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Properties

Label 5800.2.a.u.1.2
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 1.2
Root \(1.58684\) 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)
 
\(f(q)\) \(=\) \(q-0.959449 q^{3} -4.10933 q^{7} -2.07946 q^{9} +5.18879 q^{11} +0.586839 q^{13} +2.10933 q^{17} +0.372610 q^{19} +3.94270 q^{21} -6.13313 q^{23} +4.87348 q^{27} +1.00000 q^{29} -1.36411 q^{31} -4.97838 q^{33} +2.71997 q^{37} -0.563043 q^{39} +5.64250 q^{41} -11.0066 q^{43} +8.36247 q^{47} +9.88660 q^{49} -2.02380 q^{51} -3.16056 q^{53} -0.357501 q^{57} -6.58684 q^{59} +2.28708 q^{61} +8.54518 q^{63} -7.61725 q^{67} +5.88443 q^{69} +14.2085 q^{71} -2.46683 q^{73} -21.3224 q^{77} +5.30681 q^{79} +1.56251 q^{81} +14.8575 q^{83} -0.959449 q^{87} -3.91890 q^{89} -2.41152 q^{91} +1.30880 q^{93} +6.27069 q^{97} -10.7899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - 7 q^{7} + 12 q^{9} - 10 q^{11} - 3 q^{13} - 3 q^{17} + 2 q^{19} + 4 q^{21} - 13 q^{23} + 4 q^{27} + 5 q^{29} + 7 q^{31} - 12 q^{33} - 10 q^{37} - q^{39} + 4 q^{41} - 17 q^{43} - 6 q^{47}+ \cdots - 96 q^{99}+O(q^{100}) \) 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
\(3\) −0.959449 −0.553938 −0.276969 − 0.960879i \(-0.589330\pi\)
−0.276969 + 0.960879i \(0.589330\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.10933 −1.55318 −0.776591 − 0.630006i \(-0.783053\pi\)
−0.776591 + 0.630006i \(0.783053\pi\)
\(8\) 0 0
\(9\) −2.07946 −0.693152
\(10\) 0 0
\(11\) 5.18879 1.56448 0.782239 − 0.622978i \(-0.214077\pi\)
0.782239 + 0.622978i \(0.214077\pi\)
\(12\) 0 0
\(13\) 0.586839 0.162760 0.0813800 − 0.996683i \(-0.474067\pi\)
0.0813800 + 0.996683i \(0.474067\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.10933 0.511588 0.255794 − 0.966731i \(-0.417663\pi\)
0.255794 + 0.966731i \(0.417663\pi\)
\(18\) 0 0
\(19\) 0.372610 0.0854827 0.0427413 − 0.999086i \(-0.486391\pi\)
0.0427413 + 0.999086i \(0.486391\pi\)
\(20\) 0 0
\(21\) 3.94270 0.860367
\(22\) 0 0
\(23\) −6.13313 −1.27885 −0.639423 − 0.768855i \(-0.720827\pi\)
−0.639423 + 0.768855i \(0.720827\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.87348 0.937902
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.36411 −0.245001 −0.122501 − 0.992468i \(-0.539091\pi\)
−0.122501 + 0.992468i \(0.539091\pi\)
\(32\) 0 0
\(33\) −4.97838 −0.866625
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.71997 0.447160 0.223580 − 0.974686i \(-0.428226\pi\)
0.223580 + 0.974686i \(0.428226\pi\)
\(38\) 0 0
\(39\) −0.563043 −0.0901590
\(40\) 0 0
\(41\) 5.64250 0.881210 0.440605 − 0.897701i \(-0.354764\pi\)
0.440605 + 0.897701i \(0.354764\pi\)
\(42\) 0 0
\(43\) −11.0066 −1.67849 −0.839246 − 0.543752i \(-0.817003\pi\)
−0.839246 + 0.543752i \(0.817003\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.36247 1.21979 0.609896 − 0.792482i \(-0.291211\pi\)
0.609896 + 0.792482i \(0.291211\pi\)
\(48\) 0 0
\(49\) 9.88660 1.41237
\(50\) 0 0
\(51\) −2.02380 −0.283388
\(52\) 0 0
\(53\) −3.16056 −0.434136 −0.217068 − 0.976156i \(-0.569649\pi\)
−0.217068 + 0.976156i \(0.569649\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.357501 −0.0473521
\(58\) 0 0
\(59\) −6.58684 −0.857533 −0.428767 − 0.903415i \(-0.641052\pi\)
−0.428767 + 0.903415i \(0.641052\pi\)
\(60\) 0 0
\(61\) 2.28708 0.292830 0.146415 − 0.989223i \(-0.453227\pi\)
0.146415 + 0.989223i \(0.453227\pi\)
\(62\) 0 0
\(63\) 8.54518 1.07659
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.61725 −0.930595 −0.465297 − 0.885154i \(-0.654053\pi\)
−0.465297 + 0.885154i \(0.654053\pi\)
\(68\) 0 0
\(69\) 5.88443 0.708402
\(70\) 0 0
\(71\) 14.2085 1.68624 0.843120 − 0.537725i \(-0.180716\pi\)
0.843120 + 0.537725i \(0.180716\pi\)
\(72\) 0 0
\(73\) −2.46683 −0.288721 −0.144360 − 0.989525i \(-0.546112\pi\)
−0.144360 + 0.989525i \(0.546112\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −21.3224 −2.42992
\(78\) 0 0
\(79\) 5.30681 0.597062 0.298531 − 0.954400i \(-0.403503\pi\)
0.298531 + 0.954400i \(0.403503\pi\)
\(80\) 0 0
\(81\) 1.56251 0.173612
\(82\) 0 0
\(83\) 14.8575 1.63083 0.815413 − 0.578880i \(-0.196510\pi\)
0.815413 + 0.578880i \(0.196510\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.959449 −0.102864
\(88\) 0 0
\(89\) −3.91890 −0.415402 −0.207701 − 0.978192i \(-0.566598\pi\)
−0.207701 + 0.978192i \(0.566598\pi\)
\(90\) 0 0
\(91\) −2.41152 −0.252796
\(92\) 0 0
\(93\) 1.30880 0.135716
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.27069 0.636692 0.318346 − 0.947975i \(-0.396873\pi\)
0.318346 + 0.947975i \(0.396873\pi\)
\(98\) 0 0
\(99\) −10.7899 −1.08442
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.1.2 5
5.4 even 2 1160.2.a.h.1.4 ✓ 5
20.19 odd 2 2320.2.a.v.1.2 5
40.19 odd 2 9280.2.a.ci.1.4 5
40.29 even 2 9280.2.a.ck.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.h.1.4 ✓ 5 5.4 even 2
2320.2.a.v.1.2 5 20.19 odd 2
5800.2.a.u.1.2 5 1.1 even 1 trivial
9280.2.a.ci.1.4 5 40.19 odd 2
9280.2.a.ck.1.2 5 40.29 even 2