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 = [2,-2,3,2,2,-3,0,-2,3] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 655.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 882.655
Dual form 882.2.e.e.373.1

$q$-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-1.00000 q^{2} +(1.50000 - 0.866025i) q^{3} +1.00000 q^{4} +(1.00000 - 1.73205i) q^{5} +(-1.50000 + 0.866025i) q^{6} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(-1.00000 + 1.73205i) q^{10} +(-0.500000 - 0.866025i) q^{11} +(1.50000 - 0.866025i) q^{12} +(-3.00000 - 5.19615i) q^{13} -3.46410i q^{15} +1.00000 q^{16} +(-2.50000 + 4.33013i) q^{17} +(-1.50000 + 2.59808i) q^{18} +(-3.50000 - 6.06218i) q^{19} +(1.00000 - 1.73205i) q^{20} +(0.500000 + 0.866025i) q^{22} +(-2.00000 + 3.46410i) q^{23} +(-1.50000 + 0.866025i) q^{24} +(0.500000 + 0.866025i) q^{25} +(3.00000 + 5.19615i) q^{26} -5.19615i q^{27} +(2.00000 - 3.46410i) q^{29} +3.46410i q^{30} +6.00000 q^{31} -1.00000 q^{32} +(-1.50000 - 0.866025i) q^{33} +(2.50000 - 4.33013i) q^{34} +(1.50000 - 2.59808i) q^{36} +(-1.00000 - 1.73205i) q^{37} +(3.50000 + 6.06218i) q^{38} +(-9.00000 - 5.19615i) q^{39} +(-1.00000 + 1.73205i) q^{40} +(1.50000 + 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-0.500000 - 0.866025i) q^{44} +(-3.00000 - 5.19615i) q^{45} +(2.00000 - 3.46410i) q^{46} +(1.50000 - 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{50} +8.66025i q^{51} +(-3.00000 - 5.19615i) q^{52} +(-6.00000 + 10.3923i) q^{53} +5.19615i q^{54} -2.00000 q^{55} +(-10.5000 - 6.06218i) q^{57} +(-2.00000 + 3.46410i) q^{58} +7.00000 q^{59} -3.46410i q^{60} +12.0000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +(1.50000 + 0.866025i) q^{66} +13.0000 q^{67} +(-2.50000 + 4.33013i) q^{68} +6.92820i q^{69} -8.00000 q^{71} +(-1.50000 + 2.59808i) q^{72} +(0.500000 - 0.866025i) q^{73} +(1.00000 + 1.73205i) q^{74} +(1.50000 + 0.866025i) q^{75} +(-3.50000 - 6.06218i) q^{76} +(9.00000 + 5.19615i) q^{78} -6.00000 q^{79} +(1.00000 - 1.73205i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-1.50000 - 2.59808i) q^{82} +(8.00000 - 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} +(-0.500000 + 0.866025i) q^{86} -6.92820i q^{87} +(0.500000 + 0.866025i) q^{88} +(-3.00000 - 5.19615i) q^{89} +(3.00000 + 5.19615i) q^{90} +(-2.00000 + 3.46410i) q^{92} +(9.00000 - 5.19615i) q^{93} -14.0000 q^{95} +(-1.50000 + 0.866025i) q^{96} +(-2.50000 + 4.33013i) q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} - 2 q^{8} + 3 q^{9} - 2 q^{10} - q^{11} + 3 q^{12} - 6 q^{13} + 2 q^{16} - 5 q^{17} - 3 q^{18} - 7 q^{19} + 2 q^{20} + q^{22} - 4 q^{23} - 3 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.50000 − 0.866025i 0.866025 − 0.500000i
\(4\) 1.00000 0.500000
\(5\) 1.00000 − 1.73205i 0.447214 − 0.774597i −0.550990 − 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) −1.50000 + 0.866025i −0.612372 + 0.353553i
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) −1.00000 + 1.73205i −0.316228 + 0.547723i
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i 0.780750 − 0.624844i \(-0.214837\pi\)
−0.931505 + 0.363727i \(0.881504\pi\)
\(12\) 1.50000 − 0.866025i 0.433013 − 0.250000i
\(13\) −3.00000 − 5.19615i −0.832050 − 1.44115i −0.896410 − 0.443227i \(-0.853834\pi\)
0.0643593 − 0.997927i \(-0.479500\pi\)
\(14\) 0 0
\(15\) − 3.46410i − 0.894427i
\(16\) 1.00000 0.250000
\(17\) −2.50000 + 4.33013i −0.606339 + 1.05021i 0.385499 + 0.922708i \(0.374029\pi\)
−0.991838 + 0.127502i \(0.959304\pi\)
\(18\) −1.50000 + 2.59808i −0.353553 + 0.612372i
\(19\) −3.50000 − 6.06218i −0.802955 − 1.39076i −0.917663 − 0.397360i \(-0.869927\pi\)
0.114708 − 0.993399i \(-0.463407\pi\)
\(20\) 1.00000 − 1.73205i 0.223607 − 0.387298i
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.106600 + 0.184637i
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 − 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) −1.50000 + 0.866025i −0.306186 + 0.176777i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 3.00000 + 5.19615i 0.588348 + 1.01905i
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 2.00000 − 3.46410i 0.371391 − 0.643268i −0.618389 − 0.785872i \(-0.712214\pi\)
0.989780 + 0.142605i \(0.0455477\pi\)
\(30\) 3.46410i 0.632456i
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.50000 − 0.866025i −0.261116 − 0.150756i
\(34\) 2.50000 − 4.33013i 0.428746 − 0.742611i
\(35\) 0 0
\(36\) 1.50000 − 2.59808i 0.250000 − 0.433013i
\(37\) −1.00000 − 1.73205i −0.164399 − 0.284747i 0.772043 − 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 3.50000 + 6.06218i 0.567775 + 0.983415i
\(39\) −9.00000 − 5.19615i −1.44115 − 0.832050i
\(40\) −1.00000 + 1.73205i −0.158114 + 0.273861i
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) −0.500000 − 0.866025i −0.0753778 − 0.130558i
\(45\) −3.00000 − 5.19615i −0.447214 − 0.774597i
\(46\) 2.00000 − 3.46410i 0.294884 − 0.510754i
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.50000 − 0.866025i 0.216506 − 0.125000i
\(49\) 0 0
\(50\) −0.500000 − 0.866025i −0.0707107 − 0.122474i
\(51\) 8.66025i 1.21268i
\(52\) −3.00000 − 5.19615i −0.416025 − 0.720577i
\(53\) −6.00000 + 10.3923i −0.824163 + 1.42749i 0.0783936 + 0.996922i \(0.475021\pi\)
−0.902557 + 0.430570i \(0.858312\pi\)
\(54\) 5.19615i 0.707107i
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −10.5000 − 6.06218i −1.39076 − 0.802955i
\(58\) −2.00000 + 3.46410i −0.262613 + 0.454859i
\(59\) 7.00000 0.911322 0.455661 − 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) − 3.46410i − 0.447214i
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 1.50000 + 0.866025i 0.184637 + 0.106600i
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) −2.50000 + 4.33013i −0.303170 + 0.525105i
\(69\) 6.92820i 0.834058i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.50000 + 2.59808i −0.176777 + 0.306186i
\(73\) 0.500000 − 0.866025i 0.0585206 − 0.101361i −0.835281 − 0.549823i \(-0.814695\pi\)
0.893801 + 0.448463i \(0.148028\pi\)
\(74\) 1.00000 + 1.73205i 0.116248 + 0.201347i
\(75\) 1.50000 + 0.866025i 0.173205 + 0.100000i
\(76\) −3.50000 − 6.06218i −0.401478 − 0.695379i
\(77\) 0 0
\(78\) 9.00000 + 5.19615i 1.01905 + 0.588348i
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 1.00000 − 1.73205i 0.111803 − 0.193649i
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) −1.50000 − 2.59808i −0.165647 − 0.286910i
\(83\) 8.00000 − 13.8564i 0.878114 − 1.52094i 0.0247060 − 0.999695i \(-0.492135\pi\)
0.853408 − 0.521243i \(-0.174532\pi\)
\(84\) 0 0
\(85\) 5.00000 + 8.66025i 0.542326 + 0.939336i
\(86\) −0.500000 + 0.866025i −0.0539164 + 0.0933859i
\(87\) − 6.92820i − 0.742781i
\(88\) 0.500000 + 0.866025i 0.0533002 + 0.0923186i
\(89\) −3.00000 − 5.19615i −0.317999 − 0.550791i 0.662071 − 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 3.00000 + 5.19615i 0.316228 + 0.547723i
\(91\) 0 0
\(92\) −2.00000 + 3.46410i −0.208514 + 0.361158i
\(93\) 9.00000 − 5.19615i 0.933257 − 0.538816i
\(94\) 0 0
\(95\) −14.0000 −1.43637
\(96\) −1.50000 + 0.866025i −0.153093 + 0.0883883i
\(97\) −2.50000 + 4.33013i −0.253837 + 0.439658i −0.964579 − 0.263795i \(-0.915026\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
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.e.655.1 2
3.2 odd 2 2646.2.e.h.2125.1 2
7.2 even 3 882.2.h.g.79.1 2
7.3 odd 6 126.2.f.b.43.1 ✓ 2
7.4 even 3 882.2.f.f.295.1 2
7.5 odd 6 882.2.h.h.79.1 2
7.6 odd 2 882.2.e.a.655.1 2
9.4 even 3 882.2.h.g.67.1 2
9.5 odd 6 2646.2.h.c.361.1 2
21.2 odd 6 2646.2.h.c.667.1 2
21.5 even 6 2646.2.h.b.667.1 2
21.11 odd 6 2646.2.f.b.883.1 2
21.17 even 6 378.2.f.b.127.1 2
21.20 even 2 2646.2.e.i.2125.1 2
28.3 even 6 1008.2.r.a.673.1 2
63.4 even 3 882.2.f.f.589.1 2
63.5 even 6 2646.2.e.i.1549.1 2
63.11 odd 6 7938.2.a.bb.1.1 1
63.13 odd 6 882.2.h.h.67.1 2
63.23 odd 6 2646.2.e.h.1549.1 2
63.25 even 3 7938.2.a.e.1.1 1
63.31 odd 6 126.2.f.b.85.1 yes 2
63.32 odd 6 2646.2.f.b.1765.1 2
63.38 even 6 1134.2.a.f.1.1 1
63.40 odd 6 882.2.e.a.373.1 2
63.41 even 6 2646.2.h.b.361.1 2
63.52 odd 6 1134.2.a.c.1.1 1
63.58 even 3 inner 882.2.e.e.373.1 2
63.59 even 6 378.2.f.b.253.1 2
84.59 odd 6 3024.2.r.c.2017.1 2
252.31 even 6 1008.2.r.a.337.1 2
252.59 odd 6 3024.2.r.c.1009.1 2
252.115 even 6 9072.2.a.t.1.1 1
252.227 odd 6 9072.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.b.43.1 ✓ 2 7.3 odd 6
126.2.f.b.85.1 yes 2 63.31 odd 6
378.2.f.b.127.1 2 21.17 even 6
378.2.f.b.253.1 2 63.59 even 6
882.2.e.a.373.1 2 63.40 odd 6
882.2.e.a.655.1 2 7.6 odd 2
882.2.e.e.373.1 2 63.58 even 3 inner
882.2.e.e.655.1 2 1.1 even 1 trivial
882.2.f.f.295.1 2 7.4 even 3
882.2.f.f.589.1 2 63.4 even 3
882.2.h.g.67.1 2 9.4 even 3
882.2.h.g.79.1 2 7.2 even 3
882.2.h.h.67.1 2 63.13 odd 6
882.2.h.h.79.1 2 7.5 odd 6
1008.2.r.a.337.1 2 252.31 even 6
1008.2.r.a.673.1 2 28.3 even 6
1134.2.a.c.1.1 1 63.52 odd 6
1134.2.a.f.1.1 1 63.38 even 6
2646.2.e.h.1549.1 2 63.23 odd 6
2646.2.e.h.2125.1 2 3.2 odd 2
2646.2.e.i.1549.1 2 63.5 even 6
2646.2.e.i.2125.1 2 21.20 even 2
2646.2.f.b.883.1 2 21.11 odd 6
2646.2.f.b.1765.1 2 63.32 odd 6
2646.2.h.b.361.1 2 63.41 even 6
2646.2.h.b.667.1 2 21.5 even 6
2646.2.h.c.361.1 2 9.5 odd 6
2646.2.h.c.667.1 2 21.2 odd 6
3024.2.r.c.1009.1 2 252.59 odd 6
3024.2.r.c.2017.1 2 84.59 odd 6
7938.2.a.e.1.1 1 63.25 even 3
7938.2.a.bb.1.1 1 63.11 odd 6
9072.2.a.f.1.1 1 252.227 odd 6
9072.2.a.t.1.1 1 252.115 even 6