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Properties

Label 1008.2.cs.n.271.1
Level $1008$
Weight $2$
Character 1008.271
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
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] = [1008,2,Mod(271,1008)] 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("1008.271"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,6,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.271
Dual form 1008.2.cs.n.703.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(3.00000 - 1.73205i) q^{11} -5.19615i q^{13} +(6.00000 - 3.46410i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(3.50000 + 6.06218i) q^{25} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 8.66025i) q^{35} +(-0.500000 + 0.866025i) q^{37} +10.3923i q^{41} -1.73205i q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +12.0000 q^{55} +(9.00000 - 15.5885i) q^{65} +(1.50000 - 0.866025i) q^{67} -3.46410i q^{71} +(7.50000 - 4.33013i) q^{73} +(6.00000 + 6.92820i) q^{77} +(-13.5000 - 7.79423i) q^{79} +6.00000 q^{83} +24.0000 q^{85} +(-6.00000 - 3.46410i) q^{89} +(13.5000 - 2.59808i) q^{91} +(-21.0000 + 12.1244i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + q^{7} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} - 5 q^{31} - 6 q^{35} - q^{37} - 6 q^{47} - 13 q^{49} + 24 q^{55} + 18 q^{65} + 3 q^{67} + 15 q^{73} + 12 q^{77} - 27 q^{79} + 12 q^{83}+ \cdots - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.00000 + 1.73205i 1.34164 + 0.774597i 0.987048 − 0.160424i \(-0.0512862\pi\)
0.354593 + 0.935021i \(0.384620\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 − 1.73205i 0.904534 − 0.522233i 0.0258656 − 0.999665i \(-0.491766\pi\)
0.878668 + 0.477432i \(0.158432\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 1.44115i −0.693375 − 0.720577i \(-0.743877\pi\)
0.693375 − 0.720577i \(-0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 − 3.46410i 1.45521 − 0.840168i 0.456444 − 0.889752i \(-0.349123\pi\)
0.998770 + 0.0495842i \(0.0157896\pi\)
\(18\) 0 0
\(19\) −3.50000 + 6.06218i −0.802955 + 1.39076i 0.114708 + 0.993399i \(0.463407\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 3.50000 + 6.06218i 0.700000 + 1.21244i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.50000 − 4.33013i −0.449013 − 0.777714i 0.549309 − 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 + 8.66025i −0.507093 + 1.46385i
\(36\) 0 0
\(37\) −0.500000 + 0.866025i −0.0821995 + 0.142374i −0.904194 − 0.427121i \(-0.859528\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923i 1.62301i 0.584349 + 0.811503i \(0.301350\pi\)
−0.584349 + 0.811503i \(0.698650\pi\)
\(42\) 0 0
\(43\) − 1.73205i − 0.264135i −0.991241 − 0.132068i \(-0.957838\pi\)
0.991241 − 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.00000 − 15.5885i 1.11631 − 1.93351i
\(66\) 0 0
\(67\) 1.50000 − 0.866025i 0.183254 − 0.105802i −0.405567 − 0.914066i \(-0.632926\pi\)
0.588821 + 0.808264i \(0.299592\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.46410i − 0.411113i −0.978645 − 0.205557i \(-0.934100\pi\)
0.978645 − 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) 7.50000 − 4.33013i 0.877809 − 0.506803i 0.00787336 − 0.999969i \(-0.497494\pi\)
0.869935 + 0.493166i \(0.164160\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 + 6.92820i 0.683763 + 0.789542i
\(78\) 0 0
\(79\) −13.5000 − 7.79423i −1.51887 − 0.876919i −0.999753 − 0.0222151i \(-0.992928\pi\)
−0.519115 − 0.854704i \(-0.673739\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 24.0000 2.60317
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 − 3.46410i −0.635999 − 0.367194i 0.147073 − 0.989126i \(-0.453015\pi\)
−0.783072 + 0.621932i \(0.786348\pi\)
\(90\) 0 0
\(91\) 13.5000 − 2.59808i 1.41518 − 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.0000 + 12.1244i −2.15455 + 1.24393i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 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 1008.2.cs.n.271.1 2
3.2 odd 2 336.2.bl.a.271.1 yes 2
4.3 odd 2 1008.2.cs.m.271.1 2
7.2 even 3 7056.2.b.l.1567.1 2
7.3 odd 6 1008.2.cs.m.703.1 2
7.5 odd 6 7056.2.b.a.1567.2 2
12.11 even 2 336.2.bl.e.271.1 yes 2
21.2 odd 6 2352.2.b.h.1567.2 2
21.5 even 6 2352.2.b.a.1567.1 2
21.11 odd 6 2352.2.bl.f.31.1 2
21.17 even 6 336.2.bl.e.31.1 yes 2
21.20 even 2 2352.2.bl.l.607.1 2
24.5 odd 2 1344.2.bl.h.1279.1 2
24.11 even 2 1344.2.bl.d.1279.1 2
28.3 even 6 inner 1008.2.cs.n.703.1 2
28.19 even 6 7056.2.b.l.1567.2 2
28.23 odd 6 7056.2.b.a.1567.1 2
84.11 even 6 2352.2.bl.l.31.1 2
84.23 even 6 2352.2.b.a.1567.2 2
84.47 odd 6 2352.2.b.h.1567.1 2
84.59 odd 6 336.2.bl.a.31.1 ✓ 2
84.83 odd 2 2352.2.bl.f.607.1 2
168.59 odd 6 1344.2.bl.h.703.1 2
168.101 even 6 1344.2.bl.d.703.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.a.31.1 ✓ 2 84.59 odd 6
336.2.bl.a.271.1 yes 2 3.2 odd 2
336.2.bl.e.31.1 yes 2 21.17 even 6
336.2.bl.e.271.1 yes 2 12.11 even 2
1008.2.cs.m.271.1 2 4.3 odd 2
1008.2.cs.m.703.1 2 7.3 odd 6
1008.2.cs.n.271.1 2 1.1 even 1 trivial
1008.2.cs.n.703.1 2 28.3 even 6 inner
1344.2.bl.d.703.1 2 168.101 even 6
1344.2.bl.d.1279.1 2 24.11 even 2
1344.2.bl.h.703.1 2 168.59 odd 6
1344.2.bl.h.1279.1 2 24.5 odd 2
2352.2.b.a.1567.1 2 21.5 even 6
2352.2.b.a.1567.2 2 84.23 even 6
2352.2.b.h.1567.1 2 84.47 odd 6
2352.2.b.h.1567.2 2 21.2 odd 6
2352.2.bl.f.31.1 2 21.11 odd 6
2352.2.bl.f.607.1 2 84.83 odd 2
2352.2.bl.l.31.1 2 84.11 even 6
2352.2.bl.l.607.1 2 21.20 even 2
7056.2.b.a.1567.1 2 28.23 odd 6
7056.2.b.a.1567.2 2 7.5 odd 6
7056.2.b.l.1567.1 2 7.2 even 3
7056.2.b.l.1567.2 2 28.19 even 6