Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1520,2,Mod(1,1520)] 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("1520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.1372611072\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1520.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.732051 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.46410 q^{9} -3.46410 q^{11} +0.732051 q^{13} +0.732051 q^{15} +3.46410 q^{17} -1.00000 q^{19} -1.46410 q^{21} -3.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} -5.46410 q^{31} -2.53590 q^{33} -2.00000 q^{35} +3.26795 q^{37} +0.535898 q^{39} -6.00000 q^{41} -8.92820 q^{43} -2.46410 q^{45} +0.928203 q^{47} -3.00000 q^{49} +2.53590 q^{51} -7.26795 q^{53} -3.46410 q^{55} -0.732051 q^{57} +6.92820 q^{59} -8.39230 q^{61} +4.92820 q^{63} +0.732051 q^{65} -3.26795 q^{67} -2.53590 q^{69} +9.46410 q^{71} -7.46410 q^{73} +0.732051 q^{75} +6.92820 q^{77} +10.9282 q^{79} +4.46410 q^{81} +3.46410 q^{83} +3.46410 q^{85} -2.53590 q^{87} -8.53590 q^{89} -1.46410 q^{91} -4.00000 q^{93} -1.00000 q^{95} +14.5885 q^{97} +8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{21} + 2 q^{25} - 8 q^{27} - 4 q^{31} - 12 q^{33} - 4 q^{35} + 10 q^{37} + 8 q^{39} - 12 q^{41} - 4 q^{43} + 2 q^{45} - 12 q^{47}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.732051 0.422650 0.211325 − 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 − 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 0.732051 0.203034 0.101517 − 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(14\) 0 0
\(15\) 0.732051 0.189015
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.46410 −0.319493
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 − 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 − 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 − 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 0 0
\(33\) −2.53590 −0.441443
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 3.26795 0.537248 0.268624 − 0.963245i \(-0.413431\pi\)
0.268624 + 0.963245i \(0.413431\pi\)
\(38\) 0 0
\(39\) 0.535898 0.0858124
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.92820 −1.36154 −0.680769 − 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) −2.46410 −0.367327
\(46\) 0 0
\(47\) 0.928203 0.135392 0.0676962 − 0.997706i \(-0.478435\pi\)
0.0676962 + 0.997706i \(0.478435\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.53590 0.355097
\(52\) 0 0
\(53\) −7.26795 −0.998330 −0.499165 − 0.866507i \(-0.666360\pi\)
−0.499165 + 0.866507i \(0.666360\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) −0.732051 −0.0969625
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 − 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 − 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 0 0
\(63\) 4.92820 0.620895
\(64\) 0 0
\(65\) 0.732051 0.0907997
\(66\) 0 0
\(67\) −3.26795 −0.399244 −0.199622 − 0.979873i \(-0.563971\pi\)
−0.199622 + 0.979873i \(0.563971\pi\)
\(68\) 0 0
\(69\) −2.53590 −0.305286
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 − 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 0 0
\(73\) −7.46410 −0.873607 −0.436804 − 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) 0 0
\(75\) 0.732051 0.0845299
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 − 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 − 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) 0 0
\(87\) −2.53590 −0.271877
\(88\) 0 0
\(89\) −8.53590 −0.904803 −0.452402 − 0.891814i \(-0.649433\pi\)
−0.452402 + 0.891814i \(0.649433\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 14.5885 1.48123 0.740617 − 0.671928i \(-0.234533\pi\)
0.740617 + 0.671928i \(0.234533\pi\)
\(98\) 0 0
\(99\) 8.53590 0.857890
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 1520.2.a.l.1.2 2
4.3 odd 2 380.2.a.d.1.1 ✓ 2
5.4 even 2 7600.2.a.bf.1.1 2
8.3 odd 2 6080.2.a.z.1.2 2
8.5 even 2 6080.2.a.bj.1.1 2
12.11 even 2 3420.2.a.h.1.1 2
20.3 even 4 1900.2.c.e.1749.2 4
20.7 even 4 1900.2.c.e.1749.3 4
20.19 odd 2 1900.2.a.d.1.2 2
76.75 even 2 7220.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.1 ✓ 2 4.3 odd 2
1520.2.a.l.1.2 2 1.1 even 1 trivial
1900.2.a.d.1.2 2 20.19 odd 2
1900.2.c.e.1749.2 4 20.3 even 4
1900.2.c.e.1749.3 4 20.7 even 4
3420.2.a.h.1.1 2 12.11 even 2
6080.2.a.z.1.2 2 8.3 odd 2
6080.2.a.bj.1.1 2 8.5 even 2
7220.2.a.h.1.2 2 76.75 even 2
7600.2.a.bf.1.1 2 5.4 even 2