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Properties

Label 1008.2.cx.i.895.10
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 895.10
Character \(\chi\) \(=\) 1008.895
Dual form 1008.2.cx.i.223.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.44146 + 0.960311i) q^{3} +(-0.675942 - 0.390255i) q^{5} +(0.645750 + 2.56574i) q^{7} +(1.15560 + 2.76850i) q^{9} +(-4.95157 + 2.85879i) q^{11} +(-3.53283 - 2.03968i) q^{13} +(-0.599575 - 1.21165i) q^{15} +2.73465i q^{17} -7.06447 q^{19} +(-1.53309 + 4.31852i) q^{21} +(6.14054 + 3.54524i) q^{23} +(-2.19540 - 3.80255i) q^{25} +(-0.992864 + 5.10041i) q^{27} +(0.910813 + 1.57757i) q^{29} +(2.25773 - 3.91050i) q^{31} +(-9.88282 - 0.634222i) q^{33} +(0.564803 - 1.98630i) q^{35} -1.69072 q^{37} +(-3.13370 - 6.33273i) q^{39} +(-2.48884 - 1.43693i) q^{41} +(7.58143 - 4.37714i) q^{43} +(0.299299 - 2.32232i) q^{45} +(6.32895 + 10.9621i) q^{47} +(-6.16602 + 3.31365i) q^{49} +(-2.62611 + 3.94188i) q^{51} +9.58809 q^{53} +4.46263 q^{55} +(-10.1831 - 6.78409i) q^{57} +(-1.86952 + 3.23810i) q^{59} +(-7.56969 + 4.37036i) q^{61} +(-6.35701 + 4.75273i) q^{63} +(1.59199 + 2.75741i) q^{65} +(11.5672 + 6.67835i) q^{67} +(5.44679 + 11.0071i) q^{69} +5.36455i q^{71} +4.91709i q^{73} +(0.487049 - 7.58948i) q^{75} +(-10.5324 - 10.8584i) q^{77} +(8.32061 - 4.80391i) q^{79} +(-6.32916 + 6.39858i) q^{81} +(-6.81322 - 11.8008i) q^{83} +(1.06721 - 1.84846i) q^{85} +(-0.202064 + 3.14867i) q^{87} -6.93549i q^{89} +(2.95196 - 10.3814i) q^{91} +(7.00971 - 3.46870i) q^{93} +(4.77517 + 2.75695i) q^{95} +(-1.45710 + 0.841255i) q^{97} +(-13.6366 - 10.4048i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{7} + 20 q^{9} + 24 q^{15} + 10 q^{21} + 18 q^{23} + 24 q^{25} - 6 q^{29} - 12 q^{37} - 12 q^{39} + 42 q^{43} + 12 q^{49} - 42 q^{51} + 96 q^{53} - 22 q^{57} + 18 q^{63} + 42 q^{65} + 36 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) 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 0.341181 − 0.939998i \(-0.389173\pi\)
−0.643471 + 0.765470i \(0.722506\pi\)
\(6\) 0 0
\(7\) 0.645750 + 2.56574i 0.244070 + 0.969758i
\(8\) 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 −0.0776112 − 0.996984i \(-0.524729\pi\)
−0.902219 + 0.431279i \(0.858063\pi\)
\(14\) 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 \(-0.800723\pi\)
−0.810350 + 0.585946i \(0.800723\pi\)
\(20\) 0 0
\(21\) −1.53309 + 4.31852i −0.334547 + 0.942379i
\(22\) 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 −0.588880 − 0.808221i \(-0.700431\pi\)
0.994379 + 0.105875i \(0.0337642\pi\)
\(32\) 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 \(-0.405379\pi\)
−0.681593 + 0.731731i \(0.738713\pi\)
\(42\) 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 \(0.207746\pi\)
0.128697 + 0.991684i \(0.458921\pi\)
\(48\) 0 0
\(49\) −6.16602 + 3.31365i −0.880859 + 0.473378i
\(50\) 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 \(-0.911593\pi\)
0.718287 + 0.695747i \(0.244926\pi\)
\(60\) 0 0
\(61\) −7.56969 + 4.37036i −0.969200 + 0.559568i −0.898992 − 0.437965i \(-0.855699\pi\)
−0.0702075 + 0.997532i \(0.522366\pi\)
\(62\) 0 0
\(63\) −6.35701 + 4.75273i −0.800908 + 0.598788i
\(64\) 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 \(0.0929075\pi\)
−0.957705 + 0.287751i \(0.907092\pi\)
\(74\) 0 0
\(75\) 0.487049 − 7.58948i 0.0562396 − 0.876358i
\(76\) 0 0
\(77\) −10.5324 − 10.8584i −1.20028 − 1.23743i
\(78\) 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 \(-0.897754\pi\)
0.201005 − 0.979590i \(-0.435579\pi\)
\(84\) 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 \(-0.693889\pi\)
0.424200 + 0.905569i \(0.360555\pi\)
\(98\) 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.895.10 yes 24
3.2 odd 2 3024.2.cx.i.2575.8 24
4.3 odd 2 1008.2.cx.j.895.3 yes 24
7.6 odd 2 inner 1008.2.cx.i.895.3 yes 24
9.2 odd 6 3024.2.cx.j.559.5 24
9.7 even 3 1008.2.cx.j.223.10 yes 24
12.11 even 2 3024.2.cx.j.2575.8 24
21.20 even 2 3024.2.cx.i.2575.5 24
28.27 even 2 1008.2.cx.j.895.10 yes 24
36.7 odd 6 inner 1008.2.cx.i.223.3 ✓ 24
36.11 even 6 3024.2.cx.i.559.5 24
63.20 even 6 3024.2.cx.j.559.8 24
63.34 odd 6 1008.2.cx.j.223.3 yes 24
84.83 odd 2 3024.2.cx.j.2575.5 24
252.83 odd 6 3024.2.cx.i.559.8 24
252.223 even 6 inner 1008.2.cx.i.223.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.3 ✓ 24 36.7 odd 6 inner
1008.2.cx.i.223.10 yes 24 252.223 even 6 inner
1008.2.cx.i.895.3 yes 24 7.6 odd 2 inner
1008.2.cx.i.895.10 yes 24 1.1 even 1 trivial
1008.2.cx.j.223.3 yes 24 63.34 odd 6
1008.2.cx.j.223.10 yes 24 9.7 even 3
1008.2.cx.j.895.3 yes 24 4.3 odd 2
1008.2.cx.j.895.10 yes 24 28.27 even 2
3024.2.cx.i.559.5 24 36.11 even 6
3024.2.cx.i.559.8 24 252.83 odd 6
3024.2.cx.i.2575.5 24 21.20 even 2
3024.2.cx.i.2575.8 24 3.2 odd 2
3024.2.cx.j.559.5 24 9.2 odd 6
3024.2.cx.j.559.8 24 63.20 even 6
3024.2.cx.j.2575.5 24 84.83 odd 2
3024.2.cx.j.2575.8 24 12.11 even 2