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Properties

Label 1950.2.a.f.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
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] = [1950,2,Mod(1,1950)] 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("1950.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,0,-1,-4,-1,1,0,0,1,1,4,0,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.5708283941\)
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 \(\chi\) \(=\) 1950.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} +5.00000 q^{19} -4.00000 q^{21} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} +1.00000 q^{39} +3.00000 q^{41} +4.00000 q^{42} +2.00000 q^{43} +9.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} +5.00000 q^{57} -3.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +5.00000 q^{67} -3.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +7.00000 q^{74} +5.00000 q^{76} -1.00000 q^{78} +11.0000 q^{79} +1.00000 q^{81} -3.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} -2.00000 q^{86} +3.00000 q^{87} +6.00000 q^{89} -4.00000 q^{91} -4.00000 q^{93} -9.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −5.00000 −0.811107
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 4.00000 0.617213
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 5.00000 0.662266
\(58\) −3.00000 −0.393919
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 4.00000 0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
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 1950.2.a.f.1.1 ✓ 1
3.2 odd 2 5850.2.a.be.1.1 1
5.2 odd 4 1950.2.e.c.1249.1 2
5.3 odd 4 1950.2.e.c.1249.2 2
5.4 even 2 1950.2.a.t.1.1 yes 1
15.2 even 4 5850.2.e.s.5149.2 2
15.8 even 4 5850.2.e.s.5149.1 2
15.14 odd 2 5850.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
1950.2.a.t.1.1 yes 1 5.4 even 2
1950.2.e.c.1249.1 2 5.2 odd 4
1950.2.e.c.1249.2 2 5.3 odd 4
5850.2.a.y.1.1 1 15.14 odd 2
5850.2.a.be.1.1 1 3.2 odd 2
5850.2.e.s.5149.1 2 15.8 even 4
5850.2.e.s.5149.2 2 15.2 even 4