Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,2,Mod(1,1100)] 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("1100.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 1100.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.79129 q^{3} +4.79129 q^{7} +0.208712 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.79129 q^{17} -2.58258 q^{19} -8.58258 q^{21} -0.791288 q^{23} +5.00000 q^{27} +2.20871 q^{29} +0.582576 q^{31} +1.79129 q^{33} -6.58258 q^{37} -1.79129 q^{39} -10.5826 q^{41} +10.0000 q^{43} +10.5826 q^{47} +15.9564 q^{49} -6.79129 q^{51} +2.37386 q^{53} +4.62614 q^{57} -1.41742 q^{59} +8.79129 q^{61} +1.00000 q^{63} +4.00000 q^{67} +1.41742 q^{69} +16.7477 q^{71} +3.20871 q^{73} -4.79129 q^{77} +16.5390 q^{79} -9.58258 q^{81} -12.9564 q^{83} -3.95644 q^{87} +3.79129 q^{89} +4.79129 q^{91} -1.04356 q^{93} +10.7913 q^{97} -0.208712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + 5 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{17} + 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} - 8 q^{31} - q^{33} - 4 q^{37} + q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots - 5 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.79129 −1.03420 −0.517100 − 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.79129 1.81094 0.905468 − 0.424414i \(-0.139520\pi\)
0.905468 + 0.424414i \(0.139520\pi\)
\(8\) 0 0
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.79129 0.919522 0.459761 − 0.888043i \(-0.347935\pi\)
0.459761 + 0.888043i \(0.347935\pi\)
\(18\) 0 0
\(19\) −2.58258 −0.592483 −0.296242 − 0.955113i \(-0.595733\pi\)
−0.296242 + 0.955113i \(0.595733\pi\)
\(20\) 0 0
\(21\) −8.58258 −1.87287
\(22\) 0 0
\(23\) −0.791288 −0.164995 −0.0824975 − 0.996591i \(-0.526290\pi\)
−0.0824975 + 0.996591i \(0.526290\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 2.20871 0.410148 0.205074 − 0.978747i \(-0.434257\pi\)
0.205074 + 0.978747i \(0.434257\pi\)
\(30\) 0 0
\(31\) 0.582576 0.104634 0.0523168 − 0.998631i \(-0.483339\pi\)
0.0523168 + 0.998631i \(0.483339\pi\)
\(32\) 0 0
\(33\) 1.79129 0.311823
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58258 −1.08217 −0.541084 − 0.840968i \(-0.681986\pi\)
−0.541084 + 0.840968i \(0.681986\pi\)
\(38\) 0 0
\(39\) −1.79129 −0.286836
\(40\) 0 0
\(41\) −10.5826 −1.65272 −0.826360 − 0.563142i \(-0.809593\pi\)
−0.826360 + 0.563142i \(0.809593\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.5826 1.54363 0.771814 − 0.635849i \(-0.219350\pi\)
0.771814 + 0.635849i \(0.219350\pi\)
\(48\) 0 0
\(49\) 15.9564 2.27949
\(50\) 0 0
\(51\) −6.79129 −0.950971
\(52\) 0 0
\(53\) 2.37386 0.326075 0.163038 − 0.986620i \(-0.447871\pi\)
0.163038 + 0.986620i \(0.447871\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.62614 0.612747
\(58\) 0 0
\(59\) −1.41742 −0.184533 −0.0922665 − 0.995734i \(-0.529411\pi\)
−0.0922665 + 0.995734i \(0.529411\pi\)
\(60\) 0 0
\(61\) 8.79129 1.12561 0.562805 − 0.826590i \(-0.309722\pi\)
0.562805 + 0.826590i \(0.309722\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.41742 0.170638
\(70\) 0 0
\(71\) 16.7477 1.98759 0.993795 − 0.111229i \(-0.0354788\pi\)
0.993795 + 0.111229i \(0.0354788\pi\)
\(72\) 0 0
\(73\) 3.20871 0.375551 0.187776 − 0.982212i \(-0.439872\pi\)
0.187776 + 0.982212i \(0.439872\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.79129 −0.546018
\(78\) 0 0
\(79\) 16.5390 1.86078 0.930392 − 0.366565i \(-0.119466\pi\)
0.930392 + 0.366565i \(0.119466\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) −12.9564 −1.42215 −0.711077 − 0.703114i \(-0.751792\pi\)
−0.711077 + 0.703114i \(0.751792\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.95644 −0.424175
\(88\) 0 0
\(89\) 3.79129 0.401876 0.200938 − 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(90\) 0 0
\(91\) 4.79129 0.502263
\(92\) 0 0
\(93\) −1.04356 −0.108212
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.7913 1.09569 0.547845 − 0.836580i \(-0.315448\pi\)
0.547845 + 0.836580i \(0.315448\pi\)
\(98\) 0 0
\(99\) −0.208712 −0.0209764
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 1100.2.a.h.1.1 yes 2
3.2 odd 2 9900.2.a.bz.1.2 2
4.3 odd 2 4400.2.a.bi.1.2 2
5.2 odd 4 1100.2.b.d.749.3 4
5.3 odd 4 1100.2.b.d.749.2 4
5.4 even 2 1100.2.a.g.1.2 ✓ 2
15.2 even 4 9900.2.c.x.5149.4 4
15.8 even 4 9900.2.c.x.5149.1 4
15.14 odd 2 9900.2.a.bh.1.1 2
20.3 even 4 4400.2.b.s.4049.3 4
20.7 even 4 4400.2.b.s.4049.2 4
20.19 odd 2 4400.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.2 ✓ 2 5.4 even 2
1100.2.a.h.1.1 yes 2 1.1 even 1 trivial
1100.2.b.d.749.2 4 5.3 odd 4
1100.2.b.d.749.3 4 5.2 odd 4
4400.2.a.bi.1.2 2 4.3 odd 2
4400.2.a.bu.1.1 2 20.19 odd 2
4400.2.b.s.4049.2 4 20.7 even 4
4400.2.b.s.4049.3 4 20.3 even 4
9900.2.a.bh.1.1 2 15.14 odd 2
9900.2.a.bz.1.2 2 3.2 odd 2
9900.2.c.x.5149.1 4 15.8 even 4
9900.2.c.x.5149.4 4 15.2 even 4