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(295,882)] 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("882.295"); 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([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,-1,-3,0,2,-6] 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 589.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 882.589
Dual form 882.2.f.b.295.1

$q$-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} -1.73205i q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(-1.50000 + 0.866025i) q^{6} +1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} +(1.00000 + 1.73205i) q^{11} +(1.50000 + 0.866025i) q^{12} +(-1.00000 + 1.73205i) q^{13} +(1.50000 + 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{18} -7.00000 q^{19} +(-0.500000 - 0.866025i) q^{20} +(1.00000 - 1.73205i) q^{22} +(-1.50000 + 2.59808i) q^{23} -1.73205i q^{24} +(2.00000 + 3.46410i) q^{25} +2.00000 q^{26} +5.19615i q^{27} +(4.00000 + 6.92820i) q^{29} -1.73205i q^{30} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.00000 - 1.73205i) q^{33} +(1.50000 - 2.59808i) q^{36} -6.00000 q^{37} +(3.50000 + 6.06218i) q^{38} +(3.00000 + 1.73205i) q^{39} +(-0.500000 + 0.866025i) q^{40} +(6.00000 - 10.3923i) q^{41} +(4.00000 + 6.92820i) q^{43} -2.00000 q^{44} +(1.50000 - 2.59808i) q^{45} +3.00000 q^{46} +(4.00000 + 6.92820i) q^{47} +(-1.50000 + 0.866025i) q^{48} +(2.00000 - 3.46410i) q^{50} +(-1.00000 - 1.73205i) q^{52} +4.00000 q^{53} +(4.50000 - 2.59808i) q^{54} -2.00000 q^{55} +12.1244i q^{57} +(4.00000 - 6.92820i) q^{58} +(-2.00000 + 3.46410i) q^{59} +(-1.50000 + 0.866025i) q^{60} +(-6.50000 - 11.2583i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(-3.00000 - 1.73205i) q^{66} +(1.00000 - 1.73205i) q^{67} +(4.50000 + 2.59808i) q^{69} -5.00000 q^{71} -3.00000 q^{72} -14.0000 q^{73} +(3.00000 + 5.19615i) q^{74} +(6.00000 - 3.46410i) q^{75} +(3.50000 - 6.06218i) q^{76} -3.46410i q^{78} +(5.50000 + 9.52628i) q^{79} +1.00000 q^{80} +9.00000 q^{81} -12.0000 q^{82} +(-6.00000 - 10.3923i) q^{83} +(4.00000 - 6.92820i) q^{86} +(12.0000 - 6.92820i) q^{87} +(1.00000 + 1.73205i) q^{88} +14.0000 q^{89} -3.00000 q^{90} +(-1.50000 - 2.59808i) q^{92} +(6.00000 + 3.46410i) q^{93} +(4.00000 - 6.92820i) q^{94} +(3.50000 - 6.06218i) q^{95} +(1.50000 + 0.866025i) q^{96} +(1.00000 + 1.73205i) q^{97} +(-3.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} - 3 q^{6} + 2 q^{8} - 6 q^{9} + 2 q^{10} + 2 q^{11} + 3 q^{12} - 2 q^{13} + 3 q^{15} - q^{16} + 3 q^{18} - 14 q^{19} - q^{20} + 2 q^{22} - 3 q^{23} + 4 q^{25} + 4 q^{26}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) − 1.73205i − 1.00000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i −0.955901 − 0.293691i \(-0.905116\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) −1.50000 + 0.866025i −0.612372 + 0.353553i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 − 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 1.50000 + 0.866025i 0.433013 + 0.250000i
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 1.50000 + 0.866025i 0.387298 + 0.223607i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.50000 + 2.59808i 0.353553 + 0.612372i
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −0.500000 − 0.866025i −0.111803 − 0.193649i
\(21\) 0 0
\(22\) 1.00000 − 1.73205i 0.213201 − 0.369274i
\(23\) −1.50000 + 2.59808i −0.312772 + 0.541736i −0.978961 − 0.204046i \(-0.934591\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(24\) − 1.73205i − 0.353553i
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 2.00000 0.392232
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 4.00000 + 6.92820i 0.742781 + 1.28654i 0.951224 + 0.308500i \(0.0998271\pi\)
−0.208443 + 0.978035i \(0.566840\pi\)
\(30\) − 1.73205i − 0.316228i
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 3.00000 − 1.73205i 0.522233 − 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.50000 − 2.59808i 0.250000 − 0.433013i
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 3.50000 + 6.06218i 0.567775 + 0.983415i
\(39\) 3.00000 + 1.73205i 0.480384 + 0.277350i
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) 6.00000 − 10.3923i 0.937043 − 1.62301i 0.166092 − 0.986110i \(-0.446885\pi\)
0.770950 − 0.636895i \(-0.219782\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.50000 − 2.59808i 0.223607 − 0.387298i
\(46\) 3.00000 0.442326
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) −1.50000 + 0.866025i −0.216506 + 0.125000i
\(49\) 0 0
\(50\) 2.00000 − 3.46410i 0.282843 − 0.489898i
\(51\) 0 0
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 4.50000 − 2.59808i 0.612372 − 0.353553i
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 12.1244i 1.60591i
\(58\) 4.00000 − 6.92820i 0.525226 − 0.909718i
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 − 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) −1.50000 + 0.866025i −0.193649 + 0.111803i
\(61\) −6.50000 − 11.2583i −0.832240 − 1.44148i −0.896258 − 0.443533i \(-0.853725\pi\)
0.0640184 − 0.997949i \(-0.479608\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 − 1.73205i −0.124035 − 0.214834i
\(66\) −3.00000 − 1.73205i −0.369274 − 0.213201i
\(67\) 1.00000 − 1.73205i 0.122169 − 0.211604i −0.798454 − 0.602056i \(-0.794348\pi\)
0.920623 + 0.390453i \(0.127682\pi\)
\(68\) 0 0
\(69\) 4.50000 + 2.59808i 0.541736 + 0.312772i
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) −3.00000 −0.353553
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 3.00000 + 5.19615i 0.348743 + 0.604040i
\(75\) 6.00000 − 3.46410i 0.692820 − 0.400000i
\(76\) 3.50000 − 6.06218i 0.401478 − 0.695379i
\(77\) 0 0
\(78\) − 3.46410i − 0.392232i
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −12.0000 −1.32518
\(83\) −6.00000 − 10.3923i −0.658586 − 1.14070i −0.980982 − 0.194099i \(-0.937822\pi\)
0.322396 − 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 − 6.92820i 0.431331 − 0.747087i
\(87\) 12.0000 − 6.92820i 1.28654 − 0.742781i
\(88\) 1.00000 + 1.73205i 0.106600 + 0.184637i
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −1.50000 − 2.59808i −0.156386 − 0.270868i
\(93\) 6.00000 + 3.46410i 0.622171 + 0.359211i
\(94\) 4.00000 − 6.92820i 0.412568 − 0.714590i
\(95\) 3.50000 − 6.06218i 0.359092 − 0.621966i
\(96\) 1.50000 + 0.866025i 0.153093 + 0.0883883i
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 − 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) −3.00000 − 5.19615i −0.301511 − 0.522233i
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.f.b.589.1 yes 2
3.2 odd 2 2646.2.f.h.1765.1 2
7.2 even 3 882.2.h.d.67.1 2
7.3 odd 6 882.2.e.j.373.1 2
7.4 even 3 882.2.e.f.373.1 2
7.5 odd 6 882.2.h.a.67.1 2
7.6 odd 2 882.2.f.c.589.1 yes 2
9.2 odd 6 2646.2.f.h.883.1 2
9.4 even 3 7938.2.a.ba.1.1 1
9.5 odd 6 7938.2.a.f.1.1 1
9.7 even 3 inner 882.2.f.b.295.1 ✓ 2
21.2 odd 6 2646.2.h.g.361.1 2
21.5 even 6 2646.2.h.j.361.1 2
21.11 odd 6 2646.2.e.d.1549.1 2
21.17 even 6 2646.2.e.a.1549.1 2
21.20 even 2 2646.2.f.f.1765.1 2
63.2 odd 6 2646.2.e.d.2125.1 2
63.11 odd 6 2646.2.h.g.667.1 2
63.13 odd 6 7938.2.a.v.1.1 1
63.16 even 3 882.2.e.f.655.1 2
63.20 even 6 2646.2.f.f.883.1 2
63.25 even 3 882.2.h.d.79.1 2
63.34 odd 6 882.2.f.c.295.1 yes 2
63.38 even 6 2646.2.h.j.667.1 2
63.41 even 6 7938.2.a.k.1.1 1
63.47 even 6 2646.2.e.a.2125.1 2
63.52 odd 6 882.2.h.a.79.1 2
63.61 odd 6 882.2.e.j.655.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.f.373.1 2 7.4 even 3
882.2.e.f.655.1 2 63.16 even 3
882.2.e.j.373.1 2 7.3 odd 6
882.2.e.j.655.1 2 63.61 odd 6
882.2.f.b.295.1 ✓ 2 9.7 even 3 inner
882.2.f.b.589.1 yes 2 1.1 even 1 trivial
882.2.f.c.295.1 yes 2 63.34 odd 6
882.2.f.c.589.1 yes 2 7.6 odd 2
882.2.h.a.67.1 2 7.5 odd 6
882.2.h.a.79.1 2 63.52 odd 6
882.2.h.d.67.1 2 7.2 even 3
882.2.h.d.79.1 2 63.25 even 3
2646.2.e.a.1549.1 2 21.17 even 6
2646.2.e.a.2125.1 2 63.47 even 6
2646.2.e.d.1549.1 2 21.11 odd 6
2646.2.e.d.2125.1 2 63.2 odd 6
2646.2.f.f.883.1 2 63.20 even 6
2646.2.f.f.1765.1 2 21.20 even 2
2646.2.f.h.883.1 2 9.2 odd 6
2646.2.f.h.1765.1 2 3.2 odd 2
2646.2.h.g.361.1 2 21.2 odd 6
2646.2.h.g.667.1 2 63.11 odd 6
2646.2.h.j.361.1 2 21.5 even 6
2646.2.h.j.667.1 2 63.38 even 6
7938.2.a.f.1.1 1 9.5 odd 6
7938.2.a.k.1.1 1 63.41 even 6
7938.2.a.v.1.1 1 63.13 odd 6
7938.2.a.ba.1.1 1 9.4 even 3