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Properties

Label 5800.2.a.u.1.4
Level $5800$
Weight $2$
Character 5800.1
Self dual yes
Analytic conductor $46.313$
Analytic rank $1$
Dimension $5$
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 = [5,0,1,0,0,0,-7,0,12,0,-10] 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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.3145252.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 11x^{3} + 9x^{2} + 22x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.4
Root \(3.90462\) 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)
 
\(f(q)\) \(=\) \(q+2.08906 q^{3} +3.25238 q^{7} +1.36419 q^{9} -5.61657 q^{11} +2.90462 q^{13} -5.25238 q^{17} -4.99368 q^{19} +6.79443 q^{21} -7.72018 q^{23} -3.41732 q^{27} +1.00000 q^{29} -4.73499 q^{31} -11.7334 q^{33} +6.62480 q^{37} +6.06794 q^{39} -4.43213 q^{41} -4.30286 q^{43} +2.19267 q^{47} +3.57799 q^{49} -10.9726 q^{51} -5.81394 q^{53} -10.4321 q^{57} -8.90462 q^{59} +13.2313 q^{61} +4.43686 q^{63} -12.1800 q^{67} -16.1279 q^{69} -1.31840 q^{71} -5.17974 q^{73} -18.2672 q^{77} +11.5294 q^{79} -11.2316 q^{81} -2.05809 q^{83} +2.08906 q^{87} +2.17813 q^{89} +9.44694 q^{91} -9.89169 q^{93} -12.9627 q^{97} -7.66205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - 7 q^{7} + 12 q^{9} - 10 q^{11} - 3 q^{13} - 3 q^{17} + 2 q^{19} + 4 q^{21} - 13 q^{23} + 4 q^{27} + 5 q^{29} + 7 q^{31} - 12 q^{33} - 10 q^{37} - q^{39} + 4 q^{41} - 17 q^{43} - 6 q^{47}+ \cdots - 96 q^{99}+O(q^{100}) \) 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
\(3\) 2.08906 1.20612 0.603061 − 0.797695i \(-0.293948\pi\)
0.603061 + 0.797695i \(0.293948\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.25238 1.22928 0.614642 − 0.788806i \(-0.289300\pi\)
0.614642 + 0.788806i \(0.289300\pi\)
\(8\) 0 0
\(9\) 1.36419 0.454729
\(10\) 0 0
\(11\) −5.61657 −1.69346 −0.846730 − 0.532023i \(-0.821432\pi\)
−0.846730 + 0.532023i \(0.821432\pi\)
\(12\) 0 0
\(13\) 2.90462 0.805597 0.402798 − 0.915289i \(-0.368038\pi\)
0.402798 + 0.915289i \(0.368038\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.25238 −1.27389 −0.636945 − 0.770909i \(-0.719802\pi\)
−0.636945 + 0.770909i \(0.719802\pi\)
\(18\) 0 0
\(19\) −4.99368 −1.14563 −0.572815 − 0.819685i \(-0.694149\pi\)
−0.572815 + 0.819685i \(0.694149\pi\)
\(20\) 0 0
\(21\) 6.79443 1.48267
\(22\) 0 0
\(23\) −7.72018 −1.60977 −0.804884 − 0.593432i \(-0.797773\pi\)
−0.804884 + 0.593432i \(0.797773\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.41732 −0.657663
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.73499 −0.850429 −0.425215 − 0.905093i \(-0.639801\pi\)
−0.425215 + 0.905093i \(0.639801\pi\)
\(32\) 0 0
\(33\) −11.7334 −2.04252
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.62480 1.08911 0.544555 − 0.838725i \(-0.316698\pi\)
0.544555 + 0.838725i \(0.316698\pi\)
\(38\) 0 0
\(39\) 6.06794 0.971648
\(40\) 0 0
\(41\) −4.43213 −0.692182 −0.346091 − 0.938201i \(-0.612491\pi\)
−0.346091 + 0.938201i \(0.612491\pi\)
\(42\) 0 0
\(43\) −4.30286 −0.656180 −0.328090 − 0.944646i \(-0.606405\pi\)
−0.328090 + 0.944646i \(0.606405\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.19267 0.319834 0.159917 − 0.987130i \(-0.448877\pi\)
0.159917 + 0.987130i \(0.448877\pi\)
\(48\) 0 0
\(49\) 3.57799 0.511141
\(50\) 0 0
\(51\) −10.9726 −1.53647
\(52\) 0 0
\(53\) −5.81394 −0.798606 −0.399303 − 0.916819i \(-0.630748\pi\)
−0.399303 + 0.916819i \(0.630748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.4321 −1.38177
\(58\) 0 0
\(59\) −8.90462 −1.15928 −0.579641 − 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(60\) 0 0
\(61\) 13.2313 1.69409 0.847044 − 0.531522i \(-0.178380\pi\)
0.847044 + 0.531522i \(0.178380\pi\)
\(62\) 0 0
\(63\) 4.43686 0.558992
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.1800 −1.48803 −0.744015 − 0.668163i \(-0.767081\pi\)
−0.744015 + 0.668163i \(0.767081\pi\)
\(68\) 0 0
\(69\) −16.1279 −1.94158
\(70\) 0 0
\(71\) −1.31840 −0.156466 −0.0782329 − 0.996935i \(-0.524928\pi\)
−0.0782329 + 0.996935i \(0.524928\pi\)
\(72\) 0 0
\(73\) −5.17974 −0.606243 −0.303122 − 0.952952i \(-0.598029\pi\)
−0.303122 + 0.952952i \(0.598029\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −18.2672 −2.08174
\(78\) 0 0
\(79\) 11.5294 1.29716 0.648581 − 0.761146i \(-0.275363\pi\)
0.648581 + 0.761146i \(0.275363\pi\)
\(80\) 0 0
\(81\) −11.2316 −1.24795
\(82\) 0 0
\(83\) −2.05809 −0.225905 −0.112952 − 0.993600i \(-0.536031\pi\)
−0.112952 + 0.993600i \(0.536031\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.08906 0.223971
\(88\) 0 0
\(89\) 2.17813 0.230881 0.115441 − 0.993314i \(-0.463172\pi\)
0.115441 + 0.993314i \(0.463172\pi\)
\(90\) 0 0
\(91\) 9.44694 0.990308
\(92\) 0 0
\(93\) −9.89169 −1.02572
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.9627 −1.31616 −0.658082 − 0.752946i \(-0.728632\pi\)
−0.658082 + 0.752946i \(0.728632\pi\)
\(98\) 0 0
\(99\) −7.66205 −0.770065
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.u.1.4 5
5.4 even 2 1160.2.a.h.1.2 ✓ 5
20.19 odd 2 2320.2.a.v.1.4 5
40.19 odd 2 9280.2.a.ci.1.2 5
40.29 even 2 9280.2.a.ck.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.h.1.2 ✓ 5 5.4 even 2
2320.2.a.v.1.4 5 20.19 odd 2
5800.2.a.u.1.4 5 1.1 even 1 trivial
9280.2.a.ci.1.2 5 40.19 odd 2
9280.2.a.ck.1.4 5 40.29 even 2