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Properties

Label 285.2.a.c.1.1
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
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] = [285,2,Mod(1,285)] 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("285.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1,-1,1] 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: \(2.27573645761\)
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 \(\chi\) \(=\) 285.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -1.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -12.0000 q^{56} +1.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} +4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{84} +2.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) −1.00000 −0.258199
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.00000 −0.320256
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) −12.0000 −1.60357
\(57\) 1.00000 0.132453
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) 16.0000 1.82337
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) 8.00000 0.862662
\(87\) 2.00000 0.214423
\(88\) −12.0000 −1.27920
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 8.00000 0.838628
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −1.00000 −0.102598
\(96\) −5.00000 −0.510310
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
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 285.2.a.c.1.1 ✓ 1
3.2 odd 2 855.2.a.a.1.1 1
4.3 odd 2 4560.2.a.w.1.1 1
5.2 odd 4 1425.2.c.f.799.2 2
5.3 odd 4 1425.2.c.f.799.1 2
5.4 even 2 1425.2.a.c.1.1 1
15.14 odd 2 4275.2.a.j.1.1 1
19.18 odd 2 5415.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
855.2.a.a.1.1 1 3.2 odd 2
1425.2.a.c.1.1 1 5.4 even 2
1425.2.c.f.799.1 2 5.3 odd 4
1425.2.c.f.799.2 2 5.2 odd 4
4275.2.a.j.1.1 1 15.14 odd 2
4560.2.a.w.1.1 1 4.3 odd 2
5415.2.a.e.1.1 1 19.18 odd 2