Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(67,882)] 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("882.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,-2,-3,2,2,0,-6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
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: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 + 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 882.79
Dual form 882.2.h.p.67.1

$q$-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} +(-1.29418 + 1.15113i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.58836 q^{5} +(-1.64400 - 0.545231i) q^{6} -1.00000 q^{8} +(0.349814 - 2.97954i) q^{9} +(-0.794182 - 1.37556i) q^{10} -1.58836 q^{11} +(-0.349814 - 1.69636i) q^{12} +(-2.40545 - 4.16635i) q^{13} +(2.05563 - 1.82841i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(2.69963 + 4.67589i) q^{17} +(2.75526 - 1.18682i) q^{18} +(3.54944 - 6.14781i) q^{19} +(0.794182 - 1.37556i) q^{20} +(-0.794182 - 1.37556i) q^{22} +0.300372 q^{23} +(1.29418 - 1.15113i) q^{24} -2.47710 q^{25} +(2.40545 - 4.16635i) q^{26} +(2.97710 + 4.25874i) q^{27} +(4.13781 - 7.16689i) q^{29} +(2.61126 + 0.866025i) q^{30} +(-1.35600 + 2.34867i) q^{31} +(0.500000 - 0.866025i) q^{32} +(2.05563 - 1.82841i) q^{33} +(-2.69963 + 4.67589i) q^{34} +(2.40545 + 1.79272i) q^{36} +(0.500000 - 0.866025i) q^{37} +7.09888 q^{38} +(7.90909 + 2.62305i) q^{39} +1.58836 q^{40} +(-2.93818 - 5.08907i) q^{41} +(-0.833104 + 1.44298i) q^{43} +(0.794182 - 1.37556i) q^{44} +(-0.555632 + 4.73259i) q^{45} +(0.150186 + 0.260130i) q^{46} +(1.33310 + 2.30900i) q^{47} +(1.64400 + 0.545231i) q^{48} +(-1.23855 - 2.14523i) q^{50} +(-8.87636 - 2.94384i) q^{51} +4.81089 q^{52} +(2.44437 + 4.23377i) q^{53} +(-2.19963 + 4.70761i) q^{54} +2.52290 q^{55} +(2.48329 + 12.0422i) q^{57} +8.27561 q^{58} +(3.23855 - 5.60933i) q^{59} +(0.555632 + 2.69443i) q^{60} +(-2.23855 - 3.87728i) q^{61} -2.71201 q^{62} +1.00000 q^{64} +(3.82072 + 6.61769i) q^{65} +(2.61126 + 0.866025i) q^{66} +(5.02654 - 8.70623i) q^{67} -5.39926 q^{68} +(-0.388736 + 0.345766i) q^{69} +12.7207 q^{71} +(-0.349814 + 2.97954i) q^{72} +(-8.02654 - 13.9024i) q^{73} +1.00000 q^{74} +(3.20582 - 2.85146i) q^{75} +(3.54944 + 6.14781i) q^{76} +(1.68292 + 8.16100i) q^{78} +(-4.19344 - 7.26325i) q^{79} +(0.794182 + 1.37556i) q^{80} +(-8.75526 - 2.08457i) q^{81} +(2.93818 - 5.08907i) q^{82} +(-1.18292 + 2.04887i) q^{83} +(-4.28799 - 7.42702i) q^{85} -1.66621 q^{86} +(2.89493 + 14.0384i) q^{87} +1.58836 q^{88} +(-1.60507 + 2.78007i) q^{89} +(-4.37636 + 1.88510i) q^{90} +(-0.150186 + 0.260130i) q^{92} +(-0.948699 - 4.60054i) q^{93} +(-1.33310 + 2.30900i) q^{94} +(-5.63781 + 9.76497i) q^{95} +(0.349814 + 1.69636i) q^{96} +(-0.712008 + 1.23323i) q^{97} +(-0.555632 + 4.73259i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 2 q^{3} - 3 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} - 4 q^{9} + q^{10} + 2 q^{11} + 4 q^{12} - 8 q^{13} + 12 q^{15} - 3 q^{16} + 4 q^{17} + 4 q^{18} + 3 q^{19} - q^{20} + q^{22} + 14 q^{23}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) −1.29418 + 1.15113i −0.747196 + 0.664603i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.58836 −0.710338 −0.355169 − 0.934802i \(-0.615577\pi\)
−0.355169 + 0.934802i \(0.615577\pi\)
\(6\) −1.64400 − 0.545231i −0.671159 − 0.222590i
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0.349814 − 2.97954i 0.116605 − 0.993178i
\(10\) −0.794182 − 1.37556i −0.251142 − 0.434991i
\(11\) −1.58836 −0.478910 −0.239455 − 0.970907i \(-0.576969\pi\)
−0.239455 + 0.970907i \(0.576969\pi\)
\(12\) −0.349814 − 1.69636i −0.100983 − 0.489696i
\(13\) −2.40545 − 4.16635i −0.667151 − 1.15554i −0.978697 − 0.205308i \(-0.934180\pi\)
0.311547 − 0.950231i \(-0.399153\pi\)
\(14\) 0 0
\(15\) 2.05563 − 1.82841i 0.530762 − 0.472093i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.69963 + 4.67589i 0.654756 + 1.13407i 0.981955 + 0.189115i \(0.0605620\pi\)
−0.327199 + 0.944955i \(0.606105\pi\)
\(18\) 2.75526 − 1.18682i 0.649421 − 0.279736i
\(19\) 3.54944 − 6.14781i 0.814298 − 1.41041i −0.0955331 − 0.995426i \(-0.530456\pi\)
0.909831 − 0.414979i \(-0.136211\pi\)
\(20\) 0.794182 − 1.37556i 0.177584 − 0.307585i
\(21\) 0 0
\(22\) −0.794182 − 1.37556i −0.169320 − 0.293271i
\(23\) 0.300372 0.0626319 0.0313159 − 0.999510i \(-0.490030\pi\)
0.0313159 + 0.999510i \(0.490030\pi\)
\(24\) 1.29418 − 1.15113i 0.264174 − 0.234973i
\(25\) −2.47710 −0.495420
\(26\) 2.40545 − 4.16635i 0.471747 − 0.817089i
\(27\) 2.97710 + 4.25874i 0.572943 + 0.819595i
\(28\) 0 0
\(29\) 4.13781 − 7.16689i 0.768371 − 1.33086i −0.170074 − 0.985431i \(-0.554401\pi\)
0.938446 − 0.345427i \(-0.112266\pi\)
\(30\) 2.61126 + 0.866025i 0.476749 + 0.158114i
\(31\) −1.35600 + 2.34867i −0.243545 + 0.421833i −0.961722 − 0.274028i \(-0.911644\pi\)
0.718176 + 0.695861i \(0.244977\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 2.05563 − 1.82841i 0.357840 − 0.318285i
\(34\) −2.69963 + 4.67589i −0.462982 + 0.801909i
\(35\) 0 0
\(36\) 2.40545 + 1.79272i 0.400908 + 0.298786i
\(37\) 0.500000 − 0.866025i 0.0821995 − 0.142374i −0.821995 − 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 7.09888 1.15159
\(39\) 7.90909 + 2.62305i 1.26647 + 0.420024i
\(40\) 1.58836 0.251142
\(41\) −2.93818 − 5.08907i −0.458866 − 0.794780i 0.540035 − 0.841643i \(-0.318411\pi\)
−0.998901 + 0.0468628i \(0.985078\pi\)
\(42\) 0 0
\(43\) −0.833104 + 1.44298i −0.127047 + 0.220052i −0.922531 − 0.385922i \(-0.873883\pi\)
0.795484 + 0.605974i \(0.207217\pi\)
\(44\) 0.794182 − 1.37556i 0.119727 − 0.207374i
\(45\) −0.555632 + 4.73259i −0.0828287 + 0.705492i
\(46\) 0.150186 + 0.260130i 0.0221437 + 0.0383540i
\(47\) 1.33310 + 2.30900i 0.194453 + 0.336803i 0.946721 − 0.322055i \(-0.104373\pi\)
−0.752268 + 0.658857i \(0.771040\pi\)
\(48\) 1.64400 + 0.545231i 0.237290 + 0.0786973i
\(49\) 0 0
\(50\) −1.23855 − 2.14523i −0.175157 − 0.303382i
\(51\) −8.87636 − 2.94384i −1.24294 − 0.412220i
\(52\) 4.81089 0.667151
\(53\) 2.44437 + 4.23377i 0.335760 + 0.581553i 0.983630 − 0.180197i \(-0.0576736\pi\)
−0.647871 + 0.761750i \(0.724340\pi\)
\(54\) −2.19963 + 4.70761i −0.299331 + 0.640625i
\(55\) 2.52290 0.340188
\(56\) 0 0
\(57\) 2.48329 + 12.0422i 0.328920 + 1.59503i
\(58\) 8.27561 1.08664
\(59\) 3.23855 − 5.60933i 0.421623 − 0.730273i −0.574475 − 0.818522i \(-0.694794\pi\)
0.996098 + 0.0882491i \(0.0281271\pi\)
\(60\) 0.555632 + 2.69443i 0.0717318 + 0.347850i
\(61\) −2.23855 − 3.87728i −0.286617 − 0.496435i 0.686383 − 0.727240i \(-0.259197\pi\)
−0.973000 + 0.230805i \(0.925864\pi\)
\(62\) −2.71201 −0.344425
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.82072 + 6.61769i 0.473902 + 0.820823i
\(66\) 2.61126 + 0.866025i 0.321424 + 0.106600i
\(67\) 5.02654 − 8.70623i 0.614090 − 1.06363i −0.376454 − 0.926435i \(-0.622857\pi\)
0.990543 − 0.137199i \(-0.0438101\pi\)
\(68\) −5.39926 −0.654756
\(69\) −0.388736 + 0.345766i −0.0467983 + 0.0416253i
\(70\) 0 0
\(71\) 12.7207 1.50967 0.754833 − 0.655917i \(-0.227718\pi\)
0.754833 + 0.655917i \(0.227718\pi\)
\(72\) −0.349814 + 2.97954i −0.0412260 + 0.351142i
\(73\) −8.02654 − 13.9024i −0.939436 − 1.62715i −0.766527 − 0.642213i \(-0.778017\pi\)
−0.172909 − 0.984938i \(-0.555317\pi\)
\(74\) 1.00000 0.116248
\(75\) 3.20582 − 2.85146i 0.370176 − 0.329258i
\(76\) 3.54944 + 6.14781i 0.407149 + 0.705203i
\(77\) 0 0
\(78\) 1.68292 + 8.16100i 0.190553 + 0.924051i
\(79\) −4.19344 − 7.26325i −0.471799 − 0.817179i 0.527681 − 0.849443i \(-0.323062\pi\)
−0.999479 + 0.0322635i \(0.989728\pi\)
\(80\) 0.794182 + 1.37556i 0.0887922 + 0.153793i
\(81\) −8.75526 − 2.08457i −0.972807 − 0.231619i
\(82\) 2.93818 − 5.08907i 0.324467 − 0.561994i
\(83\) −1.18292 + 2.04887i −0.129842 + 0.224893i −0.923615 − 0.383321i \(-0.874780\pi\)
0.793773 + 0.608214i \(0.208114\pi\)
\(84\) 0 0
\(85\) −4.28799 − 7.42702i −0.465098 − 0.805573i
\(86\) −1.66621 −0.179672
\(87\) 2.89493 + 14.0384i 0.310369 + 1.50507i
\(88\) 1.58836 0.169320
\(89\) −1.60507 + 2.78007i −0.170138 + 0.294687i −0.938468 − 0.345367i \(-0.887755\pi\)
0.768330 + 0.640054i \(0.221088\pi\)
\(90\) −4.37636 + 1.88510i −0.461308 + 0.198707i
\(91\) 0 0
\(92\) −0.150186 + 0.260130i −0.0156580 + 0.0271204i
\(93\) −0.948699 − 4.60054i −0.0983755 − 0.477053i
\(94\) −1.33310 + 2.30900i −0.137499 + 0.238156i
\(95\) −5.63781 + 9.76497i −0.578427 + 1.00186i
\(96\) 0.349814 + 1.69636i 0.0357027 + 0.173134i
\(97\) −0.712008 + 1.23323i −0.0722934 + 0.125216i −0.899906 − 0.436084i \(-0.856365\pi\)
0.827613 + 0.561300i \(0.189698\pi\)
\(98\) 0 0
\(99\) −0.555632 + 4.73259i −0.0558431 + 0.475643i
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 882.2.h.p.79.1 6
3.2 odd 2 2646.2.h.o.667.3 6
7.2 even 3 882.2.f.o.295.1 6
7.3 odd 6 126.2.e.c.25.1 ✓ 6
7.4 even 3 882.2.e.o.655.3 6
7.5 odd 6 882.2.f.n.295.3 6
7.6 odd 2 126.2.h.d.79.3 yes 6
9.4 even 3 882.2.e.o.373.3 6
9.5 odd 6 2646.2.e.p.1549.1 6
21.2 odd 6 2646.2.f.m.883.1 6
21.5 even 6 2646.2.f.l.883.3 6
21.11 odd 6 2646.2.e.p.2125.1 6
21.17 even 6 378.2.e.d.235.3 6
21.20 even 2 378.2.h.c.289.1 6
28.3 even 6 1008.2.q.g.529.3 6
28.27 even 2 1008.2.t.h.961.1 6
63.2 odd 6 7938.2.a.bz.1.3 3
63.4 even 3 inner 882.2.h.p.67.1 6
63.5 even 6 2646.2.f.l.1765.3 6
63.13 odd 6 126.2.e.c.121.1 yes 6
63.16 even 3 7938.2.a.bw.1.1 3
63.20 even 6 1134.2.g.l.163.3 6
63.23 odd 6 2646.2.f.m.1765.1 6
63.31 odd 6 126.2.h.d.67.3 yes 6
63.32 odd 6 2646.2.h.o.361.3 6
63.34 odd 6 1134.2.g.m.163.1 6
63.38 even 6 1134.2.g.l.487.3 6
63.40 odd 6 882.2.f.n.589.3 6
63.41 even 6 378.2.e.d.37.3 6
63.47 even 6 7938.2.a.ca.1.1 3
63.52 odd 6 1134.2.g.m.487.1 6
63.58 even 3 882.2.f.o.589.1 6
63.59 even 6 378.2.h.c.361.1 6
63.61 odd 6 7938.2.a.bv.1.3 3
84.59 odd 6 3024.2.q.g.2881.3 6
84.83 odd 2 3024.2.t.h.289.1 6
252.31 even 6 1008.2.t.h.193.1 6
252.59 odd 6 3024.2.t.h.1873.1 6
252.139 even 6 1008.2.q.g.625.3 6
252.167 odd 6 3024.2.q.g.2305.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.1 ✓ 6 7.3 odd 6
126.2.e.c.121.1 yes 6 63.13 odd 6
126.2.h.d.67.3 yes 6 63.31 odd 6
126.2.h.d.79.3 yes 6 7.6 odd 2
378.2.e.d.37.3 6 63.41 even 6
378.2.e.d.235.3 6 21.17 even 6
378.2.h.c.289.1 6 21.20 even 2
378.2.h.c.361.1 6 63.59 even 6
882.2.e.o.373.3 6 9.4 even 3
882.2.e.o.655.3 6 7.4 even 3
882.2.f.n.295.3 6 7.5 odd 6
882.2.f.n.589.3 6 63.40 odd 6
882.2.f.o.295.1 6 7.2 even 3
882.2.f.o.589.1 6 63.58 even 3
882.2.h.p.67.1 6 63.4 even 3 inner
882.2.h.p.79.1 6 1.1 even 1 trivial
1008.2.q.g.529.3 6 28.3 even 6
1008.2.q.g.625.3 6 252.139 even 6
1008.2.t.h.193.1 6 252.31 even 6
1008.2.t.h.961.1 6 28.27 even 2
1134.2.g.l.163.3 6 63.20 even 6
1134.2.g.l.487.3 6 63.38 even 6
1134.2.g.m.163.1 6 63.34 odd 6
1134.2.g.m.487.1 6 63.52 odd 6
2646.2.e.p.1549.1 6 9.5 odd 6
2646.2.e.p.2125.1 6 21.11 odd 6
2646.2.f.l.883.3 6 21.5 even 6
2646.2.f.l.1765.3 6 63.5 even 6
2646.2.f.m.883.1 6 21.2 odd 6
2646.2.f.m.1765.1 6 63.23 odd 6
2646.2.h.o.361.3 6 63.32 odd 6
2646.2.h.o.667.3 6 3.2 odd 2
3024.2.q.g.2305.3 6 252.167 odd 6
3024.2.q.g.2881.3 6 84.59 odd 6
3024.2.t.h.289.1 6 84.83 odd 2
3024.2.t.h.1873.1 6 252.59 odd 6
7938.2.a.bv.1.3 3 63.61 odd 6
7938.2.a.bw.1.1 3 63.16 even 3
7938.2.a.bz.1.3 3 63.2 odd 6
7938.2.a.ca.1.1 3 63.47 even 6