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Properties

Label 1600.4.a.r.1.1
Level $1600$
Weight $4$
Character 1600.1
Self dual yes
Analytic conductor $94.403$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [1600,4,Mod(1,1600)] 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("1600.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-2,0,0,0,-6,0,-23,0,-60,0,50] 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: \(94.4030560092\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -6.00000 q^{7} -23.0000 q^{9} -60.0000 q^{11} +50.0000 q^{13} +30.0000 q^{17} -40.0000 q^{19} +12.0000 q^{21} -178.000 q^{23} +100.000 q^{27} -166.000 q^{29} +20.0000 q^{31} +120.000 q^{33} +10.0000 q^{37} -100.000 q^{39} -250.000 q^{41} +142.000 q^{43} -214.000 q^{47} -307.000 q^{49} -60.0000 q^{51} +490.000 q^{53} +80.0000 q^{57} +800.000 q^{59} -250.000 q^{61} +138.000 q^{63} -774.000 q^{67} +356.000 q^{69} +100.000 q^{71} +230.000 q^{73} +360.000 q^{77} -1320.00 q^{79} +421.000 q^{81} +982.000 q^{83} +332.000 q^{87} +874.000 q^{89} -300.000 q^{91} -40.0000 q^{93} +310.000 q^{97} +1380.00 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −0.384900 −0.192450 − 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 − 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −60.0000 −1.64461 −0.822304 − 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 0 0
\(13\) 50.0000 1.06673 0.533366 − 0.845885i \(-0.320927\pi\)
0.533366 + 0.845885i \(0.320927\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.0000 0.428004 0.214002 − 0.976833i \(-0.431350\pi\)
0.214002 + 0.976833i \(0.431350\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) 12.0000 0.124696
\(22\) 0 0
\(23\) −178.000 −1.61372 −0.806860 − 0.590743i \(-0.798835\pi\)
−0.806860 + 0.590743i \(0.798835\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 − 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 − 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) 120.000 0.633010
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 0.0444322 0.0222161 − 0.999753i \(-0.492928\pi\)
0.0222161 + 0.999753i \(0.492928\pi\)
\(38\) 0 0
\(39\) −100.000 −0.410585
\(40\) 0 0
\(41\) −250.000 −0.952279 −0.476140 − 0.879370i \(-0.657964\pi\)
−0.476140 + 0.879370i \(0.657964\pi\)
\(42\) 0 0
\(43\) 142.000 0.503600 0.251800 − 0.967779i \(-0.418977\pi\)
0.251800 + 0.967779i \(0.418977\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −214.000 −0.664151 −0.332076 − 0.943253i \(-0.607749\pi\)
−0.332076 + 0.943253i \(0.607749\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) −60.0000 −0.164739
\(52\) 0 0
\(53\) 490.000 1.26994 0.634969 − 0.772538i \(-0.281013\pi\)
0.634969 + 0.772538i \(0.281013\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 80.0000 0.185899
\(58\) 0 0
\(59\) 800.000 1.76527 0.882637 − 0.470056i \(-0.155766\pi\)
0.882637 + 0.470056i \(0.155766\pi\)
\(60\) 0 0
\(61\) −250.000 −0.524741 −0.262371 − 0.964967i \(-0.584504\pi\)
−0.262371 + 0.964967i \(0.584504\pi\)
\(62\) 0 0
\(63\) 138.000 0.275974
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −774.000 −1.41133 −0.705665 − 0.708545i \(-0.749352\pi\)
−0.705665 + 0.708545i \(0.749352\pi\)
\(68\) 0 0
\(69\) 356.000 0.621121
\(70\) 0 0
\(71\) 100.000 0.167152 0.0835762 − 0.996501i \(-0.473366\pi\)
0.0835762 + 0.996501i \(0.473366\pi\)
\(72\) 0 0
\(73\) 230.000 0.368760 0.184380 − 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 360.000 0.532803
\(78\) 0 0
\(79\) −1320.00 −1.87989 −0.939947 − 0.341321i \(-0.889126\pi\)
−0.939947 + 0.341321i \(0.889126\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 982.000 1.29866 0.649328 − 0.760508i \(-0.275050\pi\)
0.649328 + 0.760508i \(0.275050\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 332.000 0.409128
\(88\) 0 0
\(89\) 874.000 1.04094 0.520471 − 0.853879i \(-0.325756\pi\)
0.520471 + 0.853879i \(0.325756\pi\)
\(90\) 0 0
\(91\) −300.000 −0.345588
\(92\) 0 0
\(93\) −40.0000 −0.0446001
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 310.000 0.324492 0.162246 − 0.986750i \(-0.448126\pi\)
0.162246 + 0.986750i \(0.448126\pi\)
\(98\) 0 0
\(99\) 1380.00 1.40096
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 1600.4.a.r.1.1 1
4.3 odd 2 1600.4.a.bj.1.1 1
5.4 even 2 320.4.a.i.1.1 1
8.3 odd 2 800.4.a.d.1.1 1
8.5 even 2 800.4.a.h.1.1 1
20.19 odd 2 320.4.a.f.1.1 1
40.3 even 4 800.4.c.e.449.1 2
40.13 odd 4 800.4.c.f.449.2 2
40.19 odd 2 160.4.a.b.1.1 yes 1
40.27 even 4 800.4.c.e.449.2 2
40.29 even 2 160.4.a.a.1.1 ✓ 1
40.37 odd 4 800.4.c.f.449.1 2
80.19 odd 4 1280.4.d.k.641.1 2
80.29 even 4 1280.4.d.f.641.2 2
80.59 odd 4 1280.4.d.k.641.2 2
80.69 even 4 1280.4.d.f.641.1 2
120.29 odd 2 1440.4.a.o.1.1 1
120.59 even 2 1440.4.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 ✓ 1 40.29 even 2
160.4.a.b.1.1 yes 1 40.19 odd 2
320.4.a.f.1.1 1 20.19 odd 2
320.4.a.i.1.1 1 5.4 even 2
800.4.a.d.1.1 1 8.3 odd 2
800.4.a.h.1.1 1 8.5 even 2
800.4.c.e.449.1 2 40.3 even 4
800.4.c.e.449.2 2 40.27 even 4
800.4.c.f.449.1 2 40.37 odd 4
800.4.c.f.449.2 2 40.13 odd 4
1280.4.d.f.641.1 2 80.69 even 4
1280.4.d.f.641.2 2 80.29 even 4
1280.4.d.k.641.1 2 80.19 odd 4
1280.4.d.k.641.2 2 80.59 odd 4
1440.4.a.n.1.1 1 120.59 even 2
1440.4.a.o.1.1 1 120.29 odd 2
1600.4.a.r.1.1 1 1.1 even 1 trivial
1600.4.a.bj.1.1 1 4.3 odd 2