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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 53.13
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.13

$q$-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+(-30.1366 - 10.7603i) q^{2} +107.688 q^{3} +(792.430 + 648.560i) q^{4} +(-2853.42 - 1274.22i) q^{5} +(-3245.35 - 1158.76i) q^{6} +(6451.25 + 6451.25i) q^{7} +(-16902.4 - 28072.2i) q^{8} -47452.3 q^{9} +(72281.2 + 69104.5i) q^{10} +(132787. + 132787. i) q^{11} +(85335.3 + 69842.2i) q^{12} -283018. q^{13} +(-125001. - 263836. i) q^{14} +(-307279. - 137219. i) q^{15} +(207316. + 1.02788e6i) q^{16} +(418468. - 418468. i) q^{17} +(1.43005e6 + 510602. i) q^{18} +(-1.90454e6 - 1.90454e6i) q^{19} +(-1.43472e6 - 2.86034e6i) q^{20} +(694722. + 694722. i) q^{21} +(-2.57293e6 - 5.43060e6i) q^{22} +(609406. - 609406. i) q^{23} +(-1.82019e6 - 3.02304e6i) q^{24} +(6.51833e6 + 7.27178e6i) q^{25} +(8.52919e6 + 3.04537e6i) q^{26} -1.14689e7 q^{27} +(928144. + 9.29618e6i) q^{28} +(1.41002e7 + 1.41002e7i) q^{29} +(7.78382e6 + 7.44173e6i) q^{30} -1.08049e7 q^{31} +(4.81250e6 - 3.32075e7i) q^{32} +(1.42996e7 + 1.42996e7i) q^{33} +(-1.71141e7 + 8.10835e6i) q^{34} +(-1.01878e7 - 2.66284e7i) q^{35} +(-3.76026e7 - 3.07756e7i) q^{36} +1.42849e7 q^{37} +(3.69029e7 + 7.78899e7i) q^{38} -3.04776e7 q^{39} +(1.24594e7 + 1.01639e8i) q^{40} +1.65015e8i q^{41} +(-1.34611e7 - 2.84120e7i) q^{42} +1.26990e8i q^{43} +(1.91042e7 + 1.91345e8i) q^{44} +(1.35401e8 + 6.04648e7i) q^{45} +(-2.49228e7 + 1.18080e7i) q^{46} +(6.43835e7 - 6.43835e7i) q^{47} +(2.23255e7 + 1.10690e8i) q^{48} -1.99238e8i q^{49} +(-1.18194e8 - 2.89286e8i) q^{50} +(4.50640e7 - 4.50640e7i) q^{51} +(-2.24272e8 - 1.83554e8i) q^{52} -3.53275e8i q^{53} +(3.45634e8 + 1.23409e8i) q^{54} +(-2.09697e8 - 5.48099e8i) q^{55} +(7.20589e7 - 2.90143e8i) q^{56} +(-2.05096e8 - 2.05096e8i) q^{57} +(-2.73210e8 - 5.76657e8i) q^{58} +(4.17022e8 - 4.17022e8i) q^{59} +(-1.54502e8 - 3.08025e8i) q^{60} +(6.81378e8 - 6.81378e8i) q^{61} +(3.25623e8 + 1.16264e8i) q^{62} +(-3.06126e8 - 3.06126e8i) q^{63} +(-5.02357e8 + 9.48978e8i) q^{64} +(8.07567e8 + 3.60628e8i) q^{65} +(-2.77074e8 - 5.84811e8i) q^{66} -1.76079e9i q^{67} +(6.03008e8 - 6.02051e7i) q^{68} +(6.56257e7 - 6.56257e7i) q^{69} +(2.04936e7 + 9.12114e8i) q^{70} -1.47906e9i q^{71} +(8.02059e8 + 1.33209e9i) q^{72} +(1.93489e9 - 1.93489e9i) q^{73} +(-4.30498e8 - 1.53710e8i) q^{74} +(7.01946e8 + 7.83084e8i) q^{75} +(-2.74007e8 - 2.74443e9i) q^{76} +1.71329e9i q^{77} +(9.18492e8 + 3.27949e8i) q^{78} -1.59844e9i q^{79} +(7.18188e8 - 3.19713e9i) q^{80} +1.56694e9 q^{81} +(1.77562e9 - 4.97300e9i) q^{82} -1.70733e9 q^{83} +(9.99500e7 + 1.00109e9i) q^{84} +(-1.72728e9 + 6.60840e8i) q^{85} +(1.36645e9 - 3.82704e9i) q^{86} +(1.51843e9 + 1.51843e9i) q^{87} +(1.48321e9 - 5.97207e9i) q^{88} -3.82303e9 q^{89} +(-3.42991e9 - 3.27917e9i) q^{90} +(-1.82582e9 - 1.82582e9i) q^{91} +(8.78148e8 - 8.76755e7i) q^{92} -1.16356e9 q^{93} +(-2.63309e9 + 1.24751e9i) q^{94} +(3.00763e9 + 7.86126e9i) q^{95} +(5.18249e8 - 3.57605e9i) q^{96} +(4.80512e9 - 4.80512e9i) q^{97} +(-2.14387e9 + 6.00436e9i) q^{98} +(-6.30107e9 - 6.30107e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.1366 − 10.7603i −0.941769 − 0.336260i
\(3\) 107.688 0.443161 0.221580 − 0.975142i \(-0.428878\pi\)
0.221580 + 0.975142i \(0.428878\pi\)
\(4\) 792.430 + 648.560i 0.773858 + 0.633359i
\(5\) −2853.42 − 1274.22i −0.913093 − 0.407752i
\(6\) −3245.35 − 1158.76i −0.417355 − 0.149017i
\(7\) 6451.25 + 6451.25i 0.383843 + 0.383843i 0.872485 − 0.488642i \(-0.162507\pi\)
−0.488642 + 0.872485i \(0.662507\pi\)
\(8\) −16902.4 − 28072.2i −0.515822 − 0.856696i
\(9\) −47452.3 −0.803609
\(10\) 72281.2 + 69104.5i 0.722812 + 0.691045i
\(11\) 132787. + 132787.i 0.824506 + 0.824506i 0.986751 − 0.162245i \(-0.0518734\pi\)
−0.162245 + 0.986751i \(0.551873\pi\)
\(12\) 85335.3 + 69842.2i 0.342943 + 0.280680i
\(13\) −283018. −0.762249 −0.381124 − 0.924524i \(-0.624463\pi\)
−0.381124 + 0.924524i \(0.624463\pi\)
\(14\) −125001. − 263836.i −0.232420 − 0.490562i
\(15\) −307279. − 137219.i −0.404647 − 0.180700i
\(16\) 207316. + 1.02788e6i 0.197712 + 0.980260i
\(17\) 418468. − 418468.i 0.294725 − 0.294725i −0.544218 − 0.838944i \(-0.683174\pi\)
0.838944 + 0.544218i \(0.183174\pi\)
\(18\) 1.43005e6 + 510602.i 0.756814 + 0.270222i
\(19\) −1.90454e6 − 1.90454e6i −0.769170 − 0.769170i 0.208790 − 0.977960i \(-0.433047\pi\)
−0.977960 + 0.208790i \(0.933047\pi\)
\(20\) −1.43472e6 − 2.86034e6i −0.448351 − 0.893858i
\(21\) 694722. + 694722.i 0.170104 + 0.170104i
\(22\) −2.57293e6 − 5.43060e6i −0.499245 − 1.05374i
\(23\) 609406. − 609406.i 0.0946820 − 0.0946820i −0.658179 − 0.752861i \(-0.728673\pi\)
0.752861 + 0.658179i \(0.228673\pi\)
\(24\) −1.82019e6 − 3.02304e6i −0.228592 − 0.379654i
\(25\) 6.51833e6 + 7.27178e6i 0.667477 + 0.744630i
\(26\) 8.52919e6 + 3.04537e6i 0.717862 + 0.256314i
\(27\) −1.14689e7 −0.799289
\(28\) 928144. + 9.29618e6i 0.0539293 + 0.540150i
\(29\) 1.41002e7 + 1.41002e7i 0.687443 + 0.687443i 0.961666 − 0.274223i \(-0.0884208\pi\)
−0.274223 + 0.961666i \(0.588421\pi\)
\(30\) 7.78382e6 + 7.44173e6i 0.320322 + 0.306244i
\(31\) −1.08049e7 −0.377409 −0.188704 − 0.982034i \(-0.560429\pi\)
−0.188704 + 0.982034i \(0.560429\pi\)
\(32\) 4.81250e6 − 3.32075e7i 0.143424 − 0.989661i
\(33\) 1.42996e7 + 1.42996e7i 0.365389 + 0.365389i
\(34\) −1.71141e7 + 8.10835e6i −0.376668 + 0.178459i
\(35\) −1.01878e7 − 2.66284e7i −0.193972 − 0.506997i
\(36\) −3.76026e7 − 3.07756e7i −0.621879 − 0.508973i
\(37\) 1.42849e7 0.206000 0.103000 − 0.994681i \(-0.467156\pi\)
0.103000 + 0.994681i \(0.467156\pi\)
\(38\) 3.69029e7 + 7.78899e7i 0.465739 + 0.983022i
\(39\) −3.04776e7 −0.337799
\(40\) 1.24594e7 + 1.01639e8i 0.121674 + 0.992570i
\(41\) 1.65015e8i 1.42431i 0.702022 + 0.712155i \(0.252281\pi\)
−0.702022 + 0.712155i \(0.747719\pi\)
\(42\) −1.34611e7 − 2.84120e7i −0.102999 − 0.217398i
\(43\) 1.26990e8i 0.863827i 0.901915 + 0.431913i \(0.142161\pi\)
−0.901915 + 0.431913i \(0.857839\pi\)
\(44\) 1.91042e7 + 1.91345e8i 0.115842 + 1.16026i
\(45\) 1.35401e8 + 6.04648e7i 0.733769 + 0.327673i
\(46\) −2.49228e7 + 1.18080e7i −0.121006 + 0.0573308i
\(47\) 6.43835e7 − 6.43835e7i 0.280728 − 0.280728i −0.552672 − 0.833399i \(-0.686392\pi\)
0.833399 + 0.552672i \(0.186392\pi\)
\(48\) 2.23255e7 + 1.10690e8i 0.0876181 + 0.434413i
\(49\) − 1.99238e8i − 0.705329i
\(50\) −1.18194e8 − 2.89286e8i −0.378219 − 0.925716i
\(51\) 4.50640e7 − 4.50640e7i 0.130611 − 0.130611i
\(52\) −2.24272e8 − 1.83554e8i −0.589872 − 0.482778i
\(53\) − 3.53275e8i − 0.844761i −0.906419 − 0.422380i \(-0.861195\pi\)
0.906419 − 0.422380i \(-0.138805\pi\)
\(54\) 3.45634e8 + 1.23409e8i 0.752745 + 0.268769i
\(55\) −2.09697e8 − 5.48099e8i −0.416657 − 1.08904i
\(56\) 7.20589e7 − 2.90143e8i 0.130842 − 0.526831i
\(57\) −2.05096e8 − 2.05096e8i −0.340866 − 0.340866i
\(58\) −2.73210e8 − 5.76657e8i −0.416252 − 0.878572i
\(59\) 4.17022e8 − 4.17022e8i 0.583309 − 0.583309i −0.352502 − 0.935811i \(-0.614669\pi\)
0.935811 + 0.352502i \(0.114669\pi\)
\(60\) −1.54502e8 − 3.08025e8i −0.198691 − 0.396123i
\(61\) 6.81378e8 − 6.81378e8i 0.806750 − 0.806750i −0.177391 − 0.984140i \(-0.556766\pi\)
0.984140 + 0.177391i \(0.0567657\pi\)
\(62\) 3.25623e8 + 1.16264e8i 0.355432 + 0.126908i
\(63\) −3.06126e8 − 3.06126e8i −0.308459 − 0.308459i
\(64\) −5.02357e8 + 9.48978e8i −0.467856 + 0.883805i
\(65\) 8.07567e8 + 3.60628e8i 0.696004 + 0.310808i
\(66\) −2.77074e8 − 5.84811e8i −0.221246 − 0.466977i
\(67\) − 1.76079e9i − 1.30417i −0.758148 − 0.652083i \(-0.773895\pi\)
0.758148 − 0.652083i \(-0.226105\pi\)
\(68\) 6.03008e8 − 6.02051e7i 0.414742 − 0.0414085i
\(69\) 6.56257e7 − 6.56257e7i 0.0419593 − 0.0419593i
\(70\) 2.04936e7 + 9.12114e8i 0.0121935 + 0.542699i
\(71\) − 1.47906e9i − 0.819771i −0.912137 − 0.409886i \(-0.865569\pi\)
0.912137 − 0.409886i \(-0.134431\pi\)
\(72\) 8.02059e8 + 1.33209e9i 0.414519 + 0.688448i
\(73\) 1.93489e9 − 1.93489e9i 0.933344 − 0.933344i −0.0645692 − 0.997913i \(-0.520567\pi\)
0.997913 + 0.0645692i \(0.0205673\pi\)
\(74\) −4.30498e8 − 1.53710e8i −0.194005 − 0.0692698i
\(75\) 7.01946e8 + 7.83084e8i 0.295800 + 0.329991i
\(76\) −2.74007e8 − 2.74443e9i −0.108067 − 1.08239i
\(77\) 1.71329e9i 0.632961i
\(78\) 9.18492e8 + 3.27949e8i 0.318128 + 0.113588i
\(79\) − 1.59844e9i − 0.519470i −0.965680 − 0.259735i \(-0.916365\pi\)
0.965680 − 0.259735i \(-0.0836353\pi\)
\(80\) 7.18188e8 − 3.19713e9i 0.219174 − 0.975686i
\(81\) 1.56694e9 0.449395
\(82\) 1.77562e9 − 4.97300e9i 0.478939 − 1.34137i
\(83\) −1.70733e9 −0.433439 −0.216719 − 0.976234i \(-0.569536\pi\)
−0.216719 + 0.976234i \(0.569536\pi\)
\(84\) 9.99500e7 + 1.00109e9i 0.0238994 + 0.239373i
\(85\) −1.72728e9 + 6.60840e8i −0.389286 + 0.148937i
\(86\) 1.36645e9 − 3.82704e9i 0.290471 − 0.813525i
\(87\) 1.51843e9 + 1.51843e9i 0.304648 + 0.304648i
\(88\) 1.48321e9 − 5.97207e9i 0.281053 − 1.13165i
\(89\) −3.82303e9 −0.684633 −0.342316 − 0.939585i \(-0.611211\pi\)
−0.342316 + 0.939585i \(0.611211\pi\)
\(90\) −3.42991e9 − 3.27917e9i −0.580858 − 0.555330i
\(91\) −1.82582e9 − 1.82582e9i −0.292584 − 0.292584i
\(92\) 8.78148e8 − 8.76755e7i 0.133238 − 0.0133027i
\(93\) −1.16356e9 −0.167253
\(94\) −2.63309e9 + 1.24751e9i −0.358778 + 0.169983i
\(95\) 3.00763e9 + 7.86126e9i 0.388693 + 1.01595i
\(96\) 5.18249e8 − 3.57605e9i 0.0635598 − 0.438579i
\(97\) 4.80512e9 − 4.80512e9i 0.559559 − 0.559559i −0.369623 − 0.929182i \(-0.620513\pi\)
0.929182 + 0.369623i \(0.120513\pi\)
\(98\) −2.14387e9 + 6.00436e9i −0.237174 + 0.664257i
\(99\) −6.30107e9 − 6.30107e9i −0.662580 − 0.662580i
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.53.13 yes 236
5.2 odd 4 80.11.i.a.37.71 yes 236
16.13 even 4 80.11.i.a.13.71 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.13 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.71 ✓ 236 16.13 even 4
80.11.i.a.37.71 yes 236 5.2 odd 4
80.11.t.a.53.13 yes 236 1.1 even 1 trivial
80.11.t.a.77.13 yes 236 80.77 odd 4 inner