Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1600,2,Mod(801,1600)] 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("1600.801"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 801.5
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1600.801
Dual form 1600.2.d.i.801.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.44949i q^{3} -1.41421 q^{7} -3.00000 q^{9} -2.00000i q^{11} +5.65685i q^{13} +4.89898 q^{17} +6.00000i q^{19} -3.46410i q^{21} -7.07107 q^{23} -6.92820i q^{29} -6.92820 q^{31} +4.89898 q^{33} +2.82843i q^{37} -13.8564 q^{39} -4.00000 q^{41} +2.44949i q^{43} -4.24264 q^{47} -5.00000 q^{49} +12.0000i q^{51} -14.6969 q^{57} -2.00000i q^{59} -3.46410i q^{61} +4.24264 q^{63} +2.44949i q^{67} -17.3205i q^{69} +6.92820 q^{71} -4.89898 q^{73} +2.82843i q^{77} -6.92820 q^{79} -9.00000 q^{81} -12.2474i q^{83} +16.9706 q^{87} +2.00000 q^{89} -8.00000i q^{91} -16.9706i q^{93} +14.6969 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9} - 32 q^{41} - 40 q^{49} - 72 q^{81} + 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 2.44949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 − 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 5.65685i 1.56893i 0.620174 + 0.784465i \(0.287062\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 − 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) −7.07107 −1.47442 −0.737210 − 0.675664i \(-0.763857\pi\)
−0.737210 + 0.675664i \(0.763857\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.92820i − 1.28654i −0.765641 − 0.643268i \(-0.777578\pi\)
0.765641 − 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 − 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) 4.89898 0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843i 0.464991i 0.972598 + 0.232495i \(0.0746890\pi\)
−0.972598 + 0.232495i \(0.925311\pi\)
\(38\) 0 0
\(39\) −13.8564 −2.21880
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 2.44949i 0.373544i 0.982403 + 0.186772i \(0.0598025\pi\)
−0.982403 + 0.186772i \(0.940197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.24264 −0.618853 −0.309426 − 0.950923i \(-0.600137\pi\)
−0.309426 + 0.950923i \(0.600137\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −14.6969 −1.94666
\(58\) 0 0
\(59\) − 2.00000i − 0.260378i −0.991489 − 0.130189i \(-0.958442\pi\)
0.991489 − 0.130189i \(-0.0415584\pi\)
\(60\) 0 0
\(61\) − 3.46410i − 0.443533i −0.975100 − 0.221766i \(-0.928818\pi\)
0.975100 − 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.44949i 0.299253i 0.988743 + 0.149626i \(0.0478071\pi\)
−0.988743 + 0.149626i \(0.952193\pi\)
\(68\) 0 0
\(69\) − 17.3205i − 2.08514i
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 − 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 − 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.82843i 0.322329i
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 − 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) − 12.2474i − 1.34433i −0.740400 − 0.672166i \(-0.765364\pi\)
0.740400 − 0.672166i \(-0.234636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 16.9706 1.81944
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) − 8.00000i − 0.838628i
\(92\) 0 0
\(93\) − 16.9706i − 1.75977i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.6969 1.49225 0.746124 − 0.665807i \(-0.231913\pi\)
0.746124 + 0.665807i \(0.231913\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
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 1600.2.d.i.801.5 8
4.3 odd 2 inner 1600.2.d.i.801.4 8
5.2 odd 4 320.2.f.b.289.5 yes 8
5.3 odd 4 320.2.f.b.289.3 yes 8
5.4 even 2 inner 1600.2.d.i.801.3 8
8.3 odd 2 inner 1600.2.d.i.801.7 8
8.5 even 2 inner 1600.2.d.i.801.2 8
15.2 even 4 2880.2.d.g.289.7 8
15.8 even 4 2880.2.d.g.289.4 8
16.3 odd 4 6400.2.a.cu.1.3 4
16.5 even 4 6400.2.a.cu.1.4 4
16.11 odd 4 6400.2.a.ct.1.1 4
16.13 even 4 6400.2.a.ct.1.2 4
20.3 even 4 320.2.f.b.289.7 yes 8
20.7 even 4 320.2.f.b.289.1 ✓ 8
20.19 odd 2 inner 1600.2.d.i.801.6 8
40.3 even 4 320.2.f.b.289.2 yes 8
40.13 odd 4 320.2.f.b.289.6 yes 8
40.19 odd 2 inner 1600.2.d.i.801.1 8
40.27 even 4 320.2.f.b.289.8 yes 8
40.29 even 2 inner 1600.2.d.i.801.8 8
40.37 odd 4 320.2.f.b.289.4 yes 8
60.23 odd 4 2880.2.d.g.289.3 8
60.47 odd 4 2880.2.d.g.289.8 8
80.3 even 4 1280.2.c.h.769.3 4
80.13 odd 4 1280.2.c.g.769.1 4
80.19 odd 4 6400.2.a.cu.1.2 4
80.27 even 4 1280.2.c.g.769.4 4
80.29 even 4 6400.2.a.ct.1.3 4
80.37 odd 4 1280.2.c.h.769.2 4
80.43 even 4 1280.2.c.g.769.2 4
80.53 odd 4 1280.2.c.h.769.4 4
80.59 odd 4 6400.2.a.ct.1.4 4
80.67 even 4 1280.2.c.h.769.1 4
80.69 even 4 6400.2.a.cu.1.1 4
80.77 odd 4 1280.2.c.g.769.3 4
120.53 even 4 2880.2.d.g.289.6 8
120.77 even 4 2880.2.d.g.289.1 8
120.83 odd 4 2880.2.d.g.289.5 8
120.107 odd 4 2880.2.d.g.289.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 ✓ 8 20.7 even 4
320.2.f.b.289.2 yes 8 40.3 even 4
320.2.f.b.289.3 yes 8 5.3 odd 4
320.2.f.b.289.4 yes 8 40.37 odd 4
320.2.f.b.289.5 yes 8 5.2 odd 4
320.2.f.b.289.6 yes 8 40.13 odd 4
320.2.f.b.289.7 yes 8 20.3 even 4
320.2.f.b.289.8 yes 8 40.27 even 4
1280.2.c.g.769.1 4 80.13 odd 4
1280.2.c.g.769.2 4 80.43 even 4
1280.2.c.g.769.3 4 80.77 odd 4
1280.2.c.g.769.4 4 80.27 even 4
1280.2.c.h.769.1 4 80.67 even 4
1280.2.c.h.769.2 4 80.37 odd 4
1280.2.c.h.769.3 4 80.3 even 4
1280.2.c.h.769.4 4 80.53 odd 4
1600.2.d.i.801.1 8 40.19 odd 2 inner
1600.2.d.i.801.2 8 8.5 even 2 inner
1600.2.d.i.801.3 8 5.4 even 2 inner
1600.2.d.i.801.4 8 4.3 odd 2 inner
1600.2.d.i.801.5 8 1.1 even 1 trivial
1600.2.d.i.801.6 8 20.19 odd 2 inner
1600.2.d.i.801.7 8 8.3 odd 2 inner
1600.2.d.i.801.8 8 40.29 even 2 inner
2880.2.d.g.289.1 8 120.77 even 4
2880.2.d.g.289.2 8 120.107 odd 4
2880.2.d.g.289.3 8 60.23 odd 4
2880.2.d.g.289.4 8 15.8 even 4
2880.2.d.g.289.5 8 120.83 odd 4
2880.2.d.g.289.6 8 120.53 even 4
2880.2.d.g.289.7 8 15.2 even 4
2880.2.d.g.289.8 8 60.47 odd 4
6400.2.a.ct.1.1 4 16.11 odd 4
6400.2.a.ct.1.2 4 16.13 even 4
6400.2.a.ct.1.3 4 80.29 even 4
6400.2.a.ct.1.4 4 80.59 odd 4
6400.2.a.cu.1.1 4 80.69 even 4
6400.2.a.cu.1.2 4 80.19 odd 4
6400.2.a.cu.1.3 4 16.3 odd 4
6400.2.a.cu.1.4 4 16.5 even 4