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 = [14,224,0,-7168,-7527] 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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} + 234028822 x^{12} + 368819651895 x^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{17}\cdot 7^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.2
Root \(-3128.01 + 5417.87i\) of defining polynomial
Character \(\chi\) \(=\) 126.109
Dual form 126.12.g.h.37.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(16.0000 + 27.7128i) q^{2} +(-512.000 + 886.810i) q^{4} +(-3666.01 - 6349.72i) q^{5} +(-1751.68 - 44432.6i) q^{7} -32768.0 q^{8} +(117312. - 203191. i) q^{10} +(272010. - 471134. i) q^{11} +985598. q^{13} +(1.20333e6 - 759466. i) q^{14} +(-524288. - 908093. i) q^{16} +(833539. - 1.44373e6i) q^{17} +(4.86727e6 + 8.43037e6i) q^{19} +7.50799e6 q^{20} +1.74086e7 q^{22} +(4.40544e6 + 7.63045e6i) q^{23} +(-2.46520e6 + 4.26984e6i) q^{25} +(1.57696e7 + 2.73137e7i) q^{26} +(4.03002e7 + 2.11961e7i) q^{28} -9.06105e7 q^{29} +(8.67871e7 - 1.50320e8i) q^{31} +(1.67772e7 - 2.90590e7i) q^{32} +5.33465e7 q^{34} +(-2.75713e8 + 1.74013e8i) q^{35} +(-7.43999e6 - 1.28864e7i) q^{37} +(-1.55753e8 + 2.69772e8i) q^{38} +(1.20128e8 + 2.08067e8i) q^{40} +3.46411e8 q^{41} -1.05810e9 q^{43} +(2.78538e8 + 4.82442e8i) q^{44} +(-1.40974e8 + 2.44174e8i) q^{46} +(-6.86120e8 - 1.18840e9i) q^{47} +(-1.97119e9 + 1.55664e8i) q^{49} -1.57773e8 q^{50} +(-5.04626e8 + 8.74039e8i) q^{52} +(6.86392e7 - 1.18887e8i) q^{53} -3.98876e9 q^{55} +(5.73991e7 + 1.45597e9i) q^{56} +(-1.44977e9 - 2.51107e9i) q^{58} +(3.17070e9 - 5.49181e9i) q^{59} +(-9.53022e8 - 1.65068e9i) q^{61} +5.55438e9 q^{62} +1.07374e9 q^{64} +(-3.61321e9 - 6.25827e9i) q^{65} +(-3.38796e9 + 5.86812e9i) q^{67} +(8.53544e8 + 1.47838e9i) q^{68} +(-9.23380e9 - 4.85657e9i) q^{70} -1.18364e10 q^{71} +(3.72263e9 - 6.44778e9i) q^{73} +(2.38080e8 - 4.12366e8i) q^{74} -9.96818e9 q^{76} +(-2.14102e10 - 1.12608e10i) q^{77} +(1.20868e10 + 2.09350e10i) q^{79} +(-3.84409e9 + 6.65816e9i) q^{80} +(5.54258e9 + 9.60003e9i) q^{82} +5.82871e9 q^{83} -1.22230e10 q^{85} +(-1.69296e10 - 2.93230e10i) q^{86} +(-8.91321e9 + 1.54381e10i) q^{88} +(1.88445e10 + 3.26396e10i) q^{89} +(-1.72646e9 - 4.37927e10i) q^{91} -9.02234e9 q^{92} +(2.19559e10 - 3.80287e10i) q^{94} +(3.56870e10 - 6.18116e10i) q^{95} -1.27781e11 q^{97} +(-3.58529e10 - 5.21366e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 224 q^{2} - 7168 q^{4} - 7527 q^{5} + 40523 q^{7} - 458752 q^{8} + 240864 q^{10} - 216279 q^{11} - 1981052 q^{13} - 1638784 q^{14} - 7340032 q^{16} - 4758882 q^{17} + 13850542 q^{19} + 15415296 q^{20}+ \cdots + 100554558944 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −3666.01 − 6349.72i −0.524637 − 0.908697i −0.999588 − 0.0286855i \(-0.990868\pi\)
0.474952 − 0.880012i \(-0.342465\pi\)
\(6\) 0 0
\(7\) −1751.68 − 44432.6i −0.0393927 − 0.999224i
\(8\) −32768.0 −0.353553
\(9\) 0 0
\(10\) 117312. − 203191.i 0.370974 − 0.642546i
\(11\) 272010. − 471134.i 0.509242 − 0.882033i −0.490701 − 0.871328i \(-0.663259\pi\)
0.999943 − 0.0107050i \(-0.00340758\pi\)
\(12\) 0 0
\(13\) 985598. 0.736227 0.368113 − 0.929781i \(-0.380004\pi\)
0.368113 + 0.929781i \(0.380004\pi\)
\(14\) 1.20333e6 − 759466.i 0.597970 − 0.377402i
\(15\) 0 0
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) 833539. − 1.44373e6i 0.142383 − 0.246614i −0.786011 − 0.618213i \(-0.787857\pi\)
0.928393 + 0.371599i \(0.121190\pi\)
\(18\) 0 0
\(19\) 4.86727e6 + 8.43037e6i 0.450963 + 0.781091i 0.998446 − 0.0557258i \(-0.0177473\pi\)
−0.547483 + 0.836817i \(0.684414\pi\)
\(20\) 7.50799e6 0.524637
\(21\) 0 0
\(22\) 1.74086e7 0.720177
\(23\) 4.40544e6 + 7.63045e6i 0.142721 + 0.247199i 0.928520 − 0.371282i \(-0.121082\pi\)
−0.785800 + 0.618481i \(0.787748\pi\)
\(24\) 0 0
\(25\) −2.46520e6 + 4.26984e6i −0.0504872 + 0.0874464i
\(26\) 1.57696e7 + 2.73137e7i 0.260295 + 0.450845i
\(27\) 0 0
\(28\) 4.03002e7 + 2.11961e7i 0.442525 + 0.232748i
\(29\) −9.06105e7 −0.820332 −0.410166 − 0.912011i \(-0.634529\pi\)
−0.410166 + 0.912011i \(0.634529\pi\)
\(30\) 0 0
\(31\) 8.67871e7 − 1.50320e8i 0.544460 − 0.943032i −0.454181 − 0.890910i \(-0.650068\pi\)
0.998641 − 0.0521227i \(-0.0165987\pi\)
\(32\) 1.67772e7 − 2.90590e7i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 5.33465e7 0.201359
\(35\) −2.75713e8 + 1.74013e8i −0.887325 + 0.560025i
\(36\) 0 0
\(37\) −7.43999e6 − 1.28864e7i −0.0176385 − 0.0305509i 0.857071 − 0.515198i \(-0.172281\pi\)
−0.874710 + 0.484647i \(0.838948\pi\)
\(38\) −1.55753e8 + 2.69772e8i −0.318879 + 0.552315i
\(39\) 0 0
\(40\) 1.20128e8 + 2.08067e8i 0.185487 + 0.321273i
\(41\) 3.46411e8 0.466961 0.233481 − 0.972361i \(-0.424988\pi\)
0.233481 + 0.972361i \(0.424988\pi\)
\(42\) 0 0
\(43\) −1.05810e9 −1.09762 −0.548809 − 0.835948i \(-0.684919\pi\)
−0.548809 + 0.835948i \(0.684919\pi\)
\(44\) 2.78538e8 + 4.82442e8i 0.254621 + 0.441017i
\(45\) 0 0
\(46\) −1.40974e8 + 2.44174e8i −0.100919 + 0.174796i
\(47\) −6.86120e8 − 1.18840e9i −0.436377 − 0.755828i 0.561030 − 0.827796i \(-0.310405\pi\)
−0.997407 + 0.0719681i \(0.977072\pi\)
\(48\) 0 0
\(49\) −1.97119e9 + 1.55664e8i −0.996896 + 0.0787243i
\(50\) −1.57773e8 −0.0713997
\(51\) 0 0
\(52\) −5.04626e8 + 8.74039e8i −0.184057 + 0.318796i
\(53\) 6.86392e7 − 1.18887e8i 0.0225453 − 0.0390495i −0.854533 − 0.519398i \(-0.826156\pi\)
0.877078 + 0.480348i \(0.159490\pi\)
\(54\) 0 0
\(55\) −3.98876e9 −1.06867
\(56\) 5.73991e7 + 1.45597e9i 0.0139274 + 0.353279i
\(57\) 0 0
\(58\) −1.44977e9 − 2.51107e9i −0.290031 − 0.502349i
\(59\) 3.17070e9 − 5.49181e9i 0.577389 − 1.00007i −0.418389 − 0.908268i \(-0.637405\pi\)
0.995778 − 0.0917989i \(-0.0292617\pi\)
\(60\) 0 0
\(61\) −9.53022e8 − 1.65068e9i −0.144474 − 0.250236i 0.784703 − 0.619872i \(-0.212816\pi\)
−0.929176 + 0.369636i \(0.879482\pi\)
\(62\) 5.55438e9 0.769983
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −3.61321e9 − 6.25827e9i −0.386252 − 0.669007i
\(66\) 0 0
\(67\) −3.38796e9 + 5.86812e9i −0.306568 + 0.530992i −0.977609 − 0.210429i \(-0.932514\pi\)
0.671041 + 0.741420i \(0.265847\pi\)
\(68\) 8.53544e8 + 1.47838e9i 0.0711913 + 0.123307i
\(69\) 0 0
\(70\) −9.23380e9 − 4.85657e9i −0.656661 − 0.345375i
\(71\) −1.18364e10 −0.778573 −0.389286 − 0.921117i \(-0.627278\pi\)
−0.389286 + 0.921117i \(0.627278\pi\)
\(72\) 0 0
\(73\) 3.72263e9 − 6.44778e9i 0.210171 − 0.364028i −0.741597 − 0.670846i \(-0.765931\pi\)
0.951768 + 0.306819i \(0.0992645\pi\)
\(74\) 2.38080e8 − 4.12366e8i 0.0124723 − 0.0216027i
\(75\) 0 0
\(76\) −9.96818e9 −0.450963
\(77\) −2.14102e10 − 1.12608e10i −0.901409 − 0.474101i
\(78\) 0 0
\(79\) 1.20868e10 + 2.09350e10i 0.441940 + 0.765463i 0.997833 − 0.0657909i \(-0.0209570\pi\)
−0.555893 + 0.831254i \(0.687624\pi\)
\(80\) −3.84409e9 + 6.65816e9i −0.131159 + 0.227174i
\(81\) 0 0
\(82\) 5.54258e9 + 9.60003e9i 0.165096 + 0.285954i
\(83\) 5.82871e9 0.162421 0.0812106 − 0.996697i \(-0.474121\pi\)
0.0812106 + 0.996697i \(0.474121\pi\)
\(84\) 0 0
\(85\) −1.22230e10 −0.298796
\(86\) −1.69296e10 − 2.93230e10i −0.388067 − 0.672151i
\(87\) 0 0
\(88\) −8.91321e9 + 1.54381e10i −0.180044 + 0.311846i
\(89\) 1.88445e10 + 3.26396e10i 0.357717 + 0.619583i 0.987579 − 0.157123i \(-0.0502220\pi\)
−0.629862 + 0.776707i \(0.716889\pi\)
\(90\) 0 0
\(91\) −1.72646e9 − 4.37927e10i −0.0290020 − 0.735655i
\(92\) −9.02234e9 −0.142721
\(93\) 0 0
\(94\) 2.19559e10 − 3.80287e10i 0.308565 − 0.534451i
\(95\) 3.56870e10 − 6.18116e10i 0.473184 − 0.819578i
\(96\) 0 0
\(97\) −1.27781e11 −1.51086 −0.755428 − 0.655232i \(-0.772571\pi\)
−0.755428 + 0.655232i \(0.772571\pi\)
\(98\) −3.58529e10 − 5.21366e10i −0.400665 − 0.582639i
\(99\) 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.h.109.2 yes 14
3.2 odd 2 126.12.g.g.109.6 yes 14
7.2 even 3 inner 126.12.g.h.37.2 yes 14
21.2 odd 6 126.12.g.g.37.6 ✓ 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.12.g.g.37.6 ✓ 14 21.2 odd 6
126.12.g.g.109.6 yes 14 3.2 odd 2
126.12.g.h.37.2 yes 14 7.2 even 3 inner
126.12.g.h.109.2 yes 14 1.1 even 1 trivial