Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(17,1008)] 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("1008.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1008.593
Dual form 1008.2.bt.c.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.09077 + 3.62132i) q^{5} +(1.62132 - 2.09077i) q^{7} +(-2.59808 + 1.50000i) q^{11} +2.44949i q^{13} +(0.507306 + 0.878680i) q^{17} +(0.878680 + 0.507306i) q^{19} +(-3.67423 - 2.12132i) q^{23} +(-6.24264 - 10.8126i) q^{25} -1.24264i q^{29} +(-4.86396 + 2.80821i) q^{31} +(4.18154 + 10.2426i) q^{35} +(-4.12132 + 7.13834i) q^{37} -2.02922 q^{41} -8.24264 q^{43} +(-0.507306 + 0.878680i) q^{47} +(-1.74264 - 6.77962i) q^{49} +(-1.07616 + 0.621320i) q^{53} -12.5446i q^{55} +(-5.76500 - 9.98528i) q^{59} +(5.12132 + 2.95680i) q^{61} +(-8.87039 - 5.12132i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.2426i q^{71} +(7.24264 - 4.18154i) q^{73} +(-1.07616 + 7.86396i) q^{77} +(-5.62132 + 9.73641i) q^{79} -3.16693 q^{83} -4.24264 q^{85} +(-5.19615 + 9.00000i) q^{89} +(5.12132 + 3.97141i) q^{91} +(-3.67423 + 2.12132i) q^{95} +3.76127i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + 24 q^{19} - 16 q^{25} + 12 q^{31} - 16 q^{37} - 32 q^{43} + 20 q^{49} + 24 q^{61} - 40 q^{67} + 24 q^{73} - 28 q^{79} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.09077 + 3.62132i −0.935021 + 1.61950i −0.160424 + 0.987048i \(0.551286\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 1.62132 − 2.09077i 0.612801 − 0.790237i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.59808 + 1.50000i −0.783349 + 0.452267i −0.837616 − 0.546259i \(-0.816051\pi\)
0.0542666 + 0.998526i \(0.482718\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.507306 + 0.878680i 0.123040 + 0.213111i 0.920965 − 0.389645i \(-0.127402\pi\)
−0.797925 + 0.602756i \(0.794069\pi\)
\(18\) 0 0
\(19\) 0.878680 + 0.507306i 0.201583 + 0.116384i 0.597394 − 0.801948i \(-0.296203\pi\)
−0.395811 + 0.918332i \(0.629536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.67423 − 2.12132i −0.766131 − 0.442326i 0.0653618 − 0.997862i \(-0.479180\pi\)
−0.831493 + 0.555536i \(0.812513\pi\)
\(24\) 0 0
\(25\) −6.24264 − 10.8126i −1.24853 − 2.16251i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.24264i − 0.230753i −0.993322 − 0.115376i \(-0.963193\pi\)
0.993322 − 0.115376i \(-0.0368074\pi\)
\(30\) 0 0
\(31\) −4.86396 + 2.80821i −0.873593 + 0.504369i −0.868541 − 0.495618i \(-0.834942\pi\)
−0.00505256 + 0.999987i \(0.501608\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.18154 + 10.2426i 0.706809 + 1.73132i
\(36\) 0 0
\(37\) −4.12132 + 7.13834i −0.677541 + 1.17354i 0.298178 + 0.954510i \(0.403621\pi\)
−0.975719 + 0.219025i \(0.929712\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.02922 −0.316912 −0.158456 − 0.987366i \(-0.550652\pi\)
−0.158456 + 0.987366i \(0.550652\pi\)
\(42\) 0 0
\(43\) −8.24264 −1.25699 −0.628495 − 0.777813i \(-0.716329\pi\)
−0.628495 + 0.777813i \(0.716329\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.507306 + 0.878680i −0.0739982 + 0.128169i −0.900650 − 0.434545i \(-0.856909\pi\)
0.826652 + 0.562713i \(0.190243\pi\)
\(48\) 0 0
\(49\) −1.74264 − 6.77962i −0.248949 − 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.07616 + 0.621320i −0.147822 + 0.0853449i −0.572087 − 0.820193i \(-0.693866\pi\)
0.424265 + 0.905538i \(0.360533\pi\)
\(54\) 0 0
\(55\) − 12.5446i − 1.69152i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.76500 − 9.98528i −0.750540 − 1.29997i −0.947561 − 0.319574i \(-0.896460\pi\)
0.197022 − 0.980399i \(-0.436873\pi\)
\(60\) 0 0
\(61\) 5.12132 + 2.95680i 0.655718 + 0.378579i 0.790643 − 0.612277i \(-0.209746\pi\)
−0.134926 + 0.990856i \(0.543080\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.87039 − 5.12132i −1.10024 − 0.635222i
\(66\) 0 0
\(67\) −5.00000 − 8.66025i −0.610847 − 1.05802i −0.991098 − 0.133135i \(-0.957496\pi\)
0.380251 − 0.924883i \(-0.375838\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2426i 1.21558i 0.794099 + 0.607789i \(0.207943\pi\)
−0.794099 + 0.607789i \(0.792057\pi\)
\(72\) 0 0
\(73\) 7.24264 − 4.18154i 0.847687 − 0.489412i −0.0121828 − 0.999926i \(-0.503878\pi\)
0.859870 + 0.510513i \(0.170545\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.07616 + 7.86396i −0.122640 + 0.896182i
\(78\) 0 0
\(79\) −5.62132 + 9.73641i −0.632448 + 1.09543i 0.354602 + 0.935017i \(0.384616\pi\)
−0.987050 + 0.160415i \(0.948717\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.16693 −0.347616 −0.173808 − 0.984780i \(-0.555607\pi\)
−0.173808 + 0.984780i \(0.555607\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 + 9.00000i −0.550791 + 0.953998i 0.447427 + 0.894321i \(0.352341\pi\)
−0.998218 + 0.0596775i \(0.980993\pi\)
\(90\) 0 0
\(91\) 5.12132 + 3.97141i 0.536860 + 0.416317i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.67423 + 2.12132i −0.376969 + 0.217643i
\(96\) 0 0
\(97\) 3.76127i 0.381900i 0.981600 + 0.190950i \(0.0611568\pi\)
−0.981600 + 0.190950i \(0.938843\pi\)
\(98\) 0 0
\(99\) 0 0
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 1008.2.bt.c.593.1 8
3.2 odd 2 inner 1008.2.bt.c.593.4 8
4.3 odd 2 126.2.k.a.89.1 yes 8
7.2 even 3 7056.2.k.f.881.7 8
7.3 odd 6 inner 1008.2.bt.c.17.4 8
7.5 odd 6 7056.2.k.f.881.1 8
12.11 even 2 126.2.k.a.89.4 yes 8
20.3 even 4 3150.2.bp.b.1349.4 8
20.7 even 4 3150.2.bp.e.1349.1 8
20.19 odd 2 3150.2.bf.a.1601.4 8
21.2 odd 6 7056.2.k.f.881.2 8
21.5 even 6 7056.2.k.f.881.8 8
21.17 even 6 inner 1008.2.bt.c.17.1 8
28.3 even 6 126.2.k.a.17.4 yes 8
28.11 odd 6 882.2.k.a.521.3 8
28.19 even 6 882.2.d.a.881.5 8
28.23 odd 6 882.2.d.a.881.8 8
28.27 even 2 882.2.k.a.215.2 8
36.7 odd 6 1134.2.l.f.215.3 8
36.11 even 6 1134.2.l.f.215.2 8
36.23 even 6 1134.2.t.e.593.1 8
36.31 odd 6 1134.2.t.e.593.4 8
60.23 odd 4 3150.2.bp.e.1349.4 8
60.47 odd 4 3150.2.bp.b.1349.1 8
60.59 even 2 3150.2.bf.a.1601.2 8
84.11 even 6 882.2.k.a.521.2 8
84.23 even 6 882.2.d.a.881.1 8
84.47 odd 6 882.2.d.a.881.4 8
84.59 odd 6 126.2.k.a.17.1 ✓ 8
84.83 odd 2 882.2.k.a.215.3 8
140.3 odd 12 3150.2.bp.b.899.1 8
140.59 even 6 3150.2.bf.a.1151.2 8
140.87 odd 12 3150.2.bp.e.899.4 8
252.31 even 6 1134.2.l.f.269.4 8
252.59 odd 6 1134.2.l.f.269.1 8
252.115 even 6 1134.2.t.e.1025.1 8
252.227 odd 6 1134.2.t.e.1025.4 8
420.59 odd 6 3150.2.bf.a.1151.4 8
420.143 even 12 3150.2.bp.e.899.1 8
420.227 even 12 3150.2.bp.b.899.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 84.59 odd 6
126.2.k.a.17.4 yes 8 28.3 even 6
126.2.k.a.89.1 yes 8 4.3 odd 2
126.2.k.a.89.4 yes 8 12.11 even 2
882.2.d.a.881.1 8 84.23 even 6
882.2.d.a.881.4 8 84.47 odd 6
882.2.d.a.881.5 8 28.19 even 6
882.2.d.a.881.8 8 28.23 odd 6
882.2.k.a.215.2 8 28.27 even 2
882.2.k.a.215.3 8 84.83 odd 2
882.2.k.a.521.2 8 84.11 even 6
882.2.k.a.521.3 8 28.11 odd 6
1008.2.bt.c.17.1 8 21.17 even 6 inner
1008.2.bt.c.17.4 8 7.3 odd 6 inner
1008.2.bt.c.593.1 8 1.1 even 1 trivial
1008.2.bt.c.593.4 8 3.2 odd 2 inner
1134.2.l.f.215.2 8 36.11 even 6
1134.2.l.f.215.3 8 36.7 odd 6
1134.2.l.f.269.1 8 252.59 odd 6
1134.2.l.f.269.4 8 252.31 even 6
1134.2.t.e.593.1 8 36.23 even 6
1134.2.t.e.593.4 8 36.31 odd 6
1134.2.t.e.1025.1 8 252.115 even 6
1134.2.t.e.1025.4 8 252.227 odd 6
3150.2.bf.a.1151.2 8 140.59 even 6
3150.2.bf.a.1151.4 8 420.59 odd 6
3150.2.bf.a.1601.2 8 60.59 even 2
3150.2.bf.a.1601.4 8 20.19 odd 2
3150.2.bp.b.899.1 8 140.3 odd 12
3150.2.bp.b.899.4 8 420.227 even 12
3150.2.bp.b.1349.1 8 60.47 odd 4
3150.2.bp.b.1349.4 8 20.3 even 4
3150.2.bp.e.899.1 8 420.143 even 12
3150.2.bp.e.899.4 8 140.87 odd 12
3150.2.bp.e.1349.1 8 20.7 even 4
3150.2.bp.e.1349.4 8 60.23 odd 4
7056.2.k.f.881.1 8 7.5 odd 6
7056.2.k.f.881.2 8 21.2 odd 6
7056.2.k.f.881.7 8 7.2 even 3
7056.2.k.f.881.8 8 21.5 even 6