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Properties

Label 3024.2.cx.j.559.5
Level $3024$
Weight $2$
Character 3024.559
Analytic conductor $24.147$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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(559,3024)] 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("3024.559"); 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([3, 0, 4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cx (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,6,0,0,0,0] 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 559.5
Character \(\chi\) \(=\) 3024.559
Dual form 3024.2.cx.j.2575.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.675942 + 0.390255i) q^{5} +(1.89912 - 1.84210i) q^{7} +(-4.95157 - 2.85879i) q^{11} +(3.53283 - 2.03968i) q^{13} -2.73465i q^{17} -7.06447 q^{19} +(6.14054 - 3.54524i) q^{23} +(-2.19540 + 3.80255i) q^{25} +(-0.910813 + 1.57757i) q^{29} +(2.25773 + 3.91050i) q^{31} +(-0.564803 + 1.98630i) q^{35} -1.69072 q^{37} +(-2.48884 + 1.43693i) q^{41} +(-7.58143 - 4.37714i) q^{43} +(-6.32895 + 10.9621i) q^{47} +(0.213305 - 6.99675i) q^{49} -9.58809 q^{53} +4.46263 q^{55} +(1.86952 + 3.23810i) q^{59} +(7.56969 + 4.37036i) q^{61} +(-1.59199 + 2.75741i) q^{65} +(-11.5672 + 6.67835i) q^{67} -5.36455i q^{71} +4.91709i q^{73} +(-14.6698 + 3.69213i) q^{77} +(-8.32061 - 4.80391i) q^{79} +(6.81322 - 11.8008i) q^{83} +(1.06721 + 1.84846i) q^{85} +6.93549i q^{89} +(2.95196 - 10.3814i) q^{91} +(4.77517 - 2.75695i) q^{95} +(1.45710 + 0.841255i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{7} + 18 q^{23} + 24 q^{25} + 6 q^{29} - 12 q^{37} - 42 q^{43} + 12 q^{49} - 96 q^{53} - 42 q^{65} - 36 q^{67} + 18 q^{77} + 60 q^{79} - 6 q^{85} - 126 q^{95}+O(q^{100}) \) 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
\(5\) −0.675942 + 0.390255i −0.302290 + 0.174527i −0.643471 − 0.765470i \(-0.722506\pi\)
0.341181 + 0.939998i \(0.389173\pi\)
\(6\) 0 0
\(7\) 1.89912 − 1.84210i 0.717799 − 0.696250i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.95157 − 2.85879i −1.49296 − 0.861958i −0.492989 − 0.870036i \(-0.664096\pi\)
−0.999967 + 0.00807732i \(0.997429\pi\)
\(12\) 0 0
\(13\) 3.53283 − 2.03968i 0.979830 − 0.565705i 0.0776112 − 0.996984i \(-0.475271\pi\)
0.902219 + 0.431279i \(0.141937\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.73465i − 0.663249i −0.943411 − 0.331625i \(-0.892403\pi\)
0.943411 − 0.331625i \(-0.107597\pi\)
\(18\) 0 0
\(19\) −7.06447 −1.62070 −0.810350 − 0.585946i \(-0.800723\pi\)
−0.810350 + 0.585946i \(0.800723\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.14054 − 3.54524i 1.28039 − 0.739234i 0.303471 − 0.952841i \(-0.401854\pi\)
0.976920 + 0.213607i \(0.0685211\pi\)
\(24\) 0 0
\(25\) −2.19540 + 3.80255i −0.439080 + 0.760510i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.910813 + 1.57757i −0.169134 + 0.292948i −0.938116 − 0.346322i \(-0.887430\pi\)
0.768982 + 0.639271i \(0.220764\pi\)
\(30\) 0 0
\(31\) 2.25773 + 3.91050i 0.405500 + 0.702346i 0.994379 − 0.105875i \(-0.0337642\pi\)
−0.588880 + 0.808221i \(0.700431\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.564803 + 1.98630i −0.0954691 + 0.335745i
\(36\) 0 0
\(37\) −1.69072 −0.277952 −0.138976 − 0.990296i \(-0.544381\pi\)
−0.138976 + 0.990296i \(0.544381\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.48884 + 1.43693i −0.388692 + 0.224411i −0.681593 − 0.731731i \(-0.738713\pi\)
0.292901 + 0.956143i \(0.405379\pi\)
\(42\) 0 0
\(43\) −7.58143 − 4.37714i −1.15616 − 0.667508i −0.205778 − 0.978599i \(-0.565972\pi\)
−0.950380 + 0.311091i \(0.899306\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.32895 + 10.9621i −0.923172 + 1.59898i −0.128697 + 0.991684i \(0.541079\pi\)
−0.794475 + 0.607296i \(0.792254\pi\)
\(48\) 0 0
\(49\) 0.213305 − 6.99675i 0.0304721 − 0.999536i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.58809 −1.31702 −0.658512 − 0.752570i \(-0.728814\pi\)
−0.658512 + 0.752570i \(0.728814\pi\)
\(54\) 0 0
\(55\) 4.46263 0.601741
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.86952 + 3.23810i 0.243391 + 0.421565i 0.961678 − 0.274182i \(-0.0884070\pi\)
−0.718287 + 0.695747i \(0.755074\pi\)
\(60\) 0 0
\(61\) 7.56969 + 4.37036i 0.969200 + 0.559568i 0.898992 − 0.437965i \(-0.144301\pi\)
0.0702075 + 0.997532i \(0.477634\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.59199 + 2.75741i −0.197462 + 0.342014i
\(66\) 0 0
\(67\) −11.5672 + 6.67835i −1.41316 + 0.815890i −0.995685 − 0.0927963i \(-0.970419\pi\)
−0.417479 + 0.908687i \(0.637086\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.36455i − 0.636656i −0.947981 − 0.318328i \(-0.896879\pi\)
0.947981 − 0.318328i \(-0.103121\pi\)
\(72\) 0 0
\(73\) 4.91709i 0.575502i 0.957705 + 0.287751i \(0.0929075\pi\)
−0.957705 + 0.287751i \(0.907092\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.6698 + 3.69213i −1.67178 + 0.420757i
\(78\) 0 0
\(79\) −8.32061 − 4.80391i −0.936142 − 0.540482i −0.0473932 − 0.998876i \(-0.515091\pi\)
−0.888749 + 0.458394i \(0.848425\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.81322 − 11.8008i 0.747848 − 1.29531i −0.201005 − 0.979590i \(-0.564421\pi\)
0.948852 − 0.315720i \(-0.102246\pi\)
\(84\) 0 0
\(85\) 1.06721 + 1.84846i 0.115755 + 0.200494i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.93549i 0.735161i 0.929992 + 0.367580i \(0.119814\pi\)
−0.929992 + 0.367580i \(0.880186\pi\)
\(90\) 0 0
\(91\) 2.95196 − 10.3814i 0.309449 − 1.08827i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.77517 − 2.75695i 0.489922 − 0.282857i
\(96\) 0 0
\(97\) 1.45710 + 0.841255i 0.147946 + 0.0854165i 0.572146 − 0.820152i \(-0.306111\pi\)
−0.424200 + 0.905569i \(0.639445\pi\)
\(98\) 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.cx.j.559.5 24
3.2 odd 2 1008.2.cx.j.223.10 yes 24
4.3 odd 2 3024.2.cx.i.559.5 24
7.6 odd 2 inner 3024.2.cx.j.559.8 24
9.4 even 3 3024.2.cx.i.2575.8 24
9.5 odd 6 1008.2.cx.i.895.10 yes 24
12.11 even 2 1008.2.cx.i.223.3 ✓ 24
21.20 even 2 1008.2.cx.j.223.3 yes 24
28.27 even 2 3024.2.cx.i.559.8 24
36.23 even 6 1008.2.cx.j.895.3 yes 24
36.31 odd 6 inner 3024.2.cx.j.2575.8 24
63.13 odd 6 3024.2.cx.i.2575.5 24
63.41 even 6 1008.2.cx.i.895.3 yes 24
84.83 odd 2 1008.2.cx.i.223.10 yes 24
252.139 even 6 inner 3024.2.cx.j.2575.5 24
252.167 odd 6 1008.2.cx.j.895.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.3 ✓ 24 12.11 even 2
1008.2.cx.i.223.10 yes 24 84.83 odd 2
1008.2.cx.i.895.3 yes 24 63.41 even 6
1008.2.cx.i.895.10 yes 24 9.5 odd 6
1008.2.cx.j.223.3 yes 24 21.20 even 2
1008.2.cx.j.223.10 yes 24 3.2 odd 2
1008.2.cx.j.895.3 yes 24 36.23 even 6
1008.2.cx.j.895.10 yes 24 252.167 odd 6
3024.2.cx.i.559.5 24 4.3 odd 2
3024.2.cx.i.559.8 24 28.27 even 2
3024.2.cx.i.2575.5 24 63.13 odd 6
3024.2.cx.i.2575.8 24 9.4 even 3
3024.2.cx.j.559.5 24 1.1 even 1 trivial
3024.2.cx.j.559.8 24 7.6 odd 2 inner
3024.2.cx.j.2575.5 24 252.139 even 6 inner
3024.2.cx.j.2575.8 24 36.31 odd 6 inner