Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,6,0,-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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.53744\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.901180 q^{3} +2.78819 q^{5} +2.28669 q^{7} -2.18787 q^{9} -0.788195 q^{11} +5.18787 q^{13} -2.51267 q^{15} +4.28669 q^{17} -3.07489 q^{19} -2.06072 q^{21} -1.00000 q^{23} +2.77403 q^{25} +4.67521 q^{27} +3.61149 q^{29} -9.24860 q^{31} +0.710305 q^{33} +6.37575 q^{35} +2.78819 q^{37} -4.67521 q^{39} -2.76426 q^{41} +12.8772 q^{43} -6.10022 q^{45} +7.05096 q^{47} -1.77103 q^{49} -3.86308 q^{51} +0.727471 q^{53} -2.19764 q^{55} +2.77103 q^{57} +12.1781 q^{59} +3.27253 q^{61} -5.00300 q^{63} +14.4648 q^{65} -3.78519 q^{67} +0.901180 q^{69} -3.89818 q^{71} +8.56662 q^{73} -2.49990 q^{75} -1.80236 q^{77} +7.95214 q^{79} +2.35042 q^{81} -1.01417 q^{83} +11.9521 q^{85} -3.25460 q^{87} -4.37575 q^{89} +11.8631 q^{91} +8.33465 q^{93} -8.57339 q^{95} -17.4335 q^{97} +1.72447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 10 q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{23} + 12 q^{25} + 14 q^{27} + 6 q^{29} - 6 q^{31} - 12 q^{35} + 6 q^{37} - 14 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) −0.901180 −0.520297 −0.260148 − 0.965569i \(-0.583771\pi\)
−0.260148 + 0.965569i \(0.583771\pi\)
\(4\) 0 0
\(5\) 2.78819 1.24692 0.623459 − 0.781856i \(-0.285727\pi\)
0.623459 + 0.781856i \(0.285727\pi\)
\(6\) 0 0
\(7\) 2.28669 0.864289 0.432145 − 0.901804i \(-0.357757\pi\)
0.432145 + 0.901804i \(0.357757\pi\)
\(8\) 0 0
\(9\) −2.18787 −0.729291
\(10\) 0 0
\(11\) −0.788195 −0.237650 −0.118825 − 0.992915i \(-0.537913\pi\)
−0.118825 + 0.992915i \(0.537913\pi\)
\(12\) 0 0
\(13\) 5.18787 1.43886 0.719429 − 0.694566i \(-0.244404\pi\)
0.719429 + 0.694566i \(0.244404\pi\)
\(14\) 0 0
\(15\) −2.51267 −0.648767
\(16\) 0 0
\(17\) 4.28669 1.03968 0.519838 − 0.854265i \(-0.325992\pi\)
0.519838 + 0.854265i \(0.325992\pi\)
\(18\) 0 0
\(19\) −3.07489 −0.705428 −0.352714 − 0.935731i \(-0.614741\pi\)
−0.352714 + 0.935731i \(0.614741\pi\)
\(20\) 0 0
\(21\) −2.06072 −0.449687
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.77403 0.554806
\(26\) 0 0
\(27\) 4.67521 0.899744
\(28\) 0 0
\(29\) 3.61149 0.670636 0.335318 − 0.942105i \(-0.391156\pi\)
0.335318 + 0.942105i \(0.391156\pi\)
\(30\) 0 0
\(31\) −9.24860 −1.66110 −0.830549 − 0.556946i \(-0.811973\pi\)
−0.830549 + 0.556946i \(0.811973\pi\)
\(32\) 0 0
\(33\) 0.710305 0.123648
\(34\) 0 0
\(35\) 6.37575 1.07770
\(36\) 0 0
\(37\) 2.78819 0.458376 0.229188 − 0.973382i \(-0.426393\pi\)
0.229188 + 0.973382i \(0.426393\pi\)
\(38\) 0 0
\(39\) −4.67521 −0.748633
\(40\) 0 0
\(41\) −2.76426 −0.431705 −0.215853 − 0.976426i \(-0.569253\pi\)
−0.215853 + 0.976426i \(0.569253\pi\)
\(42\) 0 0
\(43\) 12.8772 1.96376 0.981881 − 0.189498i \(-0.0606862\pi\)
0.981881 + 0.189498i \(0.0606862\pi\)
\(44\) 0 0
\(45\) −6.10022 −0.909367
\(46\) 0 0
\(47\) 7.05096 1.02849 0.514244 − 0.857644i \(-0.328073\pi\)
0.514244 + 0.857644i \(0.328073\pi\)
\(48\) 0 0
\(49\) −1.77103 −0.253004
\(50\) 0 0
\(51\) −3.86308 −0.540940
\(52\) 0 0
\(53\) 0.727471 0.0999258 0.0499629 − 0.998751i \(-0.484090\pi\)
0.0499629 + 0.998751i \(0.484090\pi\)
\(54\) 0 0
\(55\) −2.19764 −0.296330
\(56\) 0 0
\(57\) 2.77103 0.367032
\(58\) 0 0
\(59\) 12.1781 1.58545 0.792727 − 0.609576i \(-0.208660\pi\)
0.792727 + 0.609576i \(0.208660\pi\)
\(60\) 0 0
\(61\) 3.27253 0.419004 0.209502 − 0.977808i \(-0.432816\pi\)
0.209502 + 0.977808i \(0.432816\pi\)
\(62\) 0 0
\(63\) −5.00300 −0.630319
\(64\) 0 0
\(65\) 14.4648 1.79414
\(66\) 0 0
\(67\) −3.78519 −0.462435 −0.231218 − 0.972902i \(-0.574271\pi\)
−0.231218 + 0.972902i \(0.574271\pi\)
\(68\) 0 0
\(69\) 0.901180 0.108489
\(70\) 0 0
\(71\) −3.89818 −0.462629 −0.231314 − 0.972879i \(-0.574303\pi\)
−0.231314 + 0.972879i \(0.574303\pi\)
\(72\) 0 0
\(73\) 8.56662 1.00265 0.501324 − 0.865260i \(-0.332847\pi\)
0.501324 + 0.865260i \(0.332847\pi\)
\(74\) 0 0
\(75\) −2.49990 −0.288664
\(76\) 0 0
\(77\) −1.80236 −0.205398
\(78\) 0 0
\(79\) 7.95214 0.894685 0.447343 − 0.894363i \(-0.352370\pi\)
0.447343 + 0.894363i \(0.352370\pi\)
\(80\) 0 0
\(81\) 2.35042 0.261158
\(82\) 0 0
\(83\) −1.01417 −0.111319 −0.0556596 − 0.998450i \(-0.517726\pi\)
−0.0556596 + 0.998450i \(0.517726\pi\)
\(84\) 0 0
\(85\) 11.9521 1.29639
\(86\) 0 0
\(87\) −3.25460 −0.348930
\(88\) 0 0
\(89\) −4.37575 −0.463828 −0.231914 − 0.972736i \(-0.574499\pi\)
−0.231914 + 0.972736i \(0.574499\pi\)
\(90\) 0 0
\(91\) 11.8631 1.24359
\(92\) 0 0
\(93\) 8.33465 0.864263
\(94\) 0 0
\(95\) −8.57339 −0.879611
\(96\) 0 0
\(97\) −17.4335 −1.77010 −0.885050 − 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(98\) 0 0
\(99\) 1.72447 0.173316
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 736.2.a.h.1.2 yes 4
3.2 odd 2 6624.2.a.be.1.2 4
4.3 odd 2 736.2.a.g.1.3 ✓ 4
8.3 odd 2 1472.2.a.z.1.2 4
8.5 even 2 1472.2.a.y.1.3 4
12.11 even 2 6624.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.g.1.3 ✓ 4 4.3 odd 2
736.2.a.h.1.2 yes 4 1.1 even 1 trivial
1472.2.a.y.1.3 4 8.5 even 2
1472.2.a.z.1.2 4 8.3 odd 2
6624.2.a.be.1.2 4 3.2 odd 2
6624.2.a.bf.1.2 4 12.11 even 2