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Properties

Label 336.6.h.b.239.32
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 239.32
Character \(\chi\) \(=\) 336.239
Dual form 336.6.h.b.239.31

$q$-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)
 
\(f(q)\) \(=\) \(q+(11.6605 + 10.3456i) q^{3} +60.0268i q^{5} -49.0000i q^{7} +(28.9356 + 241.271i) q^{9} +595.559 q^{11} -1171.32 q^{13} +(-621.016 + 699.944i) q^{15} +546.441i q^{17} +1026.70i q^{19} +(506.936 - 571.366i) q^{21} +132.143 q^{23} -478.218 q^{25} +(-2158.70 + 3112.70i) q^{27} +4610.62i q^{29} -2766.38i q^{31} +(6944.53 + 6161.44i) q^{33} +2941.31 q^{35} -7219.40 q^{37} +(-13658.2 - 12118.1i) q^{39} +6431.81i q^{41} +6852.22i q^{43} +(-14482.7 + 1736.91i) q^{45} -23988.1 q^{47} -2401.00 q^{49} +(-5653.28 + 6371.78i) q^{51} -30034.6i q^{53} +35749.5i q^{55} +(-10621.9 + 11971.9i) q^{57} -4422.95 q^{59} +49598.2 q^{61} +(11822.3 - 1417.84i) q^{63} -70310.7i q^{65} -25936.4i q^{67} +(1540.86 + 1367.11i) q^{69} -20754.8 q^{71} -39745.8 q^{73} +(-5576.27 - 4947.47i) q^{75} -29182.4i q^{77} +51274.0i q^{79} +(-57374.5 + 13962.6i) q^{81} +18429.8 q^{83} -32801.1 q^{85} +(-47699.8 + 53762.3i) q^{87} -46758.1i q^{89} +57394.8i q^{91} +(28619.9 - 32257.4i) q^{93} -61629.7 q^{95} -138257. q^{97} +(17232.9 + 143691. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85}+ \cdots - 276400 q^{97}+O(q^{100}) \) 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
\(3\) 11.6605 + 10.3456i 0.748023 + 0.663673i
\(4\) 0 0
\(5\) 60.0268i 1.07379i 0.843648 + 0.536896i \(0.180403\pi\)
−0.843648 + 0.536896i \(0.819597\pi\)
\(6\) 0 0
\(7\) − 49.0000i − 0.377964i
\(8\) 0 0
\(9\) 28.9356 + 241.271i 0.119077 + 0.992885i
\(10\) 0 0
\(11\) 595.559 1.48403 0.742016 − 0.670382i \(-0.233870\pi\)
0.742016 + 0.670382i \(0.233870\pi\)
\(12\) 0 0
\(13\) −1171.32 −1.92229 −0.961143 − 0.276051i \(-0.910974\pi\)
−0.961143 + 0.276051i \(0.910974\pi\)
\(14\) 0 0
\(15\) −621.016 + 699.944i −0.712647 + 0.803221i
\(16\) 0 0
\(17\) 546.441i 0.458586i 0.973357 + 0.229293i \(0.0736414\pi\)
−0.973357 + 0.229293i \(0.926359\pi\)
\(18\) 0 0
\(19\) 1026.70i 0.652471i 0.945289 + 0.326235i \(0.105780\pi\)
−0.945289 + 0.326235i \(0.894220\pi\)
\(20\) 0 0
\(21\) 506.936 − 571.366i 0.250845 − 0.282726i
\(22\) 0 0
\(23\) 132.143 0.0520866 0.0260433 − 0.999661i \(-0.491709\pi\)
0.0260433 + 0.999661i \(0.491709\pi\)
\(24\) 0 0
\(25\) −478.218 −0.153030
\(26\) 0 0
\(27\) −2158.70 + 3112.70i −0.569879 + 0.821729i
\(28\) 0 0
\(29\) 4610.62i 1.01804i 0.860755 + 0.509020i \(0.169992\pi\)
−0.860755 + 0.509020i \(0.830008\pi\)
\(30\) 0 0
\(31\) − 2766.38i − 0.517019i −0.966009 − 0.258510i \(-0.916769\pi\)
0.966009 − 0.258510i \(-0.0832314\pi\)
\(32\) 0 0
\(33\) 6944.53 + 6161.44i 1.11009 + 0.984912i
\(34\) 0 0
\(35\) 2941.31 0.405855
\(36\) 0 0
\(37\) −7219.40 −0.866955 −0.433478 − 0.901164i \(-0.642714\pi\)
−0.433478 + 0.901164i \(0.642714\pi\)
\(38\) 0 0
\(39\) −13658.2 − 12118.1i −1.43791 − 1.27577i
\(40\) 0 0
\(41\) 6431.81i 0.597549i 0.954324 + 0.298774i \(0.0965778\pi\)
−0.954324 + 0.298774i \(0.903422\pi\)
\(42\) 0 0
\(43\) 6852.22i 0.565146i 0.959246 + 0.282573i \(0.0911879\pi\)
−0.959246 + 0.282573i \(0.908812\pi\)
\(44\) 0 0
\(45\) −14482.7 + 1736.91i −1.06615 + 0.127863i
\(46\) 0 0
\(47\) −23988.1 −1.58398 −0.791992 − 0.610532i \(-0.790956\pi\)
−0.791992 + 0.610532i \(0.790956\pi\)
\(48\) 0 0
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) −5653.28 + 6371.78i −0.304351 + 0.343033i
\(52\) 0 0
\(53\) − 30034.6i − 1.46870i −0.678772 − 0.734349i \(-0.737487\pi\)
0.678772 − 0.734349i \(-0.262513\pi\)
\(54\) 0 0
\(55\) 35749.5i 1.59354i
\(56\) 0 0
\(57\) −10621.9 + 11971.9i −0.433027 + 0.488063i
\(58\) 0 0
\(59\) −4422.95 −0.165418 −0.0827088 − 0.996574i \(-0.526357\pi\)
−0.0827088 + 0.996574i \(0.526357\pi\)
\(60\) 0 0
\(61\) 49598.2 1.70664 0.853318 − 0.521391i \(-0.174587\pi\)
0.853318 + 0.521391i \(0.174587\pi\)
\(62\) 0 0
\(63\) 11822.3 − 1417.84i 0.375275 − 0.0450067i
\(64\) 0 0
\(65\) − 70310.7i − 2.06414i
\(66\) 0 0
\(67\) − 25936.4i − 0.705865i −0.935649 − 0.352933i \(-0.885184\pi\)
0.935649 − 0.352933i \(-0.114816\pi\)
\(68\) 0 0
\(69\) 1540.86 + 1367.11i 0.0389620 + 0.0345685i
\(70\) 0 0
\(71\) −20754.8 −0.488621 −0.244310 − 0.969697i \(-0.578562\pi\)
−0.244310 + 0.969697i \(0.578562\pi\)
\(72\) 0 0
\(73\) −39745.8 −0.872939 −0.436470 − 0.899719i \(-0.643771\pi\)
−0.436470 + 0.899719i \(0.643771\pi\)
\(74\) 0 0
\(75\) −5576.27 − 4947.47i −0.114470 − 0.101562i
\(76\) 0 0
\(77\) − 29182.4i − 0.560911i
\(78\) 0 0
\(79\) 51274.0i 0.924336i 0.886792 + 0.462168i \(0.152928\pi\)
−0.886792 + 0.462168i \(0.847072\pi\)
\(80\) 0 0
\(81\) −57374.5 + 13962.6i −0.971642 + 0.236459i
\(82\) 0 0
\(83\) 18429.8 0.293646 0.146823 − 0.989163i \(-0.453095\pi\)
0.146823 + 0.989163i \(0.453095\pi\)
\(84\) 0 0
\(85\) −32801.1 −0.492426
\(86\) 0 0
\(87\) −47699.8 + 53762.3i −0.675645 + 0.761517i
\(88\) 0 0
\(89\) − 46758.1i − 0.625723i −0.949799 − 0.312862i \(-0.898712\pi\)
0.949799 − 0.312862i \(-0.101288\pi\)
\(90\) 0 0
\(91\) 57394.8i 0.726556i
\(92\) 0 0
\(93\) 28619.9 − 32257.4i 0.343132 − 0.386742i
\(94\) 0 0
\(95\) −61629.7 −0.700618
\(96\) 0 0
\(97\) −138257. −1.49196 −0.745980 − 0.665968i \(-0.768019\pi\)
−0.745980 + 0.665968i \(0.768019\pi\)
\(98\) 0 0
\(99\) 17232.9 + 143691.i 0.176713 + 1.47347i
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.239.32 yes 40
3.2 odd 2 inner 336.6.h.b.239.10 yes 40
4.3 odd 2 inner 336.6.h.b.239.9 ✓ 40
12.11 even 2 inner 336.6.h.b.239.31 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.6.h.b.239.9 ✓ 40 4.3 odd 2 inner
336.6.h.b.239.10 yes 40 3.2 odd 2 inner
336.6.h.b.239.31 yes 40 12.11 even 2 inner
336.6.h.b.239.32 yes 40 1.1 even 1 trivial