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Properties

Label 336.2.h.b.239.6
Level $336$
Weight $2$
Character 336.239
Analytic conductor $2.683$
Analytic rank $0$
Dimension $8$
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] = [336,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.6
Root \(0.500000 + 0.564882i\) of defining polynomial
Character \(\chi\) \(=\) 336.239
Dual form 336.2.h.b.239.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.06488 + 1.36603i) q^{3} +2.12976i q^{5} +1.00000i q^{7} +(-0.732051 + 2.90931i) q^{9} -5.81863 q^{11} +4.19615 q^{13} +(-2.90931 + 2.26795i) q^{15} -5.81863i q^{17} +2.73205i q^{19} +(-1.36603 + 1.06488i) q^{21} +4.25953 q^{23} +0.464102 q^{25} +(-4.75374 + 2.09808i) q^{27} +5.81863i q^{29} -2.53590i q^{31} +(-6.19615 - 7.94839i) q^{33} -2.12976 q^{35} +11.4641 q^{37} +(4.46841 + 5.73205i) q^{39} +1.55910i q^{41} -2.00000i q^{43} +(-6.19615 - 1.55910i) q^{45} -1.00000 q^{49} +(7.94839 - 6.19615i) q^{51} -1.55910i q^{53} -12.3923i q^{55} +(-3.73205 + 2.90931i) q^{57} +9.50749 q^{59} -1.26795 q^{61} +(-2.90931 - 0.732051i) q^{63} +8.93682i q^{65} -3.46410i q^{67} +(4.53590 + 5.81863i) q^{69} +1.55910 q^{71} -11.4641 q^{73} +(0.494214 + 0.633975i) q^{75} -5.81863i q^{77} -12.0000i q^{79} +(-7.92820 - 4.25953i) q^{81} +9.50749 q^{83} +12.3923 q^{85} +(-7.94839 + 6.19615i) q^{87} -13.1963i q^{89} +4.19615i q^{91} +(3.46410 - 2.70043i) q^{93} -5.81863 q^{95} -4.92820 q^{97} +(4.25953 - 16.9282i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) \) 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
\(3\) 1.06488 + 1.36603i 0.614810 + 0.788675i
\(4\) 0 0
\(5\) 2.12976i 0.952460i 0.879321 + 0.476230i \(0.157997\pi\)
−0.879321 + 0.476230i \(0.842003\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −0.732051 + 2.90931i −0.244017 + 0.969771i
\(10\) 0 0
\(11\) −5.81863 −1.75438 −0.877191 − 0.480142i \(-0.840585\pi\)
−0.877191 + 0.480142i \(0.840585\pi\)
\(12\) 0 0
\(13\) 4.19615 1.16380 0.581902 − 0.813259i \(-0.302309\pi\)
0.581902 + 0.813259i \(0.302309\pi\)
\(14\) 0 0
\(15\) −2.90931 + 2.26795i −0.751181 + 0.585582i
\(16\) 0 0
\(17\) − 5.81863i − 1.41122i −0.708598 − 0.705612i \(-0.750672\pi\)
0.708598 − 0.705612i \(-0.249328\pi\)
\(18\) 0 0
\(19\) 2.73205i 0.626775i 0.949625 + 0.313388i \(0.101464\pi\)
−0.949625 + 0.313388i \(0.898536\pi\)
\(20\) 0 0
\(21\) −1.36603 + 1.06488i −0.298091 + 0.232376i
\(22\) 0 0
\(23\) 4.25953 0.888173 0.444087 − 0.895984i \(-0.353528\pi\)
0.444087 + 0.895984i \(0.353528\pi\)
\(24\) 0 0
\(25\) 0.464102 0.0928203
\(26\) 0 0
\(27\) −4.75374 + 2.09808i −0.914858 + 0.403775i
\(28\) 0 0
\(29\) 5.81863i 1.08049i 0.841507 + 0.540246i \(0.181669\pi\)
−0.841507 + 0.540246i \(0.818331\pi\)
\(30\) 0 0
\(31\) − 2.53590i − 0.455461i −0.973724 − 0.227730i \(-0.926870\pi\)
0.973724 − 0.227730i \(-0.0731305\pi\)
\(32\) 0 0
\(33\) −6.19615 − 7.94839i −1.07861 − 1.38364i
\(34\) 0 0
\(35\) −2.12976 −0.359996
\(36\) 0 0
\(37\) 11.4641 1.88469 0.942343 − 0.334648i \(-0.108617\pi\)
0.942343 + 0.334648i \(0.108617\pi\)
\(38\) 0 0
\(39\) 4.46841 + 5.73205i 0.715518 + 0.917863i
\(40\) 0 0
\(41\) 1.55910i 0.243490i 0.992561 + 0.121745i \(0.0388490\pi\)
−0.992561 + 0.121745i \(0.961151\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) −6.19615 − 1.55910i −0.923668 − 0.232416i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.94839 − 6.19615i 1.11300 − 0.867635i
\(52\) 0 0
\(53\) − 1.55910i − 0.214158i −0.994250 − 0.107079i \(-0.965850\pi\)
0.994250 − 0.107079i \(-0.0341498\pi\)
\(54\) 0 0
\(55\) − 12.3923i − 1.67098i
\(56\) 0 0
\(57\) −3.73205 + 2.90931i −0.494322 + 0.385348i
\(58\) 0 0
\(59\) 9.50749 1.23777 0.618885 − 0.785482i \(-0.287585\pi\)
0.618885 + 0.785482i \(0.287585\pi\)
\(60\) 0 0
\(61\) −1.26795 −0.162344 −0.0811721 − 0.996700i \(-0.525866\pi\)
−0.0811721 + 0.996700i \(0.525866\pi\)
\(62\) 0 0
\(63\) −2.90931 − 0.732051i −0.366539 − 0.0922297i
\(64\) 0 0
\(65\) 8.93682i 1.10848i
\(66\) 0 0
\(67\) − 3.46410i − 0.423207i −0.977356 − 0.211604i \(-0.932131\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 4.53590 + 5.81863i 0.546058 + 0.700480i
\(70\) 0 0
\(71\) 1.55910 0.185031 0.0925153 − 0.995711i \(-0.470509\pi\)
0.0925153 + 0.995711i \(0.470509\pi\)
\(72\) 0 0
\(73\) −11.4641 −1.34177 −0.670886 − 0.741561i \(-0.734086\pi\)
−0.670886 + 0.741561i \(0.734086\pi\)
\(74\) 0 0
\(75\) 0.494214 + 0.633975i 0.0570669 + 0.0732051i
\(76\) 0 0
\(77\) − 5.81863i − 0.663094i
\(78\) 0 0
\(79\) − 12.0000i − 1.35011i −0.737769 − 0.675053i \(-0.764121\pi\)
0.737769 − 0.675053i \(-0.235879\pi\)
\(80\) 0 0
\(81\) −7.92820 − 4.25953i −0.880911 − 0.473281i
\(82\) 0 0
\(83\) 9.50749 1.04358 0.521791 − 0.853073i \(-0.325264\pi\)
0.521791 + 0.853073i \(0.325264\pi\)
\(84\) 0 0
\(85\) 12.3923 1.34413
\(86\) 0 0
\(87\) −7.94839 + 6.19615i −0.852157 + 0.664297i
\(88\) 0 0
\(89\) − 13.1963i − 1.39881i −0.714726 − 0.699405i \(-0.753448\pi\)
0.714726 − 0.699405i \(-0.246552\pi\)
\(90\) 0 0
\(91\) 4.19615i 0.439876i
\(92\) 0 0
\(93\) 3.46410 − 2.70043i 0.359211 − 0.280022i
\(94\) 0 0
\(95\) −5.81863 −0.596978
\(96\) 0 0
\(97\) −4.92820 −0.500383 −0.250192 − 0.968196i \(-0.580494\pi\)
−0.250192 + 0.968196i \(0.580494\pi\)
\(98\) 0 0
\(99\) 4.25953 − 16.9282i 0.428099 − 1.70135i
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 336.2.h.b.239.6 yes 8
3.2 odd 2 inner 336.2.h.b.239.4 yes 8
4.3 odd 2 inner 336.2.h.b.239.3 ✓ 8
7.6 odd 2 2352.2.h.o.2255.3 8
8.3 odd 2 1344.2.h.g.575.6 8
8.5 even 2 1344.2.h.g.575.3 8
12.11 even 2 inner 336.2.h.b.239.5 yes 8
21.20 even 2 2352.2.h.o.2255.5 8
24.5 odd 2 1344.2.h.g.575.5 8
24.11 even 2 1344.2.h.g.575.4 8
28.27 even 2 2352.2.h.o.2255.6 8
84.83 odd 2 2352.2.h.o.2255.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.h.b.239.3 ✓ 8 4.3 odd 2 inner
336.2.h.b.239.4 yes 8 3.2 odd 2 inner
336.2.h.b.239.5 yes 8 12.11 even 2 inner
336.2.h.b.239.6 yes 8 1.1 even 1 trivial
1344.2.h.g.575.3 8 8.5 even 2
1344.2.h.g.575.4 8 24.11 even 2
1344.2.h.g.575.5 8 24.5 odd 2
1344.2.h.g.575.6 8 8.3 odd 2
2352.2.h.o.2255.3 8 7.6 odd 2
2352.2.h.o.2255.4 8 84.83 odd 2
2352.2.h.o.2255.5 8 21.20 even 2
2352.2.h.o.2255.6 8 28.27 even 2