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Properties

Label 5000.2.a.r.1.11
Level $5000$
Weight $2$
Character 5000.1
Self dual yes
Analytic conductor $39.925$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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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] = [5000,2,Mod(1,5000)] 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("5000.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5000 = 2^{3} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,4,0,0,0,8,0,20,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9252010106\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{3} \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(1.56513\) of defining polynomial
Character \(\chi\) \(=\) 5000.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.56513 q^{3} +0.0338937 q^{7} -0.550359 q^{9} +5.79645 q^{11} +3.11995 q^{13} +4.09048 q^{17} -0.00975477 q^{19} +0.0530481 q^{21} +2.32290 q^{23} -5.55678 q^{27} +5.42482 q^{29} -4.69793 q^{31} +9.07222 q^{33} -10.6567 q^{37} +4.88313 q^{39} +5.88332 q^{41} +0.480100 q^{43} +11.3463 q^{47} -6.99885 q^{49} +6.40215 q^{51} +6.98236 q^{53} -0.0152675 q^{57} -7.09155 q^{59} -5.31851 q^{61} -0.0186537 q^{63} -2.91997 q^{67} +3.63565 q^{69} +15.6928 q^{71} -7.17390 q^{73} +0.196463 q^{77} -0.358138 q^{79} -7.04603 q^{81} +3.21582 q^{83} +8.49057 q^{87} -15.2424 q^{89} +0.105746 q^{91} -7.35289 q^{93} +2.14700 q^{97} -3.19013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 8 q^{7} + 20 q^{9} - 12 q^{11} + 10 q^{13} + 8 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} + 22 q^{27} + 16 q^{29} - 2 q^{31} + 24 q^{33} + 22 q^{37} - 4 q^{39} + 20 q^{41} + 26 q^{43} + 24 q^{47}+ \cdots - 46 q^{99}+O(q^{100}) \) 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
\(3\) 1.56513 0.903630 0.451815 − 0.892112i \(-0.350777\pi\)
0.451815 + 0.892112i \(0.350777\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.0338937 0.0128106 0.00640530 − 0.999979i \(-0.497961\pi\)
0.00640530 + 0.999979i \(0.497961\pi\)
\(8\) 0 0
\(9\) −0.550359 −0.183453
\(10\) 0 0
\(11\) 5.79645 1.74770 0.873848 − 0.486198i \(-0.161617\pi\)
0.873848 + 0.486198i \(0.161617\pi\)
\(12\) 0 0
\(13\) 3.11995 0.865317 0.432659 − 0.901558i \(-0.357576\pi\)
0.432659 + 0.901558i \(0.357576\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.09048 0.992088 0.496044 − 0.868298i \(-0.334786\pi\)
0.496044 + 0.868298i \(0.334786\pi\)
\(18\) 0 0
\(19\) −0.00975477 −0.00223790 −0.00111895 − 0.999999i \(-0.500356\pi\)
−0.00111895 + 0.999999i \(0.500356\pi\)
\(20\) 0 0
\(21\) 0.0530481 0.0115760
\(22\) 0 0
\(23\) 2.32290 0.484358 0.242179 − 0.970232i \(-0.422138\pi\)
0.242179 + 0.970232i \(0.422138\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.55678 −1.06940
\(28\) 0 0
\(29\) 5.42482 1.00736 0.503682 − 0.863889i \(-0.331978\pi\)
0.503682 + 0.863889i \(0.331978\pi\)
\(30\) 0 0
\(31\) −4.69793 −0.843773 −0.421887 − 0.906649i \(-0.638632\pi\)
−0.421887 + 0.906649i \(0.638632\pi\)
\(32\) 0 0
\(33\) 9.07222 1.57927
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.6567 −1.75196 −0.875978 − 0.482351i \(-0.839783\pi\)
−0.875978 + 0.482351i \(0.839783\pi\)
\(38\) 0 0
\(39\) 4.88313 0.781927
\(40\) 0 0
\(41\) 5.88332 0.918821 0.459410 − 0.888224i \(-0.348061\pi\)
0.459410 + 0.888224i \(0.348061\pi\)
\(42\) 0 0
\(43\) 0.480100 0.0732146 0.0366073 − 0.999330i \(-0.488345\pi\)
0.0366073 + 0.999330i \(0.488345\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.3463 1.65503 0.827514 − 0.561445i \(-0.189755\pi\)
0.827514 + 0.561445i \(0.189755\pi\)
\(48\) 0 0
\(49\) −6.99885 −0.999836
\(50\) 0 0
\(51\) 6.40215 0.896480
\(52\) 0 0
\(53\) 6.98236 0.959101 0.479550 − 0.877514i \(-0.340800\pi\)
0.479550 + 0.877514i \(0.340800\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.0152675 −0.00202223
\(58\) 0 0
\(59\) −7.09155 −0.923242 −0.461621 − 0.887077i \(-0.652732\pi\)
−0.461621 + 0.887077i \(0.652732\pi\)
\(60\) 0 0
\(61\) −5.31851 −0.680965 −0.340483 − 0.940251i \(-0.610590\pi\)
−0.340483 + 0.940251i \(0.610590\pi\)
\(62\) 0 0
\(63\) −0.0186537 −0.00235014
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.91997 −0.356732 −0.178366 − 0.983964i \(-0.557081\pi\)
−0.178366 + 0.983964i \(0.557081\pi\)
\(68\) 0 0
\(69\) 3.63565 0.437681
\(70\) 0 0
\(71\) 15.6928 1.86239 0.931195 − 0.364520i \(-0.118767\pi\)
0.931195 + 0.364520i \(0.118767\pi\)
\(72\) 0 0
\(73\) −7.17390 −0.839642 −0.419821 − 0.907607i \(-0.637907\pi\)
−0.419821 + 0.907607i \(0.637907\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.196463 0.0223891
\(78\) 0 0
\(79\) −0.358138 −0.0402936 −0.0201468 − 0.999797i \(-0.506413\pi\)
−0.0201468 + 0.999797i \(0.506413\pi\)
\(80\) 0 0
\(81\) −7.04603 −0.782892
\(82\) 0 0
\(83\) 3.21582 0.352982 0.176491 − 0.984302i \(-0.443525\pi\)
0.176491 + 0.984302i \(0.443525\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.49057 0.910284
\(88\) 0 0
\(89\) −15.2424 −1.61569 −0.807845 − 0.589395i \(-0.799366\pi\)
−0.807845 + 0.589395i \(0.799366\pi\)
\(90\) 0 0
\(91\) 0.105746 0.0110852
\(92\) 0 0
\(93\) −7.35289 −0.762459
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.14700 0.217995 0.108998 − 0.994042i \(-0.465236\pi\)
0.108998 + 0.994042i \(0.465236\pi\)
\(98\) 0 0
\(99\) −3.19013 −0.320620
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 5000.2.a.r.1.11 16
4.3 odd 2 10000.2.a.bq.1.6 16
5.4 even 2 5000.2.a.q.1.6 16
20.19 odd 2 10000.2.a.br.1.11 16
25.2 odd 20 1000.2.q.c.649.3 32
25.9 even 10 1000.2.m.e.401.6 32
25.11 even 5 1000.2.m.d.601.3 32
25.12 odd 20 200.2.q.a.169.6 yes 32
25.13 odd 20 1000.2.q.c.849.3 32
25.14 even 10 1000.2.m.e.601.6 32
25.16 even 5 1000.2.m.d.401.3 32
25.23 odd 20 200.2.q.a.129.6 ✓ 32
100.23 even 20 400.2.y.d.129.3 32
100.87 even 20 400.2.y.d.369.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.q.a.129.6 ✓ 32 25.23 odd 20
200.2.q.a.169.6 yes 32 25.12 odd 20
400.2.y.d.129.3 32 100.23 even 20
400.2.y.d.369.3 32 100.87 even 20
1000.2.m.d.401.3 32 25.16 even 5
1000.2.m.d.601.3 32 25.11 even 5
1000.2.m.e.401.6 32 25.9 even 10
1000.2.m.e.601.6 32 25.14 even 10
1000.2.q.c.649.3 32 25.2 odd 20
1000.2.q.c.849.3 32 25.13 odd 20
5000.2.a.q.1.6 16 5.4 even 2
5000.2.a.r.1.11 16 1.1 even 1 trivial
10000.2.a.bq.1.6 16 4.3 odd 2
10000.2.a.br.1.11 16 20.19 odd 2