Properties

Label 285.2.a.a.1.1
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} +2.00000 q^{42} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -6.00000 q^{56} +1.00000 q^{57} -4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} -8.00000 q^{69} -2.00000 q^{70} +16.0000 q^{71} +3.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{84} +6.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -18.0000 q^{88} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -12.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 6.00000 1.27920
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) −6.00000 −1.04447
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000 0.308607
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.00000 −0.149071
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) −6.00000 −0.801784
\(57\) 1.00000 0.132453
\(58\) −4.00000 −0.525226
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) −8.00000 −0.963087
\(70\) −2.00000 −0.239046
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 3.00000 0.353553
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 6.00000 0.650791
\(86\) 2.00000 0.215666
\(87\) 4.00000 0.428845
\(88\) −18.0000 −1.91881
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −1.00000 −0.102598
\(96\) −5.00000 −0.510310
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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 285.2.a.a.1.1 ✓ 1
3.2 odd 2 855.2.a.c.1.1 1
4.3 odd 2 4560.2.a.h.1.1 1
5.2 odd 4 1425.2.c.c.799.1 2
5.3 odd 4 1425.2.c.c.799.2 2
5.4 even 2 1425.2.a.g.1.1 1
15.14 odd 2 4275.2.a.h.1.1 1
19.18 odd 2 5415.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
855.2.a.c.1.1 1 3.2 odd 2
1425.2.a.g.1.1 1 5.4 even 2
1425.2.c.c.799.1 2 5.2 odd 4
1425.2.c.c.799.2 2 5.3 odd 4
4275.2.a.h.1.1 1 15.14 odd 2
4560.2.a.h.1.1 1 4.3 odd 2
5415.2.a.h.1.1 1 19.18 odd 2