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Properties

Label 882.2.e.s.373.3
Level $882$
Weight $2$
Character 882.373
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(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 = [8,8,0,8,0,0,0,8,0] 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.3
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 882.373
Dual form 882.2.e.s.655.3

$q$-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+1.00000 q^{2} +(0.448288 + 1.67303i) q^{3} +1.00000 q^{4} +(0.517638 + 0.896575i) q^{5} +(0.448288 + 1.67303i) q^{6} +1.00000 q^{8} +(-2.59808 + 1.50000i) q^{9} +(0.517638 + 0.896575i) q^{10} +(0.133975 - 0.232051i) q^{11} +(0.448288 + 1.67303i) q^{12} +(-0.896575 + 1.55291i) q^{13} +(-1.26795 + 1.26795i) q^{15} +1.00000 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} +(-2.73205 - 4.73205i) q^{23} +(0.448288 + 1.67303i) q^{24} +(1.96410 - 3.40192i) q^{25} +(-0.896575 + 1.55291i) q^{26} +(-3.67423 - 3.67423i) q^{27} +(2.00000 + 3.46410i) q^{29} +(-1.26795 + 1.26795i) q^{30} +6.69213 q^{31} +1.00000 q^{32} +(0.448288 + 0.120118i) q^{33} +(3.41542 + 5.91567i) q^{34} +(-2.59808 + 1.50000i) q^{36} +(-3.73205 + 6.46410i) q^{37} +(-2.19067 + 3.79435i) q^{38} +(-3.00000 - 0.803848i) q^{39} +(0.517638 + 0.896575i) q^{40} +(4.31199 - 7.46859i) q^{41} +(-0.133975 - 0.232051i) q^{43} +(0.133975 - 0.232051i) q^{44} +(-2.68973 - 1.55291i) q^{45} +(-2.73205 - 4.73205i) q^{46} +0.757875 q^{47} +(0.448288 + 1.67303i) q^{48} +(1.96410 - 3.40192i) q^{50} +(-8.36603 + 8.36603i) q^{51} +(-0.896575 + 1.55291i) q^{52} +(-5.46410 - 9.46410i) q^{53} +(-3.67423 - 3.67423i) q^{54} +0.277401 q^{55} +(-7.33013 - 1.96410i) q^{57} +(2.00000 + 3.46410i) q^{58} -1.27551 q^{59} +(-1.26795 + 1.26795i) q^{60} -12.6264 q^{61} +6.69213 q^{62} +1.00000 q^{64} -1.85641 q^{65} +(0.448288 + 0.120118i) q^{66} +12.4641 q^{67} +(3.41542 + 5.91567i) q^{68} +(6.69213 - 6.69213i) q^{69} +9.46410 q^{71} +(-2.59808 + 1.50000i) q^{72} +(-2.70831 - 4.69093i) q^{73} +(-3.73205 + 6.46410i) q^{74} +(6.57201 + 1.76097i) q^{75} +(-2.19067 + 3.79435i) q^{76} +(-3.00000 - 0.803848i) q^{78} +8.92820 q^{79} +(0.517638 + 0.896575i) q^{80} +(4.50000 - 7.79423i) q^{81} +(4.31199 - 7.46859i) q^{82} +(-3.29530 - 5.70762i) q^{83} +(-3.53590 + 6.12436i) q^{85} +(-0.133975 - 0.232051i) q^{86} +(-4.89898 + 4.89898i) q^{87} +(0.133975 - 0.232051i) q^{88} +(-3.53553 + 6.12372i) q^{89} +(-2.68973 - 1.55291i) q^{90} +(-2.73205 - 4.73205i) q^{92} +(3.00000 + 11.1962i) q^{93} +0.757875 q^{94} -4.53590 q^{95} +(0.448288 + 1.67303i) q^{96} +(9.07227 + 15.7136i) q^{97} +0.803848i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{11} - 24 q^{15} + 8 q^{16} + 8 q^{22} - 8 q^{23} - 12 q^{25} + 16 q^{29} - 24 q^{30} + 8 q^{32} - 16 q^{37} - 24 q^{39} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 12 q^{50}+ \cdots - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.448288 + 1.67303i 0.258819 + 0.965926i
\(4\) 1.00000 0.500000
\(5\) 0.517638 + 0.896575i 0.231495 + 0.400961i 0.958248 − 0.285938i \(-0.0923050\pi\)
−0.726753 + 0.686898i \(0.758972\pi\)
\(6\) 0.448288 + 1.67303i 0.183013 + 0.683013i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.59808 + 1.50000i −0.866025 + 0.500000i
\(10\) 0.517638 + 0.896575i 0.163692 + 0.283522i
\(11\) 0.133975 − 0.232051i 0.0403949 − 0.0699660i −0.845121 − 0.534575i \(-0.820472\pi\)
0.885516 + 0.464609i \(0.153805\pi\)
\(12\) 0.448288 + 1.67303i 0.129410 + 0.482963i
\(13\) −0.896575 + 1.55291i −0.248665 + 0.430701i −0.963156 − 0.268944i \(-0.913325\pi\)
0.714490 + 0.699645i \(0.246659\pi\)
\(14\) 0 0
\(15\) −1.26795 + 1.26795i −0.327383 + 0.327383i
\(16\) 1.00000 0.250000
\(17\) 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
\(23\) −2.73205 − 4.73205i −0.569672 − 0.986701i −0.996598 − 0.0824143i \(-0.973737\pi\)
0.426926 − 0.904286i \(-0.359596\pi\)
\(24\) 0.448288 + 1.67303i 0.0915064 + 0.341506i
\(25\) 1.96410 − 3.40192i 0.392820 − 0.680385i
\(26\) −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 0.989780 − 0.142605i \(-0.0455477\pi\)
−0.618389 + 0.785872i \(0.712214\pi\)
\(30\) −1.26795 + 1.26795i −0.231495 + 0.231495i
\(31\) 6.69213 1.20194 0.600971 − 0.799271i \(-0.294781\pi\)
0.600971 + 0.799271i \(0.294781\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.448288 + 0.120118i 0.0780369 + 0.0209099i
\(34\) 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 \(0.956409\pi\)
\(38\) −2.19067 + 3.79435i −0.355374 + 0.615525i
\(39\) −3.00000 − 0.803848i −0.480384 − 0.128719i
\(40\) 0.517638 + 0.896575i 0.0818458 + 0.141761i
\(41\) 4.31199 − 7.46859i 0.673420 − 1.16640i −0.303508 − 0.952829i \(-0.598158\pi\)
0.976928 − 0.213569i \(-0.0685087\pi\)
\(42\) 0 0
\(43\) −0.133975 − 0.232051i −0.0204309 − 0.0353874i 0.855629 − 0.517589i \(-0.173170\pi\)
−0.876060 + 0.482202i \(0.839837\pi\)
\(44\) 0.133975 − 0.232051i 0.0201974 − 0.0349830i
\(45\) −2.68973 − 1.55291i −0.400961 − 0.231495i
\(46\) −2.73205 − 4.73205i −0.402819 − 0.697703i
\(47\) 0.757875 0.110547 0.0552737 − 0.998471i \(-0.482397\pi\)
0.0552737 + 0.998471i \(0.482397\pi\)
\(48\) 0.448288 + 1.67303i 0.0647048 + 0.241481i
\(49\) 0 0
\(50\) 1.96410 − 3.40192i 0.277766 − 0.481105i
\(51\) −8.36603 + 8.36603i −1.17148 + 1.17148i
\(52\) −0.896575 + 1.55291i −0.124333 + 0.215350i
\(53\) −5.46410 − 9.46410i −0.750552 − 1.29999i −0.947555 − 0.319592i \(-0.896454\pi\)
0.197003 − 0.980403i \(-0.436879\pi\)
\(54\) −3.67423 − 3.67423i −0.500000 − 0.500000i
\(55\) 0.277401 0.0374048
\(56\) 0 0
\(57\) −7.33013 − 1.96410i −0.970899 − 0.260152i
\(58\) 2.00000 + 3.46410i 0.262613 + 0.454859i
\(59\) −1.27551 −0.166058 −0.0830288 − 0.996547i \(-0.526459\pi\)
−0.0830288 + 0.996547i \(0.526459\pi\)
\(60\) −1.26795 + 1.26795i −0.163692 + 0.163692i
\(61\) −12.6264 −1.61664 −0.808322 − 0.588741i \(-0.799624\pi\)
−0.808322 + 0.588741i \(0.799624\pi\)
\(62\) 6.69213 0.849901
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.85641 −0.230259
\(66\) 0.448288 + 0.120118i 0.0551804 + 0.0147855i
\(67\) 12.4641 1.52273 0.761366 − 0.648322i \(-0.224529\pi\)
0.761366 + 0.648322i \(0.224529\pi\)
\(68\) 3.41542 + 5.91567i 0.414180 + 0.717381i
\(69\) 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 \(0.310189\pi\)
\(72\) −2.59808 + 1.50000i −0.306186 + 0.176777i
\(73\) −2.70831 − 4.69093i −0.316984 − 0.549032i 0.662874 − 0.748731i \(-0.269337\pi\)
−0.979857 + 0.199700i \(0.936003\pi\)
\(74\) −3.73205 + 6.46410i −0.433842 + 0.751437i
\(75\) 6.57201 + 1.76097i 0.758871 + 0.203339i
\(76\) −2.19067 + 3.79435i −0.251287 + 0.435242i
\(77\) 0 0
\(78\) −3.00000 − 0.803848i −0.339683 − 0.0910178i
\(79\) 8.92820 1.00450 0.502251 − 0.864722i \(-0.332505\pi\)
0.502251 + 0.864722i \(0.332505\pi\)
\(80\) 0.517638 + 0.896575i 0.0578737 + 0.100240i
\(81\) 4.50000 − 7.79423i 0.500000 − 0.866025i
\(82\) 4.31199 − 7.46859i 0.476180 − 0.824768i
\(83\) −3.29530 − 5.70762i −0.361706 − 0.626493i 0.626536 − 0.779393i \(-0.284472\pi\)
−0.988242 + 0.152900i \(0.951139\pi\)
\(84\) 0 0
\(85\) −3.53590 + 6.12436i −0.383522 + 0.664280i
\(86\) −0.133975 − 0.232051i −0.0144469 − 0.0250227i
\(87\) −4.89898 + 4.89898i −0.525226 + 0.525226i
\(88\) 0.133975 − 0.232051i 0.0142817 − 0.0247367i
\(89\) −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
\(93\) 3.00000 + 11.1962i 0.311086 + 1.16099i
\(94\) 0.757875 0.0781688
\(95\) −4.53590 −0.465373
\(96\) 0.448288 + 1.67303i 0.0457532 + 0.170753i
\(97\) 9.07227 + 15.7136i 0.921149 + 1.59548i 0.797640 + 0.603134i \(0.206082\pi\)
0.123510 + 0.992343i \(0.460585\pi\)
\(98\) 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.e.s.373.3 8
3.2 odd 2 2646.2.e.q.1549.2 8
7.2 even 3 882.2.f.q.589.1 yes 8
7.3 odd 6 882.2.h.q.67.2 8
7.4 even 3 882.2.h.q.67.3 8
7.5 odd 6 882.2.f.q.589.4 yes 8
7.6 odd 2 inner 882.2.e.s.373.2 8
9.2 odd 6 2646.2.h.t.667.3 8
9.7 even 3 882.2.h.q.79.4 8
21.2 odd 6 2646.2.f.r.1765.2 8
21.5 even 6 2646.2.f.r.1765.3 8
21.11 odd 6 2646.2.h.t.361.3 8
21.17 even 6 2646.2.h.t.361.2 8
21.20 even 2 2646.2.e.q.1549.3 8
63.2 odd 6 2646.2.f.r.883.2 8
63.5 even 6 7938.2.a.ci.1.2 4
63.11 odd 6 2646.2.e.q.2125.2 8
63.16 even 3 882.2.f.q.295.1 ✓ 8
63.20 even 6 2646.2.h.t.667.2 8
63.23 odd 6 7938.2.a.ci.1.3 4
63.25 even 3 inner 882.2.e.s.655.3 8
63.34 odd 6 882.2.h.q.79.1 8
63.38 even 6 2646.2.e.q.2125.3 8
63.40 odd 6 7938.2.a.cp.1.3 4
63.47 even 6 2646.2.f.r.883.3 8
63.52 odd 6 inner 882.2.e.s.655.2 8
63.58 even 3 7938.2.a.cp.1.2 4
63.61 odd 6 882.2.f.q.295.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.2 8 7.6 odd 2 inner
882.2.e.s.373.3 8 1.1 even 1 trivial
882.2.e.s.655.2 8 63.52 odd 6 inner
882.2.e.s.655.3 8 63.25 even 3 inner
882.2.f.q.295.1 ✓ 8 63.16 even 3
882.2.f.q.295.4 yes 8 63.61 odd 6
882.2.f.q.589.1 yes 8 7.2 even 3
882.2.f.q.589.4 yes 8 7.5 odd 6
882.2.h.q.67.2 8 7.3 odd 6
882.2.h.q.67.3 8 7.4 even 3
882.2.h.q.79.1 8 63.34 odd 6
882.2.h.q.79.4 8 9.7 even 3
2646.2.e.q.1549.2 8 3.2 odd 2
2646.2.e.q.1549.3 8 21.20 even 2
2646.2.e.q.2125.2 8 63.11 odd 6
2646.2.e.q.2125.3 8 63.38 even 6
2646.2.f.r.883.2 8 63.2 odd 6
2646.2.f.r.883.3 8 63.47 even 6
2646.2.f.r.1765.2 8 21.2 odd 6
2646.2.f.r.1765.3 8 21.5 even 6
2646.2.h.t.361.2 8 21.17 even 6
2646.2.h.t.361.3 8 21.11 odd 6
2646.2.h.t.667.2 8 63.20 even 6
2646.2.h.t.667.3 8 9.2 odd 6
7938.2.a.ci.1.2 4 63.5 even 6
7938.2.a.ci.1.3 4 63.23 odd 6
7938.2.a.cp.1.2 4 63.58 even 3
7938.2.a.cp.1.3 4 63.40 odd 6