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 1.1
Root \(-2.44705\) 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)
 
\(f(q)\) \(=\) \(q-2.77617 q^{5} -2.49525 q^{7} +4.39886 q^{11} -4.76424 q^{13} -2.61318 q^{17} +4.10600 q^{19} -1.98807 q^{23} +2.70714 q^{25} -4.38693 q^{29} -2.65824 q^{31} +6.92724 q^{35} +1.00000 q^{37} +5.61075 q^{41} -11.1300 q^{43} +10.6012 q^{47} -0.773748 q^{49} +7.81314 q^{53} -12.2120 q^{55} +3.16542 q^{59} +2.00000 q^{61} +13.2264 q^{65} +3.94300 q^{67} +10.6012 q^{71} +10.0476 q^{73} -10.9762 q^{77} -14.8702 q^{79} +6.52838 q^{83} +7.25463 q^{85} +12.2239 q^{89} +11.8879 q^{91} -11.3990 q^{95} +15.5285 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 7 q^{7} + q^{11} + 5 q^{13} + 3 q^{17} - 3 q^{19} + 3 q^{23} + 11 q^{25} + 12 q^{29} - 8 q^{31} + 6 q^{35} + 5 q^{37} + 10 q^{41} - 8 q^{43} + 24 q^{47} + 14 q^{49} + 13 q^{53} - 14 q^{55}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(5\) −2.77617 −1.24154 −0.620771 − 0.783992i \(-0.713180\pi\)
−0.620771 + 0.783992i \(0.713180\pi\)
\(6\) 0 0
\(7\) −2.49525 −0.943114 −0.471557 − 0.881835i \(-0.656308\pi\)
−0.471557 + 0.881835i \(0.656308\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.39886 1.32631 0.663153 − 0.748484i \(-0.269218\pi\)
0.663153 + 0.748484i \(0.269218\pi\)
\(12\) 0 0
\(13\) −4.76424 −1.32136 −0.660681 − 0.750667i \(-0.729732\pi\)
−0.660681 + 0.750667i \(0.729732\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.61318 −0.633789 −0.316894 − 0.948461i \(-0.602640\pi\)
−0.316894 + 0.948461i \(0.602640\pi\)
\(18\) 0 0
\(19\) 4.10600 0.941981 0.470990 − 0.882138i \(-0.343897\pi\)
0.470990 + 0.882138i \(0.343897\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.98807 −0.414540 −0.207270 − 0.978284i \(-0.566458\pi\)
−0.207270 + 0.978284i \(0.566458\pi\)
\(24\) 0 0
\(25\) 2.70714 0.541428
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.38693 −0.814632 −0.407316 − 0.913287i \(-0.633535\pi\)
−0.407316 + 0.913287i \(0.633535\pi\)
\(30\) 0 0
\(31\) −2.65824 −0.477434 −0.238717 − 0.971089i \(-0.576727\pi\)
−0.238717 + 0.971089i \(0.576727\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.92724 1.17092
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.61075 0.876252 0.438126 − 0.898914i \(-0.355642\pi\)
0.438126 + 0.898914i \(0.355642\pi\)
\(42\) 0 0
\(43\) −11.1300 −1.69730 −0.848652 − 0.528951i \(-0.822585\pi\)
−0.848652 + 0.528951i \(0.822585\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.6012 1.54635 0.773175 − 0.634193i \(-0.218667\pi\)
0.773175 + 0.634193i \(0.218667\pi\)
\(48\) 0 0
\(49\) −0.773748 −0.110535
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.81314 1.07322 0.536608 − 0.843831i \(-0.319705\pi\)
0.536608 + 0.843831i \(0.319705\pi\)
\(54\) 0 0
\(55\) −12.2120 −1.64667
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.16542 0.412103 0.206051 − 0.978541i \(-0.433939\pi\)
0.206051 + 0.978541i \(0.433939\pi\)
\(60\) 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
\(65\) 13.2264 1.64053
\(66\) 0 0
\(67\) 3.94300 0.481714 0.240857 − 0.970561i \(-0.422571\pi\)
0.240857 + 0.970561i \(0.422571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.6012 1.25814 0.629068 − 0.777350i \(-0.283437\pi\)
0.629068 + 0.777350i \(0.283437\pi\)
\(72\) 0 0
\(73\) 10.0476 1.17598 0.587991 − 0.808868i \(-0.299919\pi\)
0.587991 + 0.808868i \(0.299919\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.9762 −1.25086
\(78\) 0 0
\(79\) −14.8702 −1.67303 −0.836516 − 0.547942i \(-0.815411\pi\)
−0.836516 + 0.547942i \(0.815411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.52838 0.716582 0.358291 − 0.933610i \(-0.383360\pi\)
0.358291 + 0.933610i \(0.383360\pi\)
\(84\) 0 0
\(85\) 7.25463 0.786876
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2239 1.29573 0.647867 − 0.761753i \(-0.275661\pi\)
0.647867 + 0.761753i \(0.275661\pi\)
\(90\) 0 0
\(91\) 11.8879 1.24620
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −11.3990 −1.16951
\(96\) 0 0
\(97\) 15.5285 1.57668 0.788339 − 0.615241i \(-0.210941\pi\)
0.788339 + 0.615241i \(0.210941\pi\)
\(98\) 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.1.1 yes 5
3.2 odd 2 2664.2.a.s.1.5 ✓ 5
4.3 odd 2 5328.2.a.bu.1.1 5
12.11 even 2 5328.2.a.bt.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2664.2.a.s.1.5 ✓ 5 3.2 odd 2
2664.2.a.t.1.1 yes 5 1.1 even 1 trivial
5328.2.a.bt.1.5 5 12.11 even 2
5328.2.a.bu.1.1 5 4.3 odd 2