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(25,126)] 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("126.25"); 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([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.2
Root \(0.500000 - 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 126.25
Dual form 126.2.e.c.121.2

$q$-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-1.00000 q^{2} +(0.933463 - 1.45899i) q^{3} +1.00000 q^{4} +(-0.296790 + 0.514055i) q^{5} +(-0.933463 + 1.45899i) q^{6} +(2.32383 - 1.26483i) q^{7} -1.00000 q^{8} +(-1.25729 - 2.72382i) q^{9} +(0.296790 - 0.514055i) q^{10} +(0.296790 + 0.514055i) q^{11} +(0.933463 - 1.45899i) q^{12} +(-1.25729 - 2.17770i) q^{13} +(-2.32383 + 1.26483i) q^{14} +(0.472958 + 0.912864i) q^{15} +1.00000 q^{16} +(1.46050 - 2.52967i) q^{17} +(1.25729 + 2.72382i) q^{18} +(2.69076 + 4.66053i) q^{19} +(-0.296790 + 0.514055i) q^{20} +(0.323832 - 4.57112i) q^{21} +(-0.296790 - 0.514055i) q^{22} +(-2.23025 + 3.86291i) q^{23} +(-0.933463 + 1.45899i) q^{24} +(2.32383 + 4.02499i) q^{25} +(1.25729 + 2.17770i) q^{26} +(-5.14766 - 0.708209i) q^{27} +(2.32383 - 1.26483i) q^{28} +(-3.09718 + 5.36447i) q^{29} +(-0.472958 - 0.912864i) q^{30} -7.86693 q^{31} -1.00000 q^{32} +(1.02704 + 0.0468383i) q^{33} +(-1.46050 + 2.52967i) q^{34} +(-0.0394951 + 1.56997i) q^{35} +(-1.25729 - 2.72382i) q^{36} +(0.500000 + 0.866025i) q^{37} +(-2.69076 - 4.66053i) q^{38} +(-4.35087 - 0.198422i) q^{39} +(0.296790 - 0.514055i) q^{40} +(-0.136673 - 0.236725i) q^{41} +(-0.323832 + 4.57112i) q^{42} +(-5.58113 + 9.66679i) q^{43} +(0.296790 + 0.514055i) q^{44} +(1.77335 + 0.162084i) q^{45} +(2.23025 - 3.86291i) q^{46} +12.1623 q^{47} +(0.933463 - 1.45899i) q^{48} +(3.80039 - 5.87852i) q^{49} +(-2.32383 - 4.02499i) q^{50} +(-2.32743 - 4.49221i) q^{51} +(-1.25729 - 2.17770i) q^{52} +(4.02704 - 6.97504i) q^{53} +(5.14766 + 0.708209i) q^{54} -0.352336 q^{55} +(-2.32383 + 1.26483i) q^{56} +(9.31138 + 0.424646i) q^{57} +(3.09718 - 5.36447i) q^{58} +8.64766 q^{59} +(0.472958 + 0.912864i) q^{60} -6.64766 q^{61} +7.86693 q^{62} +(-6.36693 - 4.73944i) q^{63} +1.00000 q^{64} +1.49261 q^{65} +(-1.02704 - 0.0468383i) q^{66} -1.91381 q^{67} +(1.46050 - 2.52967i) q^{68} +(3.55408 + 6.85980i) q^{69} +(0.0394951 - 1.56997i) q^{70} -14.4107 q^{71} +(1.25729 + 2.72382i) q^{72} +(3.95691 - 6.85356i) q^{73} +(-0.500000 - 0.866025i) q^{74} +(8.04163 + 0.366739i) q^{75} +(2.69076 + 4.66053i) q^{76} +(1.33988 + 0.819187i) q^{77} +(4.35087 + 0.198422i) q^{78} -9.24844 q^{79} +(-0.296790 + 0.514055i) q^{80} +(-5.83842 + 6.84929i) q^{81} +(0.136673 + 0.236725i) q^{82} +(3.85087 - 6.66991i) q^{83} +(0.323832 - 4.57112i) q^{84} +(0.866926 + 1.50156i) q^{85} +(5.58113 - 9.66679i) q^{86} +(4.93560 + 9.52628i) q^{87} +(-0.296790 - 0.514055i) q^{88} +(-6.21780 - 10.7695i) q^{89} +(-1.77335 - 0.162084i) q^{90} +(-5.67617 - 3.47033i) q^{91} +(-2.23025 + 3.86291i) q^{92} +(-7.34348 + 11.4778i) q^{93} -12.1623 q^{94} -3.19436 q^{95} +(-0.933463 + 1.45899i) q^{96} +(5.86693 - 10.1618i) q^{97} +(-3.80039 + 5.87852i) q^{98} +(1.02704 - 1.45472i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 6 q^{8} + 8 q^{9} - q^{10} - q^{11} + 2 q^{12} + 8 q^{13} - 2 q^{14} + 12 q^{15} + 6 q^{16} - 4 q^{17} - 8 q^{18} - 3 q^{19} + q^{20}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.933463 − 1.45899i 0.538935 − 0.842347i
\(4\) 1.00000 0.500000
\(5\) −0.296790 + 0.514055i −0.132728 + 0.229892i −0.924727 − 0.380630i \(-0.875707\pi\)
0.791999 + 0.610522i \(0.209040\pi\)
\(6\) −0.933463 + 1.45899i −0.381085 + 0.595630i
\(7\) 2.32383 − 1.26483i 0.878326 − 0.478062i
\(8\) −1.00000 −0.353553
\(9\) −1.25729 − 2.72382i −0.419098 − 0.907941i
\(10\) 0.296790 − 0.514055i 0.0938531 − 0.162558i
\(11\) 0.296790 + 0.514055i 0.0894855 + 0.154993i 0.907294 − 0.420497i \(-0.138144\pi\)
−0.817808 + 0.575491i \(0.804811\pi\)
\(12\) 0.933463 − 1.45899i 0.269467 − 0.421174i
\(13\) −1.25729 − 2.17770i −0.348711 − 0.603985i 0.637310 − 0.770608i \(-0.280047\pi\)
−0.986021 + 0.166623i \(0.946714\pi\)
\(14\) −2.32383 + 1.26483i −0.621070 + 0.338041i
\(15\) 0.472958 + 0.912864i 0.122117 + 0.235700i
\(16\) 1.00000 0.250000
\(17\) 1.46050 − 2.52967i 0.354224 − 0.613535i −0.632760 − 0.774348i \(-0.718078\pi\)
0.986985 + 0.160813i \(0.0514116\pi\)
\(18\) 1.25729 + 2.72382i 0.296347 + 0.642011i
\(19\) 2.69076 + 4.66053i 0.617302 + 1.06920i 0.989976 + 0.141236i \(0.0451077\pi\)
−0.372674 + 0.927962i \(0.621559\pi\)
\(20\) −0.296790 + 0.514055i −0.0663642 + 0.114946i
\(21\) 0.323832 − 4.57112i 0.0706659 − 0.997500i
\(22\) −0.296790 − 0.514055i −0.0632758 − 0.109597i
\(23\) −2.23025 + 3.86291i −0.465040 + 0.805473i −0.999203 − 0.0399086i \(-0.987293\pi\)
0.534164 + 0.845381i \(0.320627\pi\)
\(24\) −0.933463 + 1.45899i −0.190542 + 0.297815i
\(25\) 2.32383 + 4.02499i 0.464766 + 0.804999i
\(26\) 1.25729 + 2.17770i 0.246576 + 0.427082i
\(27\) −5.14766 − 0.708209i −0.990668 − 0.136295i
\(28\) 2.32383 − 1.26483i 0.439163 − 0.239031i
\(29\) −3.09718 + 5.36447i −0.575132 + 0.996157i 0.420896 + 0.907109i \(0.361716\pi\)
−0.996027 + 0.0890480i \(0.971618\pi\)
\(30\) −0.472958 − 0.912864i −0.0863499 − 0.166665i
\(31\) −7.86693 −1.41294 −0.706471 − 0.707742i \(-0.749714\pi\)
−0.706471 + 0.707742i \(0.749714\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.02704 + 0.0468383i 0.178785 + 0.00815350i
\(34\) −1.46050 + 2.52967i −0.250475 + 0.433835i
\(35\) −0.0394951 + 1.56997i −0.00667590 + 0.265373i
\(36\) −1.25729 − 2.72382i −0.209549 − 0.453970i
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.69076 − 4.66053i −0.436498 − 0.756038i
\(39\) −4.35087 − 0.198422i −0.696697 − 0.0317729i
\(40\) 0.296790 − 0.514055i 0.0469266 − 0.0812792i
\(41\) −0.136673 − 0.236725i −0.0213448 − 0.0369702i 0.855156 − 0.518371i \(-0.173461\pi\)
−0.876500 + 0.481401i \(0.840128\pi\)
\(42\) −0.323832 + 4.57112i −0.0499683 + 0.705339i
\(43\) −5.58113 + 9.66679i −0.851114 + 1.47417i 0.0290902 + 0.999577i \(0.490739\pi\)
−0.880204 + 0.474596i \(0.842594\pi\)
\(44\) 0.296790 + 0.514055i 0.0447427 + 0.0774967i
\(45\) 1.77335 + 0.162084i 0.264355 + 0.0241621i
\(46\) 2.23025 − 3.86291i 0.328833 − 0.569555i
\(47\) 12.1623 1.77405 0.887023 − 0.461724i \(-0.152769\pi\)
0.887023 + 0.461724i \(0.152769\pi\)
\(48\) 0.933463 − 1.45899i 0.134734 − 0.210587i
\(49\) 3.80039 − 5.87852i 0.542913 − 0.839789i
\(50\) −2.32383 − 4.02499i −0.328639 − 0.569220i
\(51\) −2.32743 − 4.49221i −0.325905 − 0.629035i
\(52\) −1.25729 − 2.17770i −0.174355 − 0.301992i
\(53\) 4.02704 − 6.97504i 0.553157 − 0.958096i −0.444888 − 0.895586i \(-0.646756\pi\)
0.998044 − 0.0625092i \(-0.0199103\pi\)
\(54\) 5.14766 + 0.708209i 0.700508 + 0.0963750i
\(55\) −0.352336 −0.0475090
\(56\) −2.32383 + 1.26483i −0.310535 + 0.169021i
\(57\) 9.31138 + 0.424646i 1.23332 + 0.0562457i
\(58\) 3.09718 − 5.36447i 0.406679 − 0.704389i
\(59\) 8.64766 1.12583 0.562915 − 0.826515i \(-0.309680\pi\)
0.562915 + 0.826515i \(0.309680\pi\)
\(60\) 0.472958 + 0.912864i 0.0610586 + 0.117850i
\(61\) −6.64766 −0.851146 −0.425573 − 0.904924i \(-0.639927\pi\)
−0.425573 + 0.904924i \(0.639927\pi\)
\(62\) 7.86693 0.999101
\(63\) −6.36693 − 4.73944i −0.802157 − 0.597113i
\(64\) 1.00000 0.125000
\(65\) 1.49261 0.185135
\(66\) −1.02704 − 0.0468383i −0.126420 − 0.00576540i
\(67\) −1.91381 −0.233809 −0.116905 − 0.993143i \(-0.537297\pi\)
−0.116905 + 0.993143i \(0.537297\pi\)
\(68\) 1.46050 − 2.52967i 0.177112 − 0.306767i
\(69\) 3.55408 + 6.85980i 0.427861 + 0.825822i
\(70\) 0.0394951 − 1.56997i 0.00472057 − 0.187647i
\(71\) −14.4107 −1.71023 −0.855117 − 0.518435i \(-0.826515\pi\)
−0.855117 + 0.518435i \(0.826515\pi\)
\(72\) 1.25729 + 2.72382i 0.148174 + 0.321006i
\(73\) 3.95691 − 6.85356i 0.463121 − 0.802149i −0.535994 − 0.844222i \(-0.680063\pi\)
0.999115 + 0.0420732i \(0.0133963\pi\)
\(74\) −0.500000 − 0.866025i −0.0581238 − 0.100673i
\(75\) 8.04163 + 0.366739i 0.928568 + 0.0423474i
\(76\) 2.69076 + 4.66053i 0.308651 + 0.534599i
\(77\) 1.33988 + 0.819187i 0.152694 + 0.0933550i
\(78\) 4.35087 + 0.198422i 0.492639 + 0.0224668i
\(79\) −9.24844 −1.04053 −0.520265 − 0.854005i \(-0.674167\pi\)
−0.520265 + 0.854005i \(0.674167\pi\)
\(80\) −0.296790 + 0.514055i −0.0331821 + 0.0574731i
\(81\) −5.83842 + 6.84929i −0.648713 + 0.761033i
\(82\) 0.136673 + 0.236725i 0.0150930 + 0.0261419i
\(83\) 3.85087 − 6.66991i 0.422688 − 0.732118i −0.573513 − 0.819196i \(-0.694420\pi\)
0.996201 + 0.0870787i \(0.0277532\pi\)
\(84\) 0.323832 − 4.57112i 0.0353329 − 0.498750i
\(85\) 0.866926 + 1.50156i 0.0940313 + 0.162867i
\(86\) 5.58113 − 9.66679i 0.601828 − 1.04240i
\(87\) 4.93560 + 9.52628i 0.529152 + 1.02132i
\(88\) −0.296790 − 0.514055i −0.0316379 − 0.0547984i
\(89\) −6.21780 − 10.7695i −0.659085 − 1.14157i −0.980853 − 0.194751i \(-0.937610\pi\)
0.321767 − 0.946819i \(-0.395723\pi\)
\(90\) −1.77335 − 0.162084i −0.186927 − 0.0170852i
\(91\) −5.67617 − 3.47033i −0.595024 − 0.363790i
\(92\) −2.23025 + 3.86291i −0.232520 + 0.402736i
\(93\) −7.34348 + 11.4778i −0.761484 + 1.19019i
\(94\) −12.1623 −1.25444
\(95\) −3.19436 −0.327734
\(96\) −0.933463 + 1.45899i −0.0952711 + 0.148907i
\(97\) 5.86693 − 10.1618i 0.595696 − 1.03178i −0.397752 − 0.917493i \(-0.630210\pi\)
0.993448 − 0.114283i \(-0.0364570\pi\)
\(98\) −3.80039 + 5.87852i −0.383897 + 0.593821i
\(99\) 1.02704 − 1.45472i 0.103222 − 0.146205i
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.e.c.25.2 ✓ 6
3.2 odd 2 378.2.e.d.235.2 6
4.3 odd 2 1008.2.q.g.529.2 6
7.2 even 3 126.2.h.d.79.2 yes 6
7.3 odd 6 882.2.f.o.295.3 6
7.4 even 3 882.2.f.n.295.1 6
7.5 odd 6 882.2.h.p.79.2 6
7.6 odd 2 882.2.e.o.655.2 6
9.2 odd 6 1134.2.g.l.487.2 6
9.4 even 3 126.2.h.d.67.2 yes 6
9.5 odd 6 378.2.h.c.361.2 6
9.7 even 3 1134.2.g.m.487.2 6
12.11 even 2 3024.2.q.g.2881.2 6
21.2 odd 6 378.2.h.c.289.2 6
21.5 even 6 2646.2.h.o.667.2 6
21.11 odd 6 2646.2.f.l.883.2 6
21.17 even 6 2646.2.f.m.883.2 6
21.20 even 2 2646.2.e.p.2125.2 6
28.23 odd 6 1008.2.t.h.961.2 6
36.23 even 6 3024.2.t.h.1873.2 6
36.31 odd 6 1008.2.t.h.193.2 6
63.2 odd 6 1134.2.g.l.163.2 6
63.4 even 3 882.2.f.n.589.1 6
63.5 even 6 2646.2.e.p.1549.2 6
63.11 odd 6 7938.2.a.ca.1.2 3
63.13 odd 6 882.2.h.p.67.2 6
63.16 even 3 1134.2.g.m.163.2 6
63.23 odd 6 378.2.e.d.37.2 6
63.25 even 3 7938.2.a.bv.1.2 3
63.31 odd 6 882.2.f.o.589.3 6
63.32 odd 6 2646.2.f.l.1765.2 6
63.38 even 6 7938.2.a.bz.1.2 3
63.40 odd 6 882.2.e.o.373.2 6
63.41 even 6 2646.2.h.o.361.2 6
63.52 odd 6 7938.2.a.bw.1.2 3
63.58 even 3 inner 126.2.e.c.121.2 yes 6
63.59 even 6 2646.2.f.m.1765.2 6
84.23 even 6 3024.2.t.h.289.2 6
252.23 even 6 3024.2.q.g.2305.2 6
252.247 odd 6 1008.2.q.g.625.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.2 ✓ 6 1.1 even 1 trivial
126.2.e.c.121.2 yes 6 63.58 even 3 inner
126.2.h.d.67.2 yes 6 9.4 even 3
126.2.h.d.79.2 yes 6 7.2 even 3
378.2.e.d.37.2 6 63.23 odd 6
378.2.e.d.235.2 6 3.2 odd 2
378.2.h.c.289.2 6 21.2 odd 6
378.2.h.c.361.2 6 9.5 odd 6
882.2.e.o.373.2 6 63.40 odd 6
882.2.e.o.655.2 6 7.6 odd 2
882.2.f.n.295.1 6 7.4 even 3
882.2.f.n.589.1 6 63.4 even 3
882.2.f.o.295.3 6 7.3 odd 6
882.2.f.o.589.3 6 63.31 odd 6
882.2.h.p.67.2 6 63.13 odd 6
882.2.h.p.79.2 6 7.5 odd 6
1008.2.q.g.529.2 6 4.3 odd 2
1008.2.q.g.625.2 6 252.247 odd 6
1008.2.t.h.193.2 6 36.31 odd 6
1008.2.t.h.961.2 6 28.23 odd 6
1134.2.g.l.163.2 6 63.2 odd 6
1134.2.g.l.487.2 6 9.2 odd 6
1134.2.g.m.163.2 6 63.16 even 3
1134.2.g.m.487.2 6 9.7 even 3
2646.2.e.p.1549.2 6 63.5 even 6
2646.2.e.p.2125.2 6 21.20 even 2
2646.2.f.l.883.2 6 21.11 odd 6
2646.2.f.l.1765.2 6 63.32 odd 6
2646.2.f.m.883.2 6 21.17 even 6
2646.2.f.m.1765.2 6 63.59 even 6
2646.2.h.o.361.2 6 63.41 even 6
2646.2.h.o.667.2 6 21.5 even 6
3024.2.q.g.2305.2 6 252.23 even 6
3024.2.q.g.2881.2 6 12.11 even 2
3024.2.t.h.289.2 6 84.23 even 6
3024.2.t.h.1873.2 6 36.23 even 6
7938.2.a.bv.1.2 3 63.25 even 3
7938.2.a.bw.1.2 3 63.52 odd 6
7938.2.a.bz.1.2 3 63.38 even 6
7938.2.a.ca.1.2 3 63.11 odd 6