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Properties

Label 512.2.k.a.497.8
Level $512$
Weight $2$
Character 512.497
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 497.8
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.8

$q$-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 \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{9}{32}\right)\) \(1\)

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 0
\(3\) 0.135634 − 0.0724978i 0.0783083 − 0.0418566i −0.431778 − 0.901980i \(-0.642114\pi\)
0.510086 + 0.860123i \(0.329614\pi\)
\(4\) 0 0
\(5\) 2.12942 + 2.59470i 0.952305 + 1.16039i 0.986956 + 0.160987i \(0.0514678\pi\)
−0.0346516 + 0.999399i \(0.511032\pi\)
\(6\) 0 0
\(7\) 1.11060 − 0.742082i 0.419769 − 0.280480i −0.327694 − 0.944784i \(-0.606272\pi\)
0.747463 + 0.664303i \(0.231272\pi\)
\(8\) 0 0
\(9\) −1.65357 + 2.47474i −0.551190 + 0.824914i
\(10\) 0 0
\(11\) 0.181497 − 0.0550566i 0.0547235 − 0.0166002i −0.262804 − 0.964849i \(-0.584647\pi\)
0.317528 + 0.948249i \(0.397147\pi\)
\(12\) 0 0
\(13\) 0.374137 + 0.307046i 0.103767 + 0.0851593i 0.684846 − 0.728688i \(-0.259869\pi\)
−0.581079 + 0.813847i \(0.697369\pi\)
\(14\) 0 0
\(15\) 0.476932 + 0.197552i 0.123143 + 0.0510076i
\(16\) 0 0
\(17\) 1.79255 − 0.742500i 0.434758 − 0.180083i −0.154561 − 0.987983i \(-0.549396\pi\)
0.589319 + 0.807901i \(0.299396\pi\)
\(18\) 0 0
\(19\) 0.487163 + 4.94625i 0.111763 + 1.13475i 0.873487 + 0.486847i \(0.161853\pi\)
−0.761724 + 0.647901i \(0.775647\pi\)
\(20\) 0 0
\(21\) 0.0968362 − 0.181168i 0.0211314 − 0.0395340i
\(22\) 0 0
\(23\) −5.40603 + 1.07533i −1.12724 + 0.224221i −0.723278 − 0.690557i \(-0.757366\pi\)
−0.403958 + 0.914778i \(0.632366\pi\)
\(24\) 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 −0.363321 − 0.931664i \(-0.618357\pi\)
0.819695 − 0.572800i \(-0.194143\pi\)
\(30\) 0 0
\(31\) −3.30297 − 3.30297i −0.593232 − 0.593232i 0.345271 − 0.938503i \(-0.387787\pi\)
−0.938503 + 0.345271i \(0.887787\pi\)
\(32\) 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 0.755694 − 0.654925i \(-0.227300\pi\)
0.613404 + 0.789770i \(0.289800\pi\)
\(38\) 0 0
\(39\) 0.0730059 + 0.0145218i 0.0116903 + 0.00232534i
\(40\) 0 0
\(41\) 0.580168 + 2.91670i 0.0906070 + 0.455512i 0.999278 + 0.0379988i \(0.0120983\pi\)
−0.908671 + 0.417513i \(0.862902\pi\)
\(42\) 0 0
\(43\) 8.36535 + 4.47137i 1.27570 + 0.681878i 0.963391 − 0.268101i \(-0.0863962\pi\)
0.312313 + 0.949979i \(0.398896\pi\)
\(44\) 0 0
\(45\) −9.94237 + 0.979238i −1.48212 + 0.145976i
\(46\) 0 0
\(47\) 3.57155 + 8.62250i 0.520965 + 1.25772i 0.937305 + 0.348511i \(0.113313\pi\)
−0.416340 + 0.909209i \(0.636687\pi\)
\(48\) 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 −0.859227 − 0.511595i \(-0.829055\pi\)
0.430194 − 0.902737i \(-0.358445\pi\)
\(54\) 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 0.448095 − 0.893986i \(-0.352103\pi\)
0.964227 − 0.265077i \(-0.0853974\pi\)
\(60\) 0 0
\(61\) 0.339850 + 0.635815i 0.0435134 + 0.0814078i 0.902738 − 0.430190i \(-0.141554\pi\)
−0.859225 + 0.511598i \(0.829054\pi\)
\(62\) 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 0.885381 − 0.464866i \(-0.153898\pi\)
−0.878414 + 0.477901i \(0.841398\pi\)
\(68\) 0 0
\(69\) −0.655283 + 0.537776i −0.0788867 + 0.0647407i
\(70\) 0 0
\(71\) −8.81399 − 13.1911i −1.04603 − 1.56549i −0.803461 − 0.595357i \(-0.797011\pi\)
−0.242567 − 0.970135i \(-0.577989\pi\)
\(72\) 0 0
\(73\) −1.93431 − 1.29247i −0.226394 − 0.151272i 0.437200 − 0.899364i \(-0.355970\pi\)
−0.663595 + 0.748092i \(0.730970\pi\)
\(74\) 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 −0.646219 − 0.763152i \(-0.723651\pi\)
0.996576 − 0.0826847i \(-0.0263494\pi\)
\(80\) 0 0
\(81\) −3.36290 − 8.11876i −0.373656 − 0.902085i
\(82\) 0 0
\(83\) −14.1505 + 1.39370i −1.55322 + 0.152979i −0.837903 − 0.545820i \(-0.816218\pi\)
−0.715318 + 0.698799i \(0.753718\pi\)
\(84\) 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 0.110936 − 0.993828i \(-0.464615\pi\)
−0.277830 + 0.960630i \(0.589615\pi\)
\(90\) 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 −0.990856 − 0.134927i \(-0.956920\pi\)
−0.134927 − 0.990856i \(-0.543080\pi\)
\(98\) 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.497.8 240
4.3 odd 2 128.2.k.a.101.8 ✓ 240
128.19 odd 32 128.2.k.a.109.8 yes 240
128.109 even 32 inner 512.2.k.a.273.8 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.8 ✓ 240 4.3 odd 2
128.2.k.a.109.8 yes 240 128.19 odd 32
512.2.k.a.273.8 240 128.109 even 32 inner
512.2.k.a.497.8 240 1.1 even 1 trivial