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Properties

Label 336.2.h.b.239.8
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 239.8
Root \(0.500000 - 1.19293i\) of defining polynomial
Character \(\chi\) \(=\) 336.239
Dual form 336.2.h.b.239.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.69293 + 0.366025i) q^{3} -3.38587i q^{5} -1.00000i q^{7} +(2.73205 + 1.23931i) q^{9} +2.47863 q^{11} -6.19615 q^{13} +(1.23931 - 5.73205i) q^{15} -2.47863i q^{17} +0.732051i q^{19} +(0.366025 - 1.69293i) q^{21} +6.77174 q^{23} -6.46410 q^{25} +(4.17156 + 3.09808i) q^{27} +2.47863i q^{29} +9.46410i q^{31} +(4.19615 + 0.907241i) q^{33} -3.38587 q^{35} +4.53590 q^{37} +(-10.4897 - 2.26795i) q^{39} +9.25036i q^{41} +2.00000i q^{43} +(4.19615 - 9.25036i) q^{45} -1.00000 q^{49} +(0.907241 - 4.19615i) q^{51} -9.25036i q^{53} -8.39230i q^{55} +(-0.267949 + 1.23931i) q^{57} -8.34312 q^{59} -4.73205 q^{61} +(1.23931 - 2.73205i) q^{63} +20.9794i q^{65} -3.46410i q^{67} +(11.4641 + 2.47863i) q^{69} -9.25036 q^{71} -4.53590 q^{73} +(-10.9433 - 2.36603i) q^{75} -2.47863i q^{77} +12.0000i q^{79} +(5.92820 + 6.77174i) q^{81} -8.34312 q^{83} -8.39230 q^{85} +(-0.907241 + 4.19615i) q^{87} -14.2076i q^{89} +6.19615i q^{91} +(-3.46410 + 16.0221i) q^{93} +2.47863 q^{95} +8.92820 q^{97} +(6.77174 + 3.07180i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) \) 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
\(3\) 1.69293 + 0.366025i 0.977416 + 0.211325i
\(4\) 0 0
\(5\) − 3.38587i − 1.51421i −0.653295 − 0.757103i \(-0.726614\pi\)
0.653295 − 0.757103i \(-0.273386\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 2.73205 + 1.23931i 0.910684 + 0.413105i
\(10\) 0 0
\(11\) 2.47863 0.747334 0.373667 − 0.927563i \(-0.378100\pi\)
0.373667 + 0.927563i \(0.378100\pi\)
\(12\) 0 0
\(13\) −6.19615 −1.71850 −0.859252 − 0.511553i \(-0.829070\pi\)
−0.859252 + 0.511553i \(0.829070\pi\)
\(14\) 0 0
\(15\) 1.23931 − 5.73205i 0.319989 − 1.48001i
\(16\) 0 0
\(17\) − 2.47863i − 0.601155i −0.953757 − 0.300578i \(-0.902821\pi\)
0.953757 − 0.300578i \(-0.0971795\pi\)
\(18\) 0 0
\(19\) 0.732051i 0.167944i 0.996468 + 0.0839720i \(0.0267606\pi\)
−0.996468 + 0.0839720i \(0.973239\pi\)
\(20\) 0 0
\(21\) 0.366025 − 1.69293i 0.0798733 − 0.369428i
\(22\) 0 0
\(23\) 6.77174 1.41200 0.706002 − 0.708210i \(-0.250497\pi\)
0.706002 + 0.708210i \(0.250497\pi\)
\(24\) 0 0
\(25\) −6.46410 −1.29282
\(26\) 0 0
\(27\) 4.17156 + 3.09808i 0.802817 + 0.596225i
\(28\) 0 0
\(29\) 2.47863i 0.460270i 0.973159 + 0.230135i \(0.0739167\pi\)
−0.973159 + 0.230135i \(0.926083\pi\)
\(30\) 0 0
\(31\) 9.46410i 1.69980i 0.526942 + 0.849901i \(0.323339\pi\)
−0.526942 + 0.849901i \(0.676661\pi\)
\(32\) 0 0
\(33\) 4.19615 + 0.907241i 0.730456 + 0.157930i
\(34\) 0 0
\(35\) −3.38587 −0.572316
\(36\) 0 0
\(37\) 4.53590 0.745697 0.372849 − 0.927892i \(-0.378381\pi\)
0.372849 + 0.927892i \(0.378381\pi\)
\(38\) 0 0
\(39\) −10.4897 − 2.26795i −1.67969 − 0.363163i
\(40\) 0 0
\(41\) 9.25036i 1.44466i 0.691546 + 0.722332i \(0.256930\pi\)
−0.691546 + 0.722332i \(0.743070\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 4.19615 − 9.25036i 0.625525 − 1.37896i
\(46\) 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
\(51\) 0.907241 − 4.19615i 0.127039 − 0.587579i
\(52\) 0 0
\(53\) − 9.25036i − 1.27064i −0.772251 − 0.635318i \(-0.780869\pi\)
0.772251 − 0.635318i \(-0.219131\pi\)
\(54\) 0 0
\(55\) − 8.39230i − 1.13162i
\(56\) 0 0
\(57\) −0.267949 + 1.23931i −0.0354907 + 0.164151i
\(58\) 0 0
\(59\) −8.34312 −1.08618 −0.543091 − 0.839674i \(-0.682746\pi\)
−0.543091 + 0.839674i \(0.682746\pi\)
\(60\) 0 0
\(61\) −4.73205 −0.605877 −0.302939 − 0.953010i \(-0.597968\pi\)
−0.302939 + 0.953010i \(0.597968\pi\)
\(62\) 0 0
\(63\) 1.23931 − 2.73205i 0.156139 − 0.344206i
\(64\) 0 0
\(65\) 20.9794i 2.60217i
\(66\) 0 0
\(67\) − 3.46410i − 0.423207i −0.977356 − 0.211604i \(-0.932131\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 11.4641 + 2.47863i 1.38012 + 0.298392i
\(70\) 0 0
\(71\) −9.25036 −1.09782 −0.548908 − 0.835883i \(-0.684956\pi\)
−0.548908 + 0.835883i \(0.684956\pi\)
\(72\) 0 0
\(73\) −4.53590 −0.530887 −0.265443 − 0.964126i \(-0.585518\pi\)
−0.265443 + 0.964126i \(0.585518\pi\)
\(74\) 0 0
\(75\) −10.9433 − 2.36603i −1.26362 − 0.273205i
\(76\) 0 0
\(77\) − 2.47863i − 0.282466i
\(78\) 0 0
\(79\) 12.0000i 1.35011i 0.737769 + 0.675053i \(0.235879\pi\)
−0.737769 + 0.675053i \(0.764121\pi\)
\(80\) 0 0
\(81\) 5.92820 + 6.77174i 0.658689 + 0.752415i
\(82\) 0 0
\(83\) −8.34312 −0.915777 −0.457888 − 0.889010i \(-0.651394\pi\)
−0.457888 + 0.889010i \(0.651394\pi\)
\(84\) 0 0
\(85\) −8.39230 −0.910273
\(86\) 0 0
\(87\) −0.907241 + 4.19615i −0.0972664 + 0.449875i
\(88\) 0 0
\(89\) − 14.2076i − 1.50600i −0.658018 − 0.753002i \(-0.728605\pi\)
0.658018 − 0.753002i \(-0.271395\pi\)
\(90\) 0 0
\(91\) 6.19615i 0.649533i
\(92\) 0 0
\(93\) −3.46410 + 16.0221i −0.359211 + 1.66141i
\(94\) 0 0
\(95\) 2.47863 0.254302
\(96\) 0 0
\(97\) 8.92820 0.906522 0.453261 − 0.891378i \(-0.350261\pi\)
0.453261 + 0.891378i \(0.350261\pi\)
\(98\) 0 0
\(99\) 6.77174 + 3.07180i 0.680585 + 0.308727i
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.239.8 yes 8
3.2 odd 2 inner 336.2.h.b.239.2 yes 8
4.3 odd 2 inner 336.2.h.b.239.1 ✓ 8
7.6 odd 2 2352.2.h.o.2255.1 8
8.3 odd 2 1344.2.h.g.575.8 8
8.5 even 2 1344.2.h.g.575.1 8
12.11 even 2 inner 336.2.h.b.239.7 yes 8
21.20 even 2 2352.2.h.o.2255.7 8
24.5 odd 2 1344.2.h.g.575.7 8
24.11 even 2 1344.2.h.g.575.2 8
28.27 even 2 2352.2.h.o.2255.8 8
84.83 odd 2 2352.2.h.o.2255.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.h.b.239.1 ✓ 8 4.3 odd 2 inner
336.2.h.b.239.2 yes 8 3.2 odd 2 inner
336.2.h.b.239.7 yes 8 12.11 even 2 inner
336.2.h.b.239.8 yes 8 1.1 even 1 trivial
1344.2.h.g.575.1 8 8.5 even 2
1344.2.h.g.575.2 8 24.11 even 2
1344.2.h.g.575.7 8 24.5 odd 2
1344.2.h.g.575.8 8 8.3 odd 2
2352.2.h.o.2255.1 8 7.6 odd 2
2352.2.h.o.2255.2 8 84.83 odd 2
2352.2.h.o.2255.7 8 21.20 even 2
2352.2.h.o.2255.8 8 28.27 even 2