Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,2,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-2,0,0,0,1] 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: \(3.57729801055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 448.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.00000 q^{13} -2.00000 q^{17} -6.00000 q^{19} -2.00000 q^{21} -8.00000 q^{23} -5.00000 q^{25} +4.00000 q^{27} -2.00000 q^{29} +4.00000 q^{31} +8.00000 q^{33} -10.0000 q^{37} -8.00000 q^{39} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} +2.00000 q^{53} +12.0000 q^{57} +10.0000 q^{59} +8.00000 q^{61} +1.00000 q^{63} -8.00000 q^{67} +16.0000 q^{69} -6.00000 q^{73} +10.0000 q^{75} -4.00000 q^{77} +16.0000 q^{79} -11.0000 q^{81} +2.00000 q^{83} +4.00000 q^{87} +18.0000 q^{89} +4.00000 q^{91} -8.00000 q^{93} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 16.0000 1.92617
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 10.0000 1.15470
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
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 448.2.a.b.1.1 1
3.2 odd 2 4032.2.a.z.1.1 1
4.3 odd 2 448.2.a.f.1.1 1
7.6 odd 2 3136.2.a.y.1.1 1
8.3 odd 2 224.2.a.a.1.1 ✓ 1
8.5 even 2 224.2.a.b.1.1 yes 1
12.11 even 2 4032.2.a.p.1.1 1
16.3 odd 4 1792.2.b.f.897.2 2
16.5 even 4 1792.2.b.b.897.2 2
16.11 odd 4 1792.2.b.f.897.1 2
16.13 even 4 1792.2.b.b.897.1 2
24.5 odd 2 2016.2.a.g.1.1 1
24.11 even 2 2016.2.a.e.1.1 1
28.27 even 2 3136.2.a.f.1.1 1
40.19 odd 2 5600.2.a.t.1.1 1
40.29 even 2 5600.2.a.c.1.1 1
56.3 even 6 1568.2.i.c.961.1 2
56.5 odd 6 1568.2.i.j.1537.1 2
56.11 odd 6 1568.2.i.k.961.1 2
56.13 odd 2 1568.2.a.b.1.1 1
56.19 even 6 1568.2.i.c.1537.1 2
56.27 even 2 1568.2.a.h.1.1 1
56.37 even 6 1568.2.i.b.1537.1 2
56.45 odd 6 1568.2.i.j.961.1 2
56.51 odd 6 1568.2.i.k.1537.1 2
56.53 even 6 1568.2.i.b.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.a.1.1 ✓ 1 8.3 odd 2
224.2.a.b.1.1 yes 1 8.5 even 2
448.2.a.b.1.1 1 1.1 even 1 trivial
448.2.a.f.1.1 1 4.3 odd 2
1568.2.a.b.1.1 1 56.13 odd 2
1568.2.a.h.1.1 1 56.27 even 2
1568.2.i.b.961.1 2 56.53 even 6
1568.2.i.b.1537.1 2 56.37 even 6
1568.2.i.c.961.1 2 56.3 even 6
1568.2.i.c.1537.1 2 56.19 even 6
1568.2.i.j.961.1 2 56.45 odd 6
1568.2.i.j.1537.1 2 56.5 odd 6
1568.2.i.k.961.1 2 56.11 odd 6
1568.2.i.k.1537.1 2 56.51 odd 6
1792.2.b.b.897.1 2 16.13 even 4
1792.2.b.b.897.2 2 16.5 even 4
1792.2.b.f.897.1 2 16.11 odd 4
1792.2.b.f.897.2 2 16.3 odd 4
2016.2.a.e.1.1 1 24.11 even 2
2016.2.a.g.1.1 1 24.5 odd 2
3136.2.a.f.1.1 1 28.27 even 2
3136.2.a.y.1.1 1 7.6 odd 2
4032.2.a.p.1.1 1 12.11 even 2
4032.2.a.z.1.1 1 3.2 odd 2
5600.2.a.c.1.1 1 40.29 even 2
5600.2.a.t.1.1 1 40.19 odd 2