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Properties

Label 117.2.a.c.1.2
Level $117$
Weight $2$
Character 117.1
Self dual yes
Analytic conductor $0.934$
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] = [117,2,Mod(1,117)] 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("117.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.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: \(0.934249703649\)
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: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 117.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} +4.41421 q^{8} -6.82843 q^{10} +2.00000 q^{11} -1.00000 q^{13} -6.82843 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} -10.8284 q^{20} +4.82843 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.41421 q^{26} -10.8284 q^{28} -2.00000 q^{29} -6.82843 q^{31} -1.58579 q^{32} +8.82843 q^{34} +8.00000 q^{35} +3.65685 q^{37} +6.82843 q^{38} -12.4853 q^{40} -10.8284 q^{41} +9.65685 q^{43} +7.65685 q^{44} +9.65685 q^{46} +0.343146 q^{47} +1.00000 q^{49} +7.24264 q^{50} -3.82843 q^{52} +2.00000 q^{53} -5.65685 q^{55} -12.4853 q^{56} -4.82843 q^{58} +3.65685 q^{59} -9.31371 q^{61} -16.4853 q^{62} -9.82843 q^{64} +2.82843 q^{65} +1.17157 q^{67} +14.0000 q^{68} +19.3137 q^{70} -2.00000 q^{71} +11.6569 q^{73} +8.82843 q^{74} +10.8284 q^{76} -5.65685 q^{77} +11.3137 q^{79} -8.48528 q^{80} -26.1421 q^{82} +7.65685 q^{83} -10.3431 q^{85} +23.3137 q^{86} +8.82843 q^{88} -9.17157 q^{89} +2.82843 q^{91} +15.3137 q^{92} +0.828427 q^{94} -8.00000 q^{95} -7.65685 q^{97} +2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 8 q^{10} + 4 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} - 4 q^{17} - 16 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 16 q^{28} - 4 q^{29} - 8 q^{31} - 6 q^{32} + 12 q^{34}+ \cdots + 2 q^{98}+O(q^{100}) \) 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 \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 − 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) −6.82843 −2.15934
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −6.82843 −1.82497
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 3.65685 0.886917 0.443459 − 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −10.8284 −2.42131
\(21\) 0 0
\(22\) 4.82843 1.02942
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) −2.41421 −0.473466
\(27\) 0 0
\(28\) −10.8284 −2.04638
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 − 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 8.82843 1.51406
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 − 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 6.82843 1.10772
\(39\) 0 0
\(40\) −12.4853 −1.97410
\(41\) −10.8284 −1.69112 −0.845558 − 0.533883i \(-0.820732\pi\)
−0.845558 + 0.533883i \(0.820732\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 − 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 7.65685 1.15431
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 0.343146 0.0500530 0.0250265 − 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.24264 1.02426
\(51\) 0 0
\(52\) −3.82843 −0.530907
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) −12.4853 −1.66842
\(57\) 0 0
\(58\) −4.82843 −0.634004
\(59\) 3.65685 0.476082 0.238041 − 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 − 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) −16.4853 −2.09363
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 2.82843 0.350823
\(66\) 0 0
\(67\) 1.17157 0.143130 0.0715652 − 0.997436i \(-0.477201\pi\)
0.0715652 + 0.997436i \(0.477201\pi\)
\(68\) 14.0000 1.69775
\(69\) 0 0
\(70\) 19.3137 2.30843
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 11.6569 1.36433 0.682166 − 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) 8.82843 1.02628
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 − 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −8.48528 −0.948683
\(81\) 0 0
\(82\) −26.1421 −2.88692
\(83\) 7.65685 0.840449 0.420224 − 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) 0 0
\(85\) −10.3431 −1.12187
\(86\) 23.3137 2.51398
\(87\) 0 0
\(88\) 8.82843 0.941113
\(89\) −9.17157 −0.972185 −0.486092 − 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 15.3137 1.59656
\(93\) 0 0
\(94\) 0.828427 0.0854457
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −7.65685 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) 2.41421 0.243872
\(99\) 0 0
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 117.2.a.c.1.2 2
3.2 odd 2 39.2.a.b.1.1 ✓ 2
4.3 odd 2 1872.2.a.w.1.1 2
5.2 odd 4 2925.2.c.u.2224.4 4
5.3 odd 4 2925.2.c.u.2224.1 4
5.4 even 2 2925.2.a.v.1.1 2
7.6 odd 2 5733.2.a.u.1.2 2
8.3 odd 2 7488.2.a.co.1.2 2
8.5 even 2 7488.2.a.cl.1.2 2
9.2 odd 6 1053.2.e.m.352.2 4
9.4 even 3 1053.2.e.e.703.1 4
9.5 odd 6 1053.2.e.m.703.2 4
9.7 even 3 1053.2.e.e.352.1 4
12.11 even 2 624.2.a.k.1.2 2
13.5 odd 4 1521.2.b.j.1351.1 4
13.8 odd 4 1521.2.b.j.1351.4 4
13.12 even 2 1521.2.a.f.1.1 2
15.2 even 4 975.2.c.h.274.1 4
15.8 even 4 975.2.c.h.274.4 4
15.14 odd 2 975.2.a.l.1.2 2
21.20 even 2 1911.2.a.h.1.1 2
24.5 odd 2 2496.2.a.bf.1.1 2
24.11 even 2 2496.2.a.bi.1.1 2
33.32 even 2 4719.2.a.p.1.2 2
39.2 even 12 507.2.j.f.316.1 8
39.5 even 4 507.2.b.e.337.4 4
39.8 even 4 507.2.b.e.337.1 4
39.11 even 12 507.2.j.f.316.4 8
39.17 odd 6 507.2.e.d.484.1 4
39.20 even 12 507.2.j.f.361.1 8
39.23 odd 6 507.2.e.d.22.1 4
39.29 odd 6 507.2.e.h.22.2 4
39.32 even 12 507.2.j.f.361.4 8
39.35 odd 6 507.2.e.h.484.2 4
39.38 odd 2 507.2.a.h.1.2 2
156.155 even 2 8112.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.1 ✓ 2 3.2 odd 2
117.2.a.c.1.2 2 1.1 even 1 trivial
507.2.a.h.1.2 2 39.38 odd 2
507.2.b.e.337.1 4 39.8 even 4
507.2.b.e.337.4 4 39.5 even 4
507.2.e.d.22.1 4 39.23 odd 6
507.2.e.d.484.1 4 39.17 odd 6
507.2.e.h.22.2 4 39.29 odd 6
507.2.e.h.484.2 4 39.35 odd 6
507.2.j.f.316.1 8 39.2 even 12
507.2.j.f.316.4 8 39.11 even 12
507.2.j.f.361.1 8 39.20 even 12
507.2.j.f.361.4 8 39.32 even 12
624.2.a.k.1.2 2 12.11 even 2
975.2.a.l.1.2 2 15.14 odd 2
975.2.c.h.274.1 4 15.2 even 4
975.2.c.h.274.4 4 15.8 even 4
1053.2.e.e.352.1 4 9.7 even 3
1053.2.e.e.703.1 4 9.4 even 3
1053.2.e.m.352.2 4 9.2 odd 6
1053.2.e.m.703.2 4 9.5 odd 6
1521.2.a.f.1.1 2 13.12 even 2
1521.2.b.j.1351.1 4 13.5 odd 4
1521.2.b.j.1351.4 4 13.8 odd 4
1872.2.a.w.1.1 2 4.3 odd 2
1911.2.a.h.1.1 2 21.20 even 2
2496.2.a.bf.1.1 2 24.5 odd 2
2496.2.a.bi.1.1 2 24.11 even 2
2925.2.a.v.1.1 2 5.4 even 2
2925.2.c.u.2224.1 4 5.3 odd 4
2925.2.c.u.2224.4 4 5.2 odd 4
4719.2.a.p.1.2 2 33.32 even 2
5733.2.a.u.1.2 2 7.6 odd 2
7488.2.a.cl.1.2 2 8.5 even 2
7488.2.a.co.1.2 2 8.3 odd 2
8112.2.a.bm.1.1 2 156.155 even 2