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 37.2
Root \(-7.45146 + 12.9063i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.8.g.d.109.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.00000 + 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(212.141 - 367.439i) q^{5} +(-30.7182 - 906.973i) q^{7} +512.000 q^{8} +(1697.13 + 2939.51i) q^{10} +(3944.43 + 6831.96i) q^{11} +6717.46 q^{13} +(6406.56 + 3415.07i) q^{14} +(-2048.00 + 3547.24i) q^{16} +(-3537.51 - 6127.14i) q^{17} +(13124.6 - 22732.4i) q^{19} -27154.0 q^{20} -63110.9 q^{22} +(6035.27 - 10453.4i) q^{23} +(-50945.0 - 88239.4i) q^{25} +(-26869.8 + 46539.9i) q^{26} +(-49286.5 + 30725.7i) q^{28} -3061.25 q^{29} +(59262.6 + 102646. i) q^{31} +(-16384.0 - 28377.9i) q^{32} +56600.1 q^{34} +(-339774. - 181119. i) q^{35} +(229876. - 398156. i) q^{37} +(104997. + 181859. i) q^{38} +(108616. - 188129. i) q^{40} -316857. q^{41} -31624.2 q^{43} +(252444. - 437245. i) q^{44} +(48282.2 + 83627.2i) q^{46} +(403277. - 698496. i) q^{47} +(-821656. + 55721.1i) q^{49} +815120. q^{50} +(-214959. - 372319. i) q^{52} +(239664. + 415110. i) q^{53} +3.34710e6 q^{55} +(-15727.7 - 464370. i) q^{56} +(12245.0 - 21209.0i) q^{58} +(337431. + 584448. i) q^{59} +(283165. - 490455. i) q^{61} -948201. q^{62} +262144. q^{64} +(1.42505e6 - 2.46825e6i) q^{65} +(-592349. - 1.02598e6i) q^{67} +(-226400. + 392137. i) q^{68} +(2.61392e6 - 1.62954e6i) q^{70} -4.61736e6 q^{71} +(-1.52400e6 - 2.63965e6i) q^{73} +(1.83901e6 + 3.18525e6i) q^{74} -1.67995e6 q^{76} +(6.07524e6 - 3.78736e6i) q^{77} +(3.45080e6 - 5.97696e6i) q^{79} +(868929. + 1.50503e6i) q^{80} +(1.26743e6 - 2.19525e6i) q^{82} +9.01927e6 q^{83} -3.00180e6 q^{85} +(126497. - 219099. i) q^{86} +(2.01955e6 + 3.49796e6i) q^{88} +(-3.50739e6 + 6.07498e6i) q^{89} +(-206348. - 6.09255e6i) q^{91} -772515. q^{92} +(3.22621e6 + 5.58796e6i) q^{94} +(-5.56852e6 - 9.64496e6i) q^{95} +8.60029e6 q^{97} +(2.90058e6 - 5.91548e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{2} - 128 q^{4} - 14 q^{5} - 1848 q^{7} + 2048 q^{8} - 112 q^{10} + 2408 q^{11} + 21448 q^{13} + 22176 q^{14} - 8192 q^{16} - 35098 q^{17} + 2408 q^{19} + 1792 q^{20} - 38528 q^{22} - 61684 q^{23}+ \cdots - 4337872 q^{98}+O(q^{100}) \) 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
\(5\) 212.141 − 367.439i 0.758978 − 1.31459i −0.184394 − 0.982852i \(-0.559032\pi\)
0.943372 − 0.331737i \(-0.107635\pi\)
\(6\) 0 0
\(7\) −30.7182 − 906.973i −0.0338495 − 0.999427i
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) 1697.13 + 2939.51i 0.536679 + 0.929555i
\(11\) 3944.43 + 6831.96i 0.893532 + 1.54764i 0.835611 + 0.549322i \(0.185114\pi\)
0.0579219 + 0.998321i \(0.481553\pi\)
\(12\) 0 0
\(13\) 6717.46 0.848014 0.424007 − 0.905659i \(-0.360623\pi\)
0.424007 + 0.905659i \(0.360623\pi\)
\(14\) 6406.56 + 3415.07i 0.623989 + 0.332622i
\(15\) 0 0
\(16\) −2048.00 + 3547.24i −0.125000 + 0.216506i
\(17\) −3537.51 − 6127.14i −0.174633 − 0.302473i 0.765401 − 0.643553i \(-0.222541\pi\)
−0.940034 + 0.341080i \(0.889207\pi\)
\(18\) 0 0
\(19\) 13124.6 − 22732.4i 0.438983 − 0.760341i −0.558628 − 0.829418i \(-0.688672\pi\)
0.997611 + 0.0690773i \(0.0220055\pi\)
\(20\) −27154.0 −0.758978
\(21\) 0 0
\(22\) −63110.9 −1.26365
\(23\) 6035.27 − 10453.4i 0.103431 − 0.179147i −0.809665 − 0.586892i \(-0.800351\pi\)
0.913096 + 0.407745i \(0.133685\pi\)
\(24\) 0 0
\(25\) −50945.0 − 88239.4i −0.652096 − 1.12946i
\(26\) −26869.8 + 46539.9i −0.299818 + 0.519301i
\(27\) 0 0
\(28\) −49286.5 + 30725.7i −0.424302 + 0.264514i
\(29\) −3061.25 −0.0233081 −0.0116540 − 0.999932i \(-0.503710\pi\)
−0.0116540 + 0.999932i \(0.503710\pi\)
\(30\) 0 0
\(31\) 59262.6 + 102646.i 0.357285 + 0.618835i 0.987506 − 0.157580i \(-0.0503692\pi\)
−0.630221 + 0.776415i \(0.717036\pi\)
\(32\) −16384.0 − 28377.9i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 56600.1 0.246968
\(35\) −339774. − 181119.i −1.33953 − 0.714045i
\(36\) 0 0
\(37\) 229876. − 398156.i 0.746083 − 1.29225i −0.203604 − 0.979053i \(-0.565266\pi\)
0.949687 − 0.313200i \(-0.101401\pi\)
\(38\) 104997. + 181859.i 0.310408 + 0.537642i
\(39\) 0 0
\(40\) 108616. − 188129.i 0.268339 − 0.464777i
\(41\) −316857. −0.717992 −0.358996 − 0.933339i \(-0.616881\pi\)
−0.358996 + 0.933339i \(0.616881\pi\)
\(42\) 0 0
\(43\) −31624.2 −0.0606569 −0.0303284 − 0.999540i \(-0.509655\pi\)
−0.0303284 + 0.999540i \(0.509655\pi\)
\(44\) 252444. − 437245.i 0.446766 − 0.773822i
\(45\) 0 0
\(46\) 48282.2 + 83627.2i 0.0731365 + 0.126676i
\(47\) 403277. − 698496.i 0.566579 − 0.981344i −0.430322 − 0.902676i \(-0.641600\pi\)
0.996901 − 0.0786682i \(-0.0250668\pi\)
\(48\) 0 0
\(49\) −821656. + 55721.1i −0.997708 + 0.0676602i
\(50\) 815120. 0.922204
\(51\) 0 0
\(52\) −214959. − 372319.i −0.212004 − 0.367201i
\(53\) 239664. + 415110.i 0.221125 + 0.382999i 0.955150 − 0.296123i \(-0.0956939\pi\)
−0.734025 + 0.679122i \(0.762361\pi\)
\(54\) 0 0
\(55\) 3.34710e6 2.71269
\(56\) −15727.7 − 464370.i −0.0119676 − 0.353351i
\(57\) 0 0
\(58\) 12245.0 − 21209.0i 0.00824065 − 0.0142732i
\(59\) 337431. + 584448.i 0.213896 + 0.370479i 0.952931 − 0.303189i \(-0.0980512\pi\)
−0.739034 + 0.673668i \(0.764718\pi\)
\(60\) 0 0
\(61\) 283165. − 490455.i 0.159729 − 0.276659i −0.775042 − 0.631910i \(-0.782271\pi\)
0.934771 + 0.355251i \(0.115605\pi\)
\(62\) −948201. −0.505277
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 1.42505e6 − 2.46825e6i 0.643625 − 1.11479i
\(66\) 0 0
\(67\) −592349. − 1.02598e6i −0.240611 − 0.416751i 0.720277 − 0.693686i \(-0.244015\pi\)
−0.960889 + 0.276935i \(0.910681\pi\)
\(68\) −226400. + 392137.i −0.0873164 + 0.151237i
\(69\) 0 0
\(70\) 2.61392e6 − 1.62954e6i 0.910856 − 0.567836i
\(71\) −4.61736e6 −1.53105 −0.765524 − 0.643407i \(-0.777520\pi\)
−0.765524 + 0.643407i \(0.777520\pi\)
\(72\) 0 0
\(73\) −1.52400e6 − 2.63965e6i −0.458517 − 0.794175i 0.540366 − 0.841430i \(-0.318286\pi\)
−0.998883 + 0.0472553i \(0.984953\pi\)
\(74\) 1.83901e6 + 3.18525e6i 0.527560 + 0.913761i
\(75\) 0 0
\(76\) −1.67995e6 −0.438983
\(77\) 6.07524e6 − 3.78736e6i 1.51651 − 0.945407i
\(78\) 0 0
\(79\) 3.45080e6 − 5.97696e6i 0.787454 − 1.36391i −0.140069 − 0.990142i \(-0.544732\pi\)
0.927522 − 0.373768i \(-0.121934\pi\)
\(80\) 868929. + 1.50503e6i 0.189745 + 0.328647i
\(81\) 0 0
\(82\) 1.26743e6 − 2.19525e6i 0.253849 − 0.439679i
\(83\) 9.01927e6 1.73140 0.865701 − 0.500562i \(-0.166873\pi\)
0.865701 + 0.500562i \(0.166873\pi\)
\(84\) 0 0
\(85\) −3.00180e6 −0.530170
\(86\) 126497. − 219099.i 0.0214455 − 0.0371446i
\(87\) 0 0
\(88\) 2.01955e6 + 3.49796e6i 0.315911 + 0.547175i
\(89\) −3.50739e6 + 6.07498e6i −0.527374 + 0.913439i 0.472116 + 0.881536i \(0.343490\pi\)
−0.999491 + 0.0319032i \(0.989843\pi\)
\(90\) 0 0
\(91\) −206348. − 6.09255e6i −0.0287049 − 0.847528i
\(92\) −772515. −0.103431
\(93\) 0 0
\(94\) 3.22621e6 + 5.58796e6i 0.400632 + 0.693915i
\(95\) −5.56852e6 − 9.64496e6i −0.666357 − 1.15416i
\(96\) 0 0
\(97\) 8.60029e6 0.956779 0.478390 − 0.878148i \(-0.341221\pi\)
0.478390 + 0.878148i \(0.341221\pi\)
\(98\) 2.90058e6 − 5.91548e6i 0.311310 − 0.634891i
\(99\) 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.37.2 4
3.2 odd 2 14.8.c.b.9.1 ✓ 4
7.4 even 3 inner 126.8.g.d.109.2 4
12.11 even 2 112.8.i.b.65.2 4
21.2 odd 6 98.8.a.f.1.2 2
21.5 even 6 98.8.a.d.1.1 2
21.11 odd 6 14.8.c.b.11.1 yes 4
21.17 even 6 98.8.c.m.67.2 4
21.20 even 2 98.8.c.m.79.2 4
84.11 even 6 112.8.i.b.81.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.1 ✓ 4 3.2 odd 2
14.8.c.b.11.1 yes 4 21.11 odd 6
98.8.a.d.1.1 2 21.5 even 6
98.8.a.f.1.2 2 21.2 odd 6
98.8.c.m.67.2 4 21.17 even 6
98.8.c.m.79.2 4 21.20 even 2
112.8.i.b.65.2 4 12.11 even 2
112.8.i.b.81.2 4 84.11 even 6
126.8.g.d.37.2 4 1.1 even 1 trivial
126.8.g.d.109.2 4 7.4 even 3 inner