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Properties

Label 128.2.k.a.101.8
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 101.8
Character \(\chi\) \(=\) 128.101
Dual form 128.2.k.a.109.8

$q$-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+(-0.399184 + 1.35671i) q^{2} +(-0.135634 + 0.0724978i) q^{3} +(-1.68130 - 1.08315i) q^{4} +(2.12942 + 2.59470i) q^{5} +(-0.0442153 - 0.212955i) q^{6} +(-1.11060 + 0.742082i) q^{7} +(2.14067 - 1.84866i) q^{8} +(-1.65357 + 2.47474i) q^{9} +(-4.37028 + 1.85323i) q^{10} +(-0.181497 + 0.0550566i) q^{11} +(0.306568 + 0.0250213i) q^{12} +(0.374137 + 0.307046i) q^{13} +(-0.563451 - 1.80299i) q^{14} +(-0.476932 - 0.197552i) q^{15} +(1.65356 + 3.64222i) q^{16} +(1.79255 - 0.742500i) q^{17} +(-2.69742 - 3.23129i) q^{18} +(-0.487163 - 4.94625i) q^{19} +(-0.769739 - 6.66897i) q^{20} +(0.0968362 - 0.181168i) q^{21} +(-0.00224473 - 0.268216i) q^{22} +(5.40603 - 1.07533i) q^{23} +(-0.156324 + 0.405934i) q^{24} +(-1.22261 + 6.14649i) q^{25} +(-0.565921 + 0.385026i) q^{26} +(0.0900899 - 0.914698i) q^{27} +(2.67105 - 0.0447117i) q^{28} +(2.45765 - 8.10178i) q^{29} +(0.458403 - 0.568197i) q^{30} +(3.30297 + 3.30297i) q^{31} +(-5.60149 + 0.789483i) q^{32} +(0.0206257 - 0.0206257i) q^{33} +(0.291795 + 2.72836i) q^{34} +(-4.29042 - 1.30149i) q^{35} +(5.46068 - 2.36973i) q^{36} +(8.32789 + 0.820226i) q^{37} +(6.90508 + 1.31353i) q^{38} +(-0.0730059 - 0.0145218i) q^{39} +(9.35510 + 1.61784i) q^{40} +(0.580168 + 2.91670i) q^{41} +(0.207136 + 0.203698i) q^{42} +(-8.36535 - 4.47137i) q^{43} +(0.364787 + 0.104022i) q^{44} +(-9.94237 + 0.979238i) q^{45} +(-0.699102 + 7.76365i) q^{46} +(-3.57155 - 8.62250i) q^{47} +(-0.488332 - 0.374128i) q^{48} +(-1.99603 + 4.81884i) q^{49} +(-7.85094 - 4.11231i) q^{50} +(-0.189301 + 0.230664i) q^{51} +(-0.296460 - 0.921485i) q^{52} +(-3.12341 - 10.2965i) q^{53} +(1.20501 + 0.487359i) q^{54} +(-0.529339 - 0.353693i) q^{55} +(-1.00558 + 3.64168i) q^{56} +(0.424668 + 0.635561i) q^{57} +(10.0107 + 6.56841i) q^{58} +(-10.8483 + 8.90293i) q^{59} +(0.587888 + 0.848734i) q^{60} +(0.339850 + 0.635815i) q^{61} +(-5.79966 + 3.16267i) q^{62} -3.97554i q^{63} +(1.16493 - 7.91473i) q^{64} +1.62460i q^{65} +(0.0197495 + 0.0362165i) q^{66} +(-0.0570291 - 0.106694i) q^{67} +(-3.81807 - 0.693240i) q^{68} +(-0.655283 + 0.537776i) q^{69} +(3.47840 - 5.30131i) q^{70} +(8.81399 + 13.1911i) q^{71} +(1.03520 + 8.35449i) q^{72} +(-1.93431 - 1.29247i) q^{73} +(-4.43717 + 10.9711i) q^{74} +(-0.279780 - 0.922310i) q^{75} +(-4.53848 + 8.84383i) q^{76} +(0.160715 - 0.195832i) q^{77} +(0.0488446 - 0.0932506i) q^{78} +(-3.11404 + 7.51796i) q^{79} +(-5.92934 + 12.0463i) q^{80} +(-3.36290 - 8.11876i) q^{81} +(-4.18870 - 0.377184i) q^{82} +(14.1505 - 1.39370i) q^{83} +(-0.359043 + 0.199710i) q^{84} +(5.74366 + 3.07005i) q^{85} +(9.40566 - 9.56442i) q^{86} +(0.254021 + 1.27705i) q^{87} +(-0.286745 + 0.453384i) q^{88} +(-1.57448 - 0.313183i) q^{89} +(2.64030 - 13.8798i) q^{90} +(-0.643371 - 0.0633666i) q^{91} +(-10.2539 - 4.04761i) q^{92} +(-0.687453 - 0.208537i) q^{93} +(13.1239 - 1.40358i) q^{94} +(11.7967 - 11.7967i) q^{95} +(0.702516 - 0.513177i) q^{96} +(-11.0877 - 11.0877i) q^{97} +(-5.74097 - 4.63163i) q^{98} +(0.163868 - 0.540199i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.399184 + 1.35671i −0.282266 + 0.959336i
\(3\) −0.135634 + 0.0724978i −0.0783083 + 0.0418566i −0.510086 − 0.860123i \(-0.670386\pi\)
0.431778 + 0.901980i \(0.357886\pi\)
\(4\) −1.68130 − 1.08315i −0.840652 − 0.541576i
\(5\) 2.12942 + 2.59470i 0.952305 + 1.16039i 0.986956 + 0.160987i \(0.0514678\pi\)
−0.0346516 + 0.999399i \(0.511032\pi\)
\(6\) −0.0442153 − 0.212955i −0.0180508 − 0.0869387i
\(7\) −1.11060 + 0.742082i −0.419769 + 0.280480i −0.747463 − 0.664303i \(-0.768728\pi\)
0.327694 + 0.944784i \(0.393728\pi\)
\(8\) 2.14067 − 1.84866i 0.756841 − 0.653599i
\(9\) −1.65357 + 2.47474i −0.551190 + 0.824914i
\(10\) −4.37028 + 1.85323i −1.38200 + 0.586043i
\(11\) −0.181497 + 0.0550566i −0.0547235 + 0.0166002i −0.317528 − 0.948249i \(-0.602853\pi\)
0.262804 + 0.964849i \(0.415353\pi\)
\(12\) 0.306568 + 0.0250213i 0.0884985 + 0.00722303i
\(13\) 0.374137 + 0.307046i 0.103767 + 0.0851593i 0.684846 − 0.728688i \(-0.259869\pi\)
−0.581079 + 0.813847i \(0.697369\pi\)
\(14\) −0.563451 − 1.80299i −0.150589 − 0.481869i
\(15\) −0.476932 − 0.197552i −0.123143 − 0.0510076i
\(16\) 1.65356 + 3.64222i 0.413391 + 0.910554i
\(17\) 1.79255 − 0.742500i 0.434758 − 0.180083i −0.154561 − 0.987983i \(-0.549396\pi\)
0.589319 + 0.807901i \(0.299396\pi\)
\(18\) −2.69742 − 3.23129i −0.635788 − 0.761622i
\(19\) −0.487163 − 4.94625i −0.111763 − 1.13475i −0.873487 − 0.486847i \(-0.838147\pi\)
0.761724 − 0.647901i \(-0.224353\pi\)
\(20\) −0.769739 − 6.66897i −0.172119 − 1.49123i
\(21\) 0.0968362 − 0.181168i 0.0211314 − 0.0395340i
\(22\) −0.00224473 − 0.268216i −0.000478577 − 0.0571839i
\(23\) 5.40603 − 1.07533i 1.12724 − 0.224221i 0.403958 − 0.914778i \(-0.367634\pi\)
0.723278 + 0.690557i \(0.242634\pi\)
\(24\) −0.156324 + 0.405934i −0.0319094 + 0.0828610i
\(25\) −1.22261 + 6.14649i −0.244523 + 1.22930i
\(26\) −0.565921 + 0.385026i −0.110986 + 0.0755098i
\(27\) 0.0900899 − 0.914698i 0.0173378 − 0.176034i
\(28\) 2.67105 − 0.0447117i 0.504781 − 0.00844971i
\(29\) 2.45765 − 8.10178i 0.456374 − 1.50446i −0.363321 − 0.931664i \(-0.618357\pi\)
0.819695 − 0.572800i \(-0.194143\pi\)
\(30\) 0.458403 − 0.568197i 0.0836926 − 0.103738i
\(31\) 3.30297 + 3.30297i 0.593232 + 0.593232i 0.938503 − 0.345271i \(-0.112213\pi\)
−0.345271 + 0.938503i \(0.612213\pi\)
\(32\) −5.60149 + 0.789483i −0.990213 + 0.139562i
\(33\) 0.0206257 − 0.0206257i 0.00359047 − 0.00359047i
\(34\) 0.291795 + 2.72836i 0.0500424 + 0.467910i
\(35\) −4.29042 − 1.30149i −0.725214 − 0.219991i
\(36\) 5.46068 − 2.36973i 0.910113 − 0.394954i
\(37\) 8.32789 + 0.820226i 1.36910 + 0.134844i 0.755694 − 0.654925i \(-0.227300\pi\)
0.613404 + 0.789770i \(0.289800\pi\)
\(38\) 6.90508 + 1.31353i 1.12015 + 0.213083i
\(39\) −0.0730059 − 0.0145218i −0.0116903 − 0.00232534i
\(40\) 9.35510 + 1.61784i 1.47917 + 0.255803i
\(41\) 0.580168 + 2.91670i 0.0906070 + 0.455512i 0.999278 + 0.0379988i \(0.0120983\pi\)
−0.908671 + 0.417513i \(0.862902\pi\)
\(42\) 0.207136 + 0.203698i 0.0319618 + 0.0314312i
\(43\) −8.36535 − 4.47137i −1.27570 − 0.681878i −0.312313 − 0.949979i \(-0.601104\pi\)
−0.963391 + 0.268101i \(0.913604\pi\)
\(44\) 0.364787 + 0.104022i 0.0549937 + 0.0156820i
\(45\) −9.94237 + 0.979238i −1.48212 + 0.145976i
\(46\) −0.699102 + 7.76365i −0.103077 + 1.14469i
\(47\) −3.57155 − 8.62250i −0.520965 − 1.25772i −0.937305 − 0.348511i \(-0.886687\pi\)
0.416340 − 0.909209i \(-0.363313\pi\)
\(48\) −0.488332 − 0.374128i −0.0704846 − 0.0540007i
\(49\) −1.99603 + 4.81884i −0.285147 + 0.688406i
\(50\) −7.85094 − 4.11231i −1.11029 − 0.581569i
\(51\) −0.189301 + 0.230664i −0.0265075 + 0.0322995i
\(52\) −0.296460 − 0.921485i −0.0411116 − 0.127787i
\(53\) −3.12341 − 10.2965i −0.429033 − 1.41433i −0.859227 − 0.511595i \(-0.829055\pi\)
0.430194 − 0.902737i \(-0.358445\pi\)
\(54\) 1.20501 + 0.487359i 0.163982 + 0.0663212i
\(55\) −0.529339 − 0.353693i −0.0713761 − 0.0476920i
\(56\) −1.00558 + 3.64168i −0.134376 + 0.486640i
\(57\) 0.424668 + 0.635561i 0.0562487 + 0.0841822i
\(58\) 10.0107 + 6.56841i 1.31447 + 0.862475i
\(59\) −10.8483 + 8.90293i −1.41232 + 1.15906i −0.448095 + 0.893986i \(0.647897\pi\)
−0.964227 + 0.265077i \(0.914603\pi\)
\(60\) 0.587888 + 0.848734i 0.0758961 + 0.109571i
\(61\) 0.339850 + 0.635815i 0.0435134 + 0.0814078i 0.902738 − 0.430190i \(-0.141554\pi\)
−0.859225 + 0.511598i \(0.829054\pi\)
\(62\) −5.79966 + 3.16267i −0.736558 + 0.401659i
\(63\) − 3.97554i − 0.500871i
\(64\) 1.16493 − 7.91473i 0.145617 − 0.989341i
\(65\) 1.62460i 0.201507i
\(66\) 0.0197495 + 0.0362165i 0.00243100 + 0.00445794i
\(67\) −0.0570291 − 0.106694i −0.00696721 − 0.0130347i 0.878414 − 0.477901i \(-0.158602\pi\)
−0.885381 + 0.464866i \(0.846102\pi\)
\(68\) −3.81807 − 0.693240i −0.463009 − 0.0840677i
\(69\) −0.655283 + 0.537776i −0.0788867 + 0.0647407i
\(70\) 3.47840 − 5.30131i 0.415749 − 0.633628i
\(71\) 8.81399 + 13.1911i 1.04603 + 1.56549i 0.803461 + 0.595357i \(0.202989\pi\)
0.242567 + 0.970135i \(0.422011\pi\)
\(72\) 1.03520 + 8.35449i 0.122000 + 0.984586i
\(73\) −1.93431 − 1.29247i −0.226394 − 0.151272i 0.437200 − 0.899364i \(-0.355970\pi\)
−0.663595 + 0.748092i \(0.730970\pi\)
\(74\) −4.43717 + 10.9711i −0.515811 + 1.27536i
\(75\) −0.279780 − 0.922310i −0.0323062 − 0.106499i
\(76\) −4.53848 + 8.84383i −0.520599 + 1.01446i
\(77\) 0.160715 − 0.195832i 0.0183152 − 0.0223171i
\(78\) 0.0488446 − 0.0932506i 0.00553056 − 0.0105586i
\(79\) −3.11404 + 7.51796i −0.350357 + 0.845837i 0.646219 + 0.763152i \(0.276349\pi\)
−0.996576 + 0.0826847i \(0.973651\pi\)
\(80\) −5.92934 + 12.0463i −0.662921 + 1.34682i
\(81\) −3.36290 − 8.11876i −0.373656 − 0.902085i
\(82\) −4.18870 − 0.377184i −0.462565 − 0.0416530i
\(83\) 14.1505 − 1.39370i 1.55322 − 0.152979i 0.715318 − 0.698799i \(-0.246282\pi\)
0.837903 + 0.545820i \(0.183782\pi\)
\(84\) −0.359043 + 0.199710i −0.0391748 + 0.0217901i
\(85\) 5.74366 + 3.07005i 0.622988 + 0.332994i
\(86\) 9.40566 − 9.56442i 1.01424 − 1.03136i
\(87\) 0.254021 + 1.27705i 0.0272339 + 0.136914i
\(88\) −0.286745 + 0.453384i −0.0305671 + 0.0483309i
\(89\) −1.57448 − 0.313183i −0.166894 − 0.0331974i 0.110936 − 0.993828i \(-0.464615\pi\)
−0.277830 + 0.960630i \(0.589615\pi\)
\(90\) 2.64030 − 13.8798i 0.278312 − 1.46306i
\(91\) −0.643371 − 0.0633666i −0.0674437 − 0.00664262i
\(92\) −10.2539 − 4.04761i −1.06905 − 0.421992i
\(93\) −0.687453 − 0.208537i −0.0712856 − 0.0216243i
\(94\) 13.1239 − 1.40358i 1.35363 − 0.144769i
\(95\) 11.7967 − 11.7967i 1.21031 − 1.21031i
\(96\) 0.702516 − 0.513177i 0.0717003 − 0.0523759i
\(97\) −11.0877 − 11.0877i −1.12578 − 1.12578i −0.990856 − 0.134927i \(-0.956920\pi\)
−0.134927 − 0.990856i \(-0.543080\pi\)
\(98\) −5.74097 − 4.63163i −0.579925 − 0.467865i
\(99\) 0.163868 − 0.540199i 0.0164693 − 0.0542920i
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.101.8 ✓ 240
4.3 odd 2 512.2.k.a.497.8 240
128.19 odd 32 512.2.k.a.273.8 240
128.109 even 32 inner 128.2.k.a.109.8 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.8 ✓ 240 1.1 even 1 trivial
128.2.k.a.109.8 yes 240 128.109 even 32 inner
512.2.k.a.273.8 240 128.19 odd 32
512.2.k.a.497.8 240 4.3 odd 2