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 1.1
Root \(-1.89761\) 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)
 
\(f(q)\) \(=\) \(q-2.49854 q^{3} +0.946044 q^{5} -4.74127 q^{7} +3.24272 q^{9} +1.05396 q^{11} -0.242724 q^{13} -2.36373 q^{15} -2.74127 q^{17} +5.79522 q^{19} +11.8463 q^{21} -1.00000 q^{23} -4.10500 q^{25} -0.606457 q^{27} +1.86519 q^{29} +10.0890 q^{31} -2.63336 q^{33} -4.48545 q^{35} +0.946044 q^{37} +0.606457 q^{39} +6.35064 q^{41} +7.20186 q^{43} +3.06776 q^{45} -9.09190 q^{47} +15.4796 q^{49} +6.84918 q^{51} +12.7923 q^{53} +0.997089 q^{55} -14.4796 q^{57} +4.51164 q^{59} -8.79231 q^{61} -15.3746 q^{63} -0.229628 q^{65} +8.42858 q^{67} +2.49854 q^{69} +4.87608 q^{71} +2.64645 q^{73} +10.2565 q^{75} -4.99709 q^{77} -6.59336 q^{79} -8.21291 q^{81} -6.05104 q^{83} -2.59336 q^{85} -4.66026 q^{87} +6.48545 q^{89} +1.15082 q^{91} -25.2078 q^{93} +5.48254 q^{95} +17.7063 q^{97} +3.41769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 10 q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{23} + 12 q^{25} + 14 q^{27} + 6 q^{29} - 6 q^{31} - 12 q^{35} + 6 q^{37} - 14 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.49854 −1.44254 −0.721268 − 0.692656i \(-0.756440\pi\)
−0.721268 + 0.692656i \(0.756440\pi\)
\(4\) 0 0
\(5\) 0.946044 0.423084 0.211542 − 0.977369i \(-0.432152\pi\)
0.211542 + 0.977369i \(0.432152\pi\)
\(6\) 0 0
\(7\) −4.74127 −1.79203 −0.896015 − 0.444023i \(-0.853551\pi\)
−0.896015 + 0.444023i \(0.853551\pi\)
\(8\) 0 0
\(9\) 3.24272 1.08091
\(10\) 0 0
\(11\) 1.05396 0.317780 0.158890 − 0.987296i \(-0.449209\pi\)
0.158890 + 0.987296i \(0.449209\pi\)
\(12\) 0 0
\(13\) −0.242724 −0.0673195 −0.0336598 − 0.999433i \(-0.510716\pi\)
−0.0336598 + 0.999433i \(0.510716\pi\)
\(14\) 0 0
\(15\) −2.36373 −0.610313
\(16\) 0 0
\(17\) −2.74127 −0.664855 −0.332428 − 0.943129i \(-0.607868\pi\)
−0.332428 + 0.943129i \(0.607868\pi\)
\(18\) 0 0
\(19\) 5.79522 1.32952 0.664758 − 0.747059i \(-0.268535\pi\)
0.664758 + 0.747059i \(0.268535\pi\)
\(20\) 0 0
\(21\) 11.8463 2.58507
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.10500 −0.821000
\(26\) 0 0
\(27\) −0.606457 −0.116713
\(28\) 0 0
\(29\) 1.86519 0.346357 0.173178 − 0.984890i \(-0.444596\pi\)
0.173178 + 0.984890i \(0.444596\pi\)
\(30\) 0 0
\(31\) 10.0890 1.81204 0.906018 − 0.423238i \(-0.139107\pi\)
0.906018 + 0.423238i \(0.139107\pi\)
\(32\) 0 0
\(33\) −2.63336 −0.458408
\(34\) 0 0
\(35\) −4.48545 −0.758179
\(36\) 0 0
\(37\) 0.946044 0.155529 0.0777643 − 0.996972i \(-0.475222\pi\)
0.0777643 + 0.996972i \(0.475222\pi\)
\(38\) 0 0
\(39\) 0.606457 0.0971108
\(40\) 0 0
\(41\) 6.35064 0.991803 0.495901 − 0.868379i \(-0.334838\pi\)
0.495901 + 0.868379i \(0.334838\pi\)
\(42\) 0 0
\(43\) 7.20186 1.09827 0.549137 − 0.835732i \(-0.314957\pi\)
0.549137 + 0.835732i \(0.314957\pi\)
\(44\) 0 0
\(45\) 3.06776 0.457315
\(46\) 0 0
\(47\) −9.09190 −1.32619 −0.663095 − 0.748535i \(-0.730758\pi\)
−0.663095 + 0.748535i \(0.730758\pi\)
\(48\) 0 0
\(49\) 15.4796 2.21138
\(50\) 0 0
\(51\) 6.84918 0.959077
\(52\) 0 0
\(53\) 12.7923 1.75716 0.878580 − 0.477596i \(-0.158492\pi\)
0.878580 + 0.477596i \(0.158492\pi\)
\(54\) 0 0
\(55\) 0.997089 0.134447
\(56\) 0 0
\(57\) −14.4796 −1.91787
\(58\) 0 0
\(59\) 4.51164 0.587366 0.293683 − 0.955903i \(-0.405119\pi\)
0.293683 + 0.955903i \(0.405119\pi\)
\(60\) 0 0
\(61\) −8.79231 −1.12574 −0.562870 − 0.826545i \(-0.690303\pi\)
−0.562870 + 0.826545i \(0.690303\pi\)
\(62\) 0 0
\(63\) −15.3746 −1.93702
\(64\) 0 0
\(65\) −0.229628 −0.0284818
\(66\) 0 0
\(67\) 8.42858 1.02972 0.514858 − 0.857276i \(-0.327845\pi\)
0.514858 + 0.857276i \(0.327845\pi\)
\(68\) 0 0
\(69\) 2.49854 0.300789
\(70\) 0 0
\(71\) 4.87608 0.578684 0.289342 − 0.957226i \(-0.406563\pi\)
0.289342 + 0.957226i \(0.406563\pi\)
\(72\) 0 0
\(73\) 2.64645 0.309744 0.154872 − 0.987935i \(-0.450504\pi\)
0.154872 + 0.987935i \(0.450504\pi\)
\(74\) 0 0
\(75\) 10.2565 1.18432
\(76\) 0 0
\(77\) −4.99709 −0.569471
\(78\) 0 0
\(79\) −6.59336 −0.741811 −0.370905 − 0.928671i \(-0.620953\pi\)
−0.370905 + 0.928671i \(0.620953\pi\)
\(80\) 0 0
\(81\) −8.21291 −0.912546
\(82\) 0 0
\(83\) −6.05104 −0.664188 −0.332094 − 0.943246i \(-0.607755\pi\)
−0.332094 + 0.943246i \(0.607755\pi\)
\(84\) 0 0
\(85\) −2.59336 −0.281289
\(86\) 0 0
\(87\) −4.66026 −0.499632
\(88\) 0 0
\(89\) 6.48545 0.687456 0.343728 − 0.939069i \(-0.388310\pi\)
0.343728 + 0.939069i \(0.388310\pi\)
\(90\) 0 0
\(91\) 1.15082 0.120639
\(92\) 0 0
\(93\) −25.2078 −2.61393
\(94\) 0 0
\(95\) 5.48254 0.562496
\(96\) 0 0
\(97\) 17.7063 1.79781 0.898903 − 0.438147i \(-0.144365\pi\)
0.898903 + 0.438147i \(0.144365\pi\)
\(98\) 0 0
\(99\) 3.41769 0.343491
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.1.1 yes 4
3.2 odd 2 6624.2.a.be.1.3 4
4.3 odd 2 736.2.a.g.1.4 ✓ 4
8.3 odd 2 1472.2.a.z.1.1 4
8.5 even 2 1472.2.a.y.1.4 4
12.11 even 2 6624.2.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.g.1.4 ✓ 4 4.3 odd 2
736.2.a.h.1.1 yes 4 1.1 even 1 trivial
1472.2.a.y.1.4 4 8.5 even 2
1472.2.a.z.1.1 4 8.3 odd 2
6624.2.a.be.1.3 4 3.2 odd 2
6624.2.a.bf.1.3 4 12.11 even 2