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Properties

Label 512.2.a.c.1.2
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $1$
Dimension $2$
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 = [2,0,0,0,0,0,-8] 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: $\mathrm{SU}(2)$

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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} -2.82843 q^{5} -4.00000 q^{7} -1.00000 q^{9} +1.41421 q^{11} +2.82843 q^{13} -4.00000 q^{15} -4.00000 q^{17} -7.07107 q^{19} -5.65685 q^{21} -4.00000 q^{23} +3.00000 q^{25} -5.65685 q^{27} +8.48528 q^{29} -8.00000 q^{31} +2.00000 q^{33} +11.3137 q^{35} +2.82843 q^{37} +4.00000 q^{39} +2.00000 q^{41} +4.24264 q^{43} +2.82843 q^{45} +9.00000 q^{49} -5.65685 q^{51} +2.82843 q^{53} -4.00000 q^{55} -10.0000 q^{57} +4.24264 q^{59} -8.48528 q^{61} +4.00000 q^{63} -8.00000 q^{65} +4.24264 q^{67} -5.65685 q^{69} +4.00000 q^{71} -4.00000 q^{73} +4.24264 q^{75} -5.65685 q^{77} +8.00000 q^{79} -5.00000 q^{81} -9.89949 q^{83} +11.3137 q^{85} +12.0000 q^{87} +12.0000 q^{89} -11.3137 q^{91} -11.3137 q^{93} +20.0000 q^{95} -4.00000 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9} - 8 q^{15} - 8 q^{17} - 8 q^{23} + 6 q^{25} - 16 q^{31} + 4 q^{33} + 8 q^{39} + 4 q^{41} + 18 q^{49} - 8 q^{55} - 20 q^{57} + 8 q^{63} - 16 q^{65} + 8 q^{71} - 8 q^{73} + 16 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(3\) 1.41421 0.816497 0.408248 − 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 − 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −7.07107 −1.62221 −0.811107 − 0.584898i \(-0.801135\pi\)
−0.811107 + 0.584898i \(0.801135\pi\)
\(20\) 0 0
\(21\) −5.65685 −1.23443
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 8.48528 1.57568 0.787839 − 0.615882i \(-0.211200\pi\)
0.787839 + 0.615882i \(0.211200\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 11.3137 1.91237
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 − 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.24264 0.646997 0.323498 − 0.946229i \(-0.395141\pi\)
0.323498 + 0.946229i \(0.395141\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) 0 0
\(53\) 2.82843 0.388514 0.194257 − 0.980951i \(-0.437770\pi\)
0.194257 + 0.980951i \(0.437770\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −10.0000 −1.32453
\(58\) 0 0
\(59\) 4.24264 0.552345 0.276172 − 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) 0 0
\(61\) −8.48528 −1.08643 −0.543214 − 0.839594i \(-0.682793\pi\)
−0.543214 + 0.839594i \(0.682793\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) −5.65685 −0.681005
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 4.24264 0.489898
\(76\) 0 0
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −9.89949 −1.08661 −0.543305 − 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) 11.3137 1.22714
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −11.3137 −1.18600
\(92\) 0 0
\(93\) −11.3137 −1.17318
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
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.c.1.2 yes 2
3.2 odd 2 4608.2.a.f.1.2 2
4.3 odd 2 512.2.a.d.1.1 yes 2
8.3 odd 2 512.2.a.d.1.2 yes 2
8.5 even 2 inner 512.2.a.c.1.1 ✓ 2
12.11 even 2 4608.2.a.m.1.2 2
16.3 odd 4 512.2.b.a.257.1 2
16.5 even 4 512.2.b.b.257.1 2
16.11 odd 4 512.2.b.a.257.2 2
16.13 even 4 512.2.b.b.257.2 2
24.5 odd 2 4608.2.a.f.1.1 2
24.11 even 2 4608.2.a.m.1.1 2
32.3 odd 8 1024.2.e.c.257.1 2
32.5 even 8 1024.2.e.b.769.1 2
32.11 odd 8 1024.2.e.c.769.1 2
32.13 even 8 1024.2.e.b.257.1 2
32.19 odd 8 1024.2.e.d.257.1 2
32.21 even 8 1024.2.e.e.769.1 2
32.27 odd 8 1024.2.e.d.769.1 2
32.29 even 8 1024.2.e.e.257.1 2
48.5 odd 4 4608.2.d.b.2305.2 2
48.11 even 4 4608.2.d.a.2305.2 2
48.29 odd 4 4608.2.d.b.2305.1 2
48.35 even 4 4608.2.d.a.2305.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.c.1.1 ✓ 2 8.5 even 2 inner
512.2.a.c.1.2 yes 2 1.1 even 1 trivial
512.2.a.d.1.1 yes 2 4.3 odd 2
512.2.a.d.1.2 yes 2 8.3 odd 2
512.2.b.a.257.1 2 16.3 odd 4
512.2.b.a.257.2 2 16.11 odd 4
512.2.b.b.257.1 2 16.5 even 4
512.2.b.b.257.2 2 16.13 even 4
1024.2.e.b.257.1 2 32.13 even 8
1024.2.e.b.769.1 2 32.5 even 8
1024.2.e.c.257.1 2 32.3 odd 8
1024.2.e.c.769.1 2 32.11 odd 8
1024.2.e.d.257.1 2 32.19 odd 8
1024.2.e.d.769.1 2 32.27 odd 8
1024.2.e.e.257.1 2 32.29 even 8
1024.2.e.e.769.1 2 32.21 even 8
4608.2.a.f.1.1 2 24.5 odd 2
4608.2.a.f.1.2 2 3.2 odd 2
4608.2.a.m.1.1 2 24.11 even 2
4608.2.a.m.1.2 2 12.11 even 2
4608.2.d.a.2305.1 2 48.35 even 4
4608.2.d.a.2305.2 2 48.11 even 4
4608.2.d.b.2305.1 2 48.29 odd 4
4608.2.d.b.2305.2 2 48.5 odd 4