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Properties

Label 588.6.a.n.1.1
Level $588$
Weight $6$
Character 588.1
Self dual yes
Analytic conductor $94.306$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-36,0,0,0,0,0,324,0,462] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
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: \( x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-22.4831\) 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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -92.8255 q^{5} +81.0000 q^{9} +140.762 q^{11} -1111.24 q^{13} +835.430 q^{15} +54.8869 q^{17} -1711.86 q^{19} -3287.91 q^{23} +5491.58 q^{25} -729.000 q^{27} -3790.72 q^{29} -4847.32 q^{31} -1266.86 q^{33} -11366.7 q^{37} +10001.2 q^{39} -10385.6 q^{41} +7137.16 q^{43} -7518.87 q^{45} -16415.1 q^{47} -493.983 q^{51} -20974.3 q^{53} -13066.3 q^{55} +15406.7 q^{57} -36211.9 q^{59} +4948.70 q^{61} +103152. q^{65} -22965.6 q^{67} +29591.2 q^{69} -26341.8 q^{71} +55387.4 q^{73} -49424.2 q^{75} -49956.6 q^{79} +6561.00 q^{81} +44858.9 q^{83} -5094.91 q^{85} +34116.5 q^{87} +127945. q^{89} +43625.9 q^{93} +158904. q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41}+ \cdots + 37422 q^{99}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −92.8255 −1.66051 −0.830257 − 0.557381i \(-0.811806\pi\)
−0.830257 + 0.557381i \(0.811806\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 140.762 0.350756 0.175378 − 0.984501i \(-0.443885\pi\)
0.175378 + 0.984501i \(0.443885\pi\)
\(12\) 0 0
\(13\) −1111.24 −1.82369 −0.911844 − 0.410537i \(-0.865341\pi\)
−0.911844 + 0.410537i \(0.865341\pi\)
\(14\) 0 0
\(15\) 835.430 0.958698
\(16\) 0 0
\(17\) 54.8869 0.0460624 0.0230312 − 0.999735i \(-0.492668\pi\)
0.0230312 + 0.999735i \(0.492668\pi\)
\(18\) 0 0
\(19\) −1711.86 −1.08789 −0.543943 − 0.839122i \(-0.683069\pi\)
−0.543943 + 0.839122i \(0.683069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3287.91 −1.29599 −0.647993 − 0.761646i \(-0.724391\pi\)
−0.647993 + 0.761646i \(0.724391\pi\)
\(24\) 0 0
\(25\) 5491.58 1.75731
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −3790.72 −0.837003 −0.418501 − 0.908216i \(-0.637445\pi\)
−0.418501 + 0.908216i \(0.637445\pi\)
\(30\) 0 0
\(31\) −4847.32 −0.905936 −0.452968 − 0.891527i \(-0.649635\pi\)
−0.452968 + 0.891527i \(0.649635\pi\)
\(32\) 0 0
\(33\) −1266.86 −0.202509
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11366.7 −1.36499 −0.682496 − 0.730889i \(-0.739106\pi\)
−0.682496 + 0.730889i \(0.739106\pi\)
\(38\) 0 0
\(39\) 10001.2 1.05291
\(40\) 0 0
\(41\) −10385.6 −0.964881 −0.482440 − 0.875929i \(-0.660249\pi\)
−0.482440 + 0.875929i \(0.660249\pi\)
\(42\) 0 0
\(43\) 7137.16 0.588646 0.294323 − 0.955706i \(-0.404906\pi\)
0.294323 + 0.955706i \(0.404906\pi\)
\(44\) 0 0
\(45\) −7518.87 −0.553505
\(46\) 0 0
\(47\) −16415.1 −1.08392 −0.541961 − 0.840404i \(-0.682318\pi\)
−0.541961 + 0.840404i \(0.682318\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −493.983 −0.0265942
\(52\) 0 0
\(53\) −20974.3 −1.02565 −0.512824 − 0.858494i \(-0.671401\pi\)
−0.512824 + 0.858494i \(0.671401\pi\)
\(54\) 0 0
\(55\) −13066.3 −0.582435
\(56\) 0 0
\(57\) 15406.7 0.628092
\(58\) 0 0
\(59\) −36211.9 −1.35432 −0.677161 − 0.735835i \(-0.736790\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(60\) 0 0
\(61\) 4948.70 0.170281 0.0851405 − 0.996369i \(-0.472866\pi\)
0.0851405 + 0.996369i \(0.472866\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 103152. 3.02826
\(66\) 0 0
\(67\) −22965.6 −0.625015 −0.312507 − 0.949915i \(-0.601169\pi\)
−0.312507 + 0.949915i \(0.601169\pi\)
\(68\) 0 0
\(69\) 29591.2 0.748238
\(70\) 0 0
\(71\) −26341.8 −0.620154 −0.310077 − 0.950711i \(-0.600355\pi\)
−0.310077 + 0.950711i \(0.600355\pi\)
\(72\) 0 0
\(73\) 55387.4 1.21648 0.608238 − 0.793755i \(-0.291877\pi\)
0.608238 + 0.793755i \(0.291877\pi\)
\(74\) 0 0
\(75\) −49424.2 −1.01458
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −49956.6 −0.900586 −0.450293 − 0.892881i \(-0.648680\pi\)
−0.450293 + 0.892881i \(0.648680\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 44858.9 0.714749 0.357374 − 0.933961i \(-0.383672\pi\)
0.357374 + 0.933961i \(0.383672\pi\)
\(84\) 0 0
\(85\) −5094.91 −0.0764873
\(86\) 0 0
\(87\) 34116.5 0.483244
\(88\) 0 0
\(89\) 127945. 1.71217 0.856087 − 0.516831i \(-0.172889\pi\)
0.856087 + 0.516831i \(0.172889\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 43625.9 0.523042
\(94\) 0 0
\(95\) 158904. 1.80645
\(96\) 0 0
\(97\) −65685.9 −0.708831 −0.354415 − 0.935088i \(-0.615320\pi\)
−0.354415 + 0.935088i \(0.615320\pi\)
\(98\) 0 0
\(99\) 11401.8 0.116919
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 588.6.a.n.1.1 4
7.2 even 3 84.6.i.c.25.4 ✓ 8
7.3 odd 6 588.6.i.o.373.1 8
7.4 even 3 84.6.i.c.37.4 yes 8
7.5 odd 6 588.6.i.o.361.1 8
7.6 odd 2 588.6.a.p.1.4 4
21.2 odd 6 252.6.k.f.109.1 8
21.11 odd 6 252.6.k.f.37.1 8
28.11 odd 6 336.6.q.i.289.4 8
28.23 odd 6 336.6.q.i.193.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.4 ✓ 8 7.2 even 3
84.6.i.c.37.4 yes 8 7.4 even 3
252.6.k.f.37.1 8 21.11 odd 6
252.6.k.f.109.1 8 21.2 odd 6
336.6.q.i.193.4 8 28.23 odd 6
336.6.q.i.289.4 8 28.11 odd 6
588.6.a.n.1.1 4 1.1 even 1 trivial
588.6.a.p.1.4 4 7.6 odd 2
588.6.i.o.361.1 8 7.5 odd 6
588.6.i.o.373.1 8 7.3 odd 6