↑

Properties

Label 1024.2.e.n.769.2
Level $1024$
Weight $2$
Character 1024.769
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

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,-12] 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 769.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.769
Dual form 1024.2.e.n.257.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{3} +(1.41421 + 1.41421i) q^{5} -2.82843i q^{7} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(4.24264 - 4.24264i) q^{13} +2.82843 q^{15} +(-3.00000 + 3.00000i) q^{19} +(-2.82843 - 2.82843i) q^{21} -8.48528i q^{23} -1.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.41421 + 1.41421i) q^{29} +5.65685 q^{31} -6.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(4.24264 + 4.24264i) q^{37} -8.48528i q^{39} -6.00000i q^{41} +(3.00000 + 3.00000i) q^{43} +(-1.41421 + 1.41421i) q^{45} -1.00000 q^{49} +(1.41421 + 1.41421i) q^{53} -8.48528i q^{55} +6.00000i q^{57} +(1.00000 + 1.00000i) q^{59} +(-4.24264 + 4.24264i) q^{61} +2.82843 q^{63} +12.0000 q^{65} +(-9.00000 + 9.00000i) q^{67} +(-8.48528 - 8.48528i) q^{69} -8.48528i q^{71} +12.0000i q^{73} +(-1.00000 - 1.00000i) q^{75} +(-8.48528 + 8.48528i) q^{77} +5.65685 q^{79} +5.00000 q^{81} +(3.00000 - 3.00000i) q^{83} +2.82843i q^{87} -12.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(5.65685 - 5.65685i) q^{93} -8.48528 q^{95} -8.00000 q^{97} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 12 q^{11} - 12 q^{19} + 16 q^{27} - 24 q^{33} + 16 q^{35} + 12 q^{43} - 4 q^{49} + 4 q^{59} + 48 q^{65} - 36 q^{67} - 4 q^{75} + 20 q^{81} + 12 q^{83} - 48 q^{91} - 32 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.00000 − 1.00000i 0.577350 − 0.577350i −0.356822 − 0.934172i \(-0.616140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 0 0
\(5\) 1.41421 + 1.41421i 0.632456 + 0.632456i 0.948683 − 0.316228i \(-0.102416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 − 0.534522i \(-0.820491\pi\)
0.845154 − 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −3.00000 − 3.00000i −0.904534 − 0.904534i 0.0912903 − 0.995824i \(-0.470901\pi\)
−0.995824 + 0.0912903i \(0.970901\pi\)
\(12\) 0 0
\(13\) 4.24264 − 4.24264i 1.17670 − 1.17670i 0.196116 − 0.980581i \(-0.437167\pi\)
0.980581 − 0.196116i \(-0.0628330\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.00000 + 3.00000i −0.688247 + 0.688247i −0.961844 − 0.273597i \(-0.911786\pi\)
0.273597 + 0.961844i \(0.411786\pi\)
\(20\) 0 0
\(21\) −2.82843 − 2.82843i −0.617213 − 0.617213i
\(22\) 0 0
\(23\) − 8.48528i − 1.76930i −0.466252 − 0.884652i \(-0.654396\pi\)
0.466252 − 0.884652i \(-0.345604\pi\)
\(24\) 0 0
\(25\) − 1.00000i − 0.200000i
\(26\) 0 0
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) −1.41421 + 1.41421i −0.262613 + 0.262613i −0.826115 − 0.563502i \(-0.809454\pi\)
0.563502 + 0.826115i \(0.309454\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 − 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 4.00000 − 4.00000i 0.676123 − 0.676123i
\(36\) 0 0
\(37\) 4.24264 + 4.24264i 0.697486 + 0.697486i 0.963868 − 0.266382i \(-0.0858282\pi\)
−0.266382 + 0.963868i \(0.585828\pi\)
\(38\) 0 0
\(39\) − 8.48528i − 1.35873i
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 3.00000 + 3.00000i 0.457496 + 0.457496i 0.897833 − 0.440337i \(-0.145141\pi\)
−0.440337 + 0.897833i \(0.645141\pi\)
\(44\) 0 0
\(45\) −1.41421 + 1.41421i −0.210819 + 0.210819i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.41421 + 1.41421i 0.194257 + 0.194257i 0.797533 − 0.603276i \(-0.206138\pi\)
−0.603276 + 0.797533i \(0.706138\pi\)
\(54\) 0 0
\(55\) − 8.48528i − 1.14416i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 1.00000 + 1.00000i 0.130189 + 0.130189i 0.769199 − 0.639010i \(-0.220656\pi\)
−0.639010 + 0.769199i \(0.720656\pi\)
\(60\) 0 0
\(61\) −4.24264 + 4.24264i −0.543214 + 0.543214i −0.924470 − 0.381255i \(-0.875492\pi\)
0.381255 + 0.924470i \(0.375492\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −9.00000 + 9.00000i −1.09952 + 1.09952i −0.105059 + 0.994466i \(0.533503\pi\)
−0.994466 + 0.105059i \(0.966497\pi\)
\(68\) 0 0
\(69\) −8.48528 − 8.48528i −1.02151 − 1.02151i
\(70\) 0 0
\(71\) − 8.48528i − 1.00702i −0.863990 − 0.503509i \(-0.832042\pi\)
0.863990 − 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 0 0
\(75\) −1.00000 − 1.00000i −0.115470 − 0.115470i
\(76\) 0 0
\(77\) −8.48528 + 8.48528i −0.966988 + 0.966988i
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 − 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 3.00000 − 3.00000i 0.329293 − 0.329293i −0.523025 − 0.852318i \(-0.675196\pi\)
0.852318 + 0.523025i \(0.175196\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.82843i 0.303239i
\(88\) 0 0
\(89\) − 12.0000i − 1.27200i −0.771690 − 0.635999i \(-0.780588\pi\)
0.771690 − 0.635999i \(-0.219412\pi\)
\(90\) 0 0
\(91\) −12.0000 − 12.0000i −1.25794 − 1.25794i
\(92\) 0 0
\(93\) 5.65685 − 5.65685i 0.586588 − 0.586588i
\(94\) 0 0
\(95\) −8.48528 −0.870572
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 3.00000 − 3.00000i 0.301511 − 0.301511i
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.n.769.2 4
4.3 odd 2 1024.2.e.h.769.2 4
8.3 odd 2 inner 1024.2.e.n.769.1 4
8.5 even 2 1024.2.e.h.769.1 4
16.3 odd 4 1024.2.e.h.257.2 4
16.5 even 4 1024.2.e.h.257.1 4
16.11 odd 4 inner 1024.2.e.n.257.1 4
16.13 even 4 inner 1024.2.e.n.257.2 4
32.3 odd 8 512.2.a.b.1.1 ✓ 2
32.5 even 8 512.2.b.e.257.4 4
32.11 odd 8 512.2.b.e.257.3 4
32.13 even 8 512.2.a.e.1.1 yes 2
32.19 odd 8 512.2.a.e.1.2 yes 2
32.21 even 8 512.2.b.e.257.1 4
32.27 odd 8 512.2.b.e.257.2 4
32.29 even 8 512.2.a.b.1.2 yes 2
96.5 odd 8 4608.2.d.j.2305.2 4
96.11 even 8 4608.2.d.j.2305.3 4
96.29 odd 8 4608.2.a.p.1.1 2
96.35 even 8 4608.2.a.p.1.2 2
96.53 odd 8 4608.2.d.j.2305.4 4
96.59 even 8 4608.2.d.j.2305.1 4
96.77 odd 8 4608.2.a.c.1.1 2
96.83 even 8 4608.2.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.b.1.1 ✓ 2 32.3 odd 8
512.2.a.b.1.2 yes 2 32.29 even 8
512.2.a.e.1.1 yes 2 32.13 even 8
512.2.a.e.1.2 yes 2 32.19 odd 8
512.2.b.e.257.1 4 32.21 even 8
512.2.b.e.257.2 4 32.27 odd 8
512.2.b.e.257.3 4 32.11 odd 8
512.2.b.e.257.4 4 32.5 even 8
1024.2.e.h.257.1 4 16.5 even 4
1024.2.e.h.257.2 4 16.3 odd 4
1024.2.e.h.769.1 4 8.5 even 2
1024.2.e.h.769.2 4 4.3 odd 2
1024.2.e.n.257.1 4 16.11 odd 4 inner
1024.2.e.n.257.2 4 16.13 even 4 inner
1024.2.e.n.769.1 4 8.3 odd 2 inner
1024.2.e.n.769.2 4 1.1 even 1 trivial
4608.2.a.c.1.1 2 96.77 odd 8
4608.2.a.c.1.2 2 96.83 even 8
4608.2.a.p.1.1 2 96.29 odd 8
4608.2.a.p.1.2 2 96.35 even 8
4608.2.d.j.2305.1 4 96.59 even 8
4608.2.d.j.2305.2 4 96.5 odd 8
4608.2.d.j.2305.3 4 96.11 even 8
4608.2.d.j.2305.4 4 96.53 odd 8