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Properties

Label 500.4.a.d.1.7
Level $500$
Weight $4$
Character 500.1
Self dual yes
Analytic conductor $29.501$
Analytic rank $0$
Dimension $8$
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] = [500,4,Mod(1,500)] 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("500.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(500, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 36x^{6} + 431x^{4} - 2016x^{2} + 2896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.95548\) of defining polynomial
Character \(\chi\) \(=\) 500.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.23253 q^{3} -23.1625 q^{7} +40.7746 q^{9} +56.4123 q^{11} +65.7814 q^{13} -20.0667 q^{17} -112.285 q^{19} -190.686 q^{21} +140.935 q^{23} +113.400 q^{27} +275.905 q^{29} +1.05669 q^{31} +464.416 q^{33} +161.870 q^{37} +541.548 q^{39} +414.221 q^{41} -267.438 q^{43} +40.1564 q^{47} +193.503 q^{49} -165.200 q^{51} -247.509 q^{53} -924.392 q^{57} -19.4661 q^{59} +26.4590 q^{61} -944.443 q^{63} +631.693 q^{67} +1160.25 q^{69} -74.9346 q^{71} -954.312 q^{73} -1306.65 q^{77} +893.251 q^{79} -167.348 q^{81} +782.870 q^{83} +2271.40 q^{87} -300.846 q^{89} -1523.66 q^{91} +8.69920 q^{93} -946.878 q^{97} +2300.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9} + 40 q^{11} - 128 q^{19} + 328 q^{21} + 452 q^{29} + 88 q^{31} + 1712 q^{39} + 1748 q^{41} + 2324 q^{49} + 2456 q^{51} + 664 q^{59} + 2316 q^{61} + 3256 q^{69} + 624 q^{71} + 2136 q^{79}+ \cdots + 880 q^{99}+O(q^{100}) \) 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
\(3\) 8.23253 1.58435 0.792176 − 0.610293i \(-0.208948\pi\)
0.792176 + 0.610293i \(0.208948\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −23.1625 −1.25066 −0.625330 − 0.780361i \(-0.715036\pi\)
−0.625330 + 0.780361i \(0.715036\pi\)
\(8\) 0 0
\(9\) 40.7746 1.51017
\(10\) 0 0
\(11\) 56.4123 1.54627 0.773135 − 0.634242i \(-0.218688\pi\)
0.773135 + 0.634242i \(0.218688\pi\)
\(12\) 0 0
\(13\) 65.7814 1.40342 0.701711 − 0.712462i \(-0.252420\pi\)
0.701711 + 0.712462i \(0.252420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20.0667 −0.286288 −0.143144 − 0.989702i \(-0.545721\pi\)
−0.143144 + 0.989702i \(0.545721\pi\)
\(18\) 0 0
\(19\) −112.285 −1.35579 −0.677895 − 0.735159i \(-0.737107\pi\)
−0.677895 + 0.735159i \(0.737107\pi\)
\(20\) 0 0
\(21\) −190.686 −1.98148
\(22\) 0 0
\(23\) 140.935 1.27769 0.638845 − 0.769335i \(-0.279412\pi\)
0.638845 + 0.769335i \(0.279412\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 113.400 0.808288
\(28\) 0 0
\(29\) 275.905 1.76670 0.883350 − 0.468714i \(-0.155282\pi\)
0.883350 + 0.468714i \(0.155282\pi\)
\(30\) 0 0
\(31\) 1.05669 0.00612214 0.00306107 − 0.999995i \(-0.499026\pi\)
0.00306107 + 0.999995i \(0.499026\pi\)
\(32\) 0 0
\(33\) 464.416 2.44983
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 161.870 0.719224 0.359612 − 0.933102i \(-0.382909\pi\)
0.359612 + 0.933102i \(0.382909\pi\)
\(38\) 0 0
\(39\) 541.548 2.22351
\(40\) 0 0
\(41\) 414.221 1.57782 0.788909 − 0.614511i \(-0.210646\pi\)
0.788909 + 0.614511i \(0.210646\pi\)
\(42\) 0 0
\(43\) −267.438 −0.948462 −0.474231 − 0.880400i \(-0.657274\pi\)
−0.474231 + 0.880400i \(0.657274\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 40.1564 0.124626 0.0623130 − 0.998057i \(-0.480152\pi\)
0.0623130 + 0.998057i \(0.480152\pi\)
\(48\) 0 0
\(49\) 193.503 0.564149
\(50\) 0 0
\(51\) −165.200 −0.453580
\(52\) 0 0
\(53\) −247.509 −0.641472 −0.320736 − 0.947169i \(-0.603930\pi\)
−0.320736 + 0.947169i \(0.603930\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −924.392 −2.14805
\(58\) 0 0
\(59\) −19.4661 −0.0429538 −0.0214769 − 0.999769i \(-0.506837\pi\)
−0.0214769 + 0.999769i \(0.506837\pi\)
\(60\) 0 0
\(61\) 26.4590 0.0555365 0.0277683 − 0.999614i \(-0.491160\pi\)
0.0277683 + 0.999614i \(0.491160\pi\)
\(62\) 0 0
\(63\) −944.443 −1.88871
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 631.693 1.15184 0.575922 − 0.817504i \(-0.304643\pi\)
0.575922 + 0.817504i \(0.304643\pi\)
\(68\) 0 0
\(69\) 1160.25 2.02431
\(70\) 0 0
\(71\) −74.9346 −0.125255 −0.0626275 − 0.998037i \(-0.519948\pi\)
−0.0626275 + 0.998037i \(0.519948\pi\)
\(72\) 0 0
\(73\) −954.312 −1.53005 −0.765026 − 0.643999i \(-0.777274\pi\)
−0.765026 + 0.643999i \(0.777274\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1306.65 −1.93386
\(78\) 0 0
\(79\) 893.251 1.27213 0.636067 − 0.771634i \(-0.280560\pi\)
0.636067 + 0.771634i \(0.280560\pi\)
\(80\) 0 0
\(81\) −167.348 −0.229558
\(82\) 0 0
\(83\) 782.870 1.03532 0.517658 − 0.855588i \(-0.326804\pi\)
0.517658 + 0.855588i \(0.326804\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2271.40 2.79907
\(88\) 0 0
\(89\) −300.846 −0.358310 −0.179155 − 0.983821i \(-0.557336\pi\)
−0.179155 + 0.983821i \(0.557336\pi\)
\(90\) 0 0
\(91\) −1523.66 −1.75520
\(92\) 0 0
\(93\) 8.69920 0.00969962
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −946.878 −0.991143 −0.495571 − 0.868567i \(-0.665041\pi\)
−0.495571 + 0.868567i \(0.665041\pi\)
\(98\) 0 0
\(99\) 2300.19 2.33513
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 500.4.a.d.1.7 yes 8
4.3 odd 2 2000.4.a.r.1.2 8
5.2 odd 4 500.4.c.b.249.2 8
5.3 odd 4 500.4.c.b.249.7 8
5.4 even 2 inner 500.4.a.d.1.2 ✓ 8
20.19 odd 2 2000.4.a.r.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.4.a.d.1.2 ✓ 8 5.4 even 2 inner
500.4.a.d.1.7 yes 8 1.1 even 1 trivial
500.4.c.b.249.2 8 5.2 odd 4
500.4.c.b.249.7 8 5.3 odd 4
2000.4.a.r.1.2 8 4.3 odd 2
2000.4.a.r.1.7 8 20.19 odd 2