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Properties

Label 128.2.k.a.101.10
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 101.10
Character \(\chi\) \(=\) 128.101
Dual form 128.2.k.a.109.10

$q$-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+(0.603519 + 1.27897i) q^{2} +(1.98529 - 1.06116i) q^{3} +(-1.27153 + 1.54377i) q^{4} +(-0.212591 - 0.259043i) q^{5} +(2.55536 + 1.89870i) q^{6} +(-0.792164 + 0.529307i) q^{7} +(-2.74182 - 0.694555i) q^{8} +(1.14862 - 1.71903i) q^{9} +(0.203005 - 0.428235i) q^{10} +(2.34047 - 0.709975i) q^{11} +(-0.886173 + 4.41413i) q^{12} +(-2.81226 - 2.30796i) q^{13} +(-1.15505 - 0.693707i) q^{14} +(-0.696942 - 0.288683i) q^{15} +(-0.766429 - 3.92589i) q^{16} +(-1.82571 + 0.756236i) q^{17} +(2.89180 + 0.431581i) q^{18} +(0.157890 + 1.60309i) q^{19} +(0.670217 + 0.00118970i) q^{20} +(-1.01100 + 1.89144i) q^{21} +(2.32056 + 2.56491i) q^{22} +(-7.27219 + 1.44653i) q^{23} +(-6.18036 + 1.53062i) q^{24} +(0.953543 - 4.79379i) q^{25} +(1.25456 - 4.98969i) q^{26} +(-0.205763 + 2.08914i) q^{27} +(0.190133 - 1.89595i) q^{28} +(1.10598 - 3.64593i) q^{29} +(-0.0514012 - 1.06559i) q^{30} +(7.16783 + 7.16783i) q^{31} +(4.55854 - 3.34959i) q^{32} +(3.89313 - 3.89313i) q^{33} +(-2.06906 - 1.87863i) q^{34} +(0.305520 + 0.0926785i) q^{35} +(1.19328 + 3.95899i) q^{36} +(0.968069 + 0.0953465i) q^{37} +(-1.95501 + 1.16943i) q^{38} +(-8.03228 - 1.59772i) q^{39} +(0.402967 + 0.857906i) q^{40} +(-2.34900 - 11.8092i) q^{41} +(-3.02926 - 0.151513i) q^{42} +(7.65676 + 4.09262i) q^{43} +(-1.87995 + 4.51590i) q^{44} +(-0.689487 + 0.0679086i) q^{45} +(-6.23897 - 8.42790i) q^{46} +(-1.74223 - 4.20612i) q^{47} +(-5.68759 - 6.98073i) q^{48} +(-2.33143 + 5.62856i) q^{49} +(6.70659 - 1.67359i) q^{50} +(-2.82209 + 3.43873i) q^{51} +(7.13882 - 1.40683i) q^{52} +(2.50640 + 8.26251i) q^{53} +(-2.79613 + 0.997674i) q^{54} +(-0.681478 - 0.455349i) q^{55} +(2.53961 - 0.901065i) q^{56} +(2.01459 + 3.01505i) q^{57} +(5.33052 - 0.785873i) q^{58} +(-2.07085 + 1.69950i) q^{59} +(1.33184 - 0.708847i) q^{60} +(3.63666 + 6.80371i) q^{61} +(-4.84152 + 13.4934i) q^{62} +1.96972i q^{63} +(7.03519 + 3.80869i) q^{64} +1.21915i q^{65} +(7.32878 + 2.62962i) q^{66} +(5.45136 + 10.1988i) q^{67} +(1.15400 - 3.78005i) q^{68} +(-12.9024 + 10.5887i) q^{69} +(0.0658543 + 0.446685i) q^{70} +(1.79285 + 2.68319i) q^{71} +(-4.34326 + 3.91549i) q^{72} +(2.04993 + 1.36972i) q^{73} +(0.462303 + 1.29568i) q^{74} +(-3.19392 - 10.5289i) q^{75} +(-2.67555 - 1.79462i) q^{76} +(-1.47825 + 1.80125i) q^{77} +(-2.80420 - 11.2373i) q^{78} +(4.58780 - 11.0759i) q^{79} +(-0.854037 + 1.03315i) q^{80} +(4.18196 + 10.0961i) q^{81} +(13.6860 - 10.1314i) q^{82} +(-7.11006 + 0.700280i) q^{83} +(-1.63443 - 3.96577i) q^{84} +(0.584028 + 0.312169i) q^{85} +(-0.613340 + 12.2627i) q^{86} +(-1.67322 - 8.41187i) q^{87} +(-6.91028 + 0.321039i) q^{88} +(-13.6682 - 2.71877i) q^{89} +(-0.502972 - 0.840849i) q^{90} +(3.44939 + 0.339735i) q^{91} +(7.01369 - 13.0659i) q^{92} +(21.8365 + 6.62402i) q^{93} +(4.32803 - 4.76673i) q^{94} +(0.381702 - 0.381702i) q^{95} +(5.49558 - 11.4873i) q^{96} +(-10.5511 - 10.5511i) q^{97} +(-8.60582 + 0.415121i) q^{98} +(1.46784 - 4.83883i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.603519 + 1.27897i 0.426753 + 0.904368i
\(3\) 1.98529 − 1.06116i 1.14621 − 0.612662i 0.214809 − 0.976656i \(-0.431087\pi\)
0.931401 + 0.363994i \(0.118587\pi\)
\(4\) −1.27153 + 1.54377i −0.635764 + 0.771883i
\(5\) −0.212591 − 0.259043i −0.0950736 − 0.115847i 0.723300 − 0.690533i \(-0.242624\pi\)
−0.818374 + 0.574686i \(0.805124\pi\)
\(6\) 2.55536 + 1.89870i 1.04322 + 0.775141i
\(7\) −0.792164 + 0.529307i −0.299410 + 0.200059i −0.696196 − 0.717852i \(-0.745126\pi\)
0.396786 + 0.917911i \(0.370126\pi\)
\(8\) −2.74182 − 0.694555i −0.969381 − 0.245562i
\(9\) 1.14862 − 1.71903i 0.382872 − 0.573009i
\(10\) 0.203005 − 0.428235i 0.0641959 − 0.135420i
\(11\) 2.34047 − 0.709975i 0.705680 − 0.214066i 0.0830081 − 0.996549i \(-0.473547\pi\)
0.622672 + 0.782483i \(0.286047\pi\)
\(12\) −0.886173 + 4.41413i −0.255816 + 1.27425i
\(13\) −2.81226 − 2.30796i −0.779980 − 0.640113i 0.157750 − 0.987479i \(-0.449576\pi\)
−0.937730 + 0.347366i \(0.887076\pi\)
\(14\) −1.15505 − 0.693707i −0.308701 − 0.185401i
\(15\) −0.696942 − 0.288683i −0.179950 − 0.0745376i
\(16\) −0.766429 − 3.92589i −0.191607 − 0.981472i
\(17\) −1.82571 + 0.756236i −0.442801 + 0.183414i −0.592933 − 0.805252i \(-0.702030\pi\)
0.150132 + 0.988666i \(0.452030\pi\)
\(18\) 2.89180 + 0.431581i 0.681603 + 0.101725i
\(19\) 0.157890 + 1.60309i 0.0362225 + 0.367773i 0.996159 + 0.0875572i \(0.0279061\pi\)
−0.959937 + 0.280216i \(0.909594\pi\)
\(20\) 0.670217 + 0.00118970i 0.149865 + 0.000266025i
\(21\) −1.01100 + 1.89144i −0.220618 + 0.412747i
\(22\) 2.32056 + 2.56491i 0.494745 + 0.546841i
\(23\) −7.27219 + 1.44653i −1.51636 + 0.301622i −0.881937 − 0.471368i \(-0.843760\pi\)
−0.634419 + 0.772990i \(0.718760\pi\)
\(24\) −6.18036 + 1.53062i −1.26156 + 0.312437i
\(25\) 0.953543 − 4.79379i 0.190709 − 0.958757i
\(26\) 1.25456 − 4.98969i 0.246040 − 0.978559i
\(27\) −0.205763 + 2.08914i −0.0395990 + 0.402056i
\(28\) 0.190133 − 1.89595i 0.0359318 − 0.358300i
\(29\) 1.10598 − 3.64593i 0.205376 − 0.677032i −0.792377 − 0.610031i \(-0.791157\pi\)
0.997753 − 0.0670011i \(-0.0213431\pi\)
\(30\) −0.0514012 − 1.06559i −0.00938454 − 0.194550i
\(31\) 7.16783 + 7.16783i 1.28738 + 1.28738i 0.936372 + 0.351008i \(0.114161\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(32\) 4.55854 − 3.34959i 0.805843 − 0.592129i
\(33\) 3.89313 − 3.89313i 0.677707 − 0.677707i
\(34\) −2.06906 − 1.87863i −0.354840 − 0.322183i
\(35\) 0.305520 + 0.0926785i 0.0516423 + 0.0156655i
\(36\) 1.19328 + 3.95899i 0.198879 + 0.659831i
\(37\) 0.968069 + 0.0953465i 0.159150 + 0.0156749i 0.177278 − 0.984161i \(-0.443271\pi\)
−0.0181285 + 0.999836i \(0.505771\pi\)
\(38\) −1.95501 + 1.16943i −0.317144 + 0.189707i
\(39\) −8.03228 − 1.59772i −1.28619 − 0.255840i
\(40\) 0.402967 + 0.857906i 0.0637147 + 0.135647i
\(41\) −2.34900 − 11.8092i −0.366852 − 1.84429i −0.517473 − 0.855699i \(-0.673127\pi\)
0.150621 − 0.988592i \(-0.451873\pi\)
\(42\) −3.02926 − 0.151513i −0.467425 − 0.0233790i
\(43\) 7.65676 + 4.09262i 1.16764 + 0.624119i 0.937141 − 0.348952i \(-0.113462\pi\)
0.230504 + 0.973071i \(0.425962\pi\)
\(44\) −1.87995 + 4.51590i −0.283412 + 0.680798i
\(45\) −0.689487 + 0.0679086i −0.102783 + 0.0101232i
\(46\) −6.23897 − 8.42790i −0.919886 − 1.24263i
\(47\) −1.74223 − 4.20612i −0.254130 − 0.613525i 0.744399 − 0.667735i \(-0.232736\pi\)
−0.998530 + 0.0542099i \(0.982736\pi\)
\(48\) −5.68759 − 6.98073i −0.820932 − 1.00758i
\(49\) −2.33143 + 5.62856i −0.333061 + 0.804080i
\(50\) 6.70659 − 1.67359i 0.948455 − 0.236681i
\(51\) −2.82209 + 3.43873i −0.395172 + 0.481518i
\(52\) 7.13882 − 1.40683i 0.989976 − 0.195092i
\(53\) 2.50640 + 8.26251i 0.344281 + 1.13494i 0.942667 + 0.333735i \(0.108309\pi\)
−0.598386 + 0.801208i \(0.704191\pi\)
\(54\) −2.79613 + 0.997674i −0.380505 + 0.135766i
\(55\) −0.681478 − 0.455349i −0.0918905 − 0.0613992i
\(56\) 2.53961 − 0.901065i 0.339369 − 0.120410i
\(57\) 2.01459 + 3.01505i 0.266839 + 0.399353i
\(58\) 5.33052 − 0.785873i 0.699931 − 0.103190i
\(59\) −2.07085 + 1.69950i −0.269601 + 0.221256i −0.759474 − 0.650538i \(-0.774543\pi\)
0.489872 + 0.871794i \(0.337043\pi\)
\(60\) 1.33184 − 0.708847i 0.171940 − 0.0915117i
\(61\) 3.63666 + 6.80371i 0.465626 + 0.871125i 0.999670 + 0.0256689i \(0.00817157\pi\)
−0.534044 + 0.845457i \(0.679328\pi\)
\(62\) −4.84152 + 13.4934i −0.614873 + 1.71366i
\(63\) 1.96972i 0.248162i
\(64\) 7.03519 + 3.80869i 0.879398 + 0.476087i
\(65\) 1.21915i 0.151217i
\(66\) 7.32878 + 2.62962i 0.902110 + 0.323684i
\(67\) 5.45136 + 10.1988i 0.665990 + 1.24598i 0.957230 + 0.289329i \(0.0934322\pi\)
−0.291240 + 0.956650i \(0.594068\pi\)
\(68\) 1.15400 − 3.78005i 0.139943 − 0.458399i
\(69\) −12.9024 + 10.5887i −1.55327 + 1.27474i
\(70\) 0.0658543 + 0.446685i 0.00787109 + 0.0533890i
\(71\) 1.79285 + 2.68319i 0.212772 + 0.318436i 0.922469 − 0.386071i \(-0.126168\pi\)
−0.709697 + 0.704507i \(0.751168\pi\)
\(72\) −4.34326 + 3.91549i −0.511858 + 0.461445i
\(73\) 2.04993 + 1.36972i 0.239926 + 0.160314i 0.669721 − 0.742613i \(-0.266414\pi\)
−0.429794 + 0.902927i \(0.641414\pi\)
\(74\) 0.462303 + 1.29568i 0.0537417 + 0.150619i
\(75\) −3.19392 − 10.5289i −0.368802 − 1.21578i
\(76\) −2.67555 − 1.79462i −0.306907 − 0.205858i
\(77\) −1.47825 + 1.80125i −0.168462 + 0.205271i
\(78\) −2.80420 − 11.2373i −0.317513 − 1.27237i
\(79\) 4.58780 − 11.0759i 0.516168 − 1.24614i −0.424072 − 0.905628i \(-0.639400\pi\)
0.940240 − 0.340512i \(-0.110600\pi\)
\(80\) −0.854037 + 1.03315i −0.0954843 + 0.115509i
\(81\) 4.18196 + 10.0961i 0.464662 + 1.12179i
\(82\) 13.6860 − 10.1314i 1.51136 − 1.11883i
\(83\) −7.11006 + 0.700280i −0.780431 + 0.0768657i −0.480384 − 0.877059i \(-0.659503\pi\)
−0.300047 + 0.953924i \(0.597003\pi\)
\(84\) −1.63443 − 3.96577i −0.178331 − 0.432701i
\(85\) 0.584028 + 0.312169i 0.0633467 + 0.0338595i
\(86\) −0.613340 + 12.2627i −0.0661382 + 1.32233i
\(87\) −1.67322 − 8.41187i −0.179389 − 0.901847i
\(88\) −6.91028 + 0.321039i −0.736639 + 0.0342229i
\(89\) −13.6682 − 2.71877i −1.44882 − 0.288189i −0.592893 − 0.805281i \(-0.702014\pi\)
−0.855930 + 0.517092i \(0.827014\pi\)
\(90\) −0.502972 − 0.840849i −0.0530179 − 0.0886333i
\(91\) 3.44939 + 0.339735i 0.361594 + 0.0356139i
\(92\) 7.01369 − 13.0659i 0.731228 − 1.36221i
\(93\) 21.8365 + 6.62402i 2.26434 + 0.686879i
\(94\) 4.32803 − 4.76673i 0.446402 − 0.491651i
\(95\) 0.381702 − 0.381702i 0.0391618 − 0.0391618i
\(96\) 5.49558 − 11.4873i 0.560891 − 1.17241i
\(97\) −10.5511 − 10.5511i −1.07131 − 1.07131i −0.997254 − 0.0740512i \(-0.976407\pi\)
−0.0740512 − 0.997254i \(-0.523593\pi\)
\(98\) −8.60582 + 0.415121i −0.869319 + 0.0419335i
\(99\) 1.46784 − 4.83883i 0.147524 − 0.486321i
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.101.10 ✓ 240
4.3 odd 2 512.2.k.a.497.3 240
128.19 odd 32 512.2.k.a.273.3 240
128.109 even 32 inner 128.2.k.a.109.10 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.10 ✓ 240 1.1 even 1 trivial
128.2.k.a.109.10 yes 240 128.109 even 32 inner
512.2.k.a.273.3 240 128.19 odd 32
512.2.k.a.497.3 240 4.3 odd 2