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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,-6,0,0,0,-162,0,90] 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: \(\Q(\sqrt{-3}, \sqrt{5569})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(-18.4064 - 31.8809i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.6.i.i.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 - 7.79423i) q^{3} +(35.8129 - 62.0297i) q^{5} +(-40.5000 + 70.1481i) q^{9} +(283.690 + 491.366i) q^{11} -831.754 q^{13} -644.632 q^{15} +(444.187 + 769.355i) q^{17} +(1457.51 - 2524.48i) q^{19} +(-1551.19 + 2686.73i) q^{23} +(-1002.62 - 1736.59i) q^{25} +729.000 q^{27} +8271.03 q^{29} +(-3514.53 - 6087.34i) q^{31} +(2553.21 - 4422.29i) q^{33} +(5070.93 - 8783.11i) q^{37} +(3742.89 + 6482.88i) q^{39} -3095.65 q^{41} +15026.2 q^{43} +(2900.84 + 5024.41i) q^{45} +(9947.68 - 17229.9i) q^{47} +(3997.68 - 6924.19i) q^{51} +(4603.21 + 7972.99i) q^{53} +40639.0 q^{55} -26235.2 q^{57} +(-5150.64 - 8921.18i) q^{59} +(-11299.6 + 19571.5i) q^{61} +(-29787.5 + 51593.5i) q^{65} +(-3209.54 - 5559.09i) q^{67} +27921.4 q^{69} -61279.0 q^{71} +(-14853.6 - 25727.1i) q^{73} +(-9023.61 + 15629.3i) q^{75} +(7815.40 - 13536.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} -1668.23 q^{83} +63630.5 q^{85} +(-37219.6 - 64466.3i) q^{87} +(37916.7 - 65673.6i) q^{89} +(-31630.7 + 54786.0i) q^{93} +(-104395. - 180818. i) q^{95} +98013.9 q^{97} -45957.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9} + 90 q^{11} - 1536 q^{13} + 108 q^{15} + 1926 q^{17} + 2248 q^{19} - 6354 q^{23} - 4906 q^{25} + 2916 q^{27} + 21144 q^{29} - 3312 q^{31} + 810 q^{33} - 2104 q^{37}+ \cdots - 14580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 35.8129 − 62.0297i 0.640640 − 1.10962i −0.344650 − 0.938731i \(-0.612002\pi\)
0.985290 − 0.170890i \(-0.0546642\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 283.690 + 491.366i 0.706907 + 1.22440i 0.965999 + 0.258546i \(0.0832435\pi\)
−0.259092 + 0.965853i \(0.583423\pi\)
\(12\) 0 0
\(13\) −831.754 −1.36501 −0.682506 − 0.730880i \(-0.739110\pi\)
−0.682506 + 0.730880i \(0.739110\pi\)
\(14\) 0 0
\(15\) −644.632 −0.739747
\(16\) 0 0
\(17\) 444.187 + 769.355i 0.372772 + 0.645661i 0.989991 − 0.141131i \(-0.0450739\pi\)
−0.617219 + 0.786792i \(0.711741\pi\)
\(18\) 0 0
\(19\) 1457.51 − 2524.48i 0.926248 − 1.60431i 0.136706 − 0.990612i \(-0.456349\pi\)
0.789542 − 0.613697i \(-0.210318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1551.19 + 2686.73i −0.611427 + 1.05902i 0.379573 + 0.925162i \(0.376071\pi\)
−0.991000 + 0.133861i \(0.957262\pi\)
\(24\) 0 0
\(25\) −1002.62 − 1736.59i −0.320839 − 0.555710i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 8271.03 1.82627 0.913134 − 0.407659i \(-0.133655\pi\)
0.913134 + 0.407659i \(0.133655\pi\)
\(30\) 0 0
\(31\) −3514.53 − 6087.34i −0.656845 − 1.13769i −0.981428 − 0.191831i \(-0.938557\pi\)
0.324583 − 0.945857i \(-0.394776\pi\)
\(32\) 0 0
\(33\) 2553.21 − 4422.29i 0.408133 − 0.706907i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5070.93 − 8783.11i 0.608952 − 1.05474i −0.382461 − 0.923972i \(-0.624923\pi\)
0.991413 − 0.130765i \(-0.0417433\pi\)
\(38\) 0 0
\(39\) 3742.89 + 6482.88i 0.394045 + 0.682506i
\(40\) 0 0
\(41\) −3095.65 −0.287602 −0.143801 − 0.989607i \(-0.545933\pi\)
−0.143801 + 0.989607i \(0.545933\pi\)
\(42\) 0 0
\(43\) 15026.2 1.23930 0.619651 − 0.784877i \(-0.287274\pi\)
0.619651 + 0.784877i \(0.287274\pi\)
\(44\) 0 0
\(45\) 2900.84 + 5024.41i 0.213547 + 0.369874i
\(46\) 0 0
\(47\) 9947.68 − 17229.9i 0.656867 − 1.13773i −0.324555 − 0.945867i \(-0.605215\pi\)
0.981422 − 0.191860i \(-0.0614520\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3997.68 − 6924.19i 0.215220 − 0.372772i
\(52\) 0 0
\(53\) 4603.21 + 7972.99i 0.225098 + 0.389881i 0.956349 − 0.292228i \(-0.0943965\pi\)
−0.731251 + 0.682108i \(0.761063\pi\)
\(54\) 0 0
\(55\) 40639.0 1.81149
\(56\) 0 0
\(57\) −26235.2 −1.06954
\(58\) 0 0
\(59\) −5150.64 − 8921.18i −0.192633 − 0.333651i 0.753489 − 0.657461i \(-0.228369\pi\)
−0.946122 + 0.323810i \(0.895036\pi\)
\(60\) 0 0
\(61\) −11299.6 + 19571.5i −0.388811 + 0.673440i −0.992290 − 0.123939i \(-0.960447\pi\)
0.603479 + 0.797379i \(0.293781\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −29787.5 + 51593.5i −0.874482 + 1.51465i
\(66\) 0 0
\(67\) −3209.54 − 5559.09i −0.0873487 − 0.151292i 0.819041 − 0.573735i \(-0.194506\pi\)
−0.906390 + 0.422443i \(0.861173\pi\)
\(68\) 0 0
\(69\) 27921.4 0.706015
\(70\) 0 0
\(71\) −61279.0 −1.44267 −0.721333 − 0.692588i \(-0.756470\pi\)
−0.721333 + 0.692588i \(0.756470\pi\)
\(72\) 0 0
\(73\) −14853.6 − 25727.1i −0.326230 − 0.565046i 0.655531 − 0.755168i \(-0.272445\pi\)
−0.981760 + 0.190122i \(0.939112\pi\)
\(74\) 0 0
\(75\) −9023.61 + 15629.3i −0.185237 + 0.320839i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7815.40 − 13536.7i 0.140891 − 0.244031i −0.786941 − 0.617028i \(-0.788337\pi\)
0.927832 + 0.372997i \(0.121670\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −1668.23 −0.0265804 −0.0132902 − 0.999912i \(-0.504231\pi\)
−0.0132902 + 0.999912i \(0.504231\pi\)
\(84\) 0 0
\(85\) 63630.5 0.955252
\(86\) 0 0
\(87\) −37219.6 − 64466.3i −0.527198 − 0.913134i
\(88\) 0 0
\(89\) 37916.7 − 65673.6i 0.507405 − 0.878852i −0.492558 − 0.870280i \(-0.663938\pi\)
0.999963 − 0.00857229i \(-0.00272868\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −31630.7 + 54786.0i −0.379229 + 0.656845i
\(94\) 0 0
\(95\) −104395. − 180818.i −1.18678 − 2.05557i
\(96\) 0 0
\(97\) 98013.9 1.05769 0.528845 − 0.848718i \(-0.322625\pi\)
0.528845 + 0.848718i \(0.322625\pi\)
\(98\) 0 0
\(99\) −45957.8 −0.471271
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.i.373.2 4
7.2 even 3 588.6.a.k.1.1 2
7.3 odd 6 588.6.i.l.361.1 4
7.4 even 3 inner 588.6.i.i.361.2 4
7.5 odd 6 84.6.a.c.1.2 ✓ 2
7.6 odd 2 588.6.i.l.373.1 4
21.5 even 6 252.6.a.h.1.1 2
28.19 even 6 336.6.a.x.1.2 2
84.47 odd 6 1008.6.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.c.1.2 ✓ 2 7.5 odd 6
252.6.a.h.1.1 2 21.5 even 6
336.6.a.x.1.2 2 28.19 even 6
588.6.a.k.1.1 2 7.2 even 3
588.6.i.i.361.2 4 7.4 even 3 inner
588.6.i.i.373.2 4 1.1 even 1 trivial
588.6.i.l.361.1 4 7.3 odd 6
588.6.i.l.373.1 4 7.6 odd 2
1008.6.a.bo.1.1 2 84.47 odd 6