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Properties

Label 1296.2.a.p.1.2
Level $1296$
Weight $2$
Character 1296.1
Self dual yes
Analytic conductor $10.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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 1.2
Root \(3.37228\) 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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{5} +1.37228 q^{7} +1.00000 q^{11} +5.37228 q^{13} -0.372281 q^{17} -6.37228 q^{19} +5.37228 q^{23} +6.37228 q^{25} -1.37228 q^{29} -0.627719 q^{31} +4.62772 q^{35} -2.74456 q^{37} +0.255437 q^{41} -9.74456 q^{43} -1.37228 q^{47} -5.11684 q^{49} +10.7446 q^{53} +3.37228 q^{55} +7.00000 q^{59} -3.37228 q^{61} +18.1168 q^{65} -7.74456 q^{67} -4.00000 q^{71} +5.11684 q^{73} +1.37228 q^{77} +0.627719 q^{79} +15.3723 q^{83} -1.25544 q^{85} -6.00000 q^{89} +7.37228 q^{91} -21.4891 q^{95} -9.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 3 q^{7} + 2 q^{11} + 5 q^{13} + 5 q^{17} - 7 q^{19} + 5 q^{23} + 7 q^{25} + 3 q^{29} - 7 q^{31} + 15 q^{35} + 6 q^{37} + 12 q^{41} - 8 q^{43} + 3 q^{47} + 7 q^{49} + 10 q^{53} + q^{55} + 14 q^{59}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(5\) 3.37228 1.50813 0.754065 − 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) 0 0
\(7\) 1.37228 0.518674 0.259337 − 0.965787i \(-0.416496\pi\)
0.259337 + 0.965787i \(0.416496\pi\)
\(8\) 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
\(13\) 5.37228 1.49000 0.745001 − 0.667063i \(-0.232449\pi\)
0.745001 + 0.667063i \(0.232449\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.372281 −0.0902915 −0.0451457 − 0.998980i \(-0.514375\pi\)
−0.0451457 + 0.998980i \(0.514375\pi\)
\(18\) 0 0
\(19\) −6.37228 −1.46190 −0.730951 − 0.682430i \(-0.760923\pi\)
−0.730951 + 0.682430i \(0.760923\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.37228 1.12020 0.560099 − 0.828426i \(-0.310763\pi\)
0.560099 + 0.828426i \(0.310763\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 − 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −0.627719 −0.112742 −0.0563708 − 0.998410i \(-0.517953\pi\)
−0.0563708 + 0.998410i \(0.517953\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.62772 0.782227
\(36\) 0 0
\(37\) −2.74456 −0.451203 −0.225602 − 0.974220i \(-0.572435\pi\)
−0.225602 + 0.974220i \(0.572435\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.255437 0.0398926 0.0199463 − 0.999801i \(-0.493650\pi\)
0.0199463 + 0.999801i \(0.493650\pi\)
\(42\) 0 0
\(43\) −9.74456 −1.48603 −0.743016 − 0.669274i \(-0.766605\pi\)
−0.743016 + 0.669274i \(0.766605\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.37228 −0.200168 −0.100084 − 0.994979i \(-0.531911\pi\)
−0.100084 + 0.994979i \(0.531911\pi\)
\(48\) 0 0
\(49\) −5.11684 −0.730978
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.7446 1.47588 0.737940 − 0.674867i \(-0.235799\pi\)
0.737940 + 0.674867i \(0.235799\pi\)
\(54\) 0 0
\(55\) 3.37228 0.454718
\(56\) 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
\(61\) −3.37228 −0.431776 −0.215888 − 0.976418i \(-0.569265\pi\)
−0.215888 + 0.976418i \(0.569265\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 18.1168 2.24712
\(66\) 0 0
\(67\) −7.74456 −0.946149 −0.473074 − 0.881022i \(-0.656856\pi\)
−0.473074 + 0.881022i \(0.656856\pi\)
\(68\) 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
\(73\) 5.11684 0.598881 0.299441 − 0.954115i \(-0.403200\pi\)
0.299441 + 0.954115i \(0.403200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.37228 0.156386
\(78\) 0 0
\(79\) 0.627719 0.0706239 0.0353119 − 0.999376i \(-0.488758\pi\)
0.0353119 + 0.999376i \(0.488758\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.3723 1.68733 0.843664 − 0.536872i \(-0.180394\pi\)
0.843664 + 0.536872i \(0.180394\pi\)
\(84\) 0 0
\(85\) −1.25544 −0.136171
\(86\) 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
\(91\) 7.37228 0.772825
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.4891 −2.20474
\(96\) 0 0
\(97\) −9.74456 −0.989410 −0.494705 − 0.869061i \(-0.664724\pi\)
−0.494705 + 0.869061i \(0.664724\pi\)
\(98\) 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.1.2 2
3.2 odd 2 1296.2.a.n.1.1 2
4.3 odd 2 648.2.a.g.1.2 2
8.3 odd 2 5184.2.a.bp.1.1 2
8.5 even 2 5184.2.a.bo.1.1 2
9.2 odd 6 144.2.i.d.49.2 4
9.4 even 3 432.2.i.d.289.1 4
9.5 odd 6 144.2.i.d.97.2 4
9.7 even 3 432.2.i.d.145.1 4
12.11 even 2 648.2.a.f.1.1 2
24.5 odd 2 5184.2.a.bs.1.2 2
24.11 even 2 5184.2.a.bt.1.2 2
36.7 odd 6 216.2.i.b.145.1 4
36.11 even 6 72.2.i.b.49.1 yes 4
36.23 even 6 72.2.i.b.25.1 ✓ 4
36.31 odd 6 216.2.i.b.73.1 4
72.5 odd 6 576.2.i.l.385.1 4
72.11 even 6 576.2.i.j.193.2 4
72.13 even 6 1728.2.i.j.1153.2 4
72.29 odd 6 576.2.i.l.193.1 4
72.43 odd 6 1728.2.i.i.577.2 4
72.59 even 6 576.2.i.j.385.2 4
72.61 even 6 1728.2.i.j.577.2 4
72.67 odd 6 1728.2.i.i.1153.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.b.25.1 ✓ 4 36.23 even 6
72.2.i.b.49.1 yes 4 36.11 even 6
144.2.i.d.49.2 4 9.2 odd 6
144.2.i.d.97.2 4 9.5 odd 6
216.2.i.b.73.1 4 36.31 odd 6
216.2.i.b.145.1 4 36.7 odd 6
432.2.i.d.145.1 4 9.7 even 3
432.2.i.d.289.1 4 9.4 even 3
576.2.i.j.193.2 4 72.11 even 6
576.2.i.j.385.2 4 72.59 even 6
576.2.i.l.193.1 4 72.29 odd 6
576.2.i.l.385.1 4 72.5 odd 6
648.2.a.f.1.1 2 12.11 even 2
648.2.a.g.1.2 2 4.3 odd 2
1296.2.a.n.1.1 2 3.2 odd 2
1296.2.a.p.1.2 2 1.1 even 1 trivial
1728.2.i.i.577.2 4 72.43 odd 6
1728.2.i.i.1153.2 4 72.67 odd 6
1728.2.i.j.577.2 4 72.61 even 6
1728.2.i.j.1153.2 4 72.13 even 6
5184.2.a.bo.1.1 2 8.5 even 2
5184.2.a.bp.1.1 2 8.3 odd 2
5184.2.a.bs.1.2 2 24.5 odd 2
5184.2.a.bt.1.2 2 24.11 even 2