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Properties

Label 1008.2.cx.j.223.1
Level $1008$
Weight $2$
Character 1008.223
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
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(223,1008)] 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("1008.223"); 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, 4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cx (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,6,0,20,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 223.1
Character \(\chi\) \(=\) 1008.223
Dual form 1008.2.cx.j.895.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.71721 - 0.226267i) q^{3} +(-1.72020 + 0.993161i) q^{5} +(-2.57392 - 0.612306i) q^{7} +(2.89761 + 0.777096i) q^{9} +(-2.20182 - 1.27122i) q^{11} +(-1.56113 + 0.901321i) q^{13} +(3.17867 - 1.31624i) q^{15} -0.0160253i q^{17} -2.71084 q^{19} +(4.28142 + 1.63385i) q^{21} +(5.44682 - 3.14472i) q^{23} +(-0.527264 + 0.913248i) q^{25} +(-4.79996 - 1.99007i) q^{27} +(0.871763 - 1.50994i) q^{29} +(3.82556 + 6.62607i) q^{31} +(3.49334 + 2.68115i) q^{33} +(5.03579 - 1.50303i) q^{35} +9.20169 q^{37} +(2.88473 - 1.19452i) q^{39} +(-0.455067 + 0.262733i) q^{41} +(-4.62463 - 2.67003i) q^{43} +(-5.75626 + 1.54102i) q^{45} +(5.99971 - 10.3918i) q^{47} +(6.25016 + 3.15205i) q^{49} +(-0.00362601 + 0.0275188i) q^{51} +6.61411 q^{53} +5.05010 q^{55} +(4.65508 + 0.613376i) q^{57} +(1.95304 + 3.38276i) q^{59} +(9.31530 + 5.37819i) q^{61} +(-6.98240 - 3.77441i) q^{63} +(1.79031 - 3.10091i) q^{65} +(-2.84876 + 1.64473i) q^{67} +(-10.0649 + 4.16770i) q^{69} +0.552490i q^{71} -9.52426i q^{73} +(1.11206 - 1.44893i) q^{75} +(4.88894 + 4.62021i) q^{77} +(6.04240 + 3.48858i) q^{79} +(7.79224 + 4.50344i) q^{81} +(5.60669 - 9.71107i) q^{83} +(0.0159157 + 0.0275668i) q^{85} +(-1.83865 + 2.39563i) q^{87} -5.31422i q^{89} +(4.57012 - 1.36404i) q^{91} +(-5.07002 - 12.2439i) q^{93} +(4.66321 - 2.69230i) q^{95} +(2.73685 + 1.58012i) q^{97} +(-5.39214 - 5.39452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{7} + 20 q^{9} - 24 q^{15} + 10 q^{21} - 18 q^{23} + 24 q^{25} - 6 q^{29} - 12 q^{37} + 12 q^{39} - 42 q^{43} + 12 q^{49} + 42 q^{51} + 96 q^{53} - 22 q^{57} - 18 q^{63} + 42 q^{65} - 36 q^{67}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.71721 − 0.226267i −0.991430 − 0.130636i
\(4\) 0 0
\(5\) −1.72020 + 0.993161i −0.769299 + 0.444155i −0.832624 − 0.553838i \(-0.813163\pi\)
0.0633256 + 0.997993i \(0.479829\pi\)
\(6\) 0 0
\(7\) −2.57392 − 0.612306i −0.972852 − 0.231430i
\(8\) 0 0
\(9\) 2.89761 + 0.777096i 0.965869 + 0.259032i
\(10\) 0 0
\(11\) −2.20182 − 1.27122i −0.663873 − 0.383287i 0.129878 − 0.991530i \(-0.458541\pi\)
−0.793751 + 0.608243i \(0.791875\pi\)
\(12\) 0 0
\(13\) −1.56113 + 0.901321i −0.432981 + 0.249981i −0.700616 − 0.713539i \(-0.747091\pi\)
0.267635 + 0.963520i \(0.413758\pi\)
\(14\) 0 0
\(15\) 3.17867 − 1.31624i 0.820729 − 0.339851i
\(16\) 0 0
\(17\) − 0.0160253i − 0.00388671i −0.999998 − 0.00194336i \(-0.999381\pi\)
0.999998 − 0.00194336i \(-0.000618590\pi\)
\(18\) 0 0
\(19\) −2.71084 −0.621910 −0.310955 − 0.950425i \(-0.600649\pi\)
−0.310955 + 0.950425i \(0.600649\pi\)
\(20\) 0 0
\(21\) 4.28142 + 1.63385i 0.934282 + 0.356536i
\(22\) 0 0
\(23\) 5.44682 − 3.14472i 1.13574 − 0.655720i 0.190367 − 0.981713i \(-0.439032\pi\)
0.945372 + 0.325993i \(0.105699\pi\)
\(24\) 0 0
\(25\) −0.527264 + 0.913248i −0.105453 + 0.182650i
\(26\) 0 0
\(27\) −4.79996 − 1.99007i −0.923753 − 0.382989i
\(28\) 0 0
\(29\) 0.871763 − 1.50994i 0.161882 − 0.280388i −0.773661 − 0.633599i \(-0.781577\pi\)
0.935544 + 0.353211i \(0.114910\pi\)
\(30\) 0 0
\(31\) 3.82556 + 6.62607i 0.687092 + 1.19008i 0.972775 + 0.231753i \(0.0744462\pi\)
−0.285683 + 0.958324i \(0.592220\pi\)
\(32\) 0 0
\(33\) 3.49334 + 2.68115i 0.608113 + 0.466728i
\(34\) 0 0
\(35\) 5.03579 − 1.50303i 0.851204 − 0.254058i
\(36\) 0 0
\(37\) 9.20169 1.51275 0.756374 − 0.654139i \(-0.226969\pi\)
0.756374 + 0.654139i \(0.226969\pi\)
\(38\) 0 0
\(39\) 2.88473 − 1.19452i 0.461927 − 0.191277i
\(40\) 0 0
\(41\) −0.455067 + 0.262733i −0.0710695 + 0.0410320i −0.535114 − 0.844780i \(-0.679731\pi\)
0.464044 + 0.885812i \(0.346398\pi\)
\(42\) 0 0
\(43\) −4.62463 − 2.67003i −0.705249 − 0.407176i 0.104050 − 0.994572i \(-0.466820\pi\)
−0.809300 + 0.587396i \(0.800153\pi\)
\(44\) 0 0
\(45\) −5.75626 + 1.54102i −0.858092 + 0.229722i
\(46\) 0 0
\(47\) 5.99971 − 10.3918i 0.875148 − 1.51580i 0.0185425 − 0.999828i \(-0.494097\pi\)
0.856605 − 0.515972i \(-0.172569\pi\)
\(48\) 0 0
\(49\) 6.25016 + 3.15205i 0.892881 + 0.450294i
\(50\) 0 0
\(51\) −0.00362601 + 0.0275188i −0.000507743 + 0.00385340i
\(52\) 0 0
\(53\) 6.61411 0.908517 0.454259 − 0.890870i \(-0.349904\pi\)
0.454259 + 0.890870i \(0.349904\pi\)
\(54\) 0 0
\(55\) 5.05010 0.680956
\(56\) 0 0
\(57\) 4.65508 + 0.613376i 0.616581 + 0.0812436i
\(58\) 0 0
\(59\) 1.95304 + 3.38276i 0.254264 + 0.440398i 0.964695 − 0.263368i \(-0.0848334\pi\)
−0.710431 + 0.703767i \(0.751500\pi\)
\(60\) 0 0
\(61\) 9.31530 + 5.37819i 1.19270 + 0.688607i 0.958918 − 0.283682i \(-0.0915560\pi\)
0.233783 + 0.972289i \(0.424889\pi\)
\(62\) 0 0
\(63\) −6.98240 − 3.77441i −0.879699 − 0.475531i
\(64\) 0 0
\(65\) 1.79031 − 3.10091i 0.222061 − 0.384621i
\(66\) 0 0
\(67\) −2.84876 + 1.64473i −0.348032 + 0.200936i −0.663818 − 0.747894i \(-0.731065\pi\)
0.315786 + 0.948830i \(0.397732\pi\)
\(68\) 0 0
\(69\) −10.0649 + 4.16770i −1.21167 + 0.501732i
\(70\) 0 0
\(71\) 0.552490i 0.0655685i 0.999462 + 0.0327842i \(0.0104374\pi\)
−0.999462 + 0.0327842i \(0.989563\pi\)
\(72\) 0 0
\(73\) − 9.52426i − 1.11473i −0.830267 − 0.557365i \(-0.811812\pi\)
0.830267 − 0.557365i \(-0.188188\pi\)
\(74\) 0 0
\(75\) 1.11206 − 1.44893i 0.128410 − 0.167309i
\(76\) 0 0
\(77\) 4.88894 + 4.62021i 0.557146 + 0.526522i
\(78\) 0 0
\(79\) 6.04240 + 3.48858i 0.679823 + 0.392496i 0.799788 − 0.600282i \(-0.204945\pi\)
−0.119966 + 0.992778i \(0.538278\pi\)
\(80\) 0 0
\(81\) 7.79224 + 4.50344i 0.865805 + 0.500382i
\(82\) 0 0
\(83\) 5.60669 − 9.71107i 0.615414 − 1.06593i −0.374897 − 0.927066i \(-0.622322\pi\)
0.990312 − 0.138862i \(-0.0443446\pi\)
\(84\) 0 0
\(85\) 0.0159157 + 0.0275668i 0.00172630 + 0.00299004i
\(86\) 0 0
\(87\) −1.83865 + 2.39563i −0.197124 + 0.256838i
\(88\) 0 0
\(89\) − 5.31422i − 0.563306i −0.959516 − 0.281653i \(-0.909117\pi\)
0.959516 − 0.281653i \(-0.0908828\pi\)
\(90\) 0 0
\(91\) 4.57012 − 1.36404i 0.479079 − 0.142990i
\(92\) 0 0
\(93\) −5.07002 − 12.2439i −0.525737 − 1.26964i
\(94\) 0 0
\(95\) 4.66321 − 2.69230i 0.478435 − 0.276224i
\(96\) 0 0
\(97\) 2.73685 + 1.58012i 0.277885 + 0.160437i 0.632466 − 0.774588i \(-0.282043\pi\)
−0.354580 + 0.935026i \(0.615376\pi\)
\(98\) 0 0
\(99\) −5.39214 − 5.39452i −0.541930 − 0.542170i
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.cx.j.223.1 yes 24
3.2 odd 2 3024.2.cx.j.559.9 24
4.3 odd 2 1008.2.cx.i.223.12 yes 24
7.6 odd 2 inner 1008.2.cx.j.223.12 yes 24
9.4 even 3 1008.2.cx.i.895.1 yes 24
9.5 odd 6 3024.2.cx.i.2575.4 24
12.11 even 2 3024.2.cx.i.559.9 24
21.20 even 2 3024.2.cx.j.559.4 24
28.27 even 2 1008.2.cx.i.223.1 ✓ 24
36.23 even 6 3024.2.cx.j.2575.4 24
36.31 odd 6 inner 1008.2.cx.j.895.12 yes 24
63.13 odd 6 1008.2.cx.i.895.12 yes 24
63.41 even 6 3024.2.cx.i.2575.9 24
84.83 odd 2 3024.2.cx.i.559.4 24
252.139 even 6 inner 1008.2.cx.j.895.1 yes 24
252.167 odd 6 3024.2.cx.j.2575.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.1 ✓ 24 28.27 even 2
1008.2.cx.i.223.12 yes 24 4.3 odd 2
1008.2.cx.i.895.1 yes 24 9.4 even 3
1008.2.cx.i.895.12 yes 24 63.13 odd 6
1008.2.cx.j.223.1 yes 24 1.1 even 1 trivial
1008.2.cx.j.223.12 yes 24 7.6 odd 2 inner
1008.2.cx.j.895.1 yes 24 252.139 even 6 inner
1008.2.cx.j.895.12 yes 24 36.31 odd 6 inner
3024.2.cx.i.559.4 24 84.83 odd 2
3024.2.cx.i.559.9 24 12.11 even 2
3024.2.cx.i.2575.4 24 9.5 odd 6
3024.2.cx.i.2575.9 24 63.41 even 6
3024.2.cx.j.559.4 24 21.20 even 2
3024.2.cx.j.559.9 24 3.2 odd 2
3024.2.cx.j.2575.4 24 36.23 even 6
3024.2.cx.j.2575.9 24 252.167 odd 6