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Properties

Label 1600.4.a.v.1.1
Level $1600$
Weight $4$
Character 1600.1
Self dual yes
Analytic conductor $94.403$
Analytic rank $1$
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,-1,0,0,0,-6,0,-26,0,-19,0,12] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -6.00000 q^{7} -26.0000 q^{9} -19.0000 q^{11} +12.0000 q^{13} +75.0000 q^{17} -91.0000 q^{19} +6.00000 q^{21} +174.000 q^{23} +53.0000 q^{27} +272.000 q^{29} +230.000 q^{31} +19.0000 q^{33} -182.000 q^{37} -12.0000 q^{39} +117.000 q^{41} -372.000 q^{43} -52.0000 q^{47} -307.000 q^{49} -75.0000 q^{51} -402.000 q^{53} +91.0000 q^{57} +312.000 q^{59} -170.000 q^{61} +156.000 q^{63} -763.000 q^{67} -174.000 q^{69} +52.0000 q^{71} +981.000 q^{73} +114.000 q^{77} -1054.00 q^{79} +649.000 q^{81} -351.000 q^{83} -272.000 q^{87} +799.000 q^{89} -72.0000 q^{91} -230.000 q^{93} -962.000 q^{97} +494.000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.192450 −0.0962250 − 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 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
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) −19.0000 −0.520792 −0.260396 − 0.965502i \(-0.583853\pi\)
−0.260396 + 0.965502i \(0.583853\pi\)
\(12\) 0 0
\(13\) 12.0000 0.256015 0.128008 − 0.991773i \(-0.459142\pi\)
0.128008 + 0.991773i \(0.459142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 75.0000 1.07001 0.535005 − 0.844849i \(-0.320310\pi\)
0.535005 + 0.844849i \(0.320310\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 − 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) 0 0
\(21\) 6.00000 0.0623480
\(22\) 0 0
\(23\) 174.000 1.57746 0.788728 − 0.614742i \(-0.210740\pi\)
0.788728 + 0.614742i \(0.210740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) 272.000 1.74169 0.870847 − 0.491554i \(-0.163571\pi\)
0.870847 + 0.491554i \(0.163571\pi\)
\(30\) 0 0
\(31\) 230.000 1.33256 0.666278 − 0.745704i \(-0.267887\pi\)
0.666278 + 0.745704i \(0.267887\pi\)
\(32\) 0 0
\(33\) 19.0000 0.100227
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −182.000 −0.808665 −0.404333 − 0.914612i \(-0.632496\pi\)
−0.404333 + 0.914612i \(0.632496\pi\)
\(38\) 0 0
\(39\) −12.0000 −0.0492702
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 − 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) −372.000 −1.31929 −0.659645 − 0.751577i \(-0.729293\pi\)
−0.659645 + 0.751577i \(0.729293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.0000 −0.161383 −0.0806913 − 0.996739i \(-0.525713\pi\)
−0.0806913 + 0.996739i \(0.525713\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) −75.0000 −0.205924
\(52\) 0 0
\(53\) −402.000 −1.04187 −0.520933 − 0.853597i \(-0.674416\pi\)
−0.520933 + 0.853597i \(0.674416\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 91.0000 0.211460
\(58\) 0 0
\(59\) 312.000 0.688457 0.344228 − 0.938886i \(-0.388141\pi\)
0.344228 + 0.938886i \(0.388141\pi\)
\(60\) 0 0
\(61\) −170.000 −0.356824 −0.178412 − 0.983956i \(-0.557096\pi\)
−0.178412 + 0.983956i \(0.557096\pi\)
\(62\) 0 0
\(63\) 156.000 0.311971
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −763.000 −1.39127 −0.695636 − 0.718394i \(-0.744878\pi\)
−0.695636 + 0.718394i \(0.744878\pi\)
\(68\) 0 0
\(69\) −174.000 −0.303582
\(70\) 0 0
\(71\) 52.0000 0.0869192 0.0434596 − 0.999055i \(-0.486162\pi\)
0.0434596 + 0.999055i \(0.486162\pi\)
\(72\) 0 0
\(73\) 981.000 1.57284 0.786420 − 0.617692i \(-0.211932\pi\)
0.786420 + 0.617692i \(0.211932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 114.000 0.168721
\(78\) 0 0
\(79\) −1054.00 −1.50107 −0.750533 − 0.660833i \(-0.770203\pi\)
−0.750533 + 0.660833i \(0.770203\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) −351.000 −0.464184 −0.232092 − 0.972694i \(-0.574557\pi\)
−0.232092 + 0.972694i \(0.574557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −272.000 −0.335189
\(88\) 0 0
\(89\) 799.000 0.951616 0.475808 − 0.879549i \(-0.342156\pi\)
0.475808 + 0.879549i \(0.342156\pi\)
\(90\) 0 0
\(91\) −72.0000 −0.0829412
\(92\) 0 0
\(93\) −230.000 −0.256450
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −962.000 −1.00697 −0.503486 − 0.864003i \(-0.667949\pi\)
−0.503486 + 0.864003i \(0.667949\pi\)
\(98\) 0 0
\(99\) 494.000 0.501504
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.v.1.1 1
4.3 odd 2 1600.4.a.bf.1.1 1
5.4 even 2 1600.4.a.be.1.1 1
8.3 odd 2 200.4.a.e.1.1 ✓ 1
8.5 even 2 400.4.a.k.1.1 1
20.19 odd 2 1600.4.a.w.1.1 1
24.11 even 2 1800.4.a.w.1.1 1
40.3 even 4 200.4.c.g.49.1 2
40.13 odd 4 400.4.c.m.49.2 2
40.19 odd 2 200.4.a.f.1.1 yes 1
40.27 even 4 200.4.c.g.49.2 2
40.29 even 2 400.4.a.j.1.1 1
40.37 odd 4 400.4.c.m.49.1 2
120.59 even 2 1800.4.a.l.1.1 1
120.83 odd 4 1800.4.f.p.649.1 2
120.107 odd 4 1800.4.f.p.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 ✓ 1 8.3 odd 2
200.4.a.f.1.1 yes 1 40.19 odd 2
200.4.c.g.49.1 2 40.3 even 4
200.4.c.g.49.2 2 40.27 even 4
400.4.a.j.1.1 1 40.29 even 2
400.4.a.k.1.1 1 8.5 even 2
400.4.c.m.49.1 2 40.37 odd 4
400.4.c.m.49.2 2 40.13 odd 4
1600.4.a.v.1.1 1 1.1 even 1 trivial
1600.4.a.w.1.1 1 20.19 odd 2
1600.4.a.be.1.1 1 5.4 even 2
1600.4.a.bf.1.1 1 4.3 odd 2
1800.4.a.l.1.1 1 120.59 even 2
1800.4.a.w.1.1 1 24.11 even 2
1800.4.f.p.649.1 2 120.83 odd 4
1800.4.f.p.649.2 2 120.107 odd 4