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(373,882)] 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("882.373"); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,-2,4,-2,-2,0,4,2] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) 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 373.2
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 882.373
Dual form 882.2.e.m.655.2

$q$-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} +(0.724745 - 1.57313i) q^{3} +1.00000 q^{4} +(-1.72474 - 2.98735i) q^{5} +(0.724745 - 1.57313i) q^{6} +1.00000 q^{8} +(-1.94949 - 2.28024i) q^{9} +(-1.72474 - 2.98735i) q^{10} +(-1.00000 + 1.73205i) q^{11} +(0.724745 - 1.57313i) q^{12} +(2.44949 - 4.24264i) q^{13} +(-5.94949 + 0.548188i) q^{15} +1.00000 q^{16} +(-1.00000 - 1.73205i) q^{17} +(-1.94949 - 2.28024i) q^{18} +(-3.72474 + 6.45145i) q^{19} +(-1.72474 - 2.98735i) q^{20} +(-1.00000 + 1.73205i) q^{22} +(0.500000 + 0.866025i) q^{23} +(0.724745 - 1.57313i) q^{24} +(-3.44949 + 5.97469i) q^{25} +(2.44949 - 4.24264i) q^{26} +(-5.00000 + 1.41421i) q^{27} +(-1.44949 - 2.51059i) q^{29} +(-5.94949 + 0.548188i) q^{30} +6.00000 q^{31} +1.00000 q^{32} +(2.00000 + 2.82843i) q^{33} +(-1.00000 - 1.73205i) q^{34} +(-1.94949 - 2.28024i) q^{36} +(3.89898 - 6.75323i) q^{37} +(-3.72474 + 6.45145i) q^{38} +(-4.89898 - 6.92820i) q^{39} +(-1.72474 - 2.98735i) q^{40} +(4.89898 - 8.48528i) q^{41} +(1.44949 + 2.51059i) q^{43} +(-1.00000 + 1.73205i) q^{44} +(-3.44949 + 9.75663i) q^{45} +(0.500000 + 0.866025i) q^{46} -9.79796 q^{47} +(0.724745 - 1.57313i) q^{48} +(-3.44949 + 5.97469i) q^{50} +(-3.44949 + 0.317837i) q^{51} +(2.44949 - 4.24264i) q^{52} +(0.550510 + 0.953512i) q^{53} +(-5.00000 + 1.41421i) q^{54} +6.89898 q^{55} +(7.44949 + 10.5352i) q^{57} +(-1.44949 - 2.51059i) q^{58} -2.00000 q^{59} +(-5.94949 + 0.548188i) q^{60} +11.4495 q^{61} +6.00000 q^{62} +1.00000 q^{64} -16.8990 q^{65} +(2.00000 + 2.82843i) q^{66} -3.10102 q^{67} +(-1.00000 - 1.73205i) q^{68} +(1.72474 - 0.158919i) q^{69} +9.89898 q^{71} +(-1.94949 - 2.28024i) q^{72} +(-1.44949 - 2.51059i) q^{73} +(3.89898 - 6.75323i) q^{74} +(6.89898 + 9.75663i) q^{75} +(-3.72474 + 6.45145i) q^{76} +(-4.89898 - 6.92820i) q^{78} +7.89898 q^{79} +(-1.72474 - 2.98735i) q^{80} +(-1.39898 + 8.89060i) q^{81} +(4.89898 - 8.48528i) q^{82} +(-1.00000 - 1.73205i) q^{83} +(-3.44949 + 5.97469i) q^{85} +(1.44949 + 2.51059i) q^{86} +(-5.00000 + 0.460702i) q^{87} +(-1.00000 + 1.73205i) q^{88} +(3.55051 - 6.14966i) q^{89} +(-3.44949 + 9.75663i) q^{90} +(0.500000 + 0.866025i) q^{92} +(4.34847 - 9.43879i) q^{93} -9.79796 q^{94} +25.6969 q^{95} +(0.724745 - 1.57313i) q^{96} +(3.44949 + 5.97469i) q^{97} +(5.89898 - 1.09638i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 14 q^{15} + 4 q^{16} - 4 q^{17} + 2 q^{18} - 10 q^{19} - 2 q^{20} - 4 q^{22} + 2 q^{23} - 2 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.724745 − 1.57313i 0.418432 − 0.908248i
\(4\) 1.00000 0.500000
\(5\) −1.72474 − 2.98735i −0.771329 − 1.33598i −0.936835 − 0.349773i \(-0.886259\pi\)
0.165505 − 0.986209i \(-0.447075\pi\)
\(6\) 0.724745 − 1.57313i 0.295876 − 0.642229i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −1.94949 − 2.28024i −0.649830 − 0.760080i
\(10\) −1.72474 − 2.98735i −0.545412 − 0.944682i
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 − 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0.724745 − 1.57313i 0.209216 − 0.454124i
\(13\) 2.44949 − 4.24264i 0.679366 − 1.17670i −0.295806 − 0.955248i \(-0.595588\pi\)
0.975172 − 0.221449i \(-0.0710785\pi\)
\(14\) 0 0
\(15\) −5.94949 + 0.548188i −1.53615 + 0.141542i
\(16\) 1.00000 0.250000
\(17\) −1.00000 − 1.73205i −0.242536 − 0.420084i 0.718900 − 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) −1.94949 − 2.28024i −0.459499 − 0.537457i
\(19\) −3.72474 + 6.45145i −0.854515 + 1.48006i 0.0225791 + 0.999745i \(0.492812\pi\)
−0.877094 + 0.480318i \(0.840521\pi\)
\(20\) −1.72474 − 2.98735i −0.385665 − 0.667991i
\(21\) 0 0
\(22\) −1.00000 + 1.73205i −0.213201 + 0.369274i
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 − 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0.724745 − 1.57313i 0.147938 − 0.321114i
\(25\) −3.44949 + 5.97469i −0.689898 + 1.19494i
\(26\) 2.44949 − 4.24264i 0.480384 − 0.832050i
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) −1.44949 − 2.51059i −0.269163 − 0.466205i 0.699483 − 0.714650i \(-0.253414\pi\)
−0.968646 + 0.248445i \(0.920081\pi\)
\(30\) −5.94949 + 0.548188i −1.08622 + 0.100085i
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 + 2.82843i 0.348155 + 0.492366i
\(34\) −1.00000 − 1.73205i −0.171499 − 0.297044i
\(35\) 0 0
\(36\) −1.94949 − 2.28024i −0.324915 − 0.380040i
\(37\) 3.89898 − 6.75323i 0.640988 − 1.11022i −0.344224 − 0.938887i \(-0.611858\pi\)
0.985213 − 0.171337i \(-0.0548086\pi\)
\(38\) −3.72474 + 6.45145i −0.604233 + 1.04656i
\(39\) −4.89898 − 6.92820i −0.784465 − 1.10940i
\(40\) −1.72474 − 2.98735i −0.272706 − 0.472341i
\(41\) 4.89898 − 8.48528i 0.765092 − 1.32518i −0.175106 − 0.984550i \(-0.556027\pi\)
0.940198 − 0.340629i \(-0.110640\pi\)
\(42\) 0 0
\(43\) 1.44949 + 2.51059i 0.221045 + 0.382861i 0.955126 − 0.296201i \(-0.0957199\pi\)
−0.734080 + 0.679062i \(0.762387\pi\)
\(44\) −1.00000 + 1.73205i −0.150756 + 0.261116i
\(45\) −3.44949 + 9.75663i −0.514220 + 1.45443i
\(46\) 0.500000 + 0.866025i 0.0737210 + 0.127688i
\(47\) −9.79796 −1.42918 −0.714590 − 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0.724745 − 1.57313i 0.104608 − 0.227062i
\(49\) 0 0
\(50\) −3.44949 + 5.97469i −0.487832 + 0.844949i
\(51\) −3.44949 + 0.317837i −0.483025 + 0.0445061i
\(52\) 2.44949 − 4.24264i 0.339683 − 0.588348i
\(53\) 0.550510 + 0.953512i 0.0756184 + 0.130975i 0.901355 − 0.433081i \(-0.142574\pi\)
−0.825737 + 0.564056i \(0.809240\pi\)
\(54\) −5.00000 + 1.41421i −0.680414 + 0.192450i
\(55\) 6.89898 0.930258
\(56\) 0 0
\(57\) 7.44949 + 10.5352i 0.986709 + 1.39542i
\(58\) −1.44949 − 2.51059i −0.190327 − 0.329657i
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) −5.94949 + 0.548188i −0.768076 + 0.0707708i
\(61\) 11.4495 1.46596 0.732978 − 0.680252i \(-0.238130\pi\)
0.732978 + 0.680252i \(0.238130\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.8990 −2.09606
\(66\) 2.00000 + 2.82843i 0.246183 + 0.348155i
\(67\) −3.10102 −0.378850 −0.189425 − 0.981895i \(-0.560662\pi\)
−0.189425 + 0.981895i \(0.560662\pi\)
\(68\) −1.00000 − 1.73205i −0.121268 − 0.210042i
\(69\) 1.72474 − 0.158919i 0.207635 − 0.0191316i
\(70\) 0 0
\(71\) 9.89898 1.17479 0.587396 − 0.809299i \(-0.300153\pi\)
0.587396 + 0.809299i \(0.300153\pi\)
\(72\) −1.94949 − 2.28024i −0.229750 − 0.268729i
\(73\) −1.44949 − 2.51059i −0.169650 − 0.293842i 0.768647 − 0.639673i \(-0.220930\pi\)
−0.938297 + 0.345831i \(0.887597\pi\)
\(74\) 3.89898 − 6.75323i 0.453247 − 0.785047i
\(75\) 6.89898 + 9.75663i 0.796626 + 1.12660i
\(76\) −3.72474 + 6.45145i −0.427258 + 0.740032i
\(77\) 0 0
\(78\) −4.89898 − 6.92820i −0.554700 − 0.784465i
\(79\) 7.89898 0.888705 0.444352 − 0.895852i \(-0.353434\pi\)
0.444352 + 0.895852i \(0.353434\pi\)
\(80\) −1.72474 − 2.98735i −0.192832 − 0.333995i
\(81\) −1.39898 + 8.89060i −0.155442 + 0.987845i
\(82\) 4.89898 − 8.48528i 0.541002 − 0.937043i
\(83\) −1.00000 − 1.73205i −0.109764 − 0.190117i 0.805910 − 0.592037i \(-0.201676\pi\)
−0.915675 + 0.401920i \(0.868343\pi\)
\(84\) 0 0
\(85\) −3.44949 + 5.97469i −0.374150 + 0.648046i
\(86\) 1.44949 + 2.51059i 0.156302 + 0.270724i
\(87\) −5.00000 + 0.460702i −0.536056 + 0.0493924i
\(88\) −1.00000 + 1.73205i −0.106600 + 0.184637i
\(89\) 3.55051 − 6.14966i 0.376353 − 0.651863i −0.614175 − 0.789170i \(-0.710511\pi\)
0.990529 + 0.137307i \(0.0438445\pi\)
\(90\) −3.44949 + 9.75663i −0.363608 + 1.02844i
\(91\) 0 0
\(92\) 0.500000 + 0.866025i 0.0521286 + 0.0902894i
\(93\) 4.34847 − 9.43879i 0.450915 − 0.978757i
\(94\) −9.79796 −1.01058
\(95\) 25.6969 2.63645
\(96\) 0.724745 − 1.57313i 0.0739690 − 0.160557i
\(97\) 3.44949 + 5.97469i 0.350243 + 0.606638i 0.986292 − 0.165011i \(-0.0527658\pi\)
−0.636049 + 0.771649i \(0.719432\pi\)
\(98\) 0 0
\(99\) 5.89898 − 1.09638i 0.592870 − 0.110190i
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.e.m.373.2 4
3.2 odd 2 2646.2.e.l.1549.2 4
7.2 even 3 126.2.f.c.85.2 yes 4
7.3 odd 6 882.2.h.l.67.2 4
7.4 even 3 882.2.h.k.67.1 4
7.5 odd 6 882.2.f.j.589.1 4
7.6 odd 2 882.2.e.n.373.1 4
9.2 odd 6 2646.2.h.m.667.1 4
9.7 even 3 882.2.h.k.79.1 4
21.2 odd 6 378.2.f.d.253.2 4
21.5 even 6 2646.2.f.k.1765.1 4
21.11 odd 6 2646.2.h.m.361.1 4
21.17 even 6 2646.2.h.n.361.2 4
21.20 even 2 2646.2.e.k.1549.1 4
28.23 odd 6 1008.2.r.e.337.1 4
63.2 odd 6 378.2.f.d.127.2 4
63.5 even 6 7938.2.a.bm.1.2 2
63.11 odd 6 2646.2.e.l.2125.2 4
63.16 even 3 126.2.f.c.43.1 ✓ 4
63.20 even 6 2646.2.h.n.667.2 4
63.23 odd 6 1134.2.a.i.1.1 2
63.25 even 3 inner 882.2.e.m.655.2 4
63.34 odd 6 882.2.h.l.79.2 4
63.38 even 6 2646.2.e.k.2125.1 4
63.40 odd 6 7938.2.a.bn.1.1 2
63.47 even 6 2646.2.f.k.883.1 4
63.52 odd 6 882.2.e.n.655.1 4
63.58 even 3 1134.2.a.p.1.2 2
63.61 odd 6 882.2.f.j.295.2 4
84.23 even 6 3024.2.r.e.1009.2 4
252.23 even 6 9072.2.a.bd.1.1 2
252.79 odd 6 1008.2.r.e.673.2 4
252.191 even 6 3024.2.r.e.2017.2 4
252.247 odd 6 9072.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.1 ✓ 4 63.16 even 3
126.2.f.c.85.2 yes 4 7.2 even 3
378.2.f.d.127.2 4 63.2 odd 6
378.2.f.d.253.2 4 21.2 odd 6
882.2.e.m.373.2 4 1.1 even 1 trivial
882.2.e.m.655.2 4 63.25 even 3 inner
882.2.e.n.373.1 4 7.6 odd 2
882.2.e.n.655.1 4 63.52 odd 6
882.2.f.j.295.2 4 63.61 odd 6
882.2.f.j.589.1 4 7.5 odd 6
882.2.h.k.67.1 4 7.4 even 3
882.2.h.k.79.1 4 9.7 even 3
882.2.h.l.67.2 4 7.3 odd 6
882.2.h.l.79.2 4 63.34 odd 6
1008.2.r.e.337.1 4 28.23 odd 6
1008.2.r.e.673.2 4 252.79 odd 6
1134.2.a.i.1.1 2 63.23 odd 6
1134.2.a.p.1.2 2 63.58 even 3
2646.2.e.k.1549.1 4 21.20 even 2
2646.2.e.k.2125.1 4 63.38 even 6
2646.2.e.l.1549.2 4 3.2 odd 2
2646.2.e.l.2125.2 4 63.11 odd 6
2646.2.f.k.883.1 4 63.47 even 6
2646.2.f.k.1765.1 4 21.5 even 6
2646.2.h.m.361.1 4 21.11 odd 6
2646.2.h.m.667.1 4 9.2 odd 6
2646.2.h.n.361.2 4 21.17 even 6
2646.2.h.n.667.2 4 63.20 even 6
3024.2.r.e.1009.2 4 84.23 even 6
3024.2.r.e.2017.2 4 252.191 even 6
7938.2.a.bm.1.2 2 63.5 even 6
7938.2.a.bn.1.1 2 63.40 odd 6
9072.2.a.bd.1.1 2 252.23 even 6
9072.2.a.bk.1.2 2 252.247 odd 6