↑

Properties

Label 128.2.k.a.5.12
Level $128$
Weight $2$
Character 128.5
Analytic conductor $1.022$
Analytic rank $0$
Dimension $240$
Inner twists $2$

Related objects

Downloads

Learn more

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 5.12
Character \(\chi\) \(=\) 128.5
Dual form 128.2.k.a.77.12

$q$-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+(0.825274 + 1.14844i) q^{2} +(-3.11090 - 0.943682i) q^{3} +(-0.637847 + 1.89556i) q^{4} +(-2.52957 - 0.249141i) q^{5} +(-1.48358 - 4.35149i) q^{6} +(-1.75170 - 2.62161i) q^{7} +(-2.70334 + 0.831825i) q^{8} +(6.29277 + 4.20469i) q^{9} +(-1.80147 - 3.11068i) q^{10} +(-1.57480 + 2.94625i) q^{11} +(3.77309 - 5.29498i) q^{12} +(0.0312066 + 0.316846i) q^{13} +(1.56514 - 4.17527i) q^{14} +(7.63415 + 3.16217i) q^{15} +(-3.18630 - 2.41816i) q^{16} +(-3.42808 + 1.41996i) q^{17} +(0.364401 + 10.6969i) q^{18} +(0.234333 - 0.285535i) q^{19} +(2.08574 - 4.63605i) q^{20} +(2.97541 + 9.80861i) q^{21} +(-4.68325 + 0.622889i) q^{22} +(0.0973467 + 0.489395i) q^{23} +(9.19482 - 0.0366289i) q^{24} +(1.43275 + 0.284991i) q^{25} +(-0.338126 + 0.297324i) q^{26} +(-9.42128 - 11.4799i) q^{27} +(6.08673 - 1.64827i) q^{28} +(4.96138 - 2.65191i) q^{29} +(2.66869 + 11.3770i) q^{30} +(-1.02952 - 1.02952i) q^{31} +(0.147545 - 5.65493i) q^{32} +(7.67938 - 7.67938i) q^{33} +(-4.45984 - 2.76510i) q^{34} +(3.77791 + 7.06797i) q^{35} +(-11.9841 + 9.24637i) q^{36} +(-1.22495 + 1.00529i) q^{37} +(0.521310 + 0.0334733i) q^{38} +(0.201921 - 1.01513i) q^{39} +(7.04555 - 1.43065i) q^{40} +(-8.45379 + 1.68156i) q^{41} +(-8.80911 + 11.5119i) q^{42} +(-4.50048 + 1.36521i) q^{43} +(-4.58031 - 4.86439i) q^{44} +(-14.8705 - 12.2039i) q^{45} +(-0.481705 + 0.515682i) q^{46} +(1.50649 + 3.63698i) q^{47} +(7.63030 + 10.5295i) q^{48} +(-1.12558 + 2.71739i) q^{49} +(0.855112 + 1.88063i) q^{50} +(12.0044 - 1.18233i) q^{51} +(-0.620506 - 0.142945i) q^{52} +(-5.27790 - 2.82110i) q^{53} +(5.40884 - 20.2938i) q^{54} +(4.71761 - 7.06041i) q^{55} +(6.91617 + 5.63000i) q^{56} +(-0.998441 + 0.667137i) q^{57} +(7.14006 + 3.50931i) q^{58} +(-1.29265 + 13.1245i) q^{59} +(-10.8635 + 12.4540i) q^{60} +(4.30925 - 14.2057i) q^{61} +(0.332710 - 2.03198i) q^{62} -23.8625i q^{63} +(6.61614 - 4.49742i) q^{64} -0.809261i q^{65} +(15.1569 + 2.48175i) q^{66} +(1.60436 - 5.28888i) q^{67} +(-0.505024 - 7.40384i) q^{68} +(0.158997 - 1.61432i) q^{69} +(-4.99936 + 10.1717i) q^{70} +(-7.40781 + 4.94974i) q^{71} +(-20.5091 - 6.13225i) q^{72} +(-4.11433 + 6.15753i) q^{73} +(-2.16545 - 0.577149i) q^{74} +(-4.18820 - 2.23864i) q^{75} +(0.391781 + 0.626320i) q^{76} +(10.4825 - 1.03244i) q^{77} +(1.33246 - 0.605862i) q^{78} +(-1.35491 + 3.27103i) q^{79} +(7.45753 + 6.91074i) q^{80} +(9.78660 + 23.6270i) q^{81} +(-8.90787 - 8.32095i) q^{82} +(10.5477 + 8.65630i) q^{83} +(-20.4907 - 0.616323i) q^{84} +(9.02534 - 2.73781i) q^{85} +(-5.28199 - 4.04188i) q^{86} +(-17.9369 + 3.56787i) q^{87} +(1.80647 - 9.27468i) q^{88} +(-1.18408 + 5.95276i) q^{89} +(1.74326 - 27.1494i) q^{90} +(0.775981 - 0.636831i) q^{91} +(-0.989770 - 0.127632i) q^{92} +(2.23119 + 4.17427i) q^{93} +(-2.93360 + 4.73162i) q^{94} +(-0.663901 + 0.663901i) q^{95} +(-5.79545 + 17.4527i) q^{96} +(-5.22348 - 5.22348i) q^{97} +(-4.04968 + 0.949924i) q^{98} +(-22.2979 + 11.9185i) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.825274 + 1.14844i 0.583557 + 0.812073i
\(3\) −3.11090 − 0.943682i −1.79608 − 0.544835i −0.797771 − 0.602961i \(-0.793988\pi\)
−0.998309 + 0.0581257i \(0.981488\pi\)
\(4\) −0.637847 + 1.89556i −0.318924 + 0.947780i
\(5\) −2.52957 − 0.249141i −1.13126 − 0.111419i −0.484984 − 0.874523i \(-0.661175\pi\)
−0.646276 + 0.763104i \(0.723675\pi\)
\(6\) −1.48358 − 4.35149i −0.605669 − 1.77649i
\(7\) −1.75170 − 2.62161i −0.662081 − 0.990874i −0.998786 − 0.0492639i \(-0.984312\pi\)
0.336705 − 0.941610i \(-0.390688\pi\)
\(8\) −2.70334 + 0.831825i −0.955776 + 0.294094i
\(9\) 6.29277 + 4.20469i 2.09759 + 1.40156i
\(10\) −1.80147 − 3.11068i −0.569674 − 0.983685i
\(11\) −1.57480 + 2.94625i −0.474821 + 0.888328i 0.524497 + 0.851412i \(0.324253\pi\)
−0.999318 + 0.0369155i \(0.988247\pi\)
\(12\) 3.77309 − 5.29498i 1.08920 − 1.52853i
\(13\) 0.0312066 + 0.316846i 0.00865516 + 0.0878773i 0.998542 − 0.0539829i \(-0.0171916\pi\)
−0.989887 + 0.141860i \(0.954692\pi\)
\(14\) 1.56514 − 4.17527i 0.418300 − 1.11589i
\(15\) 7.63415 + 3.16217i 1.97113 + 0.816468i
\(16\) −3.18630 − 2.41816i −0.796576 − 0.604539i
\(17\) −3.42808 + 1.41996i −0.831431 + 0.344390i −0.757469 − 0.652871i \(-0.773564\pi\)
−0.0739617 + 0.997261i \(0.523564\pi\)
\(18\) 0.364401 + 10.6969i 0.0858901 + 2.52129i
\(19\) 0.234333 − 0.285535i 0.0537596 − 0.0655063i −0.745436 − 0.666577i \(-0.767759\pi\)
0.799196 + 0.601070i \(0.205259\pi\)
\(20\) 2.08574 − 4.63605i 0.466387 − 1.03665i
\(21\) 2.97541 + 9.80861i 0.649288 + 2.14041i
\(22\) −4.68325 + 0.622889i −0.998471 + 0.132800i
\(23\) 0.0973467 + 0.489395i 0.0202982 + 0.102046i 0.989606 − 0.143803i \(-0.0459332\pi\)
−0.969308 + 0.245849i \(0.920933\pi\)
\(24\) 9.19482 − 0.0366289i 1.87688 − 0.00747685i
\(25\) 1.43275 + 0.284991i 0.286549 + 0.0569982i
\(26\) −0.338126 + 0.297324i −0.0663120 + 0.0583100i
\(27\) −9.42128 − 11.4799i −1.81313 − 2.20930i
\(28\) 6.08673 − 1.64827i 1.15028 − 0.311494i
\(29\) 4.96138 − 2.65191i 0.921304 − 0.492447i 0.0586712 − 0.998277i \(-0.481314\pi\)
0.862633 + 0.505830i \(0.168814\pi\)
\(30\) 2.66869 + 11.3770i 0.487234 + 2.07715i
\(31\) −1.02952 − 1.02952i −0.184907 − 0.184907i 0.608583 − 0.793490i \(-0.291738\pi\)
−0.793490 + 0.608583i \(0.791738\pi\)
\(32\) 0.147545 − 5.65493i 0.0260826 − 0.999660i
\(33\) 7.67938 − 7.67938i 1.33681 − 1.33681i
\(34\) −4.45984 − 2.76510i −0.764856 − 0.474211i
\(35\) 3.77791 + 7.06797i 0.638583 + 1.19470i
\(36\) −11.9841 + 9.24637i −1.99735 + 1.54106i
\(37\) −1.22495 + 1.00529i −0.201381 + 0.165269i −0.729679 − 0.683790i \(-0.760330\pi\)
0.528298 + 0.849059i \(0.322830\pi\)
\(38\) 0.521310 + 0.0334733i 0.0845677 + 0.00543009i
\(39\) 0.201921 − 1.01513i 0.0323333 − 0.162550i
\(40\) 7.04555 − 1.43065i 1.11400 − 0.226205i
\(41\) −8.45379 + 1.68156i −1.32026 + 0.262616i −0.804416 − 0.594066i \(-0.797522\pi\)
−0.515844 + 0.856683i \(0.672522\pi\)
\(42\) −8.80911 + 11.5119i −1.35928 + 1.77632i
\(43\) −4.50048 + 1.36521i −0.686317 + 0.208192i −0.614137 − 0.789200i \(-0.710496\pi\)
−0.0721800 + 0.997392i \(0.522996\pi\)
\(44\) −4.58031 − 4.86439i −0.690508 − 0.733335i
\(45\) −14.8705 − 12.2039i −2.21676 − 1.81925i
\(46\) −0.481705 + 0.515682i −0.0710235 + 0.0760331i
\(47\) 1.50649 + 3.63698i 0.219744 + 0.530508i 0.994854 − 0.101318i \(-0.0323058\pi\)
−0.775110 + 0.631826i \(0.782306\pi\)
\(48\) 7.63030 + 10.5295i 1.10134 + 1.51980i
\(49\) −1.12558 + 2.71739i −0.160797 + 0.388198i
\(50\) 0.855112 + 1.88063i 0.120931 + 0.265961i
\(51\) 12.0044 − 1.18233i 1.68095 − 0.165559i
\(52\) −0.620506 − 0.142945i −0.0860487 − 0.0198229i
\(53\) −5.27790 − 2.82110i −0.724975 − 0.387507i 0.0672499 − 0.997736i \(-0.478578\pi\)
−0.792225 + 0.610229i \(0.791078\pi\)
\(54\) 5.40884 − 20.2938i 0.736050 − 2.76164i
\(55\) 4.71761 − 7.06041i 0.636123 − 0.952025i
\(56\) 6.91617 + 5.63000i 0.924212 + 0.752340i
\(57\) −0.998441 + 0.667137i −0.132247 + 0.0883645i
\(58\) 7.14006 + 3.50931i 0.937536 + 0.460795i
\(59\) −1.29265 + 13.1245i −0.168289 + 1.70867i 0.426252 + 0.904605i \(0.359834\pi\)
−0.594541 + 0.804065i \(0.702666\pi\)
\(60\) −10.8635 + 12.4540i −1.40247 + 1.60781i
\(61\) 4.30925 − 14.2057i 0.551743 − 1.81885i −0.0219176 − 0.999760i \(-0.506977\pi\)
0.573661 − 0.819093i \(-0.305523\pi\)
\(62\) 0.332710 − 2.03198i 0.0422542 − 0.258061i
\(63\) − 23.8625i − 3.00640i
\(64\) 6.61614 − 4.49742i 0.827017 − 0.562177i
\(65\) − 0.809261i − 0.100376i
\(66\) 15.1569 + 2.48175i 1.86569 + 0.305482i
\(67\) 1.60436 − 5.28888i 0.196004 − 0.646140i −0.802753 − 0.596311i \(-0.796632\pi\)
0.998758 − 0.0498287i \(-0.0158675\pi\)
\(68\) −0.505024 − 7.40384i −0.0612432 − 0.897848i
\(69\) 0.158997 − 1.61432i 0.0191410 − 0.194342i
\(70\) −4.99936 + 10.1717i −0.597538 + 1.21575i
\(71\) −7.40781 + 4.94974i −0.879146 + 0.587426i −0.911156 − 0.412061i \(-0.864809\pi\)
0.0320106 + 0.999488i \(0.489809\pi\)
\(72\) −20.5091 − 6.13225i −2.41702 − 0.722693i
\(73\) −4.11433 + 6.15753i −0.481546 + 0.720685i −0.990103 − 0.140344i \(-0.955179\pi\)
0.508557 + 0.861028i \(0.330179\pi\)
\(74\) −2.16545 − 0.577149i −0.251728 − 0.0670922i
\(75\) −4.18820 − 2.23864i −0.483611 − 0.258496i
\(76\) 0.391781 + 0.626320i 0.0449404 + 0.0718438i
\(77\) 10.4825 − 1.03244i 1.19459 − 0.117657i
\(78\) 1.33246 − 0.605862i 0.150871 − 0.0686003i
\(79\) −1.35491 + 3.27103i −0.152439 + 0.368020i −0.981589 − 0.191006i \(-0.938825\pi\)
0.829150 + 0.559026i \(0.188825\pi\)
\(80\) 7.45753 + 6.91074i 0.833777 + 0.772645i
\(81\) 9.78660 + 23.6270i 1.08740 + 2.62522i
\(82\) −8.90787 − 8.32095i −0.983710 − 0.918896i
\(83\) 10.5477 + 8.65630i 1.15776 + 0.950152i 0.999307 − 0.0372140i \(-0.0118483\pi\)
0.158456 + 0.987366i \(0.449348\pi\)
\(84\) −20.4907 − 0.616323i −2.23572 − 0.0672464i
\(85\) 9.02534 − 2.73781i 0.978936 − 0.296957i
\(86\) −5.28199 − 4.04188i −0.569571 − 0.435847i
\(87\) −17.9369 + 3.56787i −1.92304 + 0.382516i
\(88\) 1.80647 − 9.27468i 0.192570 − 0.988685i
\(89\) −1.18408 + 5.95276i −0.125512 + 0.630992i 0.865900 + 0.500217i \(0.166746\pi\)
−0.991412 + 0.130775i \(0.958254\pi\)
\(90\) 1.74326 − 27.1494i 0.183756 − 2.86180i
\(91\) 0.775981 − 0.636831i 0.0813449 − 0.0667581i
\(92\) −0.989770 − 0.127632i −0.103191 − 0.0133066i
\(93\) 2.23119 + 4.17427i 0.231364 + 0.432851i
\(94\) −2.93360 + 4.73162i −0.302578 + 0.488029i
\(95\) −0.663901 + 0.663901i −0.0681148 + 0.0681148i
\(96\) −5.79545 + 17.4527i −0.591496 + 1.78126i
\(97\) −5.22348 − 5.22348i −0.530364 − 0.530364i 0.390316 − 0.920681i \(-0.372366\pi\)
−0.920681 + 0.390316i \(0.872366\pi\)
\(98\) −4.04968 + 0.949924i −0.409079 + 0.0959568i
\(99\) −22.2979 + 11.9185i −2.24103 + 1.19785i
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.5.12 ✓ 240
4.3 odd 2 512.2.k.a.113.15 240
128.51 odd 32 512.2.k.a.145.15 240
128.77 even 32 inner 128.2.k.a.77.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.5.12 ✓ 240 1.1 even 1 trivial
128.2.k.a.77.12 yes 240 128.77 even 32 inner
512.2.k.a.113.15 240 4.3 odd 2
512.2.k.a.145.15 240 128.51 odd 32