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Properties

Label 336.4.q.e.193.1
Level $336$
Weight $4$
Character 336.193
Analytic conductor $19.825$
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] = [336,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,3] 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: \(19.8246417619\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.193
Dual form 336.4.q.e.289.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} +(1.50000 + 2.59808i) q^{5} +(3.50000 + 18.1865i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(-7.50000 + 12.9904i) q^{11} -64.0000 q^{13} +9.00000 q^{15} +(-42.0000 + 72.7461i) q^{17} +(-8.00000 - 13.8564i) q^{19} +(52.5000 + 18.1865i) q^{21} +(-42.0000 - 72.7461i) q^{23} +(58.0000 - 100.459i) q^{25} -27.0000 q^{27} -297.000 q^{29} +(-126.500 + 219.104i) q^{31} +(22.5000 + 38.9711i) q^{33} +(-42.0000 + 36.3731i) q^{35} +(158.000 + 273.664i) q^{37} +(-96.0000 + 166.277i) q^{39} +360.000 q^{41} -26.0000 q^{43} +(13.5000 - 23.3827i) q^{45} +(-15.0000 - 25.9808i) q^{47} +(-318.500 + 127.306i) q^{49} +(126.000 + 218.238i) q^{51} +(-181.500 + 314.367i) q^{53} -45.0000 q^{55} -48.0000 q^{57} +(-7.50000 + 12.9904i) q^{59} +(59.0000 + 102.191i) q^{61} +(126.000 - 109.119i) q^{63} +(-96.0000 - 166.277i) q^{65} +(-185.000 + 320.429i) q^{67} -252.000 q^{69} +342.000 q^{71} +(-181.000 + 313.501i) q^{73} +(-174.000 - 301.377i) q^{75} +(-262.500 - 90.9327i) q^{77} +(233.500 + 404.434i) q^{79} +(-40.5000 + 70.1481i) q^{81} -477.000 q^{83} -252.000 q^{85} +(-445.500 + 771.629i) q^{87} +(-453.000 - 784.619i) q^{89} +(-224.000 - 1163.94i) q^{91} +(379.500 + 657.313i) q^{93} +(24.0000 - 41.5692i) q^{95} +503.000 q^{97} +135.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 3 q^{5} + 7 q^{7} - 9 q^{9} - 15 q^{11} - 128 q^{13} + 18 q^{15} - 84 q^{17} - 16 q^{19} + 105 q^{21} - 84 q^{23} + 116 q^{25} - 54 q^{27} - 594 q^{29} - 253 q^{31} + 45 q^{33} - 84 q^{35}+ \cdots + 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(3\) 1.50000 − 2.59808i 0.288675 − 0.500000i
\(4\) 0 0
\(5\) 1.50000 + 2.59808i 0.134164 + 0.232379i 0.925278 − 0.379290i \(-0.123832\pi\)
−0.791114 + 0.611669i \(0.790498\pi\)
\(6\) 0 0
\(7\) 3.50000 + 18.1865i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −7.50000 + 12.9904i −0.205576 + 0.356068i −0.950316 − 0.311287i \(-0.899240\pi\)
0.744740 + 0.667355i \(0.232573\pi\)
\(12\) 0 0
\(13\) −64.0000 −1.36542 −0.682708 − 0.730691i \(-0.739198\pi\)
−0.682708 + 0.730691i \(0.739198\pi\)
\(14\) 0 0
\(15\) 9.00000 0.154919
\(16\) 0 0
\(17\) −42.0000 + 72.7461i −0.599206 + 1.03785i 0.393733 + 0.919225i \(0.371183\pi\)
−0.992939 + 0.118630i \(0.962150\pi\)
\(18\) 0 0
\(19\) −8.00000 − 13.8564i −0.0965961 − 0.167309i 0.813678 − 0.581317i \(-0.197462\pi\)
−0.910274 + 0.414007i \(0.864129\pi\)
\(20\) 0 0
\(21\) 52.5000 + 18.1865i 0.545545 + 0.188982i
\(22\) 0 0
\(23\) −42.0000 − 72.7461i −0.380765 − 0.659505i 0.610406 − 0.792088i \(-0.291006\pi\)
−0.991172 + 0.132583i \(0.957673\pi\)
\(24\) 0 0
\(25\) 58.0000 − 100.459i 0.464000 − 0.803672i
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −297.000 −1.90178 −0.950888 − 0.309535i \(-0.899827\pi\)
−0.950888 + 0.309535i \(0.899827\pi\)
\(30\) 0 0
\(31\) −126.500 + 219.104i −0.732906 + 1.26943i 0.222731 + 0.974880i \(0.428503\pi\)
−0.955636 + 0.294550i \(0.904830\pi\)
\(32\) 0 0
\(33\) 22.5000 + 38.9711i 0.118689 + 0.205576i
\(34\) 0 0
\(35\) −42.0000 + 36.3731i −0.202837 + 0.175662i
\(36\) 0 0
\(37\) 158.000 + 273.664i 0.702028 + 1.21595i 0.967753 + 0.251900i \(0.0810553\pi\)
−0.265725 + 0.964049i \(0.585611\pi\)
\(38\) 0 0
\(39\) −96.0000 + 166.277i −0.394162 + 0.682708i
\(40\) 0 0
\(41\) 360.000 1.37128 0.685641 − 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 0 0
\(43\) −26.0000 −0.0922084 −0.0461042 − 0.998937i \(-0.514681\pi\)
−0.0461042 + 0.998937i \(0.514681\pi\)
\(44\) 0 0
\(45\) 13.5000 − 23.3827i 0.0447214 − 0.0774597i
\(46\) 0 0
\(47\) −15.0000 − 25.9808i −0.0465527 − 0.0806316i 0.841810 − 0.539774i \(-0.181490\pi\)
−0.888363 + 0.459142i \(0.848157\pi\)
\(48\) 0 0
\(49\) −318.500 + 127.306i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 126.000 + 218.238i 0.345952 + 0.599206i
\(52\) 0 0
\(53\) −181.500 + 314.367i −0.470395 + 0.814748i −0.999427 − 0.0338538i \(-0.989222\pi\)
0.529032 + 0.848602i \(0.322555\pi\)
\(54\) 0 0
\(55\) −45.0000 −0.110324
\(56\) 0 0
\(57\) −48.0000 −0.111540
\(58\) 0 0
\(59\) −7.50000 + 12.9904i −0.0165494 + 0.0286645i −0.874182 − 0.485599i \(-0.838601\pi\)
0.857632 + 0.514264i \(0.171935\pi\)
\(60\) 0 0
\(61\) 59.0000 + 102.191i 0.123839 + 0.214495i 0.921279 − 0.388903i \(-0.127146\pi\)
−0.797440 + 0.603399i \(0.793813\pi\)
\(62\) 0 0
\(63\) 126.000 − 109.119i 0.251976 − 0.218218i
\(64\) 0 0
\(65\) −96.0000 − 166.277i −0.183190 − 0.317294i
\(66\) 0 0
\(67\) −185.000 + 320.429i −0.337334 + 0.584279i −0.983930 − 0.178553i \(-0.942858\pi\)
0.646597 + 0.762832i \(0.276192\pi\)
\(68\) 0 0
\(69\) −252.000 −0.439670
\(70\) 0 0
\(71\) 342.000 0.571661 0.285831 − 0.958280i \(-0.407731\pi\)
0.285831 + 0.958280i \(0.407731\pi\)
\(72\) 0 0
\(73\) −181.000 + 313.501i −0.290198 + 0.502638i −0.973856 − 0.227165i \(-0.927054\pi\)
0.683658 + 0.729802i \(0.260388\pi\)
\(74\) 0 0
\(75\) −174.000 − 301.377i −0.267891 − 0.464000i
\(76\) 0 0
\(77\) −262.500 − 90.9327i −0.388502 − 0.134581i
\(78\) 0 0
\(79\) 233.500 + 404.434i 0.332542 + 0.575979i 0.983010 − 0.183555i \(-0.0587604\pi\)
−0.650468 + 0.759534i \(0.725427\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −477.000 −0.630814 −0.315407 − 0.948957i \(-0.602141\pi\)
−0.315407 + 0.948957i \(0.602141\pi\)
\(84\) 0 0
\(85\) −252.000 −0.321568
\(86\) 0 0
\(87\) −445.500 + 771.629i −0.548996 + 0.950888i
\(88\) 0 0
\(89\) −453.000 − 784.619i −0.539527 − 0.934488i −0.998929 − 0.0462600i \(-0.985270\pi\)
0.459402 − 0.888228i \(-0.348064\pi\)
\(90\) 0 0
\(91\) −224.000 − 1163.94i −0.258039 − 1.34081i
\(92\) 0 0
\(93\) 379.500 + 657.313i 0.423143 + 0.732906i
\(94\) 0 0
\(95\) 24.0000 − 41.5692i 0.0259195 − 0.0448938i
\(96\) 0 0
\(97\) 503.000 0.526515 0.263257 − 0.964726i \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(98\) 0 0
\(99\) 135.000 0.137051
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 336.4.q.e.193.1 2
4.3 odd 2 21.4.e.a.4.1 ✓ 2
7.2 even 3 inner 336.4.q.e.289.1 2
7.3 odd 6 2352.4.a.bd.1.1 1
7.4 even 3 2352.4.a.i.1.1 1
12.11 even 2 63.4.e.a.46.1 2
28.3 even 6 147.4.a.a.1.1 1
28.11 odd 6 147.4.a.b.1.1 1
28.19 even 6 147.4.e.h.79.1 2
28.23 odd 6 21.4.e.a.16.1 yes 2
28.27 even 2 147.4.e.h.67.1 2
84.11 even 6 441.4.a.l.1.1 1
84.23 even 6 63.4.e.a.37.1 2
84.47 odd 6 441.4.e.c.226.1 2
84.59 odd 6 441.4.a.k.1.1 1
84.83 odd 2 441.4.e.c.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 ✓ 2 4.3 odd 2
21.4.e.a.16.1 yes 2 28.23 odd 6
63.4.e.a.37.1 2 84.23 even 6
63.4.e.a.46.1 2 12.11 even 2
147.4.a.a.1.1 1 28.3 even 6
147.4.a.b.1.1 1 28.11 odd 6
147.4.e.h.67.1 2 28.27 even 2
147.4.e.h.79.1 2 28.19 even 6
336.4.q.e.193.1 2 1.1 even 1 trivial
336.4.q.e.289.1 2 7.2 even 3 inner
441.4.a.k.1.1 1 84.59 odd 6
441.4.a.l.1.1 1 84.11 even 6
441.4.e.c.226.1 2 84.47 odd 6
441.4.e.c.361.1 2 84.83 odd 2
2352.4.a.i.1.1 1 7.4 even 3
2352.4.a.bd.1.1 1 7.3 odd 6