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Properties

Label 1024.2.e.i.257.1
Level $1024$
Weight $2$
Character 1024.257
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,8,0,0] 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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.257
Dual form 1024.2.e.i.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)
 
\(f(q)\) \(=\) \(q+(-1.41421 - 1.41421i) q^{3} +(1.41421 - 1.41421i) q^{5} +4.00000i q^{7} +1.00000i q^{9} +(1.41421 - 1.41421i) q^{11} +(1.41421 + 1.41421i) q^{13} -4.00000 q^{15} +2.00000 q^{17} +(1.41421 + 1.41421i) q^{19} +(5.65685 - 5.65685i) q^{21} +4.00000i q^{23} +1.00000i q^{25} +(-2.82843 + 2.82843i) q^{27} +(4.24264 + 4.24264i) q^{29} -4.00000 q^{33} +(5.65685 + 5.65685i) q^{35} +(7.07107 - 7.07107i) q^{37} -4.00000i q^{39} -6.00000i q^{41} +(-4.24264 + 4.24264i) q^{43} +(1.41421 + 1.41421i) q^{45} +8.00000 q^{47} -9.00000 q^{49} +(-2.82843 - 2.82843i) q^{51} +(4.24264 - 4.24264i) q^{53} -4.00000i q^{55} -4.00000i q^{57} +(9.89949 - 9.89949i) q^{59} +(-1.41421 - 1.41421i) q^{61} -4.00000 q^{63} +4.00000 q^{65} +(-7.07107 - 7.07107i) q^{67} +(5.65685 - 5.65685i) q^{69} -12.0000i q^{71} +14.0000i q^{73} +(1.41421 - 1.41421i) q^{75} +(5.65685 + 5.65685i) q^{77} +8.00000 q^{79} +11.0000 q^{81} +(-4.24264 - 4.24264i) q^{83} +(2.82843 - 2.82843i) q^{85} -12.0000i q^{87} +2.00000i q^{89} +(-5.65685 + 5.65685i) q^{91} +4.00000 q^{95} -2.00000 q^{97} +(1.41421 + 1.41421i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{15} + 8 q^{17} - 16 q^{33} + 32 q^{47} - 36 q^{49} - 16 q^{63} + 16 q^{65} + 32 q^{79} + 44 q^{81} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.41421 − 1.41421i −0.816497 − 0.816497i 0.169102 − 0.985599i \(-0.445913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 1.41421 − 1.41421i 0.632456 − 0.632456i −0.316228 − 0.948683i \(-0.602416\pi\)
0.948683 + 0.316228i \(0.102416\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.41421 − 1.41421i 0.426401 − 0.426401i −0.460999 − 0.887401i \(-0.652509\pi\)
0.887401 + 0.460999i \(0.152509\pi\)
\(12\) 0 0
\(13\) 1.41421 + 1.41421i 0.392232 + 0.392232i 0.875482 − 0.483250i \(-0.160544\pi\)
−0.483250 + 0.875482i \(0.660544\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.41421 + 1.41421i 0.324443 + 0.324443i 0.850469 − 0.526026i \(-0.176318\pi\)
−0.526026 + 0.850469i \(0.676318\pi\)
\(20\) 0 0
\(21\) 5.65685 − 5.65685i 1.23443 − 1.23443i
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) −2.82843 + 2.82843i −0.544331 + 0.544331i
\(28\) 0 0
\(29\) 4.24264 + 4.24264i 0.787839 + 0.787839i 0.981140 − 0.193301i \(-0.0619194\pi\)
−0.193301 + 0.981140i \(0.561919\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 5.65685 + 5.65685i 0.956183 + 0.956183i
\(36\) 0 0
\(37\) 7.07107 − 7.07107i 1.16248 − 1.16248i 0.178545 − 0.983932i \(-0.442861\pi\)
0.983932 − 0.178545i \(-0.0571389\pi\)
\(38\) 0 0
\(39\) − 4.00000i − 0.640513i
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) −4.24264 + 4.24264i −0.646997 + 0.646997i −0.952266 − 0.305269i \(-0.901253\pi\)
0.305269 + 0.952266i \(0.401253\pi\)
\(44\) 0 0
\(45\) 1.41421 + 1.41421i 0.210819 + 0.210819i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −2.82843 − 2.82843i −0.396059 − 0.396059i
\(52\) 0 0
\(53\) 4.24264 − 4.24264i 0.582772 − 0.582772i −0.352892 − 0.935664i \(-0.614802\pi\)
0.935664 + 0.352892i \(0.114802\pi\)
\(54\) 0 0
\(55\) − 4.00000i − 0.539360i
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 9.89949 − 9.89949i 1.28880 − 1.28880i 0.353291 − 0.935513i \(-0.385063\pi\)
0.935513 − 0.353291i \(-0.114937\pi\)
\(60\) 0 0
\(61\) −1.41421 − 1.41421i −0.181071 − 0.181071i 0.610751 − 0.791823i \(-0.290868\pi\)
−0.791823 + 0.610751i \(0.790868\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −7.07107 − 7.07107i −0.863868 − 0.863868i 0.127917 − 0.991785i \(-0.459171\pi\)
−0.991785 + 0.127917i \(0.959171\pi\)
\(68\) 0 0
\(69\) 5.65685 − 5.65685i 0.681005 − 0.681005i
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 1.41421 − 1.41421i 0.163299 − 0.163299i
\(76\) 0 0
\(77\) 5.65685 + 5.65685i 0.644658 + 0.644658i
\(78\) 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
\(83\) −4.24264 − 4.24264i −0.465690 − 0.465690i 0.434825 − 0.900515i \(-0.356810\pi\)
−0.900515 + 0.434825i \(0.856810\pi\)
\(84\) 0 0
\(85\) 2.82843 − 2.82843i 0.306786 − 0.306786i
\(86\) 0 0
\(87\) − 12.0000i − 1.28654i
\(88\) 0 0
\(89\) 2.00000i 0.212000i 0.994366 + 0.106000i \(0.0338043\pi\)
−0.994366 + 0.106000i \(0.966196\pi\)
\(90\) 0 0
\(91\) −5.65685 + 5.65685i −0.592999 + 0.592999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 1.41421 + 1.41421i 0.142134 + 0.142134i
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.i.257.1 4
4.3 odd 2 1024.2.e.m.257.2 4
8.3 odd 2 1024.2.e.m.257.1 4
8.5 even 2 inner 1024.2.e.i.257.2 4
16.3 odd 4 1024.2.e.m.769.1 4
16.5 even 4 inner 1024.2.e.i.769.1 4
16.11 odd 4 1024.2.e.m.769.2 4
16.13 even 4 inner 1024.2.e.i.769.2 4
32.3 odd 8 256.2.b.a.129.2 2
32.5 even 8 128.2.a.d.1.1 yes 1
32.11 odd 8 128.2.a.c.1.1 yes 1
32.13 even 8 256.2.b.c.129.2 2
32.19 odd 8 256.2.b.a.129.1 2
32.21 even 8 128.2.a.a.1.1 ✓ 1
32.27 odd 8 128.2.a.b.1.1 yes 1
32.29 even 8 256.2.b.c.129.1 2
96.5 odd 8 1152.2.a.c.1.1 1
96.11 even 8 1152.2.a.r.1.1 1
96.29 odd 8 2304.2.d.r.1153.1 2
96.35 even 8 2304.2.d.b.1153.1 2
96.53 odd 8 1152.2.a.m.1.1 1
96.59 even 8 1152.2.a.h.1.1 1
96.77 odd 8 2304.2.d.r.1153.2 2
96.83 even 8 2304.2.d.b.1153.2 2
160.27 even 8 3200.2.c.k.2049.2 2
160.37 odd 8 3200.2.c.e.2049.1 2
160.43 even 8 3200.2.c.f.2049.2 2
160.53 odd 8 3200.2.c.l.2049.1 2
160.59 odd 8 3200.2.a.u.1.1 1
160.69 even 8 3200.2.a.h.1.1 1
160.107 even 8 3200.2.c.f.2049.1 2
160.117 odd 8 3200.2.c.l.2049.2 2
160.123 even 8 3200.2.c.k.2049.1 2
160.133 odd 8 3200.2.c.e.2049.2 2
160.139 odd 8 3200.2.a.e.1.1 1
160.149 even 8 3200.2.a.x.1.1 1
224.27 even 8 6272.2.a.g.1.1 1
224.69 odd 8 6272.2.a.a.1.1 1
224.139 even 8 6272.2.a.b.1.1 1
224.181 odd 8 6272.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 ✓ 1 32.21 even 8
128.2.a.b.1.1 yes 1 32.27 odd 8
128.2.a.c.1.1 yes 1 32.11 odd 8
128.2.a.d.1.1 yes 1 32.5 even 8
256.2.b.a.129.1 2 32.19 odd 8
256.2.b.a.129.2 2 32.3 odd 8
256.2.b.c.129.1 2 32.29 even 8
256.2.b.c.129.2 2 32.13 even 8
1024.2.e.i.257.1 4 1.1 even 1 trivial
1024.2.e.i.257.2 4 8.5 even 2 inner
1024.2.e.i.769.1 4 16.5 even 4 inner
1024.2.e.i.769.2 4 16.13 even 4 inner
1024.2.e.m.257.1 4 8.3 odd 2
1024.2.e.m.257.2 4 4.3 odd 2
1024.2.e.m.769.1 4 16.3 odd 4
1024.2.e.m.769.2 4 16.11 odd 4
1152.2.a.c.1.1 1 96.5 odd 8
1152.2.a.h.1.1 1 96.59 even 8
1152.2.a.m.1.1 1 96.53 odd 8
1152.2.a.r.1.1 1 96.11 even 8
2304.2.d.b.1153.1 2 96.35 even 8
2304.2.d.b.1153.2 2 96.83 even 8
2304.2.d.r.1153.1 2 96.29 odd 8
2304.2.d.r.1153.2 2 96.77 odd 8
3200.2.a.e.1.1 1 160.139 odd 8
3200.2.a.h.1.1 1 160.69 even 8
3200.2.a.u.1.1 1 160.59 odd 8
3200.2.a.x.1.1 1 160.149 even 8
3200.2.c.e.2049.1 2 160.37 odd 8
3200.2.c.e.2049.2 2 160.133 odd 8
3200.2.c.f.2049.1 2 160.107 even 8
3200.2.c.f.2049.2 2 160.43 even 8
3200.2.c.k.2049.1 2 160.123 even 8
3200.2.c.k.2049.2 2 160.27 even 8
3200.2.c.l.2049.1 2 160.53 odd 8
3200.2.c.l.2049.2 2 160.117 odd 8
6272.2.a.a.1.1 1 224.69 odd 8
6272.2.a.b.1.1 1 224.139 even 8
6272.2.a.g.1.1 1 224.27 even 8
6272.2.a.h.1.1 1 224.181 odd 8