↑

Properties

Label 2646.2.h.n.361.1
Level $2646$
Weight $2$
Character 2646.361
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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(361,2646)] 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("2646.361"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,4,0,0,-4,0,2,-8,0,0] 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2646.361
Dual form 2646.2.h.n.667.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.44949 q^{5} -1.00000 q^{8} +(-0.724745 + 1.25529i) q^{10} -2.00000 q^{11} +(2.44949 - 4.24264i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(1.27526 + 2.20881i) q^{19} +(0.724745 + 1.25529i) q^{20} +(-1.00000 + 1.73205i) q^{22} +1.00000 q^{23} -2.89898 q^{25} +(-2.44949 - 4.24264i) q^{26} +(-3.44949 - 5.97469i) q^{29} +(3.00000 + 5.19615i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.00000 + 1.73205i) q^{34} +(-5.89898 - 10.2173i) q^{37} +2.55051 q^{38} +1.44949 q^{40} +(-4.89898 + 8.48528i) q^{41} +(-3.44949 - 5.97469i) q^{43} +(1.00000 + 1.73205i) q^{44} +(0.500000 - 0.866025i) q^{46} +(-4.89898 + 8.48528i) q^{47} +(-1.44949 + 2.51059i) q^{50} -4.89898 q^{52} +(-5.44949 + 9.43879i) q^{53} +2.89898 q^{55} -6.89898 q^{58} +(1.00000 + 1.73205i) q^{59} +(3.27526 - 5.67291i) q^{61} +6.00000 q^{62} +1.00000 q^{64} +(-3.55051 + 6.14966i) q^{65} +(6.44949 + 11.1708i) q^{67} +2.00000 q^{68} -0.101021 q^{71} +(-3.44949 + 5.97469i) q^{73} -11.7980 q^{74} +(1.27526 - 2.20881i) q^{76} +(0.949490 - 1.64456i) q^{79} +(0.724745 - 1.25529i) q^{80} +(4.89898 + 8.48528i) q^{82} +(-1.00000 - 1.73205i) q^{83} +(1.44949 - 2.51059i) q^{85} -6.89898 q^{86} +2.00000 q^{88} +(8.44949 + 14.6349i) q^{89} +(-0.500000 - 0.866025i) q^{92} +(4.89898 + 8.48528i) q^{94} +(-1.84847 - 3.20164i) q^{95} +(1.44949 + 2.51059i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 4 q^{8} + 2 q^{10} - 8 q^{11} - 2 q^{16} - 4 q^{17} + 10 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} + 8 q^{25} - 4 q^{29} + 12 q^{31} + 2 q^{32} + 4 q^{34} - 4 q^{37} + 20 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.44949 −0.648232 −0.324116 − 0.946017i \(-0.605067\pi\)
−0.324116 + 0.946017i \(0.605067\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.724745 + 1.25529i −0.229184 + 0.396959i
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.44949 − 4.24264i 0.679366 − 1.17670i −0.295806 − 0.955248i \(-0.595588\pi\)
0.975172 − 0.221449i \(-0.0710785\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 − 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) 1.27526 + 2.20881i 0.292564 + 0.506735i 0.974415 − 0.224756i \(-0.0721584\pi\)
−0.681852 + 0.731491i \(0.738825\pi\)
\(20\) 0.724745 + 1.25529i 0.162058 + 0.280692i
\(21\) 0 0
\(22\) −1.00000 + 1.73205i −0.213201 + 0.369274i
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −2.89898 −0.579796
\(26\) −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
\(31\) 3.00000 + 5.19615i 0.538816 + 0.933257i 0.998968 + 0.0454165i \(0.0144615\pi\)
−0.460152 + 0.887840i \(0.652205\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 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 −0.696165 − 0.717881i \(-0.745112\pi\)
−0.273621 − 0.961838i \(-0.588221\pi\)
\(38\) 2.55051 0.413747
\(39\) 0 0
\(40\) 1.44949 0.229184
\(41\) −4.89898 + 8.48528i −0.765092 + 1.32518i 0.175106 + 0.984550i \(0.443973\pi\)
−0.940198 + 0.340629i \(0.889360\pi\)
\(42\) 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
\(47\) −4.89898 + 8.48528i −0.714590 + 1.23771i 0.248528 + 0.968625i \(0.420053\pi\)
−0.963118 + 0.269081i \(0.913280\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.44949 + 2.51059i −0.204989 + 0.355051i
\(51\) 0 0
\(52\) −4.89898 −0.679366
\(53\) −5.44949 + 9.43879i −0.748545 + 1.29652i 0.199975 + 0.979801i \(0.435914\pi\)
−0.948520 + 0.316717i \(0.897419\pi\)
\(54\) 0 0
\(55\) 2.89898 0.390898
\(56\) 0 0
\(57\) 0 0
\(58\) −6.89898 −0.905880
\(59\) 1.00000 + 1.73205i 0.130189 + 0.225494i 0.923749 − 0.382998i \(-0.125108\pi\)
−0.793560 + 0.608492i \(0.791775\pi\)
\(60\) 0 0
\(61\) 3.27526 − 5.67291i 0.419353 − 0.726341i −0.576521 − 0.817082i \(-0.695590\pi\)
0.995875 + 0.0907408i \(0.0289235\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.55051 + 6.14966i −0.440387 + 0.762772i
\(66\) 0 0
\(67\) 6.44949 + 11.1708i 0.787931 + 1.36474i 0.927233 + 0.374486i \(0.122181\pi\)
−0.139302 + 0.990250i \(0.544486\pi\)
\(68\) 2.00000 0.242536
\(69\) 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 \(-0.965621\pi\)
0.590441 + 0.807081i \(0.298954\pi\)
\(74\) −11.7980 −1.37148
\(75\) 0 0
\(76\) 1.27526 − 2.20881i 0.146282 − 0.253368i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.949490 − 1.64456i 0.106826 − 0.185028i −0.807657 − 0.589653i \(-0.799265\pi\)
0.914483 + 0.404625i \(0.132598\pi\)
\(80\) 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 0.805910 − 0.592037i \(-0.201676\pi\)
−0.915675 + 0.401920i \(0.868343\pi\)
\(84\) 0 0
\(85\) 1.44949 − 2.51059i 0.157219 − 0.272312i
\(86\) −6.89898 −0.743936
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 8.44949 + 14.6349i 0.895644 + 1.55130i 0.833005 + 0.553265i \(0.186618\pi\)
0.0626387 + 0.998036i \(0.480048\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.500000 − 0.866025i −0.0521286 − 0.0902894i
\(93\) 0 0
\(94\) 4.89898 + 8.48528i 0.505291 + 0.875190i
\(95\) −1.84847 − 3.20164i −0.189649 − 0.328482i
\(96\) 0 0
\(97\) 1.44949 + 2.51059i 0.147173 + 0.254912i 0.930182 − 0.367099i \(-0.119649\pi\)
−0.783008 + 0.622011i \(0.786316\pi\)
\(98\) 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.h.n.361.1 4
3.2 odd 2 882.2.h.l.67.1 4
7.2 even 3 2646.2.e.k.1549.2 4
7.3 odd 6 378.2.f.d.253.1 4
7.4 even 3 2646.2.f.k.1765.2 4
7.5 odd 6 2646.2.e.l.1549.1 4
7.6 odd 2 2646.2.h.m.361.2 4
9.2 odd 6 882.2.e.n.655.2 4
9.7 even 3 2646.2.e.k.2125.2 4
21.2 odd 6 882.2.e.n.373.2 4
21.5 even 6 882.2.e.m.373.1 4
21.11 odd 6 882.2.f.j.589.2 4
21.17 even 6 126.2.f.c.85.1 yes 4
21.20 even 2 882.2.h.k.67.2 4
28.3 even 6 3024.2.r.e.1009.1 4
63.2 odd 6 882.2.h.l.79.1 4
63.4 even 3 7938.2.a.bm.1.1 2
63.11 odd 6 882.2.f.j.295.1 4
63.16 even 3 inner 2646.2.h.n.667.1 4
63.20 even 6 882.2.e.m.655.1 4
63.25 even 3 2646.2.f.k.883.2 4
63.31 odd 6 1134.2.a.i.1.2 2
63.32 odd 6 7938.2.a.bn.1.2 2
63.34 odd 6 2646.2.e.l.2125.1 4
63.38 even 6 126.2.f.c.43.2 ✓ 4
63.47 even 6 882.2.h.k.79.2 4
63.52 odd 6 378.2.f.d.127.1 4
63.59 even 6 1134.2.a.p.1.1 2
63.61 odd 6 2646.2.h.m.667.2 4
84.59 odd 6 1008.2.r.e.337.2 4
252.31 even 6 9072.2.a.bd.1.2 2
252.59 odd 6 9072.2.a.bk.1.1 2
252.115 even 6 3024.2.r.e.2017.1 4
252.227 odd 6 1008.2.r.e.673.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.2 ✓ 4 63.38 even 6
126.2.f.c.85.1 yes 4 21.17 even 6
378.2.f.d.127.1 4 63.52 odd 6
378.2.f.d.253.1 4 7.3 odd 6
882.2.e.m.373.1 4 21.5 even 6
882.2.e.m.655.1 4 63.20 even 6
882.2.e.n.373.2 4 21.2 odd 6
882.2.e.n.655.2 4 9.2 odd 6
882.2.f.j.295.1 4 63.11 odd 6
882.2.f.j.589.2 4 21.11 odd 6
882.2.h.k.67.2 4 21.20 even 2
882.2.h.k.79.2 4 63.47 even 6
882.2.h.l.67.1 4 3.2 odd 2
882.2.h.l.79.1 4 63.2 odd 6
1008.2.r.e.337.2 4 84.59 odd 6
1008.2.r.e.673.1 4 252.227 odd 6
1134.2.a.i.1.2 2 63.31 odd 6
1134.2.a.p.1.1 2 63.59 even 6
2646.2.e.k.1549.2 4 7.2 even 3
2646.2.e.k.2125.2 4 9.7 even 3
2646.2.e.l.1549.1 4 7.5 odd 6
2646.2.e.l.2125.1 4 63.34 odd 6
2646.2.f.k.883.2 4 63.25 even 3
2646.2.f.k.1765.2 4 7.4 even 3
2646.2.h.m.361.2 4 7.6 odd 2
2646.2.h.m.667.2 4 63.61 odd 6
2646.2.h.n.361.1 4 1.1 even 1 trivial
2646.2.h.n.667.1 4 63.16 even 3 inner
3024.2.r.e.1009.1 4 28.3 even 6
3024.2.r.e.2017.1 4 252.115 even 6
7938.2.a.bm.1.1 2 63.4 even 3
7938.2.a.bn.1.2 2 63.32 odd 6
9072.2.a.bd.1.2 2 252.31 even 6
9072.2.a.bk.1.1 2 252.59 odd 6