Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(1,588)] 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("588.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,108,0,-196] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(183.682394985\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 655x^{2} - 5704x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-11.2707\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +129.570 q^{5} +729.000 q^{9} -4519.97 q^{11} -7889.70 q^{13} +3498.38 q^{15} +7086.56 q^{17} +14073.6 q^{19} +69012.2 q^{23} -61336.7 q^{25} +19683.0 q^{27} +47362.7 q^{29} +170141. q^{31} -122039. q^{33} +266644. q^{37} -213022. q^{39} -574470. q^{41} +10913.1 q^{43} +94456.4 q^{45} -153793. q^{47} +191337. q^{51} -1.66613e6 q^{53} -585652. q^{55} +379987. q^{57} -411829. q^{59} -2.67276e6 q^{61} -1.02227e6 q^{65} +1.11127e6 q^{67} +1.86333e6 q^{69} +55167.2 q^{71} -3.15505e6 q^{73} -1.65609e6 q^{75} -428065. q^{79} +531441. q^{81} +754782. q^{83} +918204. q^{85} +1.27879e6 q^{87} +1.18486e7 q^{89} +4.59382e6 q^{93} +1.82351e6 q^{95} -7.12850e6 q^{97} -3.29506e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{3} - 196 q^{5} + 2916 q^{9} - 406 q^{11} - 1974 q^{13} - 5292 q^{15} - 7436 q^{17} + 15874 q^{19} + 6788 q^{23} - 69898 q^{25} + 78732 q^{27} - 94544 q^{29} - 55890 q^{31} - 10962 q^{33} - 93742 q^{37}+ \cdots - 295974 q^{99}+O(q^{100}) \) 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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 129.570 0.463563 0.231782 − 0.972768i \(-0.425545\pi\)
0.231782 + 0.972768i \(0.425545\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −4519.97 −1.02391 −0.511955 − 0.859012i \(-0.671078\pi\)
−0.511955 + 0.859012i \(0.671078\pi\)
\(12\) 0 0
\(13\) −7889.70 −0.995999 −0.497999 − 0.867177i \(-0.665932\pi\)
−0.497999 + 0.867177i \(0.665932\pi\)
\(14\) 0 0
\(15\) 3498.38 0.267638
\(16\) 0 0
\(17\) 7086.56 0.349836 0.174918 − 0.984583i \(-0.444034\pi\)
0.174918 + 0.984583i \(0.444034\pi\)
\(18\) 0 0
\(19\) 14073.6 0.470726 0.235363 − 0.971908i \(-0.424372\pi\)
0.235363 + 0.971908i \(0.424372\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 69012.2 1.18271 0.591355 − 0.806411i \(-0.298593\pi\)
0.591355 + 0.806411i \(0.298593\pi\)
\(24\) 0 0
\(25\) −61336.7 −0.785109
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 47362.7 0.360615 0.180307 − 0.983610i \(-0.442291\pi\)
0.180307 + 0.983610i \(0.442291\pi\)
\(30\) 0 0
\(31\) 170141. 1.02576 0.512878 − 0.858462i \(-0.328579\pi\)
0.512878 + 0.858462i \(0.328579\pi\)
\(32\) 0 0
\(33\) −122039. −0.591154
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 266644. 0.865416 0.432708 − 0.901534i \(-0.357558\pi\)
0.432708 + 0.901534i \(0.357558\pi\)
\(38\) 0 0
\(39\) −213022. −0.575040
\(40\) 0 0
\(41\) −574470. −1.30174 −0.650869 − 0.759190i \(-0.725595\pi\)
−0.650869 + 0.759190i \(0.725595\pi\)
\(42\) 0 0
\(43\) 10913.1 0.0209320 0.0104660 − 0.999945i \(-0.496669\pi\)
0.0104660 + 0.999945i \(0.496669\pi\)
\(44\) 0 0
\(45\) 94456.4 0.154521
\(46\) 0 0
\(47\) −153793. −0.216070 −0.108035 − 0.994147i \(-0.534456\pi\)
−0.108035 + 0.994147i \(0.534456\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 191337. 0.201978
\(52\) 0 0
\(53\) −1.66613e6 −1.53725 −0.768623 − 0.639703i \(-0.779058\pi\)
−0.768623 + 0.639703i \(0.779058\pi\)
\(54\) 0 0
\(55\) −585652. −0.474647
\(56\) 0 0
\(57\) 379987. 0.271774
\(58\) 0 0
\(59\) −411829. −0.261057 −0.130528 − 0.991445i \(-0.541667\pi\)
−0.130528 + 0.991445i \(0.541667\pi\)
\(60\) 0 0
\(61\) −2.67276e6 −1.50767 −0.753833 − 0.657066i \(-0.771797\pi\)
−0.753833 + 0.657066i \(0.771797\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.02227e6 −0.461708
\(66\) 0 0
\(67\) 1.11127e6 0.451397 0.225698 − 0.974197i \(-0.427534\pi\)
0.225698 + 0.974197i \(0.427534\pi\)
\(68\) 0 0
\(69\) 1.86333e6 0.682838
\(70\) 0 0
\(71\) 55167.2 0.0182927 0.00914633 − 0.999958i \(-0.497089\pi\)
0.00914633 + 0.999958i \(0.497089\pi\)
\(72\) 0 0
\(73\) −3.15505e6 −0.949240 −0.474620 − 0.880191i \(-0.657414\pi\)
−0.474620 + 0.880191i \(0.657414\pi\)
\(74\) 0 0
\(75\) −1.65609e6 −0.453283
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −428065. −0.0976821 −0.0488411 − 0.998807i \(-0.515553\pi\)
−0.0488411 + 0.998807i \(0.515553\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 754782. 0.144893 0.0724466 − 0.997372i \(-0.476919\pi\)
0.0724466 + 0.997372i \(0.476919\pi\)
\(84\) 0 0
\(85\) 918204. 0.162171
\(86\) 0 0
\(87\) 1.27879e6 0.208201
\(88\) 0 0
\(89\) 1.18486e7 1.78156 0.890781 − 0.454433i \(-0.150158\pi\)
0.890781 + 0.454433i \(0.150158\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.59382e6 0.592220
\(94\) 0 0
\(95\) 1.82351e6 0.218211
\(96\) 0 0
\(97\) −7.12850e6 −0.793044 −0.396522 − 0.918025i \(-0.629783\pi\)
−0.396522 + 0.918025i \(0.629783\pi\)
\(98\) 0 0
\(99\) −3.29506e6 −0.341303
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 588.8.a.j.1.3 4
7.2 even 3 588.8.i.o.361.2 8
7.3 odd 6 84.8.i.a.37.3 yes 8
7.4 even 3 588.8.i.o.373.2 8
7.5 odd 6 84.8.i.a.25.3 ✓ 8
7.6 odd 2 588.8.a.i.1.2 4
21.5 even 6 252.8.k.b.109.2 8
21.17 even 6 252.8.k.b.37.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.3 ✓ 8 7.5 odd 6
84.8.i.a.37.3 yes 8 7.3 odd 6
252.8.k.b.37.2 8 21.17 even 6
252.8.k.b.109.2 8 21.5 even 6
588.8.a.i.1.2 4 7.6 odd 2
588.8.a.j.1.3 4 1.1 even 1 trivial
588.8.i.o.361.2 8 7.2 even 3
588.8.i.o.373.2 8 7.4 even 3