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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3200,2,Mod(449,3200)] 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("3200.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.o.449.3

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843 q^{3} +5.00000 q^{9} +2.82843i q^{11} +6.00000i q^{17} +8.48528i q^{19} +5.65685 q^{27} +8.00000i q^{33} -6.00000 q^{41} -8.48528 q^{43} +7.00000 q^{49} +16.9706i q^{51} +24.0000i q^{57} -14.1421i q^{59} +8.48528 q^{67} +2.00000i q^{73} +1.00000 q^{81} +2.82843 q^{83} +18.0000 q^{89} -10.0000i q^{97} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{9} - 24 q^{41} + 28 q^{49} + 4 q^{81} + 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 2.82843 1.63299 0.816497 − 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 8.48528i 1.94666i 0.229416 + 0.973329i \(0.426318\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 8.00000i 1.39262i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 − 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 16.9706i 2.37635i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.0000i 3.17888i
\(58\) 0 0
\(59\) − 14.1421i − 1.84115i −0.390567 − 0.920575i \(-0.627721\pi\)
0.390567 − 0.920575i \(-0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528 1.03664 0.518321 − 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.82843 0.310460 0.155230 − 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 14.1421i 1.42134i
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 3200.2.f.o.449.4 4
4.3 odd 2 inner 3200.2.f.o.449.1 4
5.2 odd 4 3200.2.d.c.1601.1 2
5.3 odd 4 128.2.b.a.65.2 yes 2
5.4 even 2 inner 3200.2.f.o.449.2 4
8.3 odd 2 CM 3200.2.f.o.449.4 4
8.5 even 2 inner 3200.2.f.o.449.1 4
15.8 even 4 1152.2.d.c.577.1 2
20.3 even 4 128.2.b.a.65.1 ✓ 2
20.7 even 4 3200.2.d.c.1601.2 2
20.19 odd 2 inner 3200.2.f.o.449.3 4
40.3 even 4 128.2.b.a.65.2 yes 2
40.13 odd 4 128.2.b.a.65.1 ✓ 2
40.19 odd 2 inner 3200.2.f.o.449.2 4
40.27 even 4 3200.2.d.c.1601.1 2
40.29 even 2 inner 3200.2.f.o.449.3 4
40.37 odd 4 3200.2.d.c.1601.2 2
60.23 odd 4 1152.2.d.c.577.2 2
80.3 even 4 256.2.a.e.1.2 2
80.13 odd 4 256.2.a.e.1.1 2
80.27 even 4 6400.2.a.by.1.2 2
80.37 odd 4 6400.2.a.by.1.1 2
80.43 even 4 256.2.a.e.1.1 2
80.53 odd 4 256.2.a.e.1.2 2
80.67 even 4 6400.2.a.by.1.1 2
80.77 odd 4 6400.2.a.by.1.2 2
120.53 even 4 1152.2.d.c.577.2 2
120.83 odd 4 1152.2.d.c.577.1 2
160.3 even 8 1024.2.e.f.769.1 2
160.13 odd 8 1024.2.e.f.769.1 2
160.43 even 8 1024.2.e.a.257.1 2
160.53 odd 8 1024.2.e.f.257.1 2
160.83 even 8 1024.2.e.a.769.1 2
160.93 odd 8 1024.2.e.a.769.1 2
160.123 even 8 1024.2.e.f.257.1 2
160.133 odd 8 1024.2.e.a.257.1 2
240.53 even 4 2304.2.a.t.1.2 2
240.83 odd 4 2304.2.a.t.1.2 2
240.173 even 4 2304.2.a.t.1.1 2
240.203 odd 4 2304.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.a.65.1 ✓ 2 20.3 even 4
128.2.b.a.65.1 ✓ 2 40.13 odd 4
128.2.b.a.65.2 yes 2 5.3 odd 4
128.2.b.a.65.2 yes 2 40.3 even 4
256.2.a.e.1.1 2 80.13 odd 4
256.2.a.e.1.1 2 80.43 even 4
256.2.a.e.1.2 2 80.3 even 4
256.2.a.e.1.2 2 80.53 odd 4
1024.2.e.a.257.1 2 160.43 even 8
1024.2.e.a.257.1 2 160.133 odd 8
1024.2.e.a.769.1 2 160.83 even 8
1024.2.e.a.769.1 2 160.93 odd 8
1024.2.e.f.257.1 2 160.53 odd 8
1024.2.e.f.257.1 2 160.123 even 8
1024.2.e.f.769.1 2 160.3 even 8
1024.2.e.f.769.1 2 160.13 odd 8
1152.2.d.c.577.1 2 15.8 even 4
1152.2.d.c.577.1 2 120.83 odd 4
1152.2.d.c.577.2 2 60.23 odd 4
1152.2.d.c.577.2 2 120.53 even 4
2304.2.a.t.1.1 2 240.173 even 4
2304.2.a.t.1.1 2 240.203 odd 4
2304.2.a.t.1.2 2 240.53 even 4
2304.2.a.t.1.2 2 240.83 odd 4
3200.2.d.c.1601.1 2 5.2 odd 4
3200.2.d.c.1601.1 2 40.27 even 4
3200.2.d.c.1601.2 2 20.7 even 4
3200.2.d.c.1601.2 2 40.37 odd 4
3200.2.f.o.449.1 4 4.3 odd 2 inner
3200.2.f.o.449.1 4 8.5 even 2 inner
3200.2.f.o.449.2 4 5.4 even 2 inner
3200.2.f.o.449.2 4 40.19 odd 2 inner
3200.2.f.o.449.3 4 20.19 odd 2 inner
3200.2.f.o.449.3 4 40.29 even 2 inner
3200.2.f.o.449.4 4 1.1 even 1 trivial
3200.2.f.o.449.4 4 8.3 odd 2 CM
6400.2.a.by.1.1 2 80.37 odd 4
6400.2.a.by.1.1 2 80.67 even 4
6400.2.a.by.1.2 2 80.27 even 4
6400.2.a.by.1.2 2 80.77 odd 4