Properties

Label 588.6.a.m.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,-36,0,0,0,0,0,324,0,0] 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: \(\Q(\sqrt{177 +28 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 87x^{2} + 88x + 1838 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(7.85863\) 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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -9.02102 q^{5} +81.0000 q^{9} +605.882 q^{11} +227.306 q^{13} +81.1892 q^{15} +1321.01 q^{17} +2419.11 q^{19} -3306.10 q^{23} -3043.62 q^{25} -729.000 q^{27} -2436.19 q^{29} -6369.36 q^{31} -5452.94 q^{33} +8419.47 q^{37} -2045.75 q^{39} +6776.47 q^{41} +6590.39 q^{43} -730.703 q^{45} +2264.36 q^{47} -11889.1 q^{51} -16794.2 q^{53} -5465.67 q^{55} -21772.0 q^{57} -14715.9 q^{59} +38644.8 q^{61} -2050.53 q^{65} +7550.27 q^{67} +29754.9 q^{69} -16467.0 q^{71} -11967.6 q^{73} +27392.6 q^{75} -13021.3 q^{79} +6561.00 q^{81} +66229.5 q^{83} -11916.9 q^{85} +21925.7 q^{87} +10484.4 q^{89} +57324.3 q^{93} -21822.9 q^{95} +91308.8 q^{97} +49076.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9} + 1872 q^{17} + 1728 q^{19} - 3648 q^{23} - 3996 q^{25} - 2916 q^{27} - 1248 q^{29} + 3888 q^{31} - 12032 q^{37} - 9072 q^{41} - 2128 q^{43} + 19872 q^{47} - 16848 q^{51} - 22248 q^{53}+ \cdots + 320544 q^{97}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −9.02102 −0.161373 −0.0806865 − 0.996740i \(-0.525711\pi\)
−0.0806865 + 0.996740i \(0.525711\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 605.882 1.50975 0.754877 − 0.655866i \(-0.227696\pi\)
0.754877 + 0.655866i \(0.227696\pi\)
\(12\) 0 0
\(13\) 227.306 0.373038 0.186519 − 0.982451i \(-0.440279\pi\)
0.186519 + 0.982451i \(0.440279\pi\)
\(14\) 0 0
\(15\) 81.1892 0.0931687
\(16\) 0 0
\(17\) 1321.01 1.10863 0.554313 − 0.832308i \(-0.312981\pi\)
0.554313 + 0.832308i \(0.312981\pi\)
\(18\) 0 0
\(19\) 2419.11 1.53735 0.768673 − 0.639641i \(-0.220917\pi\)
0.768673 + 0.639641i \(0.220917\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3306.10 −1.30316 −0.651578 − 0.758581i \(-0.725893\pi\)
−0.651578 + 0.758581i \(0.725893\pi\)
\(24\) 0 0
\(25\) −3043.62 −0.973959
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −2436.19 −0.537918 −0.268959 − 0.963152i \(-0.586680\pi\)
−0.268959 + 0.963152i \(0.586680\pi\)
\(30\) 0 0
\(31\) −6369.36 −1.19040 −0.595198 − 0.803579i \(-0.702926\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(32\) 0 0
\(33\) −5452.94 −0.871657
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8419.47 1.01107 0.505534 − 0.862806i \(-0.331295\pi\)
0.505534 + 0.862806i \(0.331295\pi\)
\(38\) 0 0
\(39\) −2045.75 −0.215373
\(40\) 0 0
\(41\) 6776.47 0.629570 0.314785 − 0.949163i \(-0.398068\pi\)
0.314785 + 0.949163i \(0.398068\pi\)
\(42\) 0 0
\(43\) 6590.39 0.543550 0.271775 − 0.962361i \(-0.412389\pi\)
0.271775 + 0.962361i \(0.412389\pi\)
\(44\) 0 0
\(45\) −730.703 −0.0537910
\(46\) 0 0
\(47\) 2264.36 0.149521 0.0747603 − 0.997202i \(-0.476181\pi\)
0.0747603 + 0.997202i \(0.476181\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −11889.1 −0.640066
\(52\) 0 0
\(53\) −16794.2 −0.821240 −0.410620 − 0.911807i \(-0.634688\pi\)
−0.410620 + 0.911807i \(0.634688\pi\)
\(54\) 0 0
\(55\) −5465.67 −0.243633
\(56\) 0 0
\(57\) −21772.0 −0.887588
\(58\) 0 0
\(59\) −14715.9 −0.550371 −0.275186 − 0.961391i \(-0.588739\pi\)
−0.275186 + 0.961391i \(0.588739\pi\)
\(60\) 0 0
\(61\) 38644.8 1.32974 0.664869 − 0.746960i \(-0.268487\pi\)
0.664869 + 0.746960i \(0.268487\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2050.53 −0.0601982
\(66\) 0 0
\(67\) 7550.27 0.205483 0.102741 − 0.994708i \(-0.467239\pi\)
0.102741 + 0.994708i \(0.467239\pi\)
\(68\) 0 0
\(69\) 29754.9 0.752378
\(70\) 0 0
\(71\) −16467.0 −0.387676 −0.193838 − 0.981034i \(-0.562094\pi\)
−0.193838 + 0.981034i \(0.562094\pi\)
\(72\) 0 0
\(73\) −11967.6 −0.262845 −0.131422 − 0.991326i \(-0.541954\pi\)
−0.131422 + 0.991326i \(0.541954\pi\)
\(74\) 0 0
\(75\) 27392.6 0.562315
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13021.3 −0.234739 −0.117370 − 0.993088i \(-0.537446\pi\)
−0.117370 + 0.993088i \(0.537446\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 66229.5 1.05525 0.527626 − 0.849477i \(-0.323082\pi\)
0.527626 + 0.849477i \(0.323082\pi\)
\(84\) 0 0
\(85\) −11916.9 −0.178902
\(86\) 0 0
\(87\) 21925.7 0.310567
\(88\) 0 0
\(89\) 10484.4 0.140303 0.0701517 − 0.997536i \(-0.477652\pi\)
0.0701517 + 0.997536i \(0.477652\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 57324.3 0.687276
\(94\) 0 0
\(95\) −21822.9 −0.248086
\(96\) 0 0
\(97\) 91308.8 0.985334 0.492667 − 0.870218i \(-0.336022\pi\)
0.492667 + 0.870218i \(0.336022\pi\)
\(98\) 0 0
\(99\) 49076.4 0.503251
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.m.1.3 ✓ 4
7.2 even 3 588.6.i.q.361.2 8
7.3 odd 6 588.6.i.p.373.3 8
7.4 even 3 588.6.i.q.373.2 8
7.5 odd 6 588.6.i.p.361.3 8
7.6 odd 2 588.6.a.o.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.m.1.3 ✓ 4 1.1 even 1 trivial
588.6.a.o.1.2 yes 4 7.6 odd 2
588.6.i.p.361.3 8 7.5 odd 6
588.6.i.p.373.3 8 7.3 odd 6
588.6.i.q.361.2 8 7.2 even 3
588.6.i.q.373.2 8 7.4 even 3