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Properties

Label 128.2.k.a.109.14
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 109.14
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.14

$q$-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+(1.27317 + 0.615660i) q^{2} +(0.731810 + 0.391160i) q^{3} +(1.24192 + 1.56768i) q^{4} +(-0.0771150 + 0.0939649i) q^{5} +(0.690896 + 0.948560i) q^{6} +(-3.30773 - 2.21015i) q^{7} +(0.616022 + 2.76053i) q^{8} +(-1.28417 - 1.92190i) q^{9} +(-0.156031 + 0.0721566i) q^{10} +(0.0222437 + 0.00674755i) q^{11} +(0.295638 + 1.63304i) q^{12} +(0.466044 - 0.382472i) q^{13} +(-2.85060 - 4.85034i) q^{14} +(-0.0931888 + 0.0386001i) q^{15} +(-0.915247 + 3.89388i) q^{16} +(4.59132 + 1.90179i) q^{17} +(-0.451732 - 3.23752i) q^{18} +(0.166080 - 1.68624i) q^{19} +(-0.243078 - 0.00419439i) q^{20} +(-1.55610 - 2.91126i) q^{21} +(0.0241658 + 0.0222853i) q^{22} +(-3.40974 - 0.678239i) q^{23} +(-0.628998 + 2.26114i) q^{24} +(0.972569 + 4.88943i) q^{25} +(0.828826 - 0.200028i) q^{26} +(-0.432000 - 4.38616i) q^{27} +(-0.643134 - 7.93031i) q^{28} +(-1.44800 - 4.77342i) q^{29} +(-0.142410 - 0.00822820i) q^{30} +(-6.39673 + 6.39673i) q^{31} +(-3.56257 + 4.39409i) q^{32} +(0.0136388 + 0.0136388i) q^{33} +(4.67467 + 5.24799i) q^{34} +(0.462752 - 0.140374i) q^{35} +(1.41808 - 4.40003i) q^{36} +(8.75672 - 0.862462i) q^{37} +(1.24960 - 2.04462i) q^{38} +(0.490664 - 0.0975990i) q^{39} +(-0.306897 - 0.154994i) q^{40} +(0.0606195 - 0.304755i) q^{41} +(-0.188834 - 4.66457i) q^{42} +(-4.91504 + 2.62714i) q^{43} +(0.0170470 + 0.0432510i) q^{44} +(0.279620 + 0.0275402i) q^{45} +(-3.92361 - 2.96275i) q^{46} +(0.167395 - 0.404127i) q^{47} +(-2.19292 + 2.49157i) q^{48} +(3.37751 + 8.15402i) q^{49} +(-1.77199 + 6.82385i) q^{50} +(2.61607 + 3.18769i) q^{51} +(1.17839 + 0.255606i) q^{52} +(1.85360 - 6.11049i) q^{53} +(2.15038 - 5.85030i) q^{54} +(-0.00234935 + 0.00156979i) q^{55} +(4.06356 - 10.4926i) q^{56} +(0.781127 - 1.16904i) q^{57} +(1.09525 - 6.96886i) q^{58} +(7.95889 + 6.53169i) q^{59} +(-0.176246 - 0.0981519i) q^{60} +(-6.59803 + 12.3440i) q^{61} +(-12.0823 + 4.20591i) q^{62} +9.19534i q^{63} +(-7.24103 + 3.40109i) q^{64} +0.0732861i q^{65} +(0.00896763 + 0.0257613i) q^{66} +(5.13746 - 9.61151i) q^{67} +(2.72068 + 9.55959i) q^{68} +(-2.22998 - 1.83010i) q^{69} +(0.675585 + 0.106178i) q^{70} +(3.99036 - 5.97200i) q^{71} +(4.51438 - 4.72893i) q^{72} +(-8.64165 + 5.77416i) q^{73} +(11.6798 + 4.29310i) q^{74} +(-1.20082 + 3.95857i) q^{75} +(2.84974 - 1.83382i) q^{76} +(-0.0586630 - 0.0714811i) q^{77} +(0.684786 + 0.177822i) q^{78} +(-0.504221 - 1.21730i) q^{79} +(-0.295309 - 0.386278i) q^{80} +(-1.25411 + 3.02768i) q^{81} +(0.264805 - 0.350684i) q^{82} +(16.2560 + 1.60107i) q^{83} +(2.63137 - 6.05504i) q^{84} +(-0.532760 + 0.284766i) q^{85} +(-7.87510 + 0.318805i) q^{86} +(0.807512 - 4.05964i) q^{87} +(-0.00492419 + 0.0655610i) q^{88} +(2.00759 - 0.399335i) q^{89} +(0.339048 + 0.207214i) q^{90} +(-2.38687 + 0.235086i) q^{91} +(-3.17137 - 6.18770i) q^{92} +(-7.18334 + 2.17904i) q^{93} +(0.461928 - 0.411464i) q^{94} +(0.145640 + 0.145640i) q^{95} +(-4.32592 + 1.82210i) q^{96} +(5.29993 - 5.29993i) q^{97} +(-0.719968 + 12.4609i) q^{98} +(-0.0155966 - 0.0514152i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27317 + 0.615660i 0.900267 + 0.435338i
\(3\) 0.731810 + 0.391160i 0.422510 + 0.225837i 0.668928 − 0.743327i \(-0.266753\pi\)
−0.246418 + 0.969164i \(0.579253\pi\)
\(4\) 1.24192 + 1.56768i 0.620962 + 0.783840i
\(5\) −0.0771150 + 0.0939649i −0.0344869 + 0.0420224i −0.789968 − 0.613148i \(-0.789903\pi\)
0.755481 + 0.655170i \(0.227403\pi\)
\(6\) 0.690896 + 0.948560i 0.282057 + 0.387248i
\(7\) −3.30773 − 2.21015i −1.25020 − 0.835360i −0.258766 − 0.965940i \(-0.583316\pi\)
−0.991438 + 0.130580i \(0.958316\pi\)
\(8\) 0.616022 + 2.76053i 0.217797 + 0.975994i
\(9\) −1.28417 − 1.92190i −0.428057 − 0.640633i
\(10\) −0.156031 + 0.0721566i −0.0493413 + 0.0228179i
\(11\) 0.0222437 + 0.00674755i 0.00670673 + 0.00203446i 0.293636 − 0.955917i \(-0.405135\pi\)
−0.286930 + 0.957952i \(0.592635\pi\)
\(12\) 0.295638 + 1.63304i 0.0853433 + 0.471417i
\(13\) 0.466044 − 0.382472i 0.129257 − 0.106079i −0.567544 − 0.823343i \(-0.692106\pi\)
0.696801 + 0.717265i \(0.254606\pi\)
\(14\) −2.85060 − 4.85034i −0.761854 − 1.29631i
\(15\) −0.0931888 + 0.0386001i −0.0240612 + 0.00996650i
\(16\) −0.915247 + 3.89388i −0.228812 + 0.973471i
\(17\) 4.59132 + 1.90179i 1.11356 + 0.461251i 0.862162 − 0.506633i \(-0.169110\pi\)
0.251396 + 0.967884i \(0.419110\pi\)
\(18\) −0.451732 − 3.23752i −0.106474 − 0.763090i
\(19\) 0.166080 − 1.68624i 0.0381013 − 0.386849i −0.957156 − 0.289572i \(-0.906487\pi\)
0.995257 − 0.0972765i \(-0.0310131\pi\)
\(20\) −0.243078 − 0.00419439i −0.0543539 − 0.000937894i
\(21\) −1.55610 − 2.91126i −0.339570 − 0.635290i
\(22\) 0.0241658 + 0.0222853i 0.00515217 + 0.00475125i
\(23\) −3.40974 − 0.678239i −0.710979 − 0.141423i −0.173665 − 0.984805i \(-0.555561\pi\)
−0.537314 + 0.843382i \(0.680561\pi\)
\(24\) −0.628998 + 2.26114i −0.128394 + 0.461554i
\(25\) 0.972569 + 4.88943i 0.194514 + 0.977887i
\(26\) 0.828826 − 0.200028i 0.162546 − 0.0392287i
\(27\) −0.432000 − 4.38616i −0.0831383 − 0.844118i
\(28\) −0.643134 − 7.93031i −0.121541 − 1.49869i
\(29\) −1.44800 − 4.77342i −0.268887 − 0.886403i −0.982564 − 0.185927i \(-0.940471\pi\)
0.713676 − 0.700476i \(-0.247029\pi\)
\(30\) −0.142410 − 0.00822820i −0.0260003 − 0.00150226i
\(31\) −6.39673 + 6.39673i −1.14889 + 1.14889i −0.162115 + 0.986772i \(0.551832\pi\)
−0.986772 + 0.162115i \(0.948168\pi\)
\(32\) −3.56257 + 4.39409i −0.629780 + 0.776774i
\(33\) 0.0136388 + 0.0136388i 0.00237421 + 0.00237421i
\(34\) 4.67467 + 5.24799i 0.801700 + 0.900023i
\(35\) 0.462752 − 0.140374i 0.0782194 − 0.0237276i
\(36\) 1.41808 − 4.40003i 0.236347 − 0.733338i
\(37\) 8.75672 − 0.862462i 1.43960 − 0.141788i 0.652112 − 0.758123i \(-0.273883\pi\)
0.787484 + 0.616335i \(0.211383\pi\)
\(38\) 1.24960 − 2.04462i 0.202711 − 0.331681i
\(39\) 0.490664 − 0.0975990i 0.0785690 − 0.0156284i
\(40\) −0.306897 − 0.154994i −0.0485247 − 0.0245066i
\(41\) 0.0606195 − 0.304755i 0.00946718 − 0.0475947i −0.975763 − 0.218830i \(-0.929776\pi\)
0.985230 + 0.171236i \(0.0547759\pi\)
\(42\) −0.188834 − 4.66457i −0.0291377 − 0.719758i
\(43\) −4.91504 + 2.62714i −0.749536 + 0.400635i −0.801485 − 0.598014i \(-0.795957\pi\)
0.0519494 + 0.998650i \(0.483457\pi\)
\(44\) 0.0170470 + 0.0432510i 0.00256993 + 0.00652033i
\(45\) 0.279620 + 0.0275402i 0.0416833 + 0.00410544i
\(46\) −3.92361 − 2.96275i −0.578505 − 0.436834i
\(47\) 0.167395 − 0.404127i 0.0244171 − 0.0589480i −0.911201 − 0.411963i \(-0.864843\pi\)
0.935618 + 0.353015i \(0.114843\pi\)
\(48\) −2.19292 + 2.49157i −0.316521 + 0.359628i
\(49\) 3.37751 + 8.15402i 0.482501 + 1.16486i
\(50\) −1.77199 + 6.82385i −0.250597 + 0.965039i
\(51\) 2.61607 + 3.18769i 0.366323 + 0.446365i
\(52\) 1.17839 + 0.255606i 0.163413 + 0.0354462i
\(53\) 1.85360 − 6.11049i 0.254611 − 0.839341i −0.732701 − 0.680551i \(-0.761741\pi\)
0.987312 − 0.158790i \(-0.0507594\pi\)
\(54\) 2.15038 − 5.85030i 0.292630 − 0.796125i
\(55\) −0.00234935 + 0.00156979i −0.000316787 + 0.000211670i
\(56\) 4.06356 − 10.4926i 0.543016 − 1.40213i
\(57\) 0.781127 − 1.16904i 0.103463 − 0.154843i
\(58\) 1.09525 − 6.96886i 0.143814 − 0.915056i
\(59\) 7.95889 + 6.53169i 1.03616 + 0.850354i 0.988908 − 0.148529i \(-0.0474538\pi\)
0.0472513 + 0.998883i \(0.484954\pi\)
\(60\) −0.176246 − 0.0981519i −0.0227533 − 0.0126714i
\(61\) −6.59803 + 12.3440i −0.844791 + 1.58049i −0.0308657 + 0.999524i \(0.509826\pi\)
−0.813925 + 0.580969i \(0.802674\pi\)
\(62\) −12.0823 + 4.20591i −1.53446 + 0.534152i
\(63\) 9.19534i 1.15850i
\(64\) −7.24103 + 3.40109i −0.905129 + 0.425137i
\(65\) 0.0732861i 0.00909002i
\(66\) 0.00896763 + 0.0257613i 0.00110384 + 0.00317100i
\(67\) 5.13746 − 9.61151i 0.627641 − 1.17423i −0.344267 − 0.938872i \(-0.611873\pi\)
0.971908 − 0.235361i \(-0.0756274\pi\)
\(68\) 2.72068 + 9.55959i 0.329930 + 1.15927i
\(69\) −2.22998 − 1.83010i −0.268458 − 0.220318i
\(70\) 0.675585 + 0.106178i 0.0807479 + 0.0126907i
\(71\) 3.99036 − 5.97200i 0.473569 − 0.708746i −0.515387 − 0.856958i \(-0.672352\pi\)
0.988956 + 0.148212i \(0.0473517\pi\)
\(72\) 4.51438 − 4.72893i 0.532025 − 0.557309i
\(73\) −8.64165 + 5.77416i −1.01143 + 0.675815i −0.946709 − 0.322089i \(-0.895615\pi\)
−0.0647187 + 0.997904i \(0.520615\pi\)
\(74\) 11.6798 + 4.29310i 1.35775 + 0.499063i
\(75\) −1.20082 + 3.95857i −0.138658 + 0.457096i
\(76\) 2.84974 − 1.83382i 0.326887 − 0.210353i
\(77\) −0.0586630 − 0.0714811i −0.00668527 − 0.00814602i
\(78\) 0.684786 + 0.177822i 0.0775367 + 0.0201344i
\(79\) −0.504221 − 1.21730i −0.0567293 − 0.136957i 0.892974 − 0.450109i \(-0.148615\pi\)
−0.949703 + 0.313153i \(0.898615\pi\)
\(80\) −0.295309 − 0.386278i −0.0330165 − 0.0431872i
\(81\) −1.25411 + 3.02768i −0.139345 + 0.336409i
\(82\) 0.264805 − 0.350684i 0.0292428 − 0.0387266i
\(83\) 16.2560 + 1.60107i 1.78433 + 0.175741i 0.935593 − 0.353081i \(-0.114866\pi\)
0.848733 + 0.528822i \(0.177366\pi\)
\(84\) 2.63137 − 6.05504i 0.287106 − 0.660659i
\(85\) −0.532760 + 0.284766i −0.0577860 + 0.0308872i
\(86\) −7.87510 + 0.318805i −0.849194 + 0.0343777i
\(87\) 0.807512 − 4.05964i 0.0865744 − 0.435239i
\(88\) −0.00492419 + 0.0655610i −0.000524921 + 0.00698883i
\(89\) 2.00759 − 0.399335i 0.212805 − 0.0423294i −0.0875366 − 0.996161i \(-0.527899\pi\)
0.300341 + 0.953832i \(0.402899\pi\)
\(90\) 0.339048 + 0.207214i 0.0357388 + 0.0218423i
\(91\) −2.38687 + 0.235086i −0.250212 + 0.0246437i
\(92\) −3.17137 − 6.18770i −0.330639 − 0.645112i
\(93\) −7.18334 + 2.17904i −0.744878 + 0.225956i
\(94\) 0.461928 − 0.411464i 0.0476442 − 0.0424393i
\(95\) 0.145640 + 0.145640i 0.0149423 + 0.0149423i
\(96\) −4.32592 + 1.82210i −0.441513 + 0.185968i
\(97\) 5.29993 − 5.29993i 0.538126 − 0.538126i −0.384852 − 0.922978i \(-0.625748\pi\)
0.922978 + 0.384852i \(0.125748\pi\)
\(98\) −0.719968 + 12.4609i −0.0727277 + 1.25874i
\(99\) −0.0155966 − 0.0514152i −0.00156752 − 0.00516742i
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.109.14 yes 240
4.3 odd 2 512.2.k.a.273.5 240
128.27 odd 32 512.2.k.a.497.5 240
128.101 even 32 inner 128.2.k.a.101.14 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.14 ✓ 240 128.101 even 32 inner
128.2.k.a.109.14 yes 240 1.1 even 1 trivial
512.2.k.a.273.5 240 4.3 odd 2
512.2.k.a.497.5 240 128.27 odd 32