Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2646,2,Mod(883,2646)] 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("2646.883"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,0,-3,-1,0,0,6,0,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1765.3
Root \(0.500000 - 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 2646.1765
Dual form 2646.2.f.l.883.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.794182 - 1.37556i) q^{5} +1.00000 q^{8} -1.58836 q^{10} +(-0.794182 - 1.37556i) q^{11} +(2.40545 - 4.16635i) q^{13} +(-0.500000 - 0.866025i) q^{16} -5.39926 q^{17} +7.09888 q^{19} +(0.794182 + 1.37556i) q^{20} +(-0.794182 + 1.37556i) q^{22} +(0.150186 - 0.260130i) q^{23} +(1.23855 + 2.14523i) q^{25} -4.81089 q^{26} +(-4.13781 - 7.16689i) q^{29} +(1.35600 - 2.34867i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(2.69963 + 4.67589i) q^{34} -1.00000 q^{37} +(-3.54944 - 6.14781i) q^{38} +(0.794182 - 1.37556i) q^{40} +(-2.93818 + 5.08907i) q^{41} +(-0.833104 - 1.44298i) q^{43} +1.58836 q^{44} -0.300372 q^{46} +(1.33310 + 2.30900i) q^{47} +(1.23855 - 2.14523i) q^{50} +(2.40545 + 4.16635i) q^{52} +4.88874 q^{53} -2.52290 q^{55} +(-4.13781 + 7.16689i) q^{58} +(3.23855 - 5.60933i) q^{59} +(2.23855 + 3.87728i) q^{61} -2.71201 q^{62} +1.00000 q^{64} +(-3.82072 - 6.61769i) q^{65} +(5.02654 - 8.70623i) q^{67} +(2.69963 - 4.67589i) q^{68} -12.7207 q^{71} -16.0531 q^{73} +(0.500000 + 0.866025i) q^{74} +(-3.54944 + 6.14781i) q^{76} +(-4.19344 - 7.26325i) q^{79} -1.58836 q^{80} +5.87636 q^{82} +(-1.18292 - 2.04887i) q^{83} +(-4.28799 + 7.42702i) q^{85} +(-0.833104 + 1.44298i) q^{86} +(-0.794182 - 1.37556i) q^{88} +3.21015 q^{89} +(0.150186 + 0.260130i) q^{92} +(1.33310 - 2.30900i) q^{94} +(5.63781 - 9.76497i) q^{95} +(0.712008 + 1.23323i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - q^{5} + 6 q^{8} + 2 q^{10} + q^{11} + 8 q^{13} - 3 q^{16} - 8 q^{17} + 6 q^{19} - q^{20} + q^{22} + 7 q^{23} + 2 q^{25} - 16 q^{26} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0.794182 − 1.37556i 0.355169 − 0.615171i −0.631978 − 0.774986i \(-0.717757\pi\)
0.987147 + 0.159816i \(0.0510900\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.58836 −0.502285
\(11\) −0.794182 − 1.37556i −0.239455 − 0.414748i 0.721103 − 0.692828i \(-0.243635\pi\)
−0.960558 + 0.278080i \(0.910302\pi\)
\(12\) 0 0
\(13\) 2.40545 − 4.16635i 0.667151 − 1.15554i −0.311547 − 0.950231i \(-0.600847\pi\)
0.978697 − 0.205308i \(-0.0658196\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −5.39926 −1.30951 −0.654756 − 0.755840i \(-0.727229\pi\)
−0.654756 + 0.755840i \(0.727229\pi\)
\(18\) 0 0
\(19\) 7.09888 1.62860 0.814298 − 0.580447i \(-0.197122\pi\)
0.814298 + 0.580447i \(0.197122\pi\)
\(20\) 0.794182 + 1.37556i 0.177584 + 0.307585i
\(21\) 0 0
\(22\) −0.794182 + 1.37556i −0.169320 + 0.293271i
\(23\) 0.150186 − 0.260130i 0.0313159 − 0.0542408i −0.849943 − 0.526875i \(-0.823364\pi\)
0.881259 + 0.472634i \(0.156697\pi\)
\(24\) 0 0
\(25\) 1.23855 + 2.14523i 0.247710 + 0.429046i
\(26\) −4.81089 −0.943494
\(27\) 0 0
\(28\) 0 0
\(29\) −4.13781 − 7.16689i −0.768371 − 1.33086i −0.938446 − 0.345427i \(-0.887734\pi\)
0.170074 − 0.985431i \(-0.445599\pi\)
\(30\) 0 0
\(31\) 1.35600 − 2.34867i 0.243545 − 0.421833i −0.718176 − 0.695861i \(-0.755023\pi\)
0.961722 + 0.274028i \(0.0883561\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 2.69963 + 4.67589i 0.462982 + 0.801909i
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −3.54944 − 6.14781i −0.575796 − 0.997307i
\(39\) 0 0
\(40\) 0.794182 − 1.37556i 0.125571 − 0.217496i
\(41\) −2.93818 + 5.08907i −0.458866 + 0.794780i −0.998901 − 0.0468628i \(-0.985078\pi\)
0.540035 + 0.841643i \(0.318411\pi\)
\(42\) 0 0
\(43\) −0.833104 − 1.44298i −0.127047 − 0.220052i 0.795484 − 0.605974i \(-0.207217\pi\)
−0.922531 + 0.385922i \(0.873883\pi\)
\(44\) 1.58836 0.239455
\(45\) 0 0
\(46\) −0.300372 −0.0442874
\(47\) 1.33310 + 2.30900i 0.194453 + 0.336803i 0.946721 − 0.322055i \(-0.104373\pi\)
−0.752268 + 0.658857i \(0.771040\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.23855 − 2.14523i 0.175157 − 0.303382i
\(51\) 0 0
\(52\) 2.40545 + 4.16635i 0.333575 + 0.577769i
\(53\) 4.88874 0.671520 0.335760 − 0.941948i \(-0.391007\pi\)
0.335760 + 0.941948i \(0.391007\pi\)
\(54\) 0 0
\(55\) −2.52290 −0.340188
\(56\) 0 0
\(57\) 0 0
\(58\) −4.13781 + 7.16689i −0.543321 + 0.941059i
\(59\) 3.23855 − 5.60933i 0.421623 − 0.730273i −0.574475 − 0.818522i \(-0.694794\pi\)
0.996098 + 0.0882491i \(0.0281271\pi\)
\(60\) 0 0
\(61\) 2.23855 + 3.87728i 0.286617 + 0.496435i 0.973000 − 0.230805i \(-0.0741360\pi\)
−0.686383 + 0.727240i \(0.740803\pi\)
\(62\) −2.71201 −0.344425
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.82072 − 6.61769i −0.473902 − 0.820823i
\(66\) 0 0
\(67\) 5.02654 − 8.70623i 0.614090 − 1.06363i −0.376454 − 0.926435i \(-0.622857\pi\)
0.990543 − 0.137199i \(-0.0438101\pi\)
\(68\) 2.69963 − 4.67589i 0.327378 − 0.567035i
\(69\) 0 0
\(70\) 0 0
\(71\) −12.7207 −1.50967 −0.754833 − 0.655917i \(-0.772282\pi\)
−0.754833 + 0.655917i \(0.772282\pi\)
\(72\) 0 0
\(73\) −16.0531 −1.87887 −0.939436 − 0.342725i \(-0.888650\pi\)
−0.939436 + 0.342725i \(0.888650\pi\)
\(74\) 0.500000 + 0.866025i 0.0581238 + 0.100673i
\(75\) 0 0
\(76\) −3.54944 + 6.14781i −0.407149 + 0.705203i
\(77\) 0 0
\(78\) 0 0
\(79\) −4.19344 − 7.26325i −0.471799 − 0.817179i 0.527681 − 0.849443i \(-0.323062\pi\)
−0.999479 + 0.0322635i \(0.989728\pi\)
\(80\) −1.58836 −0.177584
\(81\) 0 0
\(82\) 5.87636 0.648935
\(83\) −1.18292 − 2.04887i −0.129842 − 0.224893i 0.793773 − 0.608214i \(-0.208114\pi\)
−0.923615 + 0.383321i \(0.874780\pi\)
\(84\) 0 0
\(85\) −4.28799 + 7.42702i −0.465098 + 0.805573i
\(86\) −0.833104 + 1.44298i −0.0898359 + 0.155600i
\(87\) 0 0
\(88\) −0.794182 − 1.37556i −0.0846601 − 0.146636i
\(89\) 3.21015 0.340275 0.170138 − 0.985420i \(-0.445579\pi\)
0.170138 + 0.985420i \(0.445579\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.150186 + 0.260130i 0.0156580 + 0.0271204i
\(93\) 0 0
\(94\) 1.33310 − 2.30900i 0.137499 − 0.238156i
\(95\) 5.63781 − 9.76497i 0.578427 − 1.00186i
\(96\) 0 0
\(97\) 0.712008 + 1.23323i 0.0722934 + 0.125216i 0.899906 − 0.436084i \(-0.143635\pi\)
−0.827613 + 0.561300i \(0.810302\pi\)
\(98\) 0 0
\(99\) 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 2646.2.f.l.1765.3 6
3.2 odd 2 882.2.f.n.589.3 6
7.2 even 3 378.2.h.c.361.1 6
7.3 odd 6 2646.2.e.p.1549.1 6
7.4 even 3 378.2.e.d.37.3 6
7.5 odd 6 2646.2.h.o.361.3 6
7.6 odd 2 2646.2.f.m.1765.1 6
9.2 odd 6 882.2.f.n.295.3 6
9.4 even 3 7938.2.a.ca.1.1 3
9.5 odd 6 7938.2.a.bv.1.3 3
9.7 even 3 inner 2646.2.f.l.883.3 6
21.2 odd 6 126.2.h.d.67.3 yes 6
21.5 even 6 882.2.h.p.67.1 6
21.11 odd 6 126.2.e.c.121.1 yes 6
21.17 even 6 882.2.e.o.373.3 6
21.20 even 2 882.2.f.o.589.1 6
28.11 odd 6 3024.2.q.g.2305.3 6
28.23 odd 6 3024.2.t.h.1873.1 6
63.2 odd 6 126.2.e.c.25.1 ✓ 6
63.4 even 3 1134.2.g.l.163.3 6
63.11 odd 6 126.2.h.d.79.3 yes 6
63.13 odd 6 7938.2.a.bz.1.3 3
63.16 even 3 378.2.e.d.235.3 6
63.20 even 6 882.2.f.o.295.1 6
63.23 odd 6 1134.2.g.m.487.1 6
63.25 even 3 378.2.h.c.289.1 6
63.32 odd 6 1134.2.g.m.163.1 6
63.34 odd 6 2646.2.f.m.883.1 6
63.38 even 6 882.2.h.p.79.1 6
63.41 even 6 7938.2.a.bw.1.1 3
63.47 even 6 882.2.e.o.655.3 6
63.52 odd 6 2646.2.h.o.667.3 6
63.58 even 3 1134.2.g.l.487.3 6
63.61 odd 6 2646.2.e.p.2125.1 6
84.11 even 6 1008.2.q.g.625.3 6
84.23 even 6 1008.2.t.h.193.1 6
252.11 even 6 1008.2.t.h.961.1 6
252.79 odd 6 3024.2.q.g.2881.3 6
252.151 odd 6 3024.2.t.h.289.1 6
252.191 even 6 1008.2.q.g.529.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.1 ✓ 6 63.2 odd 6
126.2.e.c.121.1 yes 6 21.11 odd 6
126.2.h.d.67.3 yes 6 21.2 odd 6
126.2.h.d.79.3 yes 6 63.11 odd 6
378.2.e.d.37.3 6 7.4 even 3
378.2.e.d.235.3 6 63.16 even 3
378.2.h.c.289.1 6 63.25 even 3
378.2.h.c.361.1 6 7.2 even 3
882.2.e.o.373.3 6 21.17 even 6
882.2.e.o.655.3 6 63.47 even 6
882.2.f.n.295.3 6 9.2 odd 6
882.2.f.n.589.3 6 3.2 odd 2
882.2.f.o.295.1 6 63.20 even 6
882.2.f.o.589.1 6 21.20 even 2
882.2.h.p.67.1 6 21.5 even 6
882.2.h.p.79.1 6 63.38 even 6
1008.2.q.g.529.3 6 252.191 even 6
1008.2.q.g.625.3 6 84.11 even 6
1008.2.t.h.193.1 6 84.23 even 6
1008.2.t.h.961.1 6 252.11 even 6
1134.2.g.l.163.3 6 63.4 even 3
1134.2.g.l.487.3 6 63.58 even 3
1134.2.g.m.163.1 6 63.32 odd 6
1134.2.g.m.487.1 6 63.23 odd 6
2646.2.e.p.1549.1 6 7.3 odd 6
2646.2.e.p.2125.1 6 63.61 odd 6
2646.2.f.l.883.3 6 9.7 even 3 inner
2646.2.f.l.1765.3 6 1.1 even 1 trivial
2646.2.f.m.883.1 6 63.34 odd 6
2646.2.f.m.1765.1 6 7.6 odd 2
2646.2.h.o.361.3 6 7.5 odd 6
2646.2.h.o.667.3 6 63.52 odd 6
3024.2.q.g.2305.3 6 28.11 odd 6
3024.2.q.g.2881.3 6 252.79 odd 6
3024.2.t.h.289.1 6 252.151 odd 6
3024.2.t.h.1873.1 6 28.23 odd 6
7938.2.a.bv.1.3 3 9.5 odd 6
7938.2.a.bw.1.1 3 63.41 even 6
7938.2.a.bz.1.3 3 63.13 odd 6
7938.2.a.ca.1.1 3 9.4 even 3