↑

Properties

Label 128.2.k.a.109.2
Level $128$
Weight $2$
Character 128.109
Analytic conductor $1.022$
Analytic rank $0$
Dimension $240$
Inner twists $2$

Related objects

Downloads

Learn more

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 109.2
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.2

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39471 − 0.234053i −0.986210 − 0.165500i
\(3\) −0.467037 − 0.249636i −0.269644 − 0.144128i 0.331034 − 0.943619i \(-0.392603\pi\)
−0.600678 + 0.799491i \(0.705103\pi\)
\(4\) 1.89044 + 0.652873i 0.945219 + 0.326436i
\(5\) 0.246231 − 0.300034i 0.110118 − 0.134179i −0.715035 − 0.699089i \(-0.753589\pi\)
0.825153 + 0.564910i \(0.191089\pi\)
\(6\) 0.592953 + 0.457482i 0.242072 + 0.186766i
\(7\) 1.89160 + 1.26393i 0.714958 + 0.477720i 0.859081 − 0.511840i \(-0.171036\pi\)
−0.144123 + 0.989560i \(0.546036\pi\)
\(8\) −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 0.968597 − 0.248636i \(-0.0799822\pi\)
0.667224 + 0.744857i \(0.267482\pi\)
\(12\) −0.719924 − 0.776838i −0.207824 − 0.224254i
\(13\) 5.51786 − 4.52839i 1.53038 − 1.25595i 0.675178 − 0.737655i \(-0.264067\pi\)
0.855202 − 0.518296i \(-0.173433\pi\)
\(14\) −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 −0.0165793 − 0.999863i \(-0.505278\pi\)
−0.718733 + 0.695286i \(0.755278\pi\)
\(18\) 1.57803 + 3.50739i 0.371945 + 0.826701i
\(19\) −0.249350 + 2.53169i −0.0572048 + 0.580810i 0.923268 + 0.384156i \(0.125508\pi\)
−0.980473 + 0.196654i \(0.936992\pi\)
\(20\) 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 −0.464586 − 0.885528i \(-0.653797\pi\)
−0.768098 + 0.640332i \(0.778797\pi\)
\(24\) 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 0.946114 − 0.323833i \(-0.104972\pi\)
−0.966577 + 0.256376i \(0.917472\pi\)
\(30\) 0.283264 − 0.0652596i 0.0517166 − 0.0119147i
\(31\) −0.158513 + 0.158513i −0.0284698 + 0.0284698i −0.721198 − 0.692729i \(-0.756408\pi\)
0.692729 + 0.721198i \(0.256408\pi\)
\(32\) −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 −0.443503 − 0.896273i \(-0.646264\pi\)
−0.260127 + 0.965574i \(0.583764\pi\)
\(38\) 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 0.758477 + 0.651700i \(0.225944\pi\)
−0.950136 + 0.311836i \(0.899056\pi\)
\(42\) 0.543407 + 1.61482i 0.0838495 + 0.249173i
\(43\) −7.59540 + 4.05983i −1.15829 + 0.619118i −0.934655 − 0.355556i \(-0.884292\pi\)
−0.223633 + 0.974674i \(0.571792\pi\)
\(44\) 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 −0.781263 − 0.624202i \(-0.785424\pi\)
0.993814 + 0.111059i \(0.0354241\pi\)
\(48\) −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 −0.693232 − 0.720715i \(-0.743814\pi\)
0.976809 − 0.214113i \(-0.0686861\pi\)
\(54\) 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 0.971319 − 0.237780i \(-0.0764197\pi\)
−0.0437162 + 0.999044i \(0.513920\pi\)
\(60\) −0.410345 + 0.0247196i −0.0529753 + 0.00319128i
\(61\) 2.08843 − 3.90718i 0.267397 − 0.500264i −0.712008 − 0.702171i \(-0.752214\pi\)
0.979405 + 0.201908i \(0.0647141\pi\)
\(62\) 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 −0.811153 − 0.584833i \(-0.801160\pi\)
0.936924 + 0.349533i \(0.113660\pi\)
\(68\) −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 −0.959296 − 0.282401i \(-0.908869\pi\)
0.628012 + 0.778204i \(0.283869\pi\)
\(72\) 0.693285 + 7.66077i 0.0817044 + 0.902830i
\(73\) −8.95387 + 5.98279i −1.04797 + 0.700232i −0.955352 − 0.295469i \(-0.904524\pi\)
−0.0926195 + 0.995702i \(0.529524\pi\)
\(74\) 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 −0.973912 − 0.226924i \(-0.927133\pi\)
0.528201 − 0.849120i \(-0.322867\pi\)
\(80\) 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 −0.520982 − 0.853568i \(-0.674434\pi\)
−0.677494 + 0.735528i \(0.736934\pi\)
\(84\) −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 −0.351024 − 0.936367i \(-0.614166\pi\)
0.0340283 + 0.999421i \(0.489166\pi\)
\(90\) 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 −0.413092 − 0.910689i \(-0.635551\pi\)
0.910689 + 0.413092i \(0.135551\pi\)
\(98\) 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.109.2 yes 240
4.3 odd 2 512.2.k.a.273.9 240
128.27 odd 32 512.2.k.a.497.9 240
128.101 even 32 inner 128.2.k.a.101.2 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.2 ✓ 240 128.101 even 32 inner
128.2.k.a.109.2 yes 240 1.1 even 1 trivial
512.2.k.a.273.9 240 4.3 odd 2
512.2.k.a.497.9 240 128.27 odd 32