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 = [4,0,0,0,0,0,0,0,-52,0,0,0,0,0,0] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.u.641.4

$q$-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)
 
\(f(q)\) \(=\) \(q-6.32456i q^{3} -5.00000i q^{5} +18.9737 q^{7} -13.0000 q^{9} -12.6491i q^{11} +38.0000i q^{13} -31.6228 q^{15} +34.0000 q^{17} +101.193i q^{19} -120.000i q^{21} +82.2192 q^{23} -25.0000 q^{25} -88.5438i q^{27} +270.000i q^{29} +341.526 q^{31} -80.0000 q^{33} -94.8683i q^{35} -206.000i q^{37} +240.333 q^{39} +270.000 q^{41} +537.587i q^{43} +65.0000i q^{45} +132.816 q^{47} +17.0000 q^{49} -215.035i q^{51} +258.000i q^{53} -63.2456 q^{55} +640.000 q^{57} -75.8947i q^{59} -250.000i q^{61} -246.658 q^{63} +190.000 q^{65} +815.868i q^{67} -520.000i q^{69} -645.105 q^{71} +1078.00 q^{73} +158.114i q^{75} -240.000i q^{77} +278.280 q^{79} -911.000 q^{81} +1106.80i q^{83} -170.000i q^{85} +1707.63 q^{87} -890.000 q^{89} +720.999i q^{91} -2160.00i q^{93} +505.964 q^{95} -254.000 q^{97} +164.438i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9} + 136 q^{17} - 100 q^{25} - 320 q^{33} + 1080 q^{41} + 68 q^{49} + 2560 q^{57} + 760 q^{65} + 4312 q^{73} - 3644 q^{81} - 3560 q^{89} - 1016 q^{97}+O(q^{100}) \) 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
\(3\) − 6.32456i − 1.21716i −0.793492 − 0.608581i \(-0.791739\pi\)
0.793492 − 0.608581i \(-0.208261\pi\)
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) 18.9737 1.02448 0.512241 − 0.858842i \(-0.328816\pi\)
0.512241 + 0.858842i \(0.328816\pi\)
\(8\) 0 0
\(9\) −13.0000 −0.481481
\(10\) 0 0
\(11\) − 12.6491i − 0.346714i −0.984859 − 0.173357i \(-0.944539\pi\)
0.984859 − 0.173357i \(-0.0554614\pi\)
\(12\) 0 0
\(13\) 38.0000i 0.810716i 0.914158 + 0.405358i \(0.132853\pi\)
−0.914158 + 0.405358i \(0.867147\pi\)
\(14\) 0 0
\(15\) −31.6228 −0.544331
\(16\) 0 0
\(17\) 34.0000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 101.193i 1.22185i 0.791687 + 0.610927i \(0.209203\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(20\) 0 0
\(21\) − 120.000i − 1.24696i
\(22\) 0 0
\(23\) 82.2192 0.745387 0.372693 − 0.927955i \(-0.378434\pi\)
0.372693 + 0.927955i \(0.378434\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 88.5438i − 0.631121i
\(28\) 0 0
\(29\) 270.000i 1.72889i 0.502729 + 0.864444i \(0.332329\pi\)
−0.502729 + 0.864444i \(0.667671\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 − 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) −80.0000 −0.422006
\(34\) 0 0
\(35\) − 94.8683i − 0.458162i
\(36\) 0 0
\(37\) − 206.000i − 0.915302i −0.889132 − 0.457651i \(-0.848691\pi\)
0.889132 − 0.457651i \(-0.151309\pi\)
\(38\) 0 0
\(39\) 240.333 0.986772
\(40\) 0 0
\(41\) 270.000 1.02846 0.514231 − 0.857652i \(-0.328078\pi\)
0.514231 + 0.857652i \(0.328078\pi\)
\(42\) 0 0
\(43\) 537.587i 1.90654i 0.302117 + 0.953271i \(0.402307\pi\)
−0.302117 + 0.953271i \(0.597693\pi\)
\(44\) 0 0
\(45\) 65.0000i 0.215325i
\(46\) 0 0
\(47\) 132.816 0.412195 0.206097 − 0.978531i \(-0.433924\pi\)
0.206097 + 0.978531i \(0.433924\pi\)
\(48\) 0 0
\(49\) 17.0000 0.0495627
\(50\) 0 0
\(51\) − 215.035i − 0.590410i
\(52\) 0 0
\(53\) 258.000i 0.668661i 0.942456 + 0.334330i \(0.108510\pi\)
−0.942456 + 0.334330i \(0.891490\pi\)
\(54\) 0 0
\(55\) −63.2456 −0.155055
\(56\) 0 0
\(57\) 640.000 1.48719
\(58\) 0 0
\(59\) − 75.8947i − 0.167469i −0.996488 − 0.0837343i \(-0.973315\pi\)
0.996488 − 0.0837343i \(-0.0266847\pi\)
\(60\) 0 0
\(61\) − 250.000i − 0.524741i −0.964967 − 0.262371i \(-0.915496\pi\)
0.964967 − 0.262371i \(-0.0845043\pi\)
\(62\) 0 0
\(63\) −246.658 −0.493269
\(64\) 0 0
\(65\) 190.000 0.362563
\(66\) 0 0
\(67\) 815.868i 1.48767i 0.668362 + 0.743837i \(0.266996\pi\)
−0.668362 + 0.743837i \(0.733004\pi\)
\(68\) 0 0
\(69\) − 520.000i − 0.907256i
\(70\) 0 0
\(71\) −645.105 −1.07831 −0.539154 − 0.842207i \(-0.681256\pi\)
−0.539154 + 0.842207i \(0.681256\pi\)
\(72\) 0 0
\(73\) 1078.00 1.72836 0.864181 − 0.503182i \(-0.167837\pi\)
0.864181 + 0.503182i \(0.167837\pi\)
\(74\) 0 0
\(75\) 158.114i 0.243432i
\(76\) 0 0
\(77\) − 240.000i − 0.355202i
\(78\) 0 0
\(79\) 278.280 0.396316 0.198158 − 0.980170i \(-0.436504\pi\)
0.198158 + 0.980170i \(0.436504\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) 1106.80i 1.46370i 0.681468 + 0.731848i \(0.261342\pi\)
−0.681468 + 0.731848i \(0.738658\pi\)
\(84\) 0 0
\(85\) − 170.000i − 0.216930i
\(86\) 0 0
\(87\) 1707.63 2.10434
\(88\) 0 0
\(89\) −890.000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 720.999i 0.830563i
\(92\) 0 0
\(93\) − 2160.00i − 2.40840i
\(94\) 0 0
\(95\) 505.964 0.546430
\(96\) 0 0
\(97\) −254.000 −0.265874 −0.132937 − 0.991124i \(-0.542441\pi\)
−0.132937 + 0.991124i \(0.542441\pi\)
\(98\) 0 0
\(99\) 164.438i 0.166936i
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.u.641.1 4
4.3 odd 2 inner 1280.4.d.u.641.3 4
8.3 odd 2 inner 1280.4.d.u.641.2 4
8.5 even 2 inner 1280.4.d.u.641.4 4
16.3 odd 4 320.4.a.p.1.1 2
16.5 even 4 160.4.a.f.1.1 ✓ 2
16.11 odd 4 160.4.a.f.1.2 yes 2
16.13 even 4 320.4.a.p.1.2 2
48.5 odd 4 1440.4.a.v.1.1 2
48.11 even 4 1440.4.a.v.1.2 2
80.19 odd 4 1600.4.a.ch.1.2 2
80.27 even 4 800.4.c.j.449.1 4
80.29 even 4 1600.4.a.ch.1.1 2
80.37 odd 4 800.4.c.j.449.4 4
80.43 even 4 800.4.c.j.449.3 4
80.53 odd 4 800.4.c.j.449.2 4
80.59 odd 4 800.4.a.p.1.1 2
80.69 even 4 800.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 ✓ 2 16.5 even 4
160.4.a.f.1.2 yes 2 16.11 odd 4
320.4.a.p.1.1 2 16.3 odd 4
320.4.a.p.1.2 2 16.13 even 4
800.4.a.p.1.1 2 80.59 odd 4
800.4.a.p.1.2 2 80.69 even 4
800.4.c.j.449.1 4 80.27 even 4
800.4.c.j.449.2 4 80.53 odd 4
800.4.c.j.449.3 4 80.43 even 4
800.4.c.j.449.4 4 80.37 odd 4
1280.4.d.u.641.1 4 1.1 even 1 trivial
1280.4.d.u.641.2 4 8.3 odd 2 inner
1280.4.d.u.641.3 4 4.3 odd 2 inner
1280.4.d.u.641.4 4 8.5 even 2 inner
1440.4.a.v.1.1 2 48.5 odd 4
1440.4.a.v.1.2 2 48.11 even 4
1600.4.a.ch.1.1 2 80.29 even 4
1600.4.a.ch.1.2 2 80.19 odd 4