Properties

Label 462.2.a.d.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
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] = [462,2,Mod(1,462)] 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("462.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(462, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.68908857338\)
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 \(\chi\) \(=\) 462.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -4.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +6.00000 q^{57} -6.00000 q^{58} -2.00000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -10.0000 q^{74} -5.00000 q^{75} +6.00000 q^{76} +1.00000 q^{77} -6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} -4.00000 q^{92} -2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −6.00000 −0.973329
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 6.00000 0.794719
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 − 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) −10.0000 −1.16248
\(75\) −5.00000 −0.577350
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) −6.00000 −0.679366
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
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 462.2.a.d.1.1 ✓ 1
3.2 odd 2 1386.2.a.j.1.1 1
4.3 odd 2 3696.2.a.j.1.1 1
7.6 odd 2 3234.2.a.b.1.1 1
11.10 odd 2 5082.2.a.ba.1.1 1
21.20 even 2 9702.2.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
1386.2.a.j.1.1 1 3.2 odd 2
3234.2.a.b.1.1 1 7.6 odd 2
3696.2.a.j.1.1 1 4.3 odd 2
5082.2.a.ba.1.1 1 11.10 odd 2
9702.2.a.bp.1.1 1 21.20 even 2