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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,2,Mod(649,1800)] 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("1800.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.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,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3730723638\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.2.f.d.649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{7} -2.00000 q^{11} -4.00000i q^{13} +2.00000i q^{17} -4.00000 q^{19} +8.00000i q^{23} -10.0000 q^{29} +4.00000 q^{31} +8.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +6.00000i q^{53} -14.0000 q^{59} -14.0000 q^{61} -4.00000i q^{67} -12.0000 q^{71} -6.00000i q^{73} -4.00000i q^{77} +12.0000 q^{79} +4.00000i q^{83} -12.0000 q^{89} +8.00000 q^{91} -14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{11} - 8 q^{19} - 20 q^{29} + 8 q^{31} + 6 q^{49} - 28 q^{59} - 28 q^{61} - 24 q^{71} + 24 q^{79} - 24 q^{89} + 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 − 0.583460i \(-0.801699\pi\)
0.812142 − 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 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
\(77\) − 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
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 − 0.710742i \(-0.748359\pi\)
0.703452 − 0.710742i \(-0.251641\pi\)
\(98\) 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 1800.2.f.d.649.2 2
3.2 odd 2 1800.2.f.h.649.2 2
4.3 odd 2 3600.2.f.q.2449.1 2
5.2 odd 4 1800.2.a.f.1.1 1
5.3 odd 4 360.2.a.d.1.1 yes 1
5.4 even 2 inner 1800.2.f.d.649.1 2
12.11 even 2 3600.2.f.g.2449.1 2
15.2 even 4 1800.2.a.i.1.1 1
15.8 even 4 360.2.a.c.1.1 ✓ 1
15.14 odd 2 1800.2.f.h.649.1 2
20.3 even 4 720.2.a.i.1.1 1
20.7 even 4 3600.2.a.bh.1.1 1
20.19 odd 2 3600.2.f.q.2449.2 2
40.3 even 4 2880.2.a.e.1.1 1
40.13 odd 4 2880.2.a.n.1.1 1
45.13 odd 12 3240.2.q.d.2161.1 2
45.23 even 12 3240.2.q.n.2161.1 2
45.38 even 12 3240.2.q.n.1081.1 2
45.43 odd 12 3240.2.q.d.1081.1 2
60.23 odd 4 720.2.a.a.1.1 1
60.47 odd 4 3600.2.a.bd.1.1 1
60.59 even 2 3600.2.f.g.2449.2 2
120.53 even 4 2880.2.a.bd.1.1 1
120.83 odd 4 2880.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.a.c.1.1 ✓ 1 15.8 even 4
360.2.a.d.1.1 yes 1 5.3 odd 4
720.2.a.a.1.1 1 60.23 odd 4
720.2.a.i.1.1 1 20.3 even 4
1800.2.a.f.1.1 1 5.2 odd 4
1800.2.a.i.1.1 1 15.2 even 4
1800.2.f.d.649.1 2 5.4 even 2 inner
1800.2.f.d.649.2 2 1.1 even 1 trivial
1800.2.f.h.649.1 2 15.14 odd 2
1800.2.f.h.649.2 2 3.2 odd 2
2880.2.a.e.1.1 1 40.3 even 4
2880.2.a.n.1.1 1 40.13 odd 4
2880.2.a.w.1.1 1 120.83 odd 4
2880.2.a.bd.1.1 1 120.53 even 4
3240.2.q.d.1081.1 2 45.43 odd 12
3240.2.q.d.2161.1 2 45.13 odd 12
3240.2.q.n.1081.1 2 45.38 even 12
3240.2.q.n.2161.1 2 45.23 even 12
3600.2.a.bd.1.1 1 60.47 odd 4
3600.2.a.bh.1.1 1 20.7 even 4
3600.2.f.g.2449.1 2 12.11 even 2
3600.2.f.g.2449.2 2 60.59 even 2
3600.2.f.q.2449.1 2 4.3 odd 2
3600.2.f.q.2449.2 2 20.19 odd 2