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Properties

Label 126.8.g.d.37.1
Level $126$
Weight $8$
Character 126.37
Analytic conductor $39.361$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [126,8,Mod(37,126)] 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("126.37"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-16,0,-128,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.3605132110\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{949})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(7.95146 - 13.7723i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.8.g.d.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.00000 + 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-219.141 + 379.563i) q^{5} +(-893.282 - 159.971i) q^{7} +512.000 q^{8} +(-1753.13 - 3036.51i) q^{10} +(-2740.43 - 4746.57i) q^{11} +4006.54 q^{13} +(4681.44 - 5548.96i) q^{14} +(-2048.00 + 3547.24i) q^{16} +(-14011.5 - 24268.6i) q^{17} +(-11920.6 + 20647.0i) q^{19} +28050.0 q^{20} +43846.9 q^{22} +(-36877.3 + 63873.3i) q^{23} +(-56983.0 - 98697.4i) q^{25} +(-16026.2 + 27758.1i) q^{26} +(19718.5 + 54629.8i) q^{28} +98721.3 q^{29} +(23743.4 + 41124.8i) q^{31} +(-16384.0 - 28377.9i) q^{32} +224184. q^{34} +(256474. - 304001. i) q^{35} +(50031.2 - 86656.7i) q^{37} +(-95364.6 - 165176. i) q^{38} +(-112200. + 194336. i) q^{40} -489123. q^{41} +299600. q^{43} +(-175388. + 303781. i) q^{44} +(-295018. - 510986. i) q^{46} +(481369. - 833756. i) q^{47} +(772362. + 285798. i) q^{49} +911728. q^{50} +(-128209. - 222065. i) q^{52} +(918933. + 1.59164e6i) q^{53} +2.40216e6 q^{55} +(-457360. - 81905.0i) q^{56} +(-394885. + 683961. i) q^{58} +(-7255.29 - 12566.5i) q^{59} +(-1.01469e6 + 1.75749e6i) q^{61} -379895. q^{62} +262144. q^{64} +(-877997. + 1.52074e6i) q^{65} +(1.48449e6 + 2.57121e6i) q^{67} +(-896736. + 1.55319e6i) q^{68} +(1.08028e6 + 2.99290e6i) q^{70} +4.34296e6 q^{71} +(-750529. - 1.29995e6i) q^{73} +(400250. + 693253. i) q^{74} +1.52583e6 q^{76} +(1.68867e6 + 4.67841e6i) q^{77} +(886182. - 1.53491e6i) q^{79} +(-897601. - 1.55469e6i) q^{80} +(1.95649e6 - 3.38874e6i) q^{82} +1.57509e6 q^{83} +1.22820e7 q^{85} +(-1.19840e6 + 2.07569e6i) q^{86} +(-1.40310e6 - 2.43024e6i) q^{88} +(4.39727e6 - 7.61629e6i) q^{89} +(-3.57897e6 - 640929. i) q^{91} +4.72029e6 q^{92} +(3.85096e6 + 6.67005e6i) q^{94} +(-5.22457e6 - 9.04922e6i) q^{95} -1.03493e7 q^{97} +(-5.06951e6 + 4.20789e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{2} - 128 q^{4} - 14 q^{5} - 1848 q^{7} + 2048 q^{8} - 112 q^{10} + 2408 q^{11} + 21448 q^{13} + 22176 q^{14} - 8192 q^{16} - 35098 q^{17} + 2408 q^{19} + 1792 q^{20} - 38528 q^{22} - 61684 q^{23}+ \cdots - 4337872 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 + 6.92820i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −32.0000 − 55.4256i −0.250000 − 0.433013i
\(5\) −219.141 + 379.563i −0.784022 + 1.35797i 0.145559 + 0.989350i \(0.453502\pi\)
−0.929581 + 0.368617i \(0.879831\pi\)
\(6\) 0 0
\(7\) −893.282 − 159.971i −0.984340 − 0.176278i
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) −1753.13 − 3036.51i −0.554388 − 0.960227i
\(11\) −2740.43 − 4746.57i −0.620790 − 1.07524i −0.989339 − 0.145632i \(-0.953478\pi\)
0.368549 − 0.929609i \(-0.379855\pi\)
\(12\) 0 0
\(13\) 4006.54 0.505787 0.252894 − 0.967494i \(-0.418618\pi\)
0.252894 + 0.967494i \(0.418618\pi\)
\(14\) 4681.44 − 5548.96i 0.455964 − 0.540459i
\(15\) 0 0
\(16\) −2048.00 + 3547.24i −0.125000 + 0.216506i
\(17\) −14011.5 − 24268.6i −0.691693 − 1.19805i −0.971283 − 0.237927i \(-0.923532\pi\)
0.279590 − 0.960119i \(-0.409801\pi\)
\(18\) 0 0
\(19\) −11920.6 + 20647.0i −0.398712 + 0.690590i −0.993567 − 0.113243i \(-0.963876\pi\)
0.594855 + 0.803833i \(0.297209\pi\)
\(20\) 28050.0 0.784022
\(21\) 0 0
\(22\) 43846.9 0.877930
\(23\) −36877.3 + 63873.3i −0.631992 + 1.09464i 0.355152 + 0.934808i \(0.384429\pi\)
−0.987144 + 0.159833i \(0.948904\pi\)
\(24\) 0 0
\(25\) −56983.0 − 98697.4i −0.729382 − 1.26333i
\(26\) −16026.2 + 27758.1i −0.178823 + 0.309730i
\(27\) 0 0
\(28\) 19718.5 + 54629.8i 0.169755 + 0.470301i
\(29\) 98721.3 0.751653 0.375827 − 0.926690i \(-0.377359\pi\)
0.375827 + 0.926690i \(0.377359\pi\)
\(30\) 0 0
\(31\) 23743.4 + 41124.8i 0.143145 + 0.247935i 0.928680 − 0.370883i \(-0.120945\pi\)
−0.785534 + 0.618818i \(0.787612\pi\)
\(32\) −16384.0 − 28377.9i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 224184. 0.978201
\(35\) 256474. − 304001.i 1.01112 − 1.19850i
\(36\) 0 0
\(37\) 50031.2 − 86656.7i 0.162381 − 0.281252i −0.773341 − 0.633990i \(-0.781416\pi\)
0.935722 + 0.352738i \(0.114749\pi\)
\(38\) −95364.6 − 165176.i −0.281932 − 0.488321i
\(39\) 0 0
\(40\) −112200. + 194336.i −0.277194 + 0.480114i
\(41\) −489123. −1.10834 −0.554172 − 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(42\) 0 0
\(43\) 299600. 0.574649 0.287324 − 0.957833i \(-0.407234\pi\)
0.287324 + 0.957833i \(0.407234\pi\)
\(44\) −175388. + 303781.i −0.310395 + 0.537620i
\(45\) 0 0
\(46\) −295018. − 510986.i −0.446886 − 0.774029i
\(47\) 481369. − 833756.i 0.676295 − 1.17138i −0.299794 − 0.954004i \(-0.596918\pi\)
0.976089 − 0.217373i \(-0.0697487\pi\)
\(48\) 0 0
\(49\) 772362. + 285798.i 0.937852 + 0.347034i
\(50\) 911728. 1.03150
\(51\) 0 0
\(52\) −128209. − 222065.i −0.126447 − 0.219012i
\(53\) 918933. + 1.59164e6i 0.847849 + 1.46852i 0.883124 + 0.469139i \(0.155436\pi\)
−0.0352758 + 0.999378i \(0.511231\pi\)
\(54\) 0 0
\(55\) 2.40216e6 1.94685
\(56\) −457360. − 81905.0i −0.348017 − 0.0623235i
\(57\) 0 0
\(58\) −394885. + 683961.i −0.265750 + 0.460292i
\(59\) −7255.29 − 12566.5i −0.00459910 − 0.00796587i 0.863717 − 0.503978i \(-0.168131\pi\)
−0.868316 + 0.496012i \(0.834797\pi\)
\(60\) 0 0
\(61\) −1.01469e6 + 1.75749e6i −0.572370 + 0.991374i 0.423952 + 0.905685i \(0.360643\pi\)
−0.996322 + 0.0856896i \(0.972691\pi\)
\(62\) −379895. −0.202438
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −877997. + 1.52074e6i −0.396549 + 0.686842i
\(66\) 0 0
\(67\) 1.48449e6 + 2.57121e6i 0.602997 + 1.04442i 0.992365 + 0.123339i \(0.0393602\pi\)
−0.389368 + 0.921082i \(0.627306\pi\)
\(68\) −896736. + 1.55319e6i −0.345846 + 0.599023i
\(69\) 0 0
\(70\) 1.08028e6 + 2.99290e6i 0.376439 + 1.04292i
\(71\) 4.34296e6 1.44006 0.720031 − 0.693942i \(-0.244128\pi\)
0.720031 + 0.693942i \(0.244128\pi\)
\(72\) 0 0
\(73\) −750529. − 1.29995e6i −0.225807 − 0.391109i 0.730754 − 0.682641i \(-0.239169\pi\)
−0.956561 + 0.291531i \(0.905835\pi\)
\(74\) 400250. + 693253.i 0.114821 + 0.198875i
\(75\) 0 0
\(76\) 1.52583e6 0.398712
\(77\) 1.68867e6 + 4.67841e6i 0.421528 + 1.16783i
\(78\) 0 0
\(79\) 886182. − 1.53491e6i 0.202222 − 0.350259i −0.747022 − 0.664799i \(-0.768517\pi\)
0.949244 + 0.314541i \(0.101850\pi\)
\(80\) −897601. − 1.55469e6i −0.196006 − 0.339492i
\(81\) 0 0
\(82\) 1.95649e6 − 3.38874e6i 0.391859 − 0.678720i
\(83\) 1.57509e6 0.302366 0.151183 − 0.988506i \(-0.451692\pi\)
0.151183 + 0.988506i \(0.451692\pi\)
\(84\) 0 0
\(85\) 1.22820e7 2.16921
\(86\) −1.19840e6 + 2.07569e6i −0.203169 + 0.351899i
\(87\) 0 0
\(88\) −1.40310e6 − 2.43024e6i −0.219483 − 0.380155i
\(89\) 4.39727e6 − 7.61629e6i 0.661177 − 1.14519i −0.319130 − 0.947711i \(-0.603391\pi\)
0.980307 − 0.197481i \(-0.0632761\pi\)
\(90\) 0 0
\(91\) −3.57897e6 − 640929.i −0.497867 − 0.0891590i
\(92\) 4.72029e6 0.631992
\(93\) 0 0
\(94\) 3.85096e6 + 6.67005e6i 0.478213 + 0.828288i
\(95\) −5.22457e6 − 9.04922e6i −0.625199 − 1.08288i
\(96\) 0 0
\(97\) −1.03493e7 −1.15135 −0.575676 − 0.817678i \(-0.695261\pi\)
−0.575676 + 0.817678i \(0.695261\pi\)
\(98\) −5.06951e6 + 4.20789e6i −0.544095 + 0.451620i
\(99\) 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 126.8.g.d.37.1 4
3.2 odd 2 14.8.c.b.9.2 ✓ 4
7.4 even 3 inner 126.8.g.d.109.1 4
12.11 even 2 112.8.i.b.65.1 4
21.2 odd 6 98.8.a.f.1.1 2
21.5 even 6 98.8.a.d.1.2 2
21.11 odd 6 14.8.c.b.11.2 yes 4
21.17 even 6 98.8.c.m.67.1 4
21.20 even 2 98.8.c.m.79.1 4
84.11 even 6 112.8.i.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.2 ✓ 4 3.2 odd 2
14.8.c.b.11.2 yes 4 21.11 odd 6
98.8.a.d.1.2 2 21.5 even 6
98.8.a.f.1.1 2 21.2 odd 6
98.8.c.m.67.1 4 21.17 even 6
98.8.c.m.79.1 4 21.20 even 2
112.8.i.b.65.1 4 12.11 even 2
112.8.i.b.81.1 4 84.11 even 6
126.8.g.d.37.1 4 1.1 even 1 trivial
126.8.g.d.109.1 4 7.4 even 3 inner