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Properties

Label 128.2.k.a.5.10
Level $128$
Weight $2$
Character 128.5
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 5.10
Character \(\chi\) \(=\) 128.5
Dual form 128.2.k.a.77.10

$q$-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+(0.441120 + 1.34366i) q^{2} +(-0.451169 - 0.136861i) q^{3} +(-1.61083 + 1.18543i) q^{4} +(2.96377 + 0.291905i) q^{5} +(-0.0151258 - 0.666588i) q^{6} +(1.40525 + 2.10310i) q^{7} +(-2.30337 - 1.64148i) q^{8} +(-2.30959 - 1.54322i) q^{9} +(0.915154 + 4.11105i) q^{10} +(-2.22364 + 4.16014i) q^{11} +(0.888994 - 0.314369i) q^{12} +(-0.256312 - 2.60238i) q^{13} +(-2.20597 + 2.81589i) q^{14} +(-1.29721 - 0.537322i) q^{15} +(1.18953 - 3.81903i) q^{16} +(5.15626 - 2.13579i) q^{17} +(1.05475 - 3.78403i) q^{18} +(2.53572 - 3.08978i) q^{19} +(-5.12015 + 3.04312i) q^{20} +(-0.346173 - 1.14118i) q^{21} +(-6.57068 - 1.15269i) q^{22} +(-1.30896 - 6.58060i) q^{23} +(0.814557 + 1.05583i) q^{24} +(3.79477 + 0.754828i) q^{25} +(3.38364 - 1.49235i) q^{26} +(1.72810 + 2.10570i) q^{27} +(-4.75669 - 1.72192i) q^{28} +(-1.07594 + 0.575103i) q^{29} +(0.149752 - 1.98003i) q^{30} +(-3.23187 - 3.23187i) q^{31} +(5.65620 - 0.0863334i) q^{32} +(1.57260 - 1.57260i) q^{33} +(5.14430 + 5.98610i) q^{34} +(3.55092 + 6.64331i) q^{35} +(5.54971 - 0.251990i) q^{36} +(-9.01766 + 7.40060i) q^{37} +(5.27016 + 2.04417i) q^{38} +(-0.240523 + 1.20919i) q^{39} +(-6.34750 - 5.53734i) q^{40} +(6.54760 - 1.30240i) q^{41} +(1.38065 - 0.968534i) q^{42} +(-7.37982 + 2.23864i) q^{43} +(-1.34964 - 9.33722i) q^{44} +(-6.39460 - 5.24791i) q^{45} +(8.26466 - 4.66163i) q^{46} +(-0.785850 - 1.89721i) q^{47} +(-1.05935 + 1.56023i) q^{48} +(0.230462 - 0.556385i) q^{49} +(0.659720 + 5.43184i) q^{50} +(-2.61865 + 0.257915i) q^{51} +(3.49780 + 3.88814i) q^{52} +(2.47545 + 1.32316i) q^{53} +(-2.06704 + 3.25084i) q^{54} +(-7.80471 + 11.6806i) q^{55} +(0.215398 - 7.15093i) q^{56} +(-1.56691 + 1.04697i) q^{57} +(-1.24736 - 1.19201i) q^{58} +(-0.126327 + 1.28262i) q^{59} +(2.72654 - 0.672214i) q^{60} +(-0.403759 + 1.33102i) q^{61} +(2.91689 - 5.76817i) q^{62} -7.02590i q^{63} +(2.61106 + 7.56190i) q^{64} -7.78765i q^{65} +(2.80673 + 1.41933i) q^{66} +(-1.81960 + 5.99841i) q^{67} +(-5.77401 + 9.55276i) q^{68} +(-0.310062 + 3.14811i) q^{69} +(-7.35995 + 7.70172i) q^{70} +(5.49906 - 3.67435i) q^{71} +(2.78668 + 7.34575i) q^{72} +(-5.68527 + 8.50861i) q^{73} +(-13.9217 - 8.85209i) q^{74} +(-1.60878 - 0.859910i) q^{75} +(-0.421895 + 7.98301i) q^{76} +(-11.8740 + 1.16948i) q^{77} +(-1.73084 + 0.210217i) q^{78} +(1.46762 - 3.54315i) q^{79} +(4.64028 - 10.9715i) q^{80} +(2.69748 + 6.51229i) q^{81} +(4.63825 + 8.22322i) q^{82} +(8.88716 + 7.29350i) q^{83} +(1.91041 + 1.42788i) q^{84} +(15.9054 - 4.82485i) q^{85} +(-6.26335 - 8.92843i) q^{86} +(0.564141 - 0.112215i) q^{87} +(11.9507 - 5.93228i) q^{88} +(-2.74125 + 13.7812i) q^{89} +(4.23061 - 10.9071i) q^{90} +(5.11289 - 4.19604i) q^{91} +(9.90933 + 9.04853i) q^{92} +(1.01581 + 1.90044i) q^{93} +(2.20255 - 1.89281i) q^{94} +(8.41720 - 8.41720i) q^{95} +(-2.56372 - 0.735160i) q^{96} +(-7.14444 - 7.14444i) q^{97} +(0.849252 + 0.0642298i) q^{98} +(11.5557 - 6.17664i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.441120 + 1.34366i 0.311919 + 0.950109i
\(3\) −0.451169 − 0.136861i −0.260483 − 0.0790165i 0.157342 − 0.987544i \(-0.449707\pi\)
−0.417825 + 0.908528i \(0.637207\pi\)
\(4\) −1.61083 + 1.18543i −0.805414 + 0.592713i
\(5\) 2.96377 + 0.291905i 1.32544 + 0.130544i 0.735841 − 0.677154i \(-0.236787\pi\)
0.589596 + 0.807699i \(0.299287\pi\)
\(6\) −0.0151258 − 0.666588i −0.00617507 − 0.272134i
\(7\) 1.40525 + 2.10310i 0.531134 + 0.794899i 0.995893 − 0.0905387i \(-0.0288589\pi\)
−0.464759 + 0.885437i \(0.653859\pi\)
\(8\) −2.30337 − 1.64148i −0.814366 − 0.580352i
\(9\) −2.30959 − 1.54322i −0.769862 − 0.514405i
\(10\) 0.915154 + 4.11105i 0.289397 + 1.30003i
\(11\) −2.22364 + 4.16014i −0.670452 + 1.25433i 0.284815 + 0.958583i \(0.408068\pi\)
−0.955267 + 0.295745i \(0.904432\pi\)
\(12\) 0.888994 − 0.314369i 0.256630 − 0.0907505i
\(13\) −0.256312 − 2.60238i −0.0710881 − 0.721769i −0.962989 − 0.269539i \(-0.913129\pi\)
0.891901 − 0.452230i \(-0.149371\pi\)
\(14\) −2.20597 + 2.81589i −0.589570 + 0.752579i
\(15\) −1.29721 − 0.537322i −0.334938 − 0.138736i
\(16\) 1.18953 − 3.81903i 0.297382 − 0.954759i
\(17\) 5.15626 − 2.13579i 1.25058 − 0.518006i 0.343572 − 0.939126i \(-0.388363\pi\)
0.907004 + 0.421121i \(0.138363\pi\)
\(18\) 1.05475 − 3.78403i 0.248607 − 0.891905i
\(19\) 2.53572 − 3.08978i 0.581734 − 0.708845i −0.396105 − 0.918205i \(-0.629638\pi\)
0.977838 + 0.209361i \(0.0671383\pi\)
\(20\) −5.12015 + 3.04312i −1.14490 + 0.680462i
\(21\) −0.346173 − 1.14118i −0.0755411 − 0.249026i
\(22\) −6.57068 − 1.15269i −1.40087 − 0.245754i
\(23\) −1.30896 − 6.58060i −0.272938 − 1.37215i −0.837355 − 0.546660i \(-0.815899\pi\)
0.564417 − 0.825490i \(-0.309101\pi\)
\(24\) 0.814557 + 1.05583i 0.166271 + 0.215520i
\(25\) 3.79477 + 0.754828i 0.758955 + 0.150966i
\(26\) 3.38364 − 1.49235i 0.663586 − 0.292675i
\(27\) 1.72810 + 2.10570i 0.332573 + 0.405242i
\(28\) −4.75669 − 1.72192i −0.898930 − 0.325412i
\(29\) −1.07594 + 0.575103i −0.199798 + 0.106794i −0.568262 − 0.822848i \(-0.692384\pi\)
0.368464 + 0.929642i \(0.379884\pi\)
\(30\) 0.149752 − 1.98003i 0.0273408 − 0.361502i
\(31\) −3.23187 − 3.23187i −0.580462 − 0.580462i 0.354568 − 0.935030i \(-0.384628\pi\)
−0.935030 + 0.354568i \(0.884628\pi\)
\(32\) 5.65620 − 0.0863334i 0.999884 − 0.0152617i
\(33\) 1.57260 − 1.57260i 0.273754 − 0.273754i
\(34\) 5.14430 + 5.98610i 0.882240 + 1.02661i
\(35\) 3.55092 + 6.64331i 0.600216 + 1.12292i
\(36\) 5.54971 − 0.251990i 0.924952 − 0.0419984i
\(37\) −9.01766 + 7.40060i −1.48249 + 1.21665i −0.561261 + 0.827639i \(0.689684\pi\)
−0.921233 + 0.389012i \(0.872816\pi\)
\(38\) 5.27016 + 2.04417i 0.854933 + 0.331609i
\(39\) −0.240523 + 1.20919i −0.0385145 + 0.193625i
\(40\) −6.34750 − 5.53734i −1.00363 − 0.875531i
\(41\) 6.54760 − 1.30240i 1.02256 − 0.203401i 0.344797 − 0.938677i \(-0.387948\pi\)
0.677767 + 0.735277i \(0.262948\pi\)
\(42\) 1.38065 − 0.968534i 0.213039 − 0.149448i
\(43\) −7.37982 + 2.23864i −1.12541 + 0.341390i −0.797500 − 0.603319i \(-0.793845\pi\)
−0.327911 + 0.944709i \(0.606345\pi\)
\(44\) −1.34964 − 9.33722i −0.203465 − 1.40764i
\(45\) −6.39460 − 5.24791i −0.953251 − 0.782313i
\(46\) 8.26466 − 4.66163i 1.21856 − 0.687320i
\(47\) −0.785850 − 1.89721i −0.114628 − 0.276737i 0.856145 − 0.516735i \(-0.172853\pi\)
−0.970773 + 0.239999i \(0.922853\pi\)
\(48\) −1.05935 + 1.56023i −0.152905 + 0.225200i
\(49\) 0.230462 − 0.556385i 0.0329232 − 0.0794836i
\(50\) 0.659720 + 5.43184i 0.0932985 + 0.768179i
\(51\) −2.61865 + 0.257915i −0.366684 + 0.0361153i
\(52\) 3.49780 + 3.88814i 0.485057 + 0.539188i
\(53\) 2.47545 + 1.32316i 0.340029 + 0.181749i 0.632569 − 0.774504i \(-0.282000\pi\)
−0.292540 + 0.956253i \(0.594500\pi\)
\(54\) −2.06704 + 3.25084i −0.281288 + 0.442383i
\(55\) −7.80471 + 11.6806i −1.05239 + 1.57501i
\(56\) 0.215398 − 7.15093i 0.0287837 − 0.955583i
\(57\) −1.56691 + 1.04697i −0.207542 + 0.138675i
\(58\) −1.24736 − 1.19201i −0.163787 − 0.156518i
\(59\) −0.126327 + 1.28262i −0.0164463 + 0.166982i −0.999899 − 0.0141804i \(-0.995486\pi\)
0.983453 + 0.181163i \(0.0579861\pi\)
\(60\) 2.72654 − 0.672214i 0.351994 − 0.0867825i
\(61\) −0.403759 + 1.33102i −0.0516961 + 0.170419i −0.978987 − 0.203922i \(-0.934631\pi\)
0.927291 + 0.374341i \(0.122131\pi\)
\(62\) 2.91689 − 5.76817i 0.370445 − 0.732559i
\(63\) − 7.02590i − 0.885181i
\(64\) 2.61106 + 7.56190i 0.326383 + 0.945238i
\(65\) − 7.78765i − 0.965939i
\(66\) 2.80673 + 1.41933i 0.345485 + 0.174707i
\(67\) −1.81960 + 5.99841i −0.222299 + 0.732822i 0.772863 + 0.634573i \(0.218824\pi\)
−0.995162 + 0.0982491i \(0.968676\pi\)
\(68\) −5.77401 + 9.55276i −0.700202 + 1.15844i
\(69\) −0.310062 + 3.14811i −0.0373270 + 0.378988i
\(70\) −7.35995 + 7.70172i −0.879682 + 0.920531i
\(71\) 5.49906 − 3.67435i 0.652618 − 0.436065i −0.184690 − 0.982797i \(-0.559128\pi\)
0.837308 + 0.546731i \(0.184128\pi\)
\(72\) 2.78668 + 7.34575i 0.328413 + 0.865705i
\(73\) −5.68527 + 8.50861i −0.665410 + 0.995857i 0.333184 + 0.942862i \(0.391877\pi\)
−0.998595 + 0.0529953i \(0.983123\pi\)
\(74\) −13.9217 − 8.85209i −1.61837 − 1.02903i
\(75\) −1.60878 − 0.859910i −0.185766 − 0.0992939i
\(76\) −0.421895 + 7.98301i −0.0483946 + 0.915714i
\(77\) −11.8740 + 1.16948i −1.35316 + 0.133275i
\(78\) −1.73084 + 0.210217i −0.195979 + 0.0238024i
\(79\) 1.46762 − 3.54315i 0.165120 − 0.398635i −0.819563 − 0.572989i \(-0.805784\pi\)
0.984683 + 0.174354i \(0.0557837\pi\)
\(80\) 4.64028 − 10.9715i 0.518799 − 1.22665i
\(81\) 2.69748 + 6.51229i 0.299720 + 0.723588i
\(82\) 4.63825 + 8.22322i 0.512209 + 0.908102i
\(83\) 8.88716 + 7.29350i 0.975492 + 0.800566i 0.979913 − 0.199424i \(-0.0639072\pi\)
−0.00442091 + 0.999990i \(0.501407\pi\)
\(84\) 1.91041 + 1.42788i 0.208443 + 0.155794i
\(85\) 15.9054 − 4.82485i 1.72518 − 0.523328i
\(86\) −6.26335 − 8.92843i −0.675394 − 0.962778i
\(87\) 0.564141 − 0.112215i 0.0604823 − 0.0120307i
\(88\) 11.9507 − 5.93228i 1.27395 − 0.632383i
\(89\) −2.74125 + 13.7812i −0.290572 + 1.46080i 0.509279 + 0.860602i \(0.329912\pi\)
−0.799850 + 0.600200i \(0.795088\pi\)
\(90\) 4.23061 − 10.9071i 0.445945 − 1.14971i
\(91\) 5.11289 − 4.19604i 0.535976 − 0.439864i
\(92\) 9.90933 + 9.04853i 1.03312 + 0.943374i
\(93\) 1.01581 + 1.90044i 0.105334 + 0.197066i
\(94\) 2.20255 − 1.89281i 0.227175 − 0.195228i
\(95\) 8.41720 − 8.41720i 0.863587 − 0.863587i
\(96\) −2.56372 − 0.735160i −0.261658 − 0.0750319i
\(97\) −7.14444 − 7.14444i −0.725408 − 0.725408i 0.244293 − 0.969701i \(-0.421444\pi\)
−0.969701 + 0.244293i \(0.921444\pi\)
\(98\) 0.849252 + 0.0642298i 0.0857874 + 0.00648819i
\(99\) 11.5557 − 6.17664i 1.16139 − 0.620775i
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.5.10 ✓ 240
4.3 odd 2 512.2.k.a.113.9 240
128.51 odd 32 512.2.k.a.145.9 240
128.77 even 32 inner 128.2.k.a.77.10 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.5.10 ✓ 240 1.1 even 1 trivial
128.2.k.a.77.10 yes 240 128.77 even 32 inner
512.2.k.a.113.9 240 4.3 odd 2
512.2.k.a.145.9 240 128.51 odd 32