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 1.4
Root \(-0.550328\) 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)
 
\(f(q)\) \(=\) \(q+3.19824 q^{5} -4.72765 q^{7} -1.62699 q^{11} +6.89538 q^{13} -7.02655 q^{17} +0.601761 q^{19} +3.69714 q^{23} +5.22875 q^{25} +7.32413 q^{29} +5.49714 q^{31} -15.1202 q^{35} +1.00000 q^{37} -0.125890 q^{41} -11.6985 q^{43} +9.32941 q^{47} +15.3507 q^{49} +6.83052 q^{53} -5.20352 q^{55} +2.92765 q^{59} +2.00000 q^{61} +22.0531 q^{65} +10.8266 q^{67} +9.32941 q^{71} +0.331166 q^{73} +7.69186 q^{77} +0.293620 q^{79} -9.49318 q^{83} -22.4726 q^{85} +10.9007 q^{89} -32.5990 q^{91} +1.92458 q^{95} -7.79076 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 7 q^{7} + q^{11} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 3 q^{23} + 11 q^{25} + 12 q^{29} - 8 q^{31} + 6 q^{35} + 5 q^{37} + 10 q^{41} - 8 q^{43} + 24 q^{47} + 14 q^{49} + 13 q^{53} - 14 q^{55}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(5\) 3.19824 1.43030 0.715149 − 0.698972i \(-0.246359\pi\)
0.715149 + 0.698972i \(0.246359\pi\)
\(6\) 0 0
\(7\) −4.72765 −1.78688 −0.893442 − 0.449179i \(-0.851717\pi\)
−0.893442 + 0.449179i \(0.851717\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.62699 −0.490557 −0.245279 − 0.969453i \(-0.578879\pi\)
−0.245279 + 0.969453i \(0.578879\pi\)
\(12\) 0 0
\(13\) 6.89538 1.91243 0.956217 − 0.292658i \(-0.0945396\pi\)
0.956217 + 0.292658i \(0.0945396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.02655 −1.70419 −0.852094 − 0.523388i \(-0.824668\pi\)
−0.852094 + 0.523388i \(0.824668\pi\)
\(18\) 0 0
\(19\) 0.601761 0.138053 0.0690267 − 0.997615i \(-0.478011\pi\)
0.0690267 + 0.997615i \(0.478011\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.69714 0.770907 0.385453 − 0.922727i \(-0.374045\pi\)
0.385453 + 0.922727i \(0.374045\pi\)
\(24\) 0 0
\(25\) 5.22875 1.04575
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.32413 1.36006 0.680029 − 0.733186i \(-0.261967\pi\)
0.680029 + 0.733186i \(0.261967\pi\)
\(30\) 0 0
\(31\) 5.49714 0.987316 0.493658 − 0.869656i \(-0.335660\pi\)
0.493658 + 0.869656i \(0.335660\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −15.1202 −2.55578
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.125890 −0.0196607 −0.00983037 − 0.999952i \(-0.503129\pi\)
−0.00983037 + 0.999952i \(0.503129\pi\)
\(42\) 0 0
\(43\) −11.6985 −1.78400 −0.891999 − 0.452038i \(-0.850697\pi\)
−0.891999 + 0.452038i \(0.850697\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.32941 1.36083 0.680417 − 0.732825i \(-0.261799\pi\)
0.680417 + 0.732825i \(0.261799\pi\)
\(48\) 0 0
\(49\) 15.3507 2.19295
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.83052 0.938243 0.469122 − 0.883134i \(-0.344571\pi\)
0.469122 + 0.883134i \(0.344571\pi\)
\(54\) 0 0
\(55\) −5.20352 −0.701643
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.92765 0.381147 0.190574 − 0.981673i \(-0.438965\pi\)
0.190574 + 0.981673i \(0.438965\pi\)
\(60\) 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
\(65\) 22.0531 2.73535
\(66\) 0 0
\(67\) 10.8266 1.32267 0.661337 − 0.750089i \(-0.269989\pi\)
0.661337 + 0.750089i \(0.269989\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.32941 1.10720 0.553599 − 0.832784i \(-0.313254\pi\)
0.553599 + 0.832784i \(0.313254\pi\)
\(72\) 0 0
\(73\) 0.331166 0.0387600 0.0193800 − 0.999812i \(-0.493831\pi\)
0.0193800 + 0.999812i \(0.493831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.69186 0.876569
\(78\) 0 0
\(79\) 0.293620 0.0330349 0.0165174 − 0.999864i \(-0.494742\pi\)
0.0165174 + 0.999864i \(0.494742\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.49318 −1.04201 −0.521006 − 0.853553i \(-0.674443\pi\)
−0.521006 + 0.853553i \(0.674443\pi\)
\(84\) 0 0
\(85\) −22.4726 −2.43750
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.9007 1.15547 0.577734 − 0.816225i \(-0.303937\pi\)
0.577734 + 0.816225i \(0.303937\pi\)
\(90\) 0 0
\(91\) −32.5990 −3.41730
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.92458 0.197457
\(96\) 0 0
\(97\) −7.79076 −0.791032 −0.395516 − 0.918459i \(-0.629434\pi\)
−0.395516 + 0.918459i \(0.629434\pi\)
\(98\) 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.1.4 yes 5
3.2 odd 2 2664.2.a.s.1.2 ✓ 5
4.3 odd 2 5328.2.a.bu.1.4 5
12.11 even 2 5328.2.a.bt.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2664.2.a.s.1.2 ✓ 5 3.2 odd 2
2664.2.a.t.1.4 yes 5 1.1 even 1 trivial
5328.2.a.bt.1.2 5 12.11 even 2
5328.2.a.bu.1.4 5 4.3 odd 2