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Properties

Label 1008.2.bh.e.95.13
Level $1008$
Weight $2$
Character 1008.95
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(95,1008)] 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("1008.95"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 95.13
Character \(\chi\) \(=\) 1008.95
Dual form 1008.2.bh.e.191.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.27304 + 1.17446i) q^{3} +1.36474i q^{5} +(-1.50769 + 2.17414i) q^{7} +(0.241269 + 2.99028i) q^{9} +0.930967 q^{11} +(0.456392 - 0.790494i) q^{13} +(-1.60284 + 1.73737i) q^{15} +(1.61193 + 0.930647i) q^{17} +(-0.643797 + 0.371696i) q^{19} +(-4.47280 + 0.997041i) q^{21} -2.47575 q^{23} +3.13749 q^{25} +(-3.20483 + 4.09012i) q^{27} +(-3.49659 + 2.01876i) q^{29} +(-3.01641 + 1.74153i) q^{31} +(1.18516 + 1.09339i) q^{33} +(-2.96713 - 2.05760i) q^{35} +(-1.98011 - 3.42965i) q^{37} +(1.50941 - 0.470316i) q^{39} +(-1.06965 - 0.617564i) q^{41} +(-7.13168 + 4.11748i) q^{43} +(-4.08095 + 0.329269i) q^{45} +(-2.54239 + 4.40356i) q^{47} +(-2.45375 - 6.55585i) q^{49} +(0.959040 + 3.07790i) q^{51} +(9.11540 + 5.26278i) q^{53} +1.27053i q^{55} +(-1.25612 - 0.282931i) q^{57} +(5.59767 + 9.69544i) q^{59} +(3.36605 - 5.83017i) q^{61} +(-6.86504 - 3.98386i) q^{63} +(1.07882 + 0.622856i) q^{65} +(7.97673 - 4.60537i) q^{67} +(-3.15173 - 2.90767i) q^{69} -8.97964 q^{71} +(7.08863 - 12.2779i) q^{73} +(3.99415 + 3.68487i) q^{75} +(-1.40361 + 2.02405i) q^{77} +(2.02703 + 1.17031i) q^{79} +(-8.88358 + 1.44292i) q^{81} +(2.71128 + 4.69607i) q^{83} +(-1.27009 + 2.19986i) q^{85} +(-6.82226 - 1.53666i) q^{87} +(15.9049 - 9.18267i) q^{89} +(1.03055 + 2.18408i) q^{91} +(-5.88538 - 1.32563i) q^{93} +(-0.507268 - 0.878614i) q^{95} +(2.70012 + 4.67674i) q^{97} +(0.224613 + 2.78385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 4 q^{9} + 2 q^{13} - 18 q^{17} + 8 q^{21} - 32 q^{25} + 12 q^{29} + 22 q^{33} - 2 q^{37} + 22 q^{45} + 2 q^{49} + 24 q^{53} - 10 q^{57} - 14 q^{61} - 78 q^{65} - 62 q^{69} - 28 q^{73} + 6 q^{77}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 1.27304 + 1.17446i 0.734991 + 0.678077i
\(4\) 0 0
\(5\) 1.36474i 0.610330i 0.952300 + 0.305165i \(0.0987116\pi\)
−0.952300 + 0.305165i \(0.901288\pi\)
\(6\) 0 0
\(7\) −1.50769 + 2.17414i −0.569853 + 0.821747i
\(8\) 0 0
\(9\) 0.241269 + 2.99028i 0.0804230 + 0.996761i
\(10\) 0 0
\(11\) 0.930967 0.280697 0.140348 − 0.990102i \(-0.455178\pi\)
0.140348 + 0.990102i \(0.455178\pi\)
\(12\) 0 0
\(13\) 0.456392 − 0.790494i 0.126580 − 0.219244i −0.795769 − 0.605600i \(-0.792933\pi\)
0.922350 + 0.386356i \(0.126267\pi\)
\(14\) 0 0
\(15\) −1.60284 + 1.73737i −0.413851 + 0.448587i
\(16\) 0 0
\(17\) 1.61193 + 0.930647i 0.390950 + 0.225715i 0.682572 − 0.730819i \(-0.260861\pi\)
−0.291622 + 0.956534i \(0.594195\pi\)
\(18\) 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
\(21\) −4.47280 + 0.997041i −0.976044 + 0.217572i
\(22\) 0 0
\(23\) −2.47575 −0.516229 −0.258114 − 0.966114i \(-0.583101\pi\)
−0.258114 + 0.966114i \(0.583101\pi\)
\(24\) 0 0
\(25\) 3.13749 0.627498
\(26\) 0 0
\(27\) −3.20483 + 4.09012i −0.616771 + 0.787143i
\(28\) 0 0
\(29\) −3.49659 + 2.01876i −0.649301 + 0.374874i −0.788188 − 0.615434i \(-0.788981\pi\)
0.138888 + 0.990308i \(0.455647\pi\)
\(30\) 0 0
\(31\) −3.01641 + 1.74153i −0.541764 + 0.312788i −0.745793 − 0.666177i \(-0.767929\pi\)
0.204030 + 0.978965i \(0.434596\pi\)
\(32\) 0 0
\(33\) 1.18516 + 1.09339i 0.206310 + 0.190334i
\(34\) 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
\(39\) 1.50941 − 0.470316i 0.241699 − 0.0753108i
\(40\) 0 0
\(41\) −1.06965 − 0.617564i −0.167052 − 0.0964473i 0.414143 − 0.910212i \(-0.364081\pi\)
−0.581195 + 0.813764i \(0.697415\pi\)
\(42\) 0 0
\(43\) −7.13168 + 4.11748i −1.08757 + 0.627909i −0.932929 − 0.360061i \(-0.882756\pi\)
−0.154642 + 0.987971i \(0.549422\pi\)
\(44\) 0 0
\(45\) −4.08095 + 0.329269i −0.608353 + 0.0490845i
\(46\) 0 0
\(47\) −2.54239 + 4.40356i −0.370846 + 0.642325i −0.989696 − 0.143185i \(-0.954266\pi\)
0.618850 + 0.785509i \(0.287599\pi\)
\(48\) 0 0
\(49\) −2.45375 − 6.55585i −0.350535 − 0.936550i
\(50\) 0 0
\(51\) 0.959040 + 3.07790i 0.134292 + 0.430993i
\(52\) 0 0
\(53\) 9.11540 + 5.26278i 1.25210 + 0.722898i 0.971525 − 0.236935i \(-0.0761429\pi\)
0.280571 + 0.959833i \(0.409476\pi\)
\(54\) 0 0
\(55\) 1.27053i 0.171318i
\(56\) 0 0
\(57\) −1.25612 − 0.282931i −0.166378 − 0.0374752i
\(58\) 0 0
\(59\) 5.59767 + 9.69544i 0.728754 + 1.26224i 0.957410 + 0.288732i \(0.0932335\pi\)
−0.228656 + 0.973507i \(0.573433\pi\)
\(60\) 0 0
\(61\) 3.36605 − 5.83017i 0.430978 − 0.746476i −0.565980 − 0.824419i \(-0.691502\pi\)
0.996958 + 0.0779431i \(0.0248352\pi\)
\(62\) 0 0
\(63\) −6.86504 − 3.98386i −0.864914 − 0.501920i
\(64\) 0 0
\(65\) 1.07882 + 0.622856i 0.133811 + 0.0772557i
\(66\) 0 0
\(67\) 7.97673 − 4.60537i 0.974513 − 0.562635i 0.0739040 − 0.997265i \(-0.476454\pi\)
0.900609 + 0.434630i \(0.143121\pi\)
\(68\) 0 0
\(69\) −3.15173 − 2.90767i −0.379423 − 0.350043i
\(70\) 0 0
\(71\) −8.97964 −1.06569 −0.532843 − 0.846214i \(-0.678877\pi\)
−0.532843 + 0.846214i \(0.678877\pi\)
\(72\) 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
\(75\) 3.99415 + 3.68487i 0.461205 + 0.425492i
\(76\) 0 0
\(77\) −1.40361 + 2.02405i −0.159956 + 0.230662i
\(78\) 0 0
\(79\) 2.02703 + 1.17031i 0.228059 + 0.131670i 0.609676 − 0.792651i \(-0.291300\pi\)
−0.381618 + 0.924320i \(0.624633\pi\)
\(80\) 0 0
\(81\) −8.88358 + 1.44292i −0.987064 + 0.160325i
\(82\) 0 0
\(83\) 2.71128 + 4.69607i 0.297601 + 0.515461i 0.975587 − 0.219615i \(-0.0704801\pi\)
−0.677985 + 0.735075i \(0.737147\pi\)
\(84\) 0 0
\(85\) −1.27009 + 2.19986i −0.137761 + 0.238608i
\(86\) 0 0
\(87\) −6.82226 − 1.53666i −0.731423 − 0.164747i
\(88\) 0 0
\(89\) 15.9049 − 9.18267i 1.68591 − 0.973361i 0.728317 − 0.685241i \(-0.240303\pi\)
0.957594 − 0.288120i \(-0.0930303\pi\)
\(90\) 0 0
\(91\) 1.03055 + 2.18408i 0.108031 + 0.228954i
\(92\) 0 0
\(93\) −5.88538 − 1.32563i −0.610285 − 0.137462i
\(94\) 0 0
\(95\) −0.507268 − 0.878614i −0.0520446 − 0.0901439i
\(96\) 0 0
\(97\) 2.70012 + 4.67674i 0.274156 + 0.474851i 0.969922 − 0.243417i \(-0.0782683\pi\)
−0.695766 + 0.718268i \(0.744935\pi\)
\(98\) 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.bh.e.95.13 yes 32
3.2 odd 2 3024.2.bh.e.2447.5 32
4.3 odd 2 inner 1008.2.bh.e.95.4 ✓ 32
7.2 even 3 1008.2.cj.e.527.2 yes 32
9.2 odd 6 1008.2.cj.e.767.15 yes 32
9.7 even 3 3024.2.cj.e.1439.6 32
12.11 even 2 3024.2.bh.e.2447.12 32
21.2 odd 6 3024.2.cj.e.2879.5 32
28.23 odd 6 1008.2.cj.e.527.15 yes 32
36.7 odd 6 3024.2.cj.e.1439.5 32
36.11 even 6 1008.2.cj.e.767.2 yes 32
63.2 odd 6 inner 1008.2.bh.e.191.4 yes 32
63.16 even 3 3024.2.bh.e.1871.12 32
84.23 even 6 3024.2.cj.e.2879.6 32
252.79 odd 6 3024.2.bh.e.1871.5 32
252.191 even 6 inner 1008.2.bh.e.191.13 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.bh.e.95.4 ✓ 32 4.3 odd 2 inner
1008.2.bh.e.95.13 yes 32 1.1 even 1 trivial
1008.2.bh.e.191.4 yes 32 63.2 odd 6 inner
1008.2.bh.e.191.13 yes 32 252.191 even 6 inner
1008.2.cj.e.527.2 yes 32 7.2 even 3
1008.2.cj.e.527.15 yes 32 28.23 odd 6
1008.2.cj.e.767.2 yes 32 36.11 even 6
1008.2.cj.e.767.15 yes 32 9.2 odd 6
3024.2.bh.e.1871.5 32 252.79 odd 6
3024.2.bh.e.1871.12 32 63.16 even 3
3024.2.bh.e.2447.5 32 3.2 odd 2
3024.2.bh.e.2447.12 32 12.11 even 2
3024.2.cj.e.1439.5 32 36.7 odd 6
3024.2.cj.e.1439.6 32 9.7 even 3
3024.2.cj.e.2879.5 32 21.2 odd 6
3024.2.cj.e.2879.6 32 84.23 even 6