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 = [2,0,0,0,2,0,2,0,0,0,2,0,2,0,0,0,0] 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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 1.1
Root \(-2.44949\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -1.44949 q^{7} +3.44949 q^{11} -3.89898 q^{13} +4.89898 q^{17} +4.00000 q^{19} +0.550510 q^{23} -4.00000 q^{25} -9.89898 q^{29} +7.44949 q^{31} -1.44949 q^{35} +8.89898 q^{37} +2.10102 q^{41} +12.3485 q^{43} -8.34847 q^{47} -4.89898 q^{49} +0.898979 q^{53} +3.44949 q^{55} +0.348469 q^{59} +1.89898 q^{61} -3.89898 q^{65} -2.34847 q^{67} +11.7980 q^{71} +4.89898 q^{73} -5.00000 q^{77} +8.55051 q^{79} +5.44949 q^{83} +4.89898 q^{85} +3.10102 q^{89} +5.65153 q^{91} +4.00000 q^{95} +5.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} + 8 q^{19} + 6 q^{23} - 8 q^{25} - 10 q^{29} + 10 q^{31} + 2 q^{35} + 8 q^{37} + 14 q^{41} + 10 q^{43} - 2 q^{47} - 8 q^{53} + 2 q^{55} - 14 q^{59} - 6 q^{61}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(7\) −1.44949 −0.547856 −0.273928 − 0.961750i \(-0.588323\pi\)
−0.273928 + 0.961750i \(0.588323\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 − 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 0 0
\(13\) −3.89898 −1.08138 −0.540691 − 0.841221i \(-0.681837\pi\)
−0.540691 + 0.841221i \(0.681837\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 − 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.550510 0.114789 0.0573947 − 0.998352i \(-0.481721\pi\)
0.0573947 + 0.998352i \(0.481721\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.89898 −1.83819 −0.919097 − 0.394031i \(-0.871080\pi\)
−0.919097 + 0.394031i \(0.871080\pi\)
\(30\) 0 0
\(31\) 7.44949 1.33797 0.668984 − 0.743277i \(-0.266729\pi\)
0.668984 + 0.743277i \(0.266729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.44949 −0.245008
\(36\) 0 0
\(37\) 8.89898 1.46298 0.731492 − 0.681850i \(-0.238825\pi\)
0.731492 + 0.681850i \(0.238825\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.10102 0.328124 0.164062 − 0.986450i \(-0.447540\pi\)
0.164062 + 0.986450i \(0.447540\pi\)
\(42\) 0 0
\(43\) 12.3485 1.88312 0.941562 − 0.336840i \(-0.109358\pi\)
0.941562 + 0.336840i \(0.109358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.34847 −1.21775 −0.608875 − 0.793266i \(-0.708379\pi\)
−0.608875 + 0.793266i \(0.708379\pi\)
\(48\) 0 0
\(49\) −4.89898 −0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.898979 0.123484 0.0617422 − 0.998092i \(-0.480334\pi\)
0.0617422 + 0.998092i \(0.480334\pi\)
\(54\) 0 0
\(55\) 3.44949 0.465129
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.348469 0.0453668 0.0226834 − 0.999743i \(-0.492779\pi\)
0.0226834 + 0.999743i \(0.492779\pi\)
\(60\) 0 0
\(61\) 1.89898 0.243139 0.121570 − 0.992583i \(-0.461207\pi\)
0.121570 + 0.992583i \(0.461207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.89898 −0.483609
\(66\) 0 0
\(67\) −2.34847 −0.286911 −0.143456 − 0.989657i \(-0.545821\pi\)
−0.143456 + 0.989657i \(0.545821\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.7980 1.40016 0.700080 − 0.714064i \(-0.253148\pi\)
0.700080 + 0.714064i \(0.253148\pi\)
\(72\) 0 0
\(73\) 4.89898 0.573382 0.286691 − 0.958023i \(-0.407445\pi\)
0.286691 + 0.958023i \(0.407445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 8.55051 0.962008 0.481004 − 0.876719i \(-0.340272\pi\)
0.481004 + 0.876719i \(0.340272\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.44949 0.598159 0.299080 − 0.954228i \(-0.403320\pi\)
0.299080 + 0.954228i \(0.403320\pi\)
\(84\) 0 0
\(85\) 4.89898 0.531369
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.10102 0.328708 0.164354 − 0.986401i \(-0.447446\pi\)
0.164354 + 0.986401i \(0.447446\pi\)
\(90\) 0 0
\(91\) 5.65153 0.592441
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 5.89898 0.598951 0.299475 − 0.954104i \(-0.403188\pi\)
0.299475 + 0.954104i \(0.403188\pi\)
\(98\) 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.s.1.1 2
3.2 odd 2 2592.2.a.n.1.1 2
4.3 odd 2 2592.2.a.o.1.2 2
8.3 odd 2 5184.2.a.bj.1.2 2
8.5 even 2 5184.2.a.bn.1.1 2
9.2 odd 6 288.2.i.e.193.2 yes 4
9.4 even 3 864.2.i.c.289.2 4
9.5 odd 6 288.2.i.e.97.1 yes 4
9.7 even 3 864.2.i.c.577.2 4
12.11 even 2 2592.2.a.j.1.2 2
24.5 odd 2 5184.2.a.by.1.1 2
24.11 even 2 5184.2.a.bu.1.2 2
36.7 odd 6 864.2.i.e.577.1 4
36.11 even 6 288.2.i.c.193.1 yes 4
36.23 even 6 288.2.i.c.97.2 ✓ 4
36.31 odd 6 864.2.i.e.289.1 4
72.5 odd 6 576.2.i.i.385.2 4
72.11 even 6 576.2.i.m.193.2 4
72.13 even 6 1728.2.i.k.1153.2 4
72.29 odd 6 576.2.i.i.193.1 4
72.43 odd 6 1728.2.i.m.577.1 4
72.59 even 6 576.2.i.m.385.1 4
72.61 even 6 1728.2.i.k.577.2 4
72.67 odd 6 1728.2.i.m.1153.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.c.97.2 ✓ 4 36.23 even 6
288.2.i.c.193.1 yes 4 36.11 even 6
288.2.i.e.97.1 yes 4 9.5 odd 6
288.2.i.e.193.2 yes 4 9.2 odd 6
576.2.i.i.193.1 4 72.29 odd 6
576.2.i.i.385.2 4 72.5 odd 6
576.2.i.m.193.2 4 72.11 even 6
576.2.i.m.385.1 4 72.59 even 6
864.2.i.c.289.2 4 9.4 even 3
864.2.i.c.577.2 4 9.7 even 3
864.2.i.e.289.1 4 36.31 odd 6
864.2.i.e.577.1 4 36.7 odd 6
1728.2.i.k.577.2 4 72.61 even 6
1728.2.i.k.1153.2 4 72.13 even 6
1728.2.i.m.577.1 4 72.43 odd 6
1728.2.i.m.1153.1 4 72.67 odd 6
2592.2.a.j.1.2 2 12.11 even 2
2592.2.a.n.1.1 2 3.2 odd 2
2592.2.a.o.1.2 2 4.3 odd 2
2592.2.a.s.1.1 2 1.1 even 1 trivial
5184.2.a.bj.1.2 2 8.3 odd 2
5184.2.a.bn.1.1 2 8.5 even 2
5184.2.a.bu.1.2 2 24.11 even 2
5184.2.a.by.1.1 2 24.5 odd 2