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Properties

Label 512.2.a.g.1.1
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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 = [4,0,0,0,0,0,0] 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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.93185\) 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)
 
\(f(q)\) \(=\) \(q-2.44949 q^{3} -3.46410 q^{5} -2.82843 q^{7} +3.00000 q^{9} -2.44949 q^{11} +3.46410 q^{13} +8.48528 q^{15} +4.00000 q^{17} +2.44949 q^{19} +6.92820 q^{21} -2.82843 q^{23} +7.00000 q^{25} -3.46410 q^{29} -5.65685 q^{31} +6.00000 q^{33} +9.79796 q^{35} +3.46410 q^{37} -8.48528 q^{39} -2.00000 q^{41} +12.2474 q^{43} -10.3923 q^{45} +11.3137 q^{47} +1.00000 q^{49} -9.79796 q^{51} -10.3923 q^{53} +8.48528 q^{55} -6.00000 q^{57} +2.44949 q^{59} +3.46410 q^{61} -8.48528 q^{63} -12.0000 q^{65} -7.34847 q^{67} +6.92820 q^{69} +2.82843 q^{71} +8.00000 q^{73} -17.1464 q^{75} +6.92820 q^{77} -5.65685 q^{79} -9.00000 q^{81} +7.34847 q^{83} -13.8564 q^{85} +8.48528 q^{87} -8.00000 q^{89} -9.79796 q^{91} +13.8564 q^{93} -8.48528 q^{95} +12.0000 q^{97} -7.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + 16 q^{17} + 28 q^{25} + 24 q^{33} - 8 q^{41} + 4 q^{49} - 24 q^{57} - 48 q^{65} + 32 q^{73} - 36 q^{81} - 32 q^{89} + 48 q^{97}+O(q^{100}) \) 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
\(3\) −2.44949 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) −3.46410 −1.54919 −0.774597 − 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 − 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 − 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 − 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 8.48528 2.19089
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 2.44949 0.561951 0.280976 − 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(20\) 0 0
\(21\) 6.92820 1.51186
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 − 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 − 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 − 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 9.79796 1.65616
\(36\) 0 0
\(37\) 3.46410 0.569495 0.284747 − 0.958603i \(-0.408090\pi\)
0.284747 + 0.958603i \(0.408090\pi\)
\(38\) 0 0
\(39\) −8.48528 −1.35873
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 12.2474 1.86772 0.933859 − 0.357641i \(-0.116419\pi\)
0.933859 + 0.357641i \(0.116419\pi\)
\(44\) 0 0
\(45\) −10.3923 −1.54919
\(46\) 0 0
\(47\) 11.3137 1.65027 0.825137 − 0.564933i \(-0.191098\pi\)
0.825137 + 0.564933i \(0.191098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.79796 −1.37199
\(52\) 0 0
\(53\) −10.3923 −1.42749 −0.713746 − 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) 8.48528 1.14416
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 2.44949 0.318896 0.159448 − 0.987206i \(-0.449029\pi\)
0.159448 + 0.987206i \(0.449029\pi\)
\(60\) 0 0
\(61\) 3.46410 0.443533 0.221766 − 0.975100i \(-0.428818\pi\)
0.221766 + 0.975100i \(0.428818\pi\)
\(62\) 0 0
\(63\) −8.48528 −1.06904
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −7.34847 −0.897758 −0.448879 − 0.893592i \(-0.648177\pi\)
−0.448879 + 0.893592i \(0.648177\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 − 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) −17.1464 −1.97990
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) −5.65685 −0.636446 −0.318223 − 0.948016i \(-0.603086\pi\)
−0.318223 + 0.948016i \(0.603086\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 7.34847 0.806599 0.403300 − 0.915068i \(-0.367863\pi\)
0.403300 + 0.915068i \(0.367863\pi\)
\(84\) 0 0
\(85\) −13.8564 −1.50294
\(86\) 0 0
\(87\) 8.48528 0.909718
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −9.79796 −1.02711
\(92\) 0 0
\(93\) 13.8564 1.43684
\(94\) 0 0
\(95\) −8.48528 −0.870572
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −7.34847 −0.738549
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.g.1.1 ✓ 4
3.2 odd 2 4608.2.a.w.1.3 4
4.3 odd 2 inner 512.2.a.g.1.3 yes 4
8.3 odd 2 inner 512.2.a.g.1.2 yes 4
8.5 even 2 inner 512.2.a.g.1.4 yes 4
12.11 even 2 4608.2.a.w.1.4 4
16.3 odd 4 512.2.b.d.257.4 4
16.5 even 4 512.2.b.d.257.3 4
16.11 odd 4 512.2.b.d.257.1 4
16.13 even 4 512.2.b.d.257.2 4
24.5 odd 2 4608.2.a.w.1.1 4
24.11 even 2 4608.2.a.w.1.2 4
32.3 odd 8 1024.2.e.p.257.4 8
32.5 even 8 1024.2.e.p.769.3 8
32.11 odd 8 1024.2.e.p.769.4 8
32.13 even 8 1024.2.e.p.257.3 8
32.19 odd 8 1024.2.e.p.257.1 8
32.21 even 8 1024.2.e.p.769.2 8
32.27 odd 8 1024.2.e.p.769.1 8
32.29 even 8 1024.2.e.p.257.2 8
48.5 odd 4 4608.2.d.d.2305.4 4
48.11 even 4 4608.2.d.d.2305.3 4
48.29 odd 4 4608.2.d.d.2305.2 4
48.35 even 4 4608.2.d.d.2305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.g.1.1 ✓ 4 1.1 even 1 trivial
512.2.a.g.1.2 yes 4 8.3 odd 2 inner
512.2.a.g.1.3 yes 4 4.3 odd 2 inner
512.2.a.g.1.4 yes 4 8.5 even 2 inner
512.2.b.d.257.1 4 16.11 odd 4
512.2.b.d.257.2 4 16.13 even 4
512.2.b.d.257.3 4 16.5 even 4
512.2.b.d.257.4 4 16.3 odd 4
1024.2.e.p.257.1 8 32.19 odd 8
1024.2.e.p.257.2 8 32.29 even 8
1024.2.e.p.257.3 8 32.13 even 8
1024.2.e.p.257.4 8 32.3 odd 8
1024.2.e.p.769.1 8 32.27 odd 8
1024.2.e.p.769.2 8 32.21 even 8
1024.2.e.p.769.3 8 32.5 even 8
1024.2.e.p.769.4 8 32.11 odd 8
4608.2.a.w.1.1 4 24.5 odd 2
4608.2.a.w.1.2 4 24.11 even 2
4608.2.a.w.1.3 4 3.2 odd 2
4608.2.a.w.1.4 4 12.11 even 2
4608.2.d.d.2305.1 4 48.35 even 4
4608.2.d.d.2305.2 4 48.29 odd 4
4608.2.d.d.2305.3 4 48.11 even 4
4608.2.d.d.2305.4 4 48.5 odd 4