Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7728,2,Mod(1,7728)] 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("7728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-5,0,-2,0,2,0,2,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.47214 q^{11} -4.61803 q^{13} +3.61803 q^{15} -1.00000 q^{17} +3.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} +1.47214 q^{29} +8.23607 q^{31} +3.47214 q^{33} +3.61803 q^{35} -7.47214 q^{37} +4.61803 q^{39} +8.70820 q^{41} +3.09017 q^{43} -3.61803 q^{45} +11.7082 q^{47} +1.00000 q^{49} +1.00000 q^{51} -5.61803 q^{53} +12.5623 q^{55} -3.00000 q^{57} +12.8541 q^{59} -10.7984 q^{61} -1.00000 q^{63} +16.7082 q^{65} -10.8541 q^{67} -1.00000 q^{69} +15.0902 q^{71} -10.7082 q^{73} -8.09017 q^{75} +3.47214 q^{77} -9.47214 q^{79} +1.00000 q^{81} +3.00000 q^{83} +3.61803 q^{85} -1.47214 q^{87} +9.85410 q^{89} +4.61803 q^{91} -8.23607 q^{93} -10.8541 q^{95} +5.00000 q^{97} -3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 7 q^{13} + 5 q^{15} - 2 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + 7 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.47214 −1.04689 −0.523444 − 0.852060i \(-0.675353\pi\)
−0.523444 + 0.852060i \(0.675353\pi\)
\(12\) 0 0
\(13\) −4.61803 −1.28081 −0.640406 − 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(14\) 0 0
\(15\) 3.61803 0.934172
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 8.09017 1.61803
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.47214 0.273369 0.136684 − 0.990615i \(-0.456355\pi\)
0.136684 + 0.990615i \(0.456355\pi\)
\(30\) 0 0
\(31\) 8.23607 1.47924 0.739621 − 0.673024i \(-0.235005\pi\)
0.739621 + 0.673024i \(0.235005\pi\)
\(32\) 0 0
\(33\) 3.47214 0.604421
\(34\) 0 0
\(35\) 3.61803 0.611559
\(36\) 0 0
\(37\) −7.47214 −1.22841 −0.614206 − 0.789146i \(-0.710524\pi\)
−0.614206 + 0.789146i \(0.710524\pi\)
\(38\) 0 0
\(39\) 4.61803 0.739477
\(40\) 0 0
\(41\) 8.70820 1.35999 0.679996 − 0.733215i \(-0.261981\pi\)
0.679996 + 0.733215i \(0.261981\pi\)
\(42\) 0 0
\(43\) 3.09017 0.471246 0.235623 − 0.971844i \(-0.424287\pi\)
0.235623 + 0.971844i \(0.424287\pi\)
\(44\) 0 0
\(45\) −3.61803 −0.539345
\(46\) 0 0
\(47\) 11.7082 1.70782 0.853909 − 0.520423i \(-0.174226\pi\)
0.853909 + 0.520423i \(0.174226\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −5.61803 −0.771696 −0.385848 − 0.922562i \(-0.626091\pi\)
−0.385848 + 0.922562i \(0.626091\pi\)
\(54\) 0 0
\(55\) 12.5623 1.69390
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 12.8541 1.67346 0.836731 − 0.547614i \(-0.184464\pi\)
0.836731 + 0.547614i \(0.184464\pi\)
\(60\) 0 0
\(61\) −10.7984 −1.38259 −0.691295 − 0.722573i \(-0.742960\pi\)
−0.691295 + 0.722573i \(0.742960\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 16.7082 2.07240
\(66\) 0 0
\(67\) −10.8541 −1.32604 −0.663020 − 0.748602i \(-0.730725\pi\)
−0.663020 + 0.748602i \(0.730725\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 15.0902 1.79087 0.895437 − 0.445189i \(-0.146863\pi\)
0.895437 + 0.445189i \(0.146863\pi\)
\(72\) 0 0
\(73\) −10.7082 −1.25330 −0.626650 − 0.779301i \(-0.715575\pi\)
−0.626650 + 0.779301i \(0.715575\pi\)
\(74\) 0 0
\(75\) −8.09017 −0.934172
\(76\) 0 0
\(77\) 3.47214 0.395687
\(78\) 0 0
\(79\) −9.47214 −1.06570 −0.532849 − 0.846210i \(-0.678879\pi\)
−0.532849 + 0.846210i \(0.678879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 3.61803 0.392431
\(86\) 0 0
\(87\) −1.47214 −0.157830
\(88\) 0 0
\(89\) 9.85410 1.04453 0.522266 − 0.852782i \(-0.325087\pi\)
0.522266 + 0.852782i \(0.325087\pi\)
\(90\) 0 0
\(91\) 4.61803 0.484102
\(92\) 0 0
\(93\) −8.23607 −0.854040
\(94\) 0 0
\(95\) −10.8541 −1.11361
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) −3.47214 −0.348963
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 7728.2.a.v.1.1 2
4.3 odd 2 483.2.a.c.1.1 ✓ 2
12.11 even 2 1449.2.a.k.1.2 2
28.27 even 2 3381.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.c.1.1 ✓ 2 4.3 odd 2
1449.2.a.k.1.2 2 12.11 even 2
3381.2.a.n.1.1 2 28.27 even 2
7728.2.a.v.1.1 2 1.1 even 1 trivial