Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2200,2,Mod(1,2200)] 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("2200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,0,0,3,0,6,0,-3] 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: yes
Analytic conductor: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) 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 1.2
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.16745 q^{3} +4.97196 q^{7} -1.63706 q^{9} -1.00000 q^{11} -0.665102 q^{13} +6.77647 q^{17} +1.00000 q^{19} +5.80451 q^{21} +2.16745 q^{23} -5.41353 q^{27} +7.97196 q^{29} -8.94392 q^{31} -1.16745 q^{33} -0.139410 q^{37} -0.776472 q^{39} -1.80451 q^{41} +2.80451 q^{43} +0.530387 q^{47} +17.7204 q^{49} +7.91119 q^{51} -6.30216 q^{53} +1.16745 q^{57} +11.4696 q^{59} +10.5810 q^{61} -8.13941 q^{63} +9.60902 q^{67} +2.53039 q^{69} -9.80921 q^{71} -7.02804 q^{73} -4.97196 q^{77} +5.50235 q^{79} -1.40884 q^{81} +13.5810 q^{83} +9.30686 q^{87} -9.58098 q^{89} -3.30686 q^{91} -10.4416 q^{93} -14.7812 q^{97} +1.63706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 3 q^{13} - 3 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 18 q^{27} + 12 q^{29} - 3 q^{31} - 3 q^{33} + 12 q^{37} + 21 q^{39} + 6 q^{41} - 3 q^{43} + 12 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.16745 0.674027 0.337014 − 0.941500i \(-0.390583\pi\)
0.337014 + 0.941500i \(0.390583\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.97196 1.87922 0.939612 − 0.342241i \(-0.111186\pi\)
0.939612 + 0.342241i \(0.111186\pi\)
\(8\) 0 0
\(9\) −1.63706 −0.545687
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.665102 −0.184466 −0.0922330 − 0.995737i \(-0.529400\pi\)
−0.0922330 + 0.995737i \(0.529400\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.77647 1.64354 0.821768 − 0.569822i \(-0.192988\pi\)
0.821768 + 0.569822i \(0.192988\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 5.80451 1.26665
\(22\) 0 0
\(23\) 2.16745 0.451944 0.225972 − 0.974134i \(-0.427444\pi\)
0.225972 + 0.974134i \(0.427444\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.41353 −1.04184
\(28\) 0 0
\(29\) 7.97196 1.48036 0.740178 − 0.672411i \(-0.234741\pi\)
0.740178 + 0.672411i \(0.234741\pi\)
\(30\) 0 0
\(31\) −8.94392 −1.60638 −0.803188 − 0.595726i \(-0.796864\pi\)
−0.803188 + 0.595726i \(0.796864\pi\)
\(32\) 0 0
\(33\) −1.16745 −0.203227
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.139410 −0.0229189 −0.0114594 − 0.999934i \(-0.503648\pi\)
−0.0114594 + 0.999934i \(0.503648\pi\)
\(38\) 0 0
\(39\) −0.776472 −0.124335
\(40\) 0 0
\(41\) −1.80451 −0.281817 −0.140909 − 0.990023i \(-0.545002\pi\)
−0.140909 + 0.990023i \(0.545002\pi\)
\(42\) 0 0
\(43\) 2.80451 0.427684 0.213842 − 0.976868i \(-0.431402\pi\)
0.213842 + 0.976868i \(0.431402\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.530387 0.0773649 0.0386824 − 0.999252i \(-0.487684\pi\)
0.0386824 + 0.999252i \(0.487684\pi\)
\(48\) 0 0
\(49\) 17.7204 2.53148
\(50\) 0 0
\(51\) 7.91119 1.10779
\(52\) 0 0
\(53\) −6.30216 −0.865669 −0.432834 − 0.901473i \(-0.642487\pi\)
−0.432834 + 0.901473i \(0.642487\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.16745 0.154632
\(58\) 0 0
\(59\) 11.4696 1.49322 0.746608 − 0.665264i \(-0.231681\pi\)
0.746608 + 0.665264i \(0.231681\pi\)
\(60\) 0 0
\(61\) 10.5810 1.35476 0.677378 − 0.735635i \(-0.263116\pi\)
0.677378 + 0.735635i \(0.263116\pi\)
\(62\) 0 0
\(63\) −8.13941 −1.02547
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.60902 1.17393 0.586965 − 0.809613i \(-0.300323\pi\)
0.586965 + 0.809613i \(0.300323\pi\)
\(68\) 0 0
\(69\) 2.53039 0.304623
\(70\) 0 0
\(71\) −9.80921 −1.16414 −0.582069 − 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(72\) 0 0
\(73\) −7.02804 −0.822570 −0.411285 − 0.911507i \(-0.634920\pi\)
−0.411285 + 0.911507i \(0.634920\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.97196 −0.566608
\(78\) 0 0
\(79\) 5.50235 0.619062 0.309531 − 0.950889i \(-0.399828\pi\)
0.309531 + 0.950889i \(0.399828\pi\)
\(80\) 0 0
\(81\) −1.40884 −0.156538
\(82\) 0 0
\(83\) 13.5810 1.49071 0.745353 − 0.666670i \(-0.232281\pi\)
0.745353 + 0.666670i \(0.232281\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.30686 0.997800
\(88\) 0 0
\(89\) −9.58098 −1.01558 −0.507791 − 0.861480i \(-0.669538\pi\)
−0.507791 + 0.861480i \(0.669538\pi\)
\(90\) 0 0
\(91\) −3.30686 −0.346653
\(92\) 0 0
\(93\) −10.4416 −1.08274
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.7812 −1.50080 −0.750400 − 0.660984i \(-0.770139\pi\)
−0.750400 + 0.660984i \(0.770139\pi\)
\(98\) 0 0
\(99\) 1.63706 0.164531
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 2200.2.a.w.1.2 yes 3
4.3 odd 2 4400.2.a.bx.1.2 3
5.2 odd 4 2200.2.b.l.1849.3 6
5.3 odd 4 2200.2.b.l.1849.4 6
5.4 even 2 2200.2.a.t.1.2 ✓ 3
20.3 even 4 4400.2.b.bc.4049.3 6
20.7 even 4 4400.2.b.bc.4049.4 6
20.19 odd 2 4400.2.a.ca.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.t.1.2 ✓ 3 5.4 even 2
2200.2.a.w.1.2 yes 3 1.1 even 1 trivial
2200.2.b.l.1849.3 6 5.2 odd 4
2200.2.b.l.1849.4 6 5.3 odd 4
4400.2.a.bx.1.2 3 4.3 odd 2
4400.2.a.ca.1.2 3 20.19 odd 2
4400.2.b.bc.4049.3 6 20.3 even 4
4400.2.b.bc.4049.4 6 20.7 even 4