Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8664,2,Mod(1,8664)] 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("8664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-1,0,-3] 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: \(69.1823883112\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 8664.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +4.70156 q^{11} -6.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} -4.70156 q^{21} +4.00000 q^{23} +2.29844 q^{25} +1.00000 q^{27} -2.00000 q^{29} -9.40312 q^{31} +4.70156 q^{33} -12.7016 q^{35} +3.40312 q^{37} -6.00000 q^{39} +3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} -0.701562 q^{47} +15.1047 q^{49} -2.70156 q^{51} +6.00000 q^{53} +12.7016 q^{55} +4.00000 q^{59} +1.29844 q^{61} -4.70156 q^{63} -16.2094 q^{65} +12.0000 q^{67} +4.00000 q^{69} +6.70156 q^{73} +2.29844 q^{75} -22.1047 q^{77} +10.8062 q^{79} +1.00000 q^{81} -10.8062 q^{83} -7.29844 q^{85} -2.00000 q^{87} +12.8062 q^{89} +28.2094 q^{91} -9.40312 q^{93} +6.00000 q^{97} +4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 12 q^{13} - q^{15} + q^{17} - 3 q^{21} + 8 q^{23} + 11 q^{25} + 2 q^{27} - 4 q^{29} - 6 q^{31} + 3 q^{33} - 19 q^{35} - 6 q^{37} - 12 q^{39} - 6 q^{41}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(5\) 2.70156 1.20818 0.604088 − 0.796918i \(-0.293538\pi\)
0.604088 + 0.796918i \(0.293538\pi\)
\(6\) 0 0
\(7\) −4.70156 −1.77702 −0.888512 − 0.458854i \(-0.848260\pi\)
−0.888512 + 0.458854i \(0.848260\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.70156 1.41757 0.708787 − 0.705422i \(-0.249243\pi\)
0.708787 + 0.705422i \(0.249243\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 2.70156 0.697540
\(16\) 0 0
\(17\) −2.70156 −0.655225 −0.327613 − 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −4.70156 −1.02596
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 2.29844 0.459688
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −9.40312 −1.68885 −0.844425 − 0.535673i \(-0.820058\pi\)
−0.844425 + 0.535673i \(0.820058\pi\)
\(32\) 0 0
\(33\) 4.70156 0.818437
\(34\) 0 0
\(35\) −12.7016 −2.14696
\(36\) 0 0
\(37\) 3.40312 0.559470 0.279735 − 0.960077i \(-0.409753\pi\)
0.279735 + 0.960077i \(0.409753\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 3.40312 0.531479 0.265739 − 0.964045i \(-0.414384\pi\)
0.265739 + 0.964045i \(0.414384\pi\)
\(42\) 0 0
\(43\) 10.1047 1.54095 0.770475 − 0.637470i \(-0.220019\pi\)
0.770475 + 0.637470i \(0.220019\pi\)
\(44\) 0 0
\(45\) 2.70156 0.402725
\(46\) 0 0
\(47\) −0.701562 −0.102333 −0.0511667 − 0.998690i \(-0.516294\pi\)
−0.0511667 + 0.998690i \(0.516294\pi\)
\(48\) 0 0
\(49\) 15.1047 2.15781
\(50\) 0 0
\(51\) −2.70156 −0.378294
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 12.7016 1.71268
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 1.29844 0.166248 0.0831240 − 0.996539i \(-0.473510\pi\)
0.0831240 + 0.996539i \(0.473510\pi\)
\(62\) 0 0
\(63\) −4.70156 −0.592341
\(64\) 0 0
\(65\) −16.2094 −2.01053
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.70156 0.784359 0.392179 − 0.919889i \(-0.371721\pi\)
0.392179 + 0.919889i \(0.371721\pi\)
\(74\) 0 0
\(75\) 2.29844 0.265401
\(76\) 0 0
\(77\) −22.1047 −2.51906
\(78\) 0 0
\(79\) 10.8062 1.21580 0.607899 − 0.794014i \(-0.292013\pi\)
0.607899 + 0.794014i \(0.292013\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.8062 −1.18614 −0.593070 − 0.805151i \(-0.702084\pi\)
−0.593070 + 0.805151i \(0.702084\pi\)
\(84\) 0 0
\(85\) −7.29844 −0.791627
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 12.8062 1.35746 0.678730 − 0.734388i \(-0.262531\pi\)
0.678730 + 0.734388i \(0.262531\pi\)
\(90\) 0 0
\(91\) 28.2094 2.95715
\(92\) 0 0
\(93\) −9.40312 −0.975059
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 4.70156 0.472525
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 8664.2.a.v.1.2 2
19.18 odd 2 456.2.a.e.1.2 ✓ 2
57.56 even 2 1368.2.a.l.1.1 2
76.75 even 2 912.2.a.o.1.2 2
152.37 odd 2 3648.2.a.bs.1.1 2
152.75 even 2 3648.2.a.bn.1.1 2
228.227 odd 2 2736.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.e.1.2 ✓ 2 19.18 odd 2
912.2.a.o.1.2 2 76.75 even 2
1368.2.a.l.1.1 2 57.56 even 2
2736.2.a.bb.1.1 2 228.227 odd 2
3648.2.a.bn.1.1 2 152.75 even 2
3648.2.a.bs.1.1 2 152.37 odd 2
8664.2.a.v.1.2 2 1.1 even 1 trivial