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Properties

Label 3024.2.r.g.2017.3
Level $3024$
Weight $2$
Character 3024.2017
Analytic conductor $24.147$
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] = [3024,2,Mod(1009,3024)] 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("3024.1009"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-5,0,-3,0,0,0,2] 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: \(24.1467615712\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2017.3
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2017
Dual form 3024.2.r.g.1009.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.590972 + 1.02359i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(1.85185 - 3.20750i) q^{11} +(-0.500000 - 0.866025i) q^{13} +6.94282 q^{17} -1.94282 q^{19} +(2.80150 + 4.85235i) q^{23} +(1.80150 - 3.12030i) q^{25} +(0.119562 - 0.207087i) q^{29} +(0.830095 + 1.43777i) q^{31} -1.18194 q^{35} -9.54583 q^{37} +(-5.09097 - 8.81782i) q^{41} +(1.11273 - 1.92730i) q^{43} +(-2.91423 + 5.04759i) q^{47} +(-0.500000 - 0.866025i) q^{49} +11.6030 q^{53} +4.37756 q^{55} +(-1.30150 - 2.25427i) q^{59} +(3.80150 - 6.58440i) q^{61} +(0.590972 - 1.02359i) q^{65} +(1.75404 + 3.03809i) q^{67} +8.60301 q^{71} +15.1488 q^{73} +(1.85185 + 3.20750i) q^{77} +(3.68878 - 6.38915i) q^{79} +(3.47141 - 6.01266i) q^{83} +(4.10301 + 7.10662i) q^{85} -2.74720 q^{89} +1.00000 q^{91} +(-1.14815 - 1.98866i) q^{95} +(-3.58414 + 6.20790i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{5} - 3 q^{7} + 2 q^{11} - 3 q^{13} + 24 q^{17} + 6 q^{19} - 6 q^{25} + q^{29} - 3 q^{31} + 10 q^{35} - 6 q^{37} - 22 q^{41} - 3 q^{43} + 9 q^{47} - 3 q^{49} + 36 q^{53} + 12 q^{55} + 9 q^{59}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0.590972 + 1.02359i 0.264291 + 0.457765i 0.967378 − 0.253339i \(-0.0815289\pi\)
−0.703087 + 0.711104i \(0.748196\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.85185 − 3.20750i 0.558353 − 0.967096i −0.439281 − 0.898350i \(-0.644767\pi\)
0.997634 − 0.0687465i \(-0.0219000\pi\)
\(12\) 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
\(15\) 0 0
\(16\) 0 0
\(17\) 6.94282 1.68388 0.841941 − 0.539570i \(-0.181413\pi\)
0.841941 + 0.539570i \(0.181413\pi\)
\(18\) 0 0
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.80150 + 4.85235i 0.584154 + 1.01178i 0.994980 + 0.100071i \(0.0319070\pi\)
−0.410826 + 0.911714i \(0.634760\pi\)
\(24\) 0 0
\(25\) 1.80150 − 3.12030i 0.360301 − 0.624060i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.119562 − 0.207087i 0.0222020 − 0.0384551i −0.854711 − 0.519104i \(-0.826266\pi\)
0.876913 + 0.480649i \(0.159599\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −5.09097 − 8.81782i −0.795076 − 1.37711i −0.922791 − 0.385301i \(-0.874097\pi\)
0.127715 − 0.991811i \(-0.459236\pi\)
\(42\) 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 0
\(46\) 0 0
\(47\) −2.91423 + 5.04759i −0.425084 + 0.736267i −0.996428 − 0.0844432i \(-0.973089\pi\)
0.571344 + 0.820711i \(0.306422\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.6030 1.59380 0.796898 − 0.604114i \(-0.206473\pi\)
0.796898 + 0.604114i \(0.206473\pi\)
\(54\) 0 0
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.30150 − 2.25427i −0.169442 − 0.293481i 0.768782 − 0.639511i \(-0.220863\pi\)
−0.938224 + 0.346029i \(0.887530\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 8.60301 1.02099 0.510495 − 0.859881i \(-0.329462\pi\)
0.510495 + 0.859881i \(0.329462\pi\)
\(72\) 0 0
\(73\) 15.1488 1.77304 0.886519 − 0.462693i \(-0.153117\pi\)
0.886519 + 0.462693i \(0.153117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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 0
\(82\) 0 0
\(83\) 3.47141 − 6.01266i 0.381037 − 0.659975i −0.610174 − 0.792267i \(-0.708900\pi\)
0.991211 + 0.132292i \(0.0422338\pi\)
\(84\) 0 0
\(85\) 4.10301 + 7.10662i 0.445034 + 0.770821i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.74720 −0.291203 −0.145602 − 0.989343i \(-0.546512\pi\)
−0.145602 + 0.989343i \(0.546512\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
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 3024.2.r.g.2017.3 6
3.2 odd 2 1008.2.r.k.673.2 6
4.3 odd 2 189.2.f.a.127.2 6
9.2 odd 6 9072.2.a.bq.1.3 3
9.4 even 3 inner 3024.2.r.g.1009.3 6
9.5 odd 6 1008.2.r.k.337.2 6
9.7 even 3 9072.2.a.cd.1.1 3
12.11 even 2 63.2.f.b.43.2 yes 6
28.3 even 6 1323.2.g.b.667.2 6
28.11 odd 6 1323.2.g.c.667.2 6
28.19 even 6 1323.2.h.e.802.2 6
28.23 odd 6 1323.2.h.d.802.2 6
28.27 even 2 1323.2.f.c.883.2 6
36.7 odd 6 567.2.a.g.1.2 3
36.11 even 6 567.2.a.d.1.2 3
36.23 even 6 63.2.f.b.22.2 ✓ 6
36.31 odd 6 189.2.f.a.64.2 6
84.11 even 6 441.2.g.e.79.2 6
84.23 even 6 441.2.h.c.214.2 6
84.47 odd 6 441.2.h.b.214.2 6
84.59 odd 6 441.2.g.d.79.2 6
84.83 odd 2 441.2.f.d.295.2 6
252.23 even 6 441.2.g.e.67.2 6
252.31 even 6 1323.2.h.e.226.2 6
252.59 odd 6 441.2.h.b.373.2 6
252.67 odd 6 1323.2.h.d.226.2 6
252.83 odd 6 3969.2.a.m.1.2 3
252.95 even 6 441.2.h.c.373.2 6
252.103 even 6 1323.2.g.b.361.2 6
252.131 odd 6 441.2.g.d.67.2 6
252.139 even 6 1323.2.f.c.442.2 6
252.167 odd 6 441.2.f.d.148.2 6
252.223 even 6 3969.2.a.p.1.2 3
252.247 odd 6 1323.2.g.c.361.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 36.23 even 6
63.2.f.b.43.2 yes 6 12.11 even 2
189.2.f.a.64.2 6 36.31 odd 6
189.2.f.a.127.2 6 4.3 odd 2
441.2.f.d.148.2 6 252.167 odd 6
441.2.f.d.295.2 6 84.83 odd 2
441.2.g.d.67.2 6 252.131 odd 6
441.2.g.d.79.2 6 84.59 odd 6
441.2.g.e.67.2 6 252.23 even 6
441.2.g.e.79.2 6 84.11 even 6
441.2.h.b.214.2 6 84.47 odd 6
441.2.h.b.373.2 6 252.59 odd 6
441.2.h.c.214.2 6 84.23 even 6
441.2.h.c.373.2 6 252.95 even 6
567.2.a.d.1.2 3 36.11 even 6
567.2.a.g.1.2 3 36.7 odd 6
1008.2.r.k.337.2 6 9.5 odd 6
1008.2.r.k.673.2 6 3.2 odd 2
1323.2.f.c.442.2 6 252.139 even 6
1323.2.f.c.883.2 6 28.27 even 2
1323.2.g.b.361.2 6 252.103 even 6
1323.2.g.b.667.2 6 28.3 even 6
1323.2.g.c.361.2 6 252.247 odd 6
1323.2.g.c.667.2 6 28.11 odd 6
1323.2.h.d.226.2 6 252.67 odd 6
1323.2.h.d.802.2 6 28.23 odd 6
1323.2.h.e.226.2 6 252.31 even 6
1323.2.h.e.802.2 6 28.19 even 6
3024.2.r.g.1009.3 6 9.4 even 3 inner
3024.2.r.g.2017.3 6 1.1 even 1 trivial
3969.2.a.m.1.2 3 252.83 odd 6
3969.2.a.p.1.2 3 252.223 even 6
9072.2.a.bq.1.3 3 9.2 odd 6
9072.2.a.cd.1.1 3 9.7 even 3