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Properties

Label 1280.4.d.f.641.1
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
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] = [1280,4,Mod(641,1280)] 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("1280.641"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-12,0,46,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(75.5224448073\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.f.641.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +5.00000i q^{5} -6.00000 q^{7} +23.0000 q^{9} -60.0000i q^{11} +50.0000i q^{13} +10.0000 q^{15} -30.0000 q^{17} +40.0000i q^{19} +12.0000i q^{21} -178.000 q^{23} -25.0000 q^{25} -100.000i q^{27} +166.000i q^{29} +20.0000 q^{31} -120.000 q^{33} -30.0000i q^{35} -10.0000i q^{37} +100.000 q^{39} +250.000 q^{41} -142.000i q^{43} +115.000i q^{45} +214.000 q^{47} -307.000 q^{49} +60.0000i q^{51} -490.000i q^{53} +300.000 q^{55} +80.0000 q^{57} +800.000i q^{59} +250.000i q^{61} -138.000 q^{63} -250.000 q^{65} -774.000i q^{67} +356.000i q^{69} -100.000 q^{71} +230.000 q^{73} +50.0000i q^{75} +360.000i q^{77} -1320.00 q^{79} +421.000 q^{81} +982.000i q^{83} -150.000i q^{85} +332.000 q^{87} -874.000 q^{89} -300.000i q^{91} -40.0000i q^{93} -200.000 q^{95} -310.000 q^{97} -1380.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{7} + 46 q^{9} + 20 q^{15} - 60 q^{17} - 356 q^{23} - 50 q^{25} + 40 q^{31} - 240 q^{33} + 200 q^{39} + 500 q^{41} + 428 q^{47} - 614 q^{49} + 600 q^{55} + 160 q^{57} - 276 q^{63} - 500 q^{65}+ \cdots - 620 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 2.00000i − 0.384900i −0.981307 − 0.192450i \(-0.938357\pi\)
0.981307 − 0.192450i \(-0.0616434\pi\)
\(4\) 0 0
\(5\) 5.00000i 0.447214i
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 − 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) − 60.0000i − 1.64461i −0.569049 − 0.822304i \(-0.692689\pi\)
0.569049 − 0.822304i \(-0.307311\pi\)
\(12\) 0 0
\(13\) 50.0000i 1.06673i 0.845885 + 0.533366i \(0.179073\pi\)
−0.845885 + 0.533366i \(0.820927\pi\)
\(14\) 0 0
\(15\) 10.0000 0.172133
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 − 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 40.0000i 0.482980i 0.970403 + 0.241490i \(0.0776362\pi\)
−0.970403 + 0.241490i \(0.922364\pi\)
\(20\) 0 0
\(21\) 12.0000i 0.124696i
\(22\) 0 0
\(23\) −178.000 −1.61372 −0.806860 − 0.590743i \(-0.798835\pi\)
−0.806860 + 0.590743i \(0.798835\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 100.000i − 0.712778i
\(28\) 0 0
\(29\) 166.000i 1.06295i 0.847075 + 0.531473i \(0.178361\pi\)
−0.847075 + 0.531473i \(0.821639\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 − 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) −120.000 −0.633010
\(34\) 0 0
\(35\) − 30.0000i − 0.144884i
\(36\) 0 0
\(37\) − 10.0000i − 0.0444322i −0.999753 − 0.0222161i \(-0.992928\pi\)
0.999753 − 0.0222161i \(-0.00707218\pi\)
\(38\) 0 0
\(39\) 100.000 0.410585
\(40\) 0 0
\(41\) 250.000 0.952279 0.476140 − 0.879370i \(-0.342036\pi\)
0.476140 + 0.879370i \(0.342036\pi\)
\(42\) 0 0
\(43\) − 142.000i − 0.503600i −0.967779 − 0.251800i \(-0.918977\pi\)
0.967779 − 0.251800i \(-0.0810225\pi\)
\(44\) 0 0
\(45\) 115.000i 0.380960i
\(46\) 0 0
\(47\) 214.000 0.664151 0.332076 − 0.943253i \(-0.392251\pi\)
0.332076 + 0.943253i \(0.392251\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 60.0000i 0.164739i
\(52\) 0 0
\(53\) − 490.000i − 1.26994i −0.772538 − 0.634969i \(-0.781013\pi\)
0.772538 − 0.634969i \(-0.218987\pi\)
\(54\) 0 0
\(55\) 300.000 0.735491
\(56\) 0 0
\(57\) 80.0000 0.185899
\(58\) 0 0
\(59\) 800.000i 1.76527i 0.470056 + 0.882637i \(0.344234\pi\)
−0.470056 + 0.882637i \(0.655766\pi\)
\(60\) 0 0
\(61\) 250.000i 0.524741i 0.964967 + 0.262371i \(0.0845043\pi\)
−0.964967 + 0.262371i \(0.915496\pi\)
\(62\) 0 0
\(63\) −138.000 −0.275974
\(64\) 0 0
\(65\) −250.000 −0.477057
\(66\) 0 0
\(67\) − 774.000i − 1.41133i −0.708545 − 0.705665i \(-0.750648\pi\)
0.708545 − 0.705665i \(-0.249352\pi\)
\(68\) 0 0
\(69\) 356.000i 0.621121i
\(70\) 0 0
\(71\) −100.000 −0.167152 −0.0835762 − 0.996501i \(-0.526634\pi\)
−0.0835762 + 0.996501i \(0.526634\pi\)
\(72\) 0 0
\(73\) 230.000 0.368760 0.184380 − 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(74\) 0 0
\(75\) 50.0000i 0.0769800i
\(76\) 0 0
\(77\) 360.000i 0.532803i
\(78\) 0 0
\(79\) −1320.00 −1.87989 −0.939947 − 0.341321i \(-0.889126\pi\)
−0.939947 + 0.341321i \(0.889126\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 982.000i 1.29866i 0.760508 + 0.649328i \(0.224950\pi\)
−0.760508 + 0.649328i \(0.775050\pi\)
\(84\) 0 0
\(85\) − 150.000i − 0.191409i
\(86\) 0 0
\(87\) 332.000 0.409128
\(88\) 0 0
\(89\) −874.000 −1.04094 −0.520471 − 0.853879i \(-0.674244\pi\)
−0.520471 + 0.853879i \(0.674244\pi\)
\(90\) 0 0
\(91\) − 300.000i − 0.345588i
\(92\) 0 0
\(93\) − 40.0000i − 0.0446001i
\(94\) 0 0
\(95\) −200.000 −0.215995
\(96\) 0 0
\(97\) −310.000 −0.324492 −0.162246 − 0.986750i \(-0.551874\pi\)
−0.162246 + 0.986750i \(0.551874\pi\)
\(98\) 0 0
\(99\) − 1380.00i − 1.40096i
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 1280.4.d.f.641.1 2
4.3 odd 2 1280.4.d.k.641.2 2
8.3 odd 2 1280.4.d.k.641.1 2
8.5 even 2 inner 1280.4.d.f.641.2 2
16.3 odd 4 320.4.a.f.1.1 1
16.5 even 4 160.4.a.a.1.1 ✓ 1
16.11 odd 4 160.4.a.b.1.1 yes 1
16.13 even 4 320.4.a.i.1.1 1
48.5 odd 4 1440.4.a.o.1.1 1
48.11 even 4 1440.4.a.n.1.1 1
80.19 odd 4 1600.4.a.bj.1.1 1
80.27 even 4 800.4.c.e.449.1 2
80.29 even 4 1600.4.a.r.1.1 1
80.37 odd 4 800.4.c.f.449.2 2
80.43 even 4 800.4.c.e.449.2 2
80.53 odd 4 800.4.c.f.449.1 2
80.59 odd 4 800.4.a.d.1.1 1
80.69 even 4 800.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 ✓ 1 16.5 even 4
160.4.a.b.1.1 yes 1 16.11 odd 4
320.4.a.f.1.1 1 16.3 odd 4
320.4.a.i.1.1 1 16.13 even 4
800.4.a.d.1.1 1 80.59 odd 4
800.4.a.h.1.1 1 80.69 even 4
800.4.c.e.449.1 2 80.27 even 4
800.4.c.e.449.2 2 80.43 even 4
800.4.c.f.449.1 2 80.53 odd 4
800.4.c.f.449.2 2 80.37 odd 4
1280.4.d.f.641.1 2 1.1 even 1 trivial
1280.4.d.f.641.2 2 8.5 even 2 inner
1280.4.d.k.641.1 2 8.3 odd 2
1280.4.d.k.641.2 2 4.3 odd 2
1440.4.a.n.1.1 1 48.11 even 4
1440.4.a.o.1.1 1 48.5 odd 4
1600.4.a.r.1.1 1 80.29 even 4
1600.4.a.bj.1.1 1 80.19 odd 4