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Properties

Label 336.6.h.b.239.38
Level $336$
Weight $6$
Character 336.239
Analytic conductor $53.889$
Analytic rank $0$
Dimension $40$
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] = [336,6,Mod(239,336)] 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("336.239"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.38
Character \(\chi\) \(=\) 336.239
Dual form 336.6.h.b.239.37

$q$-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)
 
\(f(q)\) \(=\) \(q+(15.4446 + 2.11320i) q^{3} +68.1623i q^{5} +49.0000i q^{7} +(234.069 + 65.2750i) q^{9} -164.006 q^{11} -181.481 q^{13} +(-144.041 + 1052.74i) q^{15} -68.1353i q^{17} +1793.71i q^{19} +(-103.547 + 756.783i) q^{21} +435.780 q^{23} -1521.09 q^{25} +(3477.15 + 1502.78i) q^{27} +1914.13i q^{29} +3130.39i q^{31} +(-2533.00 - 346.577i) q^{33} -3339.95 q^{35} -8701.65 q^{37} +(-2802.89 - 383.506i) q^{39} -9365.85i q^{41} -2079.37i q^{43} +(-4449.29 + 15954.7i) q^{45} -24390.1 q^{47} -2401.00 q^{49} +(143.984 - 1052.32i) q^{51} +14672.3i q^{53} -11179.0i q^{55} +(-3790.47 + 27703.0i) q^{57} -20279.7 q^{59} -12977.9 q^{61} +(-3198.47 + 11469.4i) q^{63} -12370.1i q^{65} -31142.9i q^{67} +(6730.44 + 920.892i) q^{69} +2135.92 q^{71} +73797.0 q^{73} +(-23492.6 - 3214.38i) q^{75} -8036.28i q^{77} +43004.6i q^{79} +(50527.4 + 30557.7i) q^{81} -99198.7 q^{83} +4644.25 q^{85} +(-4044.94 + 29562.8i) q^{87} -19290.7i q^{89} -8892.56i q^{91} +(-6615.15 + 48347.5i) q^{93} -122263. q^{95} +176015. q^{97} +(-38388.6 - 10705.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85}+ \cdots - 276400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 15.4446 + 2.11320i 0.990769 + 0.135562i
\(4\) 0 0
\(5\) 68.1623i 1.21932i 0.792662 + 0.609662i \(0.208695\pi\)
−0.792662 + 0.609662i \(0.791305\pi\)
\(6\) 0 0
\(7\) 49.0000i 0.377964i
\(8\) 0 0
\(9\) 234.069 + 65.2750i 0.963246 + 0.268621i
\(10\) 0 0
\(11\) −164.006 −0.408674 −0.204337 − 0.978901i \(-0.565504\pi\)
−0.204337 + 0.978901i \(0.565504\pi\)
\(12\) 0 0
\(13\) −181.481 −0.297833 −0.148916 − 0.988850i \(-0.547579\pi\)
−0.148916 + 0.988850i \(0.547579\pi\)
\(14\) 0 0
\(15\) −144.041 + 1052.74i −0.165294 + 1.20807i
\(16\) 0 0
\(17\) − 68.1353i − 0.0571807i −0.999591 − 0.0285904i \(-0.990898\pi\)
0.999591 − 0.0285904i \(-0.00910183\pi\)
\(18\) 0 0
\(19\) 1793.71i 1.13990i 0.821678 + 0.569952i \(0.193038\pi\)
−0.821678 + 0.569952i \(0.806962\pi\)
\(20\) 0 0
\(21\) −103.547 + 756.783i −0.0512376 + 0.374475i
\(22\) 0 0
\(23\) 435.780 0.171770 0.0858852 − 0.996305i \(-0.472628\pi\)
0.0858852 + 0.996305i \(0.472628\pi\)
\(24\) 0 0
\(25\) −1521.09 −0.486750
\(26\) 0 0
\(27\) 3477.15 + 1502.78i 0.917939 + 0.396721i
\(28\) 0 0
\(29\) 1914.13i 0.422645i 0.977416 + 0.211322i \(0.0677770\pi\)
−0.977416 + 0.211322i \(0.932223\pi\)
\(30\) 0 0
\(31\) 3130.39i 0.585051i 0.956258 + 0.292526i \(0.0944957\pi\)
−0.956258 + 0.292526i \(0.905504\pi\)
\(32\) 0 0
\(33\) −2533.00 − 346.577i −0.404902 − 0.0554007i
\(34\) 0 0
\(35\) −3339.95 −0.460861
\(36\) 0 0
\(37\) −8701.65 −1.04495 −0.522477 − 0.852653i \(-0.674992\pi\)
−0.522477 + 0.852653i \(0.674992\pi\)
\(38\) 0 0
\(39\) −2802.89 − 383.506i −0.295083 − 0.0403748i
\(40\) 0 0
\(41\) − 9365.85i − 0.870137i −0.900397 − 0.435068i \(-0.856724\pi\)
0.900397 − 0.435068i \(-0.143276\pi\)
\(42\) 0 0
\(43\) − 2079.37i − 0.171499i −0.996317 − 0.0857493i \(-0.972672\pi\)
0.996317 − 0.0857493i \(-0.0273284\pi\)
\(44\) 0 0
\(45\) −4449.29 + 15954.7i −0.327536 + 1.17451i
\(46\) 0 0
\(47\) −24390.1 −1.61053 −0.805265 − 0.592916i \(-0.797977\pi\)
−0.805265 + 0.592916i \(0.797977\pi\)
\(48\) 0 0
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) 143.984 − 1052.32i 0.00775153 − 0.0566529i
\(52\) 0 0
\(53\) 14672.3i 0.717477i 0.933438 + 0.358739i \(0.116793\pi\)
−0.933438 + 0.358739i \(0.883207\pi\)
\(54\) 0 0
\(55\) − 11179.0i − 0.498306i
\(56\) 0 0
\(57\) −3790.47 + 27703.0i −0.154528 + 1.12938i
\(58\) 0 0
\(59\) −20279.7 −0.758457 −0.379228 − 0.925303i \(-0.623811\pi\)
−0.379228 + 0.925303i \(0.623811\pi\)
\(60\) 0 0
\(61\) −12977.9 −0.446559 −0.223280 − 0.974754i \(-0.571676\pi\)
−0.223280 + 0.974754i \(0.571676\pi\)
\(62\) 0 0
\(63\) −3198.47 + 11469.4i −0.101529 + 0.364073i
\(64\) 0 0
\(65\) − 12370.1i − 0.363154i
\(66\) 0 0
\(67\) − 31142.9i − 0.847564i −0.905764 − 0.423782i \(-0.860702\pi\)
0.905764 − 0.423782i \(-0.139298\pi\)
\(68\) 0 0
\(69\) 6730.44 + 920.892i 0.170185 + 0.0232855i
\(70\) 0 0
\(71\) 2135.92 0.0502851 0.0251425 − 0.999684i \(-0.491996\pi\)
0.0251425 + 0.999684i \(0.491996\pi\)
\(72\) 0 0
\(73\) 73797.0 1.62081 0.810404 − 0.585872i \(-0.199248\pi\)
0.810404 + 0.585872i \(0.199248\pi\)
\(74\) 0 0
\(75\) −23492.6 − 3214.38i −0.482257 − 0.0659848i
\(76\) 0 0
\(77\) − 8036.28i − 0.154464i
\(78\) 0 0
\(79\) 43004.6i 0.775259i 0.921815 + 0.387630i \(0.126706\pi\)
−0.921815 + 0.387630i \(0.873294\pi\)
\(80\) 0 0
\(81\) 50527.4 + 30557.7i 0.855685 + 0.517497i
\(82\) 0 0
\(83\) −99198.7 −1.58056 −0.790279 − 0.612747i \(-0.790064\pi\)
−0.790279 + 0.612747i \(0.790064\pi\)
\(84\) 0 0
\(85\) 4644.25 0.0697218
\(86\) 0 0
\(87\) −4044.94 + 29562.8i −0.0572946 + 0.418743i
\(88\) 0 0
\(89\) − 19290.7i − 0.258150i −0.991635 − 0.129075i \(-0.958799\pi\)
0.991635 − 0.129075i \(-0.0412009\pi\)
\(90\) 0 0
\(91\) − 8892.56i − 0.112570i
\(92\) 0 0
\(93\) −6615.15 + 48347.5i −0.0793108 + 0.579651i
\(94\) 0 0
\(95\) −122263. −1.38991
\(96\) 0 0
\(97\) 176015. 1.89942 0.949708 − 0.313138i \(-0.101380\pi\)
0.949708 + 0.313138i \(0.101380\pi\)
\(98\) 0 0
\(99\) −38388.6 − 10705.5i −0.393654 − 0.109779i
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.h.b.239.38 yes 40
3.2 odd 2 inner 336.6.h.b.239.4 yes 40
4.3 odd 2 inner 336.6.h.b.239.3 ✓ 40
12.11 even 2 inner 336.6.h.b.239.37 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.6.h.b.239.3 ✓ 40 4.3 odd 2 inner
336.6.h.b.239.4 yes 40 3.2 odd 2 inner
336.6.h.b.239.37 yes 40 12.11 even 2 inner
336.6.h.b.239.38 yes 40 1.1 even 1 trivial