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Properties

Label 1008.2.cj.e.767.15
Level $1008$
Weight $2$
Character 1008.767
Analytic conductor $8.049$
Analytic rank $0$
Dimension $32$
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] = [1008,2,Mod(527,1008)] 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("1008.527"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 767.15
Character \(\chi\) \(=\) 1008.767
Dual form 1008.2.cj.e.527.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.65364 + 0.515254i) q^{3} +(1.18190 + 0.682369i) q^{5} +(2.63670 + 0.218628i) q^{7} +(2.46903 + 1.70409i) q^{9} +(0.465483 + 0.806241i) q^{11} +(0.456392 + 0.790494i) q^{13} +(1.60284 + 1.73737i) q^{15} +(-1.61193 - 0.930647i) q^{17} +(-0.643797 + 0.371696i) q^{19} +(4.24750 + 1.72010i) q^{21} +(-1.23787 + 2.14406i) q^{23} +(-1.56874 - 2.71714i) q^{25} +(3.20483 + 4.09012i) q^{27} +(-3.49659 - 2.01876i) q^{29} -3.48305i q^{31} +(0.354321 + 1.57307i) q^{33} +(2.96713 + 2.05760i) q^{35} +(-1.98011 - 3.42965i) q^{37} +(0.347401 + 1.54235i) q^{39} +(-1.06965 + 0.617564i) q^{41} +(7.13168 + 4.11748i) q^{43} +(1.75532 + 3.69884i) q^{45} -5.08479 q^{47} +(6.90440 + 1.15292i) q^{49} +(-2.18602 - 2.36950i) q^{51} +(-9.11540 - 5.26278i) q^{53} +1.27053i q^{55} +(-1.25612 + 0.282931i) q^{57} +11.1953 q^{59} -6.73209 q^{61} +(6.13753 + 5.03297i) q^{63} +1.24571i q^{65} +9.21074i q^{67} +(-3.15173 + 2.90767i) q^{69} +8.97964 q^{71} +(7.08863 - 12.2779i) q^{73} +(-1.19411 - 5.30147i) q^{75} +(1.05107 + 2.22758i) q^{77} -2.34061i q^{79} +(3.19218 + 8.41487i) q^{81} +(-2.71128 + 4.69607i) q^{83} +(-1.27009 - 2.19986i) q^{85} +(-4.74192 - 5.13992i) q^{87} +(-15.9049 + 9.18267i) q^{89} +(1.03055 + 2.18408i) q^{91} +(1.79466 - 5.75971i) q^{93} -1.01454 q^{95} +(2.70012 - 4.67674i) q^{97} +(-0.224613 + 2.78385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 4 q^{9} + 2 q^{13} + 18 q^{17} - 4 q^{21} + 16 q^{25} + 12 q^{29} - 8 q^{33} - 2 q^{37} - 56 q^{45} + 2 q^{49} - 24 q^{53} - 10 q^{57} + 28 q^{61} - 62 q^{69} - 28 q^{73} - 42 q^{77} + 4 q^{81} + 12 q^{85}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 1.65364 + 0.515254i 0.954727 + 0.297482i
\(4\) 0 0
\(5\) 1.18190 + 0.682369i 0.528561 + 0.305165i 0.740430 − 0.672133i \(-0.234622\pi\)
−0.211869 + 0.977298i \(0.567955\pi\)
\(6\) 0 0
\(7\) 2.63670 + 0.218628i 0.996580 + 0.0826337i
\(8\) 0 0
\(9\) 2.46903 + 1.70409i 0.823009 + 0.568029i
\(10\) 0 0
\(11\) 0.465483 + 0.806241i 0.140348 + 0.243091i 0.927628 − 0.373506i \(-0.121844\pi\)
−0.787279 + 0.616596i \(0.788511\pi\)
\(12\) 0 0
\(13\) 0.456392 + 0.790494i 0.126580 + 0.219244i 0.922350 − 0.386356i \(-0.126267\pi\)
−0.795769 + 0.605600i \(0.792933\pi\)
\(14\) 0 0
\(15\) 1.60284 + 1.73737i 0.413851 + 0.448587i
\(16\) 0 0
\(17\) −1.61193 − 0.930647i −0.390950 − 0.225715i 0.291622 − 0.956534i \(-0.405805\pi\)
−0.682572 + 0.730819i \(0.739139\pi\)
\(18\) 0 0
\(19\) −0.643797 + 0.371696i −0.147697 + 0.0852729i −0.572027 − 0.820235i \(-0.693843\pi\)
0.424330 + 0.905507i \(0.360510\pi\)
\(20\) 0 0
\(21\) 4.24750 + 1.72010i 0.926880 + 0.375357i
\(22\) 0 0
\(23\) −1.23787 + 2.14406i −0.258114 + 0.447067i −0.965737 − 0.259524i \(-0.916434\pi\)
0.707622 + 0.706591i \(0.249768\pi\)
\(24\) 0 0
\(25\) −1.56874 − 2.71714i −0.313749 − 0.543429i
\(26\) 0 0
\(27\) 3.20483 + 4.09012i 0.616771 + 0.787143i
\(28\) 0 0
\(29\) −3.49659 − 2.01876i −0.649301 − 0.374874i 0.138888 − 0.990308i \(-0.455647\pi\)
−0.788188 + 0.615434i \(0.788981\pi\)
\(30\) 0 0
\(31\) − 3.48305i − 0.625575i −0.949823 − 0.312788i \(-0.898737\pi\)
0.949823 − 0.312788i \(-0.101263\pi\)
\(32\) 0 0
\(33\) 0.354321 + 1.57307i 0.0616794 + 0.273837i
\(34\) 0 0
\(35\) 2.96713 + 2.05760i 0.501536 + 0.347798i
\(36\) 0 0
\(37\) −1.98011 − 3.42965i −0.325528 − 0.563831i 0.656091 − 0.754682i \(-0.272209\pi\)
−0.981619 + 0.190851i \(0.938875\pi\)
\(38\) 0 0
\(39\) 0.347401 + 1.54235i 0.0556287 + 0.246973i
\(40\) 0 0
\(41\) −1.06965 + 0.617564i −0.167052 + 0.0964473i −0.581195 − 0.813764i \(-0.697415\pi\)
0.414143 + 0.910212i \(0.364081\pi\)
\(42\) 0 0
\(43\) 7.13168 + 4.11748i 1.08757 + 0.627909i 0.932929 − 0.360061i \(-0.117244\pi\)
0.154642 + 0.987971i \(0.450578\pi\)
\(44\) 0 0
\(45\) 1.75532 + 3.69884i 0.261668 + 0.551391i
\(46\) 0 0
\(47\) −5.08479 −0.741693 −0.370846 − 0.928694i \(-0.620932\pi\)
−0.370846 + 0.928694i \(0.620932\pi\)
\(48\) 0 0
\(49\) 6.90440 + 1.15292i 0.986343 + 0.164702i
\(50\) 0 0
\(51\) −2.18602 − 2.36950i −0.306104 − 0.331797i
\(52\) 0 0
\(53\) −9.11540 − 5.26278i −1.25210 − 0.722898i −0.280571 − 0.959833i \(-0.590524\pi\)
−0.971525 + 0.236935i \(0.923857\pi\)
\(54\) 0 0
\(55\) 1.27053i 0.171318i
\(56\) 0 0
\(57\) −1.25612 + 0.282931i −0.166378 + 0.0374752i
\(58\) 0 0
\(59\) 11.1953 1.45751 0.728754 − 0.684776i \(-0.240100\pi\)
0.728754 + 0.684776i \(0.240100\pi\)
\(60\) 0 0
\(61\) −6.73209 −0.861956 −0.430978 − 0.902362i \(-0.641831\pi\)
−0.430978 + 0.902362i \(0.641831\pi\)
\(62\) 0 0
\(63\) 6.13753 + 5.03297i 0.773256 + 0.634094i
\(64\) 0 0
\(65\) 1.24571i 0.154511i
\(66\) 0 0
\(67\) 9.21074i 1.12527i 0.826705 + 0.562635i \(0.190213\pi\)
−0.826705 + 0.562635i \(0.809787\pi\)
\(68\) 0 0
\(69\) −3.15173 + 2.90767i −0.379423 + 0.350043i
\(70\) 0 0
\(71\) 8.97964 1.06569 0.532843 − 0.846214i \(-0.321123\pi\)
0.532843 + 0.846214i \(0.321123\pi\)
\(72\) 0 0
\(73\) 7.08863 − 12.2779i 0.829661 − 1.43702i −0.0686431 − 0.997641i \(-0.521867\pi\)
0.898304 − 0.439374i \(-0.144800\pi\)
\(74\) 0 0
\(75\) −1.19411 − 5.30147i −0.137884 − 0.612161i
\(76\) 0 0
\(77\) 1.05107 + 2.22758i 0.119781 + 0.253857i
\(78\) 0 0
\(79\) − 2.34061i − 0.263339i −0.991294 − 0.131670i \(-0.957966\pi\)
0.991294 − 0.131670i \(-0.0420338\pi\)
\(80\) 0 0
\(81\) 3.19218 + 8.41487i 0.354687 + 0.934985i
\(82\) 0 0
\(83\) −2.71128 + 4.69607i −0.297601 + 0.515461i −0.975587 − 0.219615i \(-0.929520\pi\)
0.677985 + 0.735075i \(0.262853\pi\)
\(84\) 0 0
\(85\) −1.27009 − 2.19986i −0.137761 − 0.238608i
\(86\) 0 0
\(87\) −4.74192 − 5.13992i −0.508387 − 0.551058i
\(88\) 0 0
\(89\) −15.9049 + 9.18267i −1.68591 + 0.973361i −0.728317 + 0.685241i \(0.759697\pi\)
−0.957594 + 0.288120i \(0.906970\pi\)
\(90\) 0 0
\(91\) 1.03055 + 2.18408i 0.108031 + 0.228954i
\(92\) 0 0
\(93\) 1.79466 − 5.75971i 0.186097 − 0.597254i
\(94\) 0 0
\(95\) −1.01454 −0.104089
\(96\) 0 0
\(97\) 2.70012 − 4.67674i 0.274156 − 0.474851i −0.695766 − 0.718268i \(-0.744935\pi\)
0.969922 + 0.243417i \(0.0782683\pi\)
\(98\) 0 0
\(99\) −0.224613 + 2.78385i −0.0225745 + 0.279788i
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 1008.2.cj.e.767.15 yes 32
3.2 odd 2 3024.2.cj.e.1439.6 32
4.3 odd 2 inner 1008.2.cj.e.767.2 yes 32
7.2 even 3 1008.2.bh.e.191.4 yes 32
9.4 even 3 3024.2.bh.e.2447.5 32
9.5 odd 6 1008.2.bh.e.95.13 yes 32
12.11 even 2 3024.2.cj.e.1439.5 32
21.2 odd 6 3024.2.bh.e.1871.12 32
28.23 odd 6 1008.2.bh.e.191.13 yes 32
36.23 even 6 1008.2.bh.e.95.4 ✓ 32
36.31 odd 6 3024.2.bh.e.2447.12 32
63.23 odd 6 inner 1008.2.cj.e.527.2 yes 32
63.58 even 3 3024.2.cj.e.2879.5 32
84.23 even 6 3024.2.bh.e.1871.5 32
252.23 even 6 inner 1008.2.cj.e.527.15 yes 32
252.247 odd 6 3024.2.cj.e.2879.6 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.bh.e.95.4 ✓ 32 36.23 even 6
1008.2.bh.e.95.13 yes 32 9.5 odd 6
1008.2.bh.e.191.4 yes 32 7.2 even 3
1008.2.bh.e.191.13 yes 32 28.23 odd 6
1008.2.cj.e.527.2 yes 32 63.23 odd 6 inner
1008.2.cj.e.527.15 yes 32 252.23 even 6 inner
1008.2.cj.e.767.2 yes 32 4.3 odd 2 inner
1008.2.cj.e.767.15 yes 32 1.1 even 1 trivial
3024.2.bh.e.1871.5 32 84.23 even 6
3024.2.bh.e.1871.12 32 21.2 odd 6
3024.2.bh.e.2447.5 32 9.4 even 3
3024.2.bh.e.2447.12 32 36.31 odd 6
3024.2.cj.e.1439.5 32 12.11 even 2
3024.2.cj.e.1439.6 32 3.2 odd 2
3024.2.cj.e.2879.5 32 63.58 even 3
3024.2.cj.e.2879.6 32 252.247 odd 6