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Properties

Label 3024.2.r.k.1009.2
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 1009.2
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1009
Dual form 3024.2.r.k.2017.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.673648 - 1.16679i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.826352 - 1.43128i) q^{11} +(1.68479 - 2.91815i) q^{13} -0.467911 q^{17} +3.22668 q^{19} +(-4.47178 + 7.74535i) q^{23} +(1.59240 + 2.75811i) q^{25} +(3.13429 + 5.42874i) q^{29} +(4.61721 - 7.99724i) q^{31} +1.34730 q^{35} +9.23442 q^{37} +(1.70574 - 2.95442i) q^{41} +(-2.20574 - 3.82045i) q^{43} +(-4.67752 - 8.10170i) q^{47} +(-0.500000 + 0.866025i) q^{49} +0.573978 q^{53} -2.22668 q^{55} +(5.19846 - 9.00400i) q^{59} +(-3.81908 - 6.61484i) q^{61} +(-2.26991 - 3.93161i) q^{65} +(0.298133 - 0.516382i) q^{67} -0.554378 q^{71} +2.04963 q^{73} +(0.826352 - 1.43128i) q^{77} +(-1.20187 - 2.08169i) q^{79} +(7.52481 + 13.0334i) q^{83} +(-0.315207 + 0.545955i) q^{85} -9.08647 q^{89} +3.36959 q^{91} +(2.17365 - 3.76487i) q^{95} +(0.949493 + 1.64457i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} - 12 q^{17} + 6 q^{19} - 12 q^{23} + 6 q^{25} + 9 q^{29} - 3 q^{31} + 6 q^{35} - 6 q^{37} - 3 q^{43} - 3 q^{47} - 3 q^{49} - 12 q^{53} + 3 q^{59} - 6 q^{61}+ \cdots + 3 q^{97}+O(q^{100}) \) 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
\(5\) 0.673648 − 1.16679i 0.301265 − 0.521806i −0.675158 − 0.737673i \(-0.735925\pi\)
0.976423 + 0.215867i \(0.0692579\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.826352 − 1.43128i −0.249154 − 0.431548i 0.714137 − 0.700006i \(-0.246819\pi\)
−0.963291 + 0.268458i \(0.913486\pi\)
\(12\) 0 0
\(13\) 1.68479 − 2.91815i 0.467277 − 0.809348i −0.532024 − 0.846729i \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.467911 −0.113485 −0.0567426 − 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 0 0
\(19\) 3.22668 0.740252 0.370126 − 0.928982i \(-0.379315\pi\)
0.370126 + 0.928982i \(0.379315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.47178 + 7.74535i −0.932431 + 1.61502i −0.153279 + 0.988183i \(0.548983\pi\)
−0.779152 + 0.626835i \(0.784350\pi\)
\(24\) 0 0
\(25\) 1.59240 + 2.75811i 0.318479 + 0.551622i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.13429 + 5.42874i 0.582022 + 1.00809i 0.995239 + 0.0974595i \(0.0310717\pi\)
−0.413217 + 0.910632i \(0.635595\pi\)
\(30\) 0 0
\(31\) 4.61721 − 7.99724i 0.829276 − 1.43635i −0.0693317 − 0.997594i \(-0.522087\pi\)
0.898607 − 0.438754i \(-0.144580\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.34730 0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.70574 − 2.95442i 0.266391 − 0.461403i −0.701536 − 0.712634i \(-0.747502\pi\)
0.967927 + 0.251231i \(0.0808353\pi\)
\(42\) 0 0
\(43\) −2.20574 − 3.82045i −0.336372 − 0.582613i 0.647376 − 0.762171i \(-0.275867\pi\)
−0.983747 + 0.179558i \(0.942533\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.67752 − 8.10170i −0.682286 − 1.18175i −0.974281 − 0.225335i \(-0.927652\pi\)
0.291995 − 0.956420i \(-0.405681\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.573978 0.0788419 0.0394210 − 0.999223i \(-0.487449\pi\)
0.0394210 + 0.999223i \(0.487449\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.19846 − 9.00400i 0.676782 − 1.17222i −0.299162 − 0.954202i \(-0.596707\pi\)
0.975945 − 0.218019i \(-0.0699595\pi\)
\(60\) 0 0
\(61\) −3.81908 − 6.61484i −0.488983 − 0.846943i 0.510937 − 0.859618i \(-0.329299\pi\)
−0.999920 + 0.0126752i \(0.995965\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.26991 − 3.93161i −0.281548 − 0.487656i
\(66\) 0 0
\(67\) 0.298133 − 0.516382i 0.0364228 − 0.0630861i −0.847239 − 0.531211i \(-0.821737\pi\)
0.883662 + 0.468125i \(0.155070\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.554378 −0.0657925 −0.0328963 − 0.999459i \(-0.510473\pi\)
−0.0328963 + 0.999459i \(0.510473\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.826352 − 1.43128i 0.0941715 − 0.163110i
\(78\) 0 0
\(79\) −1.20187 − 2.08169i −0.135221 − 0.234209i 0.790461 − 0.612512i \(-0.209841\pi\)
−0.925682 + 0.378303i \(0.876508\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.52481 + 13.0334i 0.825956 + 1.43060i 0.901187 + 0.433431i \(0.142697\pi\)
−0.0752309 + 0.997166i \(0.523969\pi\)
\(84\) 0 0
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.08647 −0.963164 −0.481582 − 0.876401i \(-0.659938\pi\)
−0.481582 + 0.876401i \(0.659938\pi\)
\(90\) 0 0
\(91\) 3.36959 0.353228
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.17365 − 3.76487i 0.223012 − 0.386267i
\(96\) 0 0
\(97\) 0.949493 + 1.64457i 0.0964064 + 0.166981i 0.910195 − 0.414181i \(-0.135932\pi\)
−0.813788 + 0.581161i \(0.802598\pi\)
\(98\) 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.1009.2 6
3.2 odd 2 1008.2.r.h.337.2 6
4.3 odd 2 189.2.f.b.64.1 6
9.2 odd 6 1008.2.r.h.673.2 6
9.4 even 3 9072.2.a.bs.1.2 3
9.5 odd 6 9072.2.a.ca.1.2 3
9.7 even 3 inner 3024.2.r.k.2017.2 6
12.11 even 2 63.2.f.a.22.3 ✓ 6
28.3 even 6 1323.2.h.b.226.3 6
28.11 odd 6 1323.2.h.c.226.3 6
28.19 even 6 1323.2.g.e.361.1 6
28.23 odd 6 1323.2.g.d.361.1 6
28.27 even 2 1323.2.f.d.442.1 6
36.7 odd 6 189.2.f.b.127.1 6
36.11 even 6 63.2.f.a.43.3 yes 6
36.23 even 6 567.2.a.h.1.1 3
36.31 odd 6 567.2.a.c.1.3 3
84.11 even 6 441.2.h.d.373.1 6
84.23 even 6 441.2.g.c.67.3 6
84.47 odd 6 441.2.g.b.67.3 6
84.59 odd 6 441.2.h.e.373.1 6
84.83 odd 2 441.2.f.c.148.3 6
252.11 even 6 441.2.g.c.79.3 6
252.47 odd 6 441.2.h.e.214.1 6
252.79 odd 6 1323.2.h.c.802.3 6
252.83 odd 6 441.2.f.c.295.3 6
252.115 even 6 1323.2.g.e.667.1 6
252.139 even 6 3969.2.a.l.1.3 3
252.151 odd 6 1323.2.g.d.667.1 6
252.167 odd 6 3969.2.a.q.1.1 3
252.187 even 6 1323.2.h.b.802.3 6
252.191 even 6 441.2.h.d.214.1 6
252.223 even 6 1323.2.f.d.883.1 6
252.227 odd 6 441.2.g.b.79.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 12.11 even 2
63.2.f.a.43.3 yes 6 36.11 even 6
189.2.f.b.64.1 6 4.3 odd 2
189.2.f.b.127.1 6 36.7 odd 6
441.2.f.c.148.3 6 84.83 odd 2
441.2.f.c.295.3 6 252.83 odd 6
441.2.g.b.67.3 6 84.47 odd 6
441.2.g.b.79.3 6 252.227 odd 6
441.2.g.c.67.3 6 84.23 even 6
441.2.g.c.79.3 6 252.11 even 6
441.2.h.d.214.1 6 252.191 even 6
441.2.h.d.373.1 6 84.11 even 6
441.2.h.e.214.1 6 252.47 odd 6
441.2.h.e.373.1 6 84.59 odd 6
567.2.a.c.1.3 3 36.31 odd 6
567.2.a.h.1.1 3 36.23 even 6
1008.2.r.h.337.2 6 3.2 odd 2
1008.2.r.h.673.2 6 9.2 odd 6
1323.2.f.d.442.1 6 28.27 even 2
1323.2.f.d.883.1 6 252.223 even 6
1323.2.g.d.361.1 6 28.23 odd 6
1323.2.g.d.667.1 6 252.151 odd 6
1323.2.g.e.361.1 6 28.19 even 6
1323.2.g.e.667.1 6 252.115 even 6
1323.2.h.b.226.3 6 28.3 even 6
1323.2.h.b.802.3 6 252.187 even 6
1323.2.h.c.226.3 6 28.11 odd 6
1323.2.h.c.802.3 6 252.79 odd 6
3024.2.r.k.1009.2 6 1.1 even 1 trivial
3024.2.r.k.2017.2 6 9.7 even 3 inner
3969.2.a.l.1.3 3 252.139 even 6
3969.2.a.q.1.1 3 252.167 odd 6
9072.2.a.bs.1.2 3 9.4 even 3
9072.2.a.ca.1.2 3 9.5 odd 6