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 1.2
Root \(1.43118\) 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)
 
\(f(q)\) \(=\) \(q-2.30644 q^{5} -3.57785 q^{7} -4.44020 q^{11} -0.354706 q^{13} +3.59095 q^{17} -7.12053 q^{19} +1.95174 q^{23} +0.319670 q^{25} +8.39193 q^{29} -9.47523 q^{31} +8.25209 q^{35} +1.00000 q^{37} -6.69838 q^{41} +11.1999 q^{43} +0.457318 q^{47} +5.80099 q^{49} -5.80086 q^{53} +10.2411 q^{55} +15.0048 q^{59} +2.00000 q^{61} +0.818108 q^{65} -13.0179 q^{67} +0.457318 q^{71} +10.1907 q^{73} +15.8864 q^{77} +0.765823 q^{79} +16.6923 q^{83} -8.28231 q^{85} -6.28932 q^{89} +1.26908 q^{91} +16.4231 q^{95} +6.70941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 7 q^{7} + q^{11} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 3 q^{23} + 11 q^{25} + 12 q^{29} - 8 q^{31} + 6 q^{35} + 5 q^{37} + 10 q^{41} - 8 q^{43} + 24 q^{47} + 14 q^{49} + 13 q^{53} - 14 q^{55}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(5\) −2.30644 −1.03147 −0.515736 − 0.856748i \(-0.672481\pi\)
−0.515736 + 0.856748i \(0.672481\pi\)
\(6\) 0 0
\(7\) −3.57785 −1.35230 −0.676149 − 0.736764i \(-0.736353\pi\)
−0.676149 + 0.736764i \(0.736353\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.44020 −1.33877 −0.669385 − 0.742916i \(-0.733442\pi\)
−0.669385 + 0.742916i \(0.733442\pi\)
\(12\) 0 0
\(13\) −0.354706 −0.0983777 −0.0491888 − 0.998789i \(-0.515664\pi\)
−0.0491888 + 0.998789i \(0.515664\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.59095 0.870932 0.435466 − 0.900205i \(-0.356584\pi\)
0.435466 + 0.900205i \(0.356584\pi\)
\(18\) 0 0
\(19\) −7.12053 −1.63356 −0.816781 − 0.576948i \(-0.804243\pi\)
−0.816781 + 0.576948i \(0.804243\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.95174 0.406965 0.203482 − 0.979079i \(-0.434774\pi\)
0.203482 + 0.979079i \(0.434774\pi\)
\(24\) 0 0
\(25\) 0.319670 0.0639341
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.39193 1.55834 0.779172 − 0.626811i \(-0.215640\pi\)
0.779172 + 0.626811i \(0.215640\pi\)
\(30\) 0 0
\(31\) −9.47523 −1.70180 −0.850901 − 0.525326i \(-0.823943\pi\)
−0.850901 + 0.525326i \(0.823943\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.25209 1.39486
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.69838 −1.04611 −0.523055 − 0.852299i \(-0.675208\pi\)
−0.523055 + 0.852299i \(0.675208\pi\)
\(42\) 0 0
\(43\) 11.1999 1.70797 0.853987 − 0.520294i \(-0.174178\pi\)
0.853987 + 0.520294i \(0.174178\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.457318 0.0667068 0.0333534 − 0.999444i \(-0.489381\pi\)
0.0333534 + 0.999444i \(0.489381\pi\)
\(48\) 0 0
\(49\) 5.80099 0.828713
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.80086 −0.796809 −0.398405 − 0.917210i \(-0.630436\pi\)
−0.398405 + 0.917210i \(0.630436\pi\)
\(54\) 0 0
\(55\) 10.2411 1.38090
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.0048 1.95346 0.976730 − 0.214471i \(-0.0688028\pi\)
0.976730 + 0.214471i \(0.0688028\pi\)
\(60\) 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
\(65\) 0.818108 0.101474
\(66\) 0 0
\(67\) −13.0179 −1.59039 −0.795196 − 0.606353i \(-0.792632\pi\)
−0.795196 + 0.606353i \(0.792632\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.457318 0.0542737 0.0271369 − 0.999632i \(-0.491361\pi\)
0.0271369 + 0.999632i \(0.491361\pi\)
\(72\) 0 0
\(73\) 10.1907 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.8864 1.81042
\(78\) 0 0
\(79\) 0.765823 0.0861618 0.0430809 − 0.999072i \(-0.486283\pi\)
0.0430809 + 0.999072i \(0.486283\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.6923 1.83222 0.916109 − 0.400930i \(-0.131313\pi\)
0.916109 + 0.400930i \(0.131313\pi\)
\(84\) 0 0
\(85\) −8.28231 −0.898342
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.28932 −0.666667 −0.333333 − 0.942809i \(-0.608173\pi\)
−0.333333 + 0.942809i \(0.608173\pi\)
\(90\) 0 0
\(91\) 1.26908 0.133036
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.4231 1.68497
\(96\) 0 0
\(97\) 6.70941 0.681238 0.340619 − 0.940202i \(-0.389363\pi\)
0.340619 + 0.940202i \(0.389363\pi\)
\(98\) 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.1.2 yes 5
3.2 odd 2 2664.2.a.s.1.4 ✓ 5
4.3 odd 2 5328.2.a.bu.1.2 5
12.11 even 2 5328.2.a.bt.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2664.2.a.s.1.4 ✓ 5 3.2 odd 2
2664.2.a.t.1.2 yes 5 1.1 even 1 trivial
5328.2.a.bt.1.4 5 12.11 even 2
5328.2.a.bu.1.2 5 4.3 odd 2