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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-18,0,90,0,0,0,-162,0,-566] 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: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2641})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 661x^{2} + 660x + 435600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(-12.5977 - 21.8198i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.6.i.j.373.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 7.79423i) q^{3} +(-3.19533 - 5.53447i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-372.758 + 645.636i) q^{11} +879.125 q^{13} +57.5159 q^{15} +(-987.446 + 1710.31i) q^{17} +(484.297 + 838.827i) q^{19} +(-903.306 - 1564.57i) q^{23} +(1542.08 - 2670.96i) q^{25} +729.000 q^{27} +547.096 q^{29} +(3164.17 - 5480.51i) q^{31} +(-3354.82 - 5810.72i) q^{33} +(7315.55 + 12670.9i) q^{37} +(-3956.06 + 6852.10i) q^{39} -14953.6 q^{41} +16676.5 q^{43} +(-258.822 + 448.292i) q^{45} +(3829.00 + 6632.03i) q^{47} +(-8887.01 - 15392.8i) q^{51} +(-11994.5 + 20775.0i) q^{53} +4764.34 q^{55} -8717.35 q^{57} +(-7693.25 + 13325.1i) q^{59} +(-13653.9 - 23649.3i) q^{61} +(-2809.10 - 4865.50i) q^{65} +(259.516 - 449.495i) q^{67} +16259.5 q^{69} +10034.8 q^{71} +(-32842.9 + 56885.5i) q^{73} +(13878.7 + 24038.6i) q^{75} +(19691.4 + 34106.5i) q^{79} +(-3280.50 + 5681.99i) q^{81} +35581.5 q^{83} +12620.9 q^{85} +(-2461.93 + 4264.19i) q^{87} +(-54356.1 - 94147.4i) q^{89} +(28477.6 + 49324.6i) q^{93} +(3094.98 - 5360.66i) q^{95} -141108. q^{97} +60386.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 90 q^{5} - 162 q^{9} - 566 q^{11} + 1872 q^{13} - 1620 q^{15} - 558 q^{17} - 324 q^{19} + 2862 q^{23} - 3082 q^{25} + 2916 q^{27} - 8912 q^{29} - 1116 q^{31} - 5094 q^{33} + 23712 q^{37}+ \cdots + 91692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −4.50000 + 7.79423i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) −3.19533 − 5.53447i −0.0571598 − 0.0990037i 0.836030 − 0.548684i \(-0.184871\pi\)
−0.893189 + 0.449681i \(0.851538\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −372.758 + 645.636i −0.928849 + 1.60881i −0.143599 + 0.989636i \(0.545867\pi\)
−0.785251 + 0.619178i \(0.787466\pi\)
\(12\) 0 0
\(13\) 879.125 1.44275 0.721377 − 0.692542i \(-0.243509\pi\)
0.721377 + 0.692542i \(0.243509\pi\)
\(14\) 0 0
\(15\) 57.5159 0.0660025
\(16\) 0 0
\(17\) −987.446 + 1710.31i −0.828688 + 1.43533i 0.0703799 + 0.997520i \(0.477579\pi\)
−0.899068 + 0.437809i \(0.855754\pi\)
\(18\) 0 0
\(19\) 484.297 + 838.827i 0.307771 + 0.533075i 0.977874 − 0.209192i \(-0.0670835\pi\)
−0.670103 + 0.742268i \(0.733750\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −903.306 − 1564.57i −0.356053 − 0.616703i 0.631244 − 0.775584i \(-0.282545\pi\)
−0.987298 + 0.158881i \(0.949211\pi\)
\(24\) 0 0
\(25\) 1542.08 − 2670.96i 0.493466 − 0.854707i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 547.096 0.120800 0.0604002 − 0.998174i \(-0.480762\pi\)
0.0604002 + 0.998174i \(0.480762\pi\)
\(30\) 0 0
\(31\) 3164.17 − 5480.51i 0.591366 − 1.02428i −0.402683 − 0.915339i \(-0.631922\pi\)
0.994049 − 0.108936i \(-0.0347444\pi\)
\(32\) 0 0
\(33\) −3354.82 − 5810.72i −0.536271 − 0.928849i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7315.55 + 12670.9i 0.878502 + 1.52161i 0.852985 + 0.521935i \(0.174790\pi\)
0.0255163 + 0.999674i \(0.491877\pi\)
\(38\) 0 0
\(39\) −3956.06 + 6852.10i −0.416487 + 0.721377i
\(40\) 0 0
\(41\) −14953.6 −1.38927 −0.694636 − 0.719361i \(-0.744435\pi\)
−0.694636 + 0.719361i \(0.744435\pi\)
\(42\) 0 0
\(43\) 16676.5 1.37542 0.687708 − 0.725987i \(-0.258617\pi\)
0.687708 + 0.725987i \(0.258617\pi\)
\(44\) 0 0
\(45\) −258.822 + 448.292i −0.0190533 + 0.0330012i
\(46\) 0 0
\(47\) 3829.00 + 6632.03i 0.252837 + 0.437927i 0.964306 − 0.264791i \(-0.0853030\pi\)
−0.711469 + 0.702718i \(0.751970\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8887.01 − 15392.8i −0.478443 − 0.828688i
\(52\) 0 0
\(53\) −11994.5 + 20775.0i −0.586531 + 1.01590i 0.408152 + 0.912914i \(0.366173\pi\)
−0.994683 + 0.102987i \(0.967160\pi\)
\(54\) 0 0
\(55\) 4764.34 0.212371
\(56\) 0 0
\(57\) −8717.35 −0.355384
\(58\) 0 0
\(59\) −7693.25 + 13325.1i −0.287727 + 0.498357i −0.973267 − 0.229678i \(-0.926233\pi\)
0.685540 + 0.728035i \(0.259566\pi\)
\(60\) 0 0
\(61\) −13653.9 − 23649.3i −0.469822 − 0.813755i 0.529583 − 0.848258i \(-0.322348\pi\)
−0.999405 + 0.0345032i \(0.989015\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2809.10 − 4865.50i −0.0824676 − 0.142838i
\(66\) 0 0
\(67\) 259.516 − 449.495i 0.00706280 − 0.0122331i −0.862472 − 0.506104i \(-0.831085\pi\)
0.869535 + 0.493871i \(0.164418\pi\)
\(68\) 0 0
\(69\) 16259.5 0.411135
\(70\) 0 0
\(71\) 10034.8 0.236245 0.118123 − 0.992999i \(-0.462312\pi\)
0.118123 + 0.992999i \(0.462312\pi\)
\(72\) 0 0
\(73\) −32842.9 + 56885.5i −0.721330 + 1.24938i 0.239137 + 0.970986i \(0.423135\pi\)
−0.960467 + 0.278394i \(0.910198\pi\)
\(74\) 0 0
\(75\) 13878.7 + 24038.6i 0.284902 + 0.493466i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 19691.4 + 34106.5i 0.354984 + 0.614850i 0.987115 − 0.160011i \(-0.0511531\pi\)
−0.632132 + 0.774861i \(0.717820\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 35581.5 0.566930 0.283465 − 0.958983i \(-0.408516\pi\)
0.283465 + 0.958983i \(0.408516\pi\)
\(84\) 0 0
\(85\) 12620.9 0.189471
\(86\) 0 0
\(87\) −2461.93 + 4264.19i −0.0348721 + 0.0604002i
\(88\) 0 0
\(89\) −54356.1 − 94147.4i −0.727399 − 1.25989i −0.957979 − 0.286839i \(-0.907395\pi\)
0.230579 − 0.973054i \(-0.425938\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 28477.6 + 49324.6i 0.341425 + 0.591366i
\(94\) 0 0
\(95\) 3094.98 − 5360.66i 0.0351843 − 0.0609410i
\(96\) 0 0
\(97\) −141108. −1.52272 −0.761362 − 0.648327i \(-0.775469\pi\)
−0.761362 + 0.648327i \(0.775469\pi\)
\(98\) 0 0
\(99\) 60386.8 0.619233
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.i.j.361.1 4
7.2 even 3 inner 588.6.i.j.373.1 4
7.3 odd 6 588.6.a.i.1.1 ✓ 2
7.4 even 3 588.6.a.j.1.2 yes 2
7.5 odd 6 588.6.i.k.373.2 4
7.6 odd 2 588.6.i.k.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.i.1.1 ✓ 2 7.3 odd 6
588.6.a.j.1.2 yes 2 7.4 even 3
588.6.i.j.361.1 4 1.1 even 1 trivial
588.6.i.j.373.1 4 7.2 even 3 inner
588.6.i.k.361.2 4 7.6 odd 2
588.6.i.k.373.2 4 7.5 odd 6