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Properties

Label 512.2.k.a.273.8
Level $512$
Weight $2$
Character 512.273
Analytic conductor $4.088$
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] = [512,2,Mod(17,512)] 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("512.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.k (of order \(32\), degree \(16\), not 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: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(15\) over \(\Q(\zeta_{32})\)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 273.8
Character \(\chi\) \(=\) 512.273
Dual form 512.2.k.a.497.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.135634 + 0.0724978i) q^{3} +(2.12942 - 2.59470i) q^{5} +(1.11060 + 0.742082i) q^{7} +(-1.65357 - 2.47474i) q^{9} +(0.181497 + 0.0550566i) q^{11} +(0.374137 - 0.307046i) q^{13} +(0.476932 - 0.197552i) q^{15} +(1.79255 + 0.742500i) q^{17} +(0.487163 - 4.94625i) q^{19} +(0.0968362 + 0.181168i) q^{21} +(-5.40603 - 1.07533i) q^{23} +(-1.22261 - 6.14649i) q^{25} +(-0.0900899 - 0.914698i) q^{27} +(2.45765 + 8.10178i) q^{29} +(-3.30297 + 3.30297i) q^{31} +(0.0206257 + 0.0206257i) q^{33} +(4.29042 - 1.30149i) q^{35} +(8.32789 - 0.820226i) q^{37} +(0.0730059 - 0.0145218i) q^{39} +(0.580168 - 2.91670i) q^{41} +(8.36535 - 4.47137i) q^{43} +(-9.94237 - 0.979238i) q^{45} +(3.57155 - 8.62250i) q^{47} +(-1.99603 - 4.81884i) q^{49} +(0.189301 + 0.230664i) q^{51} +(-3.12341 + 10.2965i) q^{53} +(0.529339 - 0.353693i) q^{55} +(0.424668 - 0.635561i) q^{57} +(10.8483 + 8.90293i) q^{59} +(0.339850 - 0.635815i) q^{61} -3.97554i q^{63} -1.62460i q^{65} +(0.0570291 - 0.106694i) q^{67} +(-0.655283 - 0.537776i) q^{69} +(-8.81399 + 13.1911i) q^{71} +(-1.93431 + 1.29247i) q^{73} +(0.279780 - 0.922310i) q^{75} +(0.160715 + 0.195832i) q^{77} +(3.11404 + 7.51796i) q^{79} +(-3.36290 + 8.11876i) q^{81} +(-14.1505 - 1.39370i) q^{83} +(5.74366 - 3.07005i) q^{85} +(-0.254021 + 1.27705i) q^{87} +(-1.57448 + 0.313183i) q^{89} +(0.643371 - 0.0633666i) q^{91} +(-0.687453 + 0.208537i) q^{93} +(-11.7967 - 11.7967i) q^{95} +(-11.0877 + 11.0877i) q^{97} +(-0.163868 - 0.540199i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{23}{32}\right)\) \(1\)

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 0
\(3\) 0.135634 + 0.0724978i 0.0783083 + 0.0418566i 0.510086 − 0.860123i \(-0.329614\pi\)
−0.431778 + 0.901980i \(0.642114\pi\)
\(4\) 0 0
\(5\) 2.12942 − 2.59470i 0.952305 − 1.16039i −0.0346516 − 0.999399i \(-0.511032\pi\)
0.986956 − 0.160987i \(-0.0514678\pi\)
\(6\) 0 0
\(7\) 1.11060 + 0.742082i 0.419769 + 0.280480i 0.747463 − 0.664303i \(-0.231272\pi\)
−0.327694 + 0.944784i \(0.606272\pi\)
\(8\) 0 0
\(9\) −1.65357 − 2.47474i −0.551190 − 0.824914i
\(10\) 0 0
\(11\) 0.181497 + 0.0550566i 0.0547235 + 0.0166002i 0.317528 − 0.948249i \(-0.397147\pi\)
−0.262804 + 0.964849i \(0.584647\pi\)
\(12\) 0 0
\(13\) 0.374137 − 0.307046i 0.103767 − 0.0851593i −0.581079 − 0.813847i \(-0.697369\pi\)
0.684846 + 0.728688i \(0.259869\pi\)
\(14\) 0 0
\(15\) 0.476932 − 0.197552i 0.123143 − 0.0510076i
\(16\) 0 0
\(17\) 1.79255 + 0.742500i 0.434758 + 0.180083i 0.589319 − 0.807901i \(-0.299396\pi\)
−0.154561 + 0.987983i \(0.549396\pi\)
\(18\) 0 0
\(19\) 0.487163 − 4.94625i 0.111763 − 1.13475i −0.761724 − 0.647901i \(-0.775647\pi\)
0.873487 − 0.486847i \(-0.161853\pi\)
\(20\) 0 0
\(21\) 0.0968362 + 0.181168i 0.0211314 + 0.0395340i
\(22\) 0 0
\(23\) −5.40603 − 1.07533i −1.12724 − 0.224221i −0.403958 − 0.914778i \(-0.632366\pi\)
−0.723278 + 0.690557i \(0.757366\pi\)
\(24\) 0 0
\(25\) −1.22261 − 6.14649i −0.244523 − 1.22930i
\(26\) 0 0
\(27\) −0.0900899 − 0.914698i −0.0173378 − 0.176034i
\(28\) 0 0
\(29\) 2.45765 + 8.10178i 0.456374 + 1.50446i 0.819695 + 0.572800i \(0.194143\pi\)
−0.363321 + 0.931664i \(0.618357\pi\)
\(30\) 0 0
\(31\) −3.30297 + 3.30297i −0.593232 + 0.593232i −0.938503 − 0.345271i \(-0.887787\pi\)
0.345271 + 0.938503i \(0.387787\pi\)
\(32\) 0 0
\(33\) 0.0206257 + 0.0206257i 0.00359047 + 0.00359047i
\(34\) 0 0
\(35\) 4.29042 − 1.30149i 0.725214 − 0.219991i
\(36\) 0 0
\(37\) 8.32789 − 0.820226i 1.36910 − 0.134844i 0.613404 − 0.789770i \(-0.289800\pi\)
0.755694 + 0.654925i \(0.227300\pi\)
\(38\) 0 0
\(39\) 0.0730059 − 0.0145218i 0.0116903 − 0.00232534i
\(40\) 0 0
\(41\) 0.580168 − 2.91670i 0.0906070 − 0.455512i −0.908671 − 0.417513i \(-0.862902\pi\)
0.999278 − 0.0379988i \(-0.0120983\pi\)
\(42\) 0 0
\(43\) 8.36535 − 4.47137i 1.27570 − 0.681878i 0.312313 − 0.949979i \(-0.398896\pi\)
0.963391 + 0.268101i \(0.0863962\pi\)
\(44\) 0 0
\(45\) −9.94237 − 0.979238i −1.48212 − 0.145976i
\(46\) 0 0
\(47\) 3.57155 − 8.62250i 0.520965 − 1.25772i −0.416340 − 0.909209i \(-0.636687\pi\)
0.937305 − 0.348511i \(-0.113313\pi\)
\(48\) 0 0
\(49\) −1.99603 − 4.81884i −0.285147 − 0.688406i
\(50\) 0 0
\(51\) 0.189301 + 0.230664i 0.0265075 + 0.0322995i
\(52\) 0 0
\(53\) −3.12341 + 10.2965i −0.429033 + 1.41433i 0.430194 + 0.902737i \(0.358445\pi\)
−0.859227 + 0.511595i \(0.829055\pi\)
\(54\) 0 0
\(55\) 0.529339 − 0.353693i 0.0713761 − 0.0476920i
\(56\) 0 0
\(57\) 0.424668 − 0.635561i 0.0562487 − 0.0841822i
\(58\) 0 0
\(59\) 10.8483 + 8.90293i 1.41232 + 1.15906i 0.964227 + 0.265077i \(0.0853974\pi\)
0.448095 + 0.893986i \(0.352103\pi\)
\(60\) 0 0
\(61\) 0.339850 − 0.635815i 0.0435134 − 0.0814078i −0.859225 − 0.511598i \(-0.829054\pi\)
0.902738 + 0.430190i \(0.141554\pi\)
\(62\) 0 0
\(63\) − 3.97554i − 0.500871i
\(64\) 0 0
\(65\) − 1.62460i − 0.201507i
\(66\) 0 0
\(67\) 0.0570291 − 0.106694i 0.00696721 − 0.0130347i −0.878414 − 0.477901i \(-0.841398\pi\)
0.885381 + 0.464866i \(0.153898\pi\)
\(68\) 0 0
\(69\) −0.655283 − 0.537776i −0.0788867 − 0.0647407i
\(70\) 0 0
\(71\) −8.81399 + 13.1911i −1.04603 + 1.56549i −0.242567 + 0.970135i \(0.577989\pi\)
−0.803461 + 0.595357i \(0.797011\pi\)
\(72\) 0 0
\(73\) −1.93431 + 1.29247i −0.226394 + 0.151272i −0.663595 − 0.748092i \(-0.730970\pi\)
0.437200 + 0.899364i \(0.355970\pi\)
\(74\) 0 0
\(75\) 0.279780 − 0.922310i 0.0323062 − 0.106499i
\(76\) 0 0
\(77\) 0.160715 + 0.195832i 0.0183152 + 0.0223171i
\(78\) 0 0
\(79\) 3.11404 + 7.51796i 0.350357 + 0.845837i 0.996576 + 0.0826847i \(0.0263494\pi\)
−0.646219 + 0.763152i \(0.723651\pi\)
\(80\) 0 0
\(81\) −3.36290 + 8.11876i −0.373656 + 0.902085i
\(82\) 0 0
\(83\) −14.1505 − 1.39370i −1.55322 − 0.152979i −0.715318 − 0.698799i \(-0.753718\pi\)
−0.837903 + 0.545820i \(0.816218\pi\)
\(84\) 0 0
\(85\) 5.74366 − 3.07005i 0.622988 − 0.332994i
\(86\) 0 0
\(87\) −0.254021 + 1.27705i −0.0272339 + 0.136914i
\(88\) 0 0
\(89\) −1.57448 + 0.313183i −0.166894 + 0.0331974i −0.277830 − 0.960630i \(-0.589615\pi\)
0.110936 + 0.993828i \(0.464615\pi\)
\(90\) 0 0
\(91\) 0.643371 − 0.0633666i 0.0674437 − 0.00664262i
\(92\) 0 0
\(93\) −0.687453 + 0.208537i −0.0712856 + 0.0216243i
\(94\) 0 0
\(95\) −11.7967 − 11.7967i −1.21031 − 1.21031i
\(96\) 0 0
\(97\) −11.0877 + 11.0877i −1.12578 + 1.12578i −0.134927 + 0.990856i \(0.543080\pi\)
−0.990856 + 0.134927i \(0.956920\pi\)
\(98\) 0 0
\(99\) −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 512.2.k.a.273.8 240
4.3 odd 2 128.2.k.a.109.8 yes 240
128.27 odd 32 128.2.k.a.101.8 ✓ 240
128.101 even 32 inner 512.2.k.a.497.8 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.8 ✓ 240 128.27 odd 32
128.2.k.a.109.8 yes 240 4.3 odd 2
512.2.k.a.273.8 240 1.1 even 1 trivial
512.2.k.a.497.8 240 128.101 even 32 inner