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Properties

Label 3024.2.r.k.1009.3
Level $3024$
Weight $2$
Character 3024.1009
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 = [6,0,0,0,3,0,3,0,0,0,-6] 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: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) 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 1009.3
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1009
Dual form 3024.2.r.k.2017.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.26604 - 2.19285i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.233956 - 0.405223i) q^{11} +(-2.91147 + 5.04282i) q^{13} -3.87939 q^{17} +2.18479 q^{19} +(0.0530334 - 0.0918566i) q^{23} +(-0.705737 - 1.22237i) q^{25} +(4.39053 + 7.60462i) q^{29} +(-3.84002 + 6.65111i) q^{31} +2.53209 q^{35} -7.68004 q^{37} +(-1.11334 + 1.92836i) q^{41} +(0.613341 + 1.06234i) q^{43} +(2.66637 + 4.61830i) q^{47} +(-0.500000 + 0.866025i) q^{49} +0.716881 q^{53} -1.18479 q^{55} +(-0.368241 + 0.637812i) q^{59} +(-0.479055 - 0.829748i) q^{61} +(7.37211 + 12.7689i) q^{65} +(-4.81908 + 8.34689i) q^{67} +13.2344 q^{71} -10.2686 q^{73} +(0.233956 - 0.405223i) q^{77} +(-6.31908 - 10.9450i) q^{79} +(1.36571 + 2.36549i) q^{83} +(-4.91147 + 8.50692i) q^{85} +8.11381 q^{89} -5.82295 q^{91} +(2.76604 - 4.79093i) q^{95} +(6.80200 + 11.7814i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} - 12 q^{17} + 6 q^{19} - 12 q^{23} + 6 q^{25} + 9 q^{29} - 3 q^{31} + 6 q^{35} - 6 q^{37} - 3 q^{43} - 3 q^{47} - 3 q^{49} - 12 q^{53} + 3 q^{59} - 6 q^{61}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.26604 − 2.19285i 0.566192 − 0.980674i −0.430745 − 0.902473i \(-0.641749\pi\)
0.996938 − 0.0782003i \(-0.0249174\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.233956 − 0.405223i −0.0705403 − 0.122179i 0.828598 − 0.559844i \(-0.189139\pi\)
−0.899138 + 0.437665i \(0.855806\pi\)
\(12\) 0 0
\(13\) −2.91147 + 5.04282i −0.807498 + 1.39863i 0.107094 + 0.994249i \(0.465845\pi\)
−0.914592 + 0.404378i \(0.867488\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.87939 −0.940889 −0.470445 − 0.882430i \(-0.655906\pi\)
−0.470445 + 0.882430i \(0.655906\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 − 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.0530334 − 0.0918566i 0.0110582 − 0.0191534i −0.860443 − 0.509546i \(-0.829813\pi\)
0.871502 + 0.490393i \(0.163147\pi\)
\(24\) 0 0
\(25\) −0.705737 − 1.22237i −0.141147 − 0.244474i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.39053 + 7.60462i 0.815301 + 1.41214i 0.909112 + 0.416552i \(0.136762\pi\)
−0.0938108 + 0.995590i \(0.529905\pi\)
\(30\) 0 0
\(31\) −3.84002 + 6.65111i −0.689688 + 1.19458i 0.282250 + 0.959341i \(0.408919\pi\)
−0.971939 + 0.235235i \(0.924414\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.53209 0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.11334 + 1.92836i −0.173875 + 0.301160i −0.939771 − 0.341804i \(-0.888962\pi\)
0.765897 + 0.642964i \(0.222295\pi\)
\(42\) 0 0
\(43\) 0.613341 + 1.06234i 0.0935336 + 0.162005i 0.908996 − 0.416806i \(-0.136850\pi\)
−0.815462 + 0.578811i \(0.803517\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.66637 + 4.61830i 0.388931 + 0.673648i 0.992306 − 0.123810i \(-0.0395112\pi\)
−0.603375 + 0.797457i \(0.706178\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.716881 0.0984712 0.0492356 − 0.998787i \(-0.484321\pi\)
0.0492356 + 0.998787i \(0.484321\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.368241 + 0.637812i −0.0479409 + 0.0830360i −0.889000 − 0.457907i \(-0.848599\pi\)
0.841059 + 0.540943i \(0.181933\pi\)
\(60\) 0 0
\(61\) −0.479055 − 0.829748i −0.0613368 − 0.106238i 0.833726 − 0.552178i \(-0.186203\pi\)
−0.895063 + 0.445939i \(0.852870\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.37211 + 12.7689i 0.914398 + 1.58378i
\(66\) 0 0
\(67\) −4.81908 + 8.34689i −0.588744 + 1.01973i 0.405653 + 0.914027i \(0.367044\pi\)
−0.994397 + 0.105708i \(0.966289\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.2344 1.57064 0.785318 − 0.619092i \(-0.212499\pi\)
0.785318 + 0.619092i \(0.212499\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 − 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.233956 − 0.405223i 0.0266617 − 0.0461794i
\(78\) 0 0
\(79\) −6.31908 − 10.9450i −0.710952 − 1.23140i −0.964500 − 0.264082i \(-0.914931\pi\)
0.253548 − 0.967323i \(-0.418402\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.36571 + 2.36549i 0.149907 + 0.259646i 0.931193 − 0.364527i \(-0.118769\pi\)
−0.781286 + 0.624173i \(0.785436\pi\)
\(84\) 0 0
\(85\) −4.91147 + 8.50692i −0.532724 + 0.922705i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.11381 0.860062 0.430031 − 0.902814i \(-0.358503\pi\)
0.430031 + 0.902814i \(0.358503\pi\)
\(90\) 0 0
\(91\) −5.82295 −0.610411
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.76604 − 4.79093i 0.283790 − 0.491539i
\(96\) 0 0
\(97\) 6.80200 + 11.7814i 0.690639 + 1.19622i 0.971629 + 0.236511i \(0.0760039\pi\)
−0.280990 + 0.959711i \(0.590663\pi\)
\(98\) 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.k.1009.3 6
3.2 odd 2 1008.2.r.h.337.1 6
4.3 odd 2 189.2.f.b.64.2 6
9.2 odd 6 1008.2.r.h.673.1 6
9.4 even 3 9072.2.a.bs.1.1 3
9.5 odd 6 9072.2.a.ca.1.3 3
9.7 even 3 inner 3024.2.r.k.2017.3 6
12.11 even 2 63.2.f.a.22.2 ✓ 6
28.3 even 6 1323.2.h.b.226.2 6
28.11 odd 6 1323.2.h.c.226.2 6
28.19 even 6 1323.2.g.e.361.2 6
28.23 odd 6 1323.2.g.d.361.2 6
28.27 even 2 1323.2.f.d.442.2 6
36.7 odd 6 189.2.f.b.127.2 6
36.11 even 6 63.2.f.a.43.2 yes 6
36.23 even 6 567.2.a.h.1.2 3
36.31 odd 6 567.2.a.c.1.2 3
84.11 even 6 441.2.h.d.373.2 6
84.23 even 6 441.2.g.c.67.2 6
84.47 odd 6 441.2.g.b.67.2 6
84.59 odd 6 441.2.h.e.373.2 6
84.83 odd 2 441.2.f.c.148.2 6
252.11 even 6 441.2.g.c.79.2 6
252.47 odd 6 441.2.h.e.214.2 6
252.79 odd 6 1323.2.h.c.802.2 6
252.83 odd 6 441.2.f.c.295.2 6
252.115 even 6 1323.2.g.e.667.2 6
252.139 even 6 3969.2.a.l.1.2 3
252.151 odd 6 1323.2.g.d.667.2 6
252.167 odd 6 3969.2.a.q.1.2 3
252.187 even 6 1323.2.h.b.802.2 6
252.191 even 6 441.2.h.d.214.2 6
252.223 even 6 1323.2.f.d.883.2 6
252.227 odd 6 441.2.g.b.79.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 12.11 even 2
63.2.f.a.43.2 yes 6 36.11 even 6
189.2.f.b.64.2 6 4.3 odd 2
189.2.f.b.127.2 6 36.7 odd 6
441.2.f.c.148.2 6 84.83 odd 2
441.2.f.c.295.2 6 252.83 odd 6
441.2.g.b.67.2 6 84.47 odd 6
441.2.g.b.79.2 6 252.227 odd 6
441.2.g.c.67.2 6 84.23 even 6
441.2.g.c.79.2 6 252.11 even 6
441.2.h.d.214.2 6 252.191 even 6
441.2.h.d.373.2 6 84.11 even 6
441.2.h.e.214.2 6 252.47 odd 6
441.2.h.e.373.2 6 84.59 odd 6
567.2.a.c.1.2 3 36.31 odd 6
567.2.a.h.1.2 3 36.23 even 6
1008.2.r.h.337.1 6 3.2 odd 2
1008.2.r.h.673.1 6 9.2 odd 6
1323.2.f.d.442.2 6 28.27 even 2
1323.2.f.d.883.2 6 252.223 even 6
1323.2.g.d.361.2 6 28.23 odd 6
1323.2.g.d.667.2 6 252.151 odd 6
1323.2.g.e.361.2 6 28.19 even 6
1323.2.g.e.667.2 6 252.115 even 6
1323.2.h.b.226.2 6 28.3 even 6
1323.2.h.b.802.2 6 252.187 even 6
1323.2.h.c.226.2 6 28.11 odd 6
1323.2.h.c.802.2 6 252.79 odd 6
3024.2.r.k.1009.3 6 1.1 even 1 trivial
3024.2.r.k.2017.3 6 9.7 even 3 inner
3969.2.a.l.1.2 3 252.139 even 6
3969.2.a.q.1.2 3 252.167 odd 6
9072.2.a.bs.1.1 3 9.4 even 3
9072.2.a.ca.1.3 3 9.5 odd 6