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Properties

Label 800.6.a.t.1.3
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
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] = [800,6,Mod(1,800)] 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("800.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-1,0,0,0,-110,0,606,0,5,0,280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.488029\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.97606 q^{3} -48.9425 q^{7} -239.095 q^{9} +150.159 q^{11} -536.057 q^{13} +406.863 q^{17} -2379.02 q^{19} +96.7132 q^{21} -2258.37 q^{23} +952.648 q^{27} +4732.24 q^{29} -1710.44 q^{31} -296.722 q^{33} -5316.74 q^{37} +1059.28 q^{39} -82.9862 q^{41} +690.449 q^{43} +11399.2 q^{47} -14411.6 q^{49} -803.985 q^{51} -12743.5 q^{53} +4701.08 q^{57} +18943.2 q^{59} -16219.3 q^{61} +11701.9 q^{63} -28609.9 q^{67} +4462.67 q^{69} +48957.0 q^{71} +37042.2 q^{73} -7349.14 q^{77} +21099.6 q^{79} +56217.6 q^{81} -13898.4 q^{83} -9351.18 q^{87} +24334.8 q^{89} +26236.0 q^{91} +3379.94 q^{93} -44276.2 q^{97} -35902.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 110 q^{7} + 606 q^{9} + 5 q^{11} + 280 q^{13} - 865 q^{17} + 4485 q^{19} - 418 q^{21} + 3946 q^{23} - 4987 q^{27} - 3252 q^{29} + 8250 q^{31} - 13465 q^{33} + 3210 q^{37} + 8800 q^{39} - 11415 q^{41}+ \cdots + 244390 q^{99}+O(q^{100}) \) 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
\(3\) −1.97606 −0.126764 −0.0633821 − 0.997989i \(-0.520189\pi\)
−0.0633821 + 0.997989i \(0.520189\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −48.9425 −0.377521 −0.188760 − 0.982023i \(-0.560447\pi\)
−0.188760 + 0.982023i \(0.560447\pi\)
\(8\) 0 0
\(9\) −239.095 −0.983931
\(10\) 0 0
\(11\) 150.159 0.374170 0.187085 − 0.982344i \(-0.440096\pi\)
0.187085 + 0.982344i \(0.440096\pi\)
\(12\) 0 0
\(13\) −536.057 −0.879737 −0.439869 − 0.898062i \(-0.644975\pi\)
−0.439869 + 0.898062i \(0.644975\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 406.863 0.341449 0.170724 − 0.985319i \(-0.445389\pi\)
0.170724 + 0.985319i \(0.445389\pi\)
\(18\) 0 0
\(19\) −2379.02 −1.51187 −0.755933 − 0.654649i \(-0.772817\pi\)
−0.755933 + 0.654649i \(0.772817\pi\)
\(20\) 0 0
\(21\) 96.7132 0.0478561
\(22\) 0 0
\(23\) −2258.37 −0.890175 −0.445087 − 0.895487i \(-0.646827\pi\)
−0.445087 + 0.895487i \(0.646827\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 952.648 0.251491
\(28\) 0 0
\(29\) 4732.24 1.04489 0.522446 − 0.852672i \(-0.325020\pi\)
0.522446 + 0.852672i \(0.325020\pi\)
\(30\) 0 0
\(31\) −1710.44 −0.319672 −0.159836 − 0.987144i \(-0.551096\pi\)
−0.159836 + 0.987144i \(0.551096\pi\)
\(32\) 0 0
\(33\) −296.722 −0.0474313
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5316.74 −0.638471 −0.319236 − 0.947675i \(-0.603426\pi\)
−0.319236 + 0.947675i \(0.603426\pi\)
\(38\) 0 0
\(39\) 1059.28 0.111519
\(40\) 0 0
\(41\) −82.9862 −0.00770986 −0.00385493 − 0.999993i \(-0.501227\pi\)
−0.00385493 + 0.999993i \(0.501227\pi\)
\(42\) 0 0
\(43\) 690.449 0.0569456 0.0284728 − 0.999595i \(-0.490936\pi\)
0.0284728 + 0.999595i \(0.490936\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11399.2 0.752714 0.376357 − 0.926475i \(-0.377177\pi\)
0.376357 + 0.926475i \(0.377177\pi\)
\(48\) 0 0
\(49\) −14411.6 −0.857478
\(50\) 0 0
\(51\) −803.985 −0.0432835
\(52\) 0 0
\(53\) −12743.5 −0.623161 −0.311581 − 0.950220i \(-0.600858\pi\)
−0.311581 + 0.950220i \(0.600858\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4701.08 0.191651
\(58\) 0 0
\(59\) 18943.2 0.708473 0.354236 − 0.935156i \(-0.384741\pi\)
0.354236 + 0.935156i \(0.384741\pi\)
\(60\) 0 0
\(61\) −16219.3 −0.558094 −0.279047 − 0.960277i \(-0.590019\pi\)
−0.279047 + 0.960277i \(0.590019\pi\)
\(62\) 0 0
\(63\) 11701.9 0.371454
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −28609.9 −0.778626 −0.389313 − 0.921106i \(-0.627288\pi\)
−0.389313 + 0.921106i \(0.627288\pi\)
\(68\) 0 0
\(69\) 4462.67 0.112842
\(70\) 0 0
\(71\) 48957.0 1.15257 0.576287 − 0.817247i \(-0.304501\pi\)
0.576287 + 0.817247i \(0.304501\pi\)
\(72\) 0 0
\(73\) 37042.2 0.813560 0.406780 − 0.913526i \(-0.366652\pi\)
0.406780 + 0.913526i \(0.366652\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7349.14 −0.141257
\(78\) 0 0
\(79\) 21099.6 0.380371 0.190185 − 0.981748i \(-0.439091\pi\)
0.190185 + 0.981748i \(0.439091\pi\)
\(80\) 0 0
\(81\) 56217.6 0.952051
\(82\) 0 0
\(83\) −13898.4 −0.221446 −0.110723 − 0.993851i \(-0.535317\pi\)
−0.110723 + 0.993851i \(0.535317\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9351.18 −0.132455
\(88\) 0 0
\(89\) 24334.8 0.325651 0.162826 − 0.986655i \(-0.447939\pi\)
0.162826 + 0.986655i \(0.447939\pi\)
\(90\) 0 0
\(91\) 26236.0 0.332119
\(92\) 0 0
\(93\) 3379.94 0.0405229
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −44276.2 −0.477795 −0.238897 − 0.971045i \(-0.576786\pi\)
−0.238897 + 0.971045i \(0.576786\pi\)
\(98\) 0 0
\(99\) −35902.2 −0.368157
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 800.6.a.t.1.3 yes 5
4.3 odd 2 800.6.a.u.1.3 yes 5
5.2 odd 4 800.6.c.o.449.6 10
5.3 odd 4 800.6.c.o.449.5 10
5.4 even 2 800.6.a.v.1.3 yes 5
20.3 even 4 800.6.c.n.449.6 10
20.7 even 4 800.6.c.n.449.5 10
20.19 odd 2 800.6.a.s.1.3 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.s.1.3 ✓ 5 20.19 odd 2
800.6.a.t.1.3 yes 5 1.1 even 1 trivial
800.6.a.u.1.3 yes 5 4.3 odd 2
800.6.a.v.1.3 yes 5 5.4 even 2
800.6.c.n.449.5 10 20.7 even 4
800.6.c.n.449.6 10 20.3 even 4
800.6.c.o.449.5 10 5.3 odd 4
800.6.c.o.449.6 10 5.2 odd 4