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Properties

Label 800.6.a.r.1.2
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
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 = [4,0,0,0,0,0,0,0,724,0,0,0,584] 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: \(4\)
Coefficient field: 4.4.81998080.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 100x^{2} - 376x - 367 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.18136\) 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)
 
\(f(q)\) \(=\) \(q-10.6653 q^{3} +176.978 q^{7} -129.252 q^{9} +400.949 q^{11} -784.755 q^{13} -740.755 q^{17} +2776.65 q^{19} -1887.52 q^{21} -176.978 q^{23} +3970.17 q^{27} +5542.53 q^{29} -5737.27 q^{31} -4276.24 q^{33} -6795.51 q^{37} +8369.64 q^{39} +18421.3 q^{41} -14068.2 q^{43} +23388.0 q^{47} +14514.2 q^{49} +7900.36 q^{51} +279.773 q^{53} -29613.8 q^{57} -40195.9 q^{59} +10555.1 q^{61} -22874.7 q^{63} -35941.5 q^{67} +1887.52 q^{69} -25260.8 q^{71} -27057.7 q^{73} +70959.1 q^{77} +32083.9 q^{79} -10934.9 q^{81} -44494.3 q^{83} -59112.7 q^{87} +80252.6 q^{89} -138884. q^{91} +61189.6 q^{93} -36409.5 q^{97} -51823.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 724 q^{9} + 584 q^{13} + 760 q^{17} + 9824 q^{21} - 168 q^{29} - 25792 q^{33} - 19736 q^{37} + 47624 q^{41} + 121348 q^{49} - 17496 q^{53} - 99840 q^{57} - 62024 q^{61} - 9824 q^{69} - 119400 q^{73}+ \cdots + 59128 q^{97}+O(q^{100}) \) 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
\(3\) −10.6653 −0.684179 −0.342089 − 0.939667i \(-0.611135\pi\)
−0.342089 + 0.939667i \(0.611135\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 176.978 1.36513 0.682565 − 0.730825i \(-0.260864\pi\)
0.682565 + 0.730825i \(0.260864\pi\)
\(8\) 0 0
\(9\) −129.252 −0.531899
\(10\) 0 0
\(11\) 400.949 0.999097 0.499548 − 0.866286i \(-0.333499\pi\)
0.499548 + 0.866286i \(0.333499\pi\)
\(12\) 0 0
\(13\) −784.755 −1.28788 −0.643940 − 0.765076i \(-0.722701\pi\)
−0.643940 + 0.765076i \(0.722701\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −740.755 −0.621659 −0.310829 − 0.950466i \(-0.600607\pi\)
−0.310829 + 0.950466i \(0.600607\pi\)
\(18\) 0 0
\(19\) 2776.65 1.76456 0.882281 − 0.470723i \(-0.156007\pi\)
0.882281 + 0.470723i \(0.156007\pi\)
\(20\) 0 0
\(21\) −1887.52 −0.933993
\(22\) 0 0
\(23\) −176.978 −0.0697589 −0.0348794 − 0.999392i \(-0.511105\pi\)
−0.0348794 + 0.999392i \(0.511105\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3970.17 1.04809
\(28\) 0 0
\(29\) 5542.53 1.22381 0.611903 − 0.790933i \(-0.290404\pi\)
0.611903 + 0.790933i \(0.290404\pi\)
\(30\) 0 0
\(31\) −5737.27 −1.07226 −0.536131 − 0.844135i \(-0.680115\pi\)
−0.536131 + 0.844135i \(0.680115\pi\)
\(32\) 0 0
\(33\) −4276.24 −0.683561
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6795.51 −0.816052 −0.408026 − 0.912970i \(-0.633783\pi\)
−0.408026 + 0.912970i \(0.633783\pi\)
\(38\) 0 0
\(39\) 8369.64 0.881140
\(40\) 0 0
\(41\) 18421.3 1.71143 0.855717 − 0.517444i \(-0.173116\pi\)
0.855717 + 0.517444i \(0.173116\pi\)
\(42\) 0 0
\(43\) −14068.2 −1.16029 −0.580147 − 0.814512i \(-0.697005\pi\)
−0.580147 + 0.814512i \(0.697005\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 23388.0 1.54436 0.772181 − 0.635402i \(-0.219166\pi\)
0.772181 + 0.635402i \(0.219166\pi\)
\(48\) 0 0
\(49\) 14514.2 0.863579
\(50\) 0 0
\(51\) 7900.36 0.425326
\(52\) 0 0
\(53\) 279.773 0.0136809 0.00684046 − 0.999977i \(-0.497823\pi\)
0.00684046 + 0.999977i \(0.497823\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −29613.8 −1.20728
\(58\) 0 0
\(59\) −40195.9 −1.50332 −0.751661 − 0.659549i \(-0.770747\pi\)
−0.751661 + 0.659549i \(0.770747\pi\)
\(60\) 0 0
\(61\) 10555.1 0.363194 0.181597 − 0.983373i \(-0.441873\pi\)
0.181597 + 0.983373i \(0.441873\pi\)
\(62\) 0 0
\(63\) −22874.7 −0.726111
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −35941.5 −0.978158 −0.489079 − 0.872239i \(-0.662667\pi\)
−0.489079 + 0.872239i \(0.662667\pi\)
\(68\) 0 0
\(69\) 1887.52 0.0477275
\(70\) 0 0
\(71\) −25260.8 −0.594705 −0.297353 − 0.954768i \(-0.596104\pi\)
−0.297353 + 0.954768i \(0.596104\pi\)
\(72\) 0 0
\(73\) −27057.7 −0.594271 −0.297135 − 0.954835i \(-0.596031\pi\)
−0.297135 + 0.954835i \(0.596031\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 70959.1 1.36390
\(78\) 0 0
\(79\) 32083.9 0.578389 0.289194 − 0.957270i \(-0.406613\pi\)
0.289194 + 0.957270i \(0.406613\pi\)
\(80\) 0 0
\(81\) −10934.9 −0.185184
\(82\) 0 0
\(83\) −44494.3 −0.708940 −0.354470 − 0.935067i \(-0.615339\pi\)
−0.354470 + 0.935067i \(0.615339\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −59112.7 −0.837303
\(88\) 0 0
\(89\) 80252.6 1.07395 0.536975 − 0.843598i \(-0.319567\pi\)
0.536975 + 0.843598i \(0.319567\pi\)
\(90\) 0 0
\(91\) −138884. −1.75812
\(92\) 0 0
\(93\) 61189.6 0.733619
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −36409.5 −0.392903 −0.196452 − 0.980514i \(-0.562942\pi\)
−0.196452 + 0.980514i \(0.562942\pi\)
\(98\) 0 0
\(99\) −51823.3 −0.531419
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.r.1.2 4
4.3 odd 2 inner 800.6.a.r.1.3 4
5.2 odd 4 800.6.c.l.449.6 8
5.3 odd 4 800.6.c.l.449.4 8
5.4 even 2 160.6.a.h.1.3 yes 4
20.3 even 4 800.6.c.l.449.5 8
20.7 even 4 800.6.c.l.449.3 8
20.19 odd 2 160.6.a.h.1.2 ✓ 4
40.19 odd 2 320.6.a.z.1.3 4
40.29 even 2 320.6.a.z.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.a.h.1.2 ✓ 4 20.19 odd 2
160.6.a.h.1.3 yes 4 5.4 even 2
320.6.a.z.1.2 4 40.29 even 2
320.6.a.z.1.3 4 40.19 odd 2
800.6.a.r.1.2 4 1.1 even 1 trivial
800.6.a.r.1.3 4 4.3 odd 2 inner
800.6.c.l.449.3 8 20.7 even 4
800.6.c.l.449.4 8 5.3 odd 4
800.6.c.l.449.5 8 20.3 even 4
800.6.c.l.449.6 8 5.2 odd 4