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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,3,Mod(1025,1728)] 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("1728.1025"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-10,0,0,0,0,0,20] 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: \(47.0845896815\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1025
Dual form 1728.3.e.g.1025.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{5} -5.00000 q^{7} +15.0000i q^{11} +10.0000 q^{13} -18.0000i q^{17} -16.0000 q^{19} -12.0000i q^{23} +16.0000 q^{25} +30.0000i q^{29} +1.00000 q^{31} +15.0000i q^{35} -20.0000 q^{37} -60.0000i q^{41} +50.0000 q^{43} -6.00000i q^{47} -24.0000 q^{49} -27.0000i q^{53} +45.0000 q^{55} +30.0000i q^{59} +76.0000 q^{61} -30.0000i q^{65} -10.0000 q^{67} -90.0000i q^{71} +65.0000 q^{73} -75.0000i q^{77} -14.0000 q^{79} -3.00000i q^{83} -54.0000 q^{85} -90.0000i q^{89} -50.0000 q^{91} +48.0000i q^{95} -85.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{7} + 20 q^{13} - 32 q^{19} + 32 q^{25} + 2 q^{31} - 40 q^{37} + 100 q^{43} - 48 q^{49} + 90 q^{55} + 152 q^{61} - 20 q^{67} + 130 q^{73} - 28 q^{79} - 108 q^{85} - 100 q^{91} - 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.00000i − 0.600000i −0.953939 − 0.300000i \(-0.903013\pi\)
0.953939 − 0.300000i \(-0.0969867\pi\)
\(6\) 0 0
\(7\) −5.00000 −0.714286 −0.357143 − 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.0000i 1.36364i 0.731522 + 0.681818i \(0.238810\pi\)
−0.731522 + 0.681818i \(0.761190\pi\)
\(12\) 0 0
\(13\) 10.0000 0.769231 0.384615 − 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 18.0000i − 1.05882i −0.848365 − 0.529412i \(-0.822413\pi\)
0.848365 − 0.529412i \(-0.177587\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 12.0000i − 0.521739i −0.965374 − 0.260870i \(-0.915991\pi\)
0.965374 − 0.260870i \(-0.0840093\pi\)
\(24\) 0 0
\(25\) 16.0000 0.640000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 30.0000i 1.03448i 0.855840 + 0.517241i \(0.173041\pi\)
−0.855840 + 0.517241i \(0.826959\pi\)
\(30\) 0 0
\(31\) 1.00000 0.0322581 0.0161290 − 0.999870i \(-0.494866\pi\)
0.0161290 + 0.999870i \(0.494866\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.0000i 0.428571i
\(36\) 0 0
\(37\) −20.0000 −0.540541 −0.270270 − 0.962784i \(-0.587113\pi\)
−0.270270 + 0.962784i \(0.587113\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 60.0000i − 1.46341i −0.681619 − 0.731707i \(-0.738724\pi\)
0.681619 − 0.731707i \(-0.261276\pi\)
\(42\) 0 0
\(43\) 50.0000 1.16279 0.581395 − 0.813621i \(-0.302507\pi\)
0.581395 + 0.813621i \(0.302507\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.127660i −0.997961 − 0.0638298i \(-0.979669\pi\)
0.997961 − 0.0638298i \(-0.0203315\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 27.0000i − 0.509434i −0.967016 − 0.254717i \(-0.918018\pi\)
0.967016 − 0.254717i \(-0.0819823\pi\)
\(54\) 0 0
\(55\) 45.0000 0.818182
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 30.0000i 0.508475i 0.967142 + 0.254237i \(0.0818244\pi\)
−0.967142 + 0.254237i \(0.918176\pi\)
\(60\) 0 0
\(61\) 76.0000 1.24590 0.622951 − 0.782261i \(-0.285934\pi\)
0.622951 + 0.782261i \(0.285934\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 30.0000i − 0.461538i
\(66\) 0 0
\(67\) −10.0000 −0.149254 −0.0746269 − 0.997212i \(-0.523777\pi\)
−0.0746269 + 0.997212i \(0.523777\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 90.0000i − 1.26761i −0.773495 − 0.633803i \(-0.781493\pi\)
0.773495 − 0.633803i \(-0.218507\pi\)
\(72\) 0 0
\(73\) 65.0000 0.890411 0.445205 − 0.895428i \(-0.353131\pi\)
0.445205 + 0.895428i \(0.353131\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 75.0000i − 0.974026i
\(78\) 0 0
\(79\) −14.0000 −0.177215 −0.0886076 − 0.996067i \(-0.528242\pi\)
−0.0886076 + 0.996067i \(0.528242\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 3.00000i − 0.0361446i −0.999837 − 0.0180723i \(-0.994247\pi\)
0.999837 − 0.0180723i \(-0.00575290\pi\)
\(84\) 0 0
\(85\) −54.0000 −0.635294
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 90.0000i − 1.01124i −0.862757 − 0.505618i \(-0.831265\pi\)
0.862757 − 0.505618i \(-0.168735\pi\)
\(90\) 0 0
\(91\) −50.0000 −0.549451
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 48.0000i 0.505263i
\(96\) 0 0
\(97\) −85.0000 −0.876289 −0.438144 − 0.898905i \(-0.644364\pi\)
−0.438144 + 0.898905i \(0.644364\pi\)
\(98\) 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 1728.3.e.g.1025.1 2
3.2 odd 2 inner 1728.3.e.g.1025.2 2
4.3 odd 2 1728.3.e.m.1025.1 2
8.3 odd 2 27.3.b.b.26.1 ✓ 2
8.5 even 2 432.3.e.c.161.2 2
12.11 even 2 1728.3.e.m.1025.2 2
24.5 odd 2 432.3.e.c.161.1 2
24.11 even 2 27.3.b.b.26.2 yes 2
40.3 even 4 675.3.d.a.674.1 2
40.19 odd 2 675.3.c.h.26.2 2
40.27 even 4 675.3.d.d.674.2 2
72.5 odd 6 1296.3.q.j.593.1 4
72.11 even 6 81.3.d.b.53.2 4
72.13 even 6 1296.3.q.j.593.2 4
72.29 odd 6 1296.3.q.j.1025.2 4
72.43 odd 6 81.3.d.b.53.1 4
72.59 even 6 81.3.d.b.26.1 4
72.61 even 6 1296.3.q.j.1025.1 4
72.67 odd 6 81.3.d.b.26.2 4
120.59 even 2 675.3.c.h.26.1 2
120.83 odd 4 675.3.d.d.674.1 2
120.107 odd 4 675.3.d.a.674.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.b.26.1 ✓ 2 8.3 odd 2
27.3.b.b.26.2 yes 2 24.11 even 2
81.3.d.b.26.1 4 72.59 even 6
81.3.d.b.26.2 4 72.67 odd 6
81.3.d.b.53.1 4 72.43 odd 6
81.3.d.b.53.2 4 72.11 even 6
432.3.e.c.161.1 2 24.5 odd 2
432.3.e.c.161.2 2 8.5 even 2
675.3.c.h.26.1 2 120.59 even 2
675.3.c.h.26.2 2 40.19 odd 2
675.3.d.a.674.1 2 40.3 even 4
675.3.d.a.674.2 2 120.107 odd 4
675.3.d.d.674.1 2 120.83 odd 4
675.3.d.d.674.2 2 40.27 even 4
1296.3.q.j.593.1 4 72.5 odd 6
1296.3.q.j.593.2 4 72.13 even 6
1296.3.q.j.1025.1 4 72.61 even 6
1296.3.q.j.1025.2 4 72.29 odd 6
1728.3.e.g.1025.1 2 1.1 even 1 trivial
1728.3.e.g.1025.2 2 3.2 odd 2 inner
1728.3.e.m.1025.1 2 4.3 odd 2
1728.3.e.m.1025.2 2 12.11 even 2