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 = [1,0,2,0,4,0,-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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +8.00000 q^{15} -2.00000 q^{17} -2.00000 q^{19} -2.00000 q^{21} -8.00000 q^{23} +11.0000 q^{25} -4.00000 q^{27} -2.00000 q^{29} -4.00000 q^{31} -4.00000 q^{35} +6.00000 q^{37} -2.00000 q^{41} +8.00000 q^{43} +4.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +10.0000 q^{53} -4.00000 q^{57} +6.00000 q^{59} -4.00000 q^{61} -1.00000 q^{63} -12.0000 q^{67} -16.0000 q^{69} -14.0000 q^{73} +22.0000 q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{83} -8.00000 q^{85} -4.00000 q^{87} +10.0000 q^{89} -8.00000 q^{93} -8.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 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
\(25\) 11.0000 2.20000
\(26\) 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 \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 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
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 22.0000 2.54034
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
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.h.1.1 1
3.2 odd 2 4032.2.a.a.1.1 1
4.3 odd 2 448.2.a.c.1.1 1
7.6 odd 2 3136.2.a.c.1.1 1
8.3 odd 2 56.2.a.b.1.1 ✓ 1
8.5 even 2 112.2.a.a.1.1 1
12.11 even 2 4032.2.a.d.1.1 1
16.3 odd 4 1792.2.b.a.897.1 2
16.5 even 4 1792.2.b.h.897.1 2
16.11 odd 4 1792.2.b.a.897.2 2
16.13 even 4 1792.2.b.h.897.2 2
24.5 odd 2 1008.2.a.m.1.1 1
24.11 even 2 504.2.a.h.1.1 1
28.27 even 2 3136.2.a.w.1.1 1
40.3 even 4 1400.2.g.b.449.2 2
40.13 odd 4 2800.2.g.g.449.1 2
40.19 odd 2 1400.2.a.a.1.1 1
40.27 even 4 1400.2.g.b.449.1 2
40.29 even 2 2800.2.a.bd.1.1 1
40.37 odd 4 2800.2.g.g.449.2 2
56.3 even 6 392.2.i.e.177.1 2
56.5 odd 6 784.2.i.b.753.1 2
56.11 odd 6 392.2.i.a.177.1 2
56.13 odd 2 784.2.a.i.1.1 1
56.19 even 6 392.2.i.e.361.1 2
56.27 even 2 392.2.a.b.1.1 1
56.37 even 6 784.2.i.j.753.1 2
56.45 odd 6 784.2.i.b.177.1 2
56.51 odd 6 392.2.i.a.361.1 2
56.53 even 6 784.2.i.j.177.1 2
88.43 even 2 6776.2.a.h.1.1 1
104.51 odd 2 9464.2.a.h.1.1 1
168.11 even 6 3528.2.s.a.3313.1 2
168.59 odd 6 3528.2.s.ba.3313.1 2
168.83 odd 2 3528.2.a.b.1.1 1
168.107 even 6 3528.2.s.a.361.1 2
168.125 even 2 7056.2.a.c.1.1 1
168.131 odd 6 3528.2.s.ba.361.1 2
280.139 even 2 9800.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 ✓ 1 8.3 odd 2
112.2.a.a.1.1 1 8.5 even 2
392.2.a.b.1.1 1 56.27 even 2
392.2.i.a.177.1 2 56.11 odd 6
392.2.i.a.361.1 2 56.51 odd 6
392.2.i.e.177.1 2 56.3 even 6
392.2.i.e.361.1 2 56.19 even 6
448.2.a.c.1.1 1 4.3 odd 2
448.2.a.h.1.1 1 1.1 even 1 trivial
504.2.a.h.1.1 1 24.11 even 2
784.2.a.i.1.1 1 56.13 odd 2
784.2.i.b.177.1 2 56.45 odd 6
784.2.i.b.753.1 2 56.5 odd 6
784.2.i.j.177.1 2 56.53 even 6
784.2.i.j.753.1 2 56.37 even 6
1008.2.a.m.1.1 1 24.5 odd 2
1400.2.a.a.1.1 1 40.19 odd 2
1400.2.g.b.449.1 2 40.27 even 4
1400.2.g.b.449.2 2 40.3 even 4
1792.2.b.a.897.1 2 16.3 odd 4
1792.2.b.a.897.2 2 16.11 odd 4
1792.2.b.h.897.1 2 16.5 even 4
1792.2.b.h.897.2 2 16.13 even 4
2800.2.a.bd.1.1 1 40.29 even 2
2800.2.g.g.449.1 2 40.13 odd 4
2800.2.g.g.449.2 2 40.37 odd 4
3136.2.a.c.1.1 1 7.6 odd 2
3136.2.a.w.1.1 1 28.27 even 2
3528.2.a.b.1.1 1 168.83 odd 2
3528.2.s.a.361.1 2 168.107 even 6
3528.2.s.a.3313.1 2 168.11 even 6
3528.2.s.ba.361.1 2 168.131 odd 6
3528.2.s.ba.3313.1 2 168.59 odd 6
4032.2.a.a.1.1 1 3.2 odd 2
4032.2.a.d.1.1 1 12.11 even 2
6776.2.a.h.1.1 1 88.43 even 2
7056.2.a.c.1.1 1 168.125 even 2
9464.2.a.h.1.1 1 104.51 odd 2
9800.2.a.bj.1.1 1 280.139 even 2