Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5184,2,Mod(1,5184)] 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("5184.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,0,0,0,0,0,0,6,0,0,0,8] 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: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5184.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.73205 q^{7} +1.73205 q^{11} +3.00000 q^{13} +4.00000 q^{17} +6.92820 q^{19} +8.66025 q^{23} -4.00000 q^{25} -1.00000 q^{29} -5.19615 q^{31} +1.73205 q^{35} +8.00000 q^{37} +5.00000 q^{41} -8.66025 q^{43} -12.1244 q^{47} -4.00000 q^{49} +8.00000 q^{53} +1.73205 q^{55} +1.73205 q^{59} +7.00000 q^{61} +3.00000 q^{65} -8.66025 q^{67} -3.46410 q^{71} -12.0000 q^{73} +3.00000 q^{77} +5.19615 q^{79} -8.66025 q^{83} +4.00000 q^{85} -4.00000 q^{89} +5.19615 q^{91} +6.92820 q^{95} -3.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{13} + 8 q^{17} - 8 q^{25} - 2 q^{29} + 16 q^{37} + 10 q^{41} - 8 q^{49} + 16 q^{53} + 14 q^{61} + 6 q^{65} - 24 q^{73} + 6 q^{77} + 8 q^{85} - 8 q^{89} - 6 q^{97}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 − 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 − 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 − 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.66025 1.80579 0.902894 − 0.429863i \(-0.141438\pi\)
0.902894 + 0.429863i \(0.141438\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −5.19615 −0.933257 −0.466628 − 0.884454i \(-0.654531\pi\)
−0.466628 + 0.884454i \(0.654531\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73205 0.292770
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −8.66025 −1.32068 −0.660338 − 0.750968i \(-0.729587\pi\)
−0.660338 + 0.750968i \(0.729587\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.1244 −1.76852 −0.884260 − 0.466996i \(-0.845336\pi\)
−0.884260 + 0.466996i \(0.845336\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 1.73205 0.233550
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.73205 0.225494 0.112747 − 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −8.66025 −1.05802 −0.529009 − 0.848616i \(-0.677436\pi\)
−0.529009 + 0.848616i \(0.677436\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 5.19615 0.584613 0.292306 − 0.956325i \(-0.405577\pi\)
0.292306 + 0.956325i \(0.405577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.66025 −0.950586 −0.475293 − 0.879827i \(-0.657658\pi\)
−0.475293 + 0.879827i \(0.657658\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 5.19615 0.544705
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.92820 0.710819
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 − 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 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 5184.2.a.bx.1.2 2
3.2 odd 2 5184.2.a.bl.1.2 2
4.3 odd 2 inner 5184.2.a.bx.1.1 2
8.3 odd 2 2592.2.a.l.1.1 2
8.5 even 2 2592.2.a.l.1.2 2
9.2 odd 6 1728.2.i.l.577.1 4
9.4 even 3 576.2.i.k.385.1 4
9.5 odd 6 1728.2.i.l.1153.1 4
9.7 even 3 576.2.i.k.193.1 4
12.11 even 2 5184.2.a.bl.1.1 2
24.5 odd 2 2592.2.a.p.1.2 2
24.11 even 2 2592.2.a.p.1.1 2
36.7 odd 6 576.2.i.k.193.2 4
36.11 even 6 1728.2.i.l.577.2 4
36.23 even 6 1728.2.i.l.1153.2 4
36.31 odd 6 576.2.i.k.385.2 4
72.5 odd 6 864.2.i.d.289.1 4
72.11 even 6 864.2.i.d.577.2 4
72.13 even 6 288.2.i.d.97.2 yes 4
72.29 odd 6 864.2.i.d.577.1 4
72.43 odd 6 288.2.i.d.193.1 yes 4
72.59 even 6 864.2.i.d.289.2 4
72.61 even 6 288.2.i.d.193.2 yes 4
72.67 odd 6 288.2.i.d.97.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.d.97.1 ✓ 4 72.67 odd 6
288.2.i.d.97.2 yes 4 72.13 even 6
288.2.i.d.193.1 yes 4 72.43 odd 6
288.2.i.d.193.2 yes 4 72.61 even 6
576.2.i.k.193.1 4 9.7 even 3
576.2.i.k.193.2 4 36.7 odd 6
576.2.i.k.385.1 4 9.4 even 3
576.2.i.k.385.2 4 36.31 odd 6
864.2.i.d.289.1 4 72.5 odd 6
864.2.i.d.289.2 4 72.59 even 6
864.2.i.d.577.1 4 72.29 odd 6
864.2.i.d.577.2 4 72.11 even 6
1728.2.i.l.577.1 4 9.2 odd 6
1728.2.i.l.577.2 4 36.11 even 6
1728.2.i.l.1153.1 4 9.5 odd 6
1728.2.i.l.1153.2 4 36.23 even 6
2592.2.a.l.1.1 2 8.3 odd 2
2592.2.a.l.1.2 2 8.5 even 2
2592.2.a.p.1.1 2 24.11 even 2
2592.2.a.p.1.2 2 24.5 odd 2
5184.2.a.bl.1.1 2 12.11 even 2
5184.2.a.bl.1.2 2 3.2 odd 2
5184.2.a.bx.1.1 2 4.3 odd 2 inner
5184.2.a.bx.1.2 2 1.1 even 1 trivial