↑

Properties

Label 1008.2.cx.c.223.1
Level $1008$
Weight $2$
Character 1008.223
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [2,0,0,0,-3,0,1,0,-6,0,-3] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 223.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.223
Dual form 1008.2.cx.c.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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +(-1.50000 + 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} -3.00000 q^{9} +(-1.50000 - 0.866025i) q^{11} +(4.50000 - 2.59808i) q^{13} +(-1.50000 - 2.59808i) q^{15} -6.92820i q^{17} +4.00000 q^{19} +(4.50000 + 0.866025i) q^{21} +(-4.50000 + 2.59808i) q^{23} +(-1.00000 + 1.73205i) q^{25} -5.19615i q^{27} +(4.50000 - 7.79423i) q^{29} +(3.50000 + 6.06218i) q^{31} +(1.50000 - 2.59808i) q^{33} +(1.50000 + 4.33013i) q^{35} +10.0000 q^{37} +(4.50000 + 7.79423i) q^{39} +(4.50000 - 2.59808i) q^{41} +(-7.50000 - 4.33013i) q^{43} +(4.50000 - 2.59808i) q^{45} +(-1.50000 + 2.59808i) q^{47} +(-6.50000 - 2.59808i) q^{49} +12.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +6.92820i q^{57} +(1.50000 + 2.59808i) q^{59} +(-1.50000 - 0.866025i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-4.50000 + 7.79423i) q^{65} +(7.50000 - 4.33013i) q^{67} +(-4.50000 - 7.79423i) q^{69} +3.46410i q^{71} -6.92820i q^{73} +(-3.00000 - 1.73205i) q^{75} +(-3.00000 + 3.46410i) q^{77} +(4.50000 + 2.59808i) q^{79} +9.00000 q^{81} +(4.50000 - 7.79423i) q^{83} +(6.00000 + 10.3923i) q^{85} +(13.5000 + 7.79423i) q^{87} -6.92820i q^{89} +(-4.50000 - 12.9904i) q^{91} +(-10.5000 + 6.06218i) q^{93} +(-6.00000 + 3.46410i) q^{95} +(4.50000 + 2.59808i) q^{97} +(4.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + q^{7} - 6 q^{9} - 3 q^{11} + 9 q^{13} - 3 q^{15} + 8 q^{19} + 9 q^{21} - 9 q^{23} - 2 q^{25} + 9 q^{29} + 7 q^{31} + 3 q^{33} + 3 q^{35} + 20 q^{37} + 9 q^{39} + 9 q^{41} - 15 q^{43} + 9 q^{45}+ \cdots + 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) −1.50000 + 0.866025i −0.670820 + 0.387298i −0.796387 − 0.604787i \(-0.793258\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 0.500000 − 2.59808i 0.188982 − 0.981981i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.50000 − 0.866025i −0.452267 − 0.261116i 0.256520 − 0.966539i \(-0.417424\pi\)
−0.708787 + 0.705422i \(0.750757\pi\)
\(12\) 0 0
\(13\) 4.50000 − 2.59808i 1.24808 − 0.720577i 0.277350 − 0.960769i \(-0.410544\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) −1.50000 − 2.59808i −0.387298 − 0.670820i
\(16\) 0 0
\(17\) − 6.92820i − 1.68034i −0.542326 − 0.840168i \(-0.682456\pi\)
0.542326 − 0.840168i \(-0.317544\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.50000 + 0.866025i 0.981981 + 0.188982i
\(22\) 0 0
\(23\) −4.50000 + 2.59808i −0.938315 + 0.541736i −0.889432 − 0.457068i \(-0.848900\pi\)
−0.0488832 + 0.998805i \(0.515566\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 4.50000 − 7.79423i 0.835629 − 1.44735i −0.0578882 − 0.998323i \(-0.518437\pi\)
0.893517 − 0.449029i \(-0.148230\pi\)
\(30\) 0 0
\(31\) 3.50000 + 6.06218i 0.628619 + 1.08880i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 1.50000 − 2.59808i 0.261116 − 0.452267i
\(34\) 0 0
\(35\) 1.50000 + 4.33013i 0.253546 + 0.731925i
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 4.50000 + 7.79423i 0.720577 + 1.24808i
\(40\) 0 0
\(41\) 4.50000 − 2.59808i 0.702782 − 0.405751i −0.105601 − 0.994409i \(-0.533677\pi\)
0.808383 + 0.588657i \(0.200343\pi\)
\(42\) 0 0
\(43\) −7.50000 − 4.33013i −1.14374 − 0.660338i −0.196385 − 0.980527i \(-0.562920\pi\)
−0.947354 + 0.320189i \(0.896254\pi\)
\(44\) 0 0
\(45\) 4.50000 − 2.59808i 0.670820 − 0.387298i
\(46\) 0 0
\(47\) −1.50000 + 2.59808i −0.218797 + 0.378968i −0.954441 − 0.298401i \(-0.903547\pi\)
0.735643 + 0.677369i \(0.236880\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 − 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) −1.50000 − 0.866025i −0.192055 − 0.110883i 0.400889 − 0.916127i \(-0.368701\pi\)
−0.592944 + 0.805243i \(0.702035\pi\)
\(62\) 0 0
\(63\) −1.50000 + 7.79423i −0.188982 + 0.981981i
\(64\) 0 0
\(65\) −4.50000 + 7.79423i −0.558156 + 0.966755i
\(66\) 0 0
\(67\) 7.50000 − 4.33013i 0.916271 − 0.529009i 0.0338274 − 0.999428i \(-0.489230\pi\)
0.882443 + 0.470418i \(0.155897\pi\)
\(68\) 0 0
\(69\) −4.50000 − 7.79423i −0.541736 − 0.938315i
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) − 6.92820i − 0.810885i −0.914121 − 0.405442i \(-0.867117\pi\)
0.914121 − 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) −3.00000 − 1.73205i −0.346410 − 0.200000i
\(76\) 0 0
\(77\) −3.00000 + 3.46410i −0.341882 + 0.394771i
\(78\) 0 0
\(79\) 4.50000 + 2.59808i 0.506290 + 0.292306i 0.731307 − 0.682048i \(-0.238911\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 4.50000 − 7.79423i 0.493939 − 0.855528i −0.506036 − 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) 0 0
\(85\) 6.00000 + 10.3923i 0.650791 + 1.12720i
\(86\) 0 0
\(87\) 13.5000 + 7.79423i 1.44735 + 0.835629i
\(88\) 0 0
\(89\) − 6.92820i − 0.734388i −0.930144 − 0.367194i \(-0.880318\pi\)
0.930144 − 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) −4.50000 − 12.9904i −0.471728 − 1.36176i
\(92\) 0 0
\(93\) −10.5000 + 6.06218i −1.08880 + 0.628619i
\(94\) 0 0
\(95\) −6.00000 + 3.46410i −0.615587 + 0.355409i
\(96\) 0 0
\(97\) 4.50000 + 2.59808i 0.456906 + 0.263795i 0.710742 − 0.703452i \(-0.248359\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 4.50000 + 2.59808i 0.452267 + 0.261116i
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.c.223.1 yes 2
3.2 odd 2 3024.2.cx.h.559.1 2
4.3 odd 2 1008.2.cx.b.223.1 ✓ 2
7.6 odd 2 1008.2.cx.f.223.1 yes 2
9.4 even 3 1008.2.cx.g.895.1 yes 2
9.5 odd 6 3024.2.cx.c.2575.1 2
12.11 even 2 3024.2.cx.e.559.1 2
21.20 even 2 3024.2.cx.b.559.1 2
28.27 even 2 1008.2.cx.g.223.1 yes 2
36.23 even 6 3024.2.cx.b.2575.1 2
36.31 odd 6 1008.2.cx.f.895.1 yes 2
63.13 odd 6 1008.2.cx.b.895.1 yes 2
63.41 even 6 3024.2.cx.e.2575.1 2
84.83 odd 2 3024.2.cx.c.559.1 2
252.139 even 6 inner 1008.2.cx.c.895.1 yes 2
252.167 odd 6 3024.2.cx.h.2575.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.b.223.1 ✓ 2 4.3 odd 2
1008.2.cx.b.895.1 yes 2 63.13 odd 6
1008.2.cx.c.223.1 yes 2 1.1 even 1 trivial
1008.2.cx.c.895.1 yes 2 252.139 even 6 inner
1008.2.cx.f.223.1 yes 2 7.6 odd 2
1008.2.cx.f.895.1 yes 2 36.31 odd 6
1008.2.cx.g.223.1 yes 2 28.27 even 2
1008.2.cx.g.895.1 yes 2 9.4 even 3
3024.2.cx.b.559.1 2 21.20 even 2
3024.2.cx.b.2575.1 2 36.23 even 6
3024.2.cx.c.559.1 2 84.83 odd 2
3024.2.cx.c.2575.1 2 9.5 odd 6
3024.2.cx.e.559.1 2 12.11 even 2
3024.2.cx.e.2575.1 2 63.41 even 6
3024.2.cx.h.559.1 2 3.2 odd 2
3024.2.cx.h.2575.1 2 252.167 odd 6