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Properties

Label 1008.2.cx.i.223.9
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 223.9
Character \(\chi\) \(=\) 1008.223
Dual form 1008.2.cx.i.895.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.21473 + 1.23468i) q^{3} +(3.37919 - 1.95098i) q^{5} +(2.55088 - 0.702155i) q^{7} +(-0.0488500 + 2.99960i) q^{9} +(-2.79983 - 1.61648i) q^{11} +(4.41092 - 2.54665i) q^{13} +(6.51363 + 1.80229i) q^{15} +5.22255i q^{17} -6.24838 q^{19} +(3.96557 + 2.29658i) q^{21} +(0.0232032 - 0.0133964i) q^{23} +(5.11261 - 8.85531i) q^{25} +(-3.76288 + 3.58340i) q^{27} +(2.19378 - 3.79973i) q^{29} +(-1.42877 - 2.47471i) q^{31} +(-1.40521 - 5.42047i) q^{33} +(7.25001 - 7.34942i) q^{35} -9.43733 q^{37} +(8.50238 + 2.35257i) q^{39} +(-3.54513 + 2.04678i) q^{41} +(2.33333 + 1.34715i) q^{43} +(5.68708 + 10.2315i) q^{45} +(-3.46834 + 6.00735i) q^{47} +(6.01396 - 3.58223i) q^{49} +(-6.44816 + 6.34400i) q^{51} -2.98191 q^{53} -12.6149 q^{55} +(-7.59011 - 7.71473i) q^{57} +(2.27240 + 3.93592i) q^{59} +(-0.402478 - 0.232371i) q^{61} +(1.98158 + 7.68592i) q^{63} +(9.93690 - 17.2112i) q^{65} +(5.69642 - 3.28883i) q^{67} +(0.0447259 + 0.0123754i) q^{69} +1.27973i q^{71} +6.76224i q^{73} +(17.1439 - 4.44441i) q^{75} +(-8.27704 - 2.15753i) q^{77} +(-2.02181 - 1.16729i) q^{79} +(-8.99523 - 0.293061i) q^{81} +(-7.18933 + 12.4523i) q^{83} +(10.1891 + 17.6480i) q^{85} +(7.35629 - 1.90705i) q^{87} -4.82670i q^{89} +(9.46359 - 9.59334i) q^{91} +(1.31989 - 4.77018i) q^{93} +(-21.1145 + 12.1904i) q^{95} +(2.00464 + 1.15738i) q^{97} +(4.98557 - 8.31940i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{7} + 20 q^{9} + 24 q^{15} + 10 q^{21} + 18 q^{23} + 24 q^{25} - 6 q^{29} - 12 q^{37} - 12 q^{39} + 42 q^{43} + 12 q^{49} - 42 q^{51} + 96 q^{53} - 22 q^{57} + 18 q^{63} + 42 q^{65} + 36 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.21473 + 1.23468i 0.701326 + 0.712841i
\(4\) 0 0
\(5\) 3.37919 − 1.95098i 1.51122 − 0.872503i 0.511305 − 0.859399i \(-0.329162\pi\)
0.999914 − 0.0131039i \(-0.00417122\pi\)
\(6\) 0 0
\(7\) 2.55088 − 0.702155i 0.964141 − 0.265390i
\(8\) 0 0
\(9\) −0.0488500 + 2.99960i −0.0162833 + 0.999867i
\(10\) 0 0
\(11\) −2.79983 − 1.61648i −0.844180 − 0.487387i 0.0145030 − 0.999895i \(-0.495383\pi\)
−0.858683 + 0.512507i \(0.828717\pi\)
\(12\) 0 0
\(13\) 4.41092 − 2.54665i 1.22337 − 0.706313i 0.257735 − 0.966216i \(-0.417024\pi\)
0.965635 + 0.259902i \(0.0836904\pi\)
\(14\) 0 0
\(15\) 6.51363 + 1.80229i 1.68181 + 0.465349i
\(16\) 0 0
\(17\) 5.22255i 1.26665i 0.773884 + 0.633327i \(0.218311\pi\)
−0.773884 + 0.633327i \(0.781689\pi\)
\(18\) 0 0
\(19\) −6.24838 −1.43348 −0.716739 − 0.697342i \(-0.754366\pi\)
−0.716739 + 0.697342i \(0.754366\pi\)
\(20\) 0 0
\(21\) 3.96557 + 2.29658i 0.865358 + 0.501154i
\(22\) 0 0
\(23\) 0.0232032 − 0.0133964i 0.00483820 − 0.00279334i −0.497579 − 0.867419i \(-0.665778\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(24\) 0 0
\(25\) 5.11261 − 8.85531i 1.02252 − 1.77106i
\(26\) 0 0
\(27\) −3.76288 + 3.58340i −0.724166 + 0.689626i
\(28\) 0 0
\(29\) 2.19378 − 3.79973i 0.407374 − 0.705592i −0.587221 − 0.809427i \(-0.699778\pi\)
0.994595 + 0.103835i \(0.0331113\pi\)
\(30\) 0 0
\(31\) −1.42877 − 2.47471i −0.256615 − 0.444471i 0.708718 − 0.705492i \(-0.249274\pi\)
−0.965333 + 0.261021i \(0.915941\pi\)
\(32\) 0 0
\(33\) −1.40521 − 5.42047i −0.244616 − 0.943583i
\(34\) 0 0
\(35\) 7.25001 − 7.34942i 1.22548 − 1.24228i
\(36\) 0 0
\(37\) −9.43733 −1.55149 −0.775744 − 0.631048i \(-0.782625\pi\)
−0.775744 + 0.631048i \(0.782625\pi\)
\(38\) 0 0
\(39\) 8.50238 + 2.35257i 1.36147 + 0.376712i
\(40\) 0 0
\(41\) −3.54513 + 2.04678i −0.553656 + 0.319653i −0.750595 − 0.660762i \(-0.770233\pi\)
0.196939 + 0.980416i \(0.436900\pi\)
\(42\) 0 0
\(43\) 2.33333 + 1.34715i 0.355829 + 0.205438i 0.667250 − 0.744834i \(-0.267471\pi\)
−0.311420 + 0.950272i \(0.600805\pi\)
\(44\) 0 0
\(45\) 5.68708 + 10.2315i 0.847780 + 1.52523i
\(46\) 0 0
\(47\) −3.46834 + 6.00735i −0.505910 + 0.876262i 0.494067 + 0.869424i \(0.335510\pi\)
−0.999977 + 0.00683782i \(0.997823\pi\)
\(48\) 0 0
\(49\) 6.01396 − 3.58223i 0.859137 − 0.511746i
\(50\) 0 0
\(51\) −6.44816 + 6.34400i −0.902922 + 0.888338i
\(52\) 0 0
\(53\) −2.98191 −0.409597 −0.204799 − 0.978804i \(-0.565654\pi\)
−0.204799 + 0.978804i \(0.565654\pi\)
\(54\) 0 0
\(55\) −12.6149 −1.70099
\(56\) 0 0
\(57\) −7.59011 − 7.71473i −1.00534 − 1.02184i
\(58\) 0 0
\(59\) 2.27240 + 3.93592i 0.295842 + 0.512413i 0.975180 − 0.221412i \(-0.0710666\pi\)
−0.679339 + 0.733825i \(0.737733\pi\)
\(60\) 0 0
\(61\) −0.402478 − 0.232371i −0.0515320 − 0.0297520i 0.474013 − 0.880518i \(-0.342805\pi\)
−0.525545 + 0.850766i \(0.676138\pi\)
\(62\) 0 0
\(63\) 1.98158 + 7.68592i 0.249655 + 0.968335i
\(64\) 0 0
\(65\) 9.93690 − 17.2112i 1.23252 − 2.13479i
\(66\) 0 0
\(67\) 5.69642 − 3.28883i 0.695929 − 0.401795i −0.109900 − 0.993943i \(-0.535053\pi\)
0.805829 + 0.592148i \(0.201720\pi\)
\(68\) 0 0
\(69\) 0.0447259 + 0.0123754i 0.00538436 + 0.00148983i
\(70\) 0 0
\(71\) 1.27973i 0.151876i 0.997113 + 0.0759381i \(0.0241951\pi\)
−0.997113 + 0.0759381i \(0.975805\pi\)
\(72\) 0 0
\(73\) 6.76224i 0.791460i 0.918367 + 0.395730i \(0.129508\pi\)
−0.918367 + 0.395730i \(0.870492\pi\)
\(74\) 0 0
\(75\) 17.1439 − 4.44441i 1.97961 − 0.513196i
\(76\) 0 0
\(77\) −8.27704 − 2.15753i −0.943256 − 0.245874i
\(78\) 0 0
\(79\) −2.02181 − 1.16729i −0.227471 − 0.131331i 0.381934 − 0.924190i \(-0.375258\pi\)
−0.609405 + 0.792859i \(0.708592\pi\)
\(80\) 0 0
\(81\) −8.99523 − 0.293061i −0.999470 − 0.0325623i
\(82\) 0 0
\(83\) −7.18933 + 12.4523i −0.789131 + 1.36682i 0.137369 + 0.990520i \(0.456135\pi\)
−0.926500 + 0.376295i \(0.877198\pi\)
\(84\) 0 0
\(85\) 10.1891 + 17.6480i 1.10516 + 1.91419i
\(86\) 0 0
\(87\) 7.35629 − 1.90705i 0.788677 − 0.204458i
\(88\) 0 0
\(89\) − 4.82670i − 0.511629i −0.966726 − 0.255815i \(-0.917656\pi\)
0.966726 − 0.255815i \(-0.0823437\pi\)
\(90\) 0 0
\(91\) 9.46359 − 9.59334i 0.992053 − 1.00566i
\(92\) 0 0
\(93\) 1.31989 − 4.77018i 0.136866 − 0.494645i
\(94\) 0 0
\(95\) −21.1145 + 12.1904i −2.16630 + 1.25071i
\(96\) 0 0
\(97\) 2.00464 + 1.15738i 0.203540 + 0.117514i 0.598306 − 0.801268i \(-0.295841\pi\)
−0.394765 + 0.918782i \(0.629174\pi\)
\(98\) 0 0
\(99\) 4.98557 − 8.31940i 0.501069 − 0.836132i
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.223.9 yes 24
3.2 odd 2 3024.2.cx.i.559.2 24
4.3 odd 2 1008.2.cx.j.223.4 yes 24
7.6 odd 2 inner 1008.2.cx.i.223.4 ✓ 24
9.4 even 3 1008.2.cx.j.895.9 yes 24
9.5 odd 6 3024.2.cx.j.2575.11 24
12.11 even 2 3024.2.cx.j.559.2 24
21.20 even 2 3024.2.cx.i.559.11 24
28.27 even 2 1008.2.cx.j.223.9 yes 24
36.23 even 6 3024.2.cx.i.2575.11 24
36.31 odd 6 inner 1008.2.cx.i.895.4 yes 24
63.13 odd 6 1008.2.cx.j.895.4 yes 24
63.41 even 6 3024.2.cx.j.2575.2 24
84.83 odd 2 3024.2.cx.j.559.11 24
252.139 even 6 inner 1008.2.cx.i.895.9 yes 24
252.167 odd 6 3024.2.cx.i.2575.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.4 ✓ 24 7.6 odd 2 inner
1008.2.cx.i.223.9 yes 24 1.1 even 1 trivial
1008.2.cx.i.895.4 yes 24 36.31 odd 6 inner
1008.2.cx.i.895.9 yes 24 252.139 even 6 inner
1008.2.cx.j.223.4 yes 24 4.3 odd 2
1008.2.cx.j.223.9 yes 24 28.27 even 2
1008.2.cx.j.895.4 yes 24 63.13 odd 6
1008.2.cx.j.895.9 yes 24 9.4 even 3
3024.2.cx.i.559.2 24 3.2 odd 2
3024.2.cx.i.559.11 24 21.20 even 2
3024.2.cx.i.2575.2 24 252.167 odd 6
3024.2.cx.i.2575.11 24 36.23 even 6
3024.2.cx.j.559.2 24 12.11 even 2
3024.2.cx.j.559.11 24 84.83 odd 2
3024.2.cx.j.2575.2 24 63.41 even 6
3024.2.cx.j.2575.11 24 9.5 odd 6