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Properties

Label 500.4.a.d.1.8
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 1.8
Root \(-2.75002\) 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)
 
\(f(q)\) \(=\) \(q+9.47155 q^{3} +28.5382 q^{7} +62.7103 q^{9} -28.1948 q^{11} +42.0273 q^{13} +107.936 q^{17} -149.387 q^{19} +270.301 q^{21} -41.5714 q^{23} +338.232 q^{27} -178.737 q^{29} -206.344 q^{31} -267.049 q^{33} +131.069 q^{37} +398.064 q^{39} +325.321 q^{41} +105.060 q^{43} +62.3468 q^{47} +471.429 q^{49} +1022.32 q^{51} -294.576 q^{53} -1414.93 q^{57} -433.066 q^{59} +129.632 q^{61} +1789.64 q^{63} +187.793 q^{67} -393.745 q^{69} +251.537 q^{71} -451.439 q^{73} -804.629 q^{77} -147.796 q^{79} +1510.41 q^{81} -598.820 q^{83} -1692.92 q^{87} -101.813 q^{89} +1199.38 q^{91} -1954.40 q^{93} +1656.29 q^{97} -1768.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9} + 40 q^{11} - 128 q^{19} + 328 q^{21} + 452 q^{29} + 88 q^{31} + 1712 q^{39} + 1748 q^{41} + 2324 q^{49} + 2456 q^{51} + 664 q^{59} + 2316 q^{61} + 3256 q^{69} + 624 q^{71} + 2136 q^{79}+ \cdots + 880 q^{99}+O(q^{100}) \) 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
\(3\) 9.47155 1.82280 0.911401 − 0.411520i \(-0.135002\pi\)
0.911401 + 0.411520i \(0.135002\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 28.5382 1.54092 0.770459 − 0.637490i \(-0.220027\pi\)
0.770459 + 0.637490i \(0.220027\pi\)
\(8\) 0 0
\(9\) 62.7103 2.32260
\(10\) 0 0
\(11\) −28.1948 −0.772823 −0.386412 − 0.922326i \(-0.626286\pi\)
−0.386412 + 0.922326i \(0.626286\pi\)
\(12\) 0 0
\(13\) 42.0273 0.896638 0.448319 − 0.893874i \(-0.352023\pi\)
0.448319 + 0.893874i \(0.352023\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 107.936 1.53990 0.769949 − 0.638106i \(-0.220282\pi\)
0.769949 + 0.638106i \(0.220282\pi\)
\(18\) 0 0
\(19\) −149.387 −1.80378 −0.901888 − 0.431970i \(-0.857818\pi\)
−0.901888 + 0.431970i \(0.857818\pi\)
\(20\) 0 0
\(21\) 270.301 2.80879
\(22\) 0 0
\(23\) −41.5714 −0.376879 −0.188440 − 0.982085i \(-0.560343\pi\)
−0.188440 + 0.982085i \(0.560343\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 338.232 2.41085
\(28\) 0 0
\(29\) −178.737 −1.14451 −0.572254 − 0.820077i \(-0.693931\pi\)
−0.572254 + 0.820077i \(0.693931\pi\)
\(30\) 0 0
\(31\) −206.344 −1.19550 −0.597749 − 0.801683i \(-0.703938\pi\)
−0.597749 + 0.801683i \(0.703938\pi\)
\(32\) 0 0
\(33\) −267.049 −1.40870
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 131.069 0.582369 0.291185 − 0.956667i \(-0.405951\pi\)
0.291185 + 0.956667i \(0.405951\pi\)
\(38\) 0 0
\(39\) 398.064 1.63439
\(40\) 0 0
\(41\) 325.321 1.23919 0.619593 − 0.784923i \(-0.287298\pi\)
0.619593 + 0.784923i \(0.287298\pi\)
\(42\) 0 0
\(43\) 105.060 0.372592 0.186296 − 0.982494i \(-0.440352\pi\)
0.186296 + 0.982494i \(0.440352\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 62.3468 0.193494 0.0967470 − 0.995309i \(-0.469156\pi\)
0.0967470 + 0.995309i \(0.469156\pi\)
\(48\) 0 0
\(49\) 471.429 1.37443
\(50\) 0 0
\(51\) 1022.32 2.80693
\(52\) 0 0
\(53\) −294.576 −0.763456 −0.381728 − 0.924275i \(-0.624671\pi\)
−0.381728 + 0.924275i \(0.624671\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1414.93 −3.28792
\(58\) 0 0
\(59\) −433.066 −0.955599 −0.477800 − 0.878469i \(-0.658565\pi\)
−0.477800 + 0.878469i \(0.658565\pi\)
\(60\) 0 0
\(61\) 129.632 0.272092 0.136046 − 0.990703i \(-0.456560\pi\)
0.136046 + 0.990703i \(0.456560\pi\)
\(62\) 0 0
\(63\) 1789.64 3.57894
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 187.793 0.342427 0.171213 − 0.985234i \(-0.445231\pi\)
0.171213 + 0.985234i \(0.445231\pi\)
\(68\) 0 0
\(69\) −393.745 −0.686976
\(70\) 0 0
\(71\) 251.537 0.420451 0.210225 − 0.977653i \(-0.432580\pi\)
0.210225 + 0.977653i \(0.432580\pi\)
\(72\) 0 0
\(73\) −451.439 −0.723793 −0.361897 − 0.932218i \(-0.617871\pi\)
−0.361897 + 0.932218i \(0.617871\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −804.629 −1.19086
\(78\) 0 0
\(79\) −147.796 −0.210485 −0.105243 − 0.994447i \(-0.533562\pi\)
−0.105243 + 0.994447i \(0.533562\pi\)
\(80\) 0 0
\(81\) 1510.41 2.07189
\(82\) 0 0
\(83\) −598.820 −0.791916 −0.395958 − 0.918269i \(-0.629587\pi\)
−0.395958 + 0.918269i \(0.629587\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1692.92 −2.08621
\(88\) 0 0
\(89\) −101.813 −0.121260 −0.0606300 − 0.998160i \(-0.519311\pi\)
−0.0606300 + 0.998160i \(0.519311\pi\)
\(90\) 0 0
\(91\) 1199.38 1.38164
\(92\) 0 0
\(93\) −1954.40 −2.17915
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1656.29 1.73372 0.866861 − 0.498549i \(-0.166134\pi\)
0.866861 + 0.498549i \(0.166134\pi\)
\(98\) 0 0
\(99\) −1768.11 −1.79496
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.1.8 yes 8
4.3 odd 2 2000.4.a.r.1.1 8
5.2 odd 4 500.4.c.b.249.1 8
5.3 odd 4 500.4.c.b.249.8 8
5.4 even 2 inner 500.4.a.d.1.1 ✓ 8
20.19 odd 2 2000.4.a.r.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.4.a.d.1.1 ✓ 8 5.4 even 2 inner
500.4.a.d.1.8 yes 8 1.1 even 1 trivial
500.4.c.b.249.1 8 5.2 odd 4
500.4.c.b.249.8 8 5.3 odd 4
2000.4.a.r.1.1 8 4.3 odd 2
2000.4.a.r.1.8 8 20.19 odd 2