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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.73205 q^{3} +1.00000 q^{5} -2.00000 q^{7} +4.46410 q^{9} +3.46410 q^{11} -2.73205 q^{13} -2.73205 q^{15} -3.46410 q^{17} -1.00000 q^{19} +5.46410 q^{21} +3.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} +1.46410 q^{31} -9.46410 q^{33} -2.00000 q^{35} +6.73205 q^{37} +7.46410 q^{39} -6.00000 q^{41} +4.92820 q^{43} +4.46410 q^{45} -12.9282 q^{47} -3.00000 q^{49} +9.46410 q^{51} -10.7321 q^{53} +3.46410 q^{55} +2.73205 q^{57} -6.92820 q^{59} +12.3923 q^{61} -8.92820 q^{63} -2.73205 q^{65} -6.73205 q^{67} -9.46410 q^{69} +2.53590 q^{71} -0.535898 q^{73} -2.73205 q^{75} -6.92820 q^{77} -2.92820 q^{79} -2.46410 q^{81} -3.46410 q^{83} -3.46410 q^{85} -9.46410 q^{87} -15.4641 q^{89} +5.46410 q^{91} -4.00000 q^{93} -1.00000 q^{95} -16.5885 q^{97} +15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{21} + 2 q^{25} - 8 q^{27} - 4 q^{31} - 12 q^{33} - 4 q^{35} + 10 q^{37} + 8 q^{39} - 12 q^{41} - 4 q^{43} + 2 q^{45} - 12 q^{47}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) −2.73205 −1.57735 −0.788675 − 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 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
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −2.73205 −0.757735 −0.378867 − 0.925451i \(-0.623686\pi\)
−0.378867 + 0.925451i \(0.623686\pi\)
\(14\) 0 0
\(15\) −2.73205 −0.705412
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 5.46410 1.19236
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 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 \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 1.46410 0.262960 0.131480 − 0.991319i \(-0.458027\pi\)
0.131480 + 0.991319i \(0.458027\pi\)
\(32\) 0 0
\(33\) −9.46410 −1.64749
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.73205 1.10674 0.553371 − 0.832935i \(-0.313341\pi\)
0.553371 + 0.832935i \(0.313341\pi\)
\(38\) 0 0
\(39\) 7.46410 1.19521
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 − 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 4.46410 0.665469
\(46\) 0 0
\(47\) −12.9282 −1.88577 −0.942886 − 0.333115i \(-0.891900\pi\)
−0.942886 + 0.333115i \(0.891900\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 9.46410 1.32524
\(52\) 0 0
\(53\) −10.7321 −1.47416 −0.737080 − 0.675805i \(-0.763796\pi\)
−0.737080 + 0.675805i \(0.763796\pi\)
\(54\) 0 0
\(55\) 3.46410 0.467099
\(56\) 0 0
\(57\) 2.73205 0.361869
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 − 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 − 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0 0
\(63\) −8.92820 −1.12485
\(64\) 0 0
\(65\) −2.73205 −0.338869
\(66\) 0 0
\(67\) −6.73205 −0.822451 −0.411225 − 0.911534i \(-0.634899\pi\)
−0.411225 + 0.911534i \(0.634899\pi\)
\(68\) 0 0
\(69\) −9.46410 −1.13934
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −0.535898 −0.0627222 −0.0313611 − 0.999508i \(-0.509984\pi\)
−0.0313611 + 0.999508i \(0.509984\pi\)
\(74\) 0 0
\(75\) −2.73205 −0.315470
\(76\) 0 0
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) −2.92820 −0.329449 −0.164724 − 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 − 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) −9.46410 −1.01466
\(88\) 0 0
\(89\) −15.4641 −1.63919 −0.819596 − 0.572942i \(-0.805802\pi\)
−0.819596 + 0.572942i \(0.805802\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −16.5885 −1.68430 −0.842151 − 0.539241i \(-0.818711\pi\)
−0.842151 + 0.539241i \(0.818711\pi\)
\(98\) 0 0
\(99\) 15.4641 1.55420
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.1.1 2
4.3 odd 2 380.2.a.d.1.2 ✓ 2
5.4 even 2 7600.2.a.bf.1.2 2
8.3 odd 2 6080.2.a.z.1.1 2
8.5 even 2 6080.2.a.bj.1.2 2
12.11 even 2 3420.2.a.h.1.2 2
20.3 even 4 1900.2.c.e.1749.4 4
20.7 even 4 1900.2.c.e.1749.1 4
20.19 odd 2 1900.2.a.d.1.1 2
76.75 even 2 7220.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.2 ✓ 2 4.3 odd 2
1520.2.a.l.1.1 2 1.1 even 1 trivial
1900.2.a.d.1.1 2 20.19 odd 2
1900.2.c.e.1749.1 4 20.7 even 4
1900.2.c.e.1749.4 4 20.3 even 4
3420.2.a.h.1.2 2 12.11 even 2
6080.2.a.z.1.1 2 8.3 odd 2
6080.2.a.bj.1.2 2 8.5 even 2
7220.2.a.h.1.1 2 76.75 even 2
7600.2.a.bf.1.2 2 5.4 even 2