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Properties

Label 128.2.k.a.117.11
Level $128$
Weight $2$
Character 128.117
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 117.11
Character \(\chi\) \(=\) 128.117
Dual form 128.2.k.a.93.11

$q$-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+(0.681839 + 1.23899i) q^{2} +(1.51314 - 0.149031i) q^{3} +(-1.07019 + 1.68958i) q^{4} +(1.30448 - 2.44050i) q^{5} +(1.21637 + 1.77315i) q^{6} +(0.196304 - 0.986888i) q^{7} +(-2.82307 - 0.173929i) q^{8} +(-0.674974 + 0.134261i) q^{9} +(3.91320 - 0.0477993i) q^{10} +(-4.03518 + 3.31159i) q^{11} +(-1.36755 + 2.71607i) q^{12} +(-2.42475 + 1.29605i) q^{13} +(1.35659 - 0.429680i) q^{14} +(1.61014 - 3.88723i) q^{15} +(-1.70939 - 3.61635i) q^{16} +(-1.15205 - 2.78128i) q^{17} +(-0.626571 - 0.744741i) q^{18} +(2.73371 + 0.829263i) q^{19} +(2.72740 + 4.81582i) q^{20} +(0.149959 - 1.52255i) q^{21} +(-6.85437 - 2.74157i) q^{22} +(4.27080 + 2.85366i) q^{23} +(-4.29763 + 0.157548i) q^{24} +(-1.47655 - 2.20981i) q^{25} +(-3.25908 - 2.12053i) q^{26} +(-5.36629 + 1.62785i) q^{27} +(1.45735 + 1.38783i) q^{28} +(6.36663 - 7.75777i) q^{29} +(5.91410 - 0.655516i) q^{30} +(4.52106 + 4.52106i) q^{31} +(3.31509 - 4.58368i) q^{32} +(-5.61226 + 5.61226i) q^{33} +(2.66047 - 3.32376i) q^{34} +(-2.15243 - 1.76645i) q^{35} +(0.495505 - 1.28411i) q^{36} +(-1.68731 - 5.56232i) q^{37} +(0.836506 + 3.95247i) q^{38} +(-3.47583 + 2.32247i) q^{39} +(-4.10711 + 6.66284i) q^{40} +(-1.13832 + 1.70361i) q^{41} +(1.98868 - 0.852341i) q^{42} +(0.815405 + 0.0803104i) q^{43} +(-1.27679 - 10.3618i) q^{44} +(-0.552824 + 1.82242i) q^{45} +(-0.623652 + 7.23722i) q^{46} +(1.24735 - 0.516668i) q^{47} +(-3.12549 - 5.21729i) q^{48} +(5.53174 + 2.29132i) q^{49} +(1.73116 - 3.33616i) q^{50} +(-2.15770 - 4.03678i) q^{51} +(0.405147 - 5.48383i) q^{52} +(-7.04052 - 8.57890i) q^{53} +(-5.67584 - 5.53885i) q^{54} +(2.81814 + 14.1678i) q^{55} +(-0.725830 + 2.75191i) q^{56} +(4.26008 + 0.847382i) q^{57} +(13.9528 + 2.59864i) q^{58} +(-5.87785 - 3.14177i) q^{59} +(4.84464 + 6.88055i) q^{60} +(0.787603 + 7.99666i) q^{61} +(-2.51891 + 8.68418i) q^{62} +0.692479i q^{63} +(7.93950 + 0.982028i) q^{64} +7.60827i q^{65} +(-10.7802 - 3.12687i) q^{66} +(1.20073 + 12.1913i) q^{67} +(5.93212 + 1.03002i) q^{68} +(6.88761 + 3.68150i) q^{69} +(0.721005 - 3.87127i) q^{70} +(-1.46122 - 0.290654i) q^{71} +(1.92885 - 0.261630i) q^{72} +(1.37184 + 6.89670i) q^{73} +(5.74118 - 5.88317i) q^{74} +(-2.56355 - 3.12370i) q^{75} +(-4.32670 + 3.73137i) q^{76} +(2.47604 + 4.63235i) q^{77} +(-5.24748 - 2.72296i) q^{78} +(-4.56188 - 1.88959i) q^{79} +(-11.0556 - 0.545678i) q^{80} +(-5.96992 + 2.47282i) q^{81} +(-2.88691 - 0.248773i) q^{82} +(2.22023 - 7.31912i) q^{83} +(2.41200 + 1.88279i) q^{84} +(-8.29055 - 0.816548i) q^{85} +(0.456472 + 1.06504i) q^{86} +(8.47746 - 12.6874i) q^{87} +(11.9676 - 8.64702i) q^{88} +(8.41696 - 5.62403i) q^{89} +(-2.63489 + 0.557652i) q^{90} +(0.803072 + 2.64737i) q^{91} +(-9.39207 + 4.16192i) q^{92} +(7.51477 + 6.16721i) q^{93} +(1.49064 + 1.19316i) q^{94} +(5.58989 - 5.58989i) q^{95} +(4.33309 - 7.42981i) q^{96} +(4.62133 + 4.62133i) q^{97} +(0.932836 + 8.41609i) q^{98} +(2.27902 - 2.77700i) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{21}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.681839 + 1.23899i 0.482133 + 0.876098i
\(3\) 1.51314 − 0.149031i 0.873612 − 0.0860432i 0.348735 − 0.937221i \(-0.386611\pi\)
0.524877 + 0.851178i \(0.324111\pi\)
\(4\) −1.07019 + 1.68958i −0.535095 + 0.844792i
\(5\) 1.30448 − 2.44050i 0.583380 − 1.09143i −0.401118 − 0.916027i \(-0.631378\pi\)
0.984497 − 0.175400i \(-0.0561218\pi\)
\(6\) 1.21637 + 1.77315i 0.496580 + 0.723885i
\(7\) 0.196304 − 0.986888i 0.0741960 − 0.373009i −0.925791 − 0.378035i \(-0.876600\pi\)
0.999987 + 0.00502632i \(0.00159993\pi\)
\(8\) −2.82307 − 0.173929i −0.998108 − 0.0614931i
\(9\) −0.674974 + 0.134261i −0.224991 + 0.0447535i
\(10\) 3.91320 − 0.0477993i 1.23746 − 0.0151155i
\(11\) −4.03518 + 3.31159i −1.21665 + 0.998481i −0.216901 + 0.976194i \(0.569595\pi\)
−0.999752 + 0.0222872i \(0.992905\pi\)
\(12\) −1.36755 + 2.71607i −0.394777 + 0.784061i
\(13\) −2.42475 + 1.29605i −0.672503 + 0.359461i −0.772041 − 0.635573i \(-0.780764\pi\)
0.0995371 + 0.995034i \(0.468264\pi\)
\(14\) 1.35659 − 0.429680i 0.362564 − 0.114837i
\(15\) 1.61014 − 3.88723i 0.415737 − 1.00368i
\(16\) −1.70939 − 3.61635i −0.427347 − 0.904088i
\(17\) −1.15205 − 2.78128i −0.279412 − 0.674560i 0.720408 − 0.693551i \(-0.243955\pi\)
−0.999820 + 0.0189905i \(0.993955\pi\)
\(18\) −0.626571 − 0.744741i −0.147684 − 0.175537i
\(19\) 2.73371 + 0.829263i 0.627157 + 0.190246i 0.587835 − 0.808981i \(-0.299980\pi\)
0.0393218 + 0.999227i \(0.487480\pi\)
\(20\) 2.72740 + 4.81582i 0.609865 + 1.07685i
\(21\) 0.149959 − 1.52255i 0.0327236 − 0.332249i
\(22\) −6.85437 − 2.74157i −1.46136 − 0.584506i
\(23\) 4.27080 + 2.85366i 0.890524 + 0.595029i 0.914456 − 0.404684i \(-0.132619\pi\)
−0.0239324 + 0.999714i \(0.507619\pi\)
\(24\) −4.29763 + 0.157548i −0.877249 + 0.0321593i
\(25\) −1.47655 − 2.20981i −0.295310 − 0.441962i
\(26\) −3.25908 − 2.12053i −0.639159 − 0.415871i
\(27\) −5.36629 + 1.62785i −1.03274 + 0.313279i
\(28\) 1.45735 + 1.38783i 0.275413 + 0.262275i
\(29\) 6.36663 − 7.75777i 1.18225 − 1.44058i 0.310544 − 0.950559i \(-0.399489\pi\)
0.871710 − 0.490022i \(-0.163011\pi\)
\(30\) 5.91410 − 0.655516i 1.07976 − 0.119680i
\(31\) 4.52106 + 4.52106i 0.812006 + 0.812006i 0.984934 − 0.172928i \(-0.0553229\pi\)
−0.172928 + 0.984934i \(0.555323\pi\)
\(32\) 3.31509 − 4.58368i 0.586031 − 0.810289i
\(33\) −5.61226 + 5.61226i −0.976969 + 0.976969i
\(34\) 2.66047 − 3.32376i 0.456267 − 0.570020i
\(35\) −2.15243 − 1.76645i −0.363827 − 0.298585i
\(36\) 0.495505 − 1.28411i 0.0825842 − 0.214018i
\(37\) −1.68731 − 5.56232i −0.277392 − 0.914440i −0.979342 − 0.202213i \(-0.935187\pi\)
0.701949 − 0.712227i \(-0.252313\pi\)
\(38\) 0.836506 + 3.95247i 0.135699 + 0.641175i
\(39\) −3.47583 + 2.32247i −0.556578 + 0.371893i
\(40\) −4.10711 + 6.66284i −0.649391 + 1.05349i
\(41\) −1.13832 + 1.70361i −0.177775 + 0.266060i −0.909646 − 0.415384i \(-0.863647\pi\)
0.731871 + 0.681443i \(0.238647\pi\)
\(42\) 1.98868 − 0.852341i 0.306860 − 0.131519i
\(43\) 0.815405 + 0.0803104i 0.124348 + 0.0122472i 0.160000 − 0.987117i \(-0.448851\pi\)
−0.0356516 + 0.999364i \(0.511351\pi\)
\(44\) −1.27679 − 10.3618i −0.192484 − 1.56210i
\(45\) −0.552824 + 1.82242i −0.0824101 + 0.271670i
\(46\) −0.623652 + 7.23722i −0.0919524 + 1.06707i
\(47\) 1.24735 − 0.516668i 0.181944 − 0.0753638i −0.289852 − 0.957072i \(-0.593606\pi\)
0.471796 + 0.881708i \(0.343606\pi\)
\(48\) −3.12549 − 5.21729i −0.451126 − 0.753051i
\(49\) 5.53174 + 2.29132i 0.790249 + 0.327332i
\(50\) 1.73116 − 3.33616i 0.244823 − 0.471805i
\(51\) −2.15770 − 4.03678i −0.302139 − 0.565262i
\(52\) 0.405147 − 5.48383i 0.0561838 − 0.760471i
\(53\) −7.04052 − 8.57890i −0.967090 − 1.17840i −0.983932 − 0.178544i \(-0.942861\pi\)
0.0168418 − 0.999858i \(-0.494639\pi\)
\(54\) −5.67584 − 5.53885i −0.772383 − 0.753742i
\(55\) 2.81814 + 14.1678i 0.379998 + 1.91038i
\(56\) −0.725830 + 2.75191i −0.0969931 + 0.367740i
\(57\) 4.26008 + 0.847382i 0.564261 + 0.112239i
\(58\) 13.9528 + 2.59864i 1.83209 + 0.341218i
\(59\) −5.87785 − 3.14177i −0.765230 − 0.409024i 0.0421078 − 0.999113i \(-0.486593\pi\)
−0.807338 + 0.590089i \(0.799093\pi\)
\(60\) 4.84464 + 6.88055i 0.625441 + 0.888275i
\(61\) 0.787603 + 7.99666i 0.100842 + 1.02387i 0.903836 + 0.427880i \(0.140739\pi\)
−0.802993 + 0.595988i \(0.796761\pi\)
\(62\) −2.51891 + 8.68418i −0.319902 + 1.10289i
\(63\) 0.692479i 0.0872442i
\(64\) 7.93950 + 0.982028i 0.992437 + 0.122754i
\(65\) 7.60827i 0.943690i
\(66\) −10.7802 − 3.12687i −1.32695 − 0.384891i
\(67\) 1.20073 + 12.1913i 0.146693 + 1.48940i 0.733509 + 0.679680i \(0.237881\pi\)
−0.586816 + 0.809720i \(0.699619\pi\)
\(68\) 5.93212 + 1.03002i 0.719375 + 0.124909i
\(69\) 6.88761 + 3.68150i 0.829170 + 0.443201i
\(70\) 0.721005 − 3.87127i 0.0861766 − 0.462706i
\(71\) −1.46122 − 0.290654i −0.173415 − 0.0344943i 0.107619 − 0.994192i \(-0.465677\pi\)
−0.281034 + 0.959698i \(0.590677\pi\)
\(72\) 1.92885 − 0.261630i 0.227317 − 0.0308334i
\(73\) 1.37184 + 6.89670i 0.160562 + 0.807198i 0.974175 + 0.225792i \(0.0724971\pi\)
−0.813614 + 0.581406i \(0.802503\pi\)
\(74\) 5.74118 − 5.88317i 0.667399 − 0.683905i
\(75\) −2.56355 − 3.12370i −0.296014 − 0.360694i
\(76\) −4.32670 + 3.73137i −0.496307 + 0.428018i
\(77\) 2.47604 + 4.63235i 0.282171 + 0.527905i
\(78\) −5.24748 − 2.72296i −0.594160 − 0.308314i
\(79\) −4.56188 − 1.88959i −0.513251 − 0.212596i 0.110998 − 0.993821i \(-0.464595\pi\)
−0.624250 + 0.781225i \(0.714595\pi\)
\(80\) −11.0556 − 0.545678i −1.23605 − 0.0610086i
\(81\) −5.96992 + 2.47282i −0.663324 + 0.274758i
\(82\) −2.88691 − 0.248773i −0.318806 − 0.0274724i
\(83\) 2.22023 − 7.31912i 0.243702 − 0.803378i −0.746700 − 0.665161i \(-0.768363\pi\)
0.990402 − 0.138217i \(-0.0441371\pi\)
\(84\) 2.41200 + 1.88279i 0.263171 + 0.205429i
\(85\) −8.29055 − 0.816548i −0.899236 − 0.0885670i
\(86\) 0.456472 + 1.06504i 0.0492226 + 0.114846i
\(87\) 8.47746 − 12.6874i 0.908879 − 1.36023i
\(88\) 11.9676 − 8.64702i 1.27575 − 0.921775i
\(89\) 8.41696 − 5.62403i 0.892196 − 0.596146i −0.0227420 − 0.999741i \(-0.507240\pi\)
0.914938 + 0.403595i \(0.132240\pi\)
\(90\) −2.63489 + 0.557652i −0.277742 + 0.0587817i
\(91\) 0.803072 + 2.64737i 0.0841848 + 0.277520i
\(92\) −9.39207 + 4.16192i −0.979191 + 0.433910i
\(93\) 7.51477 + 6.16721i 0.779246 + 0.639510i
\(94\) 1.49064 + 1.19316i 0.153747 + 0.123066i
\(95\) 5.58989 − 5.58989i 0.573510 − 0.573510i
\(96\) 4.33309 − 7.42981i 0.442244 − 0.758302i
\(97\) 4.62133 + 4.62133i 0.469225 + 0.469225i 0.901663 − 0.432439i \(-0.142347\pi\)
−0.432439 + 0.901663i \(0.642347\pi\)
\(98\) 0.932836 + 8.41609i 0.0942307 + 0.850153i
\(99\) 2.27902 − 2.77700i 0.229051 − 0.279099i
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.117.11 yes 240
4.3 odd 2 512.2.k.a.305.4 240
128.35 odd 32 512.2.k.a.465.4 240
128.93 even 32 inner 128.2.k.a.93.11 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.93.11 ✓ 240 128.93 even 32 inner
128.2.k.a.117.11 yes 240 1.1 even 1 trivial
512.2.k.a.305.4 240 4.3 odd 2
512.2.k.a.465.4 240 128.35 odd 32