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Properties

Label 128.2.k.a.109.10
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 109.10
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.10

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.603519 − 1.27897i 0.426753 − 0.904368i
\(3\) 1.98529 + 1.06116i 1.14621 + 0.612662i 0.931401 − 0.363994i \(-0.118587\pi\)
0.214809 + 0.976656i \(0.431087\pi\)
\(4\) −1.27153 − 1.54377i −0.635764 − 0.771883i
\(5\) −0.212591 + 0.259043i −0.0950736 + 0.115847i −0.818374 − 0.574686i \(-0.805124\pi\)
0.723300 + 0.690533i \(0.242624\pi\)
\(6\) 2.55536 − 1.89870i 1.04322 − 0.775141i
\(7\) −0.792164 − 0.529307i −0.299410 − 0.200059i 0.396786 − 0.917911i \(-0.370126\pi\)
−0.696196 + 0.717852i \(0.745126\pi\)
\(8\) −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 0.622672 − 0.782483i \(-0.286047\pi\)
0.0830081 + 0.996549i \(0.473547\pi\)
\(12\) −0.886173 − 4.41413i −0.255816 − 1.27425i
\(13\) −2.81226 + 2.30796i −0.779980 + 0.640113i −0.937730 − 0.347366i \(-0.887076\pi\)
0.157750 + 0.987479i \(0.449576\pi\)
\(14\) −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 0.150132 − 0.988666i \(-0.452030\pi\)
−0.592933 + 0.805252i \(0.702030\pi\)
\(18\) 2.89180 − 0.431581i 0.681603 − 0.101725i
\(19\) 0.157890 − 1.60309i 0.0362225 − 0.367773i −0.959937 − 0.280216i \(-0.909594\pi\)
0.996159 − 0.0875572i \(-0.0279061\pi\)
\(20\) 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 −0.634419 − 0.772990i \(-0.718760\pi\)
−0.881937 + 0.471368i \(0.843760\pi\)
\(24\) −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 0.997753 + 0.0670011i \(0.0213431\pi\)
−0.792377 + 0.610031i \(0.791157\pi\)
\(30\) −0.0514012 + 1.06559i −0.00938454 + 0.194550i
\(31\) 7.16783 − 7.16783i 1.28738 − 1.28738i 0.351008 − 0.936372i \(-0.385839\pi\)
0.936372 − 0.351008i \(-0.114161\pi\)
\(32\) 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 −0.0181285 − 0.999836i \(-0.505771\pi\)
0.177278 + 0.984161i \(0.443271\pi\)
\(38\) −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 0.150621 + 0.988592i \(0.451873\pi\)
−0.517473 + 0.855699i \(0.673127\pi\)
\(42\) −3.02926 + 0.151513i −0.467425 + 0.0233790i
\(43\) 7.65676 − 4.09262i 1.16764 − 0.624119i 0.230504 − 0.973071i \(-0.425962\pi\)
0.937141 + 0.348952i \(0.113462\pi\)
\(44\) −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 −0.998530 − 0.0542099i \(-0.982736\pi\)
0.744399 + 0.667735i \(0.232736\pi\)
\(48\) −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 −0.598386 − 0.801208i \(-0.704191\pi\)
0.942667 − 0.333735i \(-0.108309\pi\)
\(54\) −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 0.489872 − 0.871794i \(-0.337043\pi\)
−0.759474 + 0.650538i \(0.774543\pi\)
\(60\) 1.33184 + 0.708847i 0.171940 + 0.0915117i
\(61\) 3.63666 − 6.80371i 0.465626 − 0.871125i −0.534044 − 0.845457i \(-0.679328\pi\)
0.999670 − 0.0256689i \(-0.00817157\pi\)
\(62\) −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 −0.291240 − 0.956650i \(-0.594068\pi\)
0.957230 − 0.289329i \(-0.0934322\pi\)
\(68\) 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 −0.709697 − 0.704507i \(-0.751168\pi\)
0.922469 + 0.386071i \(0.126168\pi\)
\(72\) −4.34326 − 3.91549i −0.511858 − 0.461445i
\(73\) 2.04993 − 1.36972i 0.239926 − 0.160314i −0.429794 − 0.902927i \(-0.641414\pi\)
0.669721 + 0.742613i \(0.266414\pi\)
\(74\) 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 0.940240 + 0.340512i \(0.110600\pi\)
−0.424072 + 0.905628i \(0.639400\pi\)
\(80\) −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 −0.300047 − 0.953924i \(-0.597003\pi\)
−0.480384 + 0.877059i \(0.659503\pi\)
\(84\) −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 −0.855930 − 0.517092i \(-0.827014\pi\)
−0.592893 + 0.805281i \(0.702014\pi\)
\(90\) −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 −0.0740512 + 0.997254i \(0.523593\pi\)
−0.997254 + 0.0740512i \(0.976407\pi\)
\(98\) −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.109.10 yes 240
4.3 odd 2 512.2.k.a.273.3 240
128.27 odd 32 512.2.k.a.497.3 240
128.101 even 32 inner 128.2.k.a.101.10 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.10 ✓ 240 128.101 even 32 inner
128.2.k.a.109.10 yes 240 1.1 even 1 trivial
512.2.k.a.273.3 240 4.3 odd 2
512.2.k.a.497.3 240 128.27 odd 32