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Properties

Label 336.2.bc.c.17.1
Level $336$
Weight $2$
Character 336.17
Analytic conductor $2.683$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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] = [336,2,Mod(17,336)] 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("336.17"); 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([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,0] 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: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 17.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.17
Dual form 336.2.bc.c.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(-0.500000 + 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +1.73205i q^{13} +(4.50000 - 2.59808i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(-7.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{37} +(-1.50000 + 2.59808i) q^{39} +5.00000 q^{43} +(-6.50000 - 2.59808i) q^{49} +9.00000 q^{57} +(6.00000 - 3.46410i) q^{61} +(-7.50000 + 2.59808i) q^{63} +(5.50000 - 9.52628i) q^{67} +(-13.5000 - 7.79423i) q^{73} +(7.50000 - 4.33013i) q^{75} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-4.50000 - 0.866025i) q^{91} +(-7.50000 - 12.9904i) q^{93} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - q^{7} + 3 q^{9} + 9 q^{19} - 6 q^{21} + 5 q^{25} - 15 q^{31} - q^{37} - 3 q^{39} + 10 q^{43} - 13 q^{49} + 18 q^{57} + 12 q^{61} - 15 q^{63} + 11 q^{67} - 27 q^{73} + 15 q^{75} - 13 q^{79}+ \cdots - 15 q^{93}+O(q^{100}) \) 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
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 4.50000 − 2.59808i 1.03237 − 0.596040i 0.114708 − 0.993399i \(-0.463407\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.46410i −0.654654 + 0.755929i
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.50000 − 4.33013i −1.34704 − 0.777714i −0.359211 − 0.933257i \(-0.616954\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 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
\(39\) −1.50000 + 2.59808i −0.240192 + 0.416025i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.00000 1.19208
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 6.00000 − 3.46410i 0.768221 − 0.443533i −0.0640184 − 0.997949i \(-0.520392\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) −7.50000 + 2.59808i −0.944911 + 0.327327i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.50000 − 9.52628i 0.671932 − 1.16382i −0.305424 − 0.952217i \(-0.598798\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −13.5000 − 7.79423i −1.58006 − 0.912245i −0.994850 − 0.101361i \(-0.967680\pi\)
−0.585206 − 0.810885i \(-0.698986\pi\)
\(74\) 0 0
\(75\) 7.50000 − 4.33013i 0.866025 − 0.500000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.50000 − 11.2583i −0.731307 − 1.26666i −0.956325 − 0.292306i \(-0.905577\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −4.50000 − 0.866025i −0.471728 − 0.0907841i
\(92\) 0 0
\(93\) −7.50000 − 12.9904i −0.777714 − 1.34704i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 336.2.bc.c.17.1 2
3.2 odd 2 CM 336.2.bc.c.17.1 2
4.3 odd 2 21.2.g.a.17.1 yes 2
7.3 odd 6 2352.2.k.c.881.2 2
7.4 even 3 2352.2.k.c.881.1 2
7.5 odd 6 inner 336.2.bc.c.257.1 2
12.11 even 2 21.2.g.a.17.1 yes 2
20.3 even 4 525.2.q.d.374.2 4
20.7 even 4 525.2.q.d.374.1 4
20.19 odd 2 525.2.t.c.101.1 2
21.5 even 6 inner 336.2.bc.c.257.1 2
21.11 odd 6 2352.2.k.c.881.1 2
21.17 even 6 2352.2.k.c.881.2 2
28.3 even 6 147.2.c.a.146.1 2
28.11 odd 6 147.2.c.a.146.2 2
28.19 even 6 21.2.g.a.5.1 ✓ 2
28.23 odd 6 147.2.g.a.68.1 2
28.27 even 2 147.2.g.a.80.1 2
36.7 odd 6 567.2.s.a.458.1 2
36.11 even 6 567.2.s.a.458.1 2
36.23 even 6 567.2.i.b.269.1 2
36.31 odd 6 567.2.i.b.269.1 2
60.23 odd 4 525.2.q.d.374.2 4
60.47 odd 4 525.2.q.d.374.1 4
60.59 even 2 525.2.t.c.101.1 2
84.11 even 6 147.2.c.a.146.2 2
84.23 even 6 147.2.g.a.68.1 2
84.47 odd 6 21.2.g.a.5.1 ✓ 2
84.59 odd 6 147.2.c.a.146.1 2
84.83 odd 2 147.2.g.a.80.1 2
140.19 even 6 525.2.t.c.26.1 2
140.47 odd 12 525.2.q.d.299.2 4
140.103 odd 12 525.2.q.d.299.1 4
252.47 odd 6 567.2.i.b.215.1 2
252.103 even 6 567.2.s.a.26.1 2
252.131 odd 6 567.2.s.a.26.1 2
252.187 even 6 567.2.i.b.215.1 2
420.47 even 12 525.2.q.d.299.2 4
420.299 odd 6 525.2.t.c.26.1 2
420.383 even 12 525.2.q.d.299.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.g.a.5.1 ✓ 2 28.19 even 6
21.2.g.a.5.1 ✓ 2 84.47 odd 6
21.2.g.a.17.1 yes 2 4.3 odd 2
21.2.g.a.17.1 yes 2 12.11 even 2
147.2.c.a.146.1 2 28.3 even 6
147.2.c.a.146.1 2 84.59 odd 6
147.2.c.a.146.2 2 28.11 odd 6
147.2.c.a.146.2 2 84.11 even 6
147.2.g.a.68.1 2 28.23 odd 6
147.2.g.a.68.1 2 84.23 even 6
147.2.g.a.80.1 2 28.27 even 2
147.2.g.a.80.1 2 84.83 odd 2
336.2.bc.c.17.1 2 1.1 even 1 trivial
336.2.bc.c.17.1 2 3.2 odd 2 CM
336.2.bc.c.257.1 2 7.5 odd 6 inner
336.2.bc.c.257.1 2 21.5 even 6 inner
525.2.q.d.299.1 4 140.103 odd 12
525.2.q.d.299.1 4 420.383 even 12
525.2.q.d.299.2 4 140.47 odd 12
525.2.q.d.299.2 4 420.47 even 12
525.2.q.d.374.1 4 20.7 even 4
525.2.q.d.374.1 4 60.47 odd 4
525.2.q.d.374.2 4 20.3 even 4
525.2.q.d.374.2 4 60.23 odd 4
525.2.t.c.26.1 2 140.19 even 6
525.2.t.c.26.1 2 420.299 odd 6
525.2.t.c.101.1 2 20.19 odd 2
525.2.t.c.101.1 2 60.59 even 2
567.2.i.b.215.1 2 252.47 odd 6
567.2.i.b.215.1 2 252.187 even 6
567.2.i.b.269.1 2 36.23 even 6
567.2.i.b.269.1 2 36.31 odd 6
567.2.s.a.26.1 2 252.103 even 6
567.2.s.a.26.1 2 252.131 odd 6
567.2.s.a.458.1 2 36.7 odd 6
567.2.s.a.458.1 2 36.11 even 6
2352.2.k.c.881.1 2 7.4 even 3
2352.2.k.c.881.1 2 21.11 odd 6
2352.2.k.c.881.2 2 7.3 odd 6
2352.2.k.c.881.2 2 21.17 even 6