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Properties

Label 2592.2.a.x.1.4
Level $2592$
Weight $2$
Character 2592.1
Self dual yes
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [2592,2,Mod(1,2592)] 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("2592.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,2,0,0,0,0,0,0,0,6,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{18 +2 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.548230\) of defining polynomial
Character \(\chi\) \(=\) 2592.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{5} +4.70285 q^{7} -0.875393 q^{11} -1.37228 q^{13} +2.37228 q^{17} +5.57825 q^{19} -4.70285 q^{23} +6.37228 q^{25} +5.37228 q^{29} -6.45364 q^{31} +15.8593 q^{35} +4.00000 q^{37} -1.00000 q^{41} -0.875393 q^{43} -4.70285 q^{47} +15.1168 q^{49} +4.00000 q^{53} -2.95207 q^{55} -8.53032 q^{59} -2.11684 q^{61} -4.62772 q^{65} -8.53032 q^{67} -9.40571 q^{71} +10.3723 q^{73} -4.11684 q^{77} +6.45364 q^{79} +2.95207 q^{83} +8.00000 q^{85} -12.7446 q^{89} -6.45364 q^{91} +18.8114 q^{95} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 6 q^{13} - 2 q^{17} + 14 q^{25} + 10 q^{29} + 16 q^{37} - 4 q^{41} + 26 q^{49} + 16 q^{53} + 26 q^{61} - 30 q^{65} + 30 q^{73} + 18 q^{77} + 32 q^{85} - 28 q^{89} + 36 q^{97}+O(q^{100}) \) 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
\(7\) 4.70285 1.77751 0.888756 − 0.458381i \(-0.151570\pi\)
0.888756 + 0.458381i \(0.151570\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.875393 −0.263941 −0.131970 − 0.991254i \(-0.542130\pi\)
−0.131970 + 0.991254i \(0.542130\pi\)
\(12\) 0 0
\(13\) −1.37228 −0.380602 −0.190301 − 0.981726i \(-0.560946\pi\)
−0.190301 + 0.981726i \(0.560946\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.37228 0.575363 0.287681 − 0.957726i \(-0.407116\pi\)
0.287681 + 0.957726i \(0.407116\pi\)
\(18\) 0 0
\(19\) 5.57825 1.27974 0.639869 − 0.768484i \(-0.278989\pi\)
0.639869 + 0.768484i \(0.278989\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.70285 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.37228 0.997608 0.498804 − 0.866715i \(-0.333773\pi\)
0.498804 + 0.866715i \(0.333773\pi\)
\(30\) 0 0
\(31\) −6.45364 −1.15911 −0.579554 − 0.814934i \(-0.696773\pi\)
−0.579554 + 0.814934i \(0.696773\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.8593 2.68072
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) −0.875393 −0.133496 −0.0667481 − 0.997770i \(-0.521262\pi\)
−0.0667481 + 0.997770i \(0.521262\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.70285 −0.685982 −0.342991 − 0.939339i \(-0.611440\pi\)
−0.342991 + 0.939339i \(0.611440\pi\)
\(48\) 0 0
\(49\) 15.1168 2.15955
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) −2.95207 −0.398057
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.53032 −1.11055 −0.555276 − 0.831666i \(-0.687388\pi\)
−0.555276 + 0.831666i \(0.687388\pi\)
\(60\) 0 0
\(61\) −2.11684 −0.271034 −0.135517 − 0.990775i \(-0.543270\pi\)
−0.135517 + 0.990775i \(0.543270\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.62772 −0.573998
\(66\) 0 0
\(67\) −8.53032 −1.04214 −0.521072 − 0.853513i \(-0.674468\pi\)
−0.521072 + 0.853513i \(0.674468\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.40571 −1.11625 −0.558126 − 0.829756i \(-0.688480\pi\)
−0.558126 + 0.829756i \(0.688480\pi\)
\(72\) 0 0
\(73\) 10.3723 1.21398 0.606992 − 0.794708i \(-0.292376\pi\)
0.606992 + 0.794708i \(0.292376\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.11684 −0.469158
\(78\) 0 0
\(79\) 6.45364 0.726091 0.363046 − 0.931771i \(-0.381737\pi\)
0.363046 + 0.931771i \(0.381737\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.95207 0.324032 0.162016 − 0.986788i \(-0.448200\pi\)
0.162016 + 0.986788i \(0.448200\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.7446 −1.35092 −0.675460 − 0.737396i \(-0.736055\pi\)
−0.675460 + 0.737396i \(0.736055\pi\)
\(90\) 0 0
\(91\) −6.45364 −0.676525
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.8114 1.93001
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 − 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 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 2592.2.a.x.1.4 4
3.2 odd 2 2592.2.a.u.1.2 4
4.3 odd 2 inner 2592.2.a.x.1.3 4
8.3 odd 2 5184.2.a.cc.1.1 4
8.5 even 2 5184.2.a.cc.1.2 4
9.2 odd 6 288.2.i.f.193.2 yes 8
9.4 even 3 864.2.i.f.289.1 8
9.5 odd 6 288.2.i.f.97.2 ✓ 8
9.7 even 3 864.2.i.f.577.1 8
12.11 even 2 2592.2.a.u.1.1 4
24.5 odd 2 5184.2.a.cf.1.4 4
24.11 even 2 5184.2.a.cf.1.3 4
36.7 odd 6 864.2.i.f.577.2 8
36.11 even 6 288.2.i.f.193.3 yes 8
36.23 even 6 288.2.i.f.97.3 yes 8
36.31 odd 6 864.2.i.f.289.2 8
72.5 odd 6 576.2.i.n.385.3 8
72.11 even 6 576.2.i.n.193.2 8
72.13 even 6 1728.2.i.n.1153.3 8
72.29 odd 6 576.2.i.n.193.3 8
72.43 odd 6 1728.2.i.n.577.4 8
72.59 even 6 576.2.i.n.385.2 8
72.61 even 6 1728.2.i.n.577.3 8
72.67 odd 6 1728.2.i.n.1153.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.f.97.2 ✓ 8 9.5 odd 6
288.2.i.f.97.3 yes 8 36.23 even 6
288.2.i.f.193.2 yes 8 9.2 odd 6
288.2.i.f.193.3 yes 8 36.11 even 6
576.2.i.n.193.2 8 72.11 even 6
576.2.i.n.193.3 8 72.29 odd 6
576.2.i.n.385.2 8 72.59 even 6
576.2.i.n.385.3 8 72.5 odd 6
864.2.i.f.289.1 8 9.4 even 3
864.2.i.f.289.2 8 36.31 odd 6
864.2.i.f.577.1 8 9.7 even 3
864.2.i.f.577.2 8 36.7 odd 6
1728.2.i.n.577.3 8 72.61 even 6
1728.2.i.n.577.4 8 72.43 odd 6
1728.2.i.n.1153.3 8 72.13 even 6
1728.2.i.n.1153.4 8 72.67 odd 6
2592.2.a.u.1.1 4 12.11 even 2
2592.2.a.u.1.2 4 3.2 odd 2
2592.2.a.x.1.3 4 4.3 odd 2 inner
2592.2.a.x.1.4 4 1.1 even 1 trivial
5184.2.a.cc.1.1 4 8.3 odd 2
5184.2.a.cc.1.2 4 8.5 even 2
5184.2.a.cf.1.3 4 24.11 even 2
5184.2.a.cf.1.4 4 24.5 odd 2