↑

Properties

Label 1296.2.a.p.1.1
Level $1296$
Weight $2$
Character 1296.1
Self dual yes
Analytic conductor $10.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,2,Mod(1,1296)] 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("1296.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,1,0,-3,0,0,0,2] 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: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 1296.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{5} -4.37228 q^{7} +1.00000 q^{11} -0.372281 q^{13} +5.37228 q^{17} -0.627719 q^{19} -0.372281 q^{23} +0.627719 q^{25} +4.37228 q^{29} -6.37228 q^{31} +10.3723 q^{35} +8.74456 q^{37} +11.7446 q^{41} +1.74456 q^{43} +4.37228 q^{47} +12.1168 q^{49} -0.744563 q^{53} -2.37228 q^{55} +7.00000 q^{59} +2.37228 q^{61} +0.883156 q^{65} +3.74456 q^{67} -4.00000 q^{71} -12.1168 q^{73} -4.37228 q^{77} +6.37228 q^{79} +9.62772 q^{83} -12.7446 q^{85} -6.00000 q^{89} +1.62772 q^{91} +1.48913 q^{95} +1.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 3 q^{7} + 2 q^{11} + 5 q^{13} + 5 q^{17} - 7 q^{19} + 5 q^{23} + 7 q^{25} + 3 q^{29} - 7 q^{31} + 15 q^{35} + 6 q^{37} + 12 q^{41} - 8 q^{43} + 3 q^{47} + 7 q^{49} + 10 q^{53} + q^{55} + 14 q^{59}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(5\) −2.37228 −1.06092 −0.530458 − 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) −4.37228 −1.65257 −0.826284 − 0.563254i \(-0.809549\pi\)
−0.826284 + 0.563254i \(0.809549\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −0.372281 −0.103252 −0.0516261 − 0.998666i \(-0.516440\pi\)
−0.0516261 + 0.998666i \(0.516440\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.37228 1.30297 0.651485 − 0.758662i \(-0.274146\pi\)
0.651485 + 0.758662i \(0.274146\pi\)
\(18\) 0 0
\(19\) −0.627719 −0.144009 −0.0720043 − 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.372281 −0.0776260 −0.0388130 − 0.999246i \(-0.512358\pi\)
−0.0388130 + 0.999246i \(0.512358\pi\)
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.37228 0.811912 0.405956 − 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(30\) 0 0
\(31\) −6.37228 −1.14450 −0.572248 − 0.820081i \(-0.693928\pi\)
−0.572248 + 0.820081i \(0.693928\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.3723 1.75324
\(36\) 0 0
\(37\) 8.74456 1.43760 0.718799 − 0.695218i \(-0.244692\pi\)
0.718799 + 0.695218i \(0.244692\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.7446 1.83419 0.917096 − 0.398666i \(-0.130527\pi\)
0.917096 + 0.398666i \(0.130527\pi\)
\(42\) 0 0
\(43\) 1.74456 0.266043 0.133022 − 0.991113i \(-0.457532\pi\)
0.133022 + 0.991113i \(0.457532\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.37228 0.637763 0.318881 − 0.947795i \(-0.396693\pi\)
0.318881 + 0.947795i \(0.396693\pi\)
\(48\) 0 0
\(49\) 12.1168 1.73098
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.744563 −0.102274 −0.0511368 − 0.998692i \(-0.516284\pi\)
−0.0511368 + 0.998692i \(0.516284\pi\)
\(54\) 0 0
\(55\) −2.37228 −0.319878
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 0.911322 0.455661 − 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) 2.37228 0.303739 0.151870 − 0.988401i \(-0.451471\pi\)
0.151870 + 0.988401i \(0.451471\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.883156 0.109542
\(66\) 0 0
\(67\) 3.74456 0.457471 0.228736 − 0.973489i \(-0.426541\pi\)
0.228736 + 0.973489i \(0.426541\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −12.1168 −1.41817 −0.709085 − 0.705123i \(-0.750892\pi\)
−0.709085 + 0.705123i \(0.750892\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.37228 −0.498268
\(78\) 0 0
\(79\) 6.37228 0.716938 0.358469 − 0.933542i \(-0.383299\pi\)
0.358469 + 0.933542i \(0.383299\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.62772 1.05678 0.528390 − 0.849002i \(-0.322796\pi\)
0.528390 + 0.849002i \(0.322796\pi\)
\(84\) 0 0
\(85\) −12.7446 −1.38234
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.62772 0.170631
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.48913 0.152781
\(96\) 0 0
\(97\) 1.74456 0.177133 0.0885667 − 0.996070i \(-0.471771\pi\)
0.0885667 + 0.996070i \(0.471771\pi\)
\(98\) 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 1296.2.a.p.1.1 2
3.2 odd 2 1296.2.a.n.1.2 2
4.3 odd 2 648.2.a.g.1.1 2
8.3 odd 2 5184.2.a.bp.1.2 2
8.5 even 2 5184.2.a.bo.1.2 2
9.2 odd 6 144.2.i.d.49.1 4
9.4 even 3 432.2.i.d.289.2 4
9.5 odd 6 144.2.i.d.97.1 4
9.7 even 3 432.2.i.d.145.2 4
12.11 even 2 648.2.a.f.1.2 2
24.5 odd 2 5184.2.a.bs.1.1 2
24.11 even 2 5184.2.a.bt.1.1 2
36.7 odd 6 216.2.i.b.145.2 4
36.11 even 6 72.2.i.b.49.2 yes 4
36.23 even 6 72.2.i.b.25.2 ✓ 4
36.31 odd 6 216.2.i.b.73.2 4
72.5 odd 6 576.2.i.l.385.2 4
72.11 even 6 576.2.i.j.193.1 4
72.13 even 6 1728.2.i.j.1153.1 4
72.29 odd 6 576.2.i.l.193.2 4
72.43 odd 6 1728.2.i.i.577.1 4
72.59 even 6 576.2.i.j.385.1 4
72.61 even 6 1728.2.i.j.577.1 4
72.67 odd 6 1728.2.i.i.1153.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.b.25.2 ✓ 4 36.23 even 6
72.2.i.b.49.2 yes 4 36.11 even 6
144.2.i.d.49.1 4 9.2 odd 6
144.2.i.d.97.1 4 9.5 odd 6
216.2.i.b.73.2 4 36.31 odd 6
216.2.i.b.145.2 4 36.7 odd 6
432.2.i.d.145.2 4 9.7 even 3
432.2.i.d.289.2 4 9.4 even 3
576.2.i.j.193.1 4 72.11 even 6
576.2.i.j.385.1 4 72.59 even 6
576.2.i.l.193.2 4 72.29 odd 6
576.2.i.l.385.2 4 72.5 odd 6
648.2.a.f.1.2 2 12.11 even 2
648.2.a.g.1.1 2 4.3 odd 2
1296.2.a.n.1.2 2 3.2 odd 2
1296.2.a.p.1.1 2 1.1 even 1 trivial
1728.2.i.i.577.1 4 72.43 odd 6
1728.2.i.i.1153.1 4 72.67 odd 6
1728.2.i.j.577.1 4 72.61 even 6
1728.2.i.j.1153.1 4 72.13 even 6
5184.2.a.bo.1.2 2 8.5 even 2
5184.2.a.bp.1.2 2 8.3 odd 2
5184.2.a.bs.1.1 2 24.5 odd 2
5184.2.a.bt.1.1 2 24.11 even 2