↑

Properties

Label 512.2.k.a.497.9
Level $512$
Weight $2$
Character 512.497
Analytic conductor $4.088$
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] = [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 497.9
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.467037 - 0.249636i) q^{3} +(0.246231 + 0.300034i) q^{5} +(-1.89160 + 1.26393i) q^{7} +(-1.51091 + 2.26123i) q^{9} +(-5.42541 + 1.64578i) q^{11} +(5.51786 + 4.52839i) q^{13} +(0.189898 + 0.0786585i) q^{15} +(-3.03177 + 1.25580i) q^{17} +(0.249350 + 2.53169i) q^{19} +(-0.567925 + 1.06251i) q^{21} +(5.91175 - 1.17592i) q^{23} +(0.946061 - 4.75617i) q^{25} +(-0.296883 + 3.01431i) q^{27} +(-0.110196 + 0.363268i) q^{29} +(0.158513 + 0.158513i) q^{31} +(-2.12302 + 2.12302i) q^{33} +(-0.844991 - 0.256325i) q^{35} +(-4.28002 - 0.421545i) q^{37} +(3.70750 + 0.737467i) q^{39} +(-1.22722 - 6.16964i) q^{41} +(7.59540 + 4.05983i) q^{43} +(-1.05048 + 0.103463i) q^{45} +(-1.45718 - 3.51794i) q^{47} +(-0.698143 + 1.68547i) q^{49} +(-1.10246 + 1.34335i) q^{51} +(2.06447 + 6.80565i) q^{53} +(-1.82969 - 1.22256i) q^{55} +(0.748459 + 1.12015i) q^{57} +(-7.12505 + 5.84738i) q^{59} +(2.08843 + 3.90718i) q^{61} -6.18702i q^{63} +2.77058i q^{65} +(-1.02948 - 1.92602i) q^{67} +(2.46745 - 2.02499i) q^{69} +(2.79145 + 4.17771i) q^{71} +(-8.95387 - 5.98279i) q^{73} +(-0.745468 - 2.45748i) q^{75} +(8.18256 - 9.97047i) q^{77} +(3.96157 - 9.56408i) q^{79} +(-2.50836 - 6.05573i) q^{81} +(10.9186 - 1.07539i) q^{83} +(-1.12330 - 0.600415i) q^{85} +(0.0392192 + 0.197168i) q^{87} +(-2.99053 - 0.594853i) q^{89} +(-16.1612 - 1.59173i) q^{91} +(0.113602 + 0.0344609i) q^{93} +(-0.698195 + 0.698195i) q^{95} +(4.90076 + 4.90076i) q^{97} +(4.47579 - 14.7547i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(3\) 0.467037 − 0.249636i 0.269644 − 0.144128i −0.331034 − 0.943619i \(-0.607397\pi\)
0.600678 + 0.799491i \(0.294897\pi\)
\(4\) 0 0
\(5\) 0.246231 + 0.300034i 0.110118 + 0.134179i 0.825153 − 0.564910i \(-0.191089\pi\)
−0.715035 + 0.699089i \(0.753589\pi\)
\(6\) 0 0
\(7\) −1.89160 + 1.26393i −0.714958 + 0.477720i −0.859081 − 0.511840i \(-0.828964\pi\)
0.144123 + 0.989560i \(0.453964\pi\)
\(8\) 0 0
\(9\) −1.51091 + 2.26123i −0.503635 + 0.753743i
\(10\) 0 0
\(11\) −5.42541 + 1.64578i −1.63582 + 0.496221i −0.968597 − 0.248636i \(-0.920018\pi\)
−0.667224 + 0.744857i \(0.732518\pi\)
\(12\) 0 0
\(13\) 5.51786 + 4.52839i 1.53038 + 1.25595i 0.855202 + 0.518296i \(0.173433\pi\)
0.675178 + 0.737655i \(0.264067\pi\)
\(14\) 0 0
\(15\) 0.189898 + 0.0786585i 0.0490315 + 0.0203095i
\(16\) 0 0
\(17\) −3.03177 + 1.25580i −0.735312 + 0.304576i −0.718733 − 0.695286i \(-0.755278\pi\)
−0.0165793 + 0.999863i \(0.505278\pi\)
\(18\) 0 0
\(19\) 0.249350 + 2.53169i 0.0572048 + 0.580810i 0.980473 + 0.196654i \(0.0630076\pi\)
−0.923268 + 0.384156i \(0.874492\pi\)
\(20\) 0 0
\(21\) −0.567925 + 1.06251i −0.123931 + 0.231859i
\(22\) 0 0
\(23\) 5.91175 − 1.17592i 1.23268 − 0.245196i 0.464586 − 0.885528i \(-0.346203\pi\)
0.768098 + 0.640332i \(0.221203\pi\)
\(24\) 0 0
\(25\) 0.946061 − 4.75617i 0.189212 − 0.951234i
\(26\) 0 0
\(27\) −0.296883 + 3.01431i −0.0571352 + 0.580104i
\(28\) 0 0
\(29\) −0.110196 + 0.363268i −0.0204629 + 0.0674571i −0.966577 − 0.256376i \(-0.917472\pi\)
0.946114 + 0.323833i \(0.104972\pi\)
\(30\) 0 0
\(31\) 0.158513 + 0.158513i 0.0284698 + 0.0284698i 0.721198 − 0.692729i \(-0.243592\pi\)
−0.692729 + 0.721198i \(0.743592\pi\)
\(32\) 0 0
\(33\) −2.12302 + 2.12302i −0.369570 + 0.369570i
\(34\) 0 0
\(35\) −0.844991 − 0.256325i −0.142830 − 0.0433269i
\(36\) 0 0
\(37\) −4.28002 − 0.421545i −0.703630 − 0.0693015i −0.260127 − 0.965574i \(-0.583764\pi\)
−0.443503 + 0.896273i \(0.646264\pi\)
\(38\) 0 0
\(39\) 3.70750 + 0.737467i 0.593675 + 0.118089i
\(40\) 0 0
\(41\) −1.22722 − 6.16964i −0.191659 − 0.963536i −0.950136 − 0.311836i \(-0.899056\pi\)
0.758477 − 0.651700i \(-0.225944\pi\)
\(42\) 0 0
\(43\) 7.59540 + 4.05983i 1.15829 + 0.619118i 0.934655 − 0.355556i \(-0.115708\pi\)
0.223633 + 0.974674i \(0.428208\pi\)
\(44\) 0 0
\(45\) −1.05048 + 0.103463i −0.156596 + 0.0154233i
\(46\) 0 0
\(47\) −1.45718 − 3.51794i −0.212551 − 0.513144i 0.781263 − 0.624202i \(-0.214576\pi\)
−0.993814 + 0.111059i \(0.964576\pi\)
\(48\) 0 0
\(49\) −0.698143 + 1.68547i −0.0997347 + 0.240781i
\(50\) 0 0
\(51\) −1.10246 + 1.34335i −0.154375 + 0.188106i
\(52\) 0 0
\(53\) 2.06447 + 6.80565i 0.283577 + 0.934828i 0.976809 + 0.214113i \(0.0686861\pi\)
−0.693232 + 0.720715i \(0.743814\pi\)
\(54\) 0 0
\(55\) −1.82969 − 1.22256i −0.246716 − 0.164850i
\(56\) 0 0
\(57\) 0.748459 + 1.12015i 0.0991358 + 0.148367i
\(58\) 0 0
\(59\) −7.12505 + 5.84738i −0.927603 + 0.761264i −0.971319 − 0.237780i \(-0.923580\pi\)
0.0437162 + 0.999044i \(0.486080\pi\)
\(60\) 0 0
\(61\) 2.08843 + 3.90718i 0.267397 + 0.500264i 0.979405 − 0.201908i \(-0.0647141\pi\)
−0.712008 + 0.702171i \(0.752214\pi\)
\(62\) 0 0
\(63\) − 6.18702i − 0.779491i
\(64\) 0 0
\(65\) 2.77058i 0.343648i
\(66\) 0 0
\(67\) −1.02948 − 1.92602i −0.125771 − 0.235300i 0.811153 − 0.584833i \(-0.198840\pi\)
−0.936924 + 0.349533i \(0.886340\pi\)
\(68\) 0 0
\(69\) 2.46745 − 2.02499i 0.297046 − 0.243780i
\(70\) 0 0
\(71\) 2.79145 + 4.17771i 0.331285 + 0.495803i 0.959296 − 0.282401i \(-0.0911310\pi\)
−0.628012 + 0.778204i \(0.716131\pi\)
\(72\) 0 0
\(73\) −8.95387 − 5.98279i −1.04797 − 0.700232i −0.0926195 − 0.995702i \(-0.529524\pi\)
−0.955352 + 0.295469i \(0.904524\pi\)
\(74\) 0 0
\(75\) −0.745468 − 2.45748i −0.0860792 − 0.283765i
\(76\) 0 0
\(77\) 8.18256 − 9.97047i 0.932489 − 1.13624i
\(78\) 0 0
\(79\) 3.96157 − 9.56408i 0.445712 − 1.07604i −0.528201 − 0.849120i \(-0.677133\pi\)
0.973912 − 0.226924i \(-0.0728669\pi\)
\(80\) 0 0
\(81\) −2.50836 − 6.05573i −0.278707 − 0.672858i
\(82\) 0 0
\(83\) 10.9186 − 1.07539i 1.19848 − 0.118040i 0.520982 − 0.853568i \(-0.325566\pi\)
0.677494 + 0.735528i \(0.263066\pi\)
\(84\) 0 0
\(85\) −1.12330 − 0.600415i −0.121839 − 0.0651242i
\(86\) 0 0
\(87\) 0.0392192 + 0.197168i 0.00420474 + 0.0211387i
\(88\) 0 0
\(89\) −2.99053 − 0.594853i −0.316995 − 0.0630543i 0.0340283 − 0.999421i \(-0.489166\pi\)
−0.351024 + 0.936367i \(0.614166\pi\)
\(90\) 0 0
\(91\) −16.1612 − 1.59173i −1.69415 − 0.166859i
\(92\) 0 0
\(93\) 0.113602 + 0.0344609i 0.0117800 + 0.00357343i
\(94\) 0 0
\(95\) −0.698195 + 0.698195i −0.0716333 + 0.0716333i
\(96\) 0 0
\(97\) 4.90076 + 4.90076i 0.497597 + 0.497597i 0.910689 − 0.413092i \(-0.135551\pi\)
−0.413092 + 0.910689i \(0.635551\pi\)
\(98\) 0 0
\(99\) 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 512.2.k.a.497.9 240
4.3 odd 2 128.2.k.a.101.2 ✓ 240
128.19 odd 32 128.2.k.a.109.2 yes 240
128.109 even 32 inner 512.2.k.a.273.9 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.2 ✓ 240 4.3 odd 2
128.2.k.a.109.2 yes 240 128.19 odd 32
512.2.k.a.273.9 240 128.109 even 32 inner
512.2.k.a.497.9 240 1.1 even 1 trivial