Properties

Label 145.2.a.b.1.2
Level $145$
Weight $2$
Character 145.1
Self dual yes
Analytic conductor $1.158$
Analytic rank $1$
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] = [145,2,Mod(1,145)] 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("145.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.15783082931\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 145.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -2.00000 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.828427 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} +0.828427 q^{11} +3.65685 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} +3.00000 q^{16} +2.82843 q^{17} +0.414214 q^{18} -4.82843 q^{19} -1.82843 q^{20} +9.65685 q^{21} +0.343146 q^{22} -3.17157 q^{23} +3.17157 q^{24} +1.00000 q^{25} -0.828427 q^{26} +4.00000 q^{27} +8.82843 q^{28} +1.00000 q^{29} -0.828427 q^{30} +6.48528 q^{31} +4.41421 q^{32} -1.65685 q^{33} +1.17157 q^{34} -4.82843 q^{35} -1.82843 q^{36} -8.48528 q^{37} -2.00000 q^{38} +4.00000 q^{39} -1.58579 q^{40} -6.00000 q^{41} +4.00000 q^{42} -6.00000 q^{43} -1.51472 q^{44} +1.00000 q^{45} -1.31371 q^{46} -11.6569 q^{47} -6.00000 q^{48} +16.3137 q^{49} +0.414214 q^{50} -5.65685 q^{51} +3.65685 q^{52} -3.65685 q^{53} +1.65685 q^{54} +0.828427 q^{55} +7.65685 q^{56} +9.65685 q^{57} +0.414214 q^{58} +3.65685 q^{60} -3.65685 q^{61} +2.68629 q^{62} -4.82843 q^{63} -4.17157 q^{64} -2.00000 q^{65} -0.686292 q^{66} +6.48528 q^{67} -5.17157 q^{68} +6.34315 q^{69} -2.00000 q^{70} -15.3137 q^{71} -1.58579 q^{72} +8.48528 q^{73} -3.51472 q^{74} -2.00000 q^{75} +8.82843 q^{76} -4.00000 q^{77} +1.65685 q^{78} -2.48528 q^{79} +3.00000 q^{80} -11.0000 q^{81} -2.48528 q^{82} +7.17157 q^{83} -17.6569 q^{84} +2.82843 q^{85} -2.48528 q^{86} -2.00000 q^{87} -1.31371 q^{88} -7.65685 q^{89} +0.414214 q^{90} +9.65685 q^{91} +5.79899 q^{92} -12.9706 q^{93} -4.82843 q^{94} -4.82843 q^{95} -8.82843 q^{96} -12.4853 q^{97} +6.75736 q^{98} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 − 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.82843 −0.914214
\(5\) 1.00000 0.447214
\(6\) −0.828427 −0.338204
\(7\) −4.82843 −1.82497 −0.912487 − 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 0.414214 0.130986
\(11\) 0.828427 0.249780 0.124890 − 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 3.65685 1.05564
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) −2.00000 −0.516398
\(16\) 3.00000 0.750000
\(17\) 2.82843 0.685994 0.342997 − 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0.414214 0.0976311
\(19\) −4.82843 −1.10772 −0.553859 − 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) −1.82843 −0.408849
\(21\) 9.65685 2.10730
\(22\) 0.343146 0.0731589
\(23\) −3.17157 −0.661319 −0.330659 − 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(24\) 3.17157 0.647395
\(25\) 1.00000 0.200000
\(26\) −0.828427 −0.162468
\(27\) 4.00000 0.769800
\(28\) 8.82843 1.66842
\(29\) 1.00000 0.185695
\(30\) −0.828427 −0.151249
\(31\) 6.48528 1.16479 0.582395 − 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) 4.41421 0.780330
\(33\) −1.65685 −0.288421
\(34\) 1.17157 0.200923
\(35\) −4.82843 −0.816153
\(36\) −1.82843 −0.304738
\(37\) −8.48528 −1.39497 −0.697486 − 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −2.00000 −0.324443
\(39\) 4.00000 0.640513
\(40\) −1.58579 −0.250735
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −1.51472 −0.228352
\(45\) 1.00000 0.149071
\(46\) −1.31371 −0.193696
\(47\) −11.6569 −1.70033 −0.850163 − 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) −6.00000 −0.866025
\(49\) 16.3137 2.33053
\(50\) 0.414214 0.0585786
\(51\) −5.65685 −0.792118
\(52\) 3.65685 0.507114
\(53\) −3.65685 −0.502308 −0.251154 − 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) 1.65685 0.225469
\(55\) 0.828427 0.111705
\(56\) 7.65685 1.02319
\(57\) 9.65685 1.27908
\(58\) 0.414214 0.0543889
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 3.65685 0.472098
\(61\) −3.65685 −0.468212 −0.234106 − 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 2.68629 0.341159
\(63\) −4.82843 −0.608325
\(64\) −4.17157 −0.521447
\(65\) −2.00000 −0.248069
\(66\) −0.686292 −0.0844766
\(67\) 6.48528 0.792303 0.396152 − 0.918185i \(-0.370345\pi\)
0.396152 + 0.918185i \(0.370345\pi\)
\(68\) −5.17157 −0.627145
\(69\) 6.34315 0.763625
\(70\) −2.00000 −0.239046
\(71\) −15.3137 −1.81740 −0.908701 − 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) −1.58579 −0.186887
\(73\) 8.48528 0.993127 0.496564 − 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −3.51472 −0.408578
\(75\) −2.00000 −0.230940
\(76\) 8.82843 1.01269
\(77\) −4.00000 −0.455842
\(78\) 1.65685 0.187602
\(79\) −2.48528 −0.279616 −0.139808 − 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) 3.00000 0.335410
\(81\) −11.0000 −1.22222
\(82\) −2.48528 −0.274453
\(83\) 7.17157 0.787182 0.393591 − 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(84\) −17.6569 −1.92652
\(85\) 2.82843 0.306786
\(86\) −2.48528 −0.267995
\(87\) −2.00000 −0.214423
\(88\) −1.31371 −0.140042
\(89\) −7.65685 −0.811625 −0.405812 − 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0.414214 0.0436619
\(91\) 9.65685 1.01231
\(92\) 5.79899 0.604586
\(93\) −12.9706 −1.34498
\(94\) −4.82843 −0.498014
\(95\) −4.82843 −0.495386
\(96\) −8.82843 −0.901048
\(97\) −12.4853 −1.26769 −0.633844 − 0.773461i \(-0.718524\pi\)
−0.633844 + 0.773461i \(0.718524\pi\)
\(98\) 6.75736 0.682596
\(99\) 0.828427 0.0832601
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 145.2.a.b.1.2 ✓ 2
3.2 odd 2 1305.2.a.n.1.1 2
4.3 odd 2 2320.2.a.k.1.2 2
5.2 odd 4 725.2.b.c.349.3 4
5.3 odd 4 725.2.b.c.349.2 4
5.4 even 2 725.2.a.c.1.1 2
7.6 odd 2 7105.2.a.e.1.2 2
8.3 odd 2 9280.2.a.w.1.2 2
8.5 even 2 9280.2.a.be.1.1 2
15.14 odd 2 6525.2.a.p.1.2 2
29.28 even 2 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 ✓ 2 1.1 even 1 trivial
725.2.a.c.1.1 2 5.4 even 2
725.2.b.c.349.2 4 5.3 odd 4
725.2.b.c.349.3 4 5.2 odd 4
1305.2.a.n.1.1 2 3.2 odd 2
2320.2.a.k.1.2 2 4.3 odd 2
4205.2.a.d.1.1 2 29.28 even 2
6525.2.a.p.1.2 2 15.14 odd 2
7105.2.a.e.1.2 2 7.6 odd 2
9280.2.a.w.1.2 2 8.3 odd 2
9280.2.a.be.1.1 2 8.5 even 2