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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 13.8
Character \(\chi\) \(=\) 80.13
Dual form 80.11.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+(-31.2442 - 6.91353i) q^{2} +290.364i q^{3} +(928.406 + 432.016i) q^{4} +(2533.30 - 1829.76i) q^{5} +(2007.44 - 9072.21i) q^{6} +(-16042.8 - 16042.8i) q^{7} +(-26020.6 - 19916.6i) q^{8} -25262.4 q^{9} +(-91801.1 + 39655.5i) q^{10} +(-151572. + 151572. i) q^{11} +(-125442. + 269576. i) q^{12} -592272. i q^{13} +(390333. + 612158. i) q^{14} +(531297. + 735579. i) q^{15} +(675300. + 802173. i) q^{16} +(-1.13065e6 + 1.13065e6i) q^{17} +(789304. + 174652. i) q^{18} +(-110804. + 110804. i) q^{19} +(3.14242e6 - 604336. i) q^{20} +(4.65826e6 - 4.65826e6i) q^{21} +(5.78364e6 - 3.68785e6i) q^{22} +(785939. - 785939. i) q^{23} +(5.78307e6 - 7.55545e6i) q^{24} +(3.06957e6 - 9.27066e6i) q^{25} +(-4.09469e6 + 1.85051e7i) q^{26} +9.81043e6i q^{27} +(-7.96349e6 - 2.18250e7i) q^{28} +(-1.39310e7 + 1.39310e7i) q^{29} +(-1.15145e7 - 2.66558e7i) q^{30} +2.83506e7 q^{31} +(-1.55534e7 - 2.97320e7i) q^{32} +(-4.40110e7 - 4.40110e7i) q^{33} +(4.31432e7 - 2.75096e7i) q^{34} +(-6.99957e7 - 1.12867e7i) q^{35} +(-2.34537e7 - 1.09138e7i) q^{36} +6.75501e7i q^{37} +(4.22805e6 - 2.69595e6i) q^{38} +1.71975e8 q^{39} +(-1.02361e8 - 2.84319e6i) q^{40} +7.63868e6i q^{41} +(-1.77749e8 + 1.13339e8i) q^{42} +1.25157e8 q^{43} +(-2.06202e8 + 7.52387e7i) q^{44} +(-6.39971e7 + 4.62241e7i) q^{45} +(-2.99897e7 + 1.91225e7i) q^{46} +(2.17758e8 - 2.17758e8i) q^{47} +(-2.32922e8 + 1.96083e8i) q^{48} +2.32268e8i q^{49} +(-1.59999e8 + 2.68433e8i) q^{50} +(-3.28301e8 - 3.28301e8i) q^{51} +(2.55871e8 - 5.49869e8i) q^{52} +6.62087e8 q^{53} +(6.78247e7 - 3.06519e8i) q^{54} +(-1.06636e8 + 6.61317e8i) q^{55} +(9.79253e7 + 7.36961e8i) q^{56} +(-3.21736e7 - 3.21736e7i) q^{57} +(5.31576e8 - 3.38951e8i) q^{58} +(5.28573e8 + 5.28573e8i) q^{59} +(1.75477e8 + 9.12445e8i) q^{60} +(-3.82556e8 - 3.82556e8i) q^{61} +(-8.85792e8 - 1.96003e8i) q^{62} +(4.05279e8 + 4.05279e8i) q^{63} +(2.80401e8 + 1.03648e9i) q^{64} +(-1.08372e9 - 1.50040e9i) q^{65} +(1.07082e9 + 1.67936e9i) q^{66} +1.71648e9 q^{67} +(-1.53817e9 + 5.61245e8i) q^{68} +(2.28209e8 + 2.28209e8i) q^{69} +(2.10893e9 + 8.36562e8i) q^{70} +3.60356e9i q^{71} +(6.57342e8 + 5.03140e8i) q^{72} +(1.14659e9 - 1.14659e9i) q^{73} +(4.67010e8 - 2.11055e9i) q^{74} +(2.69187e9 + 8.91294e8i) q^{75} +(-1.50741e8 + 5.50022e7i) q^{76} +4.86327e9 q^{77} +(-5.37322e9 - 1.18895e9i) q^{78} +3.22442e9i q^{79} +(3.17852e9 + 7.96506e8i) q^{80} -4.34031e9 q^{81} +(5.28103e7 - 2.38665e8i) q^{82} +5.37975e9i q^{83} +(6.33720e9 - 2.31231e9i) q^{84} +(-7.95456e8 + 4.93311e9i) q^{85} +(-3.91045e9 - 8.65280e8i) q^{86} +(-4.04507e9 - 4.04507e9i) q^{87} +(6.96278e9 - 9.25194e8i) q^{88} -8.02081e8 q^{89} +(2.31911e9 - 1.00179e9i) q^{90} +(-9.50171e9 + 9.50171e9i) q^{91} +(1.06921e9 - 3.90132e8i) q^{92} +8.23199e9i q^{93} +(-8.30916e9 + 5.29821e9i) q^{94} +(-7.79549e7 + 4.83446e8i) q^{95} +(8.63311e9 - 4.51615e9i) q^{96} +(-3.06849e9 + 3.06849e9i) q^{97} +(1.60579e9 - 7.25704e9i) q^{98} +(3.82906e9 - 3.82906e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.2442 − 6.91353i −0.976383 − 0.216048i
\(3\) 290.364i 1.19491i 0.801901 + 0.597457i \(0.203822\pi\)
−0.801901 + 0.597457i \(0.796178\pi\)
\(4\) 928.406 + 432.016i 0.906647 + 0.421891i
\(5\) 2533.30 − 1829.76i 0.810655 − 0.585524i
\(6\) 2007.44 − 9072.21i 0.258159 − 1.16669i
\(7\) −16042.8 − 16042.8i −0.954531 − 0.954531i 0.0444790 − 0.999010i \(-0.485837\pi\)
−0.999010 + 0.0444790i \(0.985837\pi\)
\(8\) −26020.6 − 19916.6i −0.794085 − 0.607806i
\(9\) −25262.4 −0.427821
\(10\) −91801.1 + 39655.5i −0.918011 + 0.396555i
\(11\) −151572. + 151572.i −0.941141 + 0.941141i −0.998362 − 0.0572202i \(-0.981776\pi\)
0.0572202 + 0.998362i \(0.481776\pi\)
\(12\) −125442. + 269576.i −0.504124 + 1.08337i
\(13\) − 592272.i − 1.59516i −0.603213 − 0.797580i \(-0.706113\pi\)
0.603213 − 0.797580i \(-0.293887\pi\)
\(14\) 390333. + 612158.i 0.725763 + 1.13821i
\(15\) 531297. + 735579.i 0.699651 + 0.968664i
\(16\) 675300. + 802173.i 0.644016 + 0.765012i
\(17\) −1.13065e6 + 1.13065e6i −0.796315 + 0.796315i −0.982512 − 0.186197i \(-0.940384\pi\)
0.186197 + 0.982512i \(0.440384\pi\)
\(18\) 789304. + 174652.i 0.417717 + 0.0924297i
\(19\) −110804. + 110804.i −0.0447496 + 0.0447496i −0.729127 − 0.684378i \(-0.760074\pi\)
0.684378 + 0.729127i \(0.260074\pi\)
\(20\) 3.14242e6 − 604336.i 0.982005 − 0.188855i
\(21\) 4.65826e6 − 4.65826e6i 1.14058 − 1.14058i
\(22\) 5.78364e6 − 3.68785e6i 1.12225 − 0.715583i
\(23\) 785939. − 785939.i 0.122110 − 0.122110i −0.643411 − 0.765521i \(-0.722481\pi\)
0.765521 + 0.643411i \(0.222481\pi\)
\(24\) 5.78307e6 − 7.55545e6i 0.726276 − 0.948864i
\(25\) 3.06957e6 − 9.27066e6i 0.314324 − 0.949316i
\(26\) −4.09469e6 + 1.85051e7i −0.344631 + 1.55749i
\(27\) 9.81043e6i 0.683705i
\(28\) −7.96349e6 − 2.18250e7i −0.462714 − 1.26813i
\(29\) −1.39310e7 + 1.39310e7i −0.679192 + 0.679192i −0.959817 − 0.280625i \(-0.909458\pi\)
0.280625 + 0.959817i \(0.409458\pi\)
\(30\) −1.15145e7 − 2.66558e7i −0.473849 − 1.09694i
\(31\) 2.83506e7 0.990269 0.495134 − 0.868816i \(-0.335119\pi\)
0.495134 + 0.868816i \(0.335119\pi\)
\(32\) −1.55534e7 − 2.97320e7i −0.463527 − 0.886083i
\(33\) −4.40110e7 − 4.40110e7i −1.12458 − 1.12458i
\(34\) 4.31432e7 − 2.75096e7i 0.949551 − 0.605466i
\(35\) −6.99957e7 − 1.12867e7i −1.33270 − 0.214895i
\(36\) −2.34537e7 − 1.09138e7i −0.387882 − 0.180494i
\(37\) 6.75501e7i 0.974131i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(38\) 4.22805e6 − 2.69595e6i 0.0533608 − 0.0340247i
\(39\) 1.71975e8 1.90608
\(40\) −1.02361e8 − 2.84319e6i −0.999614 − 0.0277655i
\(41\) 7.63868e6i 0.0659324i 0.999456 + 0.0329662i \(0.0104954\pi\)
−0.999456 + 0.0329662i \(0.989505\pi\)
\(42\) −1.77749e8 + 1.13339e8i −1.36007 + 0.867225i
\(43\) 1.25157e8 0.851362 0.425681 − 0.904873i \(-0.360035\pi\)
0.425681 + 0.904873i \(0.360035\pi\)
\(44\) −2.06202e8 + 7.52387e7i −1.25034 + 0.456224i
\(45\) −6.39971e7 + 4.62241e7i −0.346815 + 0.250499i
\(46\) −2.99897e7 + 1.91225e7i −0.145607 + 0.0928442i
\(47\) 2.17758e8 − 2.17758e8i 0.949477 − 0.949477i −0.0493062 − 0.998784i \(-0.515701\pi\)
0.998784 + 0.0493062i \(0.0157010\pi\)
\(48\) −2.32922e8 + 1.96083e8i −0.914124 + 0.769544i
\(49\) 2.32268e8i 0.822260i
\(50\) −1.59999e8 + 2.68433e8i −0.511998 + 0.858986i
\(51\) −3.28301e8 − 3.28301e8i −0.951529 − 0.951529i
\(52\) 2.55871e8 − 5.49869e8i 0.672984 − 1.44625i
\(53\) 6.62087e8 1.58320 0.791600 − 0.611039i \(-0.209248\pi\)
0.791600 + 0.611039i \(0.209248\pi\)
\(54\) 6.78247e7 − 3.06519e8i 0.147713 − 0.667558i
\(55\) −1.06636e8 + 6.61317e8i −0.211881 + 1.31400i
\(56\) 9.79253e7 + 7.36961e8i 0.177809 + 1.33815i
\(57\) −3.21736e7 − 3.21736e7i −0.0534719 − 0.0534719i
\(58\) 5.31576e8 − 3.38951e8i 0.809889 − 0.516413i
\(59\) 5.28573e8 + 5.28573e8i 0.739341 + 0.739341i 0.972450 − 0.233110i \(-0.0748902\pi\)
−0.233110 + 0.972450i \(0.574890\pi\)
\(60\) 1.75477e8 + 9.12445e8i 0.225665 + 1.17341i
\(61\) −3.82556e8 − 3.82556e8i −0.452945 − 0.452945i 0.443386 − 0.896331i \(-0.353777\pi\)
−0.896331 + 0.443386i \(0.853777\pi\)
\(62\) −8.85792e8 − 1.96003e8i −0.966882 − 0.213946i
\(63\) 4.05279e8 + 4.05279e8i 0.408368 + 0.408368i
\(64\) 2.80401e8 + 1.03648e9i 0.261143 + 0.965300i
\(65\) −1.08372e9 − 1.50040e9i −0.934004 − 1.29313i
\(66\) 1.07082e9 + 1.67936e9i 0.855060 + 1.34099i
\(67\) 1.71648e9 1.27135 0.635674 − 0.771958i \(-0.280722\pi\)
0.635674 + 0.771958i \(0.280722\pi\)
\(68\) −1.53817e9 + 5.61245e8i −1.05793 + 0.386018i
\(69\) 2.28209e8 + 2.28209e8i 0.145911 + 0.145911i
\(70\) 2.10893e9 + 8.36562e8i 1.25479 + 0.497746i
\(71\) 3.60356e9i 1.99728i 0.0521099 + 0.998641i \(0.483405\pi\)
−0.0521099 + 0.998641i \(0.516595\pi\)
\(72\) 6.57342e8 + 5.03140e8i 0.339726 + 0.260032i
\(73\) 1.14659e9 − 1.14659e9i 0.553090 − 0.553090i −0.374241 − 0.927331i \(-0.622097\pi\)
0.927331 + 0.374241i \(0.122097\pi\)
\(74\) 4.67010e8 − 2.11055e9i 0.210459 − 0.951124i
\(75\) 2.69187e9 + 8.91294e8i 1.13435 + 0.375590i
\(76\) −1.50741e8 + 5.50022e7i −0.0594515 + 0.0216926i
\(77\) 4.86327e9 1.79670
\(78\) −5.37322e9 − 1.18895e9i −1.86106 − 0.411805i
\(79\) 3.22442e9i 1.04789i 0.851752 + 0.523945i \(0.175540\pi\)
−0.851752 + 0.523945i \(0.824460\pi\)
\(80\) 3.17852e9 + 7.96506e8i 0.970008 + 0.243074i
\(81\) −4.34031e9 −1.24479
\(82\) 5.28103e7 − 2.38665e8i 0.0142446 − 0.0643753i
\(83\) 5.37975e9i 1.36575i 0.730535 + 0.682876i \(0.239271\pi\)
−0.730535 + 0.682876i \(0.760729\pi\)
\(84\) 6.33720e9 − 2.31231e9i 1.51531 − 0.552904i
\(85\) −7.95456e8 + 4.93311e9i −0.179276 + 1.11180i
\(86\) −3.91045e9 − 8.65280e8i −0.831255 − 0.183935i
\(87\) −4.04507e9 − 4.04507e9i −0.811576 − 0.811576i
\(88\) 6.96278e9 − 9.25194e8i 1.31938 − 0.175315i
\(89\) −8.02081e8 −0.143638 −0.0718188 − 0.997418i \(-0.522880\pi\)
−0.0718188 + 0.997418i \(0.522880\pi\)
\(90\) 2.31911e9 − 1.00179e9i 0.392744 − 0.169654i
\(91\) −9.50171e9 + 9.50171e9i −1.52263 + 1.52263i
\(92\) 1.06921e9 − 3.90132e8i 0.162227 − 0.0591933i
\(93\) 8.23199e9i 1.18329i
\(94\) −8.30916e9 + 5.29821e9i −1.13219 + 0.721921i
\(95\) −7.79549e7 + 4.83446e8i −0.0100745 + 0.0624784i
\(96\) 8.63311e9 − 4.51615e9i 1.05879 − 0.553875i
\(97\) −3.06849e9 + 3.06849e9i −0.357327 + 0.357327i −0.862827 − 0.505500i \(-0.831308\pi\)
0.505500 + 0.862827i \(0.331308\pi\)
\(98\) 1.60579e9 − 7.25704e9i 0.177648 − 0.802840i
\(99\) 3.82906e9 − 3.82906e9i 0.402640 − 0.402640i
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.13.8 ✓ 236
5.2 odd 4 80.11.t.a.77.67 yes 236
16.5 even 4 80.11.t.a.53.67 yes 236
80.37 odd 4 inner 80.11.i.a.37.8 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.8 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.8 yes 236 80.37 odd 4 inner
80.11.t.a.53.67 yes 236 16.5 even 4
80.11.t.a.77.67 yes 236 5.2 odd 4