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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,-412] 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: \(106.203438010\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
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 1800.4.f.v.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)
 
\(f(q)\) \(=\) \(q+20.0000i q^{7} +56.0000 q^{11} +86.0000i q^{13} +106.000i q^{17} -4.00000 q^{19} +136.000i q^{23} -206.000 q^{29} -152.000 q^{31} +282.000i q^{37} +246.000 q^{41} -412.000i q^{43} -40.0000i q^{47} -57.0000 q^{49} -126.000i q^{53} +56.0000 q^{59} -2.00000 q^{61} -388.000i q^{67} +672.000 q^{71} -1170.00i q^{73} +1120.00i q^{77} -408.000 q^{79} +668.000i q^{83} +66.0000 q^{89} -1720.00 q^{91} -926.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 112 q^{11} - 8 q^{19} - 412 q^{29} - 304 q^{31} + 492 q^{41} - 114 q^{49} + 112 q^{59} - 4 q^{61} + 1344 q^{71} - 816 q^{79} + 132 q^{89} - 3440 q^{91}+O(q^{100}) \) 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
\(7\) 20.0000i 1.07990i 0.841698 + 0.539949i \(0.181557\pi\)
−0.841698 + 0.539949i \(0.818443\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 56.0000 1.53497 0.767483 − 0.641069i \(-0.221509\pi\)
0.767483 + 0.641069i \(0.221509\pi\)
\(12\) 0 0
\(13\) 86.0000i 1.83478i 0.397992 + 0.917389i \(0.369707\pi\)
−0.397992 + 0.917389i \(0.630293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 106.000i 1.51228i 0.654409 + 0.756140i \(0.272917\pi\)
−0.654409 + 0.756140i \(0.727083\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 − 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 136.000i 1.23295i 0.787373 + 0.616477i \(0.211441\pi\)
−0.787373 + 0.616477i \(0.788559\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −206.000 −1.31908 −0.659539 − 0.751671i \(-0.729248\pi\)
−0.659539 + 0.751671i \(0.729248\pi\)
\(30\) 0 0
\(31\) −152.000 −0.880645 −0.440323 − 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 282.000i 1.25299i 0.779427 + 0.626493i \(0.215510\pi\)
−0.779427 + 0.626493i \(0.784490\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 246.000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 412.000i − 1.46115i −0.682833 − 0.730575i \(-0.739252\pi\)
0.682833 − 0.730575i \(-0.260748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 40.0000i − 0.124140i −0.998072 − 0.0620702i \(-0.980230\pi\)
0.998072 − 0.0620702i \(-0.0197703\pi\)
\(48\) 0 0
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 126.000i − 0.326555i −0.986580 − 0.163278i \(-0.947793\pi\)
0.986580 − 0.163278i \(-0.0522066\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 56.0000 0.123569 0.0617846 − 0.998090i \(-0.480321\pi\)
0.0617846 + 0.998090i \(0.480321\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.00419793 −0.00209897 − 0.999998i \(-0.500668\pi\)
−0.00209897 + 0.999998i \(0.500668\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 388.000i − 0.707489i −0.935342 − 0.353744i \(-0.884908\pi\)
0.935342 − 0.353744i \(-0.115092\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 672.000 1.12326 0.561632 − 0.827387i \(-0.310174\pi\)
0.561632 + 0.827387i \(0.310174\pi\)
\(72\) 0 0
\(73\) − 1170.00i − 1.87586i −0.346818 − 0.937932i \(-0.612738\pi\)
0.346818 − 0.937932i \(-0.387262\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1120.00i 1.65761i
\(78\) 0 0
\(79\) −408.000 −0.581058 −0.290529 − 0.956866i \(-0.593831\pi\)
−0.290529 + 0.956866i \(0.593831\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 668.000i 0.883404i 0.897162 + 0.441702i \(0.145625\pi\)
−0.897162 + 0.441702i \(0.854375\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 66.0000 0.0786066 0.0393033 − 0.999227i \(-0.487486\pi\)
0.0393033 + 0.999227i \(0.487486\pi\)
\(90\) 0 0
\(91\) −1720.00 −1.98137
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 926.000i − 0.969289i −0.874711 − 0.484645i \(-0.838949\pi\)
0.874711 − 0.484645i \(-0.161051\pi\)
\(98\) 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 1800.4.f.v.649.2 2
3.2 odd 2 600.4.f.a.49.2 2
5.2 odd 4 1800.4.a.f.1.1 1
5.3 odd 4 360.4.a.n.1.1 1
5.4 even 2 inner 1800.4.f.v.649.1 2
12.11 even 2 1200.4.f.t.49.1 2
15.2 even 4 600.4.a.i.1.1 1
15.8 even 4 120.4.a.b.1.1 ✓ 1
15.14 odd 2 600.4.f.a.49.1 2
20.3 even 4 720.4.a.q.1.1 1
60.23 odd 4 240.4.a.g.1.1 1
60.47 odd 4 1200.4.a.p.1.1 1
60.59 even 2 1200.4.f.t.49.2 2
120.53 even 4 960.4.a.bj.1.1 1
120.83 odd 4 960.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 ✓ 1 15.8 even 4
240.4.a.g.1.1 1 60.23 odd 4
360.4.a.n.1.1 1 5.3 odd 4
600.4.a.i.1.1 1 15.2 even 4
600.4.f.a.49.1 2 15.14 odd 2
600.4.f.a.49.2 2 3.2 odd 2
720.4.a.q.1.1 1 20.3 even 4
960.4.a.k.1.1 1 120.83 odd 4
960.4.a.bj.1.1 1 120.53 even 4
1200.4.a.p.1.1 1 60.47 odd 4
1200.4.f.t.49.1 2 12.11 even 2
1200.4.f.t.49.2 2 60.59 even 2
1800.4.a.f.1.1 1 5.2 odd 4
1800.4.f.v.649.1 2 5.4 even 2 inner
1800.4.f.v.649.2 2 1.1 even 1 trivial