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Properties

Label 5800.2.a.r.1.3
Level $5800$
Weight $2$
Character 5800.1
Self dual yes
Analytic conductor $46.313$
Analytic rank $0$
Dimension $3$
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] = [5800,2,Mod(1,5800)] 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("5800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,2,0,0,0,-2,0,3,0,-4] 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: \(46.3132331723\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 5800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.90321 q^{3} +1.52543 q^{7} +5.42864 q^{9} +0.903212 q^{11} -2.42864 q^{13} -2.28100 q^{17} -0.474572 q^{19} +4.42864 q^{21} +2.90321 q^{23} +7.05086 q^{27} +1.00000 q^{29} +5.33185 q^{31} +2.62222 q^{33} +3.52543 q^{37} -7.05086 q^{39} +4.62222 q^{41} +12.7096 q^{43} +1.65878 q^{47} -4.67307 q^{49} -6.62222 q^{51} +2.13335 q^{53} -1.37778 q^{57} -7.18421 q^{59} -7.67307 q^{61} +8.28100 q^{63} +2.04593 q^{67} +8.42864 q^{69} +3.18421 q^{71} +8.90321 q^{73} +1.37778 q^{77} +5.19850 q^{79} +4.18421 q^{81} +15.6271 q^{83} +2.90321 q^{87} +11.8064 q^{89} -3.70471 q^{91} +15.4795 q^{93} -3.39207 q^{97} +4.90321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 2 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 8 q^{19} + 2 q^{23} + 8 q^{27} + 3 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{37} - 8 q^{39} + 14 q^{41} + 18 q^{43} - 2 q^{47} - q^{49} - 20 q^{51} + 6 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) 2.90321 1.67617 0.838085 − 0.545540i \(-0.183675\pi\)
0.838085 + 0.545540i \(0.183675\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.52543 0.576557 0.288279 − 0.957547i \(-0.406917\pi\)
0.288279 + 0.957547i \(0.406917\pi\)
\(8\) 0 0
\(9\) 5.42864 1.80955
\(10\) 0 0
\(11\) 0.903212 0.272329 0.136164 − 0.990686i \(-0.456522\pi\)
0.136164 + 0.990686i \(0.456522\pi\)
\(12\) 0 0
\(13\) −2.42864 −0.673583 −0.336792 − 0.941579i \(-0.609342\pi\)
−0.336792 + 0.941579i \(0.609342\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.28100 −0.553223 −0.276611 − 0.960982i \(-0.589211\pi\)
−0.276611 + 0.960982i \(0.589211\pi\)
\(18\) 0 0
\(19\) −0.474572 −0.108874 −0.0544372 − 0.998517i \(-0.517336\pi\)
−0.0544372 + 0.998517i \(0.517336\pi\)
\(20\) 0 0
\(21\) 4.42864 0.966408
\(22\) 0 0
\(23\) 2.90321 0.605362 0.302681 − 0.953092i \(-0.402118\pi\)
0.302681 + 0.953092i \(0.402118\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 7.05086 1.35694
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.33185 0.957629 0.478814 − 0.877916i \(-0.341067\pi\)
0.478814 + 0.877916i \(0.341067\pi\)
\(32\) 0 0
\(33\) 2.62222 0.456469
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.52543 0.579577 0.289788 − 0.957091i \(-0.406415\pi\)
0.289788 + 0.957091i \(0.406415\pi\)
\(38\) 0 0
\(39\) −7.05086 −1.12904
\(40\) 0 0
\(41\) 4.62222 0.721869 0.360934 − 0.932591i \(-0.382458\pi\)
0.360934 + 0.932591i \(0.382458\pi\)
\(42\) 0 0
\(43\) 12.7096 1.93820 0.969101 − 0.246666i \(-0.0793349\pi\)
0.969101 + 0.246666i \(0.0793349\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.65878 0.241958 0.120979 − 0.992655i \(-0.461397\pi\)
0.120979 + 0.992655i \(0.461397\pi\)
\(48\) 0 0
\(49\) −4.67307 −0.667582
\(50\) 0 0
\(51\) −6.62222 −0.927296
\(52\) 0 0
\(53\) 2.13335 0.293039 0.146519 − 0.989208i \(-0.453193\pi\)
0.146519 + 0.989208i \(0.453193\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.37778 −0.182492
\(58\) 0 0
\(59\) −7.18421 −0.935304 −0.467652 − 0.883913i \(-0.654900\pi\)
−0.467652 + 0.883913i \(0.654900\pi\)
\(60\) 0 0
\(61\) −7.67307 −0.982436 −0.491218 − 0.871037i \(-0.663448\pi\)
−0.491218 + 0.871037i \(0.663448\pi\)
\(62\) 0 0
\(63\) 8.28100 1.04331
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.04593 0.249950 0.124975 − 0.992160i \(-0.460115\pi\)
0.124975 + 0.992160i \(0.460115\pi\)
\(68\) 0 0
\(69\) 8.42864 1.01469
\(70\) 0 0
\(71\) 3.18421 0.377896 0.188948 − 0.981987i \(-0.439492\pi\)
0.188948 + 0.981987i \(0.439492\pi\)
\(72\) 0 0
\(73\) 8.90321 1.04204 0.521021 − 0.853544i \(-0.325551\pi\)
0.521021 + 0.853544i \(0.325551\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.37778 0.157013
\(78\) 0 0
\(79\) 5.19850 0.584877 0.292438 − 0.956284i \(-0.405533\pi\)
0.292438 + 0.956284i \(0.405533\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) 15.6271 1.71530 0.857651 − 0.514233i \(-0.171923\pi\)
0.857651 + 0.514233i \(0.171923\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.90321 0.311257
\(88\) 0 0
\(89\) 11.8064 1.25148 0.625739 − 0.780032i \(-0.284797\pi\)
0.625739 + 0.780032i \(0.284797\pi\)
\(90\) 0 0
\(91\) −3.70471 −0.388360
\(92\) 0 0
\(93\) 15.4795 1.60515
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.39207 −0.344413 −0.172206 − 0.985061i \(-0.555090\pi\)
−0.172206 + 0.985061i \(0.555090\pi\)
\(98\) 0 0
\(99\) 4.90321 0.492791
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 5800.2.a.r.1.3 3
5.4 even 2 1160.2.a.e.1.1 ✓ 3
20.19 odd 2 2320.2.a.r.1.3 3
40.19 odd 2 9280.2.a.bl.1.1 3
40.29 even 2 9280.2.a.bv.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.e.1.1 ✓ 3 5.4 even 2
2320.2.a.r.1.3 3 20.19 odd 2
5800.2.a.r.1.3 3 1.1 even 1 trivial
9280.2.a.bl.1.1 3 40.19 odd 2
9280.2.a.bv.1.3 3 40.29 even 2