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Properties

Label 2000.4.a.r.1.2
Level $2000$
Weight $4$
Character 2000.1
Self dual yes
Analytic conductor $118.004$
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] = [2000,4,Mod(1,2000)] 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("2000.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(118.003820011\)
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: no (minimal twist has level 500)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.95548\) of defining polynomial
Character \(\chi\) \(=\) 2000.1

$q$-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 \(-0.791052\pi\)
−0.792176 + 0.610293i \(0.791052\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 23.1625 1.25066 0.625330 − 0.780361i \(-0.284964\pi\)
0.625330 + 0.780361i \(0.284964\pi\)
\(8\) 0 0
\(9\) 40.7746 1.51017
\(10\) 0 0
\(11\) −56.4123 −1.54627 −0.773135 − 0.634242i \(-0.781312\pi\)
−0.773135 + 0.634242i \(0.781312\pi\)
\(12\) 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 \(-0.262893\pi\)
0.677895 + 0.735159i \(0.262893\pi\)
\(20\) 0 0
\(21\) −190.686 −1.98148
\(22\) 0 0
\(23\) −140.935 −1.27769 −0.638845 − 0.769335i \(-0.720588\pi\)
−0.638845 + 0.769335i \(0.720588\pi\)
\(24\) 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 \(-0.500974\pi\)
−0.00306107 + 0.999995i \(0.500974\pi\)
\(32\) 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 \(-0.342726\pi\)
0.474231 + 0.880400i \(0.342726\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −40.1564 −0.124626 −0.0623130 − 0.998057i \(-0.519848\pi\)
−0.0623130 + 0.998057i \(0.519848\pi\)
\(48\) 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 \(-0.493163\pi\)
0.0214769 + 0.999769i \(0.493163\pi\)
\(60\) 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 \(-0.695357\pi\)
−0.575922 + 0.817504i \(0.695357\pi\)
\(68\) 0 0
\(69\) 1160.25 2.02431
\(70\) 0 0
\(71\) 74.9346 0.125255 0.0626275 − 0.998037i \(-0.480052\pi\)
0.0626275 + 0.998037i \(0.480052\pi\)
\(72\) 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 \(-0.719440\pi\)
−0.636067 + 0.771634i \(0.719440\pi\)
\(80\) 0 0
\(81\) −167.348 −0.229558
\(82\) 0 0
\(83\) −782.870 −1.03532 −0.517658 − 0.855588i \(-0.673196\pi\)
−0.517658 + 0.855588i \(0.673196\pi\)
\(84\) 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 2000.4.a.r.1.2 8
4.3 odd 2 500.4.a.d.1.7 yes 8
5.4 even 2 inner 2000.4.a.r.1.7 8
20.3 even 4 500.4.c.b.249.7 8
20.7 even 4 500.4.c.b.249.2 8
20.19 odd 2 500.4.a.d.1.2 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.4.a.d.1.2 ✓ 8 20.19 odd 2
500.4.a.d.1.7 yes 8 4.3 odd 2
500.4.c.b.249.2 8 20.7 even 4
500.4.c.b.249.7 8 20.3 even 4
2000.4.a.r.1.2 8 1.1 even 1 trivial
2000.4.a.r.1.7 8 5.4 even 2 inner