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Properties

Label 126.2.k.a.17.2
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 17.2
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 126.17
Dual form 126.2.k.a.89.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.358719 + 0.621320i) q^{5} +(2.62132 + 0.358719i) q^{7} +1.00000i q^{8} +(-0.621320 - 0.358719i) q^{10} +(2.59808 + 1.50000i) q^{11} +2.44949i q^{13} +(-2.44949 + 1.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.95680 - 5.12132i) q^{17} +(-5.12132 + 2.95680i) q^{19} +0.717439 q^{20} -3.00000 q^{22} +(-3.67423 + 2.12132i) q^{23} +(2.24264 - 3.88437i) q^{25} +(-1.22474 - 2.12132i) q^{26} +(1.62132 - 2.09077i) q^{28} -7.24264i q^{29} +(-7.86396 - 4.54026i) q^{31} +(0.866025 + 0.500000i) q^{32} +5.91359i q^{34} +(0.717439 + 1.75736i) q^{35} +(0.121320 + 0.210133i) q^{37} +(2.95680 - 5.12132i) q^{38} +(-0.621320 + 0.358719i) q^{40} -11.8272 q^{41} -0.242641 q^{43} +(2.59808 - 1.50000i) q^{44} +(2.12132 - 3.67423i) q^{46} +(2.95680 + 5.12132i) q^{47} +(6.74264 + 1.88064i) q^{49} +4.48528i q^{50} +(2.12132 + 1.22474i) q^{52} +(6.27231 + 3.62132i) q^{53} +2.15232i q^{55} +(-0.358719 + 2.62132i) q^{56} +(3.62132 + 6.27231i) q^{58} +(-4.03295 + 6.98528i) q^{59} +(0.878680 - 0.507306i) q^{61} +9.08052 q^{62} -1.00000 q^{64} +(-1.52192 + 0.878680i) q^{65} +(5.00000 - 8.66025i) q^{67} +(-2.95680 - 5.12132i) q^{68} +(-1.50000 - 1.16320i) q^{70} +1.75736i q^{71} +(-1.24264 - 0.717439i) q^{73} +(-0.210133 - 0.121320i) q^{74} +5.91359i q^{76} +(6.27231 + 4.86396i) q^{77} +(1.37868 + 2.38794i) q^{79} +(0.358719 - 0.621320i) q^{80} +(10.2426 - 5.91359i) q^{82} -6.63103 q^{83} +4.24264 q^{85} +(0.210133 - 0.121320i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(-5.19615 - 9.00000i) q^{89} +(-0.878680 + 6.42090i) q^{91} +4.24264i q^{92} +(-5.12132 - 2.95680i) q^{94} +(-3.67423 - 2.12132i) q^{95} -13.5592i q^{97} +(-6.77962 + 1.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7} + 12 q^{10} - 4 q^{16} - 24 q^{19} - 24 q^{22} - 16 q^{25} - 4 q^{28} - 12 q^{31} - 16 q^{37} + 12 q^{40} + 32 q^{43} + 20 q^{49} + 12 q^{58} + 24 q^{61} - 8 q^{64} + 40 q^{67} - 12 q^{70}+ \cdots - 24 q^{94}+O(q^{100}) \) 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
\(5\) 0.358719 + 0.621320i 0.160424 + 0.277863i 0.935021 − 0.354593i \(-0.115380\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 2.62132 + 0.358719i 0.990766 + 0.135583i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −0.621320 − 0.358719i −0.196479 − 0.113437i
\(11\) 2.59808 + 1.50000i 0.783349 + 0.452267i 0.837616 − 0.546259i \(-0.183949\pi\)
−0.0542666 + 0.998526i \(0.517282\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) −2.44949 + 1.00000i −0.654654 + 0.267261i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.95680 − 5.12132i 0.717128 − 1.24210i −0.245005 − 0.969522i \(-0.578789\pi\)
0.962133 − 0.272581i \(-0.0878772\pi\)
\(18\) 0 0
\(19\) −5.12132 + 2.95680i −1.17491 + 0.678335i −0.954832 − 0.297146i \(-0.903965\pi\)
−0.220080 + 0.975482i \(0.570632\pi\)
\(20\) 0.717439 0.160424
\(21\) 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
\(25\) 2.24264 − 3.88437i 0.448528 − 0.776874i
\(26\) −1.22474 − 2.12132i −0.240192 − 0.416025i
\(27\) 0 0
\(28\) 1.62132 − 2.09077i 0.306401 − 0.395118i
\(29\) − 7.24264i − 1.34492i −0.740131 − 0.672462i \(-0.765237\pi\)
0.740131 − 0.672462i \(-0.234763\pi\)
\(30\) 0 0
\(31\) −7.86396 − 4.54026i −1.41241 − 0.815455i −0.416794 − 0.909001i \(-0.636846\pi\)
−0.995615 + 0.0935461i \(0.970180\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 5.91359i 1.01417i
\(35\) 0.717439 + 1.75736i 0.121269 + 0.297048i
\(36\) 0 0
\(37\) 0.121320 + 0.210133i 0.0199449 + 0.0345457i 0.875826 − 0.482628i \(-0.160318\pi\)
−0.855881 + 0.517173i \(0.826984\pi\)
\(38\) 2.95680 − 5.12132i 0.479656 − 0.830788i
\(39\) 0 0
\(40\) −0.621320 + 0.358719i −0.0982394 + 0.0567185i
\(41\) −11.8272 −1.84710 −0.923548 − 0.383483i \(-0.874724\pi\)
−0.923548 + 0.383483i \(0.874724\pi\)
\(42\) 0 0
\(43\) −0.242641 −0.0370024 −0.0185012 − 0.999829i \(-0.505889\pi\)
−0.0185012 + 0.999829i \(0.505889\pi\)
\(44\) 2.59808 − 1.50000i 0.391675 − 0.226134i
\(45\) 0 0
\(46\) 2.12132 − 3.67423i 0.312772 − 0.541736i
\(47\) 2.95680 + 5.12132i 0.431293 + 0.747021i 0.996985 − 0.0775953i \(-0.0247242\pi\)
−0.565692 + 0.824617i \(0.691391\pi\)
\(48\) 0 0
\(49\) 6.74264 + 1.88064i 0.963234 + 0.268662i
\(50\) 4.48528i 0.634315i
\(51\) 0 0
\(52\) 2.12132 + 1.22474i 0.294174 + 0.169842i
\(53\) 6.27231 + 3.62132i 0.861568 + 0.497427i 0.864537 − 0.502569i \(-0.167612\pi\)
−0.00296896 + 0.999996i \(0.500945\pi\)
\(54\) 0 0
\(55\) 2.15232i 0.290218i
\(56\) −0.358719 + 2.62132i −0.0479359 + 0.350289i
\(57\) 0 0
\(58\) 3.62132 + 6.27231i 0.475503 + 0.823595i
\(59\) −4.03295 + 6.98528i −0.525046 + 0.909406i 0.474529 + 0.880240i \(0.342619\pi\)
−0.999575 + 0.0291661i \(0.990715\pi\)
\(60\) 0 0
\(61\) 0.878680 − 0.507306i 0.112503 − 0.0649539i −0.442692 − 0.896674i \(-0.645977\pi\)
0.555196 + 0.831720i \(0.312643\pi\)
\(62\) 9.08052 1.15323
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −1.52192 + 0.878680i −0.188771 + 0.108987i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) −2.95680 − 5.12132i −0.358564 − 0.621051i
\(69\) 0 0
\(70\) −1.50000 − 1.16320i −0.179284 − 0.139029i
\(71\) 1.75736i 0.208560i 0.994548 + 0.104280i \(0.0332538\pi\)
−0.994548 + 0.104280i \(0.966746\pi\)
\(72\) 0 0
\(73\) −1.24264 − 0.717439i −0.145440 − 0.0839699i 0.425514 − 0.904952i \(-0.360093\pi\)
−0.570954 + 0.820982i \(0.693427\pi\)
\(74\) −0.210133 − 0.121320i −0.0244275 − 0.0141032i
\(75\) 0 0
\(76\) 5.91359i 0.678335i
\(77\) 6.27231 + 4.86396i 0.714796 + 0.554300i
\(78\) 0 0
\(79\) 1.37868 + 2.38794i 0.155114 + 0.268665i 0.933100 − 0.359616i \(-0.117092\pi\)
−0.777987 + 0.628281i \(0.783759\pi\)
\(80\) 0.358719 − 0.621320i 0.0401061 − 0.0694657i
\(81\) 0 0
\(82\) 10.2426 − 5.91359i 1.13111 − 0.653047i
\(83\) −6.63103 −0.727850 −0.363925 − 0.931428i \(-0.618564\pi\)
−0.363925 + 0.931428i \(0.618564\pi\)
\(84\) 0 0
\(85\) 4.24264 0.460179
\(86\) 0.210133 − 0.121320i 0.0226592 − 0.0130823i
\(87\) 0 0
\(88\) −1.50000 + 2.59808i −0.159901 + 0.276956i
\(89\) −5.19615 − 9.00000i −0.550791 − 0.953998i −0.998218 − 0.0596775i \(-0.980993\pi\)
0.447427 − 0.894321i \(-0.352341\pi\)
\(90\) 0 0
\(91\) −0.878680 + 6.42090i −0.0921107 + 0.673093i
\(92\) 4.24264i 0.442326i
\(93\) 0 0
\(94\) −5.12132 − 2.95680i −0.528224 − 0.304970i
\(95\) −3.67423 − 2.12132i −0.376969 − 0.217643i
\(96\) 0 0
\(97\) − 13.5592i − 1.37673i −0.725364 − 0.688366i \(-0.758328\pi\)
0.725364 − 0.688366i \(-0.241672\pi\)
\(98\) −6.77962 + 1.74264i −0.684845 + 0.176033i
\(99\) 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.17.2 ✓ 8
3.2 odd 2 inner 126.2.k.a.17.3 yes 8
4.3 odd 2 1008.2.bt.c.17.3 8
5.2 odd 4 3150.2.bp.b.899.2 8
5.3 odd 4 3150.2.bp.e.899.3 8
5.4 even 2 3150.2.bf.a.1151.3 8
7.2 even 3 882.2.k.a.215.4 8
7.3 odd 6 882.2.d.a.881.3 8
7.4 even 3 882.2.d.a.881.2 8
7.5 odd 6 inner 126.2.k.a.89.3 yes 8
7.6 odd 2 882.2.k.a.521.1 8
9.2 odd 6 1134.2.t.e.1025.2 8
9.4 even 3 1134.2.l.f.269.2 8
9.5 odd 6 1134.2.l.f.269.3 8
9.7 even 3 1134.2.t.e.1025.3 8
12.11 even 2 1008.2.bt.c.17.2 8
15.2 even 4 3150.2.bp.e.899.2 8
15.8 even 4 3150.2.bp.b.899.3 8
15.14 odd 2 3150.2.bf.a.1151.1 8
21.2 odd 6 882.2.k.a.215.1 8
21.5 even 6 inner 126.2.k.a.89.2 yes 8
21.11 odd 6 882.2.d.a.881.7 8
21.17 even 6 882.2.d.a.881.6 8
21.20 even 2 882.2.k.a.521.4 8
28.3 even 6 7056.2.k.f.881.6 8
28.11 odd 6 7056.2.k.f.881.4 8
28.19 even 6 1008.2.bt.c.593.2 8
35.12 even 12 3150.2.bp.b.1349.3 8
35.19 odd 6 3150.2.bf.a.1601.1 8
35.33 even 12 3150.2.bp.e.1349.2 8
63.5 even 6 1134.2.t.e.593.3 8
63.40 odd 6 1134.2.t.e.593.2 8
63.47 even 6 1134.2.l.f.215.4 8
63.61 odd 6 1134.2.l.f.215.1 8
84.11 even 6 7056.2.k.f.881.5 8
84.47 odd 6 1008.2.bt.c.593.3 8
84.59 odd 6 7056.2.k.f.881.3 8
105.47 odd 12 3150.2.bp.e.1349.3 8
105.68 odd 12 3150.2.bp.b.1349.2 8
105.89 even 6 3150.2.bf.a.1601.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.2 ✓ 8 1.1 even 1 trivial
126.2.k.a.17.3 yes 8 3.2 odd 2 inner
126.2.k.a.89.2 yes 8 21.5 even 6 inner
126.2.k.a.89.3 yes 8 7.5 odd 6 inner
882.2.d.a.881.2 8 7.4 even 3
882.2.d.a.881.3 8 7.3 odd 6
882.2.d.a.881.6 8 21.17 even 6
882.2.d.a.881.7 8 21.11 odd 6
882.2.k.a.215.1 8 21.2 odd 6
882.2.k.a.215.4 8 7.2 even 3
882.2.k.a.521.1 8 7.6 odd 2
882.2.k.a.521.4 8 21.20 even 2
1008.2.bt.c.17.2 8 12.11 even 2
1008.2.bt.c.17.3 8 4.3 odd 2
1008.2.bt.c.593.2 8 28.19 even 6
1008.2.bt.c.593.3 8 84.47 odd 6
1134.2.l.f.215.1 8 63.61 odd 6
1134.2.l.f.215.4 8 63.47 even 6
1134.2.l.f.269.2 8 9.4 even 3
1134.2.l.f.269.3 8 9.5 odd 6
1134.2.t.e.593.2 8 63.40 odd 6
1134.2.t.e.593.3 8 63.5 even 6
1134.2.t.e.1025.2 8 9.2 odd 6
1134.2.t.e.1025.3 8 9.7 even 3
3150.2.bf.a.1151.1 8 15.14 odd 2
3150.2.bf.a.1151.3 8 5.4 even 2
3150.2.bf.a.1601.1 8 35.19 odd 6
3150.2.bf.a.1601.3 8 105.89 even 6
3150.2.bp.b.899.2 8 5.2 odd 4
3150.2.bp.b.899.3 8 15.8 even 4
3150.2.bp.b.1349.2 8 105.68 odd 12
3150.2.bp.b.1349.3 8 35.12 even 12
3150.2.bp.e.899.2 8 15.2 even 4
3150.2.bp.e.899.3 8 5.3 odd 4
3150.2.bp.e.1349.2 8 35.33 even 12
3150.2.bp.e.1349.3 8 105.47 odd 12
7056.2.k.f.881.3 8 84.59 odd 6
7056.2.k.f.881.4 8 28.11 odd 6
7056.2.k.f.881.5 8 84.11 even 6
7056.2.k.f.881.6 8 28.3 even 6