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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,4,Mod(49,600)] 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("600.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.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,-18,0,-112,0,0,0,0,0,0,0,-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: \(35.4011460034\)
Analytic rank: \(1\)
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 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 600.49
Dual form 600.4.f.a.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} -56.0000 q^{11} -86.0000i q^{13} +106.000i q^{17} -4.00000 q^{19} -60.0000 q^{21} +136.000i q^{23} +27.0000i q^{27} +206.000 q^{29} -152.000 q^{31} +168.000i q^{33} -282.000i q^{37} -258.000 q^{39} -246.000 q^{41} +412.000i q^{43} -40.0000i q^{47} -57.0000 q^{49} +318.000 q^{51} -126.000i q^{53} +12.0000i q^{57} -56.0000 q^{59} -2.00000 q^{61} +180.000i q^{63} +388.000i q^{67} +408.000 q^{69} -672.000 q^{71} +1170.00i q^{73} +1120.00i q^{77} -408.000 q^{79} +81.0000 q^{81} +668.000i q^{83} -618.000i q^{87} -66.0000 q^{89} -1720.00 q^{91} +456.000i q^{93} +926.000i q^{97} +504.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 112 q^{11} - 8 q^{19} - 120 q^{21} + 412 q^{29} - 304 q^{31} - 516 q^{39} - 492 q^{41} - 114 q^{49} + 636 q^{51} - 112 q^{59} - 4 q^{61} + 816 q^{69} - 1344 q^{71} - 816 q^{79} + 162 q^{81}+ \cdots + 1008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 20.0000i − 1.07990i −0.841698 − 0.539949i \(-0.818443\pi\)
0.841698 − 0.539949i \(-0.181557\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −56.0000 −1.53497 −0.767483 − 0.641069i \(-0.778491\pi\)
−0.767483 + 0.641069i \(0.778491\pi\)
\(12\) 0 0
\(13\) − 86.0000i − 1.83478i −0.397992 − 0.917389i \(-0.630293\pi\)
0.397992 − 0.917389i \(-0.369707\pi\)
\(14\) 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
\(21\) −60.0000 −0.623480
\(22\) 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
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 206.000 1.31908 0.659539 − 0.751671i \(-0.270752\pi\)
0.659539 + 0.751671i \(0.270752\pi\)
\(30\) 0 0
\(31\) −152.000 −0.880645 −0.440323 − 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) 0 0
\(33\) 168.000i 0.886214i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 282.000i − 1.25299i −0.779427 − 0.626493i \(-0.784490\pi\)
0.779427 − 0.626493i \(-0.215510\pi\)
\(38\) 0 0
\(39\) −258.000 −1.05931
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 412.000i 1.46115i 0.682833 + 0.730575i \(0.260748\pi\)
−0.682833 + 0.730575i \(0.739252\pi\)
\(44\) 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
\(51\) 318.000 0.873116
\(52\) 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
\(57\) 12.0000i 0.0278849i
\(58\) 0 0
\(59\) −56.0000 −0.123569 −0.0617846 − 0.998090i \(-0.519679\pi\)
−0.0617846 + 0.998090i \(0.519679\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.00419793 −0.00209897 − 0.999998i \(-0.500668\pi\)
−0.00209897 + 0.999998i \(0.500668\pi\)
\(62\) 0 0
\(63\) 180.000i 0.359966i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 388.000i 0.707489i 0.935342 + 0.353744i \(0.115092\pi\)
−0.935342 + 0.353744i \(0.884908\pi\)
\(68\) 0 0
\(69\) 408.000 0.711847
\(70\) 0 0
\(71\) −672.000 −1.12326 −0.561632 − 0.827387i \(-0.689826\pi\)
−0.561632 + 0.827387i \(0.689826\pi\)
\(72\) 0 0
\(73\) 1170.00i 1.87586i 0.346818 + 0.937932i \(0.387262\pi\)
−0.346818 + 0.937932i \(0.612738\pi\)
\(74\) 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
\(81\) 81.0000 0.111111
\(82\) 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
\(87\) − 618.000i − 0.761570i
\(88\) 0 0
\(89\) −66.0000 −0.0786066 −0.0393033 − 0.999227i \(-0.512514\pi\)
−0.0393033 + 0.999227i \(0.512514\pi\)
\(90\) 0 0
\(91\) −1720.00 −1.98137
\(92\) 0 0
\(93\) 456.000i 0.508441i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 926.000i 0.969289i 0.874711 + 0.484645i \(0.161051\pi\)
−0.874711 + 0.484645i \(0.838949\pi\)
\(98\) 0 0
\(99\) 504.000 0.511656
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 600.4.f.a.49.1 2
3.2 odd 2 1800.4.f.v.649.1 2
4.3 odd 2 1200.4.f.t.49.2 2
5.2 odd 4 120.4.a.b.1.1 ✓ 1
5.3 odd 4 600.4.a.i.1.1 1
5.4 even 2 inner 600.4.f.a.49.2 2
15.2 even 4 360.4.a.n.1.1 1
15.8 even 4 1800.4.a.f.1.1 1
15.14 odd 2 1800.4.f.v.649.2 2
20.3 even 4 1200.4.a.p.1.1 1
20.7 even 4 240.4.a.g.1.1 1
20.19 odd 2 1200.4.f.t.49.1 2
40.27 even 4 960.4.a.k.1.1 1
40.37 odd 4 960.4.a.bj.1.1 1
60.47 odd 4 720.4.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 ✓ 1 5.2 odd 4
240.4.a.g.1.1 1 20.7 even 4
360.4.a.n.1.1 1 15.2 even 4
600.4.a.i.1.1 1 5.3 odd 4
600.4.f.a.49.1 2 1.1 even 1 trivial
600.4.f.a.49.2 2 5.4 even 2 inner
720.4.a.q.1.1 1 60.47 odd 4
960.4.a.k.1.1 1 40.27 even 4
960.4.a.bj.1.1 1 40.37 odd 4
1200.4.a.p.1.1 1 20.3 even 4
1200.4.f.t.49.1 2 20.19 odd 2
1200.4.f.t.49.2 2 4.3 odd 2
1800.4.a.f.1.1 1 15.8 even 4
1800.4.f.v.649.1 2 3.2 odd 2
1800.4.f.v.649.2 2 15.14 odd 2