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Properties

Label 2646.2.h.t.361.3
Level $2646$
Weight $2$
Character 2646.361
Analytic conductor $21.128$
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] = [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 = [8,4,0,-4,0,0,0,-8,0,0,16,0,0] 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: \(8\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 882)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.3
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2646.361
Dual form 2646.2.h.t.667.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.03528 q^{5} -1.00000 q^{8} +(0.517638 - 0.896575i) q^{10} +0.267949 q^{11} +(-0.896575 + 1.55291i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-3.41542 + 5.91567i) q^{17} +(-2.19067 - 3.79435i) q^{19} +(-0.517638 - 0.896575i) q^{20} +(0.133975 - 0.232051i) q^{22} -5.46410 q^{23} -3.92820 q^{25} +(0.896575 + 1.55291i) q^{26} +(-2.00000 - 3.46410i) q^{29} +(-3.34607 - 5.79555i) q^{31} +(0.500000 + 0.866025i) q^{32} +(3.41542 + 5.91567i) q^{34} +(-3.73205 - 6.46410i) q^{37} -4.38134 q^{38} -1.03528 q^{40} +(-4.31199 + 7.46859i) q^{41} +(-0.133975 - 0.232051i) q^{43} +(-0.133975 - 0.232051i) q^{44} +(-2.73205 + 4.73205i) q^{46} +(0.378937 - 0.656339i) q^{47} +(-1.96410 + 3.40192i) q^{50} +1.79315 q^{52} +(5.46410 - 9.46410i) q^{53} +0.277401 q^{55} -4.00000 q^{58} +(-0.637756 - 1.10463i) q^{59} +(6.31319 - 10.9348i) q^{61} -6.69213 q^{62} +1.00000 q^{64} +(-0.928203 + 1.60770i) q^{65} +(-6.23205 - 10.7942i) q^{67} +6.83083 q^{68} -9.46410 q^{71} +(-2.70831 + 4.69093i) q^{73} -7.46410 q^{74} +(-2.19067 + 3.79435i) q^{76} +(-4.46410 + 7.73205i) q^{79} +(-0.517638 + 0.896575i) q^{80} +(4.31199 + 7.46859i) q^{82} +(3.29530 + 5.70762i) q^{83} +(-3.53590 + 6.12436i) q^{85} -0.267949 q^{86} -0.267949 q^{88} +(3.53553 + 6.12372i) q^{89} +(2.73205 + 4.73205i) q^{92} +(-0.378937 - 0.656339i) q^{94} +(-2.26795 - 3.92820i) q^{95} +(9.07227 + 15.7136i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 16 q^{11} - 4 q^{16} + 8 q^{22} - 16 q^{23} + 24 q^{25} - 16 q^{29} + 4 q^{32} - 16 q^{37} - 8 q^{43} - 8 q^{44} - 8 q^{46} + 12 q^{50} + 16 q^{53} - 32 q^{58} + 8 q^{64}+ \cdots - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.03528 0.462990 0.231495 − 0.972836i \(-0.425638\pi\)
0.231495 + 0.972836i \(0.425638\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.517638 − 0.896575i 0.163692 − 0.283522i
\(11\) 0.267949 0.0807897 0.0403949 − 0.999184i \(-0.487138\pi\)
0.0403949 + 0.999184i \(0.487138\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.41542 + 5.91567i −0.828360 + 1.43476i 0.0709642 + 0.997479i \(0.477392\pi\)
−0.899324 + 0.437283i \(0.855941\pi\)
\(18\) 0 0
\(19\) −2.19067 − 3.79435i −0.502574 − 0.870484i −0.999996 − 0.00297513i \(-0.999053\pi\)
0.497421 − 0.867509i \(-0.334280\pi\)
\(20\) −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 \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(24\) 0 0
\(25\) −3.92820 −0.785641
\(26\) 0.896575 + 1.55291i 0.175833 + 0.304552i
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 − 3.46410i −0.371391 − 0.643268i 0.618389 − 0.785872i \(-0.287786\pi\)
−0.989780 + 0.142605i \(0.954452\pi\)
\(30\) 0 0
\(31\) −3.34607 − 5.79555i −0.600971 − 1.04091i −0.992674 − 0.120821i \(-0.961447\pi\)
0.391703 − 0.920092i \(-0.371886\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.41542 + 5.91567i 0.585739 + 1.01453i
\(35\) 0 0
\(36\) 0 0
\(37\) −3.73205 − 6.46410i −0.613545 − 1.06269i −0.990638 − 0.136516i \(-0.956409\pi\)
0.377092 − 0.926176i \(-0.376924\pi\)
\(38\) −4.38134 −0.710747
\(39\) 0 0
\(40\) −1.03528 −0.163692
\(41\) −4.31199 + 7.46859i −0.673420 + 1.16640i 0.303508 + 0.952829i \(0.401842\pi\)
−0.976928 + 0.213569i \(0.931491\pi\)
\(42\) 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
\(45\) 0 0
\(46\) −2.73205 + 4.73205i −0.402819 + 0.697703i
\(47\) 0.378937 − 0.656339i 0.0552737 − 0.0957369i −0.837065 − 0.547104i \(-0.815730\pi\)
0.892338 + 0.451367i \(0.149064\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.96410 + 3.40192i −0.277766 + 0.481105i
\(51\) 0 0
\(52\) 1.79315 0.248665
\(53\) 5.46410 − 9.46410i 0.750552 − 1.29999i −0.197003 − 0.980403i \(-0.563121\pi\)
0.947555 − 0.319592i \(-0.103546\pi\)
\(54\) 0 0
\(55\) 0.277401 0.0374048
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −0.637756 − 1.10463i −0.0830288 − 0.143810i 0.821521 − 0.570179i \(-0.193126\pi\)
−0.904550 + 0.426369i \(0.859793\pi\)
\(60\) 0 0
\(61\) 6.31319 − 10.9348i 0.808322 − 1.40005i −0.105704 − 0.994398i \(-0.533710\pi\)
0.914026 − 0.405656i \(-0.132957\pi\)
\(62\) −6.69213 −0.849901
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.928203 + 1.60770i −0.115129 + 0.199410i
\(66\) 0 0
\(67\) −6.23205 − 10.7942i −0.761366 − 1.31872i −0.942146 − 0.335201i \(-0.891196\pi\)
0.180780 − 0.983524i \(-0.442138\pi\)
\(68\) 6.83083 0.828360
\(69\) 0 0
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 − 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) −2.70831 + 4.69093i −0.316984 + 0.549032i −0.979857 − 0.199700i \(-0.936003\pi\)
0.662874 + 0.748731i \(0.269337\pi\)
\(74\) −7.46410 −0.867684
\(75\) 0 0
\(76\) −2.19067 + 3.79435i −0.251287 + 0.435242i
\(77\) 0 0
\(78\) 0 0
\(79\) −4.46410 + 7.73205i −0.502251 + 0.869924i 0.497746 + 0.867323i \(0.334161\pi\)
−0.999997 + 0.00260080i \(0.999172\pi\)
\(80\) −0.517638 + 0.896575i −0.0578737 + 0.100240i
\(81\) 0 0
\(82\) 4.31199 + 7.46859i 0.476180 + 0.824768i
\(83\) 3.29530 + 5.70762i 0.361706 + 0.626493i 0.988242 − 0.152900i \(-0.0488611\pi\)
−0.626536 + 0.779393i \(0.715528\pi\)
\(84\) 0 0
\(85\) −3.53590 + 6.12436i −0.383522 + 0.664280i
\(86\) −0.267949 −0.0288937
\(87\) 0 0
\(88\) −0.267949 −0.0285635
\(89\) 3.53553 + 6.12372i 0.374766 + 0.649113i 0.990292 − 0.139003i \(-0.0443898\pi\)
−0.615526 + 0.788116i \(0.711056\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.73205 + 4.73205i 0.284836 + 0.493350i
\(93\) 0 0
\(94\) −0.378937 − 0.656339i −0.0390844 − 0.0676962i
\(95\) −2.26795 − 3.92820i −0.232687 − 0.403025i
\(96\) 0 0
\(97\) 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 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.t.361.3 8
3.2 odd 2 882.2.h.q.67.3 8
7.2 even 3 2646.2.e.q.1549.2 8
7.3 odd 6 2646.2.f.r.1765.3 8
7.4 even 3 2646.2.f.r.1765.2 8
7.5 odd 6 2646.2.e.q.1549.3 8
7.6 odd 2 inner 2646.2.h.t.361.2 8
9.2 odd 6 882.2.e.s.655.3 8
9.7 even 3 2646.2.e.q.2125.2 8
21.2 odd 6 882.2.e.s.373.3 8
21.5 even 6 882.2.e.s.373.2 8
21.11 odd 6 882.2.f.q.589.1 yes 8
21.17 even 6 882.2.f.q.589.4 yes 8
21.20 even 2 882.2.h.q.67.2 8
63.2 odd 6 882.2.h.q.79.4 8
63.4 even 3 7938.2.a.ci.1.3 4
63.11 odd 6 882.2.f.q.295.1 ✓ 8
63.16 even 3 inner 2646.2.h.t.667.3 8
63.20 even 6 882.2.e.s.655.2 8
63.25 even 3 2646.2.f.r.883.2 8
63.31 odd 6 7938.2.a.ci.1.2 4
63.32 odd 6 7938.2.a.cp.1.2 4
63.34 odd 6 2646.2.e.q.2125.3 8
63.38 even 6 882.2.f.q.295.4 yes 8
63.47 even 6 882.2.h.q.79.1 8
63.52 odd 6 2646.2.f.r.883.3 8
63.59 even 6 7938.2.a.cp.1.3 4
63.61 odd 6 inner 2646.2.h.t.667.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.2 8 21.5 even 6
882.2.e.s.373.3 8 21.2 odd 6
882.2.e.s.655.2 8 63.20 even 6
882.2.e.s.655.3 8 9.2 odd 6
882.2.f.q.295.1 ✓ 8 63.11 odd 6
882.2.f.q.295.4 yes 8 63.38 even 6
882.2.f.q.589.1 yes 8 21.11 odd 6
882.2.f.q.589.4 yes 8 21.17 even 6
882.2.h.q.67.2 8 21.20 even 2
882.2.h.q.67.3 8 3.2 odd 2
882.2.h.q.79.1 8 63.47 even 6
882.2.h.q.79.4 8 63.2 odd 6
2646.2.e.q.1549.2 8 7.2 even 3
2646.2.e.q.1549.3 8 7.5 odd 6
2646.2.e.q.2125.2 8 9.7 even 3
2646.2.e.q.2125.3 8 63.34 odd 6
2646.2.f.r.883.2 8 63.25 even 3
2646.2.f.r.883.3 8 63.52 odd 6
2646.2.f.r.1765.2 8 7.4 even 3
2646.2.f.r.1765.3 8 7.3 odd 6
2646.2.h.t.361.2 8 7.6 odd 2 inner
2646.2.h.t.361.3 8 1.1 even 1 trivial
2646.2.h.t.667.2 8 63.61 odd 6 inner
2646.2.h.t.667.3 8 63.16 even 3 inner
7938.2.a.ci.1.2 4 63.31 odd 6
7938.2.a.ci.1.3 4 63.4 even 3
7938.2.a.cp.1.2 4 63.32 odd 6
7938.2.a.cp.1.3 4 63.59 even 6