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Properties

Label 500.4.a.d.1.3
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 1.3
Root \(-1.59904\) 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)
 
\(f(q)\) \(=\) \(q-3.53797 q^{3} -31.5688 q^{7} -14.4828 q^{9} -32.9959 q^{11} -18.4676 q^{13} -26.8411 q^{17} +142.895 q^{19} +111.689 q^{21} -147.357 q^{23} +146.765 q^{27} +62.9377 q^{29} -46.1387 q^{31} +116.739 q^{33} +276.166 q^{37} +65.3379 q^{39} -198.592 q^{41} -482.250 q^{43} -190.838 q^{47} +653.589 q^{49} +94.9630 q^{51} +655.970 q^{53} -505.558 q^{57} +690.817 q^{59} +358.003 q^{61} +457.204 q^{63} -835.505 q^{67} +521.346 q^{69} -466.719 q^{71} -410.538 q^{73} +1041.64 q^{77} +727.478 q^{79} -128.214 q^{81} +524.855 q^{83} -222.672 q^{87} +830.390 q^{89} +583.001 q^{91} +163.237 q^{93} +910.375 q^{97} +477.873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9} + 40 q^{11} - 128 q^{19} + 328 q^{21} + 452 q^{29} + 88 q^{31} + 1712 q^{39} + 1748 q^{41} + 2324 q^{49} + 2456 q^{51} + 664 q^{59} + 2316 q^{61} + 3256 q^{69} + 624 q^{71} + 2136 q^{79}+ \cdots + 880 q^{99}+O(q^{100}) \) 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
\(3\) −3.53797 −0.680882 −0.340441 − 0.940266i \(-0.610576\pi\)
−0.340441 + 0.940266i \(0.610576\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −31.5688 −1.70455 −0.852277 − 0.523090i \(-0.824779\pi\)
−0.852277 + 0.523090i \(0.824779\pi\)
\(8\) 0 0
\(9\) −14.4828 −0.536399
\(10\) 0 0
\(11\) −32.9959 −0.904423 −0.452211 − 0.891911i \(-0.649365\pi\)
−0.452211 + 0.891911i \(0.649365\pi\)
\(12\) 0 0
\(13\) −18.4676 −0.394000 −0.197000 − 0.980404i \(-0.563120\pi\)
−0.197000 + 0.980404i \(0.563120\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −26.8411 −0.382937 −0.191468 − 0.981499i \(-0.561325\pi\)
−0.191468 + 0.981499i \(0.561325\pi\)
\(18\) 0 0
\(19\) 142.895 1.72539 0.862694 − 0.505726i \(-0.168775\pi\)
0.862694 + 0.505726i \(0.168775\pi\)
\(20\) 0 0
\(21\) 111.689 1.16060
\(22\) 0 0
\(23\) −147.357 −1.33592 −0.667960 − 0.744197i \(-0.732832\pi\)
−0.667960 + 0.744197i \(0.732832\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 146.765 1.04611
\(28\) 0 0
\(29\) 62.9377 0.403008 0.201504 − 0.979488i \(-0.435417\pi\)
0.201504 + 0.979488i \(0.435417\pi\)
\(30\) 0 0
\(31\) −46.1387 −0.267315 −0.133657 − 0.991028i \(-0.542672\pi\)
−0.133657 + 0.991028i \(0.542672\pi\)
\(32\) 0 0
\(33\) 116.739 0.615805
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 276.166 1.22707 0.613533 − 0.789669i \(-0.289748\pi\)
0.613533 + 0.789669i \(0.289748\pi\)
\(38\) 0 0
\(39\) 65.3379 0.268267
\(40\) 0 0
\(41\) −198.592 −0.756460 −0.378230 − 0.925712i \(-0.623467\pi\)
−0.378230 + 0.925712i \(0.623467\pi\)
\(42\) 0 0
\(43\) −482.250 −1.71029 −0.855144 − 0.518390i \(-0.826532\pi\)
−0.855144 + 0.518390i \(0.826532\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −190.838 −0.592267 −0.296133 − 0.955147i \(-0.595697\pi\)
−0.296133 + 0.955147i \(0.595697\pi\)
\(48\) 0 0
\(49\) 653.589 1.90551
\(50\) 0 0
\(51\) 94.9630 0.260735
\(52\) 0 0
\(53\) 655.970 1.70008 0.850041 − 0.526716i \(-0.176577\pi\)
0.850041 + 0.526716i \(0.176577\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −505.558 −1.17479
\(58\) 0 0
\(59\) 690.817 1.52435 0.762176 − 0.647370i \(-0.224131\pi\)
0.762176 + 0.647370i \(0.224131\pi\)
\(60\) 0 0
\(61\) 358.003 0.751436 0.375718 − 0.926734i \(-0.377396\pi\)
0.375718 + 0.926734i \(0.377396\pi\)
\(62\) 0 0
\(63\) 457.204 0.914322
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −835.505 −1.52348 −0.761740 − 0.647883i \(-0.775655\pi\)
−0.761740 + 0.647883i \(0.775655\pi\)
\(68\) 0 0
\(69\) 521.346 0.909604
\(70\) 0 0
\(71\) −466.719 −0.780131 −0.390066 − 0.920787i \(-0.627548\pi\)
−0.390066 + 0.920787i \(0.627548\pi\)
\(72\) 0 0
\(73\) −410.538 −0.658217 −0.329108 − 0.944292i \(-0.606748\pi\)
−0.329108 + 0.944292i \(0.606748\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1041.64 1.54164
\(78\) 0 0
\(79\) 727.478 1.03605 0.518024 − 0.855366i \(-0.326668\pi\)
0.518024 + 0.855366i \(0.326668\pi\)
\(80\) 0 0
\(81\) −128.214 −0.175877
\(82\) 0 0
\(83\) 524.855 0.694100 0.347050 − 0.937847i \(-0.387183\pi\)
0.347050 + 0.937847i \(0.387183\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −222.672 −0.274401
\(88\) 0 0
\(89\) 830.390 0.989002 0.494501 − 0.869177i \(-0.335351\pi\)
0.494501 + 0.869177i \(0.335351\pi\)
\(90\) 0 0
\(91\) 583.001 0.671594
\(92\) 0 0
\(93\) 163.237 0.182010
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 910.375 0.952934 0.476467 − 0.879192i \(-0.341917\pi\)
0.476467 + 0.879192i \(0.341917\pi\)
\(98\) 0 0
\(99\) 477.873 0.485131
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.1.3 ✓ 8
4.3 odd 2 2000.4.a.r.1.6 8
5.2 odd 4 500.4.c.b.249.6 8
5.3 odd 4 500.4.c.b.249.3 8
5.4 even 2 inner 500.4.a.d.1.6 yes 8
20.19 odd 2 2000.4.a.r.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.4.a.d.1.3 ✓ 8 1.1 even 1 trivial
500.4.a.d.1.6 yes 8 5.4 even 2 inner
500.4.c.b.249.3 8 5.3 odd 4
500.4.c.b.249.6 8 5.2 odd 4
2000.4.a.r.1.3 8 20.19 odd 2
2000.4.a.r.1.6 8 4.3 odd 2