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 = [3,0,0,0,1,0,7,0,0,0,0] 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: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 296)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) 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)
 
\(f(q)\) \(=\) \(q-0.462598 q^{5} -0.323404 q^{7} -5.58242 q^{11} +6.58242 q^{13} +6.64681 q^{17} -2.64681 q^{19} -4.86081 q^{23} -4.78600 q^{25} +4.86081 q^{29} +6.46260 q^{31} +0.149606 q^{35} -1.00000 q^{37} +0.815790 q^{41} -1.87122 q^{43} +1.11982 q^{47} -6.89541 q^{49} +12.6918 q^{53} +2.58242 q^{55} +0.128782 q^{59} -1.10941 q^{61} -3.04502 q^{65} +13.4778 q^{67} +8.04502 q^{71} +3.58242 q^{73} +1.80538 q^{77} -3.78600 q^{79} +14.6918 q^{83} -3.07480 q^{85} -2.14961 q^{89} -2.12878 q^{91} +1.22441 q^{95} +1.05398 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{5} + 7 q^{7} + 3 q^{13} + 4 q^{17} + 8 q^{19} - 9 q^{23} - 4 q^{25} + 9 q^{29} + 17 q^{31} + 10 q^{35} - 3 q^{37} + 16 q^{41} - 4 q^{43} - 11 q^{47} + 8 q^{49} + 3 q^{53} - 9 q^{55} + 2 q^{59}+ \cdots + 18 q^{95}+O(q^{100}) \) 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
\(5\) −0.462598 −0.206880 −0.103440 − 0.994636i \(-0.532985\pi\)
−0.103440 + 0.994636i \(0.532985\pi\)
\(6\) 0 0
\(7\) −0.323404 −0.122235 −0.0611177 − 0.998131i \(-0.519466\pi\)
−0.0611177 + 0.998131i \(0.519466\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.58242 −1.68316 −0.841581 − 0.540131i \(-0.818375\pi\)
−0.841581 + 0.540131i \(0.818375\pi\)
\(12\) 0 0
\(13\) 6.58242 1.82563 0.912817 − 0.408369i \(-0.133902\pi\)
0.912817 + 0.408369i \(0.133902\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.64681 1.61209 0.806044 − 0.591856i \(-0.201604\pi\)
0.806044 + 0.591856i \(0.201604\pi\)
\(18\) 0 0
\(19\) −2.64681 −0.607220 −0.303610 − 0.952796i \(-0.598192\pi\)
−0.303610 + 0.952796i \(0.598192\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.86081 −1.01355 −0.506774 − 0.862079i \(-0.669162\pi\)
−0.506774 + 0.862079i \(0.669162\pi\)
\(24\) 0 0
\(25\) −4.78600 −0.957201
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.86081 0.902629 0.451314 − 0.892365i \(-0.350955\pi\)
0.451314 + 0.892365i \(0.350955\pi\)
\(30\) 0 0
\(31\) 6.46260 1.16072 0.580358 − 0.814361i \(-0.302912\pi\)
0.580358 + 0.814361i \(0.302912\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.149606 0.0252881
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.815790 0.127405 0.0637025 − 0.997969i \(-0.479709\pi\)
0.0637025 + 0.997969i \(0.479709\pi\)
\(42\) 0 0
\(43\) −1.87122 −0.285358 −0.142679 − 0.989769i \(-0.545572\pi\)
−0.142679 + 0.989769i \(0.545572\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.11982 0.163342 0.0816712 − 0.996659i \(-0.473974\pi\)
0.0816712 + 0.996659i \(0.473974\pi\)
\(48\) 0 0
\(49\) −6.89541 −0.985059
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.6918 1.74336 0.871678 − 0.490079i \(-0.163032\pi\)
0.871678 + 0.490079i \(0.163032\pi\)
\(54\) 0 0
\(55\) 2.58242 0.348213
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.128782 0.0167660 0.00838299 − 0.999965i \(-0.497332\pi\)
0.00838299 + 0.999965i \(0.497332\pi\)
\(60\) 0 0
\(61\) −1.10941 −0.142045 −0.0710225 − 0.997475i \(-0.522626\pi\)
−0.0710225 + 0.997475i \(0.522626\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.04502 −0.377688
\(66\) 0 0
\(67\) 13.4778 1.64658 0.823289 − 0.567622i \(-0.192136\pi\)
0.823289 + 0.567622i \(0.192136\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.04502 0.954768 0.477384 − 0.878695i \(-0.341585\pi\)
0.477384 + 0.878695i \(0.341585\pi\)
\(72\) 0 0
\(73\) 3.58242 0.419290 0.209645 − 0.977778i \(-0.432769\pi\)
0.209645 + 0.977778i \(0.432769\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.80538 0.205742
\(78\) 0 0
\(79\) −3.78600 −0.425959 −0.212979 − 0.977057i \(-0.568317\pi\)
−0.212979 + 0.977057i \(0.568317\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.6918 1.61264 0.806319 − 0.591481i \(-0.201457\pi\)
0.806319 + 0.591481i \(0.201457\pi\)
\(84\) 0 0
\(85\) −3.07480 −0.333509
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.14961 −0.227858 −0.113929 − 0.993489i \(-0.536344\pi\)
−0.113929 + 0.993489i \(0.536344\pi\)
\(90\) 0 0
\(91\) −2.12878 −0.223157
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.22441 0.125622
\(96\) 0 0
\(97\) 1.05398 0.107015 0.0535077 − 0.998567i \(-0.482960\pi\)
0.0535077 + 0.998567i \(0.482960\pi\)
\(98\) 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.p.1.2 3
3.2 odd 2 296.2.a.c.1.2 ✓ 3
4.3 odd 2 5328.2.a.bn.1.2 3
12.11 even 2 592.2.a.i.1.2 3
15.14 odd 2 7400.2.a.k.1.2 3
24.5 odd 2 2368.2.a.bb.1.2 3
24.11 even 2 2368.2.a.be.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.a.c.1.2 ✓ 3 3.2 odd 2
592.2.a.i.1.2 3 12.11 even 2
2368.2.a.bb.1.2 3 24.5 odd 2
2368.2.a.be.1.2 3 24.11 even 2
2664.2.a.p.1.2 3 1.1 even 1 trivial
5328.2.a.bn.1.2 3 4.3 odd 2
7400.2.a.k.1.2 3 15.14 odd 2