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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 13.6
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.6

$q$-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+(-31.2876 + 6.71444i) q^{2} +412.535i q^{3} +(933.833 - 420.158i) q^{4} +(-1073.57 + 2934.80i) q^{5} +(-2769.94 - 12907.2i) q^{6} +(-8110.53 - 8110.53i) q^{7} +(-26396.3 + 19415.9i) q^{8} -111136. q^{9} +(13884.1 - 99031.5i) q^{10} +(-73301.0 + 73301.0i) q^{11} +(173330. + 385239. i) q^{12} -41242.5i q^{13} +(308217. + 199302. i) q^{14} +(-1.21071e6 - 442887. i) q^{15} +(695511. - 784714. i) q^{16} +(-187631. + 187631. i) q^{17} +(3.47719e6 - 746217. i) q^{18} +(3.28300e6 - 3.28300e6i) q^{19} +(230541. + 3.19168e6i) q^{20} +(3.34588e6 - 3.34588e6i) q^{21} +(1.80124e6 - 2.78559e6i) q^{22} +(-5.79621e6 + 5.79621e6i) q^{23} +(-8.00974e6 - 1.08894e7i) q^{24} +(-7.46050e6 - 6.30146e6i) q^{25} +(276920. + 1.29038e6i) q^{26} -2.14878e7i q^{27} +(-1.09816e7 - 4.16617e6i) q^{28} +(1.52635e7 - 1.52635e7i) q^{29} +(4.08540e7 + 5.72766e6i) q^{30} -2.81147e6 q^{31} +(-1.64920e7 + 2.92218e7i) q^{32} +(-3.02392e7 - 3.02392e7i) q^{33} +(4.61070e6 - 7.13037e6i) q^{34} +(3.25100e7 - 1.50955e7i) q^{35} +(-1.03783e8 + 4.66947e7i) q^{36} -8.04673e7i q^{37} +(-8.06738e7 + 1.24761e8i) q^{38} +1.70140e7 q^{39} +(-2.86434e7 - 9.83123e7i) q^{40} +3.18823e7i q^{41} +(-8.22189e7 + 1.27150e8i) q^{42} -1.78259e8 q^{43} +(-3.76529e7 + 9.92489e7i) q^{44} +(1.19313e8 - 3.26163e8i) q^{45} +(1.42431e8 - 2.20268e8i) q^{46} +(-2.36978e7 + 2.36978e7i) q^{47} +(3.23722e8 + 2.86923e8i) q^{48} -1.50914e8i q^{49} +(2.75732e8 + 1.47065e8i) q^{50} +(-7.74044e7 - 7.74044e7i) q^{51} +(-1.73283e7 - 3.85136e7i) q^{52} -2.74958e8 q^{53} +(1.44279e8 + 6.72303e8i) q^{54} +(-1.36430e8 - 2.93818e8i) q^{55} +(3.71561e8 + 5.66147e7i) q^{56} +(1.35435e9 + 1.35435e9i) q^{57} +(-3.75073e8 + 5.80045e8i) q^{58} +(-3.84489e7 - 3.84489e7i) q^{59} +(-1.31668e9 + 9.51062e7i) q^{60} +(7.89799e8 + 7.89799e8i) q^{61} +(8.79642e7 - 1.88774e7i) q^{62} +(9.01373e8 + 9.01373e8i) q^{63} +(3.19787e8 - 1.02502e9i) q^{64} +(1.21038e8 + 4.42768e7i) q^{65} +(1.14915e9 + 7.43075e8i) q^{66} +2.69082e9 q^{67} +(-9.63814e7 + 2.54051e8i) q^{68} +(-2.39114e9 - 2.39114e9i) q^{69} +(-9.15805e8 + 6.90590e8i) q^{70} -3.07860e9i q^{71} +(2.93358e9 - 2.15781e9i) q^{72} +(4.41138e7 - 4.41138e7i) q^{73} +(5.40292e8 + 2.51763e9i) q^{74} +(2.59957e9 - 3.07772e9i) q^{75} +(1.68639e9 - 4.44515e9i) q^{76} +1.18902e9 q^{77} +(-5.32327e8 + 1.14239e8i) q^{78} +2.34860e9i q^{79} +(1.55630e9 + 2.88364e9i) q^{80} +2.30199e9 q^{81} +(-2.14072e8 - 9.97522e8i) q^{82} +5.69229e9i q^{83} +(1.71869e9 - 4.53029e9i) q^{84} +(-3.49224e8 - 7.52096e8i) q^{85} +(5.57730e9 - 1.19691e9i) q^{86} +(6.29674e9 + 6.29674e9i) q^{87} +(5.11670e8 - 3.35808e9i) q^{88} -5.15223e9 q^{89} +(-1.54302e9 + 1.10060e10i) q^{90} +(-3.34498e8 + 3.34498e8i) q^{91} +(-2.97737e9 + 7.84801e9i) q^{92} -1.15983e9i q^{93} +(5.82331e8 - 9.00566e8i) q^{94} +(6.11041e9 + 1.31595e10i) q^{95} +(-1.20550e10 - 6.80352e9i) q^{96} +(-6.57499e9 + 6.57499e9i) q^{97} +(1.01330e9 + 4.72174e9i) q^{98} +(8.14640e9 - 8.14640e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.2876 + 6.71444i −0.977739 + 0.209826i
\(3\) 412.535i 1.69768i 0.528654 + 0.848838i \(0.322697\pi\)
−0.528654 + 0.848838i \(0.677303\pi\)
\(4\) 933.833 − 420.158i 0.911946 − 0.410310i
\(5\) −1073.57 + 2934.80i −0.343544 + 0.939137i
\(6\) −2769.94 − 12907.2i −0.356217 − 1.65988i
\(7\) −8110.53 − 8110.53i −0.482568 − 0.482568i 0.423383 − 0.905951i \(-0.360843\pi\)
−0.905951 + 0.423383i \(0.860843\pi\)
\(8\) −26396.3 + 19415.9i −0.805551 + 0.592526i
\(9\) −111136. −1.88210
\(10\) 13884.1 − 99031.5i 0.138841 − 0.990315i
\(11\) −73301.0 + 73301.0i −0.455142 + 0.455142i −0.897057 − 0.441915i \(-0.854299\pi\)
0.441915 + 0.897057i \(0.354299\pi\)
\(12\) 173330. + 385239.i 0.696574 + 1.54819i
\(13\) − 41242.5i − 0.111078i −0.998457 − 0.0555390i \(-0.982312\pi\)
0.998457 − 0.0555390i \(-0.0176877\pi\)
\(14\) 308217. + 199302.i 0.573081 + 0.370570i
\(15\) −1.21071e6 − 442887.i −1.59435 − 0.583226i
\(16\) 695511. − 784714.i 0.663291 − 0.748362i
\(17\) −187631. + 187631.i −0.132148 + 0.132148i −0.770087 − 0.637939i \(-0.779787\pi\)
0.637939 + 0.770087i \(0.279787\pi\)
\(18\) 3.47719e6 − 746217.i 1.84020 − 0.394914i
\(19\) 3.28300e6 − 3.28300e6i 1.32588 − 1.32588i 0.416943 − 0.908932i \(-0.363101\pi\)
0.908932 − 0.416943i \(-0.136899\pi\)
\(20\) 230541. + 3.19168e6i 0.0720440 + 0.997401i
\(21\) 3.34588e6 − 3.34588e6i 0.819244 − 0.819244i
\(22\) 1.80124e6 − 2.78559e6i 0.349509 − 0.540510i
\(23\) −5.79621e6 + 5.79621e6i −0.900544 + 0.900544i −0.995483 − 0.0949392i \(-0.969734\pi\)
0.0949392 + 0.995483i \(0.469734\pi\)
\(24\) −8.00974e6 − 1.08894e7i −1.00592 − 1.36756i
\(25\) −7.46050e6 − 6.30146e6i −0.763955 − 0.645269i
\(26\) 276920. + 1.29038e6i 0.0233071 + 0.108605i
\(27\) − 2.14878e7i − 1.49752i
\(28\) −1.09816e7 − 4.16617e6i −0.638079 − 0.242074i
\(29\) 1.52635e7 − 1.52635e7i 0.744157 − 0.744157i −0.229218 − 0.973375i \(-0.573617\pi\)
0.973375 + 0.229218i \(0.0736169\pi\)
\(30\) 4.08540e7 + 5.72766e6i 1.68123 + 0.235706i
\(31\) −2.81147e6 −0.0982030 −0.0491015 − 0.998794i \(-0.515636\pi\)
−0.0491015 + 0.998794i \(0.515636\pi\)
\(32\) −1.64920e7 + 2.92218e7i −0.491499 + 0.870878i
\(33\) −3.02392e7 − 3.02392e7i −0.772683 − 0.772683i
\(34\) 4.61070e6 − 7.13037e6i 0.101478 − 0.156934i
\(35\) 3.25100e7 − 1.50955e7i 0.618981 − 0.287414i
\(36\) −1.03783e8 + 4.66947e7i −1.71637 + 0.772246i
\(37\) − 8.04673e7i − 1.16041i −0.814471 − 0.580204i \(-0.802973\pi\)
0.814471 − 0.580204i \(-0.197027\pi\)
\(38\) −8.06738e7 + 1.24761e8i −1.01816 + 1.57456i
\(39\) 1.70140e7 0.188574
\(40\) −2.86434e7 − 9.83123e7i −0.279721 − 0.960081i
\(41\) 3.18823e7i 0.275189i 0.990489 + 0.137594i \(0.0439370\pi\)
−0.990489 + 0.137594i \(0.956063\pi\)
\(42\) −8.22189e7 + 1.27150e8i −0.629108 + 0.972906i
\(43\) −1.78259e8 −1.21258 −0.606288 − 0.795245i \(-0.707342\pi\)
−0.606288 + 0.795245i \(0.707342\pi\)
\(44\) −3.76529e7 + 9.92489e7i −0.228315 + 0.601814i
\(45\) 1.19313e8 − 3.26163e8i 0.646584 − 1.76755i
\(46\) 1.42431e8 − 2.20268e8i 0.691539 − 1.06945i
\(47\) −2.36978e7 + 2.36978e7i −0.103328 + 0.103328i −0.756881 − 0.653553i \(-0.773278\pi\)
0.653553 + 0.756881i \(0.273278\pi\)
\(48\) 3.23722e8 + 2.86923e8i 1.27048 + 1.12605i
\(49\) − 1.50914e8i − 0.534255i
\(50\) 2.75732e8 + 1.47065e8i 0.882343 + 0.470607i
\(51\) −7.74044e7 − 7.74044e7i −0.224344 − 0.224344i
\(52\) −1.73283e7 − 3.85136e7i −0.0455764 − 0.101297i
\(53\) −2.74958e8 −0.657487 −0.328743 − 0.944419i \(-0.606625\pi\)
−0.328743 + 0.944419i \(0.606625\pi\)
\(54\) 1.44279e8 + 6.72303e8i 0.314219 + 1.46419i
\(55\) −1.36430e8 − 2.93818e8i −0.271079 − 0.583801i
\(56\) 3.71561e8 + 5.66147e7i 0.674668 + 0.102799i
\(57\) 1.35435e9 + 1.35435e9i 2.25091 + 2.25091i
\(58\) −3.75073e8 + 5.80045e8i −0.571447 + 0.883735i
\(59\) −3.84489e7 − 3.84489e7i −0.0537804 − 0.0537804i 0.679705 − 0.733486i \(-0.262108\pi\)
−0.733486 + 0.679705i \(0.762108\pi\)
\(60\) −1.31668e9 + 9.51062e7i −1.69326 + 0.122307i
\(61\) 7.89799e8 + 7.89799e8i 0.935120 + 0.935120i 0.998020 − 0.0629000i \(-0.0200349\pi\)
−0.0629000 + 0.998020i \(0.520035\pi\)
\(62\) 8.79642e7 − 1.88774e7i 0.0960169 − 0.0206056i
\(63\) 9.01373e8 + 9.01373e8i 0.908243 + 0.908243i
\(64\) 3.19787e8 − 1.02502e9i 0.297825 − 0.954620i
\(65\) 1.21038e8 + 4.42768e7i 0.104317 + 0.0381601i
\(66\) 1.14915e9 + 7.43075e8i 0.917611 + 0.593353i
\(67\) 2.69082e9 1.99301 0.996506 − 0.0835174i \(-0.0266154\pi\)
0.996506 + 0.0835174i \(0.0266154\pi\)
\(68\) −9.63814e7 + 2.54051e8i −0.0662901 + 0.174733i
\(69\) −2.39114e9 − 2.39114e9i −1.52883 − 1.52883i
\(70\) −9.15805e8 + 6.90590e8i −0.544895 + 0.410894i
\(71\) − 3.07860e9i − 1.70632i −0.521647 − 0.853162i \(-0.674682\pi\)
0.521647 − 0.853162i \(-0.325318\pi\)
\(72\) 2.93358e9 − 2.15781e9i 1.51613 − 1.11519i
\(73\) 4.41138e7 − 4.41138e7i 0.0212794 − 0.0212794i −0.696387 − 0.717666i \(-0.745210\pi\)
0.717666 + 0.696387i \(0.245210\pi\)
\(74\) 5.40292e8 + 2.51763e9i 0.243484 + 1.13458i
\(75\) 2.59957e9 − 3.07772e9i 1.09546 − 1.29695i
\(76\) 1.68639e9 − 4.44515e9i 0.665107 − 1.75315i
\(77\) 1.18902e9 0.439274
\(78\) −5.32327e8 + 1.14239e8i −0.184376 + 0.0395678i
\(79\) 2.34860e9i 0.763262i 0.924315 + 0.381631i \(0.124638\pi\)
−0.924315 + 0.381631i \(0.875362\pi\)
\(80\) 1.55630e9 + 2.88364e9i 0.474944 + 0.880016i
\(81\) 2.30199e9 0.660205
\(82\) −2.14072e8 − 9.97522e8i −0.0577418 − 0.269063i
\(83\) 5.69229e9i 1.44510i 0.691321 + 0.722548i \(0.257029\pi\)
−0.691321 + 0.722548i \(0.742971\pi\)
\(84\) 1.71869e9 − 4.53029e9i 0.410962 − 1.08325i
\(85\) −3.49224e8 − 7.52096e8i −0.0787064 − 0.169504i
\(86\) 5.57730e9 − 1.19691e9i 1.18558 − 0.254430i
\(87\) 6.29674e9 + 6.29674e9i 1.26334 + 1.26334i
\(88\) 5.11670e8 − 3.35808e9i 0.0969564 − 0.636323i
\(89\) −5.15223e9 −0.922667 −0.461334 − 0.887227i \(-0.652629\pi\)
−0.461334 + 0.887227i \(0.652629\pi\)
\(90\) −1.54302e9 + 1.10060e10i −0.261312 + 1.86387i
\(91\) −3.34498e8 + 3.34498e8i −0.0536027 + 0.0536027i
\(92\) −2.97737e9 + 7.84801e9i −0.451745 + 1.19075i
\(93\) − 1.15983e9i − 0.166717i
\(94\) 5.82331e8 − 9.00566e8i 0.0793471 − 0.122709i
\(95\) 6.11041e9 + 1.31595e10i 0.789682 + 1.70068i
\(96\) −1.20550e10 − 6.80352e9i −1.47847 − 0.834406i
\(97\) −6.57499e9 + 6.57499e9i −0.765661 + 0.765661i −0.977339 − 0.211679i \(-0.932107\pi\)
0.211679 + 0.977339i \(0.432107\pi\)
\(98\) 1.01330e9 + 4.72174e9i 0.112101 + 0.522362i
\(99\) 8.14640e9 − 8.14640e9i 0.856623 − 0.856623i
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.13.6 ✓ 236
5.2 odd 4 80.11.t.a.77.53 yes 236
16.5 even 4 80.11.t.a.53.53 yes 236
80.37 odd 4 inner 80.11.i.a.37.6 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.6 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.6 yes 236 80.37 odd 4 inner
80.11.t.a.53.53 yes 236 16.5 even 4
80.11.t.a.77.53 yes 236 5.2 odd 4