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Properties

Label 1008.2.t.d.961.1
Level $1008$
Weight $2$
Character 1008.961
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(193,1008)] 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("1008.193"); 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([0, 0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.961
Dual form 1008.2.t.d.193.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} -1.00000 q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} -5.00000 q^{11} +(2.50000 + 4.33013i) q^{13} +(-1.50000 - 0.866025i) q^{15} +(-1.50000 - 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} +(-3.00000 + 3.46410i) q^{21} -3.00000 q^{23} -4.00000 q^{25} +5.19615i q^{27} +(0.500000 - 0.866025i) q^{29} +(-7.50000 - 4.33013i) q^{33} +(0.500000 - 2.59808i) q^{35} +(-1.50000 + 2.59808i) q^{37} +8.66025i q^{39} +(2.50000 + 4.33013i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(-1.50000 - 2.59808i) q^{45} +(-6.50000 - 2.59808i) q^{49} -5.19615i q^{51} +(4.50000 + 7.79423i) q^{53} +5.00000 q^{55} +(1.50000 - 0.866025i) q^{57} +(7.00000 + 12.1244i) q^{61} +(-7.50000 + 2.59808i) q^{63} +(-2.50000 - 4.33013i) q^{65} +(2.00000 - 3.46410i) q^{67} +(-4.50000 - 2.59808i) q^{69} +12.0000 q^{71} +(-1.50000 - 2.59808i) q^{73} +(-6.00000 - 3.46410i) q^{75} +(2.50000 - 12.9904i) q^{77} +(4.00000 + 6.92820i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-4.50000 + 7.79423i) q^{83} +(1.50000 + 2.59808i) q^{85} +(1.50000 - 0.866025i) q^{87} +(6.50000 - 11.2583i) q^{89} +(-12.5000 + 4.33013i) q^{91} +(-0.500000 + 0.866025i) q^{95} +(4.50000 - 7.79423i) q^{97} +(-7.50000 - 12.9904i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 10 q^{11} + 5 q^{13} - 3 q^{15} - 3 q^{17} + q^{19} - 6 q^{21} - 6 q^{23} - 8 q^{25} + q^{29} - 15 q^{33} + q^{35} - 3 q^{37} + 5 q^{41} - q^{43} - 3 q^{45}+ \cdots - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 2.50000 + 4.33013i 0.693375 + 1.20096i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.50000 − 0.866025i −0.387298 − 0.223607i
\(16\) 0 0
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 0 0
\(19\) 0.500000 − 0.866025i 0.114708 − 0.198680i −0.802955 − 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.46410i −0.654654 + 0.755929i
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0.500000 − 0.866025i 0.0928477 − 0.160817i −0.815861 − 0.578249i \(-0.803736\pi\)
0.908708 + 0.417432i \(0.137070\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) −7.50000 − 4.33013i −1.30558 − 0.753778i
\(34\) 0 0
\(35\) 0.500000 − 2.59808i 0.0845154 − 0.439155i
\(36\) 0 0
\(37\) −1.50000 + 2.59808i −0.246598 + 0.427121i −0.962580 − 0.270998i \(-0.912646\pi\)
0.715981 + 0.698119i \(0.245980\pi\)
\(38\) 0 0
\(39\) 8.66025i 1.38675i
\(40\) 0 0
\(41\) 2.50000 + 4.33013i 0.390434 + 0.676252i 0.992507 − 0.122189i \(-0.0389915\pi\)
−0.602072 + 0.798441i \(0.705658\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.0762493 + 0.132068i −0.901629 − 0.432511i \(-0.857628\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) 0 0
\(45\) −1.50000 − 2.59808i −0.223607 − 0.387298i
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) − 5.19615i − 0.727607i
\(52\) 0 0
\(53\) 4.50000 + 7.79423i 0.618123 + 1.07062i 0.989828 + 0.142269i \(0.0454398\pi\)
−0.371706 + 0.928351i \(0.621227\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 1.50000 − 0.866025i 0.198680 − 0.114708i
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 7.00000 + 12.1244i 0.896258 + 1.55236i 0.832240 + 0.554416i \(0.187058\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) −7.50000 + 2.59808i −0.944911 + 0.327327i
\(64\) 0 0
\(65\) −2.50000 − 4.33013i −0.310087 − 0.537086i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) −4.50000 − 2.59808i −0.541736 − 0.312772i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.50000 − 2.59808i −0.175562 − 0.304082i 0.764794 − 0.644275i \(-0.222841\pi\)
−0.940356 + 0.340193i \(0.889507\pi\)
\(74\) 0 0
\(75\) −6.00000 − 3.46410i −0.692820 − 0.400000i
\(76\) 0 0
\(77\) 2.50000 − 12.9904i 0.284901 − 1.48039i
\(78\) 0 0
\(79\) 4.00000 + 6.92820i 0.450035 + 0.779484i 0.998388 − 0.0567635i \(-0.0180781\pi\)
−0.548352 + 0.836247i \(0.684745\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 − 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) 0 0
\(85\) 1.50000 + 2.59808i 0.162698 + 0.281801i
\(86\) 0 0
\(87\) 1.50000 − 0.866025i 0.160817 − 0.0928477i
\(88\) 0 0
\(89\) 6.50000 − 11.2583i 0.688999 − 1.19338i −0.283164 − 0.959072i \(-0.591384\pi\)
0.972162 − 0.234309i \(-0.0752827\pi\)
\(90\) 0 0
\(91\) −12.5000 + 4.33013i −1.31036 + 0.453921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) 4.50000 − 7.79423i 0.456906 − 0.791384i −0.541890 − 0.840450i \(-0.682291\pi\)
0.998796 + 0.0490655i \(0.0156243\pi\)
\(98\) 0 0
\(99\) −7.50000 − 12.9904i −0.753778 − 1.30558i
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.t.d.961.1 2
3.2 odd 2 3024.2.t.d.289.1 2
4.3 odd 2 63.2.g.a.16.1 yes 2
7.4 even 3 1008.2.q.c.529.1 2
9.4 even 3 1008.2.q.c.625.1 2
9.5 odd 6 3024.2.q.b.2305.1 2
12.11 even 2 189.2.g.a.100.1 2
21.11 odd 6 3024.2.q.b.2881.1 2
28.3 even 6 441.2.h.a.214.1 2
28.11 odd 6 63.2.h.a.25.1 yes 2
28.19 even 6 441.2.f.a.295.1 2
28.23 odd 6 441.2.f.b.295.1 2
28.27 even 2 441.2.g.a.79.1 2
36.7 odd 6 567.2.e.a.163.1 2
36.11 even 6 567.2.e.b.163.1 2
36.23 even 6 189.2.h.a.37.1 2
36.31 odd 6 63.2.h.a.58.1 yes 2
63.4 even 3 inner 1008.2.t.d.193.1 2
63.32 odd 6 3024.2.t.d.1873.1 2
84.11 even 6 189.2.h.a.46.1 2
84.23 even 6 1323.2.f.a.883.1 2
84.47 odd 6 1323.2.f.b.883.1 2
84.59 odd 6 1323.2.h.a.802.1 2
84.83 odd 2 1323.2.g.a.667.1 2
252.11 even 6 567.2.e.b.487.1 2
252.23 even 6 1323.2.f.a.442.1 2
252.31 even 6 441.2.g.a.67.1 2
252.47 odd 6 3969.2.a.a.1.1 1
252.59 odd 6 1323.2.g.a.361.1 2
252.67 odd 6 63.2.g.a.4.1 ✓ 2
252.79 odd 6 3969.2.a.d.1.1 1
252.95 even 6 189.2.g.a.172.1 2
252.103 even 6 441.2.f.a.148.1 2
252.131 odd 6 1323.2.f.b.442.1 2
252.139 even 6 441.2.h.a.373.1 2
252.151 odd 6 567.2.e.a.487.1 2
252.167 odd 6 1323.2.h.a.226.1 2
252.187 even 6 3969.2.a.f.1.1 1
252.191 even 6 3969.2.a.c.1.1 1
252.247 odd 6 441.2.f.b.148.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.g.a.4.1 ✓ 2 252.67 odd 6
63.2.g.a.16.1 yes 2 4.3 odd 2
63.2.h.a.25.1 yes 2 28.11 odd 6
63.2.h.a.58.1 yes 2 36.31 odd 6
189.2.g.a.100.1 2 12.11 even 2
189.2.g.a.172.1 2 252.95 even 6
189.2.h.a.37.1 2 36.23 even 6
189.2.h.a.46.1 2 84.11 even 6
441.2.f.a.148.1 2 252.103 even 6
441.2.f.a.295.1 2 28.19 even 6
441.2.f.b.148.1 2 252.247 odd 6
441.2.f.b.295.1 2 28.23 odd 6
441.2.g.a.67.1 2 252.31 even 6
441.2.g.a.79.1 2 28.27 even 2
441.2.h.a.214.1 2 28.3 even 6
441.2.h.a.373.1 2 252.139 even 6
567.2.e.a.163.1 2 36.7 odd 6
567.2.e.a.487.1 2 252.151 odd 6
567.2.e.b.163.1 2 36.11 even 6
567.2.e.b.487.1 2 252.11 even 6
1008.2.q.c.529.1 2 7.4 even 3
1008.2.q.c.625.1 2 9.4 even 3
1008.2.t.d.193.1 2 63.4 even 3 inner
1008.2.t.d.961.1 2 1.1 even 1 trivial
1323.2.f.a.442.1 2 252.23 even 6
1323.2.f.a.883.1 2 84.23 even 6
1323.2.f.b.442.1 2 252.131 odd 6
1323.2.f.b.883.1 2 84.47 odd 6
1323.2.g.a.361.1 2 252.59 odd 6
1323.2.g.a.667.1 2 84.83 odd 2
1323.2.h.a.226.1 2 252.167 odd 6
1323.2.h.a.802.1 2 84.59 odd 6
3024.2.q.b.2305.1 2 9.5 odd 6
3024.2.q.b.2881.1 2 21.11 odd 6
3024.2.t.d.289.1 2 3.2 odd 2
3024.2.t.d.1873.1 2 63.32 odd 6
3969.2.a.a.1.1 1 252.47 odd 6
3969.2.a.c.1.1 1 252.191 even 6
3969.2.a.d.1.1 1 252.79 odd 6
3969.2.a.f.1.1 1 252.187 even 6