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Properties

Label 126.2.k.a.17.4
Level $126$
Weight $2$
Character 126.17
Analytic conductor $1.006$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [126,2,Mod(17,126)] 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("126.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00611506547\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.4
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 126.17
Dual form 126.2.k.a.89.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.09077 + 3.62132i) q^{5} +(-1.62132 - 2.09077i) q^{7} -1.00000i q^{8} +(3.62132 + 2.09077i) q^{10} +(-2.59808 - 1.50000i) q^{11} -2.44949i q^{13} +(-2.44949 - 1.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.507306 + 0.878680i) q^{17} +(-0.878680 + 0.507306i) q^{19} +4.18154 q^{20} -3.00000 q^{22} +(-3.67423 + 2.12132i) q^{23} +(-6.24264 + 10.8126i) q^{25} +(-1.22474 - 2.12132i) q^{26} +(-2.62132 + 0.358719i) q^{28} -1.24264i q^{29} +(4.86396 + 2.80821i) q^{31} +(-0.866025 - 0.500000i) q^{32} +1.01461i q^{34} +(4.18154 - 10.2426i) q^{35} +(-4.12132 - 7.13834i) q^{37} +(-0.507306 + 0.878680i) q^{38} +(3.62132 - 2.09077i) q^{40} +2.02922 q^{41} +8.24264 q^{43} +(-2.59808 + 1.50000i) q^{44} +(-2.12132 + 3.67423i) q^{46} +(-0.507306 - 0.878680i) q^{47} +(-1.74264 + 6.77962i) q^{49} +12.4853i q^{50} +(-2.12132 - 1.22474i) q^{52} +(1.07616 + 0.621320i) q^{53} -12.5446i q^{55} +(-2.09077 + 1.62132i) q^{56} +(-0.621320 - 1.07616i) q^{58} +(-5.76500 + 9.98528i) q^{59} +(5.12132 - 2.95680i) q^{61} +5.61642 q^{62} -1.00000 q^{64} +(8.87039 - 5.12132i) q^{65} +(5.00000 - 8.66025i) q^{67} +(0.507306 + 0.878680i) q^{68} +(-1.50000 - 10.9612i) q^{70} -10.2426i q^{71} +(7.24264 + 4.18154i) q^{73} +(-7.13834 - 4.12132i) q^{74} +1.01461i q^{76} +(1.07616 + 7.86396i) q^{77} +(5.62132 + 9.73641i) q^{79} +(2.09077 - 3.62132i) q^{80} +(1.75736 - 1.01461i) q^{82} -3.16693 q^{83} -4.24264 q^{85} +(7.13834 - 4.12132i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(5.19615 + 9.00000i) q^{89} +(-5.12132 + 3.97141i) q^{91} +4.24264i q^{92} +(-0.878680 - 0.507306i) q^{94} +(-3.67423 - 2.12132i) q^{95} -3.76127i q^{97} +(1.88064 + 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7} + 12 q^{10} - 4 q^{16} - 24 q^{19} - 24 q^{22} - 16 q^{25} - 4 q^{28} - 12 q^{31} - 16 q^{37} + 12 q^{40} + 32 q^{43} + 20 q^{49} + 12 q^{58} + 24 q^{61} - 8 q^{64} + 40 q^{67} - 12 q^{70}+ \cdots - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 2.09077 + 3.62132i 0.935021 + 1.61950i 0.774597 + 0.632456i \(0.217953\pi\)
0.160424 + 0.987048i \(0.448714\pi\)
\(6\) 0 0
\(7\) −1.62132 − 2.09077i −0.612801 − 0.790237i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 3.62132 + 2.09077i 1.14516 + 0.661160i
\(11\) −2.59808 − 1.50000i −0.783349 − 0.452267i 0.0542666 − 0.998526i \(-0.482718\pi\)
−0.837616 + 0.546259i \(0.816051\pi\)
\(12\) 0 0
\(13\) − 2.44949i − 0.679366i −0.940540 − 0.339683i \(-0.889680\pi\)
0.940540 − 0.339683i \(-0.110320\pi\)
\(14\) −2.44949 − 1.00000i −0.654654 − 0.267261i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −0.507306 + 0.878680i −0.123040 + 0.213111i −0.920965 − 0.389645i \(-0.872598\pi\)
0.797925 + 0.602756i \(0.205931\pi\)
\(18\) 0 0
\(19\) −0.878680 + 0.507306i −0.201583 + 0.116384i −0.597394 − 0.801948i \(-0.703797\pi\)
0.395811 + 0.918332i \(0.370464\pi\)
\(20\) 4.18154 0.935021
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −3.67423 + 2.12132i −0.766131 + 0.442326i −0.831493 − 0.555536i \(-0.812513\pi\)
0.0653618 + 0.997862i \(0.479180\pi\)
\(24\) 0 0
\(25\) −6.24264 + 10.8126i −1.24853 + 2.16251i
\(26\) −1.22474 − 2.12132i −0.240192 − 0.416025i
\(27\) 0 0
\(28\) −2.62132 + 0.358719i −0.495383 + 0.0677916i
\(29\) − 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 \(-0.165058\pi\)
0.00505256 + 0.999987i \(0.498392\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 1.01461i 0.174005i
\(35\) 4.18154 − 10.2426i 0.706809 − 1.73132i
\(36\) 0 0
\(37\) −4.12132 − 7.13834i −0.677541 − 1.17354i −0.975719 − 0.219025i \(-0.929712\pi\)
0.298178 − 0.954510i \(-0.403621\pi\)
\(38\) −0.507306 + 0.878680i −0.0822959 + 0.142541i
\(39\) 0 0
\(40\) 3.62132 − 2.09077i 0.572581 − 0.330580i
\(41\) 2.02922 0.316912 0.158456 − 0.987366i \(-0.449348\pi\)
0.158456 + 0.987366i \(0.449348\pi\)
\(42\) 0 0
\(43\) 8.24264 1.25699 0.628495 − 0.777813i \(-0.283671\pi\)
0.628495 + 0.777813i \(0.283671\pi\)
\(44\) −2.59808 + 1.50000i −0.391675 + 0.226134i
\(45\) 0 0
\(46\) −2.12132 + 3.67423i −0.312772 + 0.541736i
\(47\) −0.507306 − 0.878680i −0.0739982 − 0.128169i 0.826652 − 0.562713i \(-0.190243\pi\)
−0.900650 + 0.434545i \(0.856909\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 12.4853i 1.76569i
\(51\) 0 0
\(52\) −2.12132 − 1.22474i −0.294174 − 0.169842i
\(53\) 1.07616 + 0.621320i 0.147822 + 0.0853449i 0.572087 − 0.820193i \(-0.306134\pi\)
−0.424265 + 0.905538i \(0.639467\pi\)
\(54\) 0 0
\(55\) − 12.5446i − 1.69152i
\(56\) −2.09077 + 1.62132i −0.279391 + 0.216658i
\(57\) 0 0
\(58\) −0.621320 − 1.07616i −0.0815834 − 0.141307i
\(59\) −5.76500 + 9.98528i −0.750540 + 1.29997i 0.197022 + 0.980399i \(0.436873\pi\)
−0.947561 + 0.319574i \(0.896460\pi\)
\(60\) 0 0
\(61\) 5.12132 − 2.95680i 0.655718 − 0.378579i −0.134926 − 0.990856i \(-0.543080\pi\)
0.790643 + 0.612277i \(0.209746\pi\)
\(62\) 5.61642 0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 8.87039 − 5.12132i 1.10024 − 0.635222i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) 0.507306 + 0.878680i 0.0615199 + 0.106556i
\(69\) 0 0
\(70\) −1.50000 − 10.9612i −0.179284 − 1.31011i
\(71\) − 10.2426i − 1.21558i −0.794099 − 0.607789i \(-0.792057\pi\)
0.794099 − 0.607789i \(-0.207943\pi\)
\(72\) 0 0
\(73\) 7.24264 + 4.18154i 0.847687 + 0.489412i 0.859870 − 0.510513i \(-0.170545\pi\)
−0.0121828 + 0.999926i \(0.503878\pi\)
\(74\) −7.13834 − 4.12132i −0.829815 − 0.479094i
\(75\) 0 0
\(76\) 1.01461i 0.116384i
\(77\) 1.07616 + 7.86396i 0.122640 + 0.896182i
\(78\) 0 0
\(79\) 5.62132 + 9.73641i 0.632448 + 1.09543i 0.987050 + 0.160415i \(0.0512831\pi\)
−0.354602 + 0.935017i \(0.615384\pi\)
\(80\) 2.09077 − 3.62132i 0.233755 − 0.404876i
\(81\) 0 0
\(82\) 1.75736 − 1.01461i 0.194068 − 0.112045i
\(83\) −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
\(86\) 7.13834 − 4.12132i 0.769747 − 0.444413i
\(87\) 0 0
\(88\) −1.50000 + 2.59808i −0.159901 + 0.276956i
\(89\) 5.19615 + 9.00000i 0.550791 + 0.953998i 0.998218 + 0.0596775i \(0.0190072\pi\)
−0.447427 + 0.894321i \(0.647659\pi\)
\(90\) 0 0
\(91\) −5.12132 + 3.97141i −0.536860 + 0.416317i
\(92\) 4.24264i 0.442326i
\(93\) 0 0
\(94\) −0.878680 − 0.507306i −0.0906289 − 0.0523246i
\(95\) −3.67423 − 2.12132i −0.376969 − 0.217643i
\(96\) 0 0
\(97\) − 3.76127i − 0.381900i −0.981600 − 0.190950i \(-0.938843\pi\)
0.981600 − 0.190950i \(-0.0611568\pi\)
\(98\) 1.88064 + 6.74264i 0.189973 + 0.681110i
\(99\) 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 126.2.k.a.17.4 yes 8
3.2 odd 2 inner 126.2.k.a.17.1 ✓ 8
4.3 odd 2 1008.2.bt.c.17.4 8
5.2 odd 4 3150.2.bp.e.899.4 8
5.3 odd 4 3150.2.bp.b.899.1 8
5.4 even 2 3150.2.bf.a.1151.2 8
7.2 even 3 882.2.k.a.215.2 8
7.3 odd 6 882.2.d.a.881.8 8
7.4 even 3 882.2.d.a.881.5 8
7.5 odd 6 inner 126.2.k.a.89.1 yes 8
7.6 odd 2 882.2.k.a.521.3 8
9.2 odd 6 1134.2.t.e.1025.4 8
9.4 even 3 1134.2.l.f.269.4 8
9.5 odd 6 1134.2.l.f.269.1 8
9.7 even 3 1134.2.t.e.1025.1 8
12.11 even 2 1008.2.bt.c.17.1 8
15.2 even 4 3150.2.bp.b.899.4 8
15.8 even 4 3150.2.bp.e.899.1 8
15.14 odd 2 3150.2.bf.a.1151.4 8
21.2 odd 6 882.2.k.a.215.3 8
21.5 even 6 inner 126.2.k.a.89.4 yes 8
21.11 odd 6 882.2.d.a.881.4 8
21.17 even 6 882.2.d.a.881.1 8
21.20 even 2 882.2.k.a.521.2 8
28.3 even 6 7056.2.k.f.881.7 8
28.11 odd 6 7056.2.k.f.881.1 8
28.19 even 6 1008.2.bt.c.593.1 8
35.12 even 12 3150.2.bp.e.1349.1 8
35.19 odd 6 3150.2.bf.a.1601.4 8
35.33 even 12 3150.2.bp.b.1349.4 8
63.5 even 6 1134.2.t.e.593.1 8
63.40 odd 6 1134.2.t.e.593.4 8
63.47 even 6 1134.2.l.f.215.2 8
63.61 odd 6 1134.2.l.f.215.3 8
84.11 even 6 7056.2.k.f.881.8 8
84.47 odd 6 1008.2.bt.c.593.4 8
84.59 odd 6 7056.2.k.f.881.2 8
105.47 odd 12 3150.2.bp.b.1349.1 8
105.68 odd 12 3150.2.bp.e.1349.4 8
105.89 even 6 3150.2.bf.a.1601.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 3.2 odd 2 inner
126.2.k.a.17.4 yes 8 1.1 even 1 trivial
126.2.k.a.89.1 yes 8 7.5 odd 6 inner
126.2.k.a.89.4 yes 8 21.5 even 6 inner
882.2.d.a.881.1 8 21.17 even 6
882.2.d.a.881.4 8 21.11 odd 6
882.2.d.a.881.5 8 7.4 even 3
882.2.d.a.881.8 8 7.3 odd 6
882.2.k.a.215.2 8 7.2 even 3
882.2.k.a.215.3 8 21.2 odd 6
882.2.k.a.521.2 8 21.20 even 2
882.2.k.a.521.3 8 7.6 odd 2
1008.2.bt.c.17.1 8 12.11 even 2
1008.2.bt.c.17.4 8 4.3 odd 2
1008.2.bt.c.593.1 8 28.19 even 6
1008.2.bt.c.593.4 8 84.47 odd 6
1134.2.l.f.215.2 8 63.47 even 6
1134.2.l.f.215.3 8 63.61 odd 6
1134.2.l.f.269.1 8 9.5 odd 6
1134.2.l.f.269.4 8 9.4 even 3
1134.2.t.e.593.1 8 63.5 even 6
1134.2.t.e.593.4 8 63.40 odd 6
1134.2.t.e.1025.1 8 9.7 even 3
1134.2.t.e.1025.4 8 9.2 odd 6
3150.2.bf.a.1151.2 8 5.4 even 2
3150.2.bf.a.1151.4 8 15.14 odd 2
3150.2.bf.a.1601.2 8 105.89 even 6
3150.2.bf.a.1601.4 8 35.19 odd 6
3150.2.bp.b.899.1 8 5.3 odd 4
3150.2.bp.b.899.4 8 15.2 even 4
3150.2.bp.b.1349.1 8 105.47 odd 12
3150.2.bp.b.1349.4 8 35.33 even 12
3150.2.bp.e.899.1 8 15.8 even 4
3150.2.bp.e.899.4 8 5.2 odd 4
3150.2.bp.e.1349.1 8 35.12 even 12
3150.2.bp.e.1349.4 8 105.68 odd 12
7056.2.k.f.881.1 8 28.11 odd 6
7056.2.k.f.881.2 8 84.59 odd 6
7056.2.k.f.881.7 8 28.3 even 6
7056.2.k.f.881.8 8 84.11 even 6