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Properties

Label 1280.2.n.j.1023.1
Level $1280$
Weight $2$
Character 1280.1023
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 = [2,0,2,0,4,0,2,0,0,0,0,0,-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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1023.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.1023
Dual form 1280.2.n.j.767.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{3} +(2.00000 + 1.00000i) q^{5} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +4.00000i q^{11} +(-3.00000 - 3.00000i) q^{13} +(3.00000 - 1.00000i) q^{15} +(-3.00000 + 3.00000i) q^{17} +6.00000 q^{19} +2.00000 q^{21} +(3.00000 - 3.00000i) q^{23} +(3.00000 + 4.00000i) q^{25} +(4.00000 + 4.00000i) q^{27} +2.00000i q^{29} +6.00000i q^{31} +(4.00000 + 4.00000i) q^{33} +(1.00000 + 3.00000i) q^{35} +(-3.00000 + 3.00000i) q^{37} -6.00000 q^{39} -6.00000 q^{41} +(3.00000 - 3.00000i) q^{43} +(-1.00000 + 2.00000i) q^{45} +(-9.00000 - 9.00000i) q^{47} -5.00000i q^{49} +6.00000i q^{51} +(-5.00000 - 5.00000i) q^{53} +(-4.00000 + 8.00000i) q^{55} +(6.00000 - 6.00000i) q^{57} +10.0000 q^{59} +12.0000 q^{61} +(-1.00000 + 1.00000i) q^{63} +(-3.00000 - 9.00000i) q^{65} +(9.00000 + 9.00000i) q^{67} -6.00000i q^{69} -6.00000i q^{71} +(-5.00000 - 5.00000i) q^{73} +(7.00000 + 1.00000i) q^{75} +(-4.00000 + 4.00000i) q^{77} +5.00000 q^{81} +(-3.00000 + 3.00000i) q^{83} +(-9.00000 + 3.00000i) q^{85} +(2.00000 + 2.00000i) q^{87} -6.00000i q^{91} +(6.00000 + 6.00000i) q^{93} +(12.0000 + 6.00000i) q^{95} +(-7.00000 + 7.00000i) q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{7} - 6 q^{13} + 6 q^{15} - 6 q^{17} + 12 q^{19} + 4 q^{21} + 6 q^{23} + 6 q^{25} + 8 q^{27} + 8 q^{33} + 2 q^{35} - 6 q^{37} - 12 q^{39} - 12 q^{41} + 6 q^{43} - 2 q^{45}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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
\(3\) 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
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 1.00000 + 1.00000i 0.377964 + 0.377964i 0.870367 − 0.492403i \(-0.163881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 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 \(-0.450222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 3.00000 − 1.00000i 0.774597 − 0.258199i
\(16\) 0 0
\(17\) −3.00000 + 3.00000i −0.727607 + 0.727607i −0.970143 − 0.242536i \(-0.922021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 3.00000 − 3.00000i 0.625543 − 0.625543i −0.321400 − 0.946943i \(-0.604153\pi\)
0.946943 + 0.321400i \(0.104153\pi\)
\(24\) 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
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 4.00000 + 4.00000i 0.696311 + 0.696311i
\(34\) 0 0
\(35\) 1.00000 + 3.00000i 0.169031 + 0.507093i
\(36\) 0 0
\(37\) −3.00000 + 3.00000i −0.493197 + 0.493197i −0.909312 − 0.416115i \(-0.863391\pi\)
0.416115 + 0.909312i \(0.363391\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 3.00000 − 3.00000i 0.457496 − 0.457496i −0.440337 − 0.897833i \(-0.645141\pi\)
0.897833 + 0.440337i \(0.145141\pi\)
\(44\) 0 0
\(45\) −1.00000 + 2.00000i −0.149071 + 0.298142i
\(46\) 0 0
\(47\) −9.00000 − 9.00000i −1.31278 − 1.31278i −0.919354 − 0.393431i \(-0.871288\pi\)
−0.393431 − 0.919354i \(-0.628712\pi\)
\(48\) 0 0
\(49\) − 5.00000i − 0.714286i
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) −5.00000 − 5.00000i −0.686803 − 0.686803i 0.274721 − 0.961524i \(-0.411414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) −4.00000 + 8.00000i −0.539360 + 1.07872i
\(56\) 0 0
\(57\) 6.00000 − 6.00000i 0.794719 − 0.794719i
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) −1.00000 + 1.00000i −0.125988 + 0.125988i
\(64\) 0 0
\(65\) −3.00000 − 9.00000i −0.372104 − 1.11631i
\(66\) 0 0
\(67\) 9.00000 + 9.00000i 1.09952 + 1.09952i 0.994466 + 0.105059i \(0.0335031\pi\)
0.105059 + 0.994466i \(0.466497\pi\)
\(68\) 0 0
\(69\) − 6.00000i − 0.722315i
\(70\) 0 0
\(71\) − 6.00000i − 0.712069i −0.934473 − 0.356034i \(-0.884129\pi\)
0.934473 − 0.356034i \(-0.115871\pi\)
\(72\) 0 0
\(73\) −5.00000 − 5.00000i −0.585206 − 0.585206i 0.351123 − 0.936329i \(-0.385800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 7.00000 + 1.00000i 0.808290 + 0.115470i
\(76\) 0 0
\(77\) −4.00000 + 4.00000i −0.455842 + 0.455842i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −3.00000 + 3.00000i −0.329293 + 0.329293i −0.852318 − 0.523025i \(-0.824804\pi\)
0.523025 + 0.852318i \(0.324804\pi\)
\(84\) 0 0
\(85\) −9.00000 + 3.00000i −0.976187 + 0.325396i
\(86\) 0 0
\(87\) 2.00000 + 2.00000i 0.214423 + 0.214423i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) − 6.00000i − 0.628971i
\(92\) 0 0
\(93\) 6.00000 + 6.00000i 0.622171 + 0.622171i
\(94\) 0 0
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) −7.00000 + 7.00000i −0.710742 + 0.710742i −0.966691 − 0.255948i \(-0.917612\pi\)
0.255948 + 0.966691i \(0.417612\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
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.j.1023.1 2
4.3 odd 2 1280.2.n.d.1023.1 2
5.2 odd 4 1280.2.n.d.767.1 2
8.3 odd 2 1280.2.n.i.1023.1 2
8.5 even 2 1280.2.n.c.1023.1 2
16.3 odd 4 320.2.o.b.223.1 yes 2
16.5 even 4 320.2.o.a.223.1 ✓ 2
16.11 odd 4 320.2.o.c.223.1 yes 2
16.13 even 4 320.2.o.d.223.1 yes 2
20.7 even 4 inner 1280.2.n.j.767.1 2
40.27 even 4 1280.2.n.c.767.1 2
40.37 odd 4 1280.2.n.i.767.1 2
80.3 even 4 1600.2.o.d.607.1 2
80.13 odd 4 1600.2.o.a.607.1 2
80.19 odd 4 1600.2.o.c.543.1 2
80.27 even 4 320.2.o.d.287.1 yes 2
80.29 even 4 1600.2.o.b.543.1 2
80.37 odd 4 320.2.o.b.287.1 yes 2
80.43 even 4 1600.2.o.b.607.1 2
80.53 odd 4 1600.2.o.c.607.1 2
80.59 odd 4 1600.2.o.a.543.1 2
80.67 even 4 320.2.o.a.287.1 yes 2
80.69 even 4 1600.2.o.d.543.1 2
80.77 odd 4 320.2.o.c.287.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.o.a.223.1 ✓ 2 16.5 even 4
320.2.o.a.287.1 yes 2 80.67 even 4
320.2.o.b.223.1 yes 2 16.3 odd 4
320.2.o.b.287.1 yes 2 80.37 odd 4
320.2.o.c.223.1 yes 2 16.11 odd 4
320.2.o.c.287.1 yes 2 80.77 odd 4
320.2.o.d.223.1 yes 2 16.13 even 4
320.2.o.d.287.1 yes 2 80.27 even 4
1280.2.n.c.767.1 2 40.27 even 4
1280.2.n.c.1023.1 2 8.5 even 2
1280.2.n.d.767.1 2 5.2 odd 4
1280.2.n.d.1023.1 2 4.3 odd 2
1280.2.n.i.767.1 2 40.37 odd 4
1280.2.n.i.1023.1 2 8.3 odd 2
1280.2.n.j.767.1 2 20.7 even 4 inner
1280.2.n.j.1023.1 2 1.1 even 1 trivial
1600.2.o.a.543.1 2 80.59 odd 4
1600.2.o.a.607.1 2 80.13 odd 4
1600.2.o.b.543.1 2 80.29 even 4
1600.2.o.b.607.1 2 80.43 even 4
1600.2.o.c.543.1 2 80.19 odd 4
1600.2.o.c.607.1 2 80.53 odd 4
1600.2.o.d.543.1 2 80.69 even 4
1600.2.o.d.607.1 2 80.3 even 4