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Properties

Label 800.6.a.p.1.3
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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 = [4,0,0,0,0,0,0,0,224,0,0,0,-664] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 93x^{2} + 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-9.63247\) 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)
 
\(f(q)\) \(=\) \(q+10.6706 q^{3} +128.626 q^{7} -129.138 q^{9} -117.955 q^{11} +389.413 q^{13} -910.965 q^{17} +675.718 q^{19} +1372.52 q^{21} -3664.38 q^{23} -3970.94 q^{27} +2856.17 q^{29} -240.535 q^{31} -1258.65 q^{33} -430.862 q^{37} +4155.28 q^{39} -6965.06 q^{41} -6745.04 q^{43} -4614.28 q^{47} -262.449 q^{49} -9720.56 q^{51} +12485.0 q^{53} +7210.33 q^{57} -21770.5 q^{59} -2827.00 q^{61} -16610.4 q^{63} -56761.3 q^{67} -39101.3 q^{69} +69097.7 q^{71} -15958.8 q^{73} -15172.0 q^{77} -63320.8 q^{79} -10991.9 q^{81} +84953.8 q^{83} +30477.1 q^{87} +10001.4 q^{89} +50088.5 q^{91} -2566.66 q^{93} -83076.8 q^{97} +15232.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 224 q^{9} - 664 q^{13} + 1540 q^{17} - 2656 q^{21} - 5608 q^{29} + 3852 q^{33} - 2464 q^{37} + 4724 q^{41} - 4012 q^{49} - 9304 q^{53} - 26700 q^{57} - 32784 q^{61} + 33176 q^{69} + 90940 q^{73} - 65872 q^{77}+ \cdots + 15752 q^{97}+O(q^{100}) \) 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
\(3\) 10.6706 0.684521 0.342260 − 0.939605i \(-0.388807\pi\)
0.342260 + 0.939605i \(0.388807\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 128.626 0.992162 0.496081 − 0.868276i \(-0.334772\pi\)
0.496081 + 0.868276i \(0.334772\pi\)
\(8\) 0 0
\(9\) −129.138 −0.531431
\(10\) 0 0
\(11\) −117.955 −0.293924 −0.146962 − 0.989142i \(-0.546949\pi\)
−0.146962 + 0.989142i \(0.546949\pi\)
\(12\) 0 0
\(13\) 389.413 0.639076 0.319538 − 0.947573i \(-0.396472\pi\)
0.319538 + 0.947573i \(0.396472\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −910.965 −0.764503 −0.382251 − 0.924058i \(-0.624851\pi\)
−0.382251 + 0.924058i \(0.624851\pi\)
\(18\) 0 0
\(19\) 675.718 0.429419 0.214710 − 0.976678i \(-0.431119\pi\)
0.214710 + 0.976678i \(0.431119\pi\)
\(20\) 0 0
\(21\) 1372.52 0.679155
\(22\) 0 0
\(23\) −3664.38 −1.44438 −0.722190 − 0.691695i \(-0.756864\pi\)
−0.722190 + 0.691695i \(0.756864\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3970.94 −1.04830
\(28\) 0 0
\(29\) 2856.17 0.630651 0.315325 − 0.948984i \(-0.397886\pi\)
0.315325 + 0.948984i \(0.397886\pi\)
\(30\) 0 0
\(31\) −240.535 −0.0449546 −0.0224773 − 0.999747i \(-0.507155\pi\)
−0.0224773 + 0.999747i \(0.507155\pi\)
\(32\) 0 0
\(33\) −1258.65 −0.201197
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −430.862 −0.0517409 −0.0258705 − 0.999665i \(-0.508236\pi\)
−0.0258705 + 0.999665i \(0.508236\pi\)
\(38\) 0 0
\(39\) 4155.28 0.437461
\(40\) 0 0
\(41\) −6965.06 −0.647091 −0.323546 − 0.946213i \(-0.604875\pi\)
−0.323546 + 0.946213i \(0.604875\pi\)
\(42\) 0 0
\(43\) −6745.04 −0.556306 −0.278153 − 0.960537i \(-0.589722\pi\)
−0.278153 + 0.960537i \(0.589722\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4614.28 −0.304691 −0.152346 − 0.988327i \(-0.548683\pi\)
−0.152346 + 0.988327i \(0.548683\pi\)
\(48\) 0 0
\(49\) −262.449 −0.0156154
\(50\) 0 0
\(51\) −9720.56 −0.523318
\(52\) 0 0
\(53\) 12485.0 0.610520 0.305260 − 0.952269i \(-0.401257\pi\)
0.305260 + 0.952269i \(0.401257\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7210.33 0.293946
\(58\) 0 0
\(59\) −21770.5 −0.814215 −0.407107 − 0.913380i \(-0.633463\pi\)
−0.407107 + 0.913380i \(0.633463\pi\)
\(60\) 0 0
\(61\) −2827.00 −0.0972751 −0.0486376 − 0.998816i \(-0.515488\pi\)
−0.0486376 + 0.998816i \(0.515488\pi\)
\(62\) 0 0
\(63\) −16610.4 −0.527266
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −56761.3 −1.54478 −0.772388 − 0.635151i \(-0.780938\pi\)
−0.772388 + 0.635151i \(0.780938\pi\)
\(68\) 0 0
\(69\) −39101.3 −0.988708
\(70\) 0 0
\(71\) 69097.7 1.62674 0.813370 − 0.581747i \(-0.197631\pi\)
0.813370 + 0.581747i \(0.197631\pi\)
\(72\) 0 0
\(73\) −15958.8 −0.350504 −0.175252 − 0.984524i \(-0.556074\pi\)
−0.175252 + 0.984524i \(0.556074\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15172.0 −0.291620
\(78\) 0 0
\(79\) −63320.8 −1.14151 −0.570754 − 0.821121i \(-0.693349\pi\)
−0.570754 + 0.821121i \(0.693349\pi\)
\(80\) 0 0
\(81\) −10991.9 −0.186150
\(82\) 0 0
\(83\) 84953.8 1.35359 0.676796 − 0.736171i \(-0.263368\pi\)
0.676796 + 0.736171i \(0.263368\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 30477.1 0.431694
\(88\) 0 0
\(89\) 10001.4 0.133839 0.0669197 − 0.997758i \(-0.478683\pi\)
0.0669197 + 0.997758i \(0.478683\pi\)
\(90\) 0 0
\(91\) 50088.5 0.634067
\(92\) 0 0
\(93\) −2566.66 −0.0307724
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −83076.8 −0.896500 −0.448250 − 0.893908i \(-0.647953\pi\)
−0.448250 + 0.893908i \(0.647953\pi\)
\(98\) 0 0
\(99\) 15232.4 0.156200
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.p.1.3 yes 4
4.3 odd 2 inner 800.6.a.p.1.2 ✓ 4
5.2 odd 4 800.6.c.m.449.3 8
5.3 odd 4 800.6.c.m.449.5 8
5.4 even 2 800.6.a.q.1.2 yes 4
20.3 even 4 800.6.c.m.449.4 8
20.7 even 4 800.6.c.m.449.6 8
20.19 odd 2 800.6.a.q.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.p.1.2 ✓ 4 4.3 odd 2 inner
800.6.a.p.1.3 yes 4 1.1 even 1 trivial
800.6.a.q.1.2 yes 4 5.4 even 2
800.6.a.q.1.3 yes 4 20.19 odd 2
800.6.c.m.449.3 8 5.2 odd 4
800.6.c.m.449.4 8 20.3 even 4
800.6.c.m.449.5 8 5.3 odd 4
800.6.c.m.449.6 8 20.7 even 4