Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1024,2,Mod(257,1024)] 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("1024.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1024, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 769.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.769
Dual form 1024.2.e.k.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.82843 - 2.82843i) q^{5} +3.00000i q^{9} +(-2.82843 + 2.82843i) q^{13} +2.00000 q^{17} +11.0000i q^{25} +(2.82843 - 2.82843i) q^{29} +(8.48528 + 8.48528i) q^{37} +10.0000i q^{41} +(8.48528 - 8.48528i) q^{45} +7.00000 q^{49} +(2.82843 + 2.82843i) q^{53} +(-8.48528 + 8.48528i) q^{61} +16.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(-5.65685 - 5.65685i) q^{85} +10.0000i q^{89} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{17} + 28 q^{49} + 64 q^{65} - 36 q^{81} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) −2.82843 − 2.82843i −1.26491 − 1.26491i −0.948683 − 0.316228i \(-0.897584\pi\)
−0.316228 − 0.948683i \(-0.602416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) −2.82843 + 2.82843i −0.784465 + 0.784465i −0.980581 − 0.196116i \(-0.937167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 11.0000i 2.20000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843 − 2.82843i 0.525226 − 0.525226i −0.393919 − 0.919145i \(-0.628881\pi\)
0.919145 + 0.393919i \(0.128881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528 + 8.48528i 1.39497 + 1.39497i 0.813733 + 0.581238i \(0.197432\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 8.48528 − 8.48528i 1.26491 − 1.26491i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.82843 + 2.82843i 0.388514 + 0.388514i 0.874157 − 0.485643i \(-0.161414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) −8.48528 + 8.48528i −1.08643 + 1.08643i −0.0905357 + 0.995893i \(0.528858\pi\)
−0.995893 + 0.0905357i \(0.971142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.0000 1.98456
\(66\) 0 0
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −5.65685 − 5.65685i −0.613572 − 0.613572i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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 1024.2.e.k.769.1 4
4.3 odd 2 CM 1024.2.e.k.769.1 4
8.3 odd 2 inner 1024.2.e.k.769.2 4
8.5 even 2 inner 1024.2.e.k.769.2 4
16.3 odd 4 inner 1024.2.e.k.257.1 4
16.5 even 4 inner 1024.2.e.k.257.2 4
16.11 odd 4 inner 1024.2.e.k.257.2 4
16.13 even 4 inner 1024.2.e.k.257.1 4
32.3 odd 8 256.2.a.c.1.1 1
32.5 even 8 128.2.b.b.65.1 ✓ 2
32.11 odd 8 128.2.b.b.65.2 yes 2
32.13 even 8 256.2.a.b.1.1 1
32.19 odd 8 256.2.a.b.1.1 1
32.21 even 8 128.2.b.b.65.2 yes 2
32.27 odd 8 128.2.b.b.65.1 ✓ 2
32.29 even 8 256.2.a.c.1.1 1
96.5 odd 8 1152.2.d.d.577.2 2
96.11 even 8 1152.2.d.d.577.1 2
96.29 odd 8 2304.2.a.a.1.1 1
96.35 even 8 2304.2.a.a.1.1 1
96.53 odd 8 1152.2.d.d.577.1 2
96.59 even 8 1152.2.d.d.577.2 2
96.77 odd 8 2304.2.a.p.1.1 1
96.83 even 8 2304.2.a.p.1.1 1
160.19 odd 8 6400.2.a.m.1.1 1
160.27 even 8 3200.2.f.d.449.1 2
160.29 even 8 6400.2.a.l.1.1 1
160.37 odd 8 3200.2.f.d.449.1 2
160.43 even 8 3200.2.f.d.449.2 2
160.53 odd 8 3200.2.f.d.449.2 2
160.59 odd 8 3200.2.d.e.1601.1 2
160.69 even 8 3200.2.d.e.1601.1 2
160.99 odd 8 6400.2.a.l.1.1 1
160.107 even 8 3200.2.f.c.449.1 2
160.109 even 8 6400.2.a.m.1.1 1
160.117 odd 8 3200.2.f.c.449.1 2
160.123 even 8 3200.2.f.c.449.2 2
160.133 odd 8 3200.2.f.c.449.2 2
160.139 odd 8 3200.2.d.e.1601.2 2
160.149 even 8 3200.2.d.e.1601.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.b.65.1 ✓ 2 32.5 even 8
128.2.b.b.65.1 ✓ 2 32.27 odd 8
128.2.b.b.65.2 yes 2 32.11 odd 8
128.2.b.b.65.2 yes 2 32.21 even 8
256.2.a.b.1.1 1 32.13 even 8
256.2.a.b.1.1 1 32.19 odd 8
256.2.a.c.1.1 1 32.3 odd 8
256.2.a.c.1.1 1 32.29 even 8
1024.2.e.k.257.1 4 16.3 odd 4 inner
1024.2.e.k.257.1 4 16.13 even 4 inner
1024.2.e.k.257.2 4 16.5 even 4 inner
1024.2.e.k.257.2 4 16.11 odd 4 inner
1024.2.e.k.769.1 4 1.1 even 1 trivial
1024.2.e.k.769.1 4 4.3 odd 2 CM
1024.2.e.k.769.2 4 8.3 odd 2 inner
1024.2.e.k.769.2 4 8.5 even 2 inner
1152.2.d.d.577.1 2 96.11 even 8
1152.2.d.d.577.1 2 96.53 odd 8
1152.2.d.d.577.2 2 96.5 odd 8
1152.2.d.d.577.2 2 96.59 even 8
2304.2.a.a.1.1 1 96.29 odd 8
2304.2.a.a.1.1 1 96.35 even 8
2304.2.a.p.1.1 1 96.77 odd 8
2304.2.a.p.1.1 1 96.83 even 8
3200.2.d.e.1601.1 2 160.59 odd 8
3200.2.d.e.1601.1 2 160.69 even 8
3200.2.d.e.1601.2 2 160.139 odd 8
3200.2.d.e.1601.2 2 160.149 even 8
3200.2.f.c.449.1 2 160.107 even 8
3200.2.f.c.449.1 2 160.117 odd 8
3200.2.f.c.449.2 2 160.123 even 8
3200.2.f.c.449.2 2 160.133 odd 8
3200.2.f.d.449.1 2 160.27 even 8
3200.2.f.d.449.1 2 160.37 odd 8
3200.2.f.d.449.2 2 160.43 even 8
3200.2.f.d.449.2 2 160.53 odd 8
6400.2.a.l.1.1 1 160.29 even 8
6400.2.a.l.1.1 1 160.99 odd 8
6400.2.a.m.1.1 1 160.19 odd 8
6400.2.a.m.1.1 1 160.109 even 8