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Properties

Label 1024.2.e.o.257.1
Level $1024$
Weight $2$
Character 1024.257
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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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 = [4,0,4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,4] 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 257.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.257
Dual form 1024.2.e.o.769.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.414214 - 0.414214i) q^{3} -2.65685i q^{9} +(-4.41421 + 4.41421i) q^{11} -5.65685 q^{17} +(5.24264 + 5.24264i) q^{19} +5.00000i q^{25} +(-2.34315 + 2.34315i) q^{27} +3.65685 q^{33} -6.00000i q^{41} +(-9.24264 + 9.24264i) q^{43} +7.00000 q^{49} +(2.34315 + 2.34315i) q^{51} -4.34315i q^{57} +(-10.0711 + 10.0711i) q^{59} +(2.75736 + 2.75736i) q^{67} +16.9706i q^{73} +(2.07107 - 2.07107i) q^{75} -6.02944 q^{81} +(-7.58579 - 7.58579i) q^{83} +5.65685i q^{89} -16.9706 q^{97} +(11.7279 + 11.7279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 12 q^{11} + 4 q^{19} - 32 q^{27} - 8 q^{33} - 20 q^{43} + 28 q^{49} + 32 q^{51} - 12 q^{59} + 28 q^{67} - 20 q^{75} - 92 q^{81} - 36 q^{83} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −0.414214 − 0.414214i −0.239146 − 0.239146i 0.577350 − 0.816497i \(-0.304087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) − 2.65685i − 0.885618i
\(10\) 0 0
\(11\) −4.41421 + 4.41421i −1.33094 + 1.33094i −0.426401 + 0.904534i \(0.640219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 − 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 5.24264 + 5.24264i 1.20274 + 1.20274i 0.973329 + 0.229416i \(0.0736815\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) −2.34315 + 2.34315i −0.450939 + 0.450939i
\(28\) 0 0
\(29\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.65685 0.636577
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) −9.24264 + 9.24264i −1.40949 + 1.40949i −0.646997 + 0.762493i \(0.723975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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
\(51\) 2.34315 + 2.34315i 0.328106 + 0.328106i
\(52\) 0 0
\(53\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.34315i − 0.575264i
\(58\) 0 0
\(59\) −10.0711 + 10.0711i −1.31114 + 1.31114i −0.390567 + 0.920575i \(0.627721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.75736 + 2.75736i 0.336865 + 0.336865i 0.855186 − 0.518321i \(-0.173443\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 16.9706i 1.98625i 0.117041 + 0.993127i \(0.462659\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 2.07107 − 2.07107i 0.239146 − 0.239146i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −6.02944 −0.669937
\(82\) 0 0
\(83\) −7.58579 − 7.58579i −0.832648 − 0.832648i 0.155230 − 0.987878i \(-0.450388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.9706 −1.72310 −0.861550 − 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 11.7279 + 11.7279i 1.17870 + 1.17870i
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.o.257.1 4
4.3 odd 2 1024.2.e.g.257.2 4
8.3 odd 2 CM 1024.2.e.o.257.1 4
8.5 even 2 1024.2.e.g.257.2 4
16.3 odd 4 inner 1024.2.e.o.769.1 4
16.5 even 4 inner 1024.2.e.o.769.1 4
16.11 odd 4 1024.2.e.g.769.2 4
16.13 even 4 1024.2.e.g.769.2 4
32.3 odd 8 512.2.b.c.257.3 4
32.5 even 8 512.2.a.f.1.1 yes 2
32.11 odd 8 512.2.a.f.1.1 yes 2
32.13 even 8 512.2.b.c.257.3 4
32.19 odd 8 512.2.b.c.257.2 4
32.21 even 8 512.2.a.a.1.2 ✓ 2
32.27 odd 8 512.2.a.a.1.2 ✓ 2
32.29 even 8 512.2.b.c.257.2 4
96.5 odd 8 4608.2.a.i.1.1 2
96.11 even 8 4608.2.a.i.1.1 2
96.29 odd 8 4608.2.d.k.2305.1 4
96.35 even 8 4608.2.d.k.2305.4 4
96.53 odd 8 4608.2.a.k.1.2 2
96.59 even 8 4608.2.a.k.1.2 2
96.77 odd 8 4608.2.d.k.2305.4 4
96.83 even 8 4608.2.d.k.2305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.a.1.2 ✓ 2 32.21 even 8
512.2.a.a.1.2 ✓ 2 32.27 odd 8
512.2.a.f.1.1 yes 2 32.5 even 8
512.2.a.f.1.1 yes 2 32.11 odd 8
512.2.b.c.257.2 4 32.19 odd 8
512.2.b.c.257.2 4 32.29 even 8
512.2.b.c.257.3 4 32.3 odd 8
512.2.b.c.257.3 4 32.13 even 8
1024.2.e.g.257.2 4 4.3 odd 2
1024.2.e.g.257.2 4 8.5 even 2
1024.2.e.g.769.2 4 16.11 odd 4
1024.2.e.g.769.2 4 16.13 even 4
1024.2.e.o.257.1 4 1.1 even 1 trivial
1024.2.e.o.257.1 4 8.3 odd 2 CM
1024.2.e.o.769.1 4 16.3 odd 4 inner
1024.2.e.o.769.1 4 16.5 even 4 inner
4608.2.a.i.1.1 2 96.5 odd 8
4608.2.a.i.1.1 2 96.11 even 8
4608.2.a.k.1.2 2 96.53 odd 8
4608.2.a.k.1.2 2 96.59 even 8
4608.2.d.k.2305.1 4 96.29 odd 8
4608.2.d.k.2305.1 4 96.83 even 8
4608.2.d.k.2305.4 4 96.35 even 8
4608.2.d.k.2305.4 4 96.77 odd 8