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Properties

Label 800.6.c.m.449.3
Level $800$
Weight $6$
Character 800.449
Analytic conductor $128.307$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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(449,800)] 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("800.449"); 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, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-448,0,0] 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: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 190x^{6} + 8881x^{4} + 20596x^{2} + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(8.63247i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.6.c.m.449.5

$q$-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)
 
\(f(q)\) \(=\) \(q-10.6706i q^{3} +128.626i q^{7} +129.138 q^{9} -117.955 q^{11} -389.413i q^{13} -910.965i q^{17} -675.718 q^{19} +1372.52 q^{21} +3664.38i q^{23} -3970.94i q^{27} -2856.17 q^{29} -240.535 q^{31} +1258.65i q^{33} -430.862i q^{37} -4155.28 q^{39} -6965.06 q^{41} +6745.04i q^{43} -4614.28i q^{47} +262.449 q^{49} -9720.56 q^{51} -12485.0i q^{53} +7210.33i q^{57} +21770.5 q^{59} -2827.00 q^{61} +16610.4i q^{63} -56761.3i q^{67} +39101.3 q^{69} +69097.7 q^{71} +15958.8i q^{73} -15172.0i q^{77} +63320.8 q^{79} -10991.9 q^{81} -84953.8i q^{83} +30477.1i q^{87} -10001.4 q^{89} +50088.5 q^{91} +2566.66i q^{93} -83076.8i q^{97} -15232.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 448 q^{9} - 5312 q^{21} + 11216 q^{29} + 9448 q^{41} + 8024 q^{49} - 65568 q^{61} - 66352 q^{69} - 281960 q^{81} + 149560 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 10.6706i − 0.684521i −0.939605 − 0.342260i \(-0.888807\pi\)
0.939605 − 0.342260i \(-0.111193\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 128.626i 0.992162i 0.868276 + 0.496081i \(0.165228\pi\)
−0.868276 + 0.496081i \(0.834772\pi\)
\(8\) 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
\(13\) − 389.413i − 0.639076i −0.947573 − 0.319538i \(-0.896472\pi\)
0.947573 − 0.319538i \(-0.103528\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 910.965i − 0.764503i −0.924058 − 0.382251i \(-0.875149\pi\)
0.924058 − 0.382251i \(-0.124851\pi\)
\(18\) 0 0
\(19\) −675.718 −0.429419 −0.214710 − 0.976678i \(-0.568881\pi\)
−0.214710 + 0.976678i \(0.568881\pi\)
\(20\) 0 0
\(21\) 1372.52 0.679155
\(22\) 0 0
\(23\) 3664.38i 1.44438i 0.691695 + 0.722190i \(0.256864\pi\)
−0.691695 + 0.722190i \(0.743136\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3970.94i − 1.04830i
\(28\) 0 0
\(29\) −2856.17 −0.630651 −0.315325 − 0.948984i \(-0.602114\pi\)
−0.315325 + 0.948984i \(0.602114\pi\)
\(30\) 0 0
\(31\) −240.535 −0.0449546 −0.0224773 − 0.999747i \(-0.507155\pi\)
−0.0224773 + 0.999747i \(0.507155\pi\)
\(32\) 0 0
\(33\) 1258.65i 0.201197i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 430.862i − 0.0517409i −0.999665 − 0.0258705i \(-0.991764\pi\)
0.999665 − 0.0258705i \(-0.00823574\pi\)
\(38\) 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
\(43\) 6745.04i 0.556306i 0.960537 + 0.278153i \(0.0897222\pi\)
−0.960537 + 0.278153i \(0.910278\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4614.28i − 0.304691i −0.988327 − 0.152346i \(-0.951317\pi\)
0.988327 − 0.152346i \(-0.0486827\pi\)
\(48\) 0 0
\(49\) 262.449 0.0156154
\(50\) 0 0
\(51\) −9720.56 −0.523318
\(52\) 0 0
\(53\) − 12485.0i − 0.610520i −0.952269 − 0.305260i \(-0.901257\pi\)
0.952269 − 0.305260i \(-0.0987433\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7210.33i 0.293946i
\(58\) 0 0
\(59\) 21770.5 0.814215 0.407107 − 0.913380i \(-0.366537\pi\)
0.407107 + 0.913380i \(0.366537\pi\)
\(60\) 0 0
\(61\) −2827.00 −0.0972751 −0.0486376 − 0.998816i \(-0.515488\pi\)
−0.0486376 + 0.998816i \(0.515488\pi\)
\(62\) 0 0
\(63\) 16610.4i 0.527266i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 56761.3i − 1.54478i −0.635151 − 0.772388i \(-0.719062\pi\)
0.635151 − 0.772388i \(-0.280938\pi\)
\(68\) 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
\(73\) 15958.8i 0.350504i 0.984524 + 0.175252i \(0.0560740\pi\)
−0.984524 + 0.175252i \(0.943926\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15172.0i − 0.291620i
\(78\) 0 0
\(79\) 63320.8 1.14151 0.570754 − 0.821121i \(-0.306651\pi\)
0.570754 + 0.821121i \(0.306651\pi\)
\(80\) 0 0
\(81\) −10991.9 −0.186150
\(82\) 0 0
\(83\) − 84953.8i − 1.35359i −0.736171 − 0.676796i \(-0.763368\pi\)
0.736171 − 0.676796i \(-0.236632\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 30477.1i 0.431694i
\(88\) 0 0
\(89\) −10001.4 −0.133839 −0.0669197 − 0.997758i \(-0.521317\pi\)
−0.0669197 + 0.997758i \(0.521317\pi\)
\(90\) 0 0
\(91\) 50088.5 0.634067
\(92\) 0 0
\(93\) 2566.66i 0.0307724i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 83076.8i − 0.896500i −0.893908 − 0.448250i \(-0.852047\pi\)
0.893908 − 0.448250i \(-0.147953\pi\)
\(98\) 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.c.m.449.3 8
4.3 odd 2 inner 800.6.c.m.449.6 8
5.2 odd 4 800.6.a.q.1.2 yes 4
5.3 odd 4 800.6.a.p.1.3 yes 4
5.4 even 2 inner 800.6.c.m.449.5 8
20.3 even 4 800.6.a.p.1.2 ✓ 4
20.7 even 4 800.6.a.q.1.3 yes 4
20.19 odd 2 inner 800.6.c.m.449.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.p.1.2 ✓ 4 20.3 even 4
800.6.a.p.1.3 yes 4 5.3 odd 4
800.6.a.q.1.2 yes 4 5.2 odd 4
800.6.a.q.1.3 yes 4 20.7 even 4
800.6.c.m.449.3 8 1.1 even 1 trivial
800.6.c.m.449.4 8 20.19 odd 2 inner
800.6.c.m.449.5 8 5.4 even 2 inner
800.6.c.m.449.6 8 4.3 odd 2 inner