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Properties

Label 882.2.f.s.589.1
Level $882$
Weight $2$
Character 882.589
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(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 = [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 589.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 882.589
Dual form 882.2.f.s.295.2

$q$-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+(0.500000 + 0.866025i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.258819 + 0.448288i) q^{5} +(0.448288 - 1.67303i) q^{6} -1.00000 q^{8} +3.00000i q^{9} -0.517638 q^{10} +(0.732051 + 1.26795i) q^{11} +(1.67303 - 0.448288i) q^{12} +(1.22474 - 2.12132i) q^{13} +(0.866025 - 0.232051i) q^{15} +(-0.500000 - 0.866025i) q^{16} -3.48477 q^{17} +(-2.59808 + 1.50000i) q^{18} +0.517638 q^{19} +(-0.258819 - 0.448288i) q^{20} +(-0.732051 + 1.26795i) q^{22} +(-3.96410 + 6.86603i) q^{23} +(1.22474 + 1.22474i) q^{24} +(2.36603 + 4.09808i) q^{25} +2.44949 q^{26} +(3.67423 - 3.67423i) q^{27} +(-1.36603 - 2.36603i) q^{29} +(0.633975 + 0.633975i) q^{30} +(-3.67423 + 6.36396i) q^{31} +(0.500000 - 0.866025i) q^{32} +(0.656339 - 2.44949i) q^{33} +(-1.74238 - 3.01790i) q^{34} +(-2.59808 - 1.50000i) q^{36} -8.00000 q^{37} +(0.258819 + 0.448288i) q^{38} +(-4.09808 + 1.09808i) q^{39} +(0.258819 - 0.448288i) q^{40} +(-2.82843 + 4.89898i) q^{41} +(6.09808 + 10.5622i) q^{43} -1.46410 q^{44} +(-1.34486 - 0.776457i) q^{45} -7.92820 q^{46} +(2.31079 + 4.00240i) q^{47} +(-0.448288 + 1.67303i) q^{48} +(-2.36603 + 4.09808i) q^{50} +(4.26795 + 4.26795i) q^{51} +(1.22474 + 2.12132i) q^{52} +6.73205 q^{53} +(5.01910 + 1.34486i) q^{54} -0.757875 q^{55} +(-0.633975 - 0.633975i) q^{57} +(1.36603 - 2.36603i) q^{58} +(-7.39924 + 12.8159i) q^{59} +(-0.232051 + 0.866025i) q^{60} +(-2.19067 - 3.79435i) q^{61} -7.34847 q^{62} +1.00000 q^{64} +(0.633975 + 1.09808i) q^{65} +(2.44949 - 0.656339i) q^{66} +(1.90192 - 3.29423i) q^{67} +(1.74238 - 3.01790i) q^{68} +(13.2641 - 3.55412i) q^{69} -0.803848 q^{71} -3.00000i q^{72} +4.62158 q^{73} +(-4.00000 - 6.92820i) q^{74} +(2.12132 - 7.91688i) q^{75} +(-0.258819 + 0.448288i) q^{76} +(-3.00000 - 3.00000i) q^{78} +(-7.06218 - 12.2321i) q^{79} +0.517638 q^{80} -9.00000 q^{81} -5.65685 q^{82} +(4.94975 + 8.57321i) q^{83} +(0.901924 - 1.56218i) q^{85} +(-6.09808 + 10.5622i) q^{86} +(-1.22474 + 4.57081i) q^{87} +(-0.732051 - 1.26795i) q^{88} -16.1112 q^{89} -1.55291i q^{90} +(-3.96410 - 6.86603i) q^{92} +(12.2942 - 3.29423i) q^{93} +(-2.31079 + 4.00240i) q^{94} +(-0.133975 + 0.232051i) q^{95} +(-1.67303 + 0.448288i) q^{96} +(0.517638 + 0.896575i) q^{97} +(-3.80385 + 2.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 8 q^{11} - 4 q^{16} + 8 q^{22} - 4 q^{23} + 12 q^{25} - 4 q^{29} + 12 q^{30} + 4 q^{32} - 64 q^{37} - 12 q^{39} + 28 q^{43} + 16 q^{44} - 8 q^{46} - 12 q^{50} + 48 q^{51}+ \cdots - 72 q^{99}+O(q^{100}) \) 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
\(3\) −1.22474 − 1.22474i −0.707107 − 0.707107i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.258819 + 0.448288i −0.115747 + 0.200480i −0.918078 − 0.396399i \(-0.870260\pi\)
0.802331 + 0.596880i \(0.203593\pi\)
\(6\) 0.448288 − 1.67303i 0.183013 − 0.683013i
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 3.00000i 1.00000i
\(10\) −0.517638 −0.163692
\(11\) 0.732051 + 1.26795i 0.220722 + 0.382301i 0.955027 − 0.296518i \(-0.0958254\pi\)
−0.734306 + 0.678819i \(0.762492\pi\)
\(12\) 1.67303 − 0.448288i 0.482963 − 0.129410i
\(13\) 1.22474 − 2.12132i 0.339683 − 0.588348i −0.644690 − 0.764444i \(-0.723014\pi\)
0.984373 + 0.176096i \(0.0563468\pi\)
\(14\) 0 0
\(15\) 0.866025 − 0.232051i 0.223607 − 0.0599153i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.48477 −0.845180 −0.422590 − 0.906321i \(-0.638879\pi\)
−0.422590 + 0.906321i \(0.638879\pi\)
\(18\) −2.59808 + 1.50000i −0.612372 + 0.353553i
\(19\) 0.517638 0.118754 0.0593772 − 0.998236i \(-0.481089\pi\)
0.0593772 + 0.998236i \(0.481089\pi\)
\(20\) −0.258819 − 0.448288i −0.0578737 − 0.100240i
\(21\) 0 0
\(22\) −0.732051 + 1.26795i −0.156074 + 0.270328i
\(23\) −3.96410 + 6.86603i −0.826572 + 1.43167i 0.0741394 + 0.997248i \(0.476379\pi\)
−0.900712 + 0.434417i \(0.856954\pi\)
\(24\) 1.22474 + 1.22474i 0.250000 + 0.250000i
\(25\) 2.36603 + 4.09808i 0.473205 + 0.819615i
\(26\) 2.44949 0.480384
\(27\) 3.67423 − 3.67423i 0.707107 − 0.707107i
\(28\) 0 0
\(29\) −1.36603 − 2.36603i −0.253665 − 0.439360i 0.710867 − 0.703326i \(-0.248303\pi\)
−0.964532 + 0.263966i \(0.914969\pi\)
\(30\) 0.633975 + 0.633975i 0.115747 + 0.115747i
\(31\) −3.67423 + 6.36396i −0.659912 + 1.14300i 0.320726 + 0.947172i \(0.396073\pi\)
−0.980638 + 0.195829i \(0.937260\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0.656339 − 2.44949i 0.114254 − 0.426401i
\(34\) −1.74238 − 3.01790i −0.298816 − 0.517565i
\(35\) 0 0
\(36\) −2.59808 − 1.50000i −0.433013 − 0.250000i
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0.258819 + 0.448288i 0.0419860 + 0.0727219i
\(39\) −4.09808 + 1.09808i −0.656217 + 0.175833i
\(40\) 0.258819 − 0.448288i 0.0409229 − 0.0708805i
\(41\) −2.82843 + 4.89898i −0.441726 + 0.765092i −0.997818 − 0.0660290i \(-0.978967\pi\)
0.556092 + 0.831121i \(0.312300\pi\)
\(42\) 0 0
\(43\) 6.09808 + 10.5622i 0.929948 + 1.61072i 0.783404 + 0.621513i \(0.213482\pi\)
0.146544 + 0.989204i \(0.453185\pi\)
\(44\) −1.46410 −0.220722
\(45\) −1.34486 − 0.776457i −0.200480 − 0.115747i
\(46\) −7.92820 −1.16895
\(47\) 2.31079 + 4.00240i 0.337063 + 0.583811i 0.983879 − 0.178836i \(-0.0572331\pi\)
−0.646816 + 0.762646i \(0.723900\pi\)
\(48\) −0.448288 + 1.67303i −0.0647048 + 0.241481i
\(49\) 0 0
\(50\) −2.36603 + 4.09808i −0.334607 + 0.579555i
\(51\) 4.26795 + 4.26795i 0.597632 + 0.597632i
\(52\) 1.22474 + 2.12132i 0.169842 + 0.294174i
\(53\) 6.73205 0.924718 0.462359 − 0.886693i \(-0.347003\pi\)
0.462359 + 0.886693i \(0.347003\pi\)
\(54\) 5.01910 + 1.34486i 0.683013 + 0.183013i
\(55\) −0.757875 −0.102192
\(56\) 0 0
\(57\) −0.633975 − 0.633975i −0.0839720 − 0.0839720i
\(58\) 1.36603 − 2.36603i 0.179368 − 0.310674i
\(59\) −7.39924 + 12.8159i −0.963299 + 1.66848i −0.249180 + 0.968457i \(0.580161\pi\)
−0.714118 + 0.700025i \(0.753172\pi\)
\(60\) −0.232051 + 0.866025i −0.0299576 + 0.111803i
\(61\) −2.19067 − 3.79435i −0.280487 − 0.485817i 0.691018 − 0.722838i \(-0.257163\pi\)
−0.971505 + 0.237020i \(0.923829\pi\)
\(62\) −7.34847 −0.933257
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.633975 + 1.09808i 0.0786349 + 0.136200i
\(66\) 2.44949 − 0.656339i 0.301511 − 0.0807897i
\(67\) 1.90192 − 3.29423i 0.232357 − 0.402454i −0.726144 − 0.687542i \(-0.758690\pi\)
0.958501 + 0.285088i \(0.0920229\pi\)
\(68\) 1.74238 − 3.01790i 0.211295 − 0.365974i
\(69\) 13.2641 − 3.55412i 1.59682 − 0.427865i
\(70\) 0 0
\(71\) −0.803848 −0.0953992 −0.0476996 − 0.998862i \(-0.515189\pi\)
−0.0476996 + 0.998862i \(0.515189\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) 4.62158 0.540915 0.270457 − 0.962732i \(-0.412825\pi\)
0.270457 + 0.962732i \(0.412825\pi\)
\(74\) −4.00000 − 6.92820i −0.464991 − 0.805387i
\(75\) 2.12132 − 7.91688i 0.244949 − 0.914162i
\(76\) −0.258819 + 0.448288i −0.0296886 + 0.0514221i
\(77\) 0 0
\(78\) −3.00000 − 3.00000i −0.339683 − 0.339683i
\(79\) −7.06218 − 12.2321i −0.794557 − 1.37621i −0.923120 − 0.384512i \(-0.874370\pi\)
0.128563 − 0.991701i \(-0.458964\pi\)
\(80\) 0.517638 0.0578737
\(81\) −9.00000 −1.00000
\(82\) −5.65685 −0.624695
\(83\) 4.94975 + 8.57321i 0.543305 + 0.941033i 0.998711 + 0.0507487i \(0.0161607\pi\)
−0.455406 + 0.890284i \(0.650506\pi\)
\(84\) 0 0
\(85\) 0.901924 − 1.56218i 0.0978274 − 0.169442i
\(86\) −6.09808 + 10.5622i −0.657572 + 1.13895i
\(87\) −1.22474 + 4.57081i −0.131306 + 0.490042i
\(88\) −0.732051 − 1.26795i −0.0780369 − 0.135164i
\(89\) −16.1112 −1.70778 −0.853889 − 0.520455i \(-0.825763\pi\)
−0.853889 + 0.520455i \(0.825763\pi\)
\(90\) − 1.55291i − 0.163692i
\(91\) 0 0
\(92\) −3.96410 − 6.86603i −0.413286 − 0.715833i
\(93\) 12.2942 − 3.29423i 1.27485 − 0.341596i
\(94\) −2.31079 + 4.00240i −0.238340 + 0.412816i
\(95\) −0.133975 + 0.232051i −0.0137455 + 0.0238079i
\(96\) −1.67303 + 0.448288i −0.170753 + 0.0457532i
\(97\) 0.517638 + 0.896575i 0.0525582 + 0.0910334i 0.891108 − 0.453792i \(-0.149929\pi\)
−0.838549 + 0.544826i \(0.816596\pi\)
\(98\) 0 0
\(99\) −3.80385 + 2.19615i −0.382301 + 0.220722i
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.s.589.1 yes 8
3.2 odd 2 2646.2.f.q.1765.3 8
7.2 even 3 882.2.h.t.67.4 8
7.3 odd 6 882.2.e.q.373.3 8
7.4 even 3 882.2.e.q.373.2 8
7.5 odd 6 882.2.h.t.67.1 8
7.6 odd 2 inner 882.2.f.s.589.4 yes 8
9.2 odd 6 2646.2.f.q.883.3 8
9.4 even 3 7938.2.a.cj.1.3 4
9.5 odd 6 7938.2.a.co.1.2 4
9.7 even 3 inner 882.2.f.s.295.2 ✓ 8
21.2 odd 6 2646.2.h.q.361.2 8
21.5 even 6 2646.2.h.q.361.3 8
21.11 odd 6 2646.2.e.t.1549.3 8
21.17 even 6 2646.2.e.t.1549.2 8
21.20 even 2 2646.2.f.q.1765.2 8
63.2 odd 6 2646.2.e.t.2125.3 8
63.11 odd 6 2646.2.h.q.667.2 8
63.13 odd 6 7938.2.a.cj.1.2 4
63.16 even 3 882.2.e.q.655.2 8
63.20 even 6 2646.2.f.q.883.2 8
63.25 even 3 882.2.h.t.79.4 8
63.34 odd 6 inner 882.2.f.s.295.3 yes 8
63.38 even 6 2646.2.h.q.667.3 8
63.41 even 6 7938.2.a.co.1.3 4
63.47 even 6 2646.2.e.t.2125.2 8
63.52 odd 6 882.2.h.t.79.1 8
63.61 odd 6 882.2.e.q.655.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.q.373.2 8 7.4 even 3
882.2.e.q.373.3 8 7.3 odd 6
882.2.e.q.655.2 8 63.16 even 3
882.2.e.q.655.3 8 63.61 odd 6
882.2.f.s.295.2 ✓ 8 9.7 even 3 inner
882.2.f.s.295.3 yes 8 63.34 odd 6 inner
882.2.f.s.589.1 yes 8 1.1 even 1 trivial
882.2.f.s.589.4 yes 8 7.6 odd 2 inner
882.2.h.t.67.1 8 7.5 odd 6
882.2.h.t.67.4 8 7.2 even 3
882.2.h.t.79.1 8 63.52 odd 6
882.2.h.t.79.4 8 63.25 even 3
2646.2.e.t.1549.2 8 21.17 even 6
2646.2.e.t.1549.3 8 21.11 odd 6
2646.2.e.t.2125.2 8 63.47 even 6
2646.2.e.t.2125.3 8 63.2 odd 6
2646.2.f.q.883.2 8 63.20 even 6
2646.2.f.q.883.3 8 9.2 odd 6
2646.2.f.q.1765.2 8 21.20 even 2
2646.2.f.q.1765.3 8 3.2 odd 2
2646.2.h.q.361.2 8 21.2 odd 6
2646.2.h.q.361.3 8 21.5 even 6
2646.2.h.q.667.2 8 63.11 odd 6
2646.2.h.q.667.3 8 63.38 even 6
7938.2.a.cj.1.2 4 63.13 odd 6
7938.2.a.cj.1.3 4 9.4 even 3
7938.2.a.co.1.2 4 9.5 odd 6
7938.2.a.co.1.3 4 63.41 even 6