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Properties

Label 1008.2.cx.i.895.3
Level $1008$
Weight $2$
Character 1008.895
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 895.3
Character \(\chi\) \(=\) 1008.895
Dual form 1008.2.cx.i.223.3

$q$-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 \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

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

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 0.643471 − 0.765470i \(-0.277494\pi\)
−0.341181 + 0.939998i \(0.610827\pi\)
\(6\) 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 −0.999967 − 0.00807732i \(-0.997429\pi\)
−0.492989 + 0.870036i \(0.664096\pi\)
\(12\) 0 0
\(13\) 3.53283 + 2.03968i 0.979830 + 0.565705i 0.902219 − 0.431279i \(-0.141937\pi\)
0.0776112 + 0.996984i \(0.475271\pi\)
\(14\) 0 0
\(15\) −0.599575 − 1.21165i −0.154810 − 0.312847i
\(16\) 0 0
\(17\) − 2.73465i − 0.663249i −0.943411 − 0.331625i \(-0.892403\pi\)
0.943411 − 0.331625i \(-0.107597\pi\)
\(18\) 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 0.976920 − 0.213607i \(-0.0685211\pi\)
0.303471 + 0.952841i \(0.401854\pi\)
\(24\) 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 0.938116 − 0.346322i \(-0.112570\pi\)
−0.768982 + 0.639271i \(0.779236\pi\)
\(30\) 0 0
\(31\) −2.25773 + 3.91050i −0.405500 + 0.702346i −0.994379 − 0.105875i \(-0.966236\pi\)
0.588880 + 0.808221i \(0.299569\pi\)
\(32\) 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 0.681593 − 0.731731i \(-0.261287\pi\)
−0.292901 + 0.956143i \(0.594621\pi\)
\(42\) 0 0
\(43\) 7.58143 − 4.37714i 1.15616 − 0.667508i 0.205778 − 0.978599i \(-0.434028\pi\)
0.950380 + 0.311091i \(0.100694\pi\)
\(44\) 0 0
\(45\) −0.299299 + 2.32232i −0.0446169 + 0.346192i
\(46\) 0 0
\(47\) −6.32895 − 10.9621i −0.923172 − 1.59898i −0.794475 − 0.607296i \(-0.792254\pi\)
−0.128697 − 0.991684i \(-0.541079\pi\)
\(48\) 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 −0.718287 − 0.695747i \(-0.755074\pi\)
0.961678 + 0.274182i \(0.0884070\pi\)
\(60\) 0 0
\(61\) 7.56969 − 4.37036i 0.969200 − 0.559568i 0.0702075 − 0.997532i \(-0.477634\pi\)
0.898992 + 0.437965i \(0.144301\pi\)
\(62\) 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 0.995685 − 0.0927963i \(-0.0295805\pi\)
0.417479 + 0.908687i \(0.362914\pi\)
\(68\) 0 0
\(69\) −5.44679 − 11.0071i −0.655717 − 1.32510i
\(70\) 0 0
\(71\) 5.36455i 0.636656i 0.947981 + 0.318328i \(0.103121\pi\)
−0.947981 + 0.318328i \(0.896879\pi\)
\(72\) 0 0
\(73\) − 4.91709i − 0.575502i −0.957705 − 0.287751i \(-0.907092\pi\)
0.957705 − 0.287751i \(-0.0929075\pi\)
\(74\) 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 0.0473932 − 0.998876i \(-0.484909\pi\)
0.888749 + 0.458394i \(0.151575\pi\)
\(80\) 0 0
\(81\) −6.32916 + 6.39858i −0.703240 + 0.710953i
\(82\) 0 0
\(83\) 6.81322 + 11.8008i 0.747848 + 1.29531i 0.948852 + 0.315720i \(0.102246\pi\)
−0.201005 + 0.979590i \(0.564421\pi\)
\(84\) 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 \(0.119814\pi\)
−0.929992 + 0.367580i \(0.880186\pi\)
\(90\) 0 0
\(91\) −2.95196 − 10.3814i −0.309449 − 1.08827i
\(92\) 0 0
\(93\) 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 −0.424200 − 0.905569i \(-0.639445\pi\)
0.572146 + 0.820152i \(0.306111\pi\)
\(98\) 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.895.3 yes 24
3.2 odd 2 3024.2.cx.i.2575.5 24
4.3 odd 2 1008.2.cx.j.895.10 yes 24
7.6 odd 2 inner 1008.2.cx.i.895.10 yes 24
9.2 odd 6 3024.2.cx.j.559.8 24
9.7 even 3 1008.2.cx.j.223.3 yes 24
12.11 even 2 3024.2.cx.j.2575.5 24
21.20 even 2 3024.2.cx.i.2575.8 24
28.27 even 2 1008.2.cx.j.895.3 yes 24
36.7 odd 6 inner 1008.2.cx.i.223.10 yes 24
36.11 even 6 3024.2.cx.i.559.8 24
63.20 even 6 3024.2.cx.j.559.5 24
63.34 odd 6 1008.2.cx.j.223.10 yes 24
84.83 odd 2 3024.2.cx.j.2575.8 24
252.83 odd 6 3024.2.cx.i.559.5 24
252.223 even 6 inner 1008.2.cx.i.223.3 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.3 ✓ 24 252.223 even 6 inner
1008.2.cx.i.223.10 yes 24 36.7 odd 6 inner
1008.2.cx.i.895.3 yes 24 1.1 even 1 trivial
1008.2.cx.i.895.10 yes 24 7.6 odd 2 inner
1008.2.cx.j.223.3 yes 24 9.7 even 3
1008.2.cx.j.223.10 yes 24 63.34 odd 6
1008.2.cx.j.895.3 yes 24 28.27 even 2
1008.2.cx.j.895.10 yes 24 4.3 odd 2
3024.2.cx.i.559.5 24 252.83 odd 6
3024.2.cx.i.559.8 24 36.11 even 6
3024.2.cx.i.2575.5 24 3.2 odd 2
3024.2.cx.i.2575.8 24 21.20 even 2
3024.2.cx.j.559.5 24 63.20 even 6
3024.2.cx.j.559.8 24 9.2 odd 6
3024.2.cx.j.2575.5 24 12.11 even 2
3024.2.cx.j.2575.8 24 84.83 odd 2