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Properties

Label 1280.2.n.a.767.1
Level $1280$
Weight $2$
Character 1280.767
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [1280,2,Mod(767,1280)] 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("1280.767"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,-2,0,-4,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 767.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.767
Dual form 1280.2.n.a.1023.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 2.00000i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 + 2.00000i) q^{7} +5.00000i q^{9} +4.00000i q^{11} +(3.00000 - 3.00000i) q^{13} +(-2.00000 + 6.00000i) q^{15} +(-3.00000 - 3.00000i) q^{17} +8.00000 q^{21} +(6.00000 + 6.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +(4.00000 - 4.00000i) q^{27} +2.00000i q^{29} +4.00000i q^{31} +(8.00000 - 8.00000i) q^{33} +(6.00000 + 2.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} -12.0000 q^{39} +(-6.00000 - 6.00000i) q^{43} +(10.0000 - 5.00000i) q^{45} +(6.00000 - 6.00000i) q^{47} -1.00000i q^{49} +12.0000i q^{51} +(3.00000 - 3.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} -8.00000 q^{59} +6.00000 q^{61} +(-10.0000 - 10.0000i) q^{63} +(-9.00000 - 3.00000i) q^{65} +(6.00000 - 6.00000i) q^{67} -24.0000i q^{69} -12.0000i q^{71} +(5.00000 - 5.00000i) q^{73} +(14.0000 - 2.00000i) q^{75} +(-8.00000 - 8.00000i) q^{77} +8.00000 q^{79} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{83} +(-3.00000 + 9.00000i) q^{85} +(4.00000 - 4.00000i) q^{87} +12.0000i q^{91} +(8.00000 - 8.00000i) q^{93} +(11.0000 + 11.0000i) q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{13} - 4 q^{15} - 6 q^{17} + 16 q^{21} + 12 q^{23} - 6 q^{25} + 8 q^{27} + 16 q^{33} + 12 q^{35} + 6 q^{37} - 24 q^{39} - 12 q^{43} + 20 q^{45} + 12 q^{47} + 6 q^{53}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(3\) −2.00000 − 2.00000i −1.15470 − 1.15470i −0.985599 − 0.169102i \(-0.945913\pi\)
−0.169102 − 0.985599i \(-0.554087\pi\)
\(4\) 0 0
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) −2.00000 + 2.00000i −0.755929 + 0.755929i −0.975579 − 0.219650i \(-0.929509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(8\) 0 0
\(9\) 5.00000i 1.66667i
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 3.00000 − 3.00000i 0.832050 − 0.832050i −0.155747 − 0.987797i \(-0.549778\pi\)
0.987797 + 0.155747i \(0.0497784\pi\)
\(14\) 0 0
\(15\) −2.00000 + 6.00000i −0.516398 + 1.54919i
\(16\) 0 0
\(17\) −3.00000 − 3.00000i −0.727607 − 0.727607i 0.242536 − 0.970143i \(-0.422021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 0 0
\(23\) 6.00000 + 6.00000i 1.25109 + 1.25109i 0.955233 + 0.295853i \(0.0956039\pi\)
0.295853 + 0.955233i \(0.404396\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 4.00000 − 4.00000i 0.769800 − 0.769800i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 8.00000 − 8.00000i 1.39262 − 1.39262i
\(34\) 0 0
\(35\) 6.00000 + 2.00000i 1.01419 + 0.338062i
\(36\) 0 0
\(37\) 3.00000 + 3.00000i 0.493197 + 0.493197i 0.909312 − 0.416115i \(-0.136609\pi\)
−0.416115 + 0.909312i \(0.636609\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −6.00000 − 6.00000i −0.914991 − 0.914991i 0.0816682 − 0.996660i \(-0.473975\pi\)
−0.996660 + 0.0816682i \(0.973975\pi\)
\(44\) 0 0
\(45\) 10.0000 − 5.00000i 1.49071 − 0.745356i
\(46\) 0 0
\(47\) 6.00000 − 6.00000i 0.875190 − 0.875190i −0.117842 − 0.993032i \(-0.537598\pi\)
0.993032 + 0.117842i \(0.0375978\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 0.142857i
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) 3.00000 − 3.00000i 0.412082 − 0.412082i −0.470381 − 0.882463i \(-0.655884\pi\)
0.882463 + 0.470381i \(0.155884\pi\)
\(54\) 0 0
\(55\) 8.00000 − 4.00000i 1.07872 − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −10.0000 − 10.0000i −1.25988 − 1.25988i
\(64\) 0 0
\(65\) −9.00000 − 3.00000i −1.11631 − 0.372104i
\(66\) 0 0
\(67\) 6.00000 − 6.00000i 0.733017 − 0.733017i −0.238200 − 0.971216i \(-0.576557\pi\)
0.971216 + 0.238200i \(0.0765572\pi\)
\(68\) 0 0
\(69\) − 24.0000i − 2.88926i
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 5.00000 − 5.00000i 0.585206 − 0.585206i −0.351123 − 0.936329i \(-0.614200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 14.0000 − 2.00000i 1.61658 − 0.230940i
\(76\) 0 0
\(77\) −8.00000 − 8.00000i −0.911685 − 0.911685i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −6.00000 − 6.00000i −0.658586 − 0.658586i 0.296460 − 0.955045i \(-0.404194\pi\)
−0.955045 + 0.296460i \(0.904194\pi\)
\(84\) 0 0
\(85\) −3.00000 + 9.00000i −0.325396 + 0.976187i
\(86\) 0 0
\(87\) 4.00000 − 4.00000i 0.428845 − 0.428845i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 12.0000i 1.25794i
\(92\) 0 0
\(93\) 8.00000 − 8.00000i 0.829561 − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000 + 11.0000i 1.11688 + 1.11688i 0.992196 + 0.124684i \(0.0397918\pi\)
0.124684 + 0.992196i \(0.460208\pi\)
\(98\) 0 0
\(99\) −20.0000 −2.01008
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 1280.2.n.a.767.1 2
4.3 odd 2 1280.2.n.k.767.1 2
5.3 odd 4 1280.2.n.k.1023.1 2
8.3 odd 2 1280.2.n.b.767.1 2
8.5 even 2 1280.2.n.l.767.1 2
16.3 odd 4 640.2.o.a.447.1 yes 2
16.5 even 4 640.2.o.b.447.1 yes 2
16.11 odd 4 640.2.o.h.447.1 yes 2
16.13 even 4 640.2.o.g.447.1 yes 2
20.3 even 4 inner 1280.2.n.a.1023.1 2
40.3 even 4 1280.2.n.l.1023.1 2
40.13 odd 4 1280.2.n.b.1023.1 2
80.3 even 4 640.2.o.b.63.1 yes 2
80.13 odd 4 640.2.o.h.63.1 yes 2
80.43 even 4 640.2.o.g.63.1 yes 2
80.53 odd 4 640.2.o.a.63.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.o.a.63.1 ✓ 2 80.53 odd 4
640.2.o.a.447.1 yes 2 16.3 odd 4
640.2.o.b.63.1 yes 2 80.3 even 4
640.2.o.b.447.1 yes 2 16.5 even 4
640.2.o.g.63.1 yes 2 80.43 even 4
640.2.o.g.447.1 yes 2 16.13 even 4
640.2.o.h.63.1 yes 2 80.13 odd 4
640.2.o.h.447.1 yes 2 16.11 odd 4
1280.2.n.a.767.1 2 1.1 even 1 trivial
1280.2.n.a.1023.1 2 20.3 even 4 inner
1280.2.n.b.767.1 2 8.3 odd 2
1280.2.n.b.1023.1 2 40.13 odd 4
1280.2.n.k.767.1 2 4.3 odd 2
1280.2.n.k.1023.1 2 5.3 odd 4
1280.2.n.l.767.1 2 8.5 even 2
1280.2.n.l.1023.1 2 40.3 even 4