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Properties

Label 336.6.h.b.239.5
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 239.5
Character \(\chi\) \(=\) 336.239
Dual form 336.6.h.b.239.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-13.6996 - 7.43773i) q^{3} -101.234i q^{5} -49.0000i q^{7} +(132.360 + 203.789i) q^{9} +563.943 q^{11} -261.477 q^{13} +(-752.954 + 1386.87i) q^{15} +2058.95i q^{17} +978.317i q^{19} +(-364.449 + 671.282i) q^{21} +2208.61 q^{23} -7123.40 q^{25} +(-297.564 - 3776.29i) q^{27} +5604.45i q^{29} +4737.20i q^{31} +(-7725.81 - 4194.46i) q^{33} -4960.48 q^{35} +10817.2 q^{37} +(3582.14 + 1944.80i) q^{39} +17703.6i q^{41} -12329.1i q^{43} +(20630.4 - 13399.4i) q^{45} -6701.70 q^{47} -2401.00 q^{49} +(15313.9 - 28206.8i) q^{51} +24845.8i q^{53} -57090.4i q^{55} +(7276.46 - 13402.6i) q^{57} -21403.3 q^{59} +4697.43 q^{61} +(9985.64 - 6485.65i) q^{63} +26470.5i q^{65} +49298.4i q^{67} +(-30257.2 - 16427.1i) q^{69} +42579.5 q^{71} -1240.19 q^{73} +(97588.0 + 52981.9i) q^{75} -27633.2i q^{77} +63235.3i q^{79} +(-24010.5 + 53947.0i) q^{81} +62227.9 q^{83} +208436. q^{85} +(41684.4 - 76779.0i) q^{87} +27544.1i q^{89} +12812.4i q^{91} +(35234.0 - 64897.9i) q^{93} +99039.3 q^{95} -178615. q^{97} +(74643.6 + 114925. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85}+ \cdots - 276400 q^{97}+O(q^{100}) \) 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
\(3\) −13.6996 − 7.43773i −0.878832 − 0.477131i
\(4\) 0 0
\(5\) − 101.234i − 1.81094i −0.424415 − 0.905468i \(-0.639520\pi\)
0.424415 − 0.905468i \(-0.360480\pi\)
\(6\) 0 0
\(7\) − 49.0000i − 0.377964i
\(8\) 0 0
\(9\) 132.360 + 203.789i 0.544692 + 0.838636i
\(10\) 0 0
\(11\) 563.943 1.40525 0.702625 − 0.711561i \(-0.252011\pi\)
0.702625 + 0.711561i \(0.252011\pi\)
\(12\) 0 0
\(13\) −261.477 −0.429117 −0.214558 − 0.976711i \(-0.568831\pi\)
−0.214558 + 0.976711i \(0.568831\pi\)
\(14\) 0 0
\(15\) −752.954 + 1386.87i −0.864053 + 1.59151i
\(16\) 0 0
\(17\) 2058.95i 1.72792i 0.503563 + 0.863959i \(0.332022\pi\)
−0.503563 + 0.863959i \(0.667978\pi\)
\(18\) 0 0
\(19\) 978.317i 0.621721i 0.950456 + 0.310861i \(0.100617\pi\)
−0.950456 + 0.310861i \(0.899383\pi\)
\(20\) 0 0
\(21\) −364.449 + 671.282i −0.180338 + 0.332167i
\(22\) 0 0
\(23\) 2208.61 0.870563 0.435282 − 0.900294i \(-0.356649\pi\)
0.435282 + 0.900294i \(0.356649\pi\)
\(24\) 0 0
\(25\) −7123.40 −2.27949
\(26\) 0 0
\(27\) −297.564 − 3776.29i −0.0785544 − 0.996910i
\(28\) 0 0
\(29\) 5604.45i 1.23748i 0.785596 + 0.618740i \(0.212356\pi\)
−0.785596 + 0.618740i \(0.787644\pi\)
\(30\) 0 0
\(31\) 4737.20i 0.885355i 0.896681 + 0.442677i \(0.145971\pi\)
−0.896681 + 0.442677i \(0.854029\pi\)
\(32\) 0 0
\(33\) −7725.81 − 4194.46i −1.23498 − 0.670488i
\(34\) 0 0
\(35\) −4960.48 −0.684469
\(36\) 0 0
\(37\) 10817.2 1.29900 0.649502 − 0.760360i \(-0.274978\pi\)
0.649502 + 0.760360i \(0.274978\pi\)
\(38\) 0 0
\(39\) 3582.14 + 1944.80i 0.377122 + 0.204745i
\(40\) 0 0
\(41\) 17703.6i 1.64476i 0.568940 + 0.822379i \(0.307354\pi\)
−0.568940 + 0.822379i \(0.692646\pi\)
\(42\) 0 0
\(43\) − 12329.1i − 1.01686i −0.861104 − 0.508429i \(-0.830226\pi\)
0.861104 − 0.508429i \(-0.169774\pi\)
\(44\) 0 0
\(45\) 20630.4 − 13399.4i 1.51872 − 0.986403i
\(46\) 0 0
\(47\) −6701.70 −0.442528 −0.221264 − 0.975214i \(-0.571018\pi\)
−0.221264 + 0.975214i \(0.571018\pi\)
\(48\) 0 0
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) 15313.9 − 28206.8i 0.824442 − 1.51855i
\(52\) 0 0
\(53\) 24845.8i 1.21496i 0.794334 + 0.607482i \(0.207820\pi\)
−0.794334 + 0.607482i \(0.792180\pi\)
\(54\) 0 0
\(55\) − 57090.4i − 2.54482i
\(56\) 0 0
\(57\) 7276.46 − 13402.6i 0.296642 − 0.546389i
\(58\) 0 0
\(59\) −21403.3 −0.800480 −0.400240 − 0.916410i \(-0.631073\pi\)
−0.400240 + 0.916410i \(0.631073\pi\)
\(60\) 0 0
\(61\) 4697.43 0.161635 0.0808176 − 0.996729i \(-0.474247\pi\)
0.0808176 + 0.996729i \(0.474247\pi\)
\(62\) 0 0
\(63\) 9985.64 − 6485.65i 0.316975 − 0.205874i
\(64\) 0 0
\(65\) 26470.5i 0.777103i
\(66\) 0 0
\(67\) 49298.4i 1.34167i 0.741606 + 0.670835i \(0.234064\pi\)
−0.741606 + 0.670835i \(0.765936\pi\)
\(68\) 0 0
\(69\) −30257.2 − 16427.1i −0.765079 − 0.415373i
\(70\) 0 0
\(71\) 42579.5 1.00243 0.501216 − 0.865322i \(-0.332886\pi\)
0.501216 + 0.865322i \(0.332886\pi\)
\(72\) 0 0
\(73\) −1240.19 −0.0272384 −0.0136192 − 0.999907i \(-0.504335\pi\)
−0.0136192 + 0.999907i \(0.504335\pi\)
\(74\) 0 0
\(75\) 97588.0 + 52981.9i 2.00329 + 1.08761i
\(76\) 0 0
\(77\) − 27633.2i − 0.531134i
\(78\) 0 0
\(79\) 63235.3i 1.13997i 0.821657 + 0.569983i \(0.193050\pi\)
−0.821657 + 0.569983i \(0.806950\pi\)
\(80\) 0 0
\(81\) −24010.5 + 53947.0i −0.406620 + 0.913597i
\(82\) 0 0
\(83\) 62227.9 0.991493 0.495746 − 0.868467i \(-0.334895\pi\)
0.495746 + 0.868467i \(0.334895\pi\)
\(84\) 0 0
\(85\) 208436. 3.12915
\(86\) 0 0
\(87\) 41684.4 − 76779.0i 0.590440 − 1.08754i
\(88\) 0 0
\(89\) 27544.1i 0.368598i 0.982870 + 0.184299i \(0.0590015\pi\)
−0.982870 + 0.184299i \(0.940998\pi\)
\(90\) 0 0
\(91\) 12812.4i 0.162191i
\(92\) 0 0
\(93\) 35234.0 − 64897.9i 0.422430 − 0.778078i
\(94\) 0 0
\(95\) 99039.3 1.12590
\(96\) 0 0
\(97\) −178615. −1.92748 −0.963739 − 0.266847i \(-0.914018\pi\)
−0.963739 + 0.266847i \(0.914018\pi\)
\(98\) 0 0
\(99\) 74643.6 + 114925.i 0.765429 + 1.17849i
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.239.5 ✓ 40
3.2 odd 2 inner 336.6.h.b.239.35 yes 40
4.3 odd 2 inner 336.6.h.b.239.36 yes 40
12.11 even 2 inner 336.6.h.b.239.6 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.6.h.b.239.5 ✓ 40 1.1 even 1 trivial
336.6.h.b.239.6 yes 40 12.11 even 2 inner
336.6.h.b.239.35 yes 40 3.2 odd 2 inner
336.6.h.b.239.36 yes 40 4.3 odd 2 inner