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Properties

Label 882.2.h.q.79.4
Level $882$
Weight $2$
Character 882.79
Analytic conductor $7.043$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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 = [8,-4,0,-4,0,0,0,8,0] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.4
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 882.79
Dual form 882.2.h.q.67.3

$q$-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} +(1.22474 + 1.22474i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.03528 q^{5} +(0.448288 - 1.67303i) q^{6} +1.00000 q^{8} +3.00000i q^{9} +(0.517638 + 0.896575i) q^{10} -0.267949 q^{11} +(-1.67303 + 0.448288i) q^{12} +(-0.896575 - 1.55291i) q^{13} +(-1.26795 - 1.26795i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(3.41542 + 5.91567i) q^{17} +(2.59808 - 1.50000i) q^{18} +(-2.19067 + 3.79435i) q^{19} +(0.517638 - 0.896575i) q^{20} +(0.133975 + 0.232051i) q^{22} +5.46410 q^{23} +(1.22474 + 1.22474i) q^{24} -3.92820 q^{25} +(-0.896575 + 1.55291i) q^{26} +(-3.67423 + 3.67423i) q^{27} +(2.00000 - 3.46410i) q^{29} +(-0.464102 + 1.73205i) q^{30} +(-3.34607 + 5.79555i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-0.328169 - 0.328169i) q^{33} +(3.41542 - 5.91567i) q^{34} +(-2.59808 - 1.50000i) q^{36} +(-3.73205 + 6.46410i) q^{37} +4.38134 q^{38} +(0.803848 - 3.00000i) q^{39} -1.03528 q^{40} +(4.31199 + 7.46859i) q^{41} +(-0.133975 + 0.232051i) q^{43} +(0.133975 - 0.232051i) q^{44} -3.10583i q^{45} +(-2.73205 - 4.73205i) q^{46} +(-0.378937 - 0.656339i) q^{47} +(0.448288 - 1.67303i) q^{48} +(1.96410 + 3.40192i) q^{50} +(-3.06218 + 11.4282i) q^{51} +1.79315 q^{52} +(-5.46410 - 9.46410i) q^{53} +(5.01910 + 1.34486i) q^{54} +0.277401 q^{55} +(-7.33013 + 1.96410i) q^{57} -4.00000 q^{58} +(0.637756 - 1.10463i) q^{59} +(1.73205 - 0.464102i) q^{60} +(6.31319 + 10.9348i) q^{61} +6.69213 q^{62} +1.00000 q^{64} +(0.928203 + 1.60770i) q^{65} +(-0.120118 + 0.448288i) q^{66} +(-6.23205 + 10.7942i) q^{67} -6.83083 q^{68} +(6.69213 + 6.69213i) q^{69} +9.46410 q^{71} +3.00000i q^{72} +(-2.70831 - 4.69093i) q^{73} +7.46410 q^{74} +(-4.81105 - 4.81105i) q^{75} +(-2.19067 - 3.79435i) q^{76} +(-3.00000 + 0.803848i) q^{78} +(-4.46410 - 7.73205i) q^{79} +(0.517638 + 0.896575i) q^{80} -9.00000 q^{81} +(4.31199 - 7.46859i) q^{82} +(-3.29530 + 5.70762i) q^{83} +(-3.53590 - 6.12436i) q^{85} +0.267949 q^{86} +(6.69213 - 1.79315i) q^{87} -0.267949 q^{88} +(-3.53553 + 6.12372i) q^{89} +(-2.68973 + 1.55291i) q^{90} +(-2.73205 + 4.73205i) q^{92} +(-11.1962 + 3.00000i) q^{93} +(-0.378937 + 0.656339i) q^{94} +(2.26795 - 3.92820i) q^{95} +(-1.67303 + 0.448288i) q^{96} +(9.07227 - 15.7136i) q^{97} -0.803848i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 16 q^{11} - 24 q^{15} - 4 q^{16} + 8 q^{22} + 16 q^{23} + 24 q^{25} + 16 q^{29} + 24 q^{30} - 4 q^{32} - 16 q^{37} + 48 q^{39} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 12 q^{50}+ \cdots + 32 q^{95}+O(q^{100}) \) 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
\(3\) 1.22474 + 1.22474i 0.707107 + 0.707107i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.03528 −0.462990 −0.231495 − 0.972836i \(-0.574362\pi\)
−0.231495 + 0.972836i \(0.574362\pi\)
\(6\) 0.448288 − 1.67303i 0.183013 − 0.683013i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 3.00000i 1.00000i
\(10\) 0.517638 + 0.896575i 0.163692 + 0.283522i
\(11\) −0.267949 −0.0807897 −0.0403949 − 0.999184i \(-0.512862\pi\)
−0.0403949 + 0.999184i \(0.512862\pi\)
\(12\) −1.67303 + 0.448288i −0.482963 + 0.129410i
\(13\) −0.896575 − 1.55291i −0.248665 − 0.430701i 0.714490 − 0.699645i \(-0.246659\pi\)
−0.963156 + 0.268944i \(0.913325\pi\)
\(14\) 0 0
\(15\) −1.26795 − 1.26795i −0.327383 − 0.327383i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.41542 + 5.91567i 0.828360 + 1.43476i 0.899324 + 0.437283i \(0.144059\pi\)
−0.0709642 + 0.997479i \(0.522608\pi\)
\(18\) 2.59808 − 1.50000i 0.612372 − 0.353553i
\(19\) −2.19067 + 3.79435i −0.502574 + 0.870484i 0.497421 + 0.867509i \(0.334280\pi\)
−0.999996 + 0.00297513i \(0.999053\pi\)
\(20\) 0.517638 − 0.896575i 0.115747 − 0.200480i
\(21\) 0 0
\(22\) 0.133975 + 0.232051i 0.0285635 + 0.0494734i
\(23\) 5.46410 1.13934 0.569672 − 0.821872i \(-0.307070\pi\)
0.569672 + 0.821872i \(0.307070\pi\)
\(24\) 1.22474 + 1.22474i 0.250000 + 0.250000i
\(25\) −3.92820 −0.785641
\(26\) −0.896575 + 1.55291i −0.175833 + 0.304552i
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) 2.00000 − 3.46410i 0.371391 − 0.643268i −0.618389 − 0.785872i \(-0.712214\pi\)
0.989780 + 0.142605i \(0.0455477\pi\)
\(30\) −0.464102 + 1.73205i −0.0847330 + 0.316228i
\(31\) −3.34607 + 5.79555i −0.600971 + 1.04091i 0.391703 + 0.920092i \(0.371886\pi\)
−0.992674 + 0.120821i \(0.961447\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −0.328169 − 0.328169i −0.0571270 − 0.0571270i
\(34\) 3.41542 − 5.91567i 0.585739 − 1.01453i
\(35\) 0 0
\(36\) −2.59808 − 1.50000i −0.433013 − 0.250000i
\(37\) −3.73205 + 6.46410i −0.613545 + 1.06269i 0.377092 + 0.926176i \(0.376924\pi\)
−0.990638 + 0.136516i \(0.956409\pi\)
\(38\) 4.38134 0.710747
\(39\) 0.803848 − 3.00000i 0.128719 − 0.480384i
\(40\) −1.03528 −0.163692
\(41\) 4.31199 + 7.46859i 0.673420 + 1.16640i 0.976928 + 0.213569i \(0.0685087\pi\)
−0.303508 + 0.952829i \(0.598158\pi\)
\(42\) 0 0
\(43\) −0.133975 + 0.232051i −0.0204309 + 0.0353874i −0.876060 − 0.482202i \(-0.839837\pi\)
0.855629 + 0.517589i \(0.173170\pi\)
\(44\) 0.133975 − 0.232051i 0.0201974 − 0.0349830i
\(45\) − 3.10583i − 0.462990i
\(46\) −2.73205 − 4.73205i −0.402819 − 0.697703i
\(47\) −0.378937 − 0.656339i −0.0552737 − 0.0957369i 0.837065 − 0.547104i \(-0.184270\pi\)
−0.892338 + 0.451367i \(0.850936\pi\)
\(48\) 0.448288 − 1.67303i 0.0647048 − 0.241481i
\(49\) 0 0
\(50\) 1.96410 + 3.40192i 0.277766 + 0.481105i
\(51\) −3.06218 + 11.4282i −0.428791 + 1.60027i
\(52\) 1.79315 0.248665
\(53\) −5.46410 − 9.46410i −0.750552 − 1.29999i −0.947555 − 0.319592i \(-0.896454\pi\)
0.197003 − 0.980403i \(-0.436879\pi\)
\(54\) 5.01910 + 1.34486i 0.683013 + 0.183013i
\(55\) 0.277401 0.0374048
\(56\) 0 0
\(57\) −7.33013 + 1.96410i −0.970899 + 0.260152i
\(58\) −4.00000 −0.525226
\(59\) 0.637756 − 1.10463i 0.0830288 − 0.143810i −0.821521 − 0.570179i \(-0.806874\pi\)
0.904550 + 0.426369i \(0.140207\pi\)
\(60\) 1.73205 − 0.464102i 0.223607 − 0.0599153i
\(61\) 6.31319 + 10.9348i 0.808322 + 1.40005i 0.914026 + 0.405656i \(0.132957\pi\)
−0.105704 + 0.994398i \(0.533710\pi\)
\(62\) 6.69213 0.849901
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.928203 + 1.60770i 0.115129 + 0.199410i
\(66\) −0.120118 + 0.448288i −0.0147855 + 0.0551804i
\(67\) −6.23205 + 10.7942i −0.761366 + 1.31872i 0.180780 + 0.983524i \(0.442138\pi\)
−0.942146 + 0.335201i \(0.891196\pi\)
\(68\) −6.83083 −0.828360
\(69\) 6.69213 + 6.69213i 0.805638 + 0.805638i
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 − 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 3.00000i 0.353553i
\(73\) −2.70831 − 4.69093i −0.316984 − 0.549032i 0.662874 − 0.748731i \(-0.269337\pi\)
−0.979857 + 0.199700i \(0.936003\pi\)
\(74\) 7.46410 0.867684
\(75\) −4.81105 − 4.81105i −0.555532 − 0.555532i
\(76\) −2.19067 − 3.79435i −0.251287 − 0.435242i
\(77\) 0 0
\(78\) −3.00000 + 0.803848i −0.339683 + 0.0910178i
\(79\) −4.46410 − 7.73205i −0.502251 − 0.869924i −0.999997 − 0.00260080i \(-0.999172\pi\)
0.497746 − 0.867323i \(-0.334161\pi\)
\(80\) 0.517638 + 0.896575i 0.0578737 + 0.100240i
\(81\) −9.00000 −1.00000
\(82\) 4.31199 − 7.46859i 0.476180 − 0.824768i
\(83\) −3.29530 + 5.70762i −0.361706 + 0.626493i −0.988242 − 0.152900i \(-0.951139\pi\)
0.626536 + 0.779393i \(0.284472\pi\)
\(84\) 0 0
\(85\) −3.53590 − 6.12436i −0.383522 − 0.664280i
\(86\) 0.267949 0.0288937
\(87\) 6.69213 − 1.79315i 0.717472 − 0.192246i
\(88\) −0.267949 −0.0285635
\(89\) −3.53553 + 6.12372i −0.374766 + 0.649113i −0.990292 − 0.139003i \(-0.955610\pi\)
0.615526 + 0.788116i \(0.288944\pi\)
\(90\) −2.68973 + 1.55291i −0.283522 + 0.163692i
\(91\) 0 0
\(92\) −2.73205 + 4.73205i −0.284836 + 0.493350i
\(93\) −11.1962 + 3.00000i −1.16099 + 0.311086i
\(94\) −0.378937 + 0.656339i −0.0390844 + 0.0676962i
\(95\) 2.26795 − 3.92820i 0.232687 − 0.403025i
\(96\) −1.67303 + 0.448288i −0.170753 + 0.0457532i
\(97\) 9.07227 − 15.7136i 0.921149 − 1.59548i 0.123510 − 0.992343i \(-0.460585\pi\)
0.797640 − 0.603134i \(-0.206082\pi\)
\(98\) 0 0
\(99\) − 0.803848i − 0.0807897i
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.q.79.4 8
3.2 odd 2 2646.2.h.t.667.3 8
7.2 even 3 882.2.f.q.295.1 ✓ 8
7.3 odd 6 882.2.e.s.655.2 8
7.4 even 3 882.2.e.s.655.3 8
7.5 odd 6 882.2.f.q.295.4 yes 8
7.6 odd 2 inner 882.2.h.q.79.1 8
9.4 even 3 882.2.e.s.373.3 8
9.5 odd 6 2646.2.e.q.1549.2 8
21.2 odd 6 2646.2.f.r.883.2 8
21.5 even 6 2646.2.f.r.883.3 8
21.11 odd 6 2646.2.e.q.2125.2 8
21.17 even 6 2646.2.e.q.2125.3 8
21.20 even 2 2646.2.h.t.667.2 8
63.2 odd 6 7938.2.a.ci.1.3 4
63.4 even 3 inner 882.2.h.q.67.3 8
63.5 even 6 2646.2.f.r.1765.3 8
63.13 odd 6 882.2.e.s.373.2 8
63.16 even 3 7938.2.a.cp.1.2 4
63.23 odd 6 2646.2.f.r.1765.2 8
63.31 odd 6 inner 882.2.h.q.67.2 8
63.32 odd 6 2646.2.h.t.361.3 8
63.40 odd 6 882.2.f.q.589.4 yes 8
63.41 even 6 2646.2.e.q.1549.3 8
63.47 even 6 7938.2.a.ci.1.2 4
63.58 even 3 882.2.f.q.589.1 yes 8
63.59 even 6 2646.2.h.t.361.2 8
63.61 odd 6 7938.2.a.cp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.2 8 63.13 odd 6
882.2.e.s.373.3 8 9.4 even 3
882.2.e.s.655.2 8 7.3 odd 6
882.2.e.s.655.3 8 7.4 even 3
882.2.f.q.295.1 ✓ 8 7.2 even 3
882.2.f.q.295.4 yes 8 7.5 odd 6
882.2.f.q.589.1 yes 8 63.58 even 3
882.2.f.q.589.4 yes 8 63.40 odd 6
882.2.h.q.67.2 8 63.31 odd 6 inner
882.2.h.q.67.3 8 63.4 even 3 inner
882.2.h.q.79.1 8 7.6 odd 2 inner
882.2.h.q.79.4 8 1.1 even 1 trivial
2646.2.e.q.1549.2 8 9.5 odd 6
2646.2.e.q.1549.3 8 63.41 even 6
2646.2.e.q.2125.2 8 21.11 odd 6
2646.2.e.q.2125.3 8 21.17 even 6
2646.2.f.r.883.2 8 21.2 odd 6
2646.2.f.r.883.3 8 21.5 even 6
2646.2.f.r.1765.2 8 63.23 odd 6
2646.2.f.r.1765.3 8 63.5 even 6
2646.2.h.t.361.2 8 63.59 even 6
2646.2.h.t.361.3 8 63.32 odd 6
2646.2.h.t.667.2 8 21.20 even 2
2646.2.h.t.667.3 8 3.2 odd 2
7938.2.a.ci.1.2 4 63.47 even 6
7938.2.a.ci.1.3 4 63.2 odd 6
7938.2.a.cp.1.2 4 63.16 even 3
7938.2.a.cp.1.3 4 63.61 odd 6