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Properties

Label 1008.2.cx.i.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.i.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)
 
\(f(q)\) \(=\) \(q+(-1.71721 - 0.226267i) q^{3} +(1.72020 - 0.993161i) q^{5} +(1.81723 + 1.92293i) q^{7} +(2.89761 + 0.777096i) q^{9} +(2.20182 + 1.27122i) q^{11} +(1.56113 - 0.901321i) q^{13} +(-3.17867 + 1.31624i) q^{15} +0.0160253i q^{17} -2.71084 q^{19} +(-2.68547 - 3.71325i) q^{21} +(-5.44682 + 3.14472i) q^{23} +(-0.527264 + 0.913248i) q^{25} +(-4.79996 - 1.99007i) q^{27} +(0.871763 - 1.50994i) q^{29} +(3.82556 + 6.62607i) q^{31} +(-3.49334 - 2.68115i) q^{33} +(5.03579 + 1.50303i) q^{35} +9.20169 q^{37} +(-2.88473 + 1.19452i) q^{39} +(0.455067 - 0.262733i) q^{41} +(4.62463 + 2.67003i) q^{43} +(5.75626 - 1.54102i) q^{45} +(5.99971 - 10.3918i) q^{47} +(-0.395322 + 6.98883i) q^{49} +(0.00362601 - 0.0275188i) q^{51} +6.61411 q^{53} +5.05010 q^{55} +(4.65508 + 0.613376i) q^{57} +(1.95304 + 3.38276i) q^{59} +(-9.31530 - 5.37819i) q^{61} +(3.77133 + 6.98406i) q^{63} +(1.79031 - 3.10091i) q^{65} +(2.84876 - 1.64473i) q^{67} +(10.0649 - 4.16770i) q^{69} -0.552490i q^{71} +9.52426i q^{73} +(1.11206 - 1.44893i) q^{75} +(1.55675 + 6.54405i) q^{77} +(-6.04240 - 3.48858i) q^{79} +(7.79224 + 4.50344i) q^{81} +(5.60669 - 9.71107i) q^{83} +(0.0159157 + 0.0275668i) q^{85} +(-1.83865 + 2.39563i) q^{87} +5.31422i q^{89} +(4.57012 + 1.36404i) q^{91} +(-5.07002 - 12.2439i) q^{93} +(-4.66321 + 2.69230i) q^{95} +(-2.73685 - 1.58012i) q^{97} +(5.39214 + 5.39452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{7} + 20 q^{9} + 24 q^{15} + 10 q^{21} + 18 q^{23} + 24 q^{25} - 6 q^{29} - 12 q^{37} - 12 q^{39} + 42 q^{43} + 12 q^{49} - 42 q^{51} + 96 q^{53} - 22 q^{57} + 18 q^{63} + 42 q^{65} + 36 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) 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 −0.0633256 − 0.997993i \(-0.520171\pi\)
0.832624 + 0.553838i \(0.186837\pi\)
\(6\) 0 0
\(7\) 1.81723 + 1.92293i 0.686850 + 0.726799i
\(8\) 0 0
\(9\) 2.89761 + 0.777096i 0.965869 + 0.259032i
\(10\) 0 0
\(11\) 2.20182 + 1.27122i 0.663873 + 0.383287i 0.793751 − 0.608243i \(-0.208125\pi\)
−0.129878 + 0.991530i \(0.541459\pi\)
\(12\) 0 0
\(13\) 1.56113 − 0.901321i 0.432981 − 0.249981i −0.267635 − 0.963520i \(-0.586242\pi\)
0.700616 + 0.713539i \(0.252909\pi\)
\(14\) 0 0
\(15\) −3.17867 + 1.31624i −0.820729 + 0.339851i
\(16\) 0 0
\(17\) 0.0160253i 0.00388671i 0.999998 + 0.00194336i \(0.000618590\pi\)
−0.999998 + 0.00194336i \(0.999381\pi\)
\(18\) 0 0
\(19\) −2.71084 −0.621910 −0.310955 − 0.950425i \(-0.600649\pi\)
−0.310955 + 0.950425i \(0.600649\pi\)
\(20\) 0 0
\(21\) −2.68547 − 3.71325i −0.586018 − 0.810298i
\(22\) 0 0
\(23\) −5.44682 + 3.14472i −1.13574 + 0.655720i −0.945372 − 0.325993i \(-0.894301\pi\)
−0.190367 + 0.981713i \(0.560968\pi\)
\(24\) 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 −0.464044 − 0.885812i \(-0.653602\pi\)
0.535114 + 0.844780i \(0.320269\pi\)
\(42\) 0 0
\(43\) 4.62463 + 2.67003i 0.705249 + 0.407176i 0.809300 − 0.587396i \(-0.199847\pi\)
−0.104050 + 0.994572i \(0.533180\pi\)
\(44\) 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
\(49\) −0.395322 + 6.98883i −0.0564746 + 0.998404i
\(50\) 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 −0.233783 − 0.972289i \(-0.575111\pi\)
−0.958918 + 0.283682i \(0.908444\pi\)
\(62\) 0 0
\(63\) 3.77133 + 6.98406i 0.475142 + 0.879909i
\(64\) 0 0
\(65\) 1.79031 − 3.10091i 0.222061 − 0.384621i
\(66\) 0 0
\(67\) 2.84876 − 1.64473i 0.348032 − 0.200936i −0.315786 − 0.948830i \(-0.602268\pi\)
0.663818 + 0.747894i \(0.268935\pi\)
\(68\) 0 0
\(69\) 10.0649 − 4.16770i 1.21167 − 0.501732i
\(70\) 0 0
\(71\) − 0.552490i − 0.0655685i −0.999462 − 0.0327842i \(-0.989563\pi\)
0.999462 − 0.0327842i \(-0.0104374\pi\)
\(72\) 0 0
\(73\) 9.52426i 1.11473i 0.830267 + 0.557365i \(0.188188\pi\)
−0.830267 + 0.557365i \(0.811812\pi\)
\(74\) 0 0
\(75\) 1.11206 − 1.44893i 0.128410 − 0.167309i
\(76\) 0 0
\(77\) 1.55675 + 6.54405i 0.177408 + 0.745763i
\(78\) 0 0
\(79\) −6.04240 − 3.48858i −0.679823 − 0.392496i 0.119966 − 0.992778i \(-0.461722\pi\)
−0.799788 + 0.600282i \(0.795055\pi\)
\(80\) 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 \(0.0908828\pi\)
−0.959516 + 0.281653i \(0.909117\pi\)
\(90\) 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 0.354580 − 0.935026i \(-0.384624\pi\)
−0.632466 + 0.774588i \(0.717957\pi\)
\(98\) 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.i.223.1 ✓ 24
3.2 odd 2 3024.2.cx.i.559.4 24
4.3 odd 2 1008.2.cx.j.223.12 yes 24
7.6 odd 2 inner 1008.2.cx.i.223.12 yes 24
9.4 even 3 1008.2.cx.j.895.1 yes 24
9.5 odd 6 3024.2.cx.j.2575.9 24
12.11 even 2 3024.2.cx.j.559.4 24
21.20 even 2 3024.2.cx.i.559.9 24
28.27 even 2 1008.2.cx.j.223.1 yes 24
36.23 even 6 3024.2.cx.i.2575.9 24
36.31 odd 6 inner 1008.2.cx.i.895.12 yes 24
63.13 odd 6 1008.2.cx.j.895.12 yes 24
63.41 even 6 3024.2.cx.j.2575.4 24
84.83 odd 2 3024.2.cx.j.559.9 24
252.139 even 6 inner 1008.2.cx.i.895.1 yes 24
252.167 odd 6 3024.2.cx.i.2575.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.1 ✓ 24 1.1 even 1 trivial
1008.2.cx.i.223.12 yes 24 7.6 odd 2 inner
1008.2.cx.i.895.1 yes 24 252.139 even 6 inner
1008.2.cx.i.895.12 yes 24 36.31 odd 6 inner
1008.2.cx.j.223.1 yes 24 28.27 even 2
1008.2.cx.j.223.12 yes 24 4.3 odd 2
1008.2.cx.j.895.1 yes 24 9.4 even 3
1008.2.cx.j.895.12 yes 24 63.13 odd 6
3024.2.cx.i.559.4 24 3.2 odd 2
3024.2.cx.i.559.9 24 21.20 even 2
3024.2.cx.i.2575.4 24 252.167 odd 6
3024.2.cx.i.2575.9 24 36.23 even 6
3024.2.cx.j.559.4 24 12.11 even 2
3024.2.cx.j.559.9 24 84.83 odd 2
3024.2.cx.j.2575.4 24 63.41 even 6
3024.2.cx.j.2575.9 24 9.5 odd 6