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(749,1100)] 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("1100.749"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-10,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.2
Root \(-1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.2.b.d.749.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.79129i q^{3} -4.79129i q^{7} -0.208712 q^{9} -1.00000 q^{11} +1.00000i q^{13} -3.79129i q^{17} +2.58258 q^{19} -8.58258 q^{21} -0.791288i q^{23} -5.00000i q^{27} -2.20871 q^{29} +0.582576 q^{31} +1.79129i q^{33} +6.58258i q^{37} +1.79129 q^{39} -10.5826 q^{41} +10.0000i q^{43} -10.5826i q^{47} -15.9564 q^{49} -6.79129 q^{51} +2.37386i q^{53} -4.62614i q^{57} +1.41742 q^{59} +8.79129 q^{61} +1.00000i q^{63} -4.00000i q^{67} -1.41742 q^{69} +16.7477 q^{71} +3.20871i q^{73} +4.79129i q^{77} -16.5390 q^{79} -9.58258 q^{81} -12.9564i q^{83} +3.95644i q^{87} -3.79129 q^{89} +4.79129 q^{91} -1.04356i q^{93} -10.7913i q^{97} +0.208712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} - 4 q^{11} - 8 q^{19} - 16 q^{21} - 18 q^{29} - 16 q^{31} - 2 q^{39} - 24 q^{41} - 18 q^{49} - 18 q^{51} + 24 q^{59} + 26 q^{61} - 24 q^{69} + 12 q^{71} - 2 q^{79} - 20 q^{81} - 6 q^{89}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.79129i − 1.03420i −0.855925 − 0.517100i \(-0.827011\pi\)
0.855925 − 0.517100i \(-0.172989\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.79129i − 1.81094i −0.424414 − 0.905468i \(-0.639520\pi\)
0.424414 − 0.905468i \(-0.360480\pi\)
\(8\) 0 0
\(9\) −0.208712 −0.0695707
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.79129i − 0.919522i −0.888043 − 0.459761i \(-0.847935\pi\)
0.888043 − 0.459761i \(-0.152065\pi\)
\(18\) 0 0
\(19\) 2.58258 0.592483 0.296242 − 0.955113i \(-0.404267\pi\)
0.296242 + 0.955113i \(0.404267\pi\)
\(20\) 0 0
\(21\) −8.58258 −1.87287
\(22\) 0 0
\(23\) − 0.791288i − 0.164995i −0.996591 − 0.0824975i \(-0.973710\pi\)
0.996591 − 0.0824975i \(-0.0262896\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −2.20871 −0.410148 −0.205074 − 0.978747i \(-0.565743\pi\)
−0.205074 + 0.978747i \(0.565743\pi\)
\(30\) 0 0
\(31\) 0.582576 0.104634 0.0523168 − 0.998631i \(-0.483339\pi\)
0.0523168 + 0.998631i \(0.483339\pi\)
\(32\) 0 0
\(33\) 1.79129i 0.311823i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.58258i 1.08217i 0.840968 + 0.541084i \(0.181986\pi\)
−0.840968 + 0.541084i \(0.818014\pi\)
\(38\) 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
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 10.5826i − 1.54363i −0.635849 − 0.771814i \(-0.719350\pi\)
0.635849 − 0.771814i \(-0.280650\pi\)
\(48\) 0 0
\(49\) −15.9564 −2.27949
\(50\) 0 0
\(51\) −6.79129 −0.950971
\(52\) 0 0
\(53\) 2.37386i 0.326075i 0.986620 + 0.163038i \(0.0521292\pi\)
−0.986620 + 0.163038i \(0.947871\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.62614i − 0.612747i
\(58\) 0 0
\(59\) 1.41742 0.184533 0.0922665 − 0.995734i \(-0.470589\pi\)
0.0922665 + 0.995734i \(0.470589\pi\)
\(60\) 0 0
\(61\) 8.79129 1.12561 0.562805 − 0.826590i \(-0.309722\pi\)
0.562805 + 0.826590i \(0.309722\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 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
\(73\) 3.20871i 0.375551i 0.982212 + 0.187776i \(0.0601278\pi\)
−0.982212 + 0.187776i \(0.939872\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.79129i 0.546018i
\(78\) 0 0
\(79\) −16.5390 −1.86078 −0.930392 − 0.366565i \(-0.880534\pi\)
−0.930392 + 0.366565i \(0.880534\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) − 12.9564i − 1.42215i −0.703114 − 0.711077i \(-0.748208\pi\)
0.703114 − 0.711077i \(-0.251792\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.95644i 0.424175i
\(88\) 0 0
\(89\) −3.79129 −0.401876 −0.200938 − 0.979604i \(-0.564399\pi\)
−0.200938 + 0.979604i \(0.564399\pi\)
\(90\) 0 0
\(91\) 4.79129 0.502263
\(92\) 0 0
\(93\) − 1.04356i − 0.108212i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.7913i − 1.09569i −0.836580 − 0.547845i \(-0.815448\pi\)
0.836580 − 0.547845i \(-0.184552\pi\)
\(98\) 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.b.d.749.2 4
3.2 odd 2 9900.2.c.x.5149.1 4
4.3 odd 2 4400.2.b.s.4049.3 4
5.2 odd 4 1100.2.a.h.1.1 yes 2
5.3 odd 4 1100.2.a.g.1.2 ✓ 2
5.4 even 2 inner 1100.2.b.d.749.3 4
15.2 even 4 9900.2.a.bz.1.2 2
15.8 even 4 9900.2.a.bh.1.1 2
15.14 odd 2 9900.2.c.x.5149.4 4
20.3 even 4 4400.2.a.bu.1.1 2
20.7 even 4 4400.2.a.bi.1.2 2
20.19 odd 2 4400.2.b.s.4049.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.2 ✓ 2 5.3 odd 4
1100.2.a.h.1.1 yes 2 5.2 odd 4
1100.2.b.d.749.2 4 1.1 even 1 trivial
1100.2.b.d.749.3 4 5.4 even 2 inner
4400.2.a.bi.1.2 2 20.7 even 4
4400.2.a.bu.1.1 2 20.3 even 4
4400.2.b.s.4049.2 4 20.19 odd 2
4400.2.b.s.4049.3 4 4.3 odd 2
9900.2.a.bh.1.1 2 15.8 even 4
9900.2.a.bz.1.2 2 15.2 even 4
9900.2.c.x.5149.1 4 3.2 odd 2
9900.2.c.x.5149.4 4 15.14 odd 2