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Properties

Label 1008.2.r.k.673.2
Level $1008$
Weight $2$
Character 1008.673
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
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(337,1008)] 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("1008.337"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,4,0,5,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 673.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 1008.673
Dual form 1008.2.r.k.337.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.619562 - 1.61745i) q^{3} +(-0.590972 - 1.02359i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-2.23229 - 2.00422i) q^{9} +(-1.85185 + 3.20750i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-2.02175 + 0.321688i) q^{15} -6.94282 q^{17} -1.94282 q^{19} +(1.09097 + 1.34528i) q^{21} +(-2.80150 - 4.85235i) q^{23} +(1.80150 - 3.12030i) q^{25} +(-4.62476 + 2.36887i) q^{27} +(-0.119562 + 0.207087i) q^{29} +(0.830095 + 1.43777i) q^{31} +(4.04063 + 4.98251i) q^{33} +1.18194 q^{35} -9.54583 q^{37} +(-1.71053 + 0.272169i) q^{39} +(5.09097 + 8.81782i) q^{41} +(1.11273 - 1.92730i) q^{43} +(-0.732287 + 3.46939i) q^{45} +(2.91423 - 5.04759i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-4.30150 + 11.2297i) q^{51} -11.6030 q^{53} +4.37756 q^{55} +(-1.20370 + 3.14241i) q^{57} +(1.30150 + 2.25427i) q^{59} +(3.80150 - 6.58440i) q^{61} +(2.85185 - 0.931107i) q^{63} +(-0.590972 + 1.02359i) q^{65} +(1.75404 + 3.03809i) q^{67} +(-9.58414 + 1.52496i) q^{69} -8.60301 q^{71} +15.1488 q^{73} +(-3.93078 - 4.84706i) q^{75} +(-1.85185 - 3.20750i) q^{77} +(3.68878 - 6.38915i) q^{79} +(0.966208 + 8.94799i) q^{81} +(-3.47141 + 6.01266i) q^{83} +(4.10301 + 7.10662i) q^{85} +(0.260877 + 0.321688i) q^{87} +2.74720 q^{89} +1.00000 q^{91} +(2.83981 - 0.451852i) q^{93} +(1.14815 + 1.98866i) q^{95} +(-3.58414 + 6.20790i) q^{97} +(10.5624 - 3.44854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 5 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 3 q^{13} - 11 q^{15} - 24 q^{17} + 6 q^{19} - 2 q^{21} - 6 q^{25} + 7 q^{27} - q^{29} - 3 q^{31} + 8 q^{33} - 10 q^{35} - 6 q^{37} - 2 q^{39} + 22 q^{41}+ \cdots + 46 q^{99}+O(q^{100}) \) 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
\(3\) 0.619562 − 1.61745i 0.357704 − 0.933835i
\(4\) 0 0
\(5\) −0.590972 − 1.02359i −0.264291 − 0.457765i 0.703087 − 0.711104i \(-0.251804\pi\)
−0.967378 + 0.253339i \(0.918471\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) −2.23229 − 2.00422i −0.744096 − 0.668073i
\(10\) 0 0
\(11\) −1.85185 + 3.20750i −0.558353 + 0.967096i 0.439281 + 0.898350i \(0.355233\pi\)
−0.997634 + 0.0687465i \(0.978100\pi\)
\(12\) 0 0
\(13\) −0.500000 − 0.866025i −0.138675 − 0.240192i 0.788320 − 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) −2.02175 + 0.321688i −0.522014 + 0.0830595i
\(16\) 0 0
\(17\) −6.94282 −1.68388 −0.841941 − 0.539570i \(-0.818587\pi\)
−0.841941 + 0.539570i \(0.818587\pi\)
\(18\) 0 0
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) 0 0
\(21\) 1.09097 + 1.34528i 0.238070 + 0.293564i
\(22\) 0 0
\(23\) −2.80150 − 4.85235i −0.584154 − 1.01178i −0.994980 − 0.100071i \(-0.968093\pi\)
0.410826 − 0.911714i \(-0.365240\pi\)
\(24\) 0 0
\(25\) 1.80150 − 3.12030i 0.360301 − 0.624060i
\(26\) 0 0
\(27\) −4.62476 + 2.36887i −0.890036 + 0.455890i
\(28\) 0 0
\(29\) −0.119562 + 0.207087i −0.0222020 + 0.0384551i −0.876913 − 0.480649i \(-0.840401\pi\)
0.854711 + 0.519104i \(0.173734\pi\)
\(30\) 0 0
\(31\) 0.830095 + 1.43777i 0.149089 + 0.258231i 0.930891 − 0.365297i \(-0.119032\pi\)
−0.781802 + 0.623527i \(0.785699\pi\)
\(32\) 0 0
\(33\) 4.04063 + 4.98251i 0.703383 + 0.867344i
\(34\) 0 0
\(35\) 1.18194 0.199785
\(36\) 0 0
\(37\) −9.54583 −1.56932 −0.784662 − 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) 0 0
\(39\) −1.71053 + 0.272169i −0.273905 + 0.0435819i
\(40\) 0 0
\(41\) 5.09097 + 8.81782i 0.795076 + 1.37711i 0.922791 + 0.385301i \(0.125903\pi\)
−0.127715 + 0.991811i \(0.540764\pi\)
\(42\) 0 0
\(43\) 1.11273 − 1.92730i 0.169689 − 0.293910i −0.768622 − 0.639704i \(-0.779057\pi\)
0.938311 + 0.345794i \(0.112390\pi\)
\(44\) 0 0
\(45\) −0.732287 + 3.46939i −0.109163 + 0.517186i
\(46\) 0 0
\(47\) 2.91423 − 5.04759i 0.425084 − 0.736267i −0.571344 − 0.820711i \(-0.693578\pi\)
0.996428 + 0.0844432i \(0.0269112\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) −4.30150 + 11.2297i −0.602331 + 1.57247i
\(52\) 0 0
\(53\) −11.6030 −1.59380 −0.796898 − 0.604114i \(-0.793527\pi\)
−0.796898 + 0.604114i \(0.793527\pi\)
\(54\) 0 0
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) −1.20370 + 3.14241i −0.159434 + 0.416223i
\(58\) 0 0
\(59\) 1.30150 + 2.25427i 0.169442 + 0.293481i 0.938224 − 0.346029i \(-0.112470\pi\)
−0.768782 + 0.639511i \(0.779137\pi\)
\(60\) 0 0
\(61\) 3.80150 − 6.58440i 0.486733 − 0.843046i −0.513151 − 0.858298i \(-0.671522\pi\)
0.999884 + 0.0152524i \(0.00485519\pi\)
\(62\) 0 0
\(63\) 2.85185 − 0.931107i 0.359299 − 0.117308i
\(64\) 0 0
\(65\) −0.590972 + 1.02359i −0.0733010 + 0.126961i
\(66\) 0 0
\(67\) 1.75404 + 3.03809i 0.214290 + 0.371161i 0.953053 − 0.302804i \(-0.0979229\pi\)
−0.738763 + 0.673966i \(0.764590\pi\)
\(68\) 0 0
\(69\) −9.58414 + 1.52496i −1.15379 + 0.183584i
\(70\) 0 0
\(71\) −8.60301 −1.02099 −0.510495 − 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(72\) 0 0
\(73\) 15.1488 1.77304 0.886519 − 0.462693i \(-0.153117\pi\)
0.886519 + 0.462693i \(0.153117\pi\)
\(74\) 0 0
\(75\) −3.93078 − 4.84706i −0.453888 − 0.559690i
\(76\) 0 0
\(77\) −1.85185 − 3.20750i −0.211038 − 0.365528i
\(78\) 0 0
\(79\) 3.68878 − 6.38915i 0.415020 − 0.718836i −0.580410 − 0.814324i \(-0.697108\pi\)
0.995431 + 0.0954881i \(0.0304412\pi\)
\(80\) 0 0
\(81\) 0.966208 + 8.94799i 0.107356 + 0.994221i
\(82\) 0 0
\(83\) −3.47141 + 6.01266i −0.381037 + 0.659975i −0.991211 − 0.132292i \(-0.957766\pi\)
0.610174 + 0.792267i \(0.291100\pi\)
\(84\) 0 0
\(85\) 4.10301 + 7.10662i 0.445034 + 0.770821i
\(86\) 0 0
\(87\) 0.260877 + 0.321688i 0.0279689 + 0.0344886i
\(88\) 0 0
\(89\) 2.74720 0.291203 0.145602 − 0.989343i \(-0.453488\pi\)
0.145602 + 0.989343i \(0.453488\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 2.83981 − 0.451852i 0.294475 − 0.0468548i
\(94\) 0 0
\(95\) 1.14815 + 1.98866i 0.117798 + 0.204032i
\(96\) 0 0
\(97\) −3.58414 + 6.20790i −0.363914 + 0.630317i −0.988601 − 0.150558i \(-0.951893\pi\)
0.624687 + 0.780875i \(0.285226\pi\)
\(98\) 0 0
\(99\) 10.5624 − 3.44854i 1.06156 − 0.346591i
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.r.k.673.2 6
3.2 odd 2 3024.2.r.g.2017.3 6
4.3 odd 2 63.2.f.b.43.2 yes 6
9.2 odd 6 9072.2.a.cd.1.1 3
9.4 even 3 inner 1008.2.r.k.337.2 6
9.5 odd 6 3024.2.r.g.1009.3 6
9.7 even 3 9072.2.a.bq.1.3 3
12.11 even 2 189.2.f.a.127.2 6
28.3 even 6 441.2.g.d.79.2 6
28.11 odd 6 441.2.g.e.79.2 6
28.19 even 6 441.2.h.b.214.2 6
28.23 odd 6 441.2.h.c.214.2 6
28.27 even 2 441.2.f.d.295.2 6
36.7 odd 6 567.2.a.d.1.2 3
36.11 even 6 567.2.a.g.1.2 3
36.23 even 6 189.2.f.a.64.2 6
36.31 odd 6 63.2.f.b.22.2 ✓ 6
84.11 even 6 1323.2.g.c.667.2 6
84.23 even 6 1323.2.h.d.802.2 6
84.47 odd 6 1323.2.h.e.802.2 6
84.59 odd 6 1323.2.g.b.667.2 6
84.83 odd 2 1323.2.f.c.883.2 6
252.23 even 6 1323.2.g.c.361.2 6
252.31 even 6 441.2.h.b.373.2 6
252.59 odd 6 1323.2.h.e.226.2 6
252.67 odd 6 441.2.h.c.373.2 6
252.83 odd 6 3969.2.a.p.1.2 3
252.95 even 6 1323.2.h.d.226.2 6
252.103 even 6 441.2.g.d.67.2 6
252.131 odd 6 1323.2.g.b.361.2 6
252.139 even 6 441.2.f.d.148.2 6
252.167 odd 6 1323.2.f.c.442.2 6
252.223 even 6 3969.2.a.m.1.2 3
252.247 odd 6 441.2.g.e.67.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 36.31 odd 6
63.2.f.b.43.2 yes 6 4.3 odd 2
189.2.f.a.64.2 6 36.23 even 6
189.2.f.a.127.2 6 12.11 even 2
441.2.f.d.148.2 6 252.139 even 6
441.2.f.d.295.2 6 28.27 even 2
441.2.g.d.67.2 6 252.103 even 6
441.2.g.d.79.2 6 28.3 even 6
441.2.g.e.67.2 6 252.247 odd 6
441.2.g.e.79.2 6 28.11 odd 6
441.2.h.b.214.2 6 28.19 even 6
441.2.h.b.373.2 6 252.31 even 6
441.2.h.c.214.2 6 28.23 odd 6
441.2.h.c.373.2 6 252.67 odd 6
567.2.a.d.1.2 3 36.7 odd 6
567.2.a.g.1.2 3 36.11 even 6
1008.2.r.k.337.2 6 9.4 even 3 inner
1008.2.r.k.673.2 6 1.1 even 1 trivial
1323.2.f.c.442.2 6 252.167 odd 6
1323.2.f.c.883.2 6 84.83 odd 2
1323.2.g.b.361.2 6 252.131 odd 6
1323.2.g.b.667.2 6 84.59 odd 6
1323.2.g.c.361.2 6 252.23 even 6
1323.2.g.c.667.2 6 84.11 even 6
1323.2.h.d.226.2 6 252.95 even 6
1323.2.h.d.802.2 6 84.23 even 6
1323.2.h.e.226.2 6 252.59 odd 6
1323.2.h.e.802.2 6 84.47 odd 6
3024.2.r.g.1009.3 6 9.5 odd 6
3024.2.r.g.2017.3 6 3.2 odd 2
3969.2.a.m.1.2 3 252.223 even 6
3969.2.a.p.1.2 3 252.83 odd 6
9072.2.a.bq.1.3 3 9.7 even 3
9072.2.a.cd.1.1 3 9.2 odd 6