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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,2,Mod(129,320)] 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("320.129"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.c (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,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 129.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.2.c.d.129.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.23607i q^{3} +2.23607 q^{5} -5.23607i q^{7} +1.47214 q^{9} +2.76393i q^{15} +6.47214 q^{21} +7.70820i q^{23} +5.00000 q^{25} +5.52786i q^{27} -6.00000 q^{29} -11.7082i q^{35} +4.47214 q^{41} -6.76393i q^{43} +3.29180 q^{45} -0.291796i q^{47} -20.4164 q^{49} -13.4164 q^{61} -7.70820i q^{63} +14.1803i q^{67} -9.52786 q^{69} +6.18034i q^{75} -2.41641 q^{81} +4.29180i q^{83} -7.41641i q^{87} -6.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 8 q^{21} + 20 q^{25} - 24 q^{29} + 40 q^{45} - 28 q^{49} - 56 q^{69} + 44 q^{81} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.23607i 0.713644i 0.934172 + 0.356822i \(0.116140\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) − 5.23607i − 1.97905i −0.144370 − 0.989524i \(-0.546115\pi\)
0.144370 − 0.989524i \(-0.453885\pi\)
\(8\) 0 0
\(9\) 1.47214 0.490712
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 2.76393i 0.713644i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.47214 1.41234
\(22\) 0 0
\(23\) 7.70820i 1.60727i 0.595121 + 0.803636i \(0.297104\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.52786i 1.06384i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 11.7082i − 1.97905i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 − 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) − 6.76393i − 1.03149i −0.856742 − 0.515745i \(-0.827515\pi\)
0.856742 − 0.515745i \(-0.172485\pi\)
\(44\) 0 0
\(45\) 3.29180 0.490712
\(46\) 0 0
\(47\) − 0.291796i − 0.0425628i −0.999774 − 0.0212814i \(-0.993225\pi\)
0.999774 − 0.0212814i \(-0.00677460\pi\)
\(48\) 0 0
\(49\) −20.4164 −2.91663
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −13.4164 −1.71780 −0.858898 − 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) − 7.70820i − 0.971142i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.1803i 1.73240i 0.499694 + 0.866202i \(0.333446\pi\)
−0.499694 + 0.866202i \(0.666554\pi\)
\(68\) 0 0
\(69\) −9.52786 −1.14702
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 6.18034i 0.713644i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) 4.29180i 0.471086i 0.971864 + 0.235543i \(0.0756868\pi\)
−0.971864 + 0.235543i \(0.924313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.41641i − 0.795122i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
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 320.2.c.d.129.3 4
3.2 odd 2 2880.2.f.w.1729.1 4
4.3 odd 2 inner 320.2.c.d.129.2 4
5.2 odd 4 1600.2.a.z.1.2 2
5.3 odd 4 1600.2.a.bd.1.1 2
5.4 even 2 inner 320.2.c.d.129.2 4
8.3 odd 2 160.2.c.b.129.3 yes 4
8.5 even 2 160.2.c.b.129.2 ✓ 4
12.11 even 2 2880.2.f.w.1729.2 4
15.14 odd 2 2880.2.f.w.1729.2 4
16.3 odd 4 1280.2.f.g.129.3 4
16.5 even 4 1280.2.f.g.129.4 4
16.11 odd 4 1280.2.f.h.129.2 4
16.13 even 4 1280.2.f.h.129.1 4
20.3 even 4 1600.2.a.z.1.2 2
20.7 even 4 1600.2.a.bd.1.1 2
20.19 odd 2 CM 320.2.c.d.129.3 4
24.5 odd 2 1440.2.f.i.289.3 4
24.11 even 2 1440.2.f.i.289.4 4
40.3 even 4 800.2.a.n.1.1 2
40.13 odd 4 800.2.a.j.1.2 2
40.19 odd 2 160.2.c.b.129.2 ✓ 4
40.27 even 4 800.2.a.j.1.2 2
40.29 even 2 160.2.c.b.129.3 yes 4
40.37 odd 4 800.2.a.n.1.1 2
60.59 even 2 2880.2.f.w.1729.1 4
80.19 odd 4 1280.2.f.h.129.1 4
80.29 even 4 1280.2.f.g.129.3 4
80.59 odd 4 1280.2.f.g.129.4 4
80.69 even 4 1280.2.f.h.129.2 4
120.29 odd 2 1440.2.f.i.289.4 4
120.53 even 4 7200.2.a.cb.1.1 2
120.59 even 2 1440.2.f.i.289.3 4
120.77 even 4 7200.2.a.cr.1.2 2
120.83 odd 4 7200.2.a.cr.1.2 2
120.107 odd 4 7200.2.a.cb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.c.b.129.2 ✓ 4 8.5 even 2
160.2.c.b.129.2 ✓ 4 40.19 odd 2
160.2.c.b.129.3 yes 4 8.3 odd 2
160.2.c.b.129.3 yes 4 40.29 even 2
320.2.c.d.129.2 4 4.3 odd 2 inner
320.2.c.d.129.2 4 5.4 even 2 inner
320.2.c.d.129.3 4 1.1 even 1 trivial
320.2.c.d.129.3 4 20.19 odd 2 CM
800.2.a.j.1.2 2 40.13 odd 4
800.2.a.j.1.2 2 40.27 even 4
800.2.a.n.1.1 2 40.3 even 4
800.2.a.n.1.1 2 40.37 odd 4
1280.2.f.g.129.3 4 16.3 odd 4
1280.2.f.g.129.3 4 80.29 even 4
1280.2.f.g.129.4 4 16.5 even 4
1280.2.f.g.129.4 4 80.59 odd 4
1280.2.f.h.129.1 4 16.13 even 4
1280.2.f.h.129.1 4 80.19 odd 4
1280.2.f.h.129.2 4 16.11 odd 4
1280.2.f.h.129.2 4 80.69 even 4
1440.2.f.i.289.3 4 24.5 odd 2
1440.2.f.i.289.3 4 120.59 even 2
1440.2.f.i.289.4 4 24.11 even 2
1440.2.f.i.289.4 4 120.29 odd 2
1600.2.a.z.1.2 2 5.2 odd 4
1600.2.a.z.1.2 2 20.3 even 4
1600.2.a.bd.1.1 2 5.3 odd 4
1600.2.a.bd.1.1 2 20.7 even 4
2880.2.f.w.1729.1 4 3.2 odd 2
2880.2.f.w.1729.1 4 60.59 even 2
2880.2.f.w.1729.2 4 12.11 even 2
2880.2.f.w.1729.2 4 15.14 odd 2
7200.2.a.cb.1.1 2 120.53 even 4
7200.2.a.cb.1.1 2 120.107 odd 4
7200.2.a.cr.1.2 2 120.77 even 4
7200.2.a.cr.1.2 2 120.83 odd 4