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Properties

Label 128.2.k.a.101.11
Level $128$
Weight $2$
Character 128.101
Analytic conductor $1.022$
Analytic rank $0$
Dimension $240$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [128,2,Mod(5,128)] 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("128.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.k (of order \(32\), degree \(16\), 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: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(15\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 101.11
Character \(\chi\) \(=\) 128.101
Dual form 128.2.k.a.109.11

$q$-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+(0.622904 - 1.26964i) q^{2} +(-1.30167 + 0.695755i) q^{3} +(-1.22398 - 1.58173i) q^{4} +(-2.50395 - 3.05107i) q^{5} +(0.0725467 + 2.08604i) q^{6} +(1.90175 - 1.27071i) q^{7} +(-2.77065 + 0.568752i) q^{8} +(-0.456450 + 0.683126i) q^{9} +(-5.43349 + 1.27859i) q^{10} +(4.47003 - 1.35597i) q^{11} +(2.69371 + 1.20729i) q^{12} +(-2.41987 - 1.98594i) q^{13} +(-0.428738 - 3.20607i) q^{14} +(5.38211 + 2.22934i) q^{15} +(-1.00374 + 3.87202i) q^{16} +(4.12129 - 1.70710i) q^{17} +(0.583001 + 1.00505i) q^{18} +(0.371315 + 3.77002i) q^{19} +(-1.76118 + 7.69503i) q^{20} +(-1.59134 + 2.97719i) q^{21} +(1.06280 - 6.51997i) q^{22} +(1.98987 - 0.395809i) q^{23} +(3.21076 - 2.66802i) q^{24} +(-2.06382 + 10.3755i) q^{25} +(-4.02878 + 1.83532i) q^{26} +(0.552861 - 5.61329i) q^{27} +(-4.33763 - 1.45273i) q^{28} +(0.870012 - 2.86805i) q^{29} +(6.18300 - 5.44468i) q^{30} +(0.377059 + 0.377059i) q^{31} +(4.29084 + 3.68629i) q^{32} +(-4.87507 + 4.87507i) q^{33} +(0.399770 - 6.29592i) q^{34} +(-8.63892 - 2.62059i) q^{35} +(1.63921 - 0.114152i) q^{36} +(1.87150 + 0.184327i) q^{37} +(5.01787 + 1.87693i) q^{38} +(4.53160 + 0.901391i) q^{39} +(8.67288 + 7.02934i) q^{40} +(-2.35748 - 11.8518i) q^{41} +(2.78872 + 3.87494i) q^{42} +(2.04620 + 1.09372i) q^{43} +(-7.61601 - 5.41070i) q^{44} +(3.22720 - 0.317851i) q^{45} +(0.736960 - 2.77297i) q^{46} +(0.807983 + 1.95064i) q^{47} +(-1.38744 - 5.73843i) q^{48} +(-0.676829 + 1.63401i) q^{49} +(11.8877 + 9.08328i) q^{50} +(-4.17683 + 5.08948i) q^{51} +(-0.179340 + 6.25834i) q^{52} +(-2.23072 - 7.35371i) q^{53} +(-6.78249 - 4.19848i) q^{54} +(-15.3299 - 10.2431i) q^{55} +(-4.54638 + 4.60232i) q^{56} +(-3.10634 - 4.64897i) q^{57} +(-3.09946 - 2.89112i) q^{58} +(-9.34237 + 7.66709i) q^{59} +(-3.06138 - 11.2417i) q^{60} +(4.03888 + 7.55620i) q^{61} +(0.713602 - 0.243858i) q^{62} +1.87915i q^{63} +(7.35304 - 3.15163i) q^{64} +12.3559i q^{65} +(3.15289 + 9.22629i) q^{66} +(3.52705 + 6.59865i) q^{67} +(-7.74455 - 4.42932i) q^{68} +(-2.31476 + 1.89967i) q^{69} +(-8.70842 + 9.33596i) q^{70} +(5.81321 + 8.70008i) q^{71} +(0.876136 - 2.15231i) q^{72} +(-4.18253 - 2.79468i) q^{73} +(1.39980 - 2.26132i) q^{74} +(-4.53243 - 14.9414i) q^{75} +(5.50868 - 5.20176i) q^{76} +(6.77784 - 8.25882i) q^{77} +(3.96719 - 5.19203i) q^{78} +(0.189443 - 0.457356i) q^{79} +(14.3271 - 6.63285i) q^{80} +(2.24261 + 5.41414i) q^{81} +(-16.5161 - 4.38940i) q^{82} +(8.89052 - 0.875640i) q^{83} +(6.65689 - 1.12695i) q^{84} +(-15.5280 - 8.29988i) q^{85} +(2.66321 - 1.91666i) q^{86} +(0.862992 + 4.33855i) q^{87} +(-11.6137 + 6.29926i) q^{88} +(5.75001 + 1.14375i) q^{89} +(1.60668 - 4.29537i) q^{90} +(-7.12555 - 0.701806i) q^{91} +(-3.06162 - 2.66297i) q^{92} +(-0.753146 - 0.228464i) q^{93} +(2.97991 + 0.189215i) q^{94} +(10.5729 - 10.5729i) q^{95} +(-8.14999 - 1.81294i) q^{96} +(0.489078 + 0.489078i) q^{97} +(1.65301 + 1.87716i) q^{98} +(-1.11405 + 3.67253i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9} - 16 q^{10} - 16 q^{11} - 16 q^{12} - 16 q^{13} - 16 q^{14} - 16 q^{15} - 16 q^{16} - 16 q^{17} - 16 q^{18}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{9}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.622904 − 1.26964i 0.440460 − 0.897772i
\(3\) −1.30167 + 0.695755i −0.751518 + 0.401695i −0.802227 − 0.597019i \(-0.796352\pi\)
0.0507096 + 0.998713i \(0.483852\pi\)
\(4\) −1.22398 − 1.58173i −0.611991 − 0.790865i
\(5\) −2.50395 − 3.05107i −1.11980 − 1.36448i −0.923346 − 0.383968i \(-0.874557\pi\)
−0.196454 − 0.980513i \(-0.562943\pi\)
\(6\) 0.0725467 + 2.08604i 0.0296171 + 0.851622i
\(7\) 1.90175 − 1.27071i 0.718794 − 0.480283i −0.141592 − 0.989925i \(-0.545222\pi\)
0.860387 + 0.509642i \(0.170222\pi\)
\(8\) −2.77065 + 0.568752i −0.979574 + 0.201084i
\(9\) −0.456450 + 0.683126i −0.152150 + 0.227709i
\(10\) −5.43349 + 1.27859i −1.71822 + 0.404327i
\(11\) 4.47003 − 1.35597i 1.34776 − 0.408840i 0.467781 − 0.883844i \(-0.345054\pi\)
0.879983 + 0.475004i \(0.157554\pi\)
\(12\) 2.69371 + 1.20729i 0.777608 + 0.348516i
\(13\) −2.41987 − 1.98594i −0.671153 − 0.550801i 0.235921 − 0.971772i \(-0.424189\pi\)
−0.907074 + 0.420972i \(0.861689\pi\)
\(14\) −0.428738 − 3.20607i −0.114585 − 0.856859i
\(15\) 5.38211 + 2.22934i 1.38965 + 0.575614i
\(16\) −1.00374 + 3.87202i −0.250935 + 0.968004i
\(17\) 4.12129 − 1.70710i 0.999561 − 0.414032i 0.177925 − 0.984044i \(-0.443062\pi\)
0.821636 + 0.570013i \(0.193062\pi\)
\(18\) 0.583001 + 1.00505i 0.137415 + 0.236893i
\(19\) 0.371315 + 3.77002i 0.0851855 + 0.864903i 0.938907 + 0.344172i \(0.111840\pi\)
−0.853721 + 0.520731i \(0.825660\pi\)
\(20\) −1.76118 + 7.69503i −0.393813 + 1.72066i
\(21\) −1.59134 + 2.97719i −0.347260 + 0.649677i
\(22\) 1.06280 − 6.51997i 0.226591 − 1.39006i
\(23\) 1.98987 − 0.395809i 0.414916 − 0.0825319i 0.0167818 − 0.999859i \(-0.494658\pi\)
0.398134 + 0.917327i \(0.369658\pi\)
\(24\) 3.21076 − 2.66802i 0.655393 − 0.544608i
\(25\) −2.06382 + 10.3755i −0.412765 + 2.07511i
\(26\) −4.02878 + 1.83532i −0.790109 + 0.359937i
\(27\) 0.552861 − 5.61329i 0.106398 − 1.08028i
\(28\) −4.33763 − 1.45273i −0.819735 − 0.274541i
\(29\) 0.870012 − 2.86805i 0.161557 − 0.532583i −0.838364 − 0.545110i \(-0.816488\pi\)
0.999922 + 0.0125277i \(0.00398778\pi\)
\(30\) 6.18300 − 5.44468i 1.12886 − 0.994059i
\(31\) 0.377059 + 0.377059i 0.0677218 + 0.0677218i 0.740156 − 0.672435i \(-0.234751\pi\)
−0.672435 + 0.740156i \(0.734751\pi\)
\(32\) 4.29084 + 3.68629i 0.758520 + 0.651649i
\(33\) −4.87507 + 4.87507i −0.848640 + 0.848640i
\(34\) 0.399770 − 6.29592i 0.0685600 − 1.07974i
\(35\) −8.63892 − 2.62059i −1.46024 − 0.442960i
\(36\) 1.63921 − 0.114152i 0.273201 − 0.0190254i
\(37\) 1.87150 + 0.184327i 0.307673 + 0.0303032i 0.250676 − 0.968071i \(-0.419347\pi\)
0.0569976 + 0.998374i \(0.481847\pi\)
\(38\) 5.01787 + 1.87693i 0.814007 + 0.304478i
\(39\) 4.53160 + 0.901391i 0.725636 + 0.144338i
\(40\) 8.67288 + 7.02934i 1.37130 + 1.11144i
\(41\) −2.35748 − 11.8518i −0.368176 − 1.85094i −0.508804 − 0.860882i \(-0.669912\pi\)
0.140628 − 0.990062i \(-0.455088\pi\)
\(42\) 2.78872 + 3.87494i 0.430308 + 0.597917i
\(43\) 2.04620 + 1.09372i 0.312042 + 0.166790i 0.619983 − 0.784616i \(-0.287140\pi\)
−0.307940 + 0.951406i \(0.599640\pi\)
\(44\) −7.61601 − 5.41070i −1.14816 − 0.815694i
\(45\) 3.22720 − 0.317851i 0.481082 − 0.0473824i
\(46\) 0.736960 − 2.77297i 0.108659 − 0.408852i
\(47\) 0.807983 + 1.95064i 0.117856 + 0.284531i 0.971789 − 0.235854i \(-0.0757887\pi\)
−0.853932 + 0.520384i \(0.825789\pi\)
\(48\) −1.38744 − 5.73843i −0.200260 − 0.828271i
\(49\) −0.676829 + 1.63401i −0.0966898 + 0.233430i
\(50\) 11.8877 + 9.08328i 1.68117 + 1.28457i
\(51\) −4.17683 + 5.08948i −0.584873 + 0.712670i
\(52\) −0.179340 + 6.25834i −0.0248700 + 0.867876i
\(53\) −2.23072 − 7.35371i −0.306413 − 1.01011i −0.966013 − 0.258493i \(-0.916774\pi\)
0.659600 − 0.751617i \(-0.270726\pi\)
\(54\) −6.78249 − 4.19848i −0.922980 − 0.571340i
\(55\) −15.3299 − 10.2431i −2.06708 − 1.38118i
\(56\) −4.54638 + 4.60232i −0.607535 + 0.615011i
\(57\) −3.10634 − 4.64897i −0.411445 − 0.615771i
\(58\) −3.09946 − 2.89112i −0.406979 − 0.379623i
\(59\) −9.34237 + 7.66709i −1.21627 + 0.998169i −0.216513 + 0.976280i \(0.569468\pi\)
−0.999760 + 0.0218897i \(0.993032\pi\)
\(60\) −3.06138 − 11.2417i −0.395223 − 1.45130i
\(61\) 4.03888 + 7.55620i 0.517125 + 0.967473i 0.995980 + 0.0895786i \(0.0285520\pi\)
−0.478855 + 0.877894i \(0.658948\pi\)
\(62\) 0.713602 − 0.243858i 0.0906275 − 0.0309701i
\(63\) 1.87915i 0.236751i
\(64\) 7.35304 − 3.15163i 0.919130 − 0.393953i
\(65\) 12.3559i 1.53256i
\(66\) 3.15289 + 9.22629i 0.388094 + 1.13568i
\(67\) 3.52705 + 6.59865i 0.430898 + 0.806154i 0.999869 − 0.0162043i \(-0.00515820\pi\)
−0.568971 + 0.822358i \(0.692658\pi\)
\(68\) −7.74455 − 4.42932i −0.939165 − 0.537134i
\(69\) −2.31476 + 1.89967i −0.278664 + 0.228694i
\(70\) −8.70842 + 9.33596i −1.04086 + 1.11586i
\(71\) 5.81321 + 8.70008i 0.689901 + 1.03251i 0.996735 + 0.0807464i \(0.0257304\pi\)
−0.306834 + 0.951763i \(0.599270\pi\)
\(72\) 0.876136 − 2.15231i 0.103254 − 0.253652i
\(73\) −4.18253 − 2.79468i −0.489528 − 0.327092i 0.286189 − 0.958173i \(-0.407612\pi\)
−0.775717 + 0.631081i \(0.782612\pi\)
\(74\) 1.39980 − 2.26132i 0.162723 − 0.262873i
\(75\) −4.53243 − 14.9414i −0.523359 − 1.72529i
\(76\) 5.50868 − 5.20176i 0.631889 − 0.596683i
\(77\) 6.77784 − 8.25882i 0.772407 − 0.941180i
\(78\) 3.96719 − 5.19203i 0.449196 − 0.587881i
\(79\) 0.189443 − 0.457356i 0.0213140 − 0.0514565i −0.912864 − 0.408263i \(-0.866135\pi\)
0.934178 + 0.356807i \(0.116135\pi\)
\(80\) 14.3271 − 6.63285i 1.60182 − 0.741575i
\(81\) 2.24261 + 5.41414i 0.249179 + 0.601571i
\(82\) −16.5161 − 4.38940i −1.82389 − 0.484728i
\(83\) 8.89052 − 0.875640i 0.975861 − 0.0961139i 0.402503 − 0.915419i \(-0.368140\pi\)
0.573358 + 0.819305i \(0.305640\pi\)
\(84\) 6.65689 − 1.12695i 0.726326 − 0.122961i
\(85\) −15.5280 − 8.29988i −1.68425 − 0.900249i
\(86\) 2.66321 − 1.91666i 0.287182 − 0.206679i
\(87\) 0.862992 + 4.33855i 0.0925225 + 0.465142i
\(88\) −11.6137 + 6.29926i −1.23802 + 0.671503i
\(89\) 5.75001 + 1.14375i 0.609500 + 0.121237i 0.490183 − 0.871619i \(-0.336930\pi\)
0.119316 + 0.992856i \(0.461930\pi\)
\(90\) 1.60668 − 4.29537i 0.169359 − 0.452772i
\(91\) −7.12555 − 0.701806i −0.746961 − 0.0735692i
\(92\) −3.06162 − 2.66297i −0.319196 − 0.277634i
\(93\) −0.753146 − 0.228464i −0.0780976 − 0.0236907i
\(94\) 2.97991 + 0.189215i 0.307355 + 0.0195160i
\(95\) 10.5729 − 10.5729i 1.08475 − 1.08475i
\(96\) −8.14999 − 1.81294i −0.831805 − 0.185032i
\(97\) 0.489078 + 0.489078i 0.0496583 + 0.0496583i 0.731500 − 0.681842i \(-0.238821\pi\)
−0.681842 + 0.731500i \(0.738821\pi\)
\(98\) 1.65301 + 1.87716i 0.166979 + 0.189622i
\(99\) −1.11405 + 3.67253i −0.111966 + 0.369103i
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 128.2.k.a.101.11 ✓ 240
4.3 odd 2 512.2.k.a.497.11 240
128.19 odd 32 512.2.k.a.273.11 240
128.109 even 32 inner 128.2.k.a.109.11 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.11 ✓ 240 1.1 even 1 trivial
128.2.k.a.109.11 yes 240 128.109 even 32 inner
512.2.k.a.273.11 240 128.19 odd 32
512.2.k.a.497.11 240 4.3 odd 2