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Properties

Label 128.2.k.a.109.11
Level $128$
Weight $2$
Character 128.109
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 109.11
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.11

$q$-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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{23}{32}\right)\) \(1\)

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 0.0507096 − 0.998713i \(-0.483852\pi\)
−0.802227 + 0.597019i \(0.796352\pi\)
\(4\) −1.22398 + 1.58173i −0.611991 + 0.790865i
\(5\) −2.50395 + 3.05107i −1.11980 + 1.36448i −0.196454 + 0.980513i \(0.562943\pi\)
−0.923346 + 0.383968i \(0.874557\pi\)
\(6\) 0.0725467 − 2.08604i 0.0296171 − 0.851622i
\(7\) 1.90175 + 1.27071i 0.718794 + 0.480283i 0.860387 − 0.509642i \(-0.170222\pi\)
−0.141592 + 0.989925i \(0.545222\pi\)
\(8\) −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 0.879983 − 0.475004i \(-0.157554\pi\)
0.467781 + 0.883844i \(0.345054\pi\)
\(12\) 2.69371 − 1.20729i 0.777608 − 0.348516i
\(13\) −2.41987 + 1.98594i −0.671153 + 0.550801i −0.907074 − 0.420972i \(-0.861689\pi\)
0.235921 + 0.971772i \(0.424189\pi\)
\(14\) −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 0.821636 − 0.570013i \(-0.193062\pi\)
0.177925 + 0.984044i \(0.443062\pi\)
\(18\) 0.583001 − 1.00505i 0.137415 − 0.236893i
\(19\) 0.371315 − 3.77002i 0.0851855 − 0.864903i −0.853721 − 0.520731i \(-0.825660\pi\)
0.938907 − 0.344172i \(-0.111840\pi\)
\(20\) −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 0.398134 − 0.917327i \(-0.369658\pi\)
0.0167818 + 0.999859i \(0.494658\pi\)
\(24\) 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 0.999922 − 0.0125277i \(-0.00398778\pi\)
−0.838364 + 0.545110i \(0.816488\pi\)
\(30\) 6.18300 + 5.44468i 1.12886 + 0.994059i
\(31\) 0.377059 − 0.377059i 0.0677218 − 0.0677218i −0.672435 − 0.740156i \(-0.734751\pi\)
0.740156 + 0.672435i \(0.234751\pi\)
\(32\) 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 0.0569976 − 0.998374i \(-0.481847\pi\)
0.250676 + 0.968071i \(0.419347\pi\)
\(38\) 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 0.140628 + 0.990062i \(0.455088\pi\)
−0.508804 + 0.860882i \(0.669912\pi\)
\(42\) 2.78872 − 3.87494i 0.430308 − 0.597917i
\(43\) 2.04620 − 1.09372i 0.312042 − 0.166790i −0.307940 − 0.951406i \(-0.599640\pi\)
0.619983 + 0.784616i \(0.287140\pi\)
\(44\) −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 −0.853932 − 0.520384i \(-0.825789\pi\)
0.971789 + 0.235854i \(0.0757887\pi\)
\(48\) −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 0.659600 + 0.751617i \(0.270726\pi\)
−0.966013 + 0.258493i \(0.916774\pi\)
\(54\) −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 −0.999760 − 0.0218897i \(-0.993032\pi\)
−0.216513 − 0.976280i \(-0.569468\pi\)
\(60\) −3.06138 + 11.2417i −0.395223 + 1.45130i
\(61\) 4.03888 − 7.55620i 0.517125 − 0.967473i −0.478855 − 0.877894i \(-0.658948\pi\)
0.995980 − 0.0895786i \(-0.0285520\pi\)
\(62\) 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 −0.568971 − 0.822358i \(-0.692658\pi\)
0.999869 + 0.0162043i \(0.00515820\pi\)
\(68\) −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 −0.306834 − 0.951763i \(-0.599270\pi\)
0.996735 − 0.0807464i \(-0.0257304\pi\)
\(72\) 0.876136 + 2.15231i 0.103254 + 0.253652i
\(73\) −4.18253 + 2.79468i −0.489528 + 0.327092i −0.775717 − 0.631081i \(-0.782612\pi\)
0.286189 + 0.958173i \(0.407612\pi\)
\(74\) 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 0.934178 − 0.356807i \(-0.116135\pi\)
−0.912864 + 0.408263i \(0.866135\pi\)
\(80\) 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 0.573358 − 0.819305i \(-0.305640\pi\)
0.402503 + 0.915419i \(0.368140\pi\)
\(84\) 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 0.119316 − 0.992856i \(-0.461930\pi\)
0.490183 + 0.871619i \(0.336930\pi\)
\(90\) 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 −0.681842 − 0.731500i \(-0.738821\pi\)
0.731500 + 0.681842i \(0.238821\pi\)
\(98\) 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.109.11 yes 240
4.3 odd 2 512.2.k.a.273.11 240
128.27 odd 32 512.2.k.a.497.11 240
128.101 even 32 inner 128.2.k.a.101.11 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.11 ✓ 240 128.101 even 32 inner
128.2.k.a.109.11 yes 240 1.1 even 1 trivial
512.2.k.a.273.11 240 4.3 odd 2
512.2.k.a.497.11 240 128.27 odd 32