Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128,0,-4096,-7504] 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: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.4
Root \(-100.397 - 173.893i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.12.g.e.109.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 - 27.7128i) q^{2} +(-512.000 - 886.810i) q^{4} +(3326.37 - 5761.45i) q^{5} +(12255.3 + 42745.0i) q^{7} -32768.0 q^{8} +(-106444. - 184366. i) q^{10} +(60907.1 + 105494. i) q^{11} -1.81677e6 q^{13} +(1.38067e6 + 344291. i) q^{14} +(-524288. + 908093. i) q^{16} +(2.70113e6 + 4.67849e6i) q^{17} +(5.10442e6 - 8.84111e6i) q^{19} -6.81241e6 q^{20} +3.89805e6 q^{22} +(1.88132e7 - 3.25854e7i) q^{23} +(2.28454e6 + 3.95694e6i) q^{25} +(-2.90684e7 + 5.03479e7i) q^{26} +(3.16320e7 - 3.27536e7i) q^{28} +1.87020e7 q^{29} +(1.22797e8 + 2.12690e8i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} +1.72872e8 q^{34} +(2.87039e8 + 7.15776e7i) q^{35} +(-3.25911e8 + 5.64495e8i) q^{37} +(-1.63341e8 - 2.82916e8i) q^{38} +(-1.08999e8 + 1.88791e8i) q^{40} +6.38428e8 q^{41} +8.54732e8 q^{43} +(6.23688e7 - 1.08026e8i) q^{44} +(-6.02023e8 - 1.04273e9i) q^{46} +(-4.23660e8 + 7.33801e8i) q^{47} +(-1.67694e9 + 1.04770e9i) q^{49} +1.46210e8 q^{50} +(9.30187e8 + 1.61113e9i) q^{52} +(-2.11267e9 - 3.65925e9i) q^{53} +8.10399e8 q^{55} +(-4.01581e8 - 1.40067e9i) q^{56} +(2.99232e8 - 5.18285e8i) q^{58} +(-1.10225e9 - 1.90916e9i) q^{59} +(4.84994e9 - 8.40035e9i) q^{61} +7.85899e9 q^{62} +1.07374e9 q^{64} +(-6.04326e9 + 1.04672e10i) q^{65} +(4.86446e9 + 8.42549e9i) q^{67} +(2.76595e9 - 4.79077e9i) q^{68} +(6.57624e9 - 6.80941e9i) q^{70} +2.83613e10 q^{71} +(1.83417e9 + 3.17688e9i) q^{73} +(1.04292e10 + 1.80638e10i) q^{74} -1.04539e10 q^{76} +(-3.76291e9 + 3.89633e9i) q^{77} +(-6.34771e9 + 1.09946e10i) q^{79} +(3.48796e9 + 6.04132e9i) q^{80} +(1.02148e10 - 1.76926e10i) q^{82} +4.03358e10 q^{83} +3.59398e10 q^{85} +(1.36757e10 - 2.36870e10i) q^{86} +(-1.99580e9 - 3.45683e9i) q^{88} +(-1.64368e10 + 2.84694e10i) q^{89} +(-2.22651e10 - 7.76579e10i) q^{91} -3.85295e10 q^{92} +(1.35571e10 + 2.34816e10i) q^{94} +(-3.39584e10 - 5.88177e10i) q^{95} +1.02279e11 q^{97} +(2.20377e9 + 6.32361e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 4096 q^{4} - 7504 q^{5} - 42224 q^{7} - 262144 q^{8} + 240128 q^{10} - 213026 q^{11} - 2609712 q^{13} - 1257536 q^{14} - 4194304 q^{16} - 8854244 q^{17} + 7232806 q^{19} + 15368192 q^{20}+ \cdots - 210290620288 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 − 27.7128i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) 3326.37 − 5761.45i 0.476032 − 0.824511i −0.523591 − 0.851970i \(-0.675408\pi\)
0.999623 + 0.0274583i \(0.00874136\pi\)
\(6\) 0 0
\(7\) 12255.3 + 42745.0i 0.275603 + 0.961271i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) −106444. − 184366.i −0.336605 − 0.583018i
\(11\) 60907.1 + 105494.i 0.114027 + 0.197501i 0.917390 − 0.397988i \(-0.130292\pi\)
−0.803363 + 0.595489i \(0.796958\pi\)
\(12\) 0 0
\(13\) −1.81677e6 −1.35710 −0.678550 − 0.734554i \(-0.737391\pi\)
−0.678550 + 0.734554i \(0.737391\pi\)
\(14\) 1.38067e6 + 344291.i 0.686097 + 0.171089i
\(15\) 0 0
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 2.70113e6 + 4.67849e6i 0.461398 + 0.799165i 0.999031 − 0.0440143i \(-0.0140147\pi\)
−0.537633 + 0.843179i \(0.680681\pi\)
\(18\) 0 0
\(19\) 5.10442e6 − 8.84111e6i 0.472935 − 0.819148i −0.526585 − 0.850122i \(-0.676528\pi\)
0.999520 + 0.0309748i \(0.00986115\pi\)
\(20\) −6.81241e6 −0.476032
\(21\) 0 0
\(22\) 3.89805e6 0.161259
\(23\) 1.88132e7 − 3.25854e7i 0.609481 − 1.05565i −0.381845 − 0.924226i \(-0.624711\pi\)
0.991326 − 0.131425i \(-0.0419554\pi\)
\(24\) 0 0
\(25\) 2.28454e6 + 3.95694e6i 0.0467873 + 0.0810381i
\(26\) −2.90684e7 + 5.03479e7i −0.479807 + 0.831051i
\(27\) 0 0
\(28\) 3.16320e7 − 3.27536e7i 0.347342 − 0.359658i
\(29\) 1.87020e7 0.169316 0.0846581 − 0.996410i \(-0.473020\pi\)
0.0846581 + 0.996410i \(0.473020\pi\)
\(30\) 0 0
\(31\) 1.22797e8 + 2.12690e8i 0.770367 + 1.33431i 0.937362 + 0.348357i \(0.113260\pi\)
−0.166995 + 0.985958i \(0.553406\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 1.72872e8 0.652515
\(35\) 2.87039e8 + 7.15776e7i 0.923775 + 0.230358i
\(36\) 0 0
\(37\) −3.25911e8 + 5.64495e8i −0.772663 + 1.33829i 0.163436 + 0.986554i \(0.447742\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(38\) −1.63341e8 − 2.82916e8i −0.334416 − 0.579225i
\(39\) 0 0
\(40\) −1.08999e8 + 1.88791e8i −0.168303 + 0.291509i
\(41\) 6.38428e8 0.860598 0.430299 − 0.902686i \(-0.358408\pi\)
0.430299 + 0.902686i \(0.358408\pi\)
\(42\) 0 0
\(43\) 8.54732e8 0.886652 0.443326 − 0.896360i \(-0.353798\pi\)
0.443326 + 0.896360i \(0.353798\pi\)
\(44\) 6.23688e7 − 1.08026e8i 0.0570135 − 0.0987503i
\(45\) 0 0
\(46\) −6.02023e8 − 1.04273e9i −0.430968 − 0.746458i
\(47\) −4.23660e8 + 7.33801e8i −0.269451 + 0.466702i −0.968720 − 0.248156i \(-0.920175\pi\)
0.699269 + 0.714858i \(0.253509\pi\)
\(48\) 0 0
\(49\) −1.67694e9 + 1.04770e9i −0.848086 + 0.529859i
\(50\) 1.46210e8 0.0661673
\(51\) 0 0
\(52\) 9.30187e8 + 1.61113e9i 0.339275 + 0.587642i
\(53\) −2.11267e9 − 3.65925e9i −0.693927 − 1.20192i −0.970541 − 0.240936i \(-0.922546\pi\)
0.276614 − 0.960981i \(-0.410788\pi\)
\(54\) 0 0
\(55\) 8.10399e8 0.217122
\(56\) −4.01581e8 − 1.40067e9i −0.0974405 − 0.339861i
\(57\) 0 0
\(58\) 2.99232e8 − 5.18285e8i 0.0598623 − 0.103685i
\(59\) −1.10225e9 − 1.90916e9i −0.200722 − 0.347661i 0.748039 − 0.663655i \(-0.230995\pi\)
−0.948761 + 0.315994i \(0.897662\pi\)
\(60\) 0 0
\(61\) 4.84994e9 − 8.40035e9i 0.735229 − 1.27345i −0.219394 − 0.975636i \(-0.570408\pi\)
0.954623 − 0.297817i \(-0.0962587\pi\)
\(62\) 7.85899e9 1.08946
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −6.04326e9 + 1.04672e10i −0.646023 + 1.11894i
\(66\) 0 0
\(67\) 4.86446e9 + 8.42549e9i 0.440173 + 0.762401i 0.997702 − 0.0677559i \(-0.0215839\pi\)
−0.557529 + 0.830157i \(0.688251\pi\)
\(68\) 2.76595e9 − 4.79077e9i 0.230699 − 0.399582i
\(69\) 0 0
\(70\) 6.57624e9 − 6.80941e9i 0.467669 − 0.484251i
\(71\) 2.83613e10 1.86554 0.932770 − 0.360472i \(-0.117384\pi\)
0.932770 + 0.360472i \(0.117384\pi\)
\(72\) 0 0
\(73\) 1.83417e9 + 3.17688e9i 0.103553 + 0.179360i 0.913146 − 0.407632i \(-0.133645\pi\)
−0.809593 + 0.586992i \(0.800312\pi\)
\(74\) 1.04292e10 + 1.80638e10i 0.546355 + 0.946315i
\(75\) 0 0
\(76\) −1.04539e10 −0.472935
\(77\) −3.76291e9 + 3.89633e9i −0.158426 + 0.164043i
\(78\) 0 0
\(79\) −6.34771e9 + 1.09946e10i −0.232096 + 0.402002i −0.958425 − 0.285345i \(-0.907892\pi\)
0.726329 + 0.687348i \(0.241225\pi\)
\(80\) 3.48796e9 + 6.04132e9i 0.119008 + 0.206128i
\(81\) 0 0
\(82\) 1.02148e10 − 1.76926e10i 0.304267 − 0.527007i
\(83\) 4.03358e10 1.12399 0.561994 − 0.827141i \(-0.310034\pi\)
0.561994 + 0.827141i \(0.310034\pi\)
\(84\) 0 0
\(85\) 3.59398e10 0.878561
\(86\) 1.36757e10 − 2.36870e10i 0.313479 − 0.542961i
\(87\) 0 0
\(88\) −1.99580e9 − 3.45683e9i −0.0403147 − 0.0698270i
\(89\) −1.64368e10 + 2.84694e10i −0.312013 + 0.540423i −0.978798 − 0.204827i \(-0.934337\pi\)
0.666785 + 0.745250i \(0.267670\pi\)
\(90\) 0 0
\(91\) −2.22651e10 − 7.76579e10i −0.374021 − 1.30454i
\(92\) −3.85295e10 −0.609481
\(93\) 0 0
\(94\) 1.35571e10 + 2.34816e10i 0.190530 + 0.330008i
\(95\) −3.39584e10 − 5.88177e10i −0.450264 − 0.779881i
\(96\) 0 0
\(97\) 1.02279e11 1.20932 0.604661 − 0.796483i \(-0.293308\pi\)
0.604661 + 0.796483i \(0.293308\pi\)
\(98\) 2.20377e9 + 6.32361e10i 0.0246276 + 0.706678i
\(99\) 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 126.12.g.e.37.4 8
3.2 odd 2 14.12.c.a.9.3 ✓ 8
7.4 even 3 inner 126.12.g.e.109.4 8
12.11 even 2 112.12.i.a.65.2 8
21.2 odd 6 98.12.a.j.1.2 4
21.5 even 6 98.12.a.l.1.3 4
21.11 odd 6 14.12.c.a.11.3 yes 8
21.17 even 6 98.12.c.l.67.2 8
21.20 even 2 98.12.c.l.79.2 8
84.11 even 6 112.12.i.a.81.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 3.2 odd 2
14.12.c.a.11.3 yes 8 21.11 odd 6
98.12.a.j.1.2 4 21.2 odd 6
98.12.a.l.1.3 4 21.5 even 6
98.12.c.l.67.2 8 21.17 even 6
98.12.c.l.79.2 8 21.20 even 2
112.12.i.a.65.2 8 12.11 even 2
112.12.i.a.81.2 8 84.11 even 6
126.12.g.e.37.4 8 1.1 even 1 trivial
126.12.g.e.109.4 8 7.4 even 3 inner