↑

Properties

Label 336.2.bl.e.31.1
Level $336$
Weight $2$
Character 336.31
Analytic conductor $2.683$
Analytic rank $0$
Dimension $2$
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] = [336,2,Mod(31,336)] 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("336.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68297350792\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.31
Dual form 336.2.bl.e.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(-3.00000 + 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 + 1.73205i) q^{11} +5.19615i q^{13} +3.46410i q^{15} +(-6.00000 - 3.46410i) q^{17} +(3.50000 + 6.06218i) q^{19} +(2.00000 + 1.73205i) q^{21} +(3.50000 - 6.06218i) q^{25} -1.00000 q^{27} +(2.50000 - 4.33013i) q^{31} +(3.00000 - 1.73205i) q^{33} +(-3.00000 - 8.66025i) q^{35} +(-0.500000 - 0.866025i) q^{37} +(4.50000 + 2.59808i) q^{39} +10.3923i q^{41} -1.73205i q^{43} +(3.00000 + 1.73205i) q^{45} +(-3.00000 - 5.19615i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-6.00000 + 3.46410i) q^{51} -12.0000 q^{55} +7.00000 q^{57} +(2.50000 - 0.866025i) q^{63} +(-9.00000 - 15.5885i) q^{65} +(-1.50000 - 0.866025i) q^{67} +3.46410i q^{71} +(7.50000 + 4.33013i) q^{73} +(-3.50000 - 6.06218i) q^{75} +(-6.00000 + 6.92820i) q^{77} +(13.5000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} +6.00000 q^{83} +24.0000 q^{85} +(6.00000 - 3.46410i) q^{89} +(-13.5000 - 2.59808i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-21.0000 - 12.1244i) q^{95} -6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 6 q^{5} - q^{7} - q^{9} + 6 q^{11} - 12 q^{17} + 7 q^{19} + 4 q^{21} + 7 q^{25} - 2 q^{27} + 5 q^{31} + 6 q^{33} - 6 q^{35} - q^{37} + 9 q^{39} + 6 q^{45} - 6 q^{47} - 13 q^{49} - 12 q^{51}+ \cdots - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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 0
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) 0 0
\(5\) −3.00000 + 1.73205i −1.34164 + 0.774597i −0.987048 − 0.160424i \(-0.948714\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 3.00000 + 1.73205i 0.904534 + 0.522233i 0.878668 − 0.477432i \(-0.158432\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) 5.19615i 1.44115i 0.693375 + 0.720577i \(0.256123\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 3.46410i 0.894427i
\(16\) 0 0
\(17\) −6.00000 − 3.46410i −1.45521 − 0.840168i −0.456444 − 0.889752i \(-0.650877\pi\)
−0.998770 + 0.0495842i \(0.984210\pi\)
\(18\) 0 0
\(19\) 3.50000 + 6.06218i 0.802955 + 1.39076i 0.917663 + 0.397360i \(0.130073\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 3.50000 − 6.06218i 0.700000 − 1.21244i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 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 \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0 0
\(33\) 3.00000 − 1.73205i 0.522233 − 0.301511i
\(34\) 0 0
\(35\) −3.00000 − 8.66025i −0.507093 − 1.46385i
\(36\) 0 0
\(37\) −0.500000 − 0.866025i −0.0821995 − 0.142374i 0.821995 − 0.569495i \(-0.192861\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 4.50000 + 2.59808i 0.720577 + 0.416025i
\(40\) 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
\(45\) 3.00000 + 1.73205i 0.447214 + 0.258199i
\(46\) 0 0
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) −6.00000 + 3.46410i −0.840168 + 0.485071i
\(52\) 0 0
\(53\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 2.50000 − 0.866025i 0.314970 − 0.109109i
\(64\) 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 \(-0.367074\pi\)
−0.588821 + 0.808264i \(0.700408\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) 7.50000 + 4.33013i 0.877809 + 0.506803i 0.869935 − 0.493166i \(-0.164160\pi\)
0.00787336 + 0.999969i \(0.497494\pi\)
\(74\) 0 0
\(75\) −3.50000 − 6.06218i −0.404145 − 0.700000i
\(76\) 0 0
\(77\) −6.00000 + 6.92820i −0.683763 + 0.789542i
\(78\) 0 0
\(79\) 13.5000 − 7.79423i 1.51887 − 0.876919i 0.519115 − 0.854704i \(-0.326261\pi\)
0.999753 − 0.0222151i \(-0.00707187\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 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 \(-0.546985\pi\)
0.783072 + 0.621932i \(0.213652\pi\)
\(90\) 0 0
\(91\) −13.5000 − 2.59808i −1.41518 − 0.272352i
\(92\) 0 0
\(93\) −2.50000 − 4.33013i −0.259238 − 0.449013i
\(94\) 0 0
\(95\) −21.0000 − 12.1244i −2.15455 − 1.24393i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) − 3.46410i − 0.348155i
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 336.2.bl.e.31.1 yes 2
3.2 odd 2 1008.2.cs.m.703.1 2
4.3 odd 2 336.2.bl.a.31.1 ✓ 2
7.2 even 3 2352.2.bl.l.607.1 2
7.3 odd 6 2352.2.b.h.1567.2 2
7.4 even 3 2352.2.b.a.1567.1 2
7.5 odd 6 336.2.bl.a.271.1 yes 2
7.6 odd 2 2352.2.bl.f.31.1 2
8.3 odd 2 1344.2.bl.h.703.1 2
8.5 even 2 1344.2.bl.d.703.1 2
12.11 even 2 1008.2.cs.n.703.1 2
21.5 even 6 1008.2.cs.n.271.1 2
21.11 odd 6 7056.2.b.a.1567.2 2
21.17 even 6 7056.2.b.l.1567.1 2
28.3 even 6 2352.2.b.a.1567.2 2
28.11 odd 6 2352.2.b.h.1567.1 2
28.19 even 6 inner 336.2.bl.e.271.1 yes 2
28.23 odd 6 2352.2.bl.f.607.1 2
28.27 even 2 2352.2.bl.l.31.1 2
56.5 odd 6 1344.2.bl.h.1279.1 2
56.19 even 6 1344.2.bl.d.1279.1 2
84.11 even 6 7056.2.b.l.1567.2 2
84.47 odd 6 1008.2.cs.m.271.1 2
84.59 odd 6 7056.2.b.a.1567.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.a.31.1 ✓ 2 4.3 odd 2
336.2.bl.a.271.1 yes 2 7.5 odd 6
336.2.bl.e.31.1 yes 2 1.1 even 1 trivial
336.2.bl.e.271.1 yes 2 28.19 even 6 inner
1008.2.cs.m.271.1 2 84.47 odd 6
1008.2.cs.m.703.1 2 3.2 odd 2
1008.2.cs.n.271.1 2 21.5 even 6
1008.2.cs.n.703.1 2 12.11 even 2
1344.2.bl.d.703.1 2 8.5 even 2
1344.2.bl.d.1279.1 2 56.19 even 6
1344.2.bl.h.703.1 2 8.3 odd 2
1344.2.bl.h.1279.1 2 56.5 odd 6
2352.2.b.a.1567.1 2 7.4 even 3
2352.2.b.a.1567.2 2 28.3 even 6
2352.2.b.h.1567.1 2 28.11 odd 6
2352.2.b.h.1567.2 2 7.3 odd 6
2352.2.bl.f.31.1 2 7.6 odd 2
2352.2.bl.f.607.1 2 28.23 odd 6
2352.2.bl.l.31.1 2 28.27 even 2
2352.2.bl.l.607.1 2 7.2 even 3
7056.2.b.a.1567.1 2 84.59 odd 6
7056.2.b.a.1567.2 2 21.11 odd 6
7056.2.b.l.1567.1 2 21.17 even 6
7056.2.b.l.1567.2 2 84.11 even 6