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 = [3,0,4,0,-2,0,-2] 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: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) 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)
 
\(f(q)\) \(=\) \(q+1.47068 q^{3} +2.24914 q^{5} +2.24914 q^{7} -0.837090 q^{9} -0.249141 q^{11} -2.77846 q^{13} +3.30777 q^{15} +4.24914 q^{17} +8.61555 q^{19} +3.30777 q^{21} +1.00000 q^{23} +0.0586332 q^{25} -5.64315 q^{27} -4.66119 q^{29} -4.08623 q^{31} -0.366407 q^{33} +5.05863 q^{35} -8.86469 q^{37} -4.08623 q^{39} -1.22154 q^{41} +7.55691 q^{43} -1.88273 q^{45} -1.96896 q^{47} -1.94137 q^{49} +6.24914 q^{51} +3.43965 q^{53} -0.560352 q^{55} +12.6707 q^{57} +12.4983 q^{59} +3.67418 q^{61} -1.88273 q^{63} -6.24914 q^{65} +10.6302 q^{67} +1.47068 q^{69} -2.52932 q^{71} -10.2181 q^{73} +0.0862308 q^{75} -0.560352 q^{77} -4.82410 q^{79} -5.78801 q^{81} -3.68879 q^{83} +9.55691 q^{85} -6.85514 q^{87} -11.3224 q^{89} -6.24914 q^{91} -6.00955 q^{93} +19.3776 q^{95} +2.86469 q^{97} +0.208553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 5 q^{9} + 8 q^{11} + 2 q^{15} + 4 q^{17} + 10 q^{19} + 2 q^{21} + 3 q^{23} + q^{25} + 16 q^{27} - 4 q^{29} + 4 q^{31} + 6 q^{33} + 16 q^{35} - 2 q^{37} + 4 q^{39} - 12 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 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
\(3\) 1.47068 0.849099 0.424550 − 0.905405i \(-0.360432\pi\)
0.424550 + 0.905405i \(0.360432\pi\)
\(4\) 0 0
\(5\) 2.24914 1.00585 0.502923 − 0.864331i \(-0.332258\pi\)
0.502923 + 0.864331i \(0.332258\pi\)
\(6\) 0 0
\(7\) 2.24914 0.850095 0.425048 − 0.905171i \(-0.360257\pi\)
0.425048 + 0.905171i \(0.360257\pi\)
\(8\) 0 0
\(9\) −0.837090 −0.279030
\(10\) 0 0
\(11\) −0.249141 −0.0751187 −0.0375593 − 0.999294i \(-0.511958\pi\)
−0.0375593 + 0.999294i \(0.511958\pi\)
\(12\) 0 0
\(13\) −2.77846 −0.770605 −0.385303 − 0.922790i \(-0.625903\pi\)
−0.385303 + 0.922790i \(0.625903\pi\)
\(14\) 0 0
\(15\) 3.30777 0.854063
\(16\) 0 0
\(17\) 4.24914 1.03057 0.515284 − 0.857019i \(-0.327686\pi\)
0.515284 + 0.857019i \(0.327686\pi\)
\(18\) 0 0
\(19\) 8.61555 1.97654 0.988271 − 0.152710i \(-0.0488000\pi\)
0.988271 + 0.152710i \(0.0488000\pi\)
\(20\) 0 0
\(21\) 3.30777 0.721815
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0.0586332 0.0117266
\(26\) 0 0
\(27\) −5.64315 −1.08602
\(28\) 0 0
\(29\) −4.66119 −0.865561 −0.432781 − 0.901499i \(-0.642468\pi\)
−0.432781 + 0.901499i \(0.642468\pi\)
\(30\) 0 0
\(31\) −4.08623 −0.733909 −0.366954 − 0.930239i \(-0.619599\pi\)
−0.366954 + 0.930239i \(0.619599\pi\)
\(32\) 0 0
\(33\) −0.366407 −0.0637832
\(34\) 0 0
\(35\) 5.05863 0.855065
\(36\) 0 0
\(37\) −8.86469 −1.45735 −0.728673 − 0.684862i \(-0.759863\pi\)
−0.728673 + 0.684862i \(0.759863\pi\)
\(38\) 0 0
\(39\) −4.08623 −0.654321
\(40\) 0 0
\(41\) −1.22154 −0.190773 −0.0953865 − 0.995440i \(-0.530409\pi\)
−0.0953865 + 0.995440i \(0.530409\pi\)
\(42\) 0 0
\(43\) 7.55691 1.15242 0.576209 − 0.817302i \(-0.304531\pi\)
0.576209 + 0.817302i \(0.304531\pi\)
\(44\) 0 0
\(45\) −1.88273 −0.280661
\(46\) 0 0
\(47\) −1.96896 −0.287203 −0.143601 − 0.989636i \(-0.545868\pi\)
−0.143601 + 0.989636i \(0.545868\pi\)
\(48\) 0 0
\(49\) −1.94137 −0.277338
\(50\) 0 0
\(51\) 6.24914 0.875055
\(52\) 0 0
\(53\) 3.43965 0.472472 0.236236 − 0.971696i \(-0.424086\pi\)
0.236236 + 0.971696i \(0.424086\pi\)
\(54\) 0 0
\(55\) −0.560352 −0.0755579
\(56\) 0 0
\(57\) 12.6707 1.67828
\(58\) 0 0
\(59\) 12.4983 1.62714 0.813569 − 0.581469i \(-0.197522\pi\)
0.813569 + 0.581469i \(0.197522\pi\)
\(60\) 0 0
\(61\) 3.67418 0.470431 0.235215 − 0.971943i \(-0.424420\pi\)
0.235215 + 0.971943i \(0.424420\pi\)
\(62\) 0 0
\(63\) −1.88273 −0.237202
\(64\) 0 0
\(65\) −6.24914 −0.775110
\(66\) 0 0
\(67\) 10.6302 1.29868 0.649340 − 0.760498i \(-0.275045\pi\)
0.649340 + 0.760498i \(0.275045\pi\)
\(68\) 0 0
\(69\) 1.47068 0.177049
\(70\) 0 0
\(71\) −2.52932 −0.300175 −0.150087 − 0.988673i \(-0.547955\pi\)
−0.150087 + 0.988673i \(0.547955\pi\)
\(72\) 0 0
\(73\) −10.2181 −1.19594 −0.597969 − 0.801519i \(-0.704026\pi\)
−0.597969 + 0.801519i \(0.704026\pi\)
\(74\) 0 0
\(75\) 0.0862308 0.00995708
\(76\) 0 0
\(77\) −0.560352 −0.0638580
\(78\) 0 0
\(79\) −4.82410 −0.542754 −0.271377 − 0.962473i \(-0.587479\pi\)
−0.271377 + 0.962473i \(0.587479\pi\)
\(80\) 0 0
\(81\) −5.78801 −0.643112
\(82\) 0 0
\(83\) −3.68879 −0.404897 −0.202449 − 0.979293i \(-0.564890\pi\)
−0.202449 + 0.979293i \(0.564890\pi\)
\(84\) 0 0
\(85\) 9.55691 1.03659
\(86\) 0 0
\(87\) −6.85514 −0.734948
\(88\) 0 0
\(89\) −11.3224 −1.20017 −0.600085 − 0.799936i \(-0.704867\pi\)
−0.600085 + 0.799936i \(0.704867\pi\)
\(90\) 0 0
\(91\) −6.24914 −0.655088
\(92\) 0 0
\(93\) −6.00955 −0.623162
\(94\) 0 0
\(95\) 19.3776 1.98810
\(96\) 0 0
\(97\) 2.86469 0.290865 0.145432 − 0.989368i \(-0.453543\pi\)
0.145432 + 0.989368i \(0.453543\pi\)
\(98\) 0 0
\(99\) 0.208553 0.0209604
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.f.1.2 yes 3
3.2 odd 2 6624.2.a.y.1.1 3
4.3 odd 2 736.2.a.e.1.2 ✓ 3
8.3 odd 2 1472.2.a.x.1.2 3
8.5 even 2 1472.2.a.w.1.2 3
12.11 even 2 6624.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.e.1.2 ✓ 3 4.3 odd 2
736.2.a.f.1.2 yes 3 1.1 even 1 trivial
1472.2.a.w.1.2 3 8.5 even 2
1472.2.a.x.1.2 3 8.3 odd 2
6624.2.a.y.1.1 3 3.2 odd 2
6624.2.a.z.1.1 3 12.11 even 2