Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3600,2,Mod(2449,3600)] 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("3600.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.7461447277\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1800)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3600.2449
Dual form 3600.2.f.s.2449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{7} +4.00000 q^{11} +1.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} +4.00000 q^{29} +5.00000 q^{31} -6.00000i q^{37} -12.0000 q^{41} +5.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} -12.0000i q^{53} -8.00000 q^{59} +7.00000 q^{61} -13.0000i q^{67} +12.0000 q^{71} +6.00000i q^{73} -4.00000i q^{77} +12.0000 q^{79} +8.00000i q^{83} +1.00000 q^{91} -13.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{11} + 2 q^{19} + 8 q^{29} + 10 q^{31} - 24 q^{41} + 12 q^{49} - 16 q^{59} + 14 q^{61} + 24 q^{71} + 24 q^{79} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 − 0.188982i \(-0.939481\pi\)
0.981981 − 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 − 0.493197i \(-0.835828\pi\)
0.869918 − 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 5.00000i 0.762493i 0.924473 + 0.381246i \(0.124505\pi\)
−0.924473 + 0.381246i \(0.875495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 − 0.824163i \(-0.691646\pi\)
0.566352 − 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 − 0.794101i \(-0.707942\pi\)
0.607785 − 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.0000i − 1.31995i −0.751288 − 0.659975i \(-0.770567\pi\)
0.751288 − 0.659975i \(-0.229433\pi\)
\(98\) 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 3600.2.f.s.2449.1 2
3.2 odd 2 3600.2.f.d.2449.1 2
4.3 odd 2 1800.2.f.b.649.2 2
5.2 odd 4 3600.2.a.y.1.1 1
5.3 odd 4 3600.2.a.r.1.1 1
5.4 even 2 inner 3600.2.f.s.2449.2 2
12.11 even 2 1800.2.f.i.649.2 2
15.2 even 4 3600.2.a.w.1.1 1
15.8 even 4 3600.2.a.p.1.1 1
15.14 odd 2 3600.2.f.d.2449.2 2
20.3 even 4 1800.2.a.o.1.1 yes 1
20.7 even 4 1800.2.a.k.1.1 ✓ 1
20.19 odd 2 1800.2.f.b.649.1 2
60.23 odd 4 1800.2.a.p.1.1 yes 1
60.47 odd 4 1800.2.a.l.1.1 yes 1
60.59 even 2 1800.2.f.i.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1800.2.a.k.1.1 ✓ 1 20.7 even 4
1800.2.a.l.1.1 yes 1 60.47 odd 4
1800.2.a.o.1.1 yes 1 20.3 even 4
1800.2.a.p.1.1 yes 1 60.23 odd 4
1800.2.f.b.649.1 2 20.19 odd 2
1800.2.f.b.649.2 2 4.3 odd 2
1800.2.f.i.649.1 2 60.59 even 2
1800.2.f.i.649.2 2 12.11 even 2
3600.2.a.p.1.1 1 15.8 even 4
3600.2.a.r.1.1 1 5.3 odd 4
3600.2.a.w.1.1 1 15.2 even 4
3600.2.a.y.1.1 1 5.2 odd 4
3600.2.f.d.2449.1 2 3.2 odd 2
3600.2.f.d.2449.2 2 15.14 odd 2
3600.2.f.s.2449.1 2 1.1 even 1 trivial
3600.2.f.s.2449.2 2 5.4 even 2 inner