Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,-3,0,-2] 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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3344.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} +3.00000 q^{15} -2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +7.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +4.00000 q^{29} +3.00000 q^{31} -1.00000 q^{33} +6.00000 q^{35} -5.00000 q^{37} -4.00000 q^{39} +4.00000 q^{41} +2.00000 q^{43} +6.00000 q^{45} -3.00000 q^{49} +2.00000 q^{51} -14.0000 q^{53} -3.00000 q^{55} -1.00000 q^{57} +5.00000 q^{59} -8.00000 q^{61} +4.00000 q^{63} -12.0000 q^{65} -7.00000 q^{67} -7.00000 q^{69} +9.00000 q^{71} +8.00000 q^{73} -4.00000 q^{75} -2.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +6.00000 q^{85} -4.00000 q^{87} -13.0000 q^{89} -8.00000 q^{91} -3.00000 q^{93} -3.00000 q^{95} -11.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 − 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 − 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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 3344.2.a.c.1.1 1
4.3 odd 2 1672.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.b.1.1 ✓ 1 4.3 odd 2
3344.2.a.c.1.1 1 1.1 even 1 trivial