Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,2,Mod(1,2664)] 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("2664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,2,0,-7,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.2721470985\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.935504.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.355205\) of defining polynomial
Character \(\chi\) \(=\) 2664.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.329694 q^{5} +2.34717 q^{7} +5.05758 q^{11} +4.20352 q^{13} +3.30706 q^{17} -2.83373 q^{19} +3.87383 q^{23} -4.89130 q^{25} +0.816253 q^{29} -0.630202 q^{31} +0.773846 q^{35} +1.00000 q^{37} +3.51344 q^{41} -4.79062 q^{43} -1.18089 q^{47} -1.49081 q^{49} -6.72503 q^{53} +1.66745 q^{55} +2.15687 q^{59} +2.00000 q^{61} +1.38587 q^{65} -5.81109 q^{67} -1.18089 q^{71} -1.00655 q^{73} +11.8710 q^{77} +1.03725 q^{79} -0.283731 q^{83} +1.09032 q^{85} +4.20638 q^{89} +9.86637 q^{91} -0.934261 q^{95} -2.40705 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 7 q^{7} + q^{11} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 3 q^{23} + 11 q^{25} + 12 q^{29} - 8 q^{31} + 6 q^{35} + 5 q^{37} + 10 q^{41} - 8 q^{43} + 24 q^{47} + 14 q^{49} + 13 q^{53} - 14 q^{55}+ \cdots + 20 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) 0.329694 0.147443 0.0737217 − 0.997279i \(-0.476512\pi\)
0.0737217 + 0.997279i \(0.476512\pi\)
\(6\) 0 0
\(7\) 2.34717 0.887146 0.443573 − 0.896238i \(-0.353711\pi\)
0.443573 + 0.896238i \(0.353711\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.05758 1.52492 0.762458 − 0.647037i \(-0.223992\pi\)
0.762458 + 0.647037i \(0.223992\pi\)
\(12\) 0 0
\(13\) 4.20352 1.16585 0.582924 − 0.812527i \(-0.301909\pi\)
0.582924 + 0.812527i \(0.301909\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.30706 0.802080 0.401040 − 0.916060i \(-0.368649\pi\)
0.401040 + 0.916060i \(0.368649\pi\)
\(18\) 0 0
\(19\) −2.83373 −0.650101 −0.325051 − 0.945697i \(-0.605381\pi\)
−0.325051 + 0.945697i \(0.605381\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.87383 0.807749 0.403875 − 0.914814i \(-0.367663\pi\)
0.403875 + 0.914814i \(0.367663\pi\)
\(24\) 0 0
\(25\) −4.89130 −0.978260
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.816253 0.151574 0.0757872 − 0.997124i \(-0.475853\pi\)
0.0757872 + 0.997124i \(0.475853\pi\)
\(30\) 0 0
\(31\) −0.630202 −0.113188 −0.0565939 − 0.998397i \(-0.518024\pi\)
−0.0565939 + 0.998397i \(0.518024\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.773846 0.130804
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.51344 0.548707 0.274354 − 0.961629i \(-0.411536\pi\)
0.274354 + 0.961629i \(0.411536\pi\)
\(42\) 0 0
\(43\) −4.79062 −0.730562 −0.365281 − 0.930897i \(-0.619027\pi\)
−0.365281 + 0.930897i \(0.619027\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.18089 −0.172251 −0.0861254 − 0.996284i \(-0.527449\pi\)
−0.0861254 + 0.996284i \(0.527449\pi\)
\(48\) 0 0
\(49\) −1.49081 −0.212973
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.72503 −0.923754 −0.461877 − 0.886944i \(-0.652824\pi\)
−0.461877 + 0.886944i \(0.652824\pi\)
\(54\) 0 0
\(55\) 1.66745 0.224839
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.15687 0.280800 0.140400 − 0.990095i \(-0.455161\pi\)
0.140400 + 0.990095i \(0.455161\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.38587 0.171897
\(66\) 0 0
\(67\) −5.81109 −0.709938 −0.354969 − 0.934878i \(-0.615509\pi\)
−0.354969 + 0.934878i \(0.615509\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.18089 −0.140146 −0.0700730 − 0.997542i \(-0.522323\pi\)
−0.0700730 + 0.997542i \(0.522323\pi\)
\(72\) 0 0
\(73\) −1.00655 −0.117808 −0.0589041 − 0.998264i \(-0.518761\pi\)
−0.0589041 + 0.998264i \(0.518761\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.8710 1.35282
\(78\) 0 0
\(79\) 1.03725 0.116700 0.0583498 − 0.998296i \(-0.481416\pi\)
0.0583498 + 0.998296i \(0.481416\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.283731 −0.0311435 −0.0155717 − 0.999879i \(-0.504957\pi\)
−0.0155717 + 0.999879i \(0.504957\pi\)
\(84\) 0 0
\(85\) 1.09032 0.118262
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.20638 0.445875 0.222938 − 0.974833i \(-0.428435\pi\)
0.222938 + 0.974833i \(0.428435\pi\)
\(90\) 0 0
\(91\) 9.86637 1.03428
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.934261 −0.0958532
\(96\) 0 0
\(97\) −2.40705 −0.244398 −0.122199 − 0.992506i \(-0.538995\pi\)
−0.122199 + 0.992506i \(0.538995\pi\)
\(98\) 0 0
\(99\) 0 0
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 2664.2.a.t.1.3 yes 5
3.2 odd 2 2664.2.a.s.1.3 ✓ 5
4.3 odd 2 5328.2.a.bu.1.3 5
12.11 even 2 5328.2.a.bt.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2664.2.a.s.1.3 ✓ 5 3.2 odd 2
2664.2.a.t.1.3 yes 5 1.1 even 1 trivial
5328.2.a.bt.1.3 5 12.11 even 2
5328.2.a.bu.1.3 5 4.3 odd 2