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 2017.1
Root \(0.500000 - 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2017
Dual form 3024.2.r.g.1009.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.79418 - 3.10761i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(1.40545 - 2.43430i) q^{11} +(-0.500000 - 0.866025i) q^{13} +4.11126 q^{17} +0.888736 q^{19} +(-2.93818 - 5.08907i) q^{23} +(-3.93818 + 6.82112i) q^{25} +(-0.849814 + 1.47192i) q^{29} +(-3.49381 - 6.05146i) q^{31} +3.58836 q^{35} +4.76509 q^{37} +(-2.70582 - 4.68661i) q^{41} +(2.60507 - 4.51212i) q^{43} +(1.33310 - 2.30900i) q^{47} +(-0.500000 - 0.866025i) q^{49} +0.123644 q^{53} -10.0865 q^{55} +(4.43818 + 7.68715i) q^{59} +(-1.93818 + 3.35702i) q^{61} +(-1.79418 + 3.10761i) q^{65} +(6.15452 + 10.6599i) q^{67} -2.87636 q^{71} -10.6414 q^{73} +(1.40545 + 2.43430i) q^{77} +(-3.54325 + 6.13709i) q^{79} +(2.05563 - 3.56046i) q^{83} +(-7.37636 - 12.7762i) q^{85} -9.60940 q^{89} +1.00000 q^{91} +(-1.59455 - 2.76185i) q^{95} +(-3.66071 + 6.34053i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{5} - 3 q^{7} + 2 q^{11} - 3 q^{13} + 24 q^{17} + 6 q^{19} - 6 q^{25} + q^{29} - 3 q^{31} + 10 q^{35} - 6 q^{37} - 22 q^{41} - 3 q^{43} + 9 q^{47} - 3 q^{49} + 36 q^{53} + 12 q^{55} + 9 q^{59}+ \cdots - 3 q^{97}+O(q^{100}) \) 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
\(5\) −1.79418 − 3.10761i −0.802383 − 1.38977i −0.918044 − 0.396479i \(-0.870232\pi\)
0.115661 − 0.993289i \(-0.463101\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.40545 − 2.43430i 0.423758 − 0.733970i −0.572546 − 0.819873i \(-0.694044\pi\)
0.996304 + 0.0859026i \(0.0273774\pi\)
\(12\) 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
\(17\) 4.11126 0.997128 0.498564 − 0.866853i \(-0.333861\pi\)
0.498564 + 0.866853i \(0.333861\pi\)
\(18\) 0 0
\(19\) 0.888736 0.203890 0.101945 − 0.994790i \(-0.467493\pi\)
0.101945 + 0.994790i \(0.467493\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.93818 − 5.08907i −0.612652 − 1.06115i −0.990792 − 0.135396i \(-0.956769\pi\)
0.378139 − 0.925749i \(-0.376564\pi\)
\(24\) 0 0
\(25\) −3.93818 + 6.82112i −0.787636 + 1.36422i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.849814 + 1.47192i −0.157807 + 0.273329i −0.934077 − 0.357071i \(-0.883776\pi\)
0.776271 + 0.630399i \(0.217109\pi\)
\(30\) 0 0
\(31\) −3.49381 − 6.05146i −0.627507 − 1.08687i −0.988050 − 0.154131i \(-0.950742\pi\)
0.360544 − 0.932742i \(-0.382591\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.58836 0.606544
\(36\) 0 0
\(37\) 4.76509 0.783376 0.391688 − 0.920098i \(-0.371891\pi\)
0.391688 + 0.920098i \(0.371891\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.70582 − 4.68661i −0.422578 − 0.731926i 0.573613 − 0.819126i \(-0.305541\pi\)
−0.996191 + 0.0872002i \(0.972208\pi\)
\(42\) 0 0
\(43\) 2.60507 − 4.51212i 0.397270 − 0.688092i −0.596118 − 0.802897i \(-0.703291\pi\)
0.993388 + 0.114805i \(0.0366243\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.33310 − 2.30900i 0.194453 − 0.336803i −0.752268 − 0.658857i \(-0.771040\pi\)
0.946721 + 0.322055i \(0.104373\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.123644 0.0169838 0.00849190 − 0.999964i \(-0.497297\pi\)
0.00849190 + 0.999964i \(0.497297\pi\)
\(54\) 0 0
\(55\) −10.0865 −1.36006
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.43818 + 7.68715i 0.577802 + 1.00078i 0.995731 + 0.0923022i \(0.0294226\pi\)
−0.417929 + 0.908479i \(0.637244\pi\)
\(60\) 0 0
\(61\) −1.93818 + 3.35702i −0.248158 + 0.429823i −0.963015 − 0.269448i \(-0.913159\pi\)
0.714857 + 0.699271i \(0.246492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.79418 + 3.10761i −0.222541 + 0.385452i
\(66\) 0 0
\(67\) 6.15452 + 10.6599i 0.751894 + 1.30232i 0.946904 + 0.321517i \(0.104193\pi\)
−0.195010 + 0.980801i \(0.562474\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.87636 −0.341361 −0.170680 − 0.985326i \(-0.554597\pi\)
−0.170680 + 0.985326i \(0.554597\pi\)
\(72\) 0 0
\(73\) −10.6414 −1.24549 −0.622744 − 0.782426i \(-0.713982\pi\)
−0.622744 + 0.782426i \(0.713982\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.40545 + 2.43430i 0.160165 + 0.277415i
\(78\) 0 0
\(79\) −3.54325 + 6.13709i −0.398647 + 0.690477i −0.993559 − 0.113314i \(-0.963853\pi\)
0.594912 + 0.803791i \(0.297187\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.05563 − 3.56046i 0.225635 − 0.390811i −0.730875 − 0.682512i \(-0.760888\pi\)
0.956510 + 0.291700i \(0.0942210\pi\)
\(84\) 0 0
\(85\) −7.37636 − 12.7762i −0.800078 − 1.38578i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.60940 −1.01859 −0.509297 − 0.860591i \(-0.670095\pi\)
−0.509297 + 0.860591i \(0.670095\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.59455 − 2.76185i −0.163598 − 0.283360i
\(96\) 0 0
\(97\) −3.66071 + 6.34053i −0.371688 + 0.643783i −0.989825 − 0.142287i \(-0.954554\pi\)
0.618137 + 0.786070i \(0.287888\pi\)
\(98\) 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.2017.1 6
3.2 odd 2 1008.2.r.k.673.1 6
4.3 odd 2 189.2.f.a.127.3 6
9.2 odd 6 9072.2.a.bq.1.1 3
9.4 even 3 inner 3024.2.r.g.1009.1 6
9.5 odd 6 1008.2.r.k.337.1 6
9.7 even 3 9072.2.a.cd.1.3 3
12.11 even 2 63.2.f.b.43.1 yes 6
28.3 even 6 1323.2.g.b.667.3 6
28.11 odd 6 1323.2.g.c.667.3 6
28.19 even 6 1323.2.h.e.802.1 6
28.23 odd 6 1323.2.h.d.802.1 6
28.27 even 2 1323.2.f.c.883.3 6
36.7 odd 6 567.2.a.g.1.1 3
36.11 even 6 567.2.a.d.1.3 3
36.23 even 6 63.2.f.b.22.1 ✓ 6
36.31 odd 6 189.2.f.a.64.3 6
84.11 even 6 441.2.g.e.79.1 6
84.23 even 6 441.2.h.c.214.3 6
84.47 odd 6 441.2.h.b.214.3 6
84.59 odd 6 441.2.g.d.79.1 6
84.83 odd 2 441.2.f.d.295.1 6
252.23 even 6 441.2.g.e.67.1 6
252.31 even 6 1323.2.h.e.226.1 6
252.59 odd 6 441.2.h.b.373.3 6
252.67 odd 6 1323.2.h.d.226.1 6
252.83 odd 6 3969.2.a.m.1.3 3
252.95 even 6 441.2.h.c.373.3 6
252.103 even 6 1323.2.g.b.361.3 6
252.131 odd 6 441.2.g.d.67.1 6
252.139 even 6 1323.2.f.c.442.3 6
252.167 odd 6 441.2.f.d.148.1 6
252.223 even 6 3969.2.a.p.1.1 3
252.247 odd 6 1323.2.g.c.361.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 ✓ 6 36.23 even 6
63.2.f.b.43.1 yes 6 12.11 even 2
189.2.f.a.64.3 6 36.31 odd 6
189.2.f.a.127.3 6 4.3 odd 2
441.2.f.d.148.1 6 252.167 odd 6
441.2.f.d.295.1 6 84.83 odd 2
441.2.g.d.67.1 6 252.131 odd 6
441.2.g.d.79.1 6 84.59 odd 6
441.2.g.e.67.1 6 252.23 even 6
441.2.g.e.79.1 6 84.11 even 6
441.2.h.b.214.3 6 84.47 odd 6
441.2.h.b.373.3 6 252.59 odd 6
441.2.h.c.214.3 6 84.23 even 6
441.2.h.c.373.3 6 252.95 even 6
567.2.a.d.1.3 3 36.11 even 6
567.2.a.g.1.1 3 36.7 odd 6
1008.2.r.k.337.1 6 9.5 odd 6
1008.2.r.k.673.1 6 3.2 odd 2
1323.2.f.c.442.3 6 252.139 even 6
1323.2.f.c.883.3 6 28.27 even 2
1323.2.g.b.361.3 6 252.103 even 6
1323.2.g.b.667.3 6 28.3 even 6
1323.2.g.c.361.3 6 252.247 odd 6
1323.2.g.c.667.3 6 28.11 odd 6
1323.2.h.d.226.1 6 252.67 odd 6
1323.2.h.d.802.1 6 28.23 odd 6
1323.2.h.e.226.1 6 252.31 even 6
1323.2.h.e.802.1 6 28.19 even 6
3024.2.r.g.1009.1 6 9.4 even 3 inner
3024.2.r.g.2017.1 6 1.1 even 1 trivial
3969.2.a.m.1.3 3 252.83 odd 6
3969.2.a.p.1.1 3 252.223 even 6
9072.2.a.bq.1.1 3 9.2 odd 6
9072.2.a.cd.1.3 3 9.7 even 3