.999807 + 0.0196693i
\(81\) −4.57255e8 −0.131139
\(82\) 1.73688e8 + 3.74047e8i 0.0468489 + 0.100892i
\(83\) − 4.47837e9i − 1.13692i −0.822711 − 0.568460i \(-0.807539\pi\)
0.822711 − 0.568460i \(-0.192461\pi\)
\(84\) −2.60989e8 − 3.09813e9i −0.0624059 − 0.740805i
\(85\) −1.26475e9 − 1.70943e9i −0.285042 − 0.385262i
\(86\) 2.76844e9 − 1.28551e9i 0.588494 − 0.273265i
\(87\) −2.70283e9 − 2.70283e9i −0.542278 − 0.542278i
\(88\) 3.35279e9 + 5.87260e9i 0.635320 + 1.11280i
\(89\) 8.10767e9 1.45193 0.725965 − 0.687731i \(-0.241393\pi\)
0.725965 + 0.687731i \(0.241393\pi\)
\(90\) 9.30128e8 + 3.15946e9i 0.157518 + 0.535058i
\(91\) 1.23656e9 − 1.23656e9i 0.198156 − 0.198156i
\(92\) 6.02957e7 + 7.15755e8i 0.00914845 + 0.108599i
\(93\) 3.80619e9i 0.547111i
\(94\) 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 −0.493578 − 0.869701i \(-0.664311\pi\)
0.869701 + 0.493578i \(0.164311\pi\)
\(98\) −9.50774e8 − 2.04755e9i −0.105183 − 0.226519i
\(99\) 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.i.a.13.17 ✓ 236
5.2 odd 4 80.11.t.a.77.43 yes 236
16.5 even 4 80.11.t.a.53.43 yes 236
80.37 odd 4 inner 80.11.i.a.37.17 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.17 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.17 yes 236 80.37 odd 4 inner
80.11.t.a.53.43 yes 236 16.5 even 4
80.11.t.a.77.43 yes 236 5.2 odd 4