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Properties

Label 512.2.a.a.1.2
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $1$
Dimension $2$
CM discriminant -8
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(1,512)] 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("512.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.08834058349\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 512.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.585786 q^{3} -2.65685 q^{9} -6.24264 q^{11} +5.65685 q^{17} -7.41421 q^{19} -5.00000 q^{25} +3.31371 q^{27} +3.65685 q^{33} -6.00000 q^{41} -13.0711 q^{43} -7.00000 q^{49} -3.31371 q^{51} +4.34315 q^{57} +14.2426 q^{59} +3.89949 q^{67} +16.9706 q^{73} +2.92893 q^{75} +6.02944 q^{81} +10.7279 q^{83} -5.65685 q^{89} -16.9706 q^{97} +16.5858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 6 q^{9} - 4 q^{11} - 12 q^{19} - 10 q^{25} - 16 q^{27} - 4 q^{33} - 12 q^{41} - 12 q^{43} - 14 q^{49} + 16 q^{51} + 20 q^{57} + 20 q^{59} - 12 q^{67} + 20 q^{75} + 46 q^{81} - 4 q^{83} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.585786 −0.338204 −0.169102 − 0.985599i \(-0.554087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.65685 −0.885618
\(10\) 0 0
\(11\) −6.24264 −1.88223 −0.941113 − 0.338091i \(-0.890219\pi\)
−0.941113 + 0.338091i \(0.890219\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 − 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) −7.41421 −1.70094 −0.850469 − 0.526026i \(-0.823682\pi\)
−0.850469 + 0.526026i \(0.823682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 3.31371 0.637723
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.65685 0.636577
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −13.0711 −1.99332 −0.996660 − 0.0816682i \(-0.973975\pi\)
−0.996660 + 0.0816682i \(0.973975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −3.31371 −0.464012
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.34315 0.575264
\(58\) 0 0
\(59\) 14.2426 1.85423 0.927117 − 0.374772i \(-0.122279\pi\)
0.927117 + 0.374772i \(0.122279\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.89949 0.476399 0.238200 − 0.971216i \(-0.423443\pi\)
0.238200 + 0.971216i \(0.423443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 16.9706 1.98625 0.993127 − 0.117041i \(-0.0373409\pi\)
0.993127 + 0.117041i \(0.0373409\pi\)
\(74\) 0 0
\(75\) 2.92893 0.338204
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 0 0
\(83\) 10.7279 1.17754 0.588771 − 0.808300i \(-0.299612\pi\)
0.588771 + 0.808300i \(0.299612\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.65685 −0.599625 −0.299813 − 0.953998i \(-0.596924\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.9706 −1.72310 −0.861550 − 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 16.5858 1.66693
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.a.a.1.2 ✓ 2
3.2 odd 2 4608.2.a.k.1.2 2
4.3 odd 2 512.2.a.f.1.1 yes 2
8.3 odd 2 CM 512.2.a.a.1.2 ✓ 2
8.5 even 2 512.2.a.f.1.1 yes 2
12.11 even 2 4608.2.a.i.1.1 2
16.3 odd 4 512.2.b.c.257.3 4
16.5 even 4 512.2.b.c.257.3 4
16.11 odd 4 512.2.b.c.257.2 4
16.13 even 4 512.2.b.c.257.2 4
24.5 odd 2 4608.2.a.i.1.1 2
24.11 even 2 4608.2.a.k.1.2 2
32.3 odd 8 1024.2.e.g.257.2 4
32.5 even 8 1024.2.e.g.769.2 4
32.11 odd 8 1024.2.e.g.769.2 4
32.13 even 8 1024.2.e.g.257.2 4
32.19 odd 8 1024.2.e.o.257.1 4
32.21 even 8 1024.2.e.o.769.1 4
32.27 odd 8 1024.2.e.o.769.1 4
32.29 even 8 1024.2.e.o.257.1 4
48.5 odd 4 4608.2.d.k.2305.4 4
48.11 even 4 4608.2.d.k.2305.1 4
48.29 odd 4 4608.2.d.k.2305.1 4
48.35 even 4 4608.2.d.k.2305.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.a.1.2 ✓ 2 1.1 even 1 trivial
512.2.a.a.1.2 ✓ 2 8.3 odd 2 CM
512.2.a.f.1.1 yes 2 4.3 odd 2
512.2.a.f.1.1 yes 2 8.5 even 2
512.2.b.c.257.2 4 16.11 odd 4
512.2.b.c.257.2 4 16.13 even 4
512.2.b.c.257.3 4 16.3 odd 4
512.2.b.c.257.3 4 16.5 even 4
1024.2.e.g.257.2 4 32.3 odd 8
1024.2.e.g.257.2 4 32.13 even 8
1024.2.e.g.769.2 4 32.5 even 8
1024.2.e.g.769.2 4 32.11 odd 8
1024.2.e.o.257.1 4 32.19 odd 8
1024.2.e.o.257.1 4 32.29 even 8
1024.2.e.o.769.1 4 32.21 even 8
1024.2.e.o.769.1 4 32.27 odd 8
4608.2.a.i.1.1 2 12.11 even 2
4608.2.a.i.1.1 2 24.5 odd 2
4608.2.a.k.1.2 2 3.2 odd 2
4608.2.a.k.1.2 2 24.11 even 2
4608.2.d.k.2305.1 4 48.11 even 4
4608.2.d.k.2305.1 4 48.29 odd 4
4608.2.d.k.2305.4 4 48.5 odd 4
4608.2.d.k.2305.4 4 48.35 even 4