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Properties

Label 588.6.a.p.1.3
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 1.3
Root \(26.2941\) 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)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} +31.9616 q^{5} +81.0000 q^{9} +260.884 q^{11} -769.735 q^{13} +287.654 q^{15} +1553.32 q^{17} -750.049 q^{19} +754.854 q^{23} -2103.46 q^{25} +729.000 q^{27} +6008.93 q^{29} +6420.04 q^{31} +2347.95 q^{33} -4775.73 q^{37} -6927.62 q^{39} +5423.27 q^{41} -11896.4 q^{43} +2588.89 q^{45} +17428.0 q^{47} +13979.9 q^{51} +37650.6 q^{53} +8338.26 q^{55} -6750.44 q^{57} -22078.1 q^{59} +8173.38 q^{61} -24602.0 q^{65} +13001.7 q^{67} +6793.69 q^{69} -12349.6 q^{71} -43600.3 q^{73} -18931.1 q^{75} +76749.5 q^{79} +6561.00 q^{81} -21893.6 q^{83} +49646.5 q^{85} +54080.4 q^{87} +136967. q^{89} +57780.4 q^{93} -23972.8 q^{95} +93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{3} + 324 q^{9} + 462 q^{11} + 602 q^{13} + 228 q^{17} + 358 q^{19} + 2148 q^{23} + 5454 q^{25} + 2916 q^{27} - 5532 q^{29} + 830 q^{31} + 4158 q^{33} + 3914 q^{37} + 5418 q^{39} + 8316 q^{41}+ \cdots + 37422 q^{99}+O(q^{100}) \) 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
\(5\) 31.9616 0.571746 0.285873 − 0.958267i \(-0.407716\pi\)
0.285873 + 0.958267i \(0.407716\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 260.884 0.650077 0.325039 − 0.945701i \(-0.394623\pi\)
0.325039 + 0.945701i \(0.394623\pi\)
\(12\) 0 0
\(13\) −769.735 −1.26323 −0.631616 − 0.775281i \(-0.717608\pi\)
−0.631616 + 0.775281i \(0.717608\pi\)
\(14\) 0 0
\(15\) 287.654 0.330098
\(16\) 0 0
\(17\) 1553.32 1.30358 0.651791 − 0.758399i \(-0.274018\pi\)
0.651791 + 0.758399i \(0.274018\pi\)
\(18\) 0 0
\(19\) −750.049 −0.476656 −0.238328 − 0.971185i \(-0.576599\pi\)
−0.238328 + 0.971185i \(0.576599\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 754.854 0.297539 0.148769 − 0.988872i \(-0.452469\pi\)
0.148769 + 0.988872i \(0.452469\pi\)
\(24\) 0 0
\(25\) −2103.46 −0.673106
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 6008.93 1.32679 0.663395 − 0.748269i \(-0.269115\pi\)
0.663395 + 0.748269i \(0.269115\pi\)
\(30\) 0 0
\(31\) 6420.04 1.19987 0.599934 − 0.800049i \(-0.295193\pi\)
0.599934 + 0.800049i \(0.295193\pi\)
\(32\) 0 0
\(33\) 2347.95 0.375322
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4775.73 −0.573502 −0.286751 − 0.958005i \(-0.592575\pi\)
−0.286751 + 0.958005i \(0.592575\pi\)
\(38\) 0 0
\(39\) −6927.62 −0.729327
\(40\) 0 0
\(41\) 5423.27 0.503850 0.251925 − 0.967747i \(-0.418936\pi\)
0.251925 + 0.967747i \(0.418936\pi\)
\(42\) 0 0
\(43\) −11896.4 −0.981171 −0.490585 − 0.871393i \(-0.663217\pi\)
−0.490585 + 0.871393i \(0.663217\pi\)
\(44\) 0 0
\(45\) 2588.89 0.190582
\(46\) 0 0
\(47\) 17428.0 1.15081 0.575404 − 0.817869i \(-0.304845\pi\)
0.575404 + 0.817869i \(0.304845\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 13979.9 0.752623
\(52\) 0 0
\(53\) 37650.6 1.84112 0.920561 − 0.390599i \(-0.127732\pi\)
0.920561 + 0.390599i \(0.127732\pi\)
\(54\) 0 0
\(55\) 8338.26 0.371679
\(56\) 0 0
\(57\) −6750.44 −0.275198
\(58\) 0 0
\(59\) −22078.1 −0.825717 −0.412859 − 0.910795i \(-0.635470\pi\)
−0.412859 + 0.910795i \(0.635470\pi\)
\(60\) 0 0
\(61\) 8173.38 0.281240 0.140620 − 0.990064i \(-0.455090\pi\)
0.140620 + 0.990064i \(0.455090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −24602.0 −0.722248
\(66\) 0 0
\(67\) 13001.7 0.353846 0.176923 − 0.984225i \(-0.443386\pi\)
0.176923 + 0.984225i \(0.443386\pi\)
\(68\) 0 0
\(69\) 6793.69 0.171784
\(70\) 0 0
\(71\) −12349.6 −0.290742 −0.145371 − 0.989377i \(-0.546438\pi\)
−0.145371 + 0.989377i \(0.546438\pi\)
\(72\) 0 0
\(73\) −43600.3 −0.957596 −0.478798 − 0.877925i \(-0.658927\pi\)
−0.478798 + 0.877925i \(0.658927\pi\)
\(74\) 0 0
\(75\) −18931.1 −0.388618
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 76749.5 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −21893.6 −0.348836 −0.174418 − 0.984672i \(-0.555804\pi\)
−0.174418 + 0.984672i \(0.555804\pi\)
\(84\) 0 0
\(85\) 49646.5 0.745318
\(86\) 0 0
\(87\) 54080.4 0.766023
\(88\) 0 0
\(89\) 136967. 1.83291 0.916454 − 0.400139i \(-0.131038\pi\)
0.916454 + 0.400139i \(0.131038\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 57780.4 0.692744
\(94\) 0 0
\(95\) −23972.8 −0.272527
\(96\) 0 0
\(97\) 93050.1 1.00412 0.502062 − 0.864832i \(-0.332575\pi\)
0.502062 + 0.864832i \(0.332575\pi\)
\(98\) 0 0
\(99\) 21131.6 0.216692
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.p.1.3 4
7.2 even 3 588.6.i.o.361.2 8
7.3 odd 6 84.6.i.c.37.3 yes 8
7.4 even 3 588.6.i.o.373.2 8
7.5 odd 6 84.6.i.c.25.3 ✓ 8
7.6 odd 2 588.6.a.n.1.2 4
21.5 even 6 252.6.k.f.109.2 8
21.17 even 6 252.6.k.f.37.2 8
28.3 even 6 336.6.q.i.289.3 8
28.19 even 6 336.6.q.i.193.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.3 ✓ 8 7.5 odd 6
84.6.i.c.37.3 yes 8 7.3 odd 6
252.6.k.f.37.2 8 21.17 even 6
252.6.k.f.109.2 8 21.5 even 6
336.6.q.i.193.3 8 28.19 even 6
336.6.q.i.289.3 8 28.3 even 6
588.6.a.n.1.2 4 7.6 odd 2
588.6.a.p.1.3 4 1.1 even 1 trivial
588.6.i.o.361.2 8 7.2 even 3
588.6.i.o.373.2 8 7.4 even 3