↑

Properties

Label 128.2.k.a.109.8
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 109.8
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.8

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.399184 − 1.35671i −0.282266 − 0.959336i
\(3\) −0.135634 − 0.0724978i −0.0783083 − 0.0418566i 0.431778 − 0.901980i \(-0.357886\pi\)
−0.510086 + 0.860123i \(0.670386\pi\)
\(4\) −1.68130 + 1.08315i −0.840652 + 0.541576i
\(5\) 2.12942 − 2.59470i 0.952305 − 1.16039i −0.0346516 − 0.999399i \(-0.511032\pi\)
0.986956 − 0.160987i \(-0.0514678\pi\)
\(6\) −0.0442153 + 0.212955i −0.0180508 + 0.0869387i
\(7\) −1.11060 − 0.742082i −0.419769 − 0.280480i 0.327694 − 0.944784i \(-0.393728\pi\)
−0.747463 + 0.664303i \(0.768728\pi\)
\(8\) 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 0.262804 − 0.964849i \(-0.415353\pi\)
−0.317528 + 0.948249i \(0.602853\pi\)
\(12\) 0.306568 − 0.0250213i 0.0884985 − 0.00722303i
\(13\) 0.374137 − 0.307046i 0.103767 − 0.0851593i −0.581079 − 0.813847i \(-0.697369\pi\)
0.684846 + 0.728688i \(0.259869\pi\)
\(14\) −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 0.589319 − 0.807901i \(-0.299396\pi\)
−0.154561 + 0.987983i \(0.549396\pi\)
\(18\) −2.69742 + 3.23129i −0.635788 + 0.761622i
\(19\) −0.487163 + 4.94625i −0.111763 + 1.13475i 0.761724 + 0.647901i \(0.224353\pi\)
−0.873487 + 0.486847i \(0.838147\pi\)
\(20\) −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 0.723278 − 0.690557i \(-0.242634\pi\)
0.403958 + 0.914778i \(0.367634\pi\)
\(24\) −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 0.819695 + 0.572800i \(0.194143\pi\)
−0.363321 + 0.931664i \(0.618357\pi\)
\(30\) 0.458403 + 0.568197i 0.0836926 + 0.103738i
\(31\) 3.30297 − 3.30297i 0.593232 − 0.593232i −0.345271 − 0.938503i \(-0.612213\pi\)
0.938503 + 0.345271i \(0.112213\pi\)
\(32\) −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 0.613404 − 0.789770i \(-0.289800\pi\)
0.755694 + 0.654925i \(0.227300\pi\)
\(38\) 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 −0.908671 − 0.417513i \(-0.862902\pi\)
0.999278 − 0.0379988i \(-0.0120983\pi\)
\(42\) 0.207136 − 0.203698i 0.0319618 − 0.0314312i
\(43\) −8.36535 + 4.47137i −1.27570 + 0.681878i −0.963391 − 0.268101i \(-0.913604\pi\)
−0.312313 + 0.949979i \(0.601104\pi\)
\(44\) 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 0.416340 + 0.909209i \(0.363313\pi\)
−0.937305 + 0.348511i \(0.886687\pi\)
\(48\) −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 0.430194 + 0.902737i \(0.358445\pi\)
−0.859227 + 0.511595i \(0.829055\pi\)
\(54\) 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 −0.964227 − 0.265077i \(-0.914603\pi\)
−0.448095 − 0.893986i \(-0.647897\pi\)
\(60\) 0.587888 − 0.848734i 0.0758961 − 0.109571i
\(61\) 0.339850 − 0.635815i 0.0435134 − 0.0814078i −0.859225 − 0.511598i \(-0.829054\pi\)
0.902738 + 0.430190i \(0.141554\pi\)
\(62\) −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 −0.885381 − 0.464866i \(-0.846102\pi\)
0.878414 + 0.477901i \(0.158602\pi\)
\(68\) −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 0.242567 − 0.970135i \(-0.422011\pi\)
0.803461 − 0.595357i \(-0.202989\pi\)
\(72\) 1.03520 − 8.35449i 0.122000 − 0.984586i
\(73\) −1.93431 + 1.29247i −0.226394 + 0.151272i −0.663595 − 0.748092i \(-0.730970\pi\)
0.437200 + 0.899364i \(0.355970\pi\)
\(74\) −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 −0.996576 − 0.0826847i \(-0.973651\pi\)
0.646219 − 0.763152i \(-0.276349\pi\)
\(80\) −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 0.837903 − 0.545820i \(-0.183782\pi\)
0.715318 + 0.698799i \(0.246282\pi\)
\(84\) −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 −0.277830 − 0.960630i \(-0.589615\pi\)
0.110936 + 0.993828i \(0.464615\pi\)
\(90\) 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 −0.134927 + 0.990856i \(0.543080\pi\)
−0.990856 + 0.134927i \(0.956920\pi\)
\(98\) −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.109.8 yes 240
4.3 odd 2 512.2.k.a.273.8 240
128.27 odd 32 512.2.k.a.497.8 240
128.101 even 32 inner 128.2.k.a.101.8 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.8 ✓ 240 128.101 even 32 inner
128.2.k.a.109.8 yes 240 1.1 even 1 trivial
512.2.k.a.273.8 240 4.3 odd 2
512.2.k.a.497.8 240 128.27 odd 32