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Properties

Label 2646.2.e.l.1549.1
Level $2646$
Weight $2$
Character 2646.1549
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [2646,2,Mod(1549,2646)] 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("2646.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,0,4,2,0,0,-4,0,-2,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.1
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2646.1549
Dual form 2646.2.e.l.2125.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.724745 - 1.25529i) q^{5} -1.00000 q^{8} +(0.724745 + 1.25529i) q^{10} +(1.00000 - 1.73205i) q^{11} +(-2.44949 + 4.24264i) q^{13} +1.00000 q^{16} +(1.00000 + 1.73205i) q^{17} +(-1.27526 + 2.20881i) q^{19} +(-0.724745 - 1.25529i) q^{20} +(-1.00000 + 1.73205i) q^{22} +(-0.500000 - 0.866025i) q^{23} +(1.44949 - 2.51059i) q^{25} +(2.44949 - 4.24264i) q^{26} +(-3.44949 - 5.97469i) q^{29} +6.00000 q^{31} -1.00000 q^{32} +(-1.00000 - 1.73205i) q^{34} +(-5.89898 + 10.2173i) q^{37} +(1.27526 - 2.20881i) q^{38} +(0.724745 + 1.25529i) q^{40} +(4.89898 - 8.48528i) q^{41} +(-3.44949 - 5.97469i) q^{43} +(1.00000 - 1.73205i) q^{44} +(0.500000 + 0.866025i) q^{46} -9.79796 q^{47} +(-1.44949 + 2.51059i) q^{50} +(-2.44949 + 4.24264i) q^{52} +(-5.44949 - 9.43879i) q^{53} -2.89898 q^{55} +(3.44949 + 5.97469i) q^{58} +2.00000 q^{59} +6.55051 q^{61} -6.00000 q^{62} +1.00000 q^{64} +7.10102 q^{65} -12.8990 q^{67} +(1.00000 + 1.73205i) q^{68} -0.101021 q^{71} +(3.44949 + 5.97469i) q^{73} +(5.89898 - 10.2173i) q^{74} +(-1.27526 + 2.20881i) q^{76} -1.89898 q^{79} +(-0.724745 - 1.25529i) q^{80} +(-4.89898 + 8.48528i) q^{82} +(1.00000 + 1.73205i) q^{83} +(1.44949 - 2.51059i) q^{85} +(3.44949 + 5.97469i) q^{86} +(-1.00000 + 1.73205i) q^{88} +(-8.44949 + 14.6349i) q^{89} +(-0.500000 - 0.866025i) q^{92} +9.79796 q^{94} +3.69694 q^{95} +(-1.44949 - 2.51059i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 2 q^{10} + 4 q^{11} + 4 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{29} + 24 q^{31} - 4 q^{32} - 4 q^{34} - 4 q^{37} + 10 q^{38}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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 0
\(4\) 1.00000 0.500000
\(5\) −0.724745 − 1.25529i −0.324116 − 0.561385i 0.657217 − 0.753701i \(-0.271733\pi\)
−0.981333 + 0.192316i \(0.938400\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.724745 + 1.25529i 0.229184 + 0.396959i
\(11\) 1.00000 − 1.73205i 0.301511 − 0.522233i −0.674967 − 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) −2.44949 + 4.24264i −0.679366 + 1.17670i 0.295806 + 0.955248i \(0.404412\pi\)
−0.975172 + 0.221449i \(0.928921\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 − 0.275029i \(-0.0886875\pi\)
−0.718900 + 0.695113i \(0.755354\pi\)
\(18\) 0 0
\(19\) −1.27526 + 2.20881i −0.292564 + 0.506735i −0.974415 − 0.224756i \(-0.927842\pi\)
0.681852 + 0.731491i \(0.261175\pi\)
\(20\) −0.724745 − 1.25529i −0.162058 − 0.280692i
\(21\) 0 0
\(22\) −1.00000 + 1.73205i −0.213201 + 0.369274i
\(23\) −0.500000 − 0.866025i −0.104257 − 0.180579i 0.809177 − 0.587565i \(-0.199913\pi\)
−0.913434 + 0.406986i \(0.866580\pi\)
\(24\) 0 0
\(25\) 1.44949 − 2.51059i 0.289898 − 0.502118i
\(26\) 2.44949 − 4.24264i 0.480384 − 0.832050i
\(27\) 0 0
\(28\) 0 0
\(29\) −3.44949 − 5.97469i −0.640554 − 1.10947i −0.985309 − 0.170780i \(-0.945371\pi\)
0.344755 − 0.938693i \(-0.387962\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 − 1.73205i −0.171499 − 0.297044i
\(35\) 0 0
\(36\) 0 0
\(37\) −5.89898 + 10.2173i −0.969786 + 1.67972i −0.273621 + 0.961838i \(0.588221\pi\)
−0.696165 + 0.717881i \(0.745112\pi\)
\(38\) 1.27526 − 2.20881i 0.206874 − 0.358316i
\(39\) 0 0
\(40\) 0.724745 + 1.25529i 0.114592 + 0.198480i
\(41\) 4.89898 − 8.48528i 0.765092 − 1.32518i −0.175106 − 0.984550i \(-0.556027\pi\)
0.940198 − 0.340629i \(-0.110640\pi\)
\(42\) 0 0
\(43\) −3.44949 − 5.97469i −0.526042 − 0.911132i −0.999540 − 0.0303367i \(-0.990342\pi\)
0.473497 − 0.880795i \(-0.342991\pi\)
\(44\) 1.00000 − 1.73205i 0.150756 − 0.261116i
\(45\) 0 0
\(46\) 0.500000 + 0.866025i 0.0737210 + 0.127688i
\(47\) −9.79796 −1.42918 −0.714590 − 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.44949 + 2.51059i −0.204989 + 0.355051i
\(51\) 0 0
\(52\) −2.44949 + 4.24264i −0.339683 + 0.588348i
\(53\) −5.44949 − 9.43879i −0.748545 − 1.29652i −0.948520 − 0.316717i \(-0.897419\pi\)
0.199975 − 0.979801i \(-0.435914\pi\)
\(54\) 0 0
\(55\) −2.89898 −0.390898
\(56\) 0 0
\(57\) 0 0
\(58\) 3.44949 + 5.97469i 0.452940 + 0.784515i
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 6.55051 0.838707 0.419353 − 0.907823i \(-0.362257\pi\)
0.419353 + 0.907823i \(0.362257\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.10102 0.880773
\(66\) 0 0
\(67\) −12.8990 −1.57586 −0.787931 − 0.615764i \(-0.788847\pi\)
−0.787931 + 0.615764i \(0.788847\pi\)
\(68\) 1.00000 + 1.73205i 0.121268 + 0.210042i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.101021 −0.0119889 −0.00599446 − 0.999982i \(-0.501908\pi\)
−0.00599446 + 0.999982i \(0.501908\pi\)
\(72\) 0 0
\(73\) 3.44949 + 5.97469i 0.403732 + 0.699285i 0.994173 − 0.107796i \(-0.0343794\pi\)
−0.590441 + 0.807081i \(0.701046\pi\)
\(74\) 5.89898 − 10.2173i 0.685742 − 1.18774i
\(75\) 0 0
\(76\) −1.27526 + 2.20881i −0.146282 + 0.253368i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.89898 −0.213652 −0.106826 − 0.994278i \(-0.534069\pi\)
−0.106826 + 0.994278i \(0.534069\pi\)
\(80\) −0.724745 − 1.25529i −0.0810289 − 0.140346i
\(81\) 0 0
\(82\) −4.89898 + 8.48528i −0.541002 + 0.937043i
\(83\) 1.00000 + 1.73205i 0.109764 + 0.190117i 0.915675 − 0.401920i \(-0.131657\pi\)
−0.805910 + 0.592037i \(0.798324\pi\)
\(84\) 0 0
\(85\) 1.44949 − 2.51059i 0.157219 − 0.272312i
\(86\) 3.44949 + 5.97469i 0.371968 + 0.644268i
\(87\) 0 0
\(88\) −1.00000 + 1.73205i −0.106600 + 0.184637i
\(89\) −8.44949 + 14.6349i −0.895644 + 1.55130i −0.0626387 + 0.998036i \(0.519952\pi\)
−0.833005 + 0.553265i \(0.813382\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.500000 − 0.866025i −0.0521286 − 0.0902894i
\(93\) 0 0
\(94\) 9.79796 1.01058
\(95\) 3.69694 0.379298
\(96\) 0 0
\(97\) −1.44949 − 2.51059i −0.147173 − 0.254912i 0.783008 − 0.622011i \(-0.213684\pi\)
−0.930182 + 0.367099i \(0.880351\pi\)
\(98\) 0 0
\(99\) 0 0
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 2646.2.e.l.1549.1 4
3.2 odd 2 882.2.e.m.373.1 4
7.2 even 3 378.2.f.d.253.1 4
7.3 odd 6 2646.2.h.n.361.1 4
7.4 even 3 2646.2.h.m.361.2 4
7.5 odd 6 2646.2.f.k.1765.2 4
7.6 odd 2 2646.2.e.k.1549.2 4
9.2 odd 6 882.2.h.k.79.2 4
9.7 even 3 2646.2.h.m.667.2 4
21.2 odd 6 126.2.f.c.85.1 yes 4
21.5 even 6 882.2.f.j.589.2 4
21.11 odd 6 882.2.h.k.67.2 4
21.17 even 6 882.2.h.l.67.1 4
21.20 even 2 882.2.e.n.373.2 4
28.23 odd 6 3024.2.r.e.1009.1 4
63.2 odd 6 126.2.f.c.43.2 ✓ 4
63.5 even 6 7938.2.a.bn.1.2 2
63.11 odd 6 882.2.e.m.655.1 4
63.16 even 3 378.2.f.d.127.1 4
63.20 even 6 882.2.h.l.79.1 4
63.23 odd 6 1134.2.a.p.1.1 2
63.25 even 3 inner 2646.2.e.l.2125.1 4
63.34 odd 6 2646.2.h.n.667.1 4
63.38 even 6 882.2.e.n.655.2 4
63.40 odd 6 7938.2.a.bm.1.1 2
63.47 even 6 882.2.f.j.295.1 4
63.52 odd 6 2646.2.e.k.2125.2 4
63.58 even 3 1134.2.a.i.1.2 2
63.61 odd 6 2646.2.f.k.883.2 4
84.23 even 6 1008.2.r.e.337.2 4
252.23 even 6 9072.2.a.bk.1.1 2
252.79 odd 6 3024.2.r.e.2017.1 4
252.191 even 6 1008.2.r.e.673.1 4
252.247 odd 6 9072.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.2 ✓ 4 63.2 odd 6
126.2.f.c.85.1 yes 4 21.2 odd 6
378.2.f.d.127.1 4 63.16 even 3
378.2.f.d.253.1 4 7.2 even 3
882.2.e.m.373.1 4 3.2 odd 2
882.2.e.m.655.1 4 63.11 odd 6
882.2.e.n.373.2 4 21.20 even 2
882.2.e.n.655.2 4 63.38 even 6
882.2.f.j.295.1 4 63.47 even 6
882.2.f.j.589.2 4 21.5 even 6
882.2.h.k.67.2 4 21.11 odd 6
882.2.h.k.79.2 4 9.2 odd 6
882.2.h.l.67.1 4 21.17 even 6
882.2.h.l.79.1 4 63.20 even 6
1008.2.r.e.337.2 4 84.23 even 6
1008.2.r.e.673.1 4 252.191 even 6
1134.2.a.i.1.2 2 63.58 even 3
1134.2.a.p.1.1 2 63.23 odd 6
2646.2.e.k.1549.2 4 7.6 odd 2
2646.2.e.k.2125.2 4 63.52 odd 6
2646.2.e.l.1549.1 4 1.1 even 1 trivial
2646.2.e.l.2125.1 4 63.25 even 3 inner
2646.2.f.k.883.2 4 63.61 odd 6
2646.2.f.k.1765.2 4 7.5 odd 6
2646.2.h.m.361.2 4 7.4 even 3
2646.2.h.m.667.2 4 9.7 even 3
2646.2.h.n.361.1 4 7.3 odd 6
2646.2.h.n.667.1 4 63.34 odd 6
3024.2.r.e.1009.1 4 28.23 odd 6
3024.2.r.e.2017.1 4 252.79 odd 6
7938.2.a.bm.1.1 2 63.40 odd 6
7938.2.a.bn.1.2 2 63.5 even 6
9072.2.a.bd.1.2 2 252.247 odd 6
9072.2.a.bk.1.1 2 252.23 even 6