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Properties

Label 112.2.i.a.81.1
Level $112$
Weight $2$
Character 112.81
Analytic conductor $0.894$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [112,2,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 112.81
Dual form 112.2.i.a.65.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(1.50000 - 2.59808i) q^{11} -6.00000 q^{13} -1.00000 q^{15} +(2.50000 - 4.33013i) q^{17} +(0.500000 + 0.866025i) q^{19} +(-2.50000 + 0.866025i) q^{21} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} -5.00000 q^{27} +2.00000 q^{29} +(-2.50000 + 4.33013i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-0.500000 + 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(3.00000 - 5.19615i) q^{39} -2.00000 q^{41} +4.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(2.50000 + 4.33013i) q^{47} +(1.00000 + 6.92820i) q^{49} +(2.50000 + 4.33013i) q^{51} +(0.500000 - 0.866025i) q^{53} +3.00000 q^{55} -1.00000 q^{57} +(7.50000 - 12.9904i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-1.00000 + 5.19615i) q^{63} +(-3.00000 - 5.19615i) q^{65} +(-4.50000 + 7.79423i) q^{67} +7.00000 q^{69} +(-3.50000 + 6.06218i) q^{73} +(2.00000 + 3.46410i) q^{75} +(7.50000 - 2.59808i) q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -12.0000 q^{83} +5.00000 q^{85} +(-1.00000 + 1.73205i) q^{87} +(-3.50000 - 6.06218i) q^{89} +(-12.0000 - 10.3923i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} + 4 q^{7} + 2 q^{9} + 3 q^{11} - 12 q^{13} - 2 q^{15} + 5 q^{17} + q^{19} - 5 q^{21} - 7 q^{23} + 4 q^{25} - 10 q^{27} + 4 q^{29} - 5 q^{31} + 3 q^{33} - q^{35} - 3 q^{37} + 6 q^{39}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 0
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i −0.973494 − 0.228714i \(-0.926548\pi\)
0.684819 + 0.728714i \(0.259881\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 − 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.50000 − 4.33013i 0.606339 − 1.05021i −0.385499 − 0.922708i \(-0.625971\pi\)
0.991838 − 0.127502i \(-0.0406959\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 − 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) −2.50000 + 0.866025i −0.545545 + 0.188982i
\(22\) 0 0
\(23\) −3.50000 − 6.06218i −0.729800 − 1.26405i −0.956967 − 0.290196i \(-0.906280\pi\)
0.227167 − 0.973856i \(-0.427054\pi\)
\(24\) 0 0
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 − 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 0 0
\(33\) 1.50000 + 2.59808i 0.261116 + 0.452267i
\(34\) 0 0
\(35\) −0.500000 + 2.59808i −0.0845154 + 0.439155i
\(36\) 0 0
\(37\) −1.50000 − 2.59808i −0.246598 − 0.427121i 0.715981 − 0.698119i \(-0.245980\pi\)
−0.962580 + 0.270998i \(0.912646\pi\)
\(38\) 0 0
\(39\) 3.00000 − 5.19615i 0.480384 − 0.832050i
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.00000 + 1.73205i −0.149071 + 0.258199i
\(46\) 0 0
\(47\) 2.50000 + 4.33013i 0.364662 + 0.631614i 0.988722 − 0.149763i \(-0.0478510\pi\)
−0.624059 + 0.781377i \(0.714518\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 2.50000 + 4.33013i 0.350070 + 0.606339i
\(52\) 0 0
\(53\) 0.500000 − 0.866025i 0.0686803 − 0.118958i −0.829640 − 0.558298i \(-0.811454\pi\)
0.898321 + 0.439340i \(0.144788\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 7.50000 − 12.9904i 0.976417 − 1.69120i 0.301239 − 0.953549i \(-0.402600\pi\)
0.675178 − 0.737655i \(-0.264067\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 − 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) −1.00000 + 5.19615i −0.125988 + 0.654654i
\(64\) 0 0
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) 0 0
\(67\) −4.50000 + 7.79423i −0.549762 + 0.952217i 0.448528 + 0.893769i \(0.351948\pi\)
−0.998290 + 0.0584478i \(0.981385\pi\)
\(68\) 0 0
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.50000 + 6.06218i −0.409644 + 0.709524i −0.994850 − 0.101361i \(-0.967680\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 2.00000 + 3.46410i 0.230940 + 0.400000i
\(76\) 0 0
\(77\) 7.50000 − 2.59808i 0.854704 − 0.296078i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 − 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) −1.00000 + 1.73205i −0.107211 + 0.185695i
\(88\) 0 0
\(89\) −3.50000 − 6.06218i −0.370999 − 0.642590i 0.618720 − 0.785611i \(-0.287651\pi\)
−0.989720 + 0.143022i \(0.954318\pi\)
\(90\) 0 0
\(91\) −12.0000 − 10.3923i −1.25794 − 1.08941i
\(92\) 0 0
\(93\) −2.50000 − 4.33013i −0.259238 − 0.449013i
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 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 112.2.i.a.81.1 2
3.2 odd 2 1008.2.s.g.865.1 2
4.3 odd 2 56.2.i.b.25.1 yes 2
7.2 even 3 inner 112.2.i.a.65.1 2
7.3 odd 6 784.2.a.c.1.1 1
7.4 even 3 784.2.a.h.1.1 1
7.5 odd 6 784.2.i.h.177.1 2
7.6 odd 2 784.2.i.h.753.1 2
8.3 odd 2 448.2.i.b.193.1 2
8.5 even 2 448.2.i.d.193.1 2
12.11 even 2 504.2.s.c.361.1 2
20.3 even 4 1400.2.bh.a.249.2 4
20.7 even 4 1400.2.bh.a.249.1 4
20.19 odd 2 1400.2.q.d.1201.1 2
21.2 odd 6 1008.2.s.g.289.1 2
21.11 odd 6 7056.2.a.bj.1.1 1
21.17 even 6 7056.2.a.u.1.1 1
28.3 even 6 392.2.a.e.1.1 1
28.11 odd 6 392.2.a.c.1.1 1
28.19 even 6 392.2.i.b.177.1 2
28.23 odd 6 56.2.i.b.9.1 ✓ 2
28.27 even 2 392.2.i.b.361.1 2
56.3 even 6 3136.2.a.i.1.1 1
56.11 odd 6 3136.2.a.u.1.1 1
56.37 even 6 448.2.i.d.65.1 2
56.45 odd 6 3136.2.a.t.1.1 1
56.51 odd 6 448.2.i.b.65.1 2
56.53 even 6 3136.2.a.j.1.1 1
84.11 even 6 3528.2.a.p.1.1 1
84.23 even 6 504.2.s.c.289.1 2
84.47 odd 6 3528.2.s.q.3313.1 2
84.59 odd 6 3528.2.a.j.1.1 1
84.83 odd 2 3528.2.s.q.361.1 2
140.23 even 12 1400.2.bh.a.849.1 4
140.39 odd 6 9800.2.a.be.1.1 1
140.59 even 6 9800.2.a.s.1.1 1
140.79 odd 6 1400.2.q.d.401.1 2
140.107 even 12 1400.2.bh.a.849.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.i.b.9.1 ✓ 2 28.23 odd 6
56.2.i.b.25.1 yes 2 4.3 odd 2
112.2.i.a.65.1 2 7.2 even 3 inner
112.2.i.a.81.1 2 1.1 even 1 trivial
392.2.a.c.1.1 1 28.11 odd 6
392.2.a.e.1.1 1 28.3 even 6
392.2.i.b.177.1 2 28.19 even 6
392.2.i.b.361.1 2 28.27 even 2
448.2.i.b.65.1 2 56.51 odd 6
448.2.i.b.193.1 2 8.3 odd 2
448.2.i.d.65.1 2 56.37 even 6
448.2.i.d.193.1 2 8.5 even 2
504.2.s.c.289.1 2 84.23 even 6
504.2.s.c.361.1 2 12.11 even 2
784.2.a.c.1.1 1 7.3 odd 6
784.2.a.h.1.1 1 7.4 even 3
784.2.i.h.177.1 2 7.5 odd 6
784.2.i.h.753.1 2 7.6 odd 2
1008.2.s.g.289.1 2 21.2 odd 6
1008.2.s.g.865.1 2 3.2 odd 2
1400.2.q.d.401.1 2 140.79 odd 6
1400.2.q.d.1201.1 2 20.19 odd 2
1400.2.bh.a.249.1 4 20.7 even 4
1400.2.bh.a.249.2 4 20.3 even 4
1400.2.bh.a.849.1 4 140.23 even 12
1400.2.bh.a.849.2 4 140.107 even 12
3136.2.a.i.1.1 1 56.3 even 6
3136.2.a.j.1.1 1 56.53 even 6
3136.2.a.t.1.1 1 56.45 odd 6
3136.2.a.u.1.1 1 56.11 odd 6
3528.2.a.j.1.1 1 84.59 odd 6
3528.2.a.p.1.1 1 84.11 even 6
3528.2.s.q.361.1 2 84.83 odd 2
3528.2.s.q.3313.1 2 84.47 odd 6
7056.2.a.u.1.1 1 21.17 even 6
7056.2.a.bj.1.1 1 21.11 odd 6
9800.2.a.s.1.1 1 140.59 even 6
9800.2.a.be.1.1 1 140.39 odd 6