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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 13.71
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.71

$q$-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+(10.7603 + 30.1366i) q^{2} +107.688i q^{3} +(-792.430 + 648.560i) q^{4} +(-1274.22 + 2853.42i) q^{5} +(-3245.35 + 1158.76i) q^{6} +(-6451.25 - 6451.25i) q^{7} +(-28072.2 - 16902.4i) q^{8} +47452.3 q^{9} +(-99703.3 - 7697.10i) q^{10} +(132787. - 132787. i) q^{11} +(-69842.2 - 85335.3i) q^{12} -283018. i q^{13} +(125001. - 263836. i) q^{14} +(-307279. - 137219. i) q^{15} +(207316. - 1.02788e6i) q^{16} +(418468. - 418468. i) q^{17} +(510602. + 1.43005e6i) q^{18} +(1.90454e6 - 1.90454e6i) q^{19} +(-840877. - 3.08754e6i) q^{20} +(694722. - 694722. i) q^{21} +(5.43060e6 + 2.57293e6i) 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} +1.14689e7i q^{27} +(9.29618e6 + 928144. i) q^{28} +(-1.41002e7 + 1.41002e7i) q^{29} +(828885. - 1.07369e7i) q^{30} -1.08049e7 q^{31} +(3.32075e7 - 4.81250e6i) q^{32} +(1.42996e7 + 1.42996e7i) q^{33} +(1.71141e7 + 8.10835e6i) q^{34} +(2.66284e7 - 1.01878e7i) q^{35} +(-3.76026e7 + 3.07756e7i) q^{36} -1.42849e7i q^{37} +(7.78899e7 + 3.69029e7i) q^{38} +3.04776e7 q^{39} +(8.40000e7 - 5.85642e7i) q^{40} -1.65015e8i q^{41} +(2.84120e7 + 1.34611e7i) q^{42} +1.26990e8 q^{43} +(-1.91042e7 + 1.91345e8i) q^{44} +(-6.04648e7 + 1.35401e8i) q^{45} +(-2.49228e7 - 1.18080e7i) q^{46} +(6.43835e7 - 6.43835e7i) q^{47} +(1.10690e8 + 2.23255e7i) q^{48} -1.99238e8i q^{49} +(1.49007e8 - 2.74687e8i) q^{50} +(4.50640e7 + 4.50640e7i) q^{51} +(1.83554e8 + 2.24272e8i) q^{52} -3.53275e8 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} +(-5.76657e8 - 2.73210e8i) q^{58} +(-4.17022e8 - 4.17022e8i) q^{59} +(3.32492e8 - 9.05524e7i) q^{60} +(6.81378e8 + 6.81378e8i) q^{61} +(-1.16264e8 - 3.25623e8i) 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} +1.76079e9 q^{67} +(-6.02051e7 + 6.03008e8i) q^{68} +(-6.56257e7 - 6.56257e7i) q^{69} +(5.93555e8 + 6.92867e8i) q^{70} +1.47906e9i q^{71} +(-1.33209e9 - 8.02059e8i) q^{72} +(-1.93489e9 + 1.93489e9i) q^{73} +(4.30498e8 - 1.53710e8i) q^{74} +(7.83084e8 - 7.01946e8i) q^{75} +(-2.74007e8 + 2.74443e9i) q^{76} -1.71329e9 q^{77} +(3.27949e8 + 9.18492e8i) q^{78} -1.59844e9i q^{79} +(2.66879e9 + 1.90130e9i) q^{80} +1.56694e9 q^{81} +(4.97300e9 - 1.77562e9i) q^{82} -1.70733e9i q^{83} +(-9.99500e7 + 1.00109e9i) q^{84} +(6.60840e8 + 1.72728e9i) q^{85} +(1.36645e9 + 3.82704e9i) q^{86} +(-1.51843e9 - 1.51843e9i) q^{87} +(-5.97207e9 + 1.48321e9i) q^{88} +3.82303e9 q^{89} +(-4.73115e9 - 3.65245e8i) q^{90} +(-1.82582e9 + 1.82582e9i) q^{91} +(8.76755e7 - 8.78148e8i) q^{92} -1.16356e9i 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} +(6.00436e9 - 2.14387e9i) q^{98} +(6.30107e9 - 6.30107e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.7603 + 30.1366i 0.336260 + 0.941769i
\(3\) 107.688i 0.443161i 0.975142 + 0.221580i \(0.0711215\pi\)
−0.975142 + 0.221580i \(0.928878\pi\)
\(4\) −792.430 + 648.560i −0.773858 + 0.633359i
\(5\) −1274.22 + 2853.42i −0.407752 + 0.913093i
\(6\) −3245.35 + 1158.76i −0.417355 + 0.149017i
\(7\) −6451.25 − 6451.25i −0.383843 − 0.383843i 0.488642 − 0.872485i \(-0.337493\pi\)
−0.872485 + 0.488642i \(0.837493\pi\)
\(8\) −28072.2 − 16902.4i −0.856696 − 0.515822i
\(9\) 47452.3 0.803609
\(10\) −99703.3 − 7697.10i −0.997033 − 0.0769710i
\(11\) 132787. − 132787.i 0.824506 − 0.824506i −0.162245 − 0.986751i \(-0.551873\pi\)
0.986751 + 0.162245i \(0.0518734\pi\)
\(12\) −69842.2 − 85335.3i −0.280680 − 0.342943i
\(13\) − 283018.i − 0.762249i −0.924524 − 0.381124i \(-0.875537\pi\)
0.924524 − 0.381124i \(-0.124463\pi\)
\(14\) 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 \(0.183174\pi\)
\(18\) 510602. + 1.43005e6i 0.270222 + 0.756814i
\(19\) 1.90454e6 − 1.90454e6i 0.769170 − 0.769170i −0.208790 − 0.977960i \(-0.566953\pi\)
0.977960 + 0.208790i \(0.0669527\pi\)
\(20\) −840877. − 3.08754e6i −0.262774 − 0.964857i
\(21\) 694722. − 694722.i 0.170104 − 0.170104i
\(22\) 5.43060e6 + 2.57293e6i 1.05374 + 0.499245i
\(23\) −609406. + 609406.i −0.0946820 + 0.0946820i −0.752861 − 0.658179i \(-0.771327\pi\)
0.658179 + 0.752861i \(0.271327\pi\)
\(24\) 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
\(27\) 1.14689e7i 0.799289i
\(28\) 9.29618e6 + 928144.i 0.540150 + 0.0539293i
\(29\) −1.41002e7 + 1.41002e7i −0.687443 + 0.687443i −0.961666 − 0.274223i \(-0.911579\pi\)
0.274223 + 0.961666i \(0.411579\pi\)
\(30\) 828885. − 1.07369e7i 0.0341105 − 0.441846i
\(31\) −1.08049e7 −0.377409 −0.188704 − 0.982034i \(-0.560429\pi\)
−0.188704 + 0.982034i \(0.560429\pi\)
\(32\) 3.32075e7 − 4.81250e6i 0.989661 − 0.143424i
\(33\) 1.42996e7 + 1.42996e7i 0.365389 + 0.365389i
\(34\) 1.71141e7 + 8.10835e6i 0.376668 + 0.178459i
\(35\) 2.66284e7 − 1.01878e7i 0.506997 − 0.193972i
\(36\) −3.76026e7 + 3.07756e7i −0.621879 + 0.508973i
\(37\) − 1.42849e7i − 0.206000i −0.994681 − 0.103000i \(-0.967156\pi\)
0.994681 − 0.103000i \(-0.0328442\pi\)
\(38\) 7.78899e7 + 3.69029e7i 0.983022 + 0.465739i
\(39\) 3.04776e7 0.337799
\(40\) 8.40000e7 − 5.85642e7i 0.820312 − 0.571916i
\(41\) − 1.65015e8i − 1.42431i −0.702022 − 0.712155i \(-0.747719\pi\)
0.702022 − 0.712155i \(-0.252281\pi\)
\(42\) 2.84120e7 + 1.34611e7i 0.217398 + 0.102999i
\(43\) 1.26990e8 0.863827 0.431913 − 0.901915i \(-0.357839\pi\)
0.431913 + 0.901915i \(0.357839\pi\)
\(44\) −1.91042e7 + 1.91345e8i −0.115842 + 1.16026i
\(45\) −6.04648e7 + 1.35401e8i −0.327673 + 0.733769i
\(46\) −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 \(0.186392\pi\)
\(48\) 1.10690e8 + 2.23255e7i 0.434413 + 0.0876181i
\(49\) − 1.99238e8i − 0.705329i
\(50\) 1.49007e8 − 2.74687e8i 0.476824 − 0.878999i
\(51\) 4.50640e7 + 4.50640e7i 0.130611 + 0.130611i
\(52\) 1.83554e8 + 2.24272e8i 0.482778 + 0.589872i
\(53\) −3.53275e8 −0.844761 −0.422380 − 0.906419i \(-0.638805\pi\)
−0.422380 + 0.906419i \(0.638805\pi\)
\(54\) −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
\(58\) −5.76657e8 − 2.73210e8i −0.878572 − 0.416252i
\(59\) −4.17022e8 − 4.17022e8i −0.583309 − 0.583309i 0.352502 − 0.935811i \(-0.385331\pi\)
−0.935811 + 0.352502i \(0.885331\pi\)
\(60\) 3.32492e8 − 9.05524e7i 0.427587 − 0.116451i
\(61\) 6.81378e8 + 6.81378e8i 0.806750 + 0.806750i 0.984140 − 0.177391i \(-0.0567657\pi\)
−0.177391 + 0.984140i \(0.556766\pi\)
\(62\) −1.16264e8 − 3.25623e8i −0.126908 − 0.355432i
\(63\) −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
\(67\) 1.76079e9 1.30417 0.652083 − 0.758148i \(-0.273895\pi\)
0.652083 + 0.758148i \(0.273895\pi\)
\(68\) −6.02051e7 + 6.03008e8i −0.0414085 + 0.414742i
\(69\) −6.56257e7 − 6.56257e7i −0.0419593 − 0.0419593i
\(70\) 5.93555e8 + 6.92867e8i 0.353159 + 0.412249i
\(71\) 1.47906e9i 0.819771i 0.912137 + 0.409886i \(0.134431\pi\)
−0.912137 + 0.409886i \(0.865569\pi\)
\(72\) −1.33209e9 − 8.02059e8i −0.688448 − 0.414519i
\(73\) −1.93489e9 + 1.93489e9i −0.933344 + 0.933344i −0.997913 − 0.0645692i \(-0.979433\pi\)
0.0645692 + 0.997913i \(0.479433\pi\)
\(74\) 4.30498e8 − 1.53710e8i 0.194005 − 0.0692698i
\(75\) 7.83084e8 − 7.01946e8i 0.329991 − 0.295800i
\(76\) −2.74007e8 + 2.74443e9i −0.108067 + 1.08239i
\(77\) −1.71329e9 −0.632961
\(78\) 3.27949e8 + 9.18492e8i 0.113588 + 0.318128i
\(79\) − 1.59844e9i − 0.519470i −0.965680 − 0.259735i \(-0.916365\pi\)
0.965680 − 0.259735i \(-0.0836353\pi\)
\(80\) 2.66879e9 + 1.90130e9i 0.814451 + 0.580232i
\(81\) 1.56694e9 0.449395
\(82\) 4.97300e9 − 1.77562e9i 1.34137 − 0.478939i
\(83\) − 1.70733e9i − 0.433439i −0.976234 − 0.216719i \(-0.930464\pi\)
0.976234 − 0.216719i \(-0.0695357\pi\)
\(84\) −9.99500e7 + 1.00109e9i −0.0238994 + 0.239373i
\(85\) 6.60840e8 + 1.72728e9i 0.148937 + 0.389286i
\(86\) 1.36645e9 + 3.82704e9i 0.290471 + 0.813525i
\(87\) −1.51843e9 − 1.51843e9i −0.304648 − 0.304648i
\(88\) −5.97207e9 + 1.48321e9i −1.13165 + 0.281053i
\(89\) 3.82303e9 0.684633 0.342316 − 0.939585i \(-0.388789\pi\)
0.342316 + 0.939585i \(0.388789\pi\)
\(90\) −4.73115e9 − 3.65245e8i −0.801224 − 0.0618545i
\(91\) −1.82582e9 + 1.82582e9i −0.292584 + 0.292584i
\(92\) 8.76755e7 − 8.78148e8i 0.0133027 − 0.133238i
\(93\) − 1.16356e9i − 0.167253i
\(94\) 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 \(0.120513\pi\)
\(98\) 6.00436e9 − 2.14387e9i 0.664257 − 0.237174i
\(99\) 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.i.a.13.71 ✓ 236
5.2 odd 4 80.11.t.a.77.13 yes 236
16.5 even 4 80.11.t.a.53.13 yes 236
80.37 odd 4 inner 80.11.i.a.37.71 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.71 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.71 yes 236 80.37 odd 4 inner
80.11.t.a.53.13 yes 236 16.5 even 4
80.11.t.a.77.13 yes 236 5.2 odd 4