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Properties

Label 5000.2.a.r.1.14
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 1.14
Root \(2.93924\) 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)
 
\(f(q)\) \(=\) \(q+2.93924 q^{3} -3.13612 q^{7} +5.63915 q^{9} -3.40276 q^{11} +2.54698 q^{13} -5.02713 q^{17} +4.82313 q^{19} -9.21783 q^{21} +7.91296 q^{23} +7.75711 q^{27} +0.354499 q^{29} -1.46843 q^{31} -10.0015 q^{33} +5.92797 q^{37} +7.48618 q^{39} +7.64057 q^{41} +9.20779 q^{43} -3.01905 q^{47} +2.83527 q^{49} -14.7760 q^{51} +8.46812 q^{53} +14.1764 q^{57} -4.32055 q^{59} +8.48693 q^{61} -17.6851 q^{63} +8.61464 q^{67} +23.2581 q^{69} +7.38387 q^{71} -2.35402 q^{73} +10.6715 q^{77} +0.203196 q^{79} +5.88257 q^{81} -8.76542 q^{83} +1.04196 q^{87} -2.99087 q^{89} -7.98763 q^{91} -4.31607 q^{93} +10.2249 q^{97} -19.1887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 8 q^{7} + 20 q^{9} - 12 q^{11} + 10 q^{13} + 8 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} + 22 q^{27} + 16 q^{29} - 2 q^{31} + 24 q^{33} + 22 q^{37} - 4 q^{39} + 20 q^{41} + 26 q^{43} + 24 q^{47}+ \cdots - 46 q^{99}+O(q^{100}) \) 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
\(3\) 2.93924 1.69697 0.848486 − 0.529217i \(-0.177514\pi\)
0.848486 + 0.529217i \(0.177514\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.13612 −1.18534 −0.592671 − 0.805444i \(-0.701927\pi\)
−0.592671 + 0.805444i \(0.701927\pi\)
\(8\) 0 0
\(9\) 5.63915 1.87972
\(10\) 0 0
\(11\) −3.40276 −1.02597 −0.512985 − 0.858397i \(-0.671460\pi\)
−0.512985 + 0.858397i \(0.671460\pi\)
\(12\) 0 0
\(13\) 2.54698 0.706404 0.353202 − 0.935547i \(-0.385093\pi\)
0.353202 + 0.935547i \(0.385093\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.02713 −1.21926 −0.609629 − 0.792687i \(-0.708682\pi\)
−0.609629 + 0.792687i \(0.708682\pi\)
\(18\) 0 0
\(19\) 4.82313 1.10650 0.553251 − 0.833015i \(-0.313387\pi\)
0.553251 + 0.833015i \(0.313387\pi\)
\(20\) 0 0
\(21\) −9.21783 −2.01149
\(22\) 0 0
\(23\) 7.91296 1.64997 0.824983 − 0.565158i \(-0.191185\pi\)
0.824983 + 0.565158i \(0.191185\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 7.75711 1.49286
\(28\) 0 0
\(29\) 0.354499 0.0658289 0.0329144 − 0.999458i \(-0.489521\pi\)
0.0329144 + 0.999458i \(0.489521\pi\)
\(30\) 0 0
\(31\) −1.46843 −0.263738 −0.131869 − 0.991267i \(-0.542098\pi\)
−0.131869 + 0.991267i \(0.542098\pi\)
\(32\) 0 0
\(33\) −10.0015 −1.74104
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.92797 0.974553 0.487276 − 0.873248i \(-0.337990\pi\)
0.487276 + 0.873248i \(0.337990\pi\)
\(38\) 0 0
\(39\) 7.48618 1.19875
\(40\) 0 0
\(41\) 7.64057 1.19326 0.596628 − 0.802518i \(-0.296507\pi\)
0.596628 + 0.802518i \(0.296507\pi\)
\(42\) 0 0
\(43\) 9.20779 1.40418 0.702088 − 0.712090i \(-0.252251\pi\)
0.702088 + 0.712090i \(0.252251\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.01905 −0.440374 −0.220187 − 0.975458i \(-0.570667\pi\)
−0.220187 + 0.975458i \(0.570667\pi\)
\(48\) 0 0
\(49\) 2.83527 0.405038
\(50\) 0 0
\(51\) −14.7760 −2.06905
\(52\) 0 0
\(53\) 8.46812 1.16319 0.581593 − 0.813480i \(-0.302430\pi\)
0.581593 + 0.813480i \(0.302430\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.1764 1.87770
\(58\) 0 0
\(59\) −4.32055 −0.562488 −0.281244 − 0.959636i \(-0.590747\pi\)
−0.281244 + 0.959636i \(0.590747\pi\)
\(60\) 0 0
\(61\) 8.48693 1.08664 0.543320 − 0.839526i \(-0.317167\pi\)
0.543320 + 0.839526i \(0.317167\pi\)
\(62\) 0 0
\(63\) −17.6851 −2.22811
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.61464 1.05245 0.526223 − 0.850347i \(-0.323608\pi\)
0.526223 + 0.850347i \(0.323608\pi\)
\(68\) 0 0
\(69\) 23.2581 2.79995
\(70\) 0 0
\(71\) 7.38387 0.876304 0.438152 − 0.898901i \(-0.355633\pi\)
0.438152 + 0.898901i \(0.355633\pi\)
\(72\) 0 0
\(73\) −2.35402 −0.275517 −0.137759 − 0.990466i \(-0.543990\pi\)
−0.137759 + 0.990466i \(0.543990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.6715 1.21613
\(78\) 0 0
\(79\) 0.203196 0.0228614 0.0114307 − 0.999935i \(-0.496361\pi\)
0.0114307 + 0.999935i \(0.496361\pi\)
\(80\) 0 0
\(81\) 5.88257 0.653619
\(82\) 0 0
\(83\) −8.76542 −0.962130 −0.481065 − 0.876685i \(-0.659750\pi\)
−0.481065 + 0.876685i \(0.659750\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.04196 0.111710
\(88\) 0 0
\(89\) −2.99087 −0.317032 −0.158516 − 0.987356i \(-0.550671\pi\)
−0.158516 + 0.987356i \(0.550671\pi\)
\(90\) 0 0
\(91\) −7.98763 −0.837331
\(92\) 0 0
\(93\) −4.31607 −0.447556
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.2249 1.03818 0.519090 − 0.854720i \(-0.326271\pi\)
0.519090 + 0.854720i \(0.326271\pi\)
\(98\) 0 0
\(99\) −19.1887 −1.92853
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.1.14 16
4.3 odd 2 10000.2.a.bq.1.3 16
5.4 even 2 5000.2.a.q.1.3 16
20.19 odd 2 10000.2.a.br.1.14 16
25.3 odd 20 1000.2.q.c.49.1 32
25.4 even 10 1000.2.m.e.201.2 32
25.6 even 5 1000.2.m.d.801.7 32
25.8 odd 20 200.2.q.a.89.8 yes 32
25.17 odd 20 1000.2.q.c.449.1 32
25.19 even 10 1000.2.m.e.801.2 32
25.21 even 5 1000.2.m.d.201.7 32
25.22 odd 20 200.2.q.a.9.8 ✓ 32
100.47 even 20 400.2.y.d.209.1 32
100.83 even 20 400.2.y.d.289.1 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.q.a.9.8 ✓ 32 25.22 odd 20
200.2.q.a.89.8 yes 32 25.8 odd 20
400.2.y.d.209.1 32 100.47 even 20
400.2.y.d.289.1 32 100.83 even 20
1000.2.m.d.201.7 32 25.21 even 5
1000.2.m.d.801.7 32 25.6 even 5
1000.2.m.e.201.2 32 25.4 even 10
1000.2.m.e.801.2 32 25.19 even 10
1000.2.q.c.49.1 32 25.3 odd 20
1000.2.q.c.449.1 32 25.17 odd 20
5000.2.a.q.1.3 16 5.4 even 2
5000.2.a.r.1.14 16 1.1 even 1 trivial
10000.2.a.bq.1.3 16 4.3 odd 2
10000.2.a.br.1.14 16 20.19 odd 2