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 1.1
Root \(-0.0147515\) 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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -160.171 q^{5} +729.000 q^{9} +3163.86 q^{11} -4771.53 q^{13} +4324.61 q^{15} -13009.5 q^{17} +44116.3 q^{19} -65066.0 q^{23} -52470.3 q^{25} -19683.0 q^{27} -246115. q^{29} +300657. q^{31} -85424.2 q^{33} +516957. q^{37} +128831. q^{39} -377844. q^{41} -71420.3 q^{43} -116765. q^{45} +1.11676e6 q^{47} +351255. q^{51} -368731. q^{53} -506758. q^{55} -1.19114e6 q^{57} +1.03195e6 q^{59} -322682. q^{61} +764260. q^{65} -23527.6 q^{67} +1.75678e6 q^{69} +2.84199e6 q^{71} +673046. q^{73} +1.41670e6 q^{75} -858896. q^{79} +531441. q^{81} +7.32276e6 q^{83} +2.08373e6 q^{85} +6.64512e6 q^{87} +4.62800e6 q^{89} -8.11773e6 q^{93} -7.06614e6 q^{95} +951821. q^{97} +2.30645e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 108 q^{3} + 196 q^{5} + 2916 q^{9} - 406 q^{11} + 1974 q^{13} - 5292 q^{15} + 7436 q^{17} - 15874 q^{19} + 6788 q^{23} - 69898 q^{25} - 78732 q^{27} - 94544 q^{29} + 55890 q^{31} + 10962 q^{33} - 93742 q^{37}+ \cdots - 295974 q^{99}+O(q^{100}) \) 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
\(5\) −160.171 −0.573045 −0.286522 − 0.958074i \(-0.592499\pi\)
−0.286522 + 0.958074i \(0.592499\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3163.86 0.716709 0.358354 − 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(12\) 0 0
\(13\) −4771.53 −0.602360 −0.301180 − 0.953567i \(-0.597380\pi\)
−0.301180 + 0.953567i \(0.597380\pi\)
\(14\) 0 0
\(15\) 4324.61 0.330847
\(16\) 0 0
\(17\) −13009.5 −0.642226 −0.321113 − 0.947041i \(-0.604057\pi\)
−0.321113 + 0.947041i \(0.604057\pi\)
\(18\) 0 0
\(19\) 44116.3 1.47557 0.737787 − 0.675033i \(-0.235871\pi\)
0.737787 + 0.675033i \(0.235871\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −65066.0 −1.11508 −0.557541 − 0.830149i \(-0.688255\pi\)
−0.557541 + 0.830149i \(0.688255\pi\)
\(24\) 0 0
\(25\) −52470.3 −0.671620
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −246115. −1.87390 −0.936949 − 0.349466i \(-0.886363\pi\)
−0.936949 + 0.349466i \(0.886363\pi\)
\(30\) 0 0
\(31\) 300657. 1.81261 0.906306 − 0.422622i \(-0.138890\pi\)
0.906306 + 0.422622i \(0.138890\pi\)
\(32\) 0 0
\(33\) −85424.2 −0.413792
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 516957. 1.67783 0.838916 − 0.544260i \(-0.183190\pi\)
0.838916 + 0.544260i \(0.183190\pi\)
\(38\) 0 0
\(39\) 128831. 0.347773
\(40\) 0 0
\(41\) −377844. −0.856188 −0.428094 − 0.903734i \(-0.640815\pi\)
−0.428094 + 0.903734i \(0.640815\pi\)
\(42\) 0 0
\(43\) −71420.3 −0.136988 −0.0684940 − 0.997652i \(-0.521819\pi\)
−0.0684940 + 0.997652i \(0.521819\pi\)
\(44\) 0 0
\(45\) −116765. −0.191015
\(46\) 0 0
\(47\) 1.11676e6 1.56898 0.784488 − 0.620144i \(-0.212926\pi\)
0.784488 + 0.620144i \(0.212926\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 351255. 0.370789
\(52\) 0 0
\(53\) −368731. −0.340208 −0.170104 − 0.985426i \(-0.554410\pi\)
−0.170104 + 0.985426i \(0.554410\pi\)
\(54\) 0 0
\(55\) −506758. −0.410706
\(56\) 0 0
\(57\) −1.19114e6 −0.851924
\(58\) 0 0
\(59\) 1.03195e6 0.654151 0.327076 − 0.944998i \(-0.393937\pi\)
0.327076 + 0.944998i \(0.393937\pi\)
\(60\) 0 0
\(61\) −322682. −0.182020 −0.0910101 − 0.995850i \(-0.529010\pi\)
−0.0910101 + 0.995850i \(0.529010\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 764260. 0.345179
\(66\) 0 0
\(67\) −23527.6 −0.00955688 −0.00477844 − 0.999989i \(-0.501521\pi\)
−0.00477844 + 0.999989i \(0.501521\pi\)
\(68\) 0 0
\(69\) 1.75678e6 0.643793
\(70\) 0 0
\(71\) 2.84199e6 0.942363 0.471181 − 0.882036i \(-0.343828\pi\)
0.471181 + 0.882036i \(0.343828\pi\)
\(72\) 0 0
\(73\) 673046. 0.202495 0.101248 − 0.994861i \(-0.467717\pi\)
0.101248 + 0.994861i \(0.467717\pi\)
\(74\) 0 0
\(75\) 1.41670e6 0.387760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −858896. −0.195995 −0.0979977 − 0.995187i \(-0.531244\pi\)
−0.0979977 + 0.995187i \(0.531244\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 7.32276e6 1.40573 0.702864 − 0.711324i \(-0.251904\pi\)
0.702864 + 0.711324i \(0.251904\pi\)
\(84\) 0 0
\(85\) 2.08373e6 0.368024
\(86\) 0 0
\(87\) 6.64512e6 1.08190
\(88\) 0 0
\(89\) 4.62800e6 0.695870 0.347935 − 0.937519i \(-0.386883\pi\)
0.347935 + 0.937519i \(0.386883\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.11773e6 −1.04651
\(94\) 0 0
\(95\) −7.06614e6 −0.845570
\(96\) 0 0
\(97\) 951821. 0.105890 0.0529449 − 0.998597i \(-0.483139\pi\)
0.0529449 + 0.998597i \(0.483139\pi\)
\(98\) 0 0
\(99\) 2.30645e6 0.238903
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.i.1.1 4
7.2 even 3 84.8.i.a.25.4 ✓ 8
7.3 odd 6 588.8.i.o.373.1 8
7.4 even 3 84.8.i.a.37.4 yes 8
7.5 odd 6 588.8.i.o.361.1 8
7.6 odd 2 588.8.a.j.1.4 4
21.2 odd 6 252.8.k.b.109.1 8
21.11 odd 6 252.8.k.b.37.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.4 ✓ 8 7.2 even 3
84.8.i.a.37.4 yes 8 7.4 even 3
252.8.k.b.37.1 8 21.11 odd 6
252.8.k.b.109.1 8 21.2 odd 6
588.8.a.i.1.1 4 1.1 even 1 trivial
588.8.a.j.1.4 4 7.6 odd 2
588.8.i.o.361.1 8 7.5 odd 6
588.8.i.o.373.1 8 7.3 odd 6