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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 53.67
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.67

$q$-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+(6.91353 + 31.2442i) q^{2} +290.364 q^{3} +(-928.406 + 432.016i) q^{4} +(1829.76 + 2533.30i) q^{5} +(2007.44 + 9072.21i) q^{6} +(16042.8 + 16042.8i) q^{7} +(-19916.6 - 26020.6i) q^{8} +25262.4 q^{9} +(-66500.9 + 74683.6i) q^{10} +(-151572. - 151572. i) q^{11} +(-269576. + 125442. i) q^{12} -592272. q^{13} +(-390333. + 612158. i) q^{14} +(531297. + 735579. i) q^{15} +(675300. - 802173. i) q^{16} +(-1.13065e6 + 1.13065e6i) q^{17} +(174652. + 789304. i) q^{18} +(110804. + 110804. i) q^{19} +(-2.79319e6 - 1.56144e6i) q^{20} +(4.65826e6 + 4.65826e6i) q^{21} +(3.68785e6 - 5.78364e6i) 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} -9.81043e6 q^{27} +(-2.18250e7 - 7.96349e6i) q^{28} +(1.39310e7 + 1.39310e7i) q^{29} +(-1.93095e7 + 2.16854e7i) q^{30} +2.83506e7 q^{31} +(2.97320e7 + 1.55534e7i) q^{32} +(-4.40110e7 - 4.40110e7i) q^{33} +(-4.31432e7 - 2.75096e7i) q^{34} +(-1.12867e7 + 6.99957e7i) q^{35} +(-2.34537e7 + 1.09138e7i) q^{36} -6.75501e7 q^{37} +(-2.69595e6 + 4.22805e6i) q^{38} -1.71975e8 q^{39} +(2.94753e7 - 9.80661e7i) q^{40} -7.63868e6i q^{41} +(-1.13339e8 + 1.77749e8i) q^{42} +1.25157e8i q^{43} +(2.06202e8 + 7.52387e7i) q^{44} +(4.62241e7 + 6.39971e7i) q^{45} +(-2.99897e7 - 1.91225e7i) q^{46} +(2.17758e8 - 2.17758e8i) q^{47} +(1.96083e8 - 2.32922e8i) q^{48} +2.32268e8i q^{49} +(-3.10876e8 - 3.18134e7i) q^{50} +(-3.28301e8 + 3.28301e8i) q^{51} +(5.49869e8 - 2.55871e8i) q^{52} +6.62087e8i 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} +(-3.38951e8 + 5.31576e8i) q^{58} +(-5.28573e8 + 5.28573e8i) q^{59} +(-8.11042e8 - 4.53387e8i) q^{60} +(-3.82556e8 + 3.82556e8i) q^{61} +(1.96003e8 + 8.85792e8i) 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} -1.71648e9i q^{67} +(5.61245e8 - 1.53817e9i) q^{68} +(-2.28209e8 + 2.28209e8i) q^{69} +(-2.26499e9 + 1.31273e8i) q^{70} -3.60356e9i q^{71} +(-5.03140e8 - 6.57342e8i) q^{72} +(-1.14659e9 + 1.14659e9i) q^{73} +(-4.67010e8 - 2.11055e9i) q^{74} +(-8.91294e8 + 2.69187e9i) q^{75} +(-1.50741e8 - 5.50022e7i) q^{76} -4.86327e9i q^{77} +(-1.18895e9 - 5.37322e9i) q^{78} +3.22442e9i q^{79} +(3.26778e9 + 2.42950e8i) q^{80} -4.34031e9 q^{81} +(2.38665e8 - 5.28103e7i) q^{82} +5.37975e9 q^{83} +(-6.33720e9 - 2.31231e9i) q^{84} +(-4.93311e9 - 7.95456e8i) q^{85} +(-3.91045e9 + 8.65280e8i) q^{86} +(4.04507e9 + 4.04507e9i) q^{87} +(-9.25194e8 + 6.96278e9i) q^{88} +8.02081e8 q^{89} +(-1.67997e9 + 1.88668e9i) q^{90} +(-9.50171e9 - 9.50171e9i) q^{91} +(3.90132e8 - 1.06921e9i) q^{92} +8.23199e9 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} +(-7.25704e9 + 1.60579e9i) q^{98} +(-3.82906e9 - 3.82906e9i) 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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.91353 + 31.2442i 0.216048 + 0.976383i
\(3\) 290.364 1.19491 0.597457 − 0.801901i \(-0.296178\pi\)
0.597457 + 0.801901i \(0.296178\pi\)
\(4\) −928.406 + 432.016i −0.906647 + 0.421891i
\(5\) 1829.76 + 2533.30i 0.585524 + 0.810655i
\(6\) 2007.44 + 9072.21i 0.258159 + 1.16669i
\(7\) 16042.8 + 16042.8i 0.954531 + 0.954531i 0.999010 − 0.0444790i \(-0.0141628\pi\)
−0.0444790 + 0.999010i \(0.514163\pi\)
\(8\) −19916.6 − 26020.6i −0.607806 − 0.794085i
\(9\) 25262.4 0.427821
\(10\) −66500.9 + 74683.6i −0.665009 + 0.746836i
\(11\) −151572. − 151572.i −0.941141 − 0.941141i 0.0572202 − 0.998362i \(-0.481776\pi\)
−0.998362 + 0.0572202i \(0.981776\pi\)
\(12\) −269576. + 125442.i −1.08337 + 0.504124i
\(13\) −592272. −1.59516 −0.797580 − 0.603213i \(-0.793887\pi\)
−0.797580 + 0.603213i \(0.793887\pi\)
\(14\) −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 \(0.440384\pi\)
\(18\) 174652. + 789304.i 0.0924297 + 0.417717i
\(19\) 110804. + 110804.i 0.0447496 + 0.0447496i 0.729127 − 0.684378i \(-0.239926\pi\)
−0.684378 + 0.729127i \(0.739926\pi\)
\(20\) −2.79319e6 − 1.56144e6i −0.872871 − 0.487951i
\(21\) 4.65826e6 + 4.65826e6i 1.14058 + 1.14058i
\(22\) 3.68785e6 − 5.78364e6i 0.715583 − 1.12225i
\(23\) −785939. + 785939.i −0.122110 + 0.122110i −0.765521 − 0.643411i \(-0.777519\pi\)
0.643411 + 0.765521i \(0.277519\pi\)
\(24\) −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
\(27\) −9.81043e6 −0.683705
\(28\) −2.18250e7 − 7.96349e6i −1.26813 − 0.462714i
\(29\) 1.39310e7 + 1.39310e7i 0.679192 + 0.679192i 0.959817 − 0.280625i \(-0.0905419\pi\)
−0.280625 + 0.959817i \(0.590542\pi\)
\(30\) −1.93095e7 + 2.16854e7i −0.794629 + 0.892405i
\(31\) 2.83506e7 0.990269 0.495134 − 0.868816i \(-0.335119\pi\)
0.495134 + 0.868816i \(0.335119\pi\)
\(32\) 2.97320e7 + 1.55534e7i 0.886083 + 0.463527i
\(33\) −4.40110e7 − 4.40110e7i −1.12458 − 1.12458i
\(34\) −4.31432e7 − 2.75096e7i −0.949551 − 0.605466i
\(35\) −1.12867e7 + 6.99957e7i −0.214895 + 1.33270i
\(36\) −2.34537e7 + 1.09138e7i −0.387882 + 0.180494i
\(37\) −6.75501e7 −0.974131 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(38\) −2.69595e6 + 4.22805e6i −0.0340247 + 0.0533608i
\(39\) −1.71975e8 −1.90608
\(40\) 2.94753e7 − 9.80661e7i 0.287845 − 0.957677i
\(41\) − 7.63868e6i − 0.0659324i −0.999456 − 0.0329662i \(-0.989505\pi\)
0.999456 − 0.0329662i \(-0.0104954\pi\)
\(42\) −1.13339e8 + 1.77749e8i −0.867225 + 1.36007i
\(43\) 1.25157e8i 0.851362i 0.904873 + 0.425681i \(0.139965\pi\)
−0.904873 + 0.425681i \(0.860035\pi\)
\(44\) 2.06202e8 + 7.52387e7i 1.25034 + 0.456224i
\(45\) 4.62241e7 + 6.39971e7i 0.250499 + 0.346815i
\(46\) −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 \(0.0157010\pi\)
\(48\) 1.96083e8 − 2.32922e8i 0.769544 − 0.914124i
\(49\) 2.32268e8i 0.822260i
\(50\) −3.10876e8 − 3.18134e7i −0.994805 − 0.101803i
\(51\) −3.28301e8 + 3.28301e8i −0.951529 + 0.951529i
\(52\) 5.49869e8 − 2.55871e8i 1.44625 − 0.672984i
\(53\) 6.62087e8i 1.58320i 0.611039 + 0.791600i \(0.290752\pi\)
−0.611039 + 0.791600i \(0.709248\pi\)
\(54\) −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
\(58\) −3.38951e8 + 5.31576e8i −0.516413 + 0.809889i
\(59\) −5.28573e8 + 5.28573e8i −0.739341 + 0.739341i −0.972450 − 0.233110i \(-0.925110\pi\)
0.233110 + 0.972450i \(0.425110\pi\)
\(60\) −8.11042e8 − 4.53387e8i −1.04301 − 0.583059i
\(61\) −3.82556e8 + 3.82556e8i −0.452945 + 0.452945i −0.896331 − 0.443386i \(-0.853777\pi\)
0.443386 + 0.896331i \(0.353777\pi\)
\(62\) 1.96003e8 + 8.85792e8i 0.213946 + 0.966882i
\(63\) 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
\(67\) − 1.71648e9i − 1.27135i −0.771958 − 0.635674i \(-0.780722\pi\)
0.771958 − 0.635674i \(-0.219278\pi\)
\(68\) 5.61245e8 − 1.53817e9i 0.386018 − 1.05793i
\(69\) −2.28209e8 + 2.28209e8i −0.145911 + 0.145911i
\(70\) −2.26499e9 + 1.31273e8i −1.34765 + 0.0781063i
\(71\) − 3.60356e9i − 1.99728i −0.0521099 − 0.998641i \(-0.516595\pi\)
0.0521099 − 0.998641i \(-0.483405\pi\)
\(72\) −5.03140e8 − 6.57342e8i −0.260032 − 0.339726i
\(73\) −1.14659e9 + 1.14659e9i −0.553090 + 0.553090i −0.927331 − 0.374241i \(-0.877903\pi\)
0.374241 + 0.927331i \(0.377903\pi\)
\(74\) −4.67010e8 − 2.11055e9i −0.210459 − 0.951124i
\(75\) −8.91294e8 + 2.69187e9i −0.375590 + 1.13435i
\(76\) −1.50741e8 − 5.50022e7i −0.0594515 − 0.0216926i
\(77\) − 4.86327e9i − 1.79670i
\(78\) −1.18895e9 − 5.37322e9i −0.411805 − 1.86106i
\(79\) 3.22442e9i 1.04789i 0.851752 + 0.523945i \(0.175540\pi\)
−0.851752 + 0.523945i \(0.824460\pi\)
\(80\) 3.26778e9 + 2.42950e8i 0.997248 + 0.0741425i
\(81\) −4.34031e9 −1.24479
\(82\) 2.38665e8 − 5.28103e7i 0.0643753 − 0.0142446i
\(83\) 5.37975e9 1.36575 0.682876 − 0.730535i \(-0.260729\pi\)
0.682876 + 0.730535i \(0.260729\pi\)
\(84\) −6.33720e9 − 2.31231e9i −1.51531 − 0.552904i
\(85\) −4.93311e9 − 7.95456e8i −1.11180 − 0.179276i
\(86\) −3.91045e9 + 8.65280e8i −0.831255 + 0.183935i
\(87\) 4.04507e9 + 4.04507e9i 0.811576 + 0.811576i
\(88\) −9.25194e8 + 6.96278e9i −0.175315 + 1.31938i
\(89\) 8.02081e8 0.143638 0.0718188 − 0.997418i \(-0.477120\pi\)
0.0718188 + 0.997418i \(0.477120\pi\)
\(90\) −1.67997e9 + 1.88668e9i −0.284504 + 0.319512i
\(91\) −9.50171e9 − 9.50171e9i −1.52263 − 1.52263i
\(92\) 3.90132e8 − 1.06921e9i 0.0591933 − 0.162227i
\(93\) 8.23199e9 1.18329
\(94\) 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 \(0.331308\pi\)
\(98\) −7.25704e9 + 1.60579e9i −0.802840 + 0.177648i
\(99\) −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.t.a.53.67 yes 236
5.2 odd 4 80.11.i.a.37.8 yes 236
16.13 even 4 80.11.i.a.13.8 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.67 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.8 ✓ 236 16.13 even 4
80.11.i.a.37.8 yes 236 5.2 odd 4
80.11.t.a.53.67 yes 236 1.1 even 1 trivial
80.11.t.a.77.67 yes 236 80.77 odd 4 inner