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Properties

Label 1008.2.cx.i.223.3
Level $1008$
Weight $2$
Character 1008.223
Analytic conductor $8.049$
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] = [1008,2,Mod(223,1008)] 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("1008.223"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cx (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,-6,0,20,0,0] 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 223.3
Character \(\chi\) \(=\) 1008.223
Dual form 1008.2.cx.i.895.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.44146 + 0.960311i) q^{3} +(0.675942 - 0.390255i) q^{5} +(-1.89912 + 1.84210i) q^{7} +(1.15560 - 2.76850i) q^{9} +(-4.95157 - 2.85879i) q^{11} +(3.53283 - 2.03968i) q^{13} +(-0.599575 + 1.21165i) q^{15} +2.73465i q^{17} +7.06447 q^{19} +(0.968507 - 4.47906i) q^{21} +(6.14054 - 3.54524i) q^{23} +(-2.19540 + 3.80255i) q^{25} +(0.992864 + 5.10041i) q^{27} +(0.910813 - 1.57757i) q^{29} +(-2.25773 - 3.91050i) q^{31} +(9.88282 - 0.634222i) q^{33} +(-0.564803 + 1.98630i) q^{35} -1.69072 q^{37} +(-3.13370 + 6.33273i) q^{39} +(2.48884 - 1.43693i) q^{41} +(7.58143 + 4.37714i) q^{43} +(-0.299299 - 2.32232i) q^{45} +(-6.32895 + 10.9621i) q^{47} +(0.213305 - 6.99675i) q^{49} +(-2.62611 - 3.94188i) q^{51} +9.58809 q^{53} -4.46263 q^{55} +(-10.1831 + 6.78409i) q^{57} +(1.86952 + 3.23810i) q^{59} +(7.56969 + 4.37036i) q^{61} +(2.90523 + 7.38645i) q^{63} +(1.59199 - 2.75741i) q^{65} +(11.5672 - 6.67835i) q^{67} +(-5.44679 + 11.0071i) q^{69} -5.36455i q^{71} +4.91709i q^{73} +(-0.487049 - 7.58948i) q^{75} +(14.6698 - 3.69213i) q^{77} +(8.32061 + 4.80391i) q^{79} +(-6.32916 - 6.39858i) q^{81} +(6.81322 - 11.8008i) q^{83} +(1.06721 + 1.84846i) q^{85} +(0.202064 + 3.14867i) q^{87} -6.93549i q^{89} +(-2.95196 + 10.3814i) q^{91} +(7.00971 + 3.46870i) q^{93} +(4.77517 - 2.75695i) q^{95} +(1.45710 + 0.841255i) q^{97} +(-13.6366 + 10.4048i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{7} + 20 q^{9} + 24 q^{15} + 10 q^{21} + 18 q^{23} + 24 q^{25} - 6 q^{29} - 12 q^{37} - 12 q^{39} + 42 q^{43} + 12 q^{49} - 42 q^{51} + 96 q^{53} - 22 q^{57} + 18 q^{63} + 42 q^{65} + 36 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −1.44146 + 0.960311i −0.832226 + 0.554436i
\(4\) 0 0
\(5\) 0.675942 − 0.390255i 0.302290 − 0.174527i −0.341181 − 0.939998i \(-0.610827\pi\)
0.643471 + 0.765470i \(0.277494\pi\)
\(6\) 0 0
\(7\) −1.89912 + 1.84210i −0.717799 + 0.696250i
\(8\) 0 0
\(9\) 1.15560 − 2.76850i 0.385201 − 0.922833i
\(10\) 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
\(15\) −0.599575 + 1.21165i −0.154810 + 0.312847i
\(16\) 0 0
\(17\) 2.73465i 0.663249i 0.943411 + 0.331625i \(0.107597\pi\)
−0.943411 + 0.331625i \(0.892403\pi\)
\(18\) 0 0
\(19\) 7.06447 1.62070 0.810350 − 0.585946i \(-0.199277\pi\)
0.810350 + 0.585946i \(0.199277\pi\)
\(20\) 0 0
\(21\) 0.968507 − 4.47906i 0.211346 − 0.977411i
\(22\) 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
\(27\) 0.992864 + 5.10041i 0.191077 + 0.981575i
\(28\) 0 0
\(29\) 0.910813 − 1.57757i 0.169134 − 0.292948i −0.768982 − 0.639271i \(-0.779236\pi\)
0.938116 + 0.346322i \(0.112570\pi\)
\(30\) 0 0
\(31\) −2.25773 − 3.91050i −0.405500 − 0.702346i 0.588880 − 0.808221i \(-0.299569\pi\)
−0.994379 + 0.105875i \(0.966236\pi\)
\(32\) 0 0
\(33\) 9.88282 − 0.634222i 1.72038 − 0.110404i
\(34\) 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
\(39\) −3.13370 + 6.33273i −0.501793 + 1.01405i
\(40\) 0 0
\(41\) 2.48884 − 1.43693i 0.388692 − 0.224411i −0.292901 − 0.956143i \(-0.594621\pi\)
0.681593 + 0.731731i \(0.261287\pi\)
\(42\) 0 0
\(43\) 7.58143 + 4.37714i 1.15616 + 0.667508i 0.950380 − 0.311091i \(-0.100694\pi\)
0.205778 + 0.978599i \(0.434028\pi\)
\(44\) 0 0
\(45\) −0.299299 − 2.32232i −0.0446169 − 0.346192i
\(46\) 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
\(51\) −2.62611 − 3.94188i −0.367729 − 0.551973i
\(52\) 0 0
\(53\) 9.58809 1.31702 0.658512 − 0.752570i \(-0.271186\pi\)
0.658512 + 0.752570i \(0.271186\pi\)
\(54\) 0 0
\(55\) −4.46263 −0.601741
\(56\) 0 0
\(57\) −10.1831 + 6.78409i −1.34879 + 0.898575i
\(58\) 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
\(63\) 2.90523 + 7.38645i 0.366025 + 0.930605i
\(64\) 0 0
\(65\) 1.59199 − 2.75741i 0.197462 − 0.342014i
\(66\) 0 0
\(67\) 11.5672 − 6.67835i 1.41316 − 0.815890i 0.417479 − 0.908687i \(-0.362914\pi\)
0.995685 + 0.0927963i \(0.0295805\pi\)
\(68\) 0 0
\(69\) −5.44679 + 11.0071i −0.655717 + 1.32510i
\(70\) 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
\(75\) −0.487049 − 7.58948i −0.0562396 − 0.876358i
\(76\) 0 0
\(77\) 14.6698 − 3.69213i 1.67178 − 0.420757i
\(78\) 0 0
\(79\) 8.32061 + 4.80391i 0.936142 + 0.540482i 0.888749 − 0.458394i \(-0.151575\pi\)
0.0473932 + 0.998876i \(0.484909\pi\)
\(80\) 0 0
\(81\) −6.32916 − 6.39858i −0.703240 − 0.710953i
\(82\) 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
\(87\) 0.202064 + 3.14867i 0.0216635 + 0.337573i
\(88\) 0 0
\(89\) − 6.93549i − 0.735161i −0.929992 − 0.367580i \(-0.880186\pi\)
0.929992 − 0.367580i \(-0.119814\pi\)
\(90\) 0 0
\(91\) −2.95196 + 10.3814i −0.309449 + 1.08827i
\(92\) 0 0
\(93\) 7.00971 + 3.46870i 0.726873 + 0.359687i
\(94\) 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
\(99\) −13.6366 + 10.4048i −1.37053 + 1.04572i
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 1008.2.cx.i.223.3 ✓ 24
3.2 odd 2 3024.2.cx.i.559.5 24
4.3 odd 2 1008.2.cx.j.223.10 yes 24
7.6 odd 2 inner 1008.2.cx.i.223.10 yes 24
9.4 even 3 1008.2.cx.j.895.3 yes 24
9.5 odd 6 3024.2.cx.j.2575.8 24
12.11 even 2 3024.2.cx.j.559.5 24
21.20 even 2 3024.2.cx.i.559.8 24
28.27 even 2 1008.2.cx.j.223.3 yes 24
36.23 even 6 3024.2.cx.i.2575.8 24
36.31 odd 6 inner 1008.2.cx.i.895.10 yes 24
63.13 odd 6 1008.2.cx.j.895.10 yes 24
63.41 even 6 3024.2.cx.j.2575.5 24
84.83 odd 2 3024.2.cx.j.559.8 24
252.139 even 6 inner 1008.2.cx.i.895.3 yes 24
252.167 odd 6 3024.2.cx.i.2575.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.3 ✓ 24 1.1 even 1 trivial
1008.2.cx.i.223.10 yes 24 7.6 odd 2 inner
1008.2.cx.i.895.3 yes 24 252.139 even 6 inner
1008.2.cx.i.895.10 yes 24 36.31 odd 6 inner
1008.2.cx.j.223.3 yes 24 28.27 even 2
1008.2.cx.j.223.10 yes 24 4.3 odd 2
1008.2.cx.j.895.3 yes 24 9.4 even 3
1008.2.cx.j.895.10 yes 24 63.13 odd 6
3024.2.cx.i.559.5 24 3.2 odd 2
3024.2.cx.i.559.8 24 21.20 even 2
3024.2.cx.i.2575.5 24 252.167 odd 6
3024.2.cx.i.2575.8 24 36.23 even 6
3024.2.cx.j.559.5 24 12.11 even 2
3024.2.cx.j.559.8 24 84.83 odd 2
3024.2.cx.j.2575.5 24 63.41 even 6
3024.2.cx.j.2575.8 24 9.5 odd 6