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Properties

Label 285.2.a.f.1.2
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
Analytic rank $0$
Dimension $2$
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 = [2,2,-2,2,-2] 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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} +3.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -1.41421 q^{11} -3.82843 q^{12} +2.58579 q^{13} +8.24264 q^{14} +1.00000 q^{15} +3.00000 q^{16} -6.82843 q^{17} +2.41421 q^{18} +1.00000 q^{19} -3.82843 q^{20} -3.41421 q^{21} -3.41421 q^{22} -3.65685 q^{23} -4.41421 q^{24} +1.00000 q^{25} +6.24264 q^{26} -1.00000 q^{27} +13.0711 q^{28} +5.07107 q^{29} +2.41421 q^{30} -10.4853 q^{31} -1.58579 q^{32} +1.41421 q^{33} -16.4853 q^{34} -3.41421 q^{35} +3.82843 q^{36} -3.07107 q^{37} +2.41421 q^{38} -2.58579 q^{39} -4.41421 q^{40} -4.58579 q^{41} -8.24264 q^{42} +3.41421 q^{43} -5.41421 q^{44} -1.00000 q^{45} -8.82843 q^{46} +11.6569 q^{47} -3.00000 q^{48} +4.65685 q^{49} +2.41421 q^{50} +6.82843 q^{51} +9.89949 q^{52} +4.00000 q^{53} -2.41421 q^{54} +1.41421 q^{55} +15.0711 q^{56} -1.00000 q^{57} +12.2426 q^{58} -8.48528 q^{59} +3.82843 q^{60} -5.65685 q^{61} -25.3137 q^{62} +3.41421 q^{63} -9.82843 q^{64} -2.58579 q^{65} +3.41421 q^{66} +12.0000 q^{67} -26.1421 q^{68} +3.65685 q^{69} -8.24264 q^{70} +12.4853 q^{71} +4.41421 q^{72} -2.00000 q^{73} -7.41421 q^{74} -1.00000 q^{75} +3.82843 q^{76} -4.82843 q^{77} -6.24264 q^{78} +11.3137 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.0711 q^{82} +6.48528 q^{83} -13.0711 q^{84} +6.82843 q^{85} +8.24264 q^{86} -5.07107 q^{87} -6.24264 q^{88} -14.7279 q^{89} -2.41421 q^{90} +8.82843 q^{91} -14.0000 q^{92} +10.4853 q^{93} +28.1421 q^{94} -1.00000 q^{95} +1.58579 q^{96} +4.24264 q^{97} +11.2426 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 8 q^{13} + 8 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} - 4 q^{21}+ \cdots + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 − 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) −2.41421 −0.985599
\(7\) 3.41421 1.29045 0.645226 − 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) −2.41421 −0.763441
\(11\) −1.41421 −0.426401 −0.213201 − 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) −3.82843 −1.10517
\(13\) 2.58579 0.717168 0.358584 − 0.933497i \(-0.383260\pi\)
0.358584 + 0.933497i \(0.383260\pi\)
\(14\) 8.24264 2.20294
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 − 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 2.41421 0.569036
\(19\) 1.00000 0.229416
\(20\) −3.82843 −0.856062
\(21\) −3.41421 −0.745042
\(22\) −3.41421 −0.727913
\(23\) −3.65685 −0.762507 −0.381253 − 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) −4.41421 −0.901048
\(25\) 1.00000 0.200000
\(26\) 6.24264 1.22428
\(27\) −1.00000 −0.192450
\(28\) 13.0711 2.47020
\(29\) 5.07107 0.941674 0.470837 − 0.882220i \(-0.343952\pi\)
0.470837 + 0.882220i \(0.343952\pi\)
\(30\) 2.41421 0.440773
\(31\) −10.4853 −1.88321 −0.941606 − 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) −1.58579 −0.280330
\(33\) 1.41421 0.246183
\(34\) −16.4853 −2.82720
\(35\) −3.41421 −0.577107
\(36\) 3.82843 0.638071
\(37\) −3.07107 −0.504880 −0.252440 − 0.967612i \(-0.581233\pi\)
−0.252440 + 0.967612i \(0.581233\pi\)
\(38\) 2.41421 0.391637
\(39\) −2.58579 −0.414057
\(40\) −4.41421 −0.697948
\(41\) −4.58579 −0.716180 −0.358090 − 0.933687i \(-0.616572\pi\)
−0.358090 + 0.933687i \(0.616572\pi\)
\(42\) −8.24264 −1.27187
\(43\) 3.41421 0.520663 0.260331 − 0.965519i \(-0.416168\pi\)
0.260331 + 0.965519i \(0.416168\pi\)
\(44\) −5.41421 −0.816223
\(45\) −1.00000 −0.149071
\(46\) −8.82843 −1.30168
\(47\) 11.6569 1.70033 0.850163 − 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) −3.00000 −0.433013
\(49\) 4.65685 0.665265
\(50\) 2.41421 0.341421
\(51\) 6.82843 0.956171
\(52\) 9.89949 1.37281
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −2.41421 −0.328533
\(55\) 1.41421 0.190693
\(56\) 15.0711 2.01396
\(57\) −1.00000 −0.132453
\(58\) 12.2426 1.60754
\(59\) −8.48528 −1.10469 −0.552345 − 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 3.82843 0.494248
\(61\) −5.65685 −0.724286 −0.362143 − 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) −25.3137 −3.21484
\(63\) 3.41421 0.430150
\(64\) −9.82843 −1.22855
\(65\) −2.58579 −0.320727
\(66\) 3.41421 0.420261
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −26.1421 −3.17020
\(69\) 3.65685 0.440234
\(70\) −8.24264 −0.985184
\(71\) 12.4853 1.48173 0.740865 − 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 4.41421 0.520220
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −7.41421 −0.861885
\(75\) −1.00000 −0.115470
\(76\) 3.82843 0.439151
\(77\) −4.82843 −0.550250
\(78\) −6.24264 −0.706840
\(79\) 11.3137 1.27289 0.636446 − 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −11.0711 −1.22259
\(83\) 6.48528 0.711852 0.355926 − 0.934514i \(-0.384165\pi\)
0.355926 + 0.934514i \(0.384165\pi\)
\(84\) −13.0711 −1.42617
\(85\) 6.82843 0.740647
\(86\) 8.24264 0.888827
\(87\) −5.07107 −0.543676
\(88\) −6.24264 −0.665468
\(89\) −14.7279 −1.56116 −0.780578 − 0.625058i \(-0.785075\pi\)
−0.780578 + 0.625058i \(0.785075\pi\)
\(90\) −2.41421 −0.254480
\(91\) 8.82843 0.925471
\(92\) −14.0000 −1.45960
\(93\) 10.4853 1.08727
\(94\) 28.1421 2.90264
\(95\) −1.00000 −0.102598
\(96\) 1.58579 0.161849
\(97\) 4.24264 0.430775 0.215387 − 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 11.2426 1.13568
\(99\) −1.41421 −0.142134
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.f.1.2 ✓ 2
3.2 odd 2 855.2.a.e.1.1 2
4.3 odd 2 4560.2.a.bj.1.1 2
5.2 odd 4 1425.2.c.j.799.4 4
5.3 odd 4 1425.2.c.j.799.1 4
5.4 even 2 1425.2.a.l.1.1 2
15.14 odd 2 4275.2.a.x.1.2 2
19.18 odd 2 5415.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 ✓ 2 1.1 even 1 trivial
855.2.a.e.1.1 2 3.2 odd 2
1425.2.a.l.1.1 2 5.4 even 2
1425.2.c.j.799.1 4 5.3 odd 4
1425.2.c.j.799.4 4 5.2 odd 4
4275.2.a.x.1.2 2 15.14 odd 2
4560.2.a.bj.1.1 2 4.3 odd 2
5415.2.a.p.1.1 2 19.18 odd 2