Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-4,0,0,0,0,0,-2,0,-2,0,0,0,-4,0,4,0,0,0,4,0,2,0,2,0, -6,0,8,0,-2,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} +1.82843 q^{13} +1.41421 q^{15} -7.65685 q^{17} +3.41421 q^{19} -2.24264 q^{23} +6.65685 q^{25} +2.41421 q^{27} -8.65685 q^{29} +4.00000 q^{31} +0.414214 q^{33} -6.58579 q^{37} -0.757359 q^{39} -2.58579 q^{41} -5.65685 q^{43} +9.65685 q^{45} -6.48528 q^{47} +3.17157 q^{51} -11.8995 q^{53} +3.41421 q^{55} -1.41421 q^{57} +8.41421 q^{59} +6.17157 q^{61} -6.24264 q^{65} -11.2426 q^{67} +0.928932 q^{69} -3.07107 q^{71} -6.58579 q^{73} -2.75736 q^{75} +4.75736 q^{79} +7.48528 q^{81} -16.1421 q^{83} +26.1421 q^{85} +3.58579 q^{87} -4.48528 q^{89} -1.65685 q^{93} -11.6569 q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{33} - 16 q^{37} - 10 q^{39} - 8 q^{41} + 8 q^{45} + 4 q^{47} + 12 q^{51}+ \cdots - 2 q^{97}+O(q^{100}) \) 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
\(3\) −0.414214 −0.239146 −0.119573 − 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 − 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 − 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 3.41421 0.783274 0.391637 − 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.24264 −0.467623 −0.233811 − 0.972282i \(-0.575120\pi\)
−0.233811 + 0.972282i \(0.575120\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 − 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0.414214 0.0721053
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58579 −1.08270 −0.541348 − 0.840798i \(-0.682086\pi\)
−0.541348 + 0.840798i \(0.682086\pi\)
\(38\) 0 0
\(39\) −0.757359 −0.121275
\(40\) 0 0
\(41\) −2.58579 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) 9.65685 1.43956
\(46\) 0 0
\(47\) −6.48528 −0.945976 −0.472988 − 0.881069i \(-0.656825\pi\)
−0.472988 + 0.881069i \(0.656825\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.17157 0.444109
\(52\) 0 0
\(53\) −11.8995 −1.63452 −0.817261 − 0.576268i \(-0.804508\pi\)
−0.817261 + 0.576268i \(0.804508\pi\)
\(54\) 0 0
\(55\) 3.41421 0.460372
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) 0 0
\(59\) 8.41421 1.09544 0.547719 − 0.836663i \(-0.315496\pi\)
0.547719 + 0.836663i \(0.315496\pi\)
\(60\) 0 0
\(61\) 6.17157 0.790189 0.395094 − 0.918640i \(-0.370712\pi\)
0.395094 + 0.918640i \(0.370712\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) −11.2426 −1.37351 −0.686754 − 0.726890i \(-0.740965\pi\)
−0.686754 + 0.726890i \(0.740965\pi\)
\(68\) 0 0
\(69\) 0.928932 0.111830
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 − 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) −6.58579 −0.770808 −0.385404 − 0.922748i \(-0.625938\pi\)
−0.385404 + 0.922748i \(0.625938\pi\)
\(74\) 0 0
\(75\) −2.75736 −0.318392
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.75736 0.535245 0.267622 − 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −16.1421 −1.77183 −0.885915 − 0.463848i \(-0.846468\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 0 0
\(87\) 3.58579 0.384437
\(88\) 0 0
\(89\) −4.48528 −0.475439 −0.237719 − 0.971334i \(-0.576400\pi\)
−0.237719 + 0.971334i \(0.576400\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.65685 −0.171808
\(94\) 0 0
\(95\) −11.6569 −1.19597
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 − 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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 8624.2.a.cc.1.1 2
4.3 odd 2 1078.2.a.t.1.2 2
7.2 even 3 1232.2.q.f.529.2 4
7.4 even 3 1232.2.q.f.177.2 4
7.6 odd 2 8624.2.a.bh.1.2 2
12.11 even 2 9702.2.a.cx.1.2 2
28.3 even 6 1078.2.e.m.177.2 4
28.11 odd 6 154.2.e.e.23.1 ✓ 4
28.19 even 6 1078.2.e.m.67.2 4
28.23 odd 6 154.2.e.e.67.1 yes 4
28.27 even 2 1078.2.a.x.1.1 2
84.11 even 6 1386.2.k.t.793.1 4
84.23 even 6 1386.2.k.t.991.1 4
84.83 odd 2 9702.2.a.ch.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 ✓ 4 28.11 odd 6
154.2.e.e.67.1 yes 4 28.23 odd 6
1078.2.a.t.1.2 2 4.3 odd 2
1078.2.a.x.1.1 2 28.27 even 2
1078.2.e.m.67.2 4 28.19 even 6
1078.2.e.m.177.2 4 28.3 even 6
1232.2.q.f.177.2 4 7.4 even 3
1232.2.q.f.529.2 4 7.2 even 3
1386.2.k.t.793.1 4 84.11 even 6
1386.2.k.t.991.1 4 84.23 even 6
8624.2.a.bh.1.2 2 7.6 odd 2
8624.2.a.cc.1.1 2 1.1 even 1 trivial
9702.2.a.ch.1.1 2 84.83 odd 2
9702.2.a.cx.1.2 2 12.11 even 2