↑

Properties

Label 800.6.a.x.1.2
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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 = [6,0,0,0,0,0,0,0,228,0,0,0,-128] 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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 207x^{4} + 419x^{3} + 5927x^{2} - 6137x - 8328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-5.29401\) 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)
 
\(f(q)\) \(=\) \(q-11.5880 q^{3} +89.6769 q^{7} -108.718 q^{9} +718.855 q^{11} +871.743 q^{13} +2090.64 q^{17} +476.947 q^{19} -1039.18 q^{21} -1330.77 q^{23} +4075.71 q^{27} +730.313 q^{29} +9788.53 q^{31} -8330.12 q^{33} -6219.08 q^{37} -10101.8 q^{39} +867.570 q^{41} +12946.8 q^{43} -26161.9 q^{47} -8765.06 q^{49} -24226.4 q^{51} +11148.4 q^{53} -5526.87 q^{57} -21407.2 q^{59} +32686.2 q^{61} -9749.45 q^{63} -37209.3 q^{67} +15420.9 q^{69} +35282.1 q^{71} +20909.5 q^{73} +64464.7 q^{77} +21560.4 q^{79} -20811.1 q^{81} -87640.0 q^{83} -8462.89 q^{87} +111553. q^{89} +78175.2 q^{91} -113430. q^{93} +18353.2 q^{97} -78152.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 228 q^{9} - 128 q^{13} - 814 q^{17} + 884 q^{21} + 4152 q^{29} + 5058 q^{33} + 348 q^{37} + 10334 q^{41} + 1454 q^{49} - 33924 q^{53} + 38450 q^{57} + 25796 q^{61} + 5796 q^{69} - 59218 q^{73} + 49052 q^{77}+ \cdots + 250052 q^{97}+O(q^{100}) \) 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
\(3\) −11.5880 −0.743372 −0.371686 − 0.928358i \(-0.621220\pi\)
−0.371686 + 0.928358i \(0.621220\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 89.6769 0.691728 0.345864 − 0.938285i \(-0.387586\pi\)
0.345864 + 0.938285i \(0.387586\pi\)
\(8\) 0 0
\(9\) −108.718 −0.447397
\(10\) 0 0
\(11\) 718.855 1.79126 0.895632 − 0.444795i \(-0.146724\pi\)
0.895632 + 0.444795i \(0.146724\pi\)
\(12\) 0 0
\(13\) 871.743 1.43064 0.715320 − 0.698797i \(-0.246281\pi\)
0.715320 + 0.698797i \(0.246281\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2090.64 1.75451 0.877257 − 0.480021i \(-0.159371\pi\)
0.877257 + 0.480021i \(0.159371\pi\)
\(18\) 0 0
\(19\) 476.947 0.303100 0.151550 − 0.988450i \(-0.451574\pi\)
0.151550 + 0.988450i \(0.451574\pi\)
\(20\) 0 0
\(21\) −1039.18 −0.514212
\(22\) 0 0
\(23\) −1330.77 −0.524544 −0.262272 − 0.964994i \(-0.584472\pi\)
−0.262272 + 0.964994i \(0.584472\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4075.71 1.07596
\(28\) 0 0
\(29\) 730.313 0.161255 0.0806277 − 0.996744i \(-0.474308\pi\)
0.0806277 + 0.996744i \(0.474308\pi\)
\(30\) 0 0
\(31\) 9788.53 1.82942 0.914710 − 0.404111i \(-0.132419\pi\)
0.914710 + 0.404111i \(0.132419\pi\)
\(32\) 0 0
\(33\) −8330.12 −1.33158
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6219.08 −0.746830 −0.373415 − 0.927664i \(-0.621813\pi\)
−0.373415 + 0.927664i \(0.621813\pi\)
\(38\) 0 0
\(39\) −10101.8 −1.06350
\(40\) 0 0
\(41\) 867.570 0.0806018 0.0403009 − 0.999188i \(-0.487168\pi\)
0.0403009 + 0.999188i \(0.487168\pi\)
\(42\) 0 0
\(43\) 12946.8 1.06780 0.533900 − 0.845548i \(-0.320726\pi\)
0.533900 + 0.845548i \(0.320726\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −26161.9 −1.72753 −0.863764 − 0.503897i \(-0.831899\pi\)
−0.863764 + 0.503897i \(0.831899\pi\)
\(48\) 0 0
\(49\) −8765.06 −0.521512
\(50\) 0 0
\(51\) −24226.4 −1.30426
\(52\) 0 0
\(53\) 11148.4 0.545160 0.272580 − 0.962133i \(-0.412123\pi\)
0.272580 + 0.962133i \(0.412123\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5526.87 −0.225316
\(58\) 0 0
\(59\) −21407.2 −0.800628 −0.400314 − 0.916378i \(-0.631099\pi\)
−0.400314 + 0.916378i \(0.631099\pi\)
\(60\) 0 0
\(61\) 32686.2 1.12471 0.562354 − 0.826896i \(-0.309896\pi\)
0.562354 + 0.826896i \(0.309896\pi\)
\(62\) 0 0
\(63\) −9749.45 −0.309477
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −37209.3 −1.01266 −0.506331 − 0.862339i \(-0.668998\pi\)
−0.506331 + 0.862339i \(0.668998\pi\)
\(68\) 0 0
\(69\) 15420.9 0.389932
\(70\) 0 0
\(71\) 35282.1 0.830633 0.415316 − 0.909677i \(-0.363671\pi\)
0.415316 + 0.909677i \(0.363671\pi\)
\(72\) 0 0
\(73\) 20909.5 0.459238 0.229619 − 0.973281i \(-0.426252\pi\)
0.229619 + 0.973281i \(0.426252\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 64464.7 1.23907
\(78\) 0 0
\(79\) 21560.4 0.388678 0.194339 − 0.980934i \(-0.437744\pi\)
0.194339 + 0.980934i \(0.437744\pi\)
\(80\) 0 0
\(81\) −20811.1 −0.352438
\(82\) 0 0
\(83\) −87640.0 −1.39639 −0.698195 − 0.715907i \(-0.746013\pi\)
−0.698195 + 0.715907i \(0.746013\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8462.89 −0.119873
\(88\) 0 0
\(89\) 111553. 1.49282 0.746410 − 0.665486i \(-0.231776\pi\)
0.746410 + 0.665486i \(0.231776\pi\)
\(90\) 0 0
\(91\) 78175.2 0.989613
\(92\) 0 0
\(93\) −113430. −1.35994
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18353.2 0.198053 0.0990266 − 0.995085i \(-0.468427\pi\)
0.0990266 + 0.995085i \(0.468427\pi\)
\(98\) 0 0
\(99\) −78152.2 −0.801407
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.x.1.2 ✓ 6
4.3 odd 2 inner 800.6.a.x.1.5 yes 6
5.2 odd 4 800.6.c.p.449.10 12
5.3 odd 4 800.6.c.p.449.4 12
5.4 even 2 800.6.a.y.1.5 yes 6
20.3 even 4 800.6.c.p.449.9 12
20.7 even 4 800.6.c.p.449.3 12
20.19 odd 2 800.6.a.y.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.x.1.2 ✓ 6 1.1 even 1 trivial
800.6.a.x.1.5 yes 6 4.3 odd 2 inner
800.6.a.y.1.2 yes 6 20.19 odd 2
800.6.a.y.1.5 yes 6 5.4 even 2
800.6.c.p.449.3 12 20.7 even 4
800.6.c.p.449.4 12 5.3 odd 4
800.6.c.p.449.9 12 20.3 even 4
800.6.c.p.449.10 12 5.2 odd 4