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Properties

Label 336.6.q.i.289.3
Level $336$
Weight $6$
Character 336.289
Analytic conductor $53.889$
Analytic rank $0$
Dimension $8$
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] = [336,6,Mod(193,336)] 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("336.193"); S:= CuspForms(chi, 6); 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, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-36,0,0] 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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.3
Root \(13.1471 + 22.7714i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.6.q.i.193.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 - 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(-85.7043 - 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(130.442 + 225.932i) q^{11} +769.735 q^{13} -287.654 q^{15} +(776.659 + 1345.21i) q^{17} +(375.024 - 649.561i) q^{19} +(-372.488 + 1105.72i) q^{21} +(377.427 - 653.723i) q^{23} +(1051.73 + 1821.65i) q^{25} +729.000 q^{27} +6008.93 q^{29} +(-3210.02 - 5559.92i) q^{31} +(1173.98 - 2033.39i) q^{33} +(-4062.06 + 817.779i) q^{35} +(2387.86 - 4135.90i) q^{37} +(-3463.81 - 5999.49i) q^{39} -5423.27 q^{41} +11896.4 q^{43} +(1294.44 + 2242.04i) q^{45} +(-8714.00 + 15093.1i) q^{47} +(-2116.54 + 16673.2i) q^{49} +(6989.93 - 12106.9i) q^{51} +(-18825.3 - 32606.4i) q^{53} +8338.26 q^{55} -6750.44 q^{57} +(11039.0 + 19120.2i) q^{59} +(4086.69 - 7078.36i) q^{61} +(10294.4 - 2072.49i) q^{63} +(12301.0 - 21305.9i) q^{65} +(6500.87 + 11259.8i) q^{67} -6793.69 q^{69} +12349.6 q^{71} +(-21800.2 - 37759.0i) q^{73} +(9465.55 - 16394.8i) q^{75} +(10797.3 - 32051.6i) q^{77} +(38374.7 - 66467.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} -21893.6 q^{83} +49646.5 q^{85} +(-27040.2 - 46835.0i) q^{87} +(68483.5 - 118617. i) q^{89} +(-65969.7 - 74873.4i) q^{91} +(-28890.2 + 50039.3i) q^{93} +(-11986.4 - 20761.0i) q^{95} -93050.1 q^{97} -21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) −4.50000 − 7.79423i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 15.9808 − 27.6796i 0.285873 − 0.495147i −0.686947 − 0.726707i \(-0.741050\pi\)
0.972821 + 0.231560i \(0.0743831\pi\)
\(6\) 0 0
\(7\) −85.7043 − 97.2716i −0.661086 − 0.750311i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 130.442 + 225.932i 0.325039 + 0.562984i 0.981520 − 0.191359i \(-0.0612893\pi\)
−0.656481 + 0.754342i \(0.727956\pi\)
\(12\) 0 0
\(13\) 769.735 1.26323 0.631616 − 0.775281i \(-0.282392\pi\)
0.631616 + 0.775281i \(0.282392\pi\)
\(14\) 0 0
\(15\) −287.654 −0.330098
\(16\) 0 0
\(17\) 776.659 + 1345.21i 0.651791 + 1.12893i 0.982688 + 0.185268i \(0.0593152\pi\)
−0.330898 + 0.943667i \(0.607351\pi\)
\(18\) 0 0
\(19\) 375.024 − 649.561i 0.238328 − 0.412797i −0.721906 − 0.691991i \(-0.756734\pi\)
0.960235 + 0.279194i \(0.0900673\pi\)
\(20\) 0 0
\(21\) −372.488 + 1105.72i −0.184316 + 0.547139i
\(22\) 0 0
\(23\) 377.427 − 653.723i 0.148769 − 0.257676i −0.782004 − 0.623274i \(-0.785802\pi\)
0.930773 + 0.365598i \(0.119135\pi\)
\(24\) 0 0
\(25\) 1051.73 + 1821.65i 0.336553 + 0.582927i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 6008.93 1.32679 0.663395 − 0.748269i \(-0.269115\pi\)
0.663395 + 0.748269i \(0.269115\pi\)
\(30\) 0 0
\(31\) −3210.02 − 5559.92i −0.599934 − 1.03912i −0.992830 − 0.119533i \(-0.961860\pi\)
0.392896 − 0.919583i \(-0.371473\pi\)
\(32\) 0 0
\(33\) 1173.98 − 2033.39i 0.187661 − 0.325039i
\(34\) 0 0
\(35\) −4062.06 + 817.779i −0.560501 + 0.112841i
\(36\) 0 0
\(37\) 2387.86 − 4135.90i 0.286751 − 0.496668i −0.686281 − 0.727336i \(-0.740758\pi\)
0.973032 + 0.230669i \(0.0740914\pi\)
\(38\) 0 0
\(39\) −3463.81 − 5999.49i −0.364664 − 0.631616i
\(40\) 0 0
\(41\) −5423.27 −0.503850 −0.251925 − 0.967747i \(-0.581064\pi\)
−0.251925 + 0.967747i \(0.581064\pi\)
\(42\) 0 0
\(43\) 11896.4 0.981171 0.490585 − 0.871393i \(-0.336783\pi\)
0.490585 + 0.871393i \(0.336783\pi\)
\(44\) 0 0
\(45\) 1294.44 + 2242.04i 0.0952911 + 0.165049i
\(46\) 0 0
\(47\) −8714.00 + 15093.1i −0.575404 + 0.996629i 0.420594 + 0.907249i \(0.361822\pi\)
−0.995998 + 0.0893798i \(0.971512\pi\)
\(48\) 0 0
\(49\) −2116.54 + 16673.2i −0.125932 + 0.992039i
\(50\) 0 0
\(51\) 6989.93 − 12106.9i 0.376311 − 0.651791i
\(52\) 0 0
\(53\) −18825.3 − 32606.4i −0.920561 − 1.59446i −0.798549 − 0.601930i \(-0.794399\pi\)
−0.122012 − 0.992529i \(-0.538935\pi\)
\(54\) 0 0
\(55\) 8338.26 0.371679
\(56\) 0 0
\(57\) −6750.44 −0.275198
\(58\) 0 0
\(59\) 11039.0 + 19120.2i 0.412859 + 0.715092i 0.995201 − 0.0978516i \(-0.0311971\pi\)
−0.582342 + 0.812944i \(0.697864\pi\)
\(60\) 0 0
\(61\) 4086.69 − 7078.36i 0.140620 − 0.243561i −0.787110 − 0.616812i \(-0.788424\pi\)
0.927730 + 0.373251i \(0.121757\pi\)
\(62\) 0 0
\(63\) 10294.4 − 2072.49i 0.326777 − 0.0657872i
\(64\) 0 0
\(65\) 12301.0 − 21305.9i 0.361124 − 0.625485i
\(66\) 0 0
\(67\) 6500.87 + 11259.8i 0.176923 + 0.306440i 0.940825 − 0.338892i \(-0.110052\pi\)
−0.763902 + 0.645332i \(0.776719\pi\)
\(68\) 0 0
\(69\) −6793.69 −0.171784
\(70\) 0 0
\(71\) 12349.6 0.290742 0.145371 − 0.989377i \(-0.453562\pi\)
0.145371 + 0.989377i \(0.453562\pi\)
\(72\) 0 0
\(73\) −21800.2 − 37759.0i −0.478798 − 0.829303i 0.520906 − 0.853614i \(-0.325594\pi\)
−0.999704 + 0.0243110i \(0.992261\pi\)
\(74\) 0 0
\(75\) 9465.55 − 16394.8i 0.194309 − 0.336553i
\(76\) 0 0
\(77\) 10797.3 − 32051.6i 0.207534 − 0.616060i
\(78\) 0 0
\(79\) 38374.7 − 66467.0i 0.691796 − 1.19823i −0.279453 − 0.960159i \(-0.590153\pi\)
0.971249 − 0.238066i \(-0.0765134\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −21893.6 −0.348836 −0.174418 − 0.984672i \(-0.555804\pi\)
−0.174418 + 0.984672i \(0.555804\pi\)
\(84\) 0 0
\(85\) 49646.5 0.745318
\(86\) 0 0
\(87\) −27040.2 − 46835.0i −0.383011 − 0.663395i
\(88\) 0 0
\(89\) 68483.5 − 118617.i 0.916454 − 1.58735i 0.111696 − 0.993742i \(-0.464372\pi\)
0.804758 − 0.593603i \(-0.202295\pi\)
\(90\) 0 0
\(91\) −65969.7 − 74873.4i −0.835104 − 0.947816i
\(92\) 0 0
\(93\) −28890.2 + 50039.3i −0.346372 + 0.599934i
\(94\) 0 0
\(95\) −11986.4 − 20761.0i −0.136263 − 0.236015i
\(96\) 0 0
\(97\) −93050.1 −1.00412 −0.502062 − 0.864832i \(-0.667425\pi\)
−0.502062 + 0.864832i \(0.667425\pi\)
\(98\) 0 0
\(99\) −21131.6 −0.216692
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 336.6.q.i.289.3 8
4.3 odd 2 84.6.i.c.37.3 yes 8
7.4 even 3 inner 336.6.q.i.193.3 8
12.11 even 2 252.6.k.f.37.2 8
28.3 even 6 588.6.i.o.361.2 8
28.11 odd 6 84.6.i.c.25.3 ✓ 8
28.19 even 6 588.6.a.p.1.3 4
28.23 odd 6 588.6.a.n.1.2 4
28.27 even 2 588.6.i.o.373.2 8
84.11 even 6 252.6.k.f.109.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.3 ✓ 8 28.11 odd 6
84.6.i.c.37.3 yes 8 4.3 odd 2
252.6.k.f.37.2 8 12.11 even 2
252.6.k.f.109.2 8 84.11 even 6
336.6.q.i.193.3 8 7.4 even 3 inner
336.6.q.i.289.3 8 1.1 even 1 trivial
588.6.a.n.1.2 4 28.23 odd 6
588.6.a.p.1.3 4 28.19 even 6
588.6.i.o.361.2 8 28.3 even 6
588.6.i.o.373.2 8 28.27 even 2