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 = [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: \(41.3944484078\)
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 1.1
Root \(0.328543\) 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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{5} -2.20979 q^{7} -5.93580 q^{11} -4.37228 q^{13} +3.37228 q^{17} -3.72601 q^{19} -2.20979 q^{23} +0.627719 q^{25} -0.372281 q^{29} -9.66181 q^{31} +5.24224 q^{35} -4.00000 q^{37} +1.00000 q^{41} +5.93580 q^{43} -2.20979 q^{47} -2.11684 q^{49} +4.00000 q^{53} +14.0814 q^{55} +10.3554 q^{59} -15.1168 q^{61} +10.3723 q^{65} -10.3554 q^{67} -4.41957 q^{71} +4.62772 q^{73} +13.1168 q^{77} +9.66181 q^{79} -14.0814 q^{83} -8.00000 q^{85} +1.25544 q^{89} +9.66181 q^{91} +8.83915 q^{95} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 6 q^{13} + 2 q^{17} + 14 q^{25} + 10 q^{29} - 16 q^{37} + 4 q^{41} + 26 q^{49} + 16 q^{53} - 26 q^{61} + 30 q^{65} + 30 q^{73} + 18 q^{77} - 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) \) 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
\(5\) −2.37228 −1.06092 −0.530458 − 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) −2.20979 −0.835221 −0.417610 − 0.908626i \(-0.637132\pi\)
−0.417610 + 0.908626i \(0.637132\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.93580 −1.78971 −0.894855 − 0.446357i \(-0.852721\pi\)
−0.894855 + 0.446357i \(0.852721\pi\)
\(12\) 0 0
\(13\) −4.37228 −1.21265 −0.606326 − 0.795216i \(-0.707357\pi\)
−0.606326 + 0.795216i \(0.707357\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.37228 0.817898 0.408949 − 0.912557i \(-0.365895\pi\)
0.408949 + 0.912557i \(0.365895\pi\)
\(18\) 0 0
\(19\) −3.72601 −0.854805 −0.427403 − 0.904061i \(-0.640571\pi\)
−0.427403 + 0.904061i \(0.640571\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.20979 −0.460772 −0.230386 − 0.973099i \(-0.573999\pi\)
−0.230386 + 0.973099i \(0.573999\pi\)
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.372281 −0.0691309 −0.0345655 − 0.999402i \(-0.511005\pi\)
−0.0345655 + 0.999402i \(0.511005\pi\)
\(30\) 0 0
\(31\) −9.66181 −1.73531 −0.867656 − 0.497165i \(-0.834374\pi\)
−0.867656 + 0.497165i \(0.834374\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.24224 0.886099
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) 5.93580 0.905201 0.452600 − 0.891714i \(-0.350496\pi\)
0.452600 + 0.891714i \(0.350496\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.20979 −0.322330 −0.161165 − 0.986927i \(-0.551525\pi\)
−0.161165 + 0.986927i \(0.551525\pi\)
\(48\) 0 0
\(49\) −2.11684 −0.302406
\(50\) 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
\(55\) 14.0814 1.89873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3554 1.34815 0.674077 − 0.738661i \(-0.264542\pi\)
0.674077 + 0.738661i \(0.264542\pi\)
\(60\) 0 0
\(61\) −15.1168 −1.93551 −0.967757 − 0.251887i \(-0.918949\pi\)
−0.967757 + 0.251887i \(0.918949\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.3723 1.28652
\(66\) 0 0
\(67\) −10.3554 −1.26511 −0.632555 − 0.774516i \(-0.717994\pi\)
−0.632555 + 0.774516i \(0.717994\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.41957 −0.524507 −0.262253 − 0.964999i \(-0.584466\pi\)
−0.262253 + 0.964999i \(0.584466\pi\)
\(72\) 0 0
\(73\) 4.62772 0.541634 0.270817 − 0.962631i \(-0.412706\pi\)
0.270817 + 0.962631i \(0.412706\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.1168 1.49480
\(78\) 0 0
\(79\) 9.66181 1.08704 0.543519 − 0.839397i \(-0.317092\pi\)
0.543519 + 0.839397i \(0.317092\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.0814 −1.54563 −0.772816 − 0.634630i \(-0.781153\pi\)
−0.772816 + 0.634630i \(0.781153\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.25544 0.133076 0.0665380 − 0.997784i \(-0.478805\pi\)
0.0665380 + 0.997784i \(0.478805\pi\)
\(90\) 0 0
\(91\) 9.66181 1.01283
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.83915 0.906877
\(96\) 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 5184.2.a.cf.1.1 4
3.2 odd 2 5184.2.a.cc.1.3 4
4.3 odd 2 inner 5184.2.a.cf.1.2 4
8.3 odd 2 2592.2.a.u.1.4 4
8.5 even 2 2592.2.a.u.1.3 4
9.2 odd 6 1728.2.i.n.577.2 8
9.4 even 3 576.2.i.n.385.1 8
9.5 odd 6 1728.2.i.n.1153.2 8
9.7 even 3 576.2.i.n.193.1 8
12.11 even 2 5184.2.a.cc.1.4 4
24.5 odd 2 2592.2.a.x.1.1 4
24.11 even 2 2592.2.a.x.1.2 4
36.7 odd 6 576.2.i.n.193.4 8
36.11 even 6 1728.2.i.n.577.1 8
36.23 even 6 1728.2.i.n.1153.1 8
36.31 odd 6 576.2.i.n.385.4 8
72.5 odd 6 864.2.i.f.289.4 8
72.11 even 6 864.2.i.f.577.3 8
72.13 even 6 288.2.i.f.97.4 yes 8
72.29 odd 6 864.2.i.f.577.4 8
72.43 odd 6 288.2.i.f.193.1 yes 8
72.59 even 6 864.2.i.f.289.3 8
72.61 even 6 288.2.i.f.193.4 yes 8
72.67 odd 6 288.2.i.f.97.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.f.97.1 ✓ 8 72.67 odd 6
288.2.i.f.97.4 yes 8 72.13 even 6
288.2.i.f.193.1 yes 8 72.43 odd 6
288.2.i.f.193.4 yes 8 72.61 even 6
576.2.i.n.193.1 8 9.7 even 3
576.2.i.n.193.4 8 36.7 odd 6
576.2.i.n.385.1 8 9.4 even 3
576.2.i.n.385.4 8 36.31 odd 6
864.2.i.f.289.3 8 72.59 even 6
864.2.i.f.289.4 8 72.5 odd 6
864.2.i.f.577.3 8 72.11 even 6
864.2.i.f.577.4 8 72.29 odd 6
1728.2.i.n.577.1 8 36.11 even 6
1728.2.i.n.577.2 8 9.2 odd 6
1728.2.i.n.1153.1 8 36.23 even 6
1728.2.i.n.1153.2 8 9.5 odd 6
2592.2.a.u.1.3 4 8.5 even 2
2592.2.a.u.1.4 4 8.3 odd 2
2592.2.a.x.1.1 4 24.5 odd 2
2592.2.a.x.1.2 4 24.11 even 2
5184.2.a.cc.1.3 4 3.2 odd 2
5184.2.a.cc.1.4 4 12.11 even 2
5184.2.a.cf.1.1 4 1.1 even 1 trivial
5184.2.a.cf.1.2 4 4.3 odd 2 inner