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Properties

Label 128.2.k.a.101.2
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 101.2
Character \(\chi\) \(=\) 128.101
Dual form 128.2.k.a.109.2

$q$-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+(-1.39471 + 0.234053i) q^{2} +(-0.467037 + 0.249636i) q^{3} +(1.89044 - 0.652873i) q^{4} +(0.246231 + 0.300034i) q^{5} +(0.592953 - 0.457482i) q^{6} +(1.89160 - 1.26393i) q^{7} +(-2.48381 + 1.35303i) q^{8} +(-1.51091 + 2.26123i) q^{9} +(-0.413645 - 0.360829i) q^{10} +(5.42541 - 1.64578i) q^{11} +(-0.719924 + 0.776838i) q^{12} +(5.51786 + 4.52839i) q^{13} +(-2.34241 + 2.20555i) q^{14} +(-0.189898 - 0.0786585i) q^{15} +(3.14751 - 2.46843i) q^{16} +(-3.03177 + 1.25580i) q^{17} +(1.57803 - 3.50739i) q^{18} +(-0.249350 - 2.53169i) q^{19} +(0.661368 + 0.406437i) q^{20} +(-0.567925 + 1.06251i) q^{21} +(-7.18167 + 3.56522i) q^{22} +(-5.91175 + 1.17592i) q^{23} +(0.822264 - 1.25196i) q^{24} +(0.946061 - 4.75617i) q^{25} +(-8.75571 - 5.02433i) q^{26} +(0.296883 - 3.01431i) q^{27} +(2.75077 - 3.62435i) q^{28} +(-0.110196 + 0.363268i) q^{29} +(0.283264 + 0.0652596i) q^{30} +(-0.158513 - 0.158513i) q^{31} +(-3.81213 + 4.17943i) q^{32} +(-2.12302 + 2.12302i) q^{33} +(3.93452 - 2.46107i) q^{34} +(0.844991 + 0.256325i) q^{35} +(-1.37998 + 5.26114i) q^{36} +(-4.28002 - 0.421545i) q^{37} +(0.940322 + 3.47262i) q^{38} +(-3.70750 - 0.737467i) q^{39} +(-1.01755 - 0.412067i) q^{40} +(-1.22722 - 6.16964i) q^{41} +(0.543407 - 1.61482i) q^{42} +(-7.59540 - 4.05983i) q^{43} +(9.18191 - 6.65334i) q^{44} +(-1.05048 + 0.103463i) q^{45} +(7.96995 - 3.02373i) q^{46} +(1.45718 + 3.51794i) q^{47} +(-0.853795 + 1.93858i) q^{48} +(-0.698143 + 1.68547i) q^{49} +(-0.206286 + 6.85491i) q^{50} +(1.10246 - 1.34335i) q^{51} +(13.3876 + 4.95819i) q^{52} +(2.06447 + 6.80565i) q^{53} +(0.291441 + 4.27358i) q^{54} +(1.82969 + 1.22256i) q^{55} +(-2.98824 + 5.69875i) q^{56} +(0.748459 + 1.12015i) q^{57} +(0.0686678 - 0.532445i) q^{58} +(7.12505 - 5.84738i) q^{59} +(-0.410345 - 0.0247196i) q^{60} +(2.08843 + 3.90718i) q^{61} +(0.258181 + 0.183980i) q^{62} +6.18702i q^{63} +(4.33861 - 6.72134i) q^{64} +2.77058i q^{65} +(2.46410 - 3.45790i) q^{66} +(1.02948 + 1.92602i) q^{67} +(-4.91150 + 4.35337i) q^{68} +(2.46745 - 2.02499i) q^{69} +(-1.23851 - 0.159727i) q^{70} +(-2.79145 - 4.17771i) q^{71} +(0.693285 - 7.66077i) q^{72} +(-8.95387 - 5.98279i) q^{73} +(6.06805 - 0.413817i) q^{74} +(0.745468 + 2.45748i) q^{75} +(-2.12425 - 4.62322i) q^{76} +(8.18256 - 9.97047i) q^{77} +(5.34349 + 0.160803i) q^{78} +(-3.96157 + 9.56408i) q^{79} +(1.51563 + 0.336555i) q^{80} +(-2.50836 - 6.05573i) q^{81} +(3.15564 + 8.31764i) q^{82} +(-10.9186 + 1.07539i) q^{83} +(-0.379942 + 2.37940i) q^{84} +(-1.12330 - 0.600415i) q^{85} +(11.5436 + 3.88456i) q^{86} +(-0.0392192 - 0.197168i) q^{87} +(-11.2489 + 11.4285i) q^{88} +(-2.99053 - 0.594853i) q^{89} +(1.44090 - 0.390168i) q^{90} +(16.1612 + 1.59173i) q^{91} +(-10.4081 + 6.08262i) q^{92} +(0.113602 + 0.0344609i) q^{93} +(-2.85572 - 4.56545i) q^{94} +(0.698195 - 0.698195i) q^{95} +(0.737067 - 2.90360i) q^{96} +(4.90076 + 4.90076i) q^{97} +(0.579219 - 2.51414i) q^{98} +(-4.47579 + 14.7547i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39471 + 0.234053i −0.986210 + 0.165500i
\(3\) −0.467037 + 0.249636i −0.269644 + 0.144128i −0.600678 − 0.799491i \(-0.705103\pi\)
0.331034 + 0.943619i \(0.392603\pi\)
\(4\) 1.89044 − 0.652873i 0.945219 − 0.326436i
\(5\) 0.246231 + 0.300034i 0.110118 + 0.134179i 0.825153 − 0.564910i \(-0.191089\pi\)
−0.715035 + 0.699089i \(0.753589\pi\)
\(6\) 0.592953 − 0.457482i 0.242072 − 0.186766i
\(7\) 1.89160 − 1.26393i 0.714958 − 0.477720i −0.144123 − 0.989560i \(-0.546036\pi\)
0.859081 + 0.511840i \(0.171036\pi\)
\(8\) −2.48381 + 1.35303i −0.878159 + 0.478369i
\(9\) −1.51091 + 2.26123i −0.503635 + 0.753743i
\(10\) −0.413645 − 0.360829i −0.130806 − 0.114104i
\(11\) 5.42541 − 1.64578i 1.63582 − 0.496221i 0.667224 − 0.744857i \(-0.267482\pi\)
0.968597 + 0.248636i \(0.0799822\pi\)
\(12\) −0.719924 + 0.776838i −0.207824 + 0.224254i
\(13\) 5.51786 + 4.52839i 1.53038 + 1.25595i 0.855202 + 0.518296i \(0.173433\pi\)
0.675178 + 0.737655i \(0.264067\pi\)
\(14\) −2.34241 + 2.20555i −0.626036 + 0.589458i
\(15\) −0.189898 − 0.0786585i −0.0490315 − 0.0203095i
\(16\) 3.14751 − 2.46843i 0.786879 − 0.617108i
\(17\) −3.03177 + 1.25580i −0.735312 + 0.304576i −0.718733 − 0.695286i \(-0.755278\pi\)
−0.0165793 + 0.999863i \(0.505278\pi\)
\(18\) 1.57803 − 3.50739i 0.371945 − 0.826701i
\(19\) −0.249350 − 2.53169i −0.0572048 − 0.580810i −0.980473 − 0.196654i \(-0.936992\pi\)
0.923268 − 0.384156i \(-0.125508\pi\)
\(20\) 0.661368 + 0.406437i 0.147886 + 0.0908821i
\(21\) −0.567925 + 1.06251i −0.123931 + 0.231859i
\(22\) −7.18167 + 3.56522i −1.53114 + 0.760107i
\(23\) −5.91175 + 1.17592i −1.23268 + 0.245196i −0.768098 − 0.640332i \(-0.778797\pi\)
−0.464586 + 0.885528i \(0.653797\pi\)
\(24\) 0.822264 − 1.25196i 0.167844 − 0.255556i
\(25\) 0.946061 − 4.75617i 0.189212 − 0.951234i
\(26\) −8.75571 − 5.02433i −1.71714 − 0.985352i
\(27\) 0.296883 − 3.01431i 0.0571352 − 0.580104i
\(28\) 2.75077 − 3.62435i 0.519847 − 0.684938i
\(29\) −0.110196 + 0.363268i −0.0204629 + 0.0674571i −0.966577 − 0.256376i \(-0.917472\pi\)
0.946114 + 0.323833i \(0.104972\pi\)
\(30\) 0.283264 + 0.0652596i 0.0517166 + 0.0119147i
\(31\) −0.158513 − 0.158513i −0.0284698 − 0.0284698i 0.692729 − 0.721198i \(-0.256408\pi\)
−0.721198 + 0.692729i \(0.756408\pi\)
\(32\) −3.81213 + 4.17943i −0.673896 + 0.738826i
\(33\) −2.12302 + 2.12302i −0.369570 + 0.369570i
\(34\) 3.93452 − 2.46107i 0.674765 − 0.422071i
\(35\) 0.844991 + 0.256325i 0.142830 + 0.0433269i
\(36\) −1.37998 + 5.26114i −0.229996 + 0.876857i
\(37\) −4.28002 − 0.421545i −0.703630 − 0.0693015i −0.260127 − 0.965574i \(-0.583764\pi\)
−0.443503 + 0.896273i \(0.646264\pi\)
\(38\) 0.940322 + 3.47262i 0.152540 + 0.563333i
\(39\) −3.70750 − 0.737467i −0.593675 − 0.118089i
\(40\) −1.01755 − 0.412067i −0.160888 − 0.0651536i
\(41\) −1.22722 − 6.16964i −0.191659 − 0.963536i −0.950136 − 0.311836i \(-0.899056\pi\)
0.758477 − 0.651700i \(-0.225944\pi\)
\(42\) 0.543407 − 1.61482i 0.0838495 − 0.249173i
\(43\) −7.59540 − 4.05983i −1.15829 − 0.619118i −0.223633 − 0.974674i \(-0.571792\pi\)
−0.934655 + 0.355556i \(0.884292\pi\)
\(44\) 9.18191 − 6.65334i 1.38423 − 1.00303i
\(45\) −1.05048 + 0.103463i −0.156596 + 0.0154233i
\(46\) 7.96995 − 3.02373i 1.17511 − 0.445825i
\(47\) 1.45718 + 3.51794i 0.212551 + 0.513144i 0.993814 − 0.111059i \(-0.0354241\pi\)
−0.781263 + 0.624202i \(0.785424\pi\)
\(48\) −0.853795 + 1.93858i −0.123235 + 0.279810i
\(49\) −0.698143 + 1.68547i −0.0997347 + 0.240781i
\(50\) −0.206286 + 6.85491i −0.0291733 + 0.969431i
\(51\) 1.10246 − 1.34335i 0.154375 − 0.188106i
\(52\) 13.3876 + 4.95819i 1.85653 + 0.687577i
\(53\) 2.06447 + 6.80565i 0.283577 + 0.934828i 0.976809 + 0.214113i \(0.0686861\pi\)
−0.693232 + 0.720715i \(0.743814\pi\)
\(54\) 0.291441 + 4.27358i 0.0396601 + 0.581560i
\(55\) 1.82969 + 1.22256i 0.246716 + 0.164850i
\(56\) −2.98824 + 5.69875i −0.399320 + 0.761527i
\(57\) 0.748459 + 1.12015i 0.0991358 + 0.148367i
\(58\) 0.0686678 − 0.532445i 0.00901652 − 0.0699135i
\(59\) 7.12505 − 5.84738i 0.927603 − 0.761264i −0.0437162 − 0.999044i \(-0.513920\pi\)
0.971319 + 0.237780i \(0.0764197\pi\)
\(60\) −0.410345 − 0.0247196i −0.0529753 − 0.00319128i
\(61\) 2.08843 + 3.90718i 0.267397 + 0.500264i 0.979405 − 0.201908i \(-0.0647141\pi\)
−0.712008 + 0.702171i \(0.752214\pi\)
\(62\) 0.258181 + 0.183980i 0.0327890 + 0.0233655i
\(63\) 6.18702i 0.779491i
\(64\) 4.33861 − 6.72134i 0.542326 − 0.840168i
\(65\) 2.77058i 0.343648i
\(66\) 2.46410 − 3.45790i 0.303310 − 0.425638i
\(67\) 1.02948 + 1.92602i 0.125771 + 0.235300i 0.936924 − 0.349533i \(-0.113660\pi\)
−0.811153 + 0.584833i \(0.801160\pi\)
\(68\) −4.91150 + 4.35337i −0.595606 + 0.527924i
\(69\) 2.46745 − 2.02499i 0.297046 − 0.243780i
\(70\) −1.23851 − 0.159727i −0.148031 − 0.0190910i
\(71\) −2.79145 − 4.17771i −0.331285 − 0.495803i 0.628012 − 0.778204i \(-0.283869\pi\)
−0.959296 + 0.282401i \(0.908869\pi\)
\(72\) 0.693285 − 7.66077i 0.0817044 − 0.902830i
\(73\) −8.95387 − 5.98279i −1.04797 − 0.700232i −0.0926195 − 0.995702i \(-0.529524\pi\)
−0.955352 + 0.295469i \(0.904524\pi\)
\(74\) 6.06805 − 0.413817i 0.705397 − 0.0481053i
\(75\) 0.745468 + 2.45748i 0.0860792 + 0.283765i
\(76\) −2.12425 − 4.62322i −0.243669 − 0.530319i
\(77\) 8.18256 − 9.97047i 0.932489 − 1.13624i
\(78\) 5.34349 + 0.160803i 0.605032 + 0.0182073i
\(79\) −3.96157 + 9.56408i −0.445712 + 1.07604i 0.528201 + 0.849120i \(0.322867\pi\)
−0.973912 + 0.226924i \(0.927133\pi\)
\(80\) 1.51563 + 0.336555i 0.169452 + 0.0376280i
\(81\) −2.50836 − 6.05573i −0.278707 − 0.672858i
\(82\) 3.15564 + 8.31764i 0.348482 + 0.918529i
\(83\) −10.9186 + 1.07539i −1.19848 + 0.118040i −0.677494 − 0.735528i \(-0.736934\pi\)
−0.520982 + 0.853568i \(0.674434\pi\)
\(84\) −0.379942 + 2.37940i −0.0414550 + 0.259614i
\(85\) −1.12330 − 0.600415i −0.121839 − 0.0651242i
\(86\) 11.5436 + 3.88456i 1.24478 + 0.418883i
\(87\) −0.0392192 − 0.197168i −0.00420474 − 0.0211387i
\(88\) −11.2489 + 11.4285i −1.19913 + 1.21829i
\(89\) −2.99053 − 0.594853i −0.316995 − 0.0630543i 0.0340283 − 0.999421i \(-0.489166\pi\)
−0.351024 + 0.936367i \(0.614166\pi\)
\(90\) 1.44090 − 0.390168i 0.151884 − 0.0411273i
\(91\) 16.1612 + 1.59173i 1.69415 + 0.166859i
\(92\) −10.4081 + 6.08262i −1.08512 + 0.634157i
\(93\) 0.113602 + 0.0344609i 0.0117800 + 0.00357343i
\(94\) −2.85572 − 4.56545i −0.294545 − 0.470890i
\(95\) 0.698195 − 0.698195i 0.0716333 − 0.0716333i
\(96\) 0.737067 − 2.90360i 0.0752265 − 0.296347i
\(97\) 4.90076 + 4.90076i 0.497597 + 0.497597i 0.910689 − 0.413092i \(-0.135551\pi\)
−0.413092 + 0.910689i \(0.635551\pi\)
\(98\) 0.579219 − 2.51414i 0.0585100 − 0.253967i
\(99\) −4.47579 + 14.7547i −0.449834 + 1.48290i
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.101.2 ✓ 240
4.3 odd 2 512.2.k.a.497.9 240
128.19 odd 32 512.2.k.a.273.9 240
128.109 even 32 inner 128.2.k.a.109.2 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.2 ✓ 240 1.1 even 1 trivial
128.2.k.a.109.2 yes 240 128.109 even 32 inner
512.2.k.a.273.9 240 128.19 odd 32
512.2.k.a.497.9 240 4.3 odd 2