Properties

Label 200.4.a.b.1.1
Level $200$
Weight $4$
Character 200.1
Self dual yes
Analytic conductor $11.800$
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] = [200,4,Mod(1,200)] 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("200.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -26.0000 q^{7} +54.0000 q^{9} -59.0000 q^{11} -28.0000 q^{13} -5.00000 q^{17} +109.000 q^{19} +234.000 q^{21} +194.000 q^{23} -243.000 q^{27} -32.0000 q^{29} +10.0000 q^{31} +531.000 q^{33} +198.000 q^{37} +252.000 q^{39} +117.000 q^{41} -388.000 q^{43} +68.0000 q^{47} +333.000 q^{49} +45.0000 q^{51} +18.0000 q^{53} -981.000 q^{57} +392.000 q^{59} -710.000 q^{61} -1404.00 q^{63} +253.000 q^{67} -1746.00 q^{69} -612.000 q^{71} +549.000 q^{73} +1534.00 q^{77} +414.000 q^{79} +729.000 q^{81} +121.000 q^{83} +288.000 q^{87} -81.0000 q^{89} +728.000 q^{91} -90.0000 q^{93} +1502.00 q^{97} -3186.00 q^{99} +O(q^{100})\)

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
\(3\) −9.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −26.0000 −1.40387 −0.701934 − 0.712242i \(-0.747680\pi\)
−0.701934 + 0.712242i \(0.747680\pi\)
\(8\) 0 0
\(9\) 54.0000 2.00000
\(10\) 0 0
\(11\) −59.0000 −1.61720 −0.808599 − 0.588361i \(-0.799774\pi\)
−0.808599 + 0.588361i \(0.799774\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.00000 −0.0713340 −0.0356670 − 0.999364i \(-0.511356\pi\)
−0.0356670 + 0.999364i \(0.511356\pi\)
\(18\) 0 0
\(19\) 109.000 1.31612 0.658061 − 0.752965i \(-0.271377\pi\)
0.658061 + 0.752965i \(0.271377\pi\)
\(20\) 0 0
\(21\) 234.000 2.43157
\(22\) 0 0
\(23\) 194.000 1.75877 0.879387 − 0.476108i \(-0.157953\pi\)
0.879387 + 0.476108i \(0.157953\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −243.000 −1.73205
\(28\) 0 0
\(29\) −32.0000 −0.204905 −0.102453 − 0.994738i \(-0.532669\pi\)
−0.102453 + 0.994738i \(0.532669\pi\)
\(30\) 0 0
\(31\) 10.0000 0.0579372 0.0289686 − 0.999580i \(-0.490778\pi\)
0.0289686 + 0.999580i \(0.490778\pi\)
\(32\) 0 0
\(33\) 531.000 2.80107
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 198.000 0.879757 0.439878 − 0.898057i \(-0.355022\pi\)
0.439878 + 0.898057i \(0.355022\pi\)
\(38\) 0 0
\(39\) 252.000 1.03467
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 − 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) −388.000 −1.37603 −0.688017 − 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 68.0000 0.211039 0.105519 − 0.994417i \(-0.466350\pi\)
0.105519 + 0.994417i \(0.466350\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) 45.0000 0.123554
\(52\) 0 0
\(53\) 18.0000 0.0466508 0.0233254 − 0.999728i \(-0.492575\pi\)
0.0233254 + 0.999728i \(0.492575\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −981.000 −2.27959
\(58\) 0 0
\(59\) 392.000 0.864984 0.432492 − 0.901638i \(-0.357634\pi\)
0.432492 + 0.901638i \(0.357634\pi\)
\(60\) 0 0
\(61\) −710.000 −1.49027 −0.745133 − 0.666916i \(-0.767614\pi\)
−0.745133 + 0.666916i \(0.767614\pi\)
\(62\) 0 0
\(63\) −1404.00 −2.80774
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 253.000 0.461326 0.230663 − 0.973034i \(-0.425910\pi\)
0.230663 + 0.973034i \(0.425910\pi\)
\(68\) 0 0
\(69\) −1746.00 −3.04629
\(70\) 0 0
\(71\) −612.000 −1.02297 −0.511486 − 0.859292i \(-0.670905\pi\)
−0.511486 + 0.859292i \(0.670905\pi\)
\(72\) 0 0
\(73\) 549.000 0.880214 0.440107 − 0.897945i \(-0.354941\pi\)
0.440107 + 0.897945i \(0.354941\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1534.00 2.27033
\(78\) 0 0
\(79\) 414.000 0.589603 0.294802 − 0.955559i \(-0.404746\pi\)
0.294802 + 0.955559i \(0.404746\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 121.000 0.160018 0.0800089 − 0.996794i \(-0.474505\pi\)
0.0800089 + 0.996794i \(0.474505\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 288.000 0.354906
\(88\) 0 0
\(89\) −81.0000 −0.0964717 −0.0482359 − 0.998836i \(-0.515360\pi\)
−0.0482359 + 0.998836i \(0.515360\pi\)
\(90\) 0 0
\(91\) 728.000 0.838628
\(92\) 0 0
\(93\) −90.0000 −0.100350
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1502.00 1.57222 0.786108 − 0.618089i \(-0.212093\pi\)
0.786108 + 0.618089i \(0.212093\pi\)
\(98\) 0 0
\(99\) −3186.00 −3.23439
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 200.4.a.b.1.1 ✓ 1
3.2 odd 2 1800.4.a.c.1.1 1
4.3 odd 2 400.4.a.t.1.1 1
5.2 odd 4 200.4.c.b.49.2 2
5.3 odd 4 200.4.c.b.49.1 2
5.4 even 2 200.4.a.j.1.1 yes 1
8.3 odd 2 1600.4.a.b.1.1 1
8.5 even 2 1600.4.a.bz.1.1 1
15.2 even 4 1800.4.f.w.649.1 2
15.8 even 4 1800.4.f.w.649.2 2
15.14 odd 2 1800.4.a.bh.1.1 1
20.3 even 4 400.4.c.b.49.2 2
20.7 even 4 400.4.c.b.49.1 2
20.19 odd 2 400.4.a.a.1.1 1
40.19 odd 2 1600.4.a.by.1.1 1
40.29 even 2 1600.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.b.1.1 ✓ 1 1.1 even 1 trivial
200.4.a.j.1.1 yes 1 5.4 even 2
200.4.c.b.49.1 2 5.3 odd 4
200.4.c.b.49.2 2 5.2 odd 4
400.4.a.a.1.1 1 20.19 odd 2
400.4.a.t.1.1 1 4.3 odd 2
400.4.c.b.49.1 2 20.7 even 4
400.4.c.b.49.2 2 20.3 even 4
1600.4.a.b.1.1 1 8.3 odd 2
1600.4.a.c.1.1 1 40.29 even 2
1600.4.a.by.1.1 1 40.19 odd 2
1600.4.a.bz.1.1 1 8.5 even 2
1800.4.a.c.1.1 1 3.2 odd 2
1800.4.a.bh.1.1 1 15.14 odd 2
1800.4.f.w.649.1 2 15.2 even 4
1800.4.f.w.649.2 2 15.8 even 4