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Properties

Label 512.2.k.a.17.9
Level $512$
Weight $2$
Character 512.17
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 17.9
Character \(\chi\) \(=\) 512.17
Dual form 512.2.k.a.241.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.337004 - 0.630491i) q^{3} +(1.52625 + 1.25256i) q^{5} +(-0.541333 - 0.361707i) q^{7} +(1.38276 + 2.06945i) q^{9} +(-0.748255 + 2.46667i) q^{11} +(0.513434 + 0.625621i) q^{13} +(1.30408 - 0.540168i) q^{15} +(2.47390 + 1.02472i) q^{17} +(5.09105 + 0.501425i) q^{19} +(-0.410485 + 0.219409i) q^{21} +(-4.52344 - 0.899768i) q^{23} +(-0.214923 - 1.08049i) q^{25} +(3.90516 - 0.384625i) q^{27} +(-3.57899 + 1.08567i) q^{29} +(3.07762 - 3.07762i) q^{31} +(1.30305 + 1.30305i) q^{33} +(-0.373149 - 1.23011i) q^{35} +(0.299694 + 3.04284i) q^{37} +(0.567478 - 0.112878i) q^{39} +(2.06036 - 10.3581i) q^{41} +(6.06607 + 11.3488i) q^{43} +(-0.481672 + 4.89049i) q^{45} +(-0.624681 + 1.50811i) q^{47} +(-2.51657 - 6.07555i) q^{49} +(1.47979 - 1.21443i) q^{51} +(7.28714 + 2.21053i) q^{53} +(-4.23167 + 2.82751i) q^{55} +(2.03185 - 3.04088i) q^{57} +(0.610163 - 0.743486i) q^{59} +(-11.5491 - 6.17314i) q^{61} -1.62042i q^{63} +1.59796i q^{65} +(-3.05529 - 1.63308i) q^{67} +(-2.09172 + 2.54876i) q^{69} +(-4.48909 + 6.71839i) q^{71} +(9.82925 - 6.56769i) q^{73} +(-0.753668 - 0.228623i) q^{75} +(1.29727 - 1.06464i) q^{77} +(-5.27026 - 12.7235i) q^{79} +(-1.78384 + 4.30657i) q^{81} +(0.0466556 - 0.473702i) q^{83} +(2.49226 + 4.66269i) q^{85} +(-0.521627 + 2.62240i) q^{87} +(-8.16694 + 1.62450i) q^{89} +(-0.0516471 - 0.524382i) q^{91} +(-0.903240 - 2.97758i) q^{93} +(7.14215 + 7.14215i) q^{95} +(-5.39716 + 5.39716i) q^{97} +(-6.13931 + 1.86234i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{7}{32}\right)\) \(1\)

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
\(3\) 0.337004 − 0.630491i 0.194570 − 0.364014i −0.765651 − 0.643256i \(-0.777583\pi\)
0.960221 + 0.279242i \(0.0900831\pi\)
\(4\) 0 0
\(5\) 1.52625 + 1.25256i 0.682559 + 0.560162i 0.910498 − 0.413514i \(-0.135699\pi\)
−0.227939 + 0.973675i \(0.573199\pi\)
\(6\) 0 0
\(7\) −0.541333 − 0.361707i −0.204605 − 0.136712i 0.449047 − 0.893508i \(-0.351764\pi\)
−0.653651 + 0.756796i \(0.726764\pi\)
\(8\) 0 0
\(9\) 1.38276 + 2.06945i 0.460921 + 0.689817i
\(10\) 0 0
\(11\) −0.748255 + 2.46667i −0.225607 + 0.743728i 0.768929 + 0.639335i \(0.220790\pi\)
−0.994536 + 0.104393i \(0.966710\pi\)
\(12\) 0 0
\(13\) 0.513434 + 0.625621i 0.142401 + 0.173516i 0.839327 − 0.543627i \(-0.182949\pi\)
−0.696926 + 0.717143i \(0.745449\pi\)
\(14\) 0 0
\(15\) 1.30408 − 0.540168i 0.336712 − 0.139471i
\(16\) 0 0
\(17\) 2.47390 + 1.02472i 0.600009 + 0.248532i 0.661950 − 0.749548i \(-0.269729\pi\)
−0.0619413 + 0.998080i \(0.519729\pi\)
\(18\) 0 0
\(19\) 5.09105 + 0.501425i 1.16797 + 0.115035i 0.663351 − 0.748309i \(-0.269134\pi\)
0.504617 + 0.863343i \(0.331634\pi\)
\(20\) 0 0
\(21\) −0.410485 + 0.219409i −0.0895751 + 0.0478789i
\(22\) 0 0
\(23\) −4.52344 − 0.899768i −0.943202 − 0.187615i −0.300544 − 0.953768i \(-0.597168\pi\)
−0.642658 + 0.766153i \(0.722168\pi\)
\(24\) 0 0
\(25\) −0.214923 − 1.08049i −0.0429845 − 0.216098i
\(26\) 0 0
\(27\) 3.90516 − 0.384625i 0.751548 − 0.0740211i
\(28\) 0 0
\(29\) −3.57899 + 1.08567i −0.664602 + 0.201605i −0.604521 − 0.796589i \(-0.706636\pi\)
−0.0600802 + 0.998194i \(0.519136\pi\)
\(30\) 0 0
\(31\) 3.07762 − 3.07762i 0.552757 − 0.552757i −0.374479 − 0.927236i \(-0.622178\pi\)
0.927236 + 0.374479i \(0.122178\pi\)
\(32\) 0 0
\(33\) 1.30305 + 1.30305i 0.226831 + 0.226831i
\(34\) 0 0
\(35\) −0.373149 − 1.23011i −0.0630737 − 0.207926i
\(36\) 0 0
\(37\) 0.299694 + 3.04284i 0.0492694 + 0.500241i 0.987875 + 0.155249i \(0.0496181\pi\)
−0.938606 + 0.344991i \(0.887882\pi\)
\(38\) 0 0
\(39\) 0.567478 − 0.112878i 0.0908692 − 0.0180750i
\(40\) 0 0
\(41\) 2.06036 − 10.3581i 0.321774 − 1.61767i −0.393844 − 0.919177i \(-0.628855\pi\)
0.715619 − 0.698491i \(-0.246145\pi\)
\(42\) 0 0
\(43\) 6.06607 + 11.3488i 0.925067 + 1.73068i 0.632748 + 0.774358i \(0.281927\pi\)
0.292319 + 0.956321i \(0.405573\pi\)
\(44\) 0 0
\(45\) −0.481672 + 4.89049i −0.0718034 + 0.729032i
\(46\) 0 0
\(47\) −0.624681 + 1.50811i −0.0911191 + 0.219981i −0.962868 − 0.269971i \(-0.912986\pi\)
0.871749 + 0.489952i \(0.162986\pi\)
\(48\) 0 0
\(49\) −2.51657 − 6.07555i −0.359511 − 0.867936i
\(50\) 0 0
\(51\) 1.47979 − 1.21443i 0.207213 − 0.170055i
\(52\) 0 0
\(53\) 7.28714 + 2.21053i 1.00097 + 0.303640i 0.747902 − 0.663810i \(-0.231062\pi\)
0.253064 + 0.967449i \(0.418562\pi\)
\(54\) 0 0
\(55\) −4.23167 + 2.82751i −0.570599 + 0.381262i
\(56\) 0 0
\(57\) 2.03185 − 3.04088i 0.269125 − 0.402774i
\(58\) 0 0
\(59\) 0.610163 − 0.743486i 0.0794365 − 0.0967937i −0.731776 − 0.681546i \(-0.761308\pi\)
0.811212 + 0.584752i \(0.198808\pi\)
\(60\) 0 0
\(61\) −11.5491 − 6.17314i −1.47871 − 0.790389i −0.482382 − 0.875961i \(-0.660228\pi\)
−0.996332 + 0.0855720i \(0.972728\pi\)
\(62\) 0 0
\(63\) − 1.62042i − 0.204153i
\(64\) 0 0
\(65\) 1.59796i 0.198203i
\(66\) 0 0
\(67\) −3.05529 − 1.63308i −0.373263 − 0.199513i 0.274089 − 0.961704i \(-0.411624\pi\)
−0.647352 + 0.762191i \(0.724124\pi\)
\(68\) 0 0
\(69\) −2.09172 + 2.54876i −0.251813 + 0.306835i
\(70\) 0 0
\(71\) −4.48909 + 6.71839i −0.532757 + 0.797327i −0.996042 − 0.0888825i \(-0.971670\pi\)
0.463286 + 0.886209i \(0.346670\pi\)
\(72\) 0 0
\(73\) 9.82925 − 6.56769i 1.15043 − 0.768690i 0.174043 − 0.984738i \(-0.444317\pi\)
0.976383 + 0.216048i \(0.0693167\pi\)
\(74\) 0 0
\(75\) −0.753668 − 0.228623i −0.0870261 − 0.0263991i
\(76\) 0 0
\(77\) 1.29727 − 1.06464i 0.147837 − 0.121327i
\(78\) 0 0
\(79\) −5.27026 − 12.7235i −0.592951 − 1.43151i −0.880640 − 0.473786i \(-0.842887\pi\)
0.287690 − 0.957724i \(-0.407113\pi\)
\(80\) 0 0
\(81\) −1.78384 + 4.30657i −0.198204 + 0.478507i
\(82\) 0 0
\(83\) 0.0466556 − 0.473702i 0.00512111 − 0.0519955i −0.992285 − 0.123981i \(-0.960434\pi\)
0.997406 + 0.0719859i \(0.0229337\pi\)
\(84\) 0 0
\(85\) 2.49226 + 4.66269i 0.270323 + 0.505740i
\(86\) 0 0
\(87\) −0.521627 + 2.62240i −0.0559243 + 0.281151i
\(88\) 0 0
\(89\) −8.16694 + 1.62450i −0.865694 + 0.172197i −0.607910 − 0.794006i \(-0.707992\pi\)
−0.257783 + 0.966203i \(0.582992\pi\)
\(90\) 0 0
\(91\) −0.0516471 − 0.524382i −0.00541409 − 0.0549702i
\(92\) 0 0
\(93\) −0.903240 − 2.97758i −0.0936616 − 0.308761i
\(94\) 0 0
\(95\) 7.14215 + 7.14215i 0.732769 + 0.732769i
\(96\) 0 0
\(97\) −5.39716 + 5.39716i −0.547998 + 0.547998i −0.925861 − 0.377863i \(-0.876659\pi\)
0.377863 + 0.925861i \(0.376659\pi\)
\(98\) 0 0
\(99\) −6.13931 + 1.86234i −0.617024 + 0.187172i
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.17.9 240
4.3 odd 2 128.2.k.a.45.5 yes 240
128.37 even 32 inner 512.2.k.a.241.9 240
128.91 odd 32 128.2.k.a.37.5 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.37.5 ✓ 240 128.91 odd 32
128.2.k.a.45.5 yes 240 4.3 odd 2
512.2.k.a.17.9 240 1.1 even 1 trivial
512.2.k.a.241.9 240 128.37 even 32 inner