↑

Properties

Label 800.6.a.v.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 $1$

Related objects

Downloads

Learn more

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 \(-0.479811\pi\)
0.0633821 + 0.997989i \(0.479811\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 48.9425 0.377521 0.188760 − 0.982023i \(-0.439553\pi\)
0.188760 + 0.982023i \(0.439553\pi\)
\(8\) 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 \(-0.355025\pi\)
0.439869 + 0.898062i \(0.355025\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −406.863 −0.341449 −0.170724 − 0.985319i \(-0.554611\pi\)
−0.170724 + 0.985319i \(0.554611\pi\)
\(18\) 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 \(-0.353173\pi\)
0.445087 + 0.895487i \(0.353173\pi\)
\(24\) 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 \(-0.396574\pi\)
0.319236 + 0.947675i \(0.396574\pi\)
\(38\) 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 \(-0.509064\pi\)
−0.0284728 + 0.999595i \(0.509064\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11399.2 −0.752714 −0.376357 − 0.926475i \(-0.622823\pi\)
−0.376357 + 0.926475i \(0.622823\pi\)
\(48\) 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 \(-0.399142\pi\)
0.311581 + 0.950220i \(0.399142\pi\)
\(54\) 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 \(-0.372712\pi\)
0.389313 + 0.921106i \(0.372712\pi\)
\(68\) 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 \(-0.633348\pi\)
−0.406780 + 0.913526i \(0.633348\pi\)
\(74\) 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 \(-0.464683\pi\)
0.110723 + 0.993851i \(0.464683\pi\)
\(84\) 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 \(-0.423214\pi\)
0.238897 + 0.971045i \(0.423214\pi\)
\(98\) 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.v.1.3 yes 5
4.3 odd 2 800.6.a.s.1.3 ✓ 5
5.2 odd 4 800.6.c.o.449.5 10
5.3 odd 4 800.6.c.o.449.6 10
5.4 even 2 800.6.a.t.1.3 yes 5
20.3 even 4 800.6.c.n.449.5 10
20.7 even 4 800.6.c.n.449.6 10
20.19 odd 2 800.6.a.u.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.s.1.3 ✓ 5 4.3 odd 2
800.6.a.t.1.3 yes 5 5.4 even 2
800.6.a.u.1.3 yes 5 20.19 odd 2
800.6.a.v.1.3 yes 5 1.1 even 1 trivial
800.6.c.n.449.5 10 20.3 even 4
800.6.c.n.449.6 10 20.7 even 4
800.6.c.o.449.5 10 5.2 odd 4
800.6.c.o.449.6 10 5.3 odd 4