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 = [2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,-14,0,0,0,0,0,0,0,0,0,-4] 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: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{7} +2.00000 q^{11} -3.00000i q^{13} +6.00000i q^{17} -7.00000 q^{19} +6.00000i q^{23} -2.00000 q^{29} +5.00000 q^{31} -10.0000i q^{37} -12.0000 q^{41} -3.00000i q^{43} +10.0000i q^{47} -2.00000 q^{49} +6.00000 q^{59} -13.0000 q^{61} +7.00000i q^{67} -4.00000 q^{71} -6.00000i q^{73} +6.00000i q^{77} -8.00000 q^{79} -6.00000i q^{83} +16.0000 q^{89} +9.00000 q^{91} +7.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{11} - 14 q^{19} - 4 q^{29} + 10 q^{31} - 24 q^{41} - 4 q^{49} + 12 q^{59} - 26 q^{61} - 8 q^{71} - 16 q^{79} + 32 q^{89} + 18 q^{91}+O(q^{100}) \) 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
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 3.00000i − 0.832050i −0.909353 − 0.416025i \(-0.863423\pi\)
0.909353 − 0.416025i \(-0.136577\pi\)
\(14\) 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
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) 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
\(43\) − 3.00000i − 0.457496i −0.973486 − 0.228748i \(-0.926537\pi\)
0.973486 − 0.228748i \(-0.0734631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 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
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 − 0.351123i \(-0.885800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 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.o.2449.2 2
3.2 odd 2 1200.2.f.c.49.1 2
4.3 odd 2 1800.2.f.e.649.1 2
5.2 odd 4 3600.2.a.i.1.1 1
5.3 odd 4 3600.2.a.bl.1.1 1
5.4 even 2 inner 3600.2.f.o.2449.1 2
12.11 even 2 600.2.f.d.49.2 2
15.2 even 4 1200.2.a.b.1.1 1
15.8 even 4 1200.2.a.q.1.1 1
15.14 odd 2 1200.2.f.c.49.2 2
20.3 even 4 1800.2.a.e.1.1 1
20.7 even 4 1800.2.a.t.1.1 1
20.19 odd 2 1800.2.f.e.649.2 2
24.5 odd 2 4800.2.f.z.3649.2 2
24.11 even 2 4800.2.f.k.3649.1 2
60.23 odd 4 600.2.a.b.1.1 ✓ 1
60.47 odd 4 600.2.a.i.1.1 yes 1
60.59 even 2 600.2.f.d.49.1 2
120.29 odd 2 4800.2.f.z.3649.1 2
120.53 even 4 4800.2.a.bd.1.1 1
120.59 even 2 4800.2.f.k.3649.2 2
120.77 even 4 4800.2.a.bs.1.1 1
120.83 odd 4 4800.2.a.bp.1.1 1
120.107 odd 4 4800.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.b.1.1 ✓ 1 60.23 odd 4
600.2.a.i.1.1 yes 1 60.47 odd 4
600.2.f.d.49.1 2 60.59 even 2
600.2.f.d.49.2 2 12.11 even 2
1200.2.a.b.1.1 1 15.2 even 4
1200.2.a.q.1.1 1 15.8 even 4
1200.2.f.c.49.1 2 3.2 odd 2
1200.2.f.c.49.2 2 15.14 odd 2
1800.2.a.e.1.1 1 20.3 even 4
1800.2.a.t.1.1 1 20.7 even 4
1800.2.f.e.649.1 2 4.3 odd 2
1800.2.f.e.649.2 2 20.19 odd 2
3600.2.a.i.1.1 1 5.2 odd 4
3600.2.a.bl.1.1 1 5.3 odd 4
3600.2.f.o.2449.1 2 5.4 even 2 inner
3600.2.f.o.2449.2 2 1.1 even 1 trivial
4800.2.a.bc.1.1 1 120.107 odd 4
4800.2.a.bd.1.1 1 120.53 even 4
4800.2.a.bp.1.1 1 120.83 odd 4
4800.2.a.bs.1.1 1 120.77 even 4
4800.2.f.k.3649.1 2 24.11 even 2
4800.2.f.k.3649.2 2 120.59 even 2
4800.2.f.z.3649.1 2 120.29 odd 2
4800.2.f.z.3649.2 2 24.5 odd 2