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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 = [8,0,-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: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 91x^{6} - 2x^{5} + 5907x^{4} - 304x^{3} + 167650x^{2} + 161744x + 3378244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.3
Root \(-3.42932 + 5.93975i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.6.i.p.373.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 7.79423i) q^{3} +(-4.51051 - 7.81243i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-302.941 + 524.709i) q^{11} -227.306 q^{13} +81.1892 q^{15} +(660.507 - 1144.03i) q^{17} +(1209.56 + 2095.01i) q^{19} +(1653.05 + 2863.17i) q^{23} +(1521.81 - 2635.85i) q^{25} +729.000 q^{27} -2436.19 q^{29} +(-3184.68 + 5516.03i) q^{31} +(-2726.47 - 4722.38i) q^{33} +(-4209.74 - 7291.48i) q^{37} +(1022.88 - 1771.68i) q^{39} -6776.47 q^{41} +6590.39 q^{43} +(-365.351 + 632.807i) q^{45} +(1132.18 + 1960.99i) q^{47} +(5944.57 + 10296.3i) q^{51} +(8397.11 - 14544.2i) q^{53} +5465.67 q^{55} -21772.0 q^{57} +(-7357.93 + 12744.3i) q^{59} +(19322.4 + 33467.4i) q^{61} +(1025.27 + 1775.81i) q^{65} +(-3775.13 + 6538.72i) q^{67} -29754.9 q^{69} -16467.0 q^{71} +(-5983.79 + 10364.2i) q^{73} +(13696.3 + 23722.7i) q^{75} +(6510.63 + 11276.7i) q^{79} +(-3280.50 + 5681.99i) q^{81} -66229.5 q^{83} -11916.9 q^{85} +(10962.9 - 18988.2i) q^{87} +(5242.20 + 9079.76i) q^{89} +(-28662.1 - 49644.3i) q^{93} +(10911.4 - 18899.1i) q^{95} -91308.8 q^{97} +49076.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{3} - 324 q^{9} + 1872 q^{17} + 1728 q^{19} + 3648 q^{23} + 3996 q^{25} + 5832 q^{27} - 2496 q^{29} + 3888 q^{31} + 12032 q^{37} + 18144 q^{41} - 4256 q^{43} + 19872 q^{47} + 16848 q^{51} + 22248 q^{53}+ \cdots - 641088 q^{97}+O(q^{100}) \) 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
\(5\) −4.51051 − 7.81243i −0.0806865 − 0.139753i 0.822859 − 0.568246i \(-0.192378\pi\)
−0.903545 + 0.428493i \(0.859045\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −302.941 + 524.709i −0.754877 + 1.30749i 0.190558 + 0.981676i \(0.438970\pi\)
−0.945435 + 0.325810i \(0.894363\pi\)
\(12\) 0 0
\(13\) −227.306 −0.373038 −0.186519 − 0.982451i \(-0.559721\pi\)
−0.186519 + 0.982451i \(0.559721\pi\)
\(14\) 0 0
\(15\) 81.1892 0.0931687
\(16\) 0 0
\(17\) 660.507 − 1144.03i 0.554313 − 0.960099i −0.443643 − 0.896203i \(-0.646314\pi\)
0.997957 − 0.0638954i \(-0.0203524\pi\)
\(18\) 0 0
\(19\) 1209.56 + 2095.01i 0.768673 + 1.33138i 0.938282 + 0.345870i \(0.112416\pi\)
−0.169609 + 0.985511i \(0.554250\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1653.05 + 2863.17i 0.651578 + 1.12857i 0.982740 + 0.184993i \(0.0592261\pi\)
−0.331162 + 0.943574i \(0.607441\pi\)
\(24\) 0 0
\(25\) 1521.81 − 2635.85i 0.486979 − 0.843473i
\(26\) 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
\(31\) −3184.68 + 5516.03i −0.595198 + 1.03091i 0.398321 + 0.917246i \(0.369593\pi\)
−0.993519 + 0.113667i \(0.963740\pi\)
\(32\) 0 0
\(33\) −2726.47 − 4722.38i −0.435829 − 0.754877i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4209.74 − 7291.48i −0.505534 − 0.875611i −0.999980 − 0.00640231i \(-0.997962\pi\)
0.494445 − 0.869209i \(-0.335371\pi\)
\(38\) 0 0
\(39\) 1022.88 − 1771.68i 0.107687 − 0.186519i
\(40\) 0 0
\(41\) −6776.47 −0.629570 −0.314785 − 0.949163i \(-0.601932\pi\)
−0.314785 + 0.949163i \(0.601932\pi\)
\(42\) 0 0
\(43\) 6590.39 0.543550 0.271775 − 0.962361i \(-0.412389\pi\)
0.271775 + 0.962361i \(0.412389\pi\)
\(44\) 0 0
\(45\) −365.351 + 632.807i −0.0268955 + 0.0465843i
\(46\) 0 0
\(47\) 1132.18 + 1960.99i 0.0747603 + 0.129489i 0.900982 − 0.433856i \(-0.142847\pi\)
−0.826222 + 0.563345i \(0.809514\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5944.57 + 10296.3i 0.320033 + 0.554313i
\(52\) 0 0
\(53\) 8397.11 − 14544.2i 0.410620 − 0.711215i −0.584338 − 0.811511i \(-0.698646\pi\)
0.994958 + 0.100296i \(0.0319790\pi\)
\(54\) 0 0
\(55\) 5465.67 0.243633
\(56\) 0 0
\(57\) −21772.0 −0.887588
\(58\) 0 0
\(59\) −7357.93 + 12744.3i −0.275186 + 0.476636i −0.970182 − 0.242378i \(-0.922073\pi\)
0.694996 + 0.719013i \(0.255406\pi\)
\(60\) 0 0
\(61\) 19322.4 + 33467.4i 0.664869 + 1.15159i 0.979321 + 0.202314i \(0.0648461\pi\)
−0.314451 + 0.949274i \(0.601821\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1025.27 + 1775.81i 0.0300991 + 0.0521331i
\(66\) 0 0
\(67\) −3775.13 + 6538.72i −0.102741 + 0.177953i −0.912813 − 0.408377i \(-0.866095\pi\)
0.810072 + 0.586331i \(0.199428\pi\)
\(68\) 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
\(73\) −5983.79 + 10364.2i −0.131422 + 0.227630i −0.924225 − 0.381848i \(-0.875288\pi\)
0.792803 + 0.609478i \(0.208621\pi\)
\(74\) 0 0
\(75\) 13696.3 + 23722.7i 0.281158 + 0.486979i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6510.63 + 11276.7i 0.117370 + 0.203290i 0.918724 − 0.394899i \(-0.129220\pi\)
−0.801355 + 0.598189i \(0.795887\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −66229.5 −1.05525 −0.527626 − 0.849477i \(-0.676918\pi\)
−0.527626 + 0.849477i \(0.676918\pi\)
\(84\) 0 0
\(85\) −11916.9 −0.178902
\(86\) 0 0
\(87\) 10962.9 − 18988.2i 0.155284 − 0.268959i
\(88\) 0 0
\(89\) 5242.20 + 9079.76i 0.0701517 + 0.121506i 0.898968 − 0.438015i \(-0.144318\pi\)
−0.828816 + 0.559521i \(0.810985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −28662.1 − 49644.3i −0.343638 − 0.595198i
\(94\) 0 0
\(95\) 10911.4 − 18899.1i 0.124043 − 0.214849i
\(96\) 0 0
\(97\) −91308.8 −0.985334 −0.492667 − 0.870218i \(-0.663978\pi\)
−0.492667 + 0.870218i \(0.663978\pi\)
\(98\) 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.i.p.361.3 8
7.2 even 3 inner 588.6.i.p.373.3 8
7.3 odd 6 588.6.a.m.1.3 ✓ 4
7.4 even 3 588.6.a.o.1.2 yes 4
7.5 odd 6 588.6.i.q.373.2 8
7.6 odd 2 588.6.i.q.361.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.m.1.3 ✓ 4 7.3 odd 6
588.6.a.o.1.2 yes 4 7.4 even 3
588.6.i.p.361.3 8 1.1 even 1 trivial
588.6.i.p.373.3 8 7.2 even 3 inner
588.6.i.q.361.2 8 7.6 odd 2
588.6.i.q.373.2 8 7.5 odd 6