Properties

Label 50.2.a.a.1.1
Level $50$
Weight $2$
Character 50.1
Self dual yes
Analytic conductor $0.399$
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] = [50,2,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.399252010106\)
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 \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +5.00000 q^{19} +2.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -5.00000 q^{27} +2.00000 q^{28} +2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} -5.00000 q^{38} -4.00000 q^{39} -3.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -3.00000 q^{44} -6.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -3.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} -2.00000 q^{56} +5.00000 q^{57} +2.00000 q^{61} -2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -13.0000 q^{67} -3.00000 q^{68} +6.00000 q^{69} +12.0000 q^{71} +2.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} +5.00000 q^{76} -6.00000 q^{77} +4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +3.00000 q^{82} -9.00000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +3.00000 q^{88} +15.0000 q^{89} -8.00000 q^{91} +6.00000 q^{92} +2.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 2.00000 0.471405
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −5.00000 −0.962250
\(28\) 2.00000 0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −5.00000 −0.811107
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) −3.00000 −0.363803
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 2.00000 0.235702
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) −6.00000 −0.683763
\(78\) 4.00000 0.452911
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 6.00000 0.625543
\(93\) 2.00000 0.207390
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) 6.00000 0.603023
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 50.2.a.a.1.1 ✓ 1
3.2 odd 2 450.2.a.g.1.1 1
4.3 odd 2 400.2.a.d.1.1 1
5.2 odd 4 50.2.b.a.49.1 2
5.3 odd 4 50.2.b.a.49.2 2
5.4 even 2 50.2.a.b.1.1 yes 1
7.6 odd 2 2450.2.a.g.1.1 1
8.3 odd 2 1600.2.a.p.1.1 1
8.5 even 2 1600.2.a.j.1.1 1
11.10 odd 2 6050.2.a.bi.1.1 1
12.11 even 2 3600.2.a.l.1.1 1
13.12 even 2 8450.2.a.v.1.1 1
15.2 even 4 450.2.c.c.199.2 2
15.8 even 4 450.2.c.c.199.1 2
15.14 odd 2 450.2.a.c.1.1 1
20.3 even 4 400.2.c.c.49.1 2
20.7 even 4 400.2.c.c.49.2 2
20.19 odd 2 400.2.a.f.1.1 1
35.13 even 4 2450.2.c.m.99.2 2
35.27 even 4 2450.2.c.m.99.1 2
35.34 odd 2 2450.2.a.bd.1.1 1
40.3 even 4 1600.2.c.h.449.2 2
40.13 odd 4 1600.2.c.i.449.1 2
40.19 odd 2 1600.2.a.i.1.1 1
40.27 even 4 1600.2.c.h.449.1 2
40.29 even 2 1600.2.a.q.1.1 1
40.37 odd 4 1600.2.c.i.449.2 2
55.54 odd 2 6050.2.a.h.1.1 1
60.23 odd 4 3600.2.f.f.2449.2 2
60.47 odd 4 3600.2.f.f.2449.1 2
60.59 even 2 3600.2.a.bc.1.1 1
65.64 even 2 8450.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
50.2.a.b.1.1 yes 1 5.4 even 2
50.2.b.a.49.1 2 5.2 odd 4
50.2.b.a.49.2 2 5.3 odd 4
400.2.a.d.1.1 1 4.3 odd 2
400.2.a.f.1.1 1 20.19 odd 2
400.2.c.c.49.1 2 20.3 even 4
400.2.c.c.49.2 2 20.7 even 4
450.2.a.c.1.1 1 15.14 odd 2
450.2.a.g.1.1 1 3.2 odd 2
450.2.c.c.199.1 2 15.8 even 4
450.2.c.c.199.2 2 15.2 even 4
1600.2.a.i.1.1 1 40.19 odd 2
1600.2.a.j.1.1 1 8.5 even 2
1600.2.a.p.1.1 1 8.3 odd 2
1600.2.a.q.1.1 1 40.29 even 2
1600.2.c.h.449.1 2 40.27 even 4
1600.2.c.h.449.2 2 40.3 even 4
1600.2.c.i.449.1 2 40.13 odd 4
1600.2.c.i.449.2 2 40.37 odd 4
2450.2.a.g.1.1 1 7.6 odd 2
2450.2.a.bd.1.1 1 35.34 odd 2
2450.2.c.m.99.1 2 35.27 even 4
2450.2.c.m.99.2 2 35.13 even 4
3600.2.a.l.1.1 1 12.11 even 2
3600.2.a.bc.1.1 1 60.59 even 2
3600.2.f.f.2449.1 2 60.47 odd 4
3600.2.f.f.2449.2 2 60.23 odd 4
6050.2.a.h.1.1 1 55.54 odd 2
6050.2.a.bi.1.1 1 11.10 odd 2
8450.2.a.d.1.1 1 65.64 even 2
8450.2.a.v.1.1 1 13.12 even 2