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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(473,888)] 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("888.473"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.br (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 569.22
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.22

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.567654 - 1.63639i) q^{3} +(0.0585783 - 0.218617i) q^{5} +(1.23175 + 2.13345i) q^{7} +(-2.35554 - 1.85780i) q^{9} +4.45126 q^{11} +(1.45800 + 0.390671i) q^{13} +(-0.324490 - 0.219956i) q^{15} +(-1.17157 + 0.313921i) q^{17} +(4.74490 + 1.27139i) q^{19} +(4.19035 - 0.804556i) q^{21} +(-3.28903 - 3.28903i) q^{23} +(4.28577 + 2.47439i) q^{25} +(-4.37722 + 2.79999i) q^{27} +(5.98185 - 5.98185i) q^{29} +(-7.24947 - 7.24947i) q^{31} +(2.52677 - 7.28399i) q^{33} +(0.538561 - 0.144307i) q^{35} +(5.17282 + 3.20030i) q^{37} +(1.46693 - 2.16410i) q^{39} +(5.48131 + 9.49391i) q^{41} +(-1.65145 + 1.65145i) q^{43} +(-0.544131 + 0.406134i) q^{45} -1.96589i q^{47} +(0.465605 - 0.806452i) q^{49} +(-0.151349 + 2.09534i) q^{51} +(-9.27137 - 5.35283i) q^{53} +(0.260747 - 0.973120i) q^{55} +(4.77395 - 7.04279i) q^{57} +(-6.54296 + 1.75318i) q^{59} +(3.26812 - 12.1968i) q^{61} +(1.06210 - 7.31376i) q^{63} +(0.170815 - 0.295860i) q^{65} +(-12.3822 + 7.14887i) q^{67} +(-7.24917 + 3.51510i) q^{69} +(3.52135 - 2.03305i) q^{71} -8.68620i q^{73} +(6.48189 - 5.60858i) q^{75} +(5.48281 + 9.49651i) q^{77} +(-2.07721 - 0.556587i) q^{79} +(2.09712 + 8.75226i) q^{81} +(1.03984 + 0.600350i) q^{83} +0.274514i q^{85} +(-6.39301 - 13.1843i) q^{87} +(3.33227 + 12.4362i) q^{89} +(0.962415 + 3.59178i) q^{91} +(-15.9781 + 7.74777i) q^{93} +(0.555896 - 0.962840i) q^{95} +(7.22624 - 7.22624i) q^{97} +(-10.4851 - 8.26956i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\)

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
\(3\) 0.567654 − 1.63639i 0.327735 − 0.944770i
\(4\) 0 0
\(5\) 0.0585783 − 0.218617i 0.0261970 − 0.0977685i −0.951590 − 0.307372i \(-0.900551\pi\)
0.977787 + 0.209603i \(0.0672172\pi\)
\(6\) 0 0
\(7\) 1.23175 + 2.13345i 0.465556 + 0.806367i 0.999226 − 0.0393258i \(-0.0125210\pi\)
−0.533670 + 0.845693i \(0.679188\pi\)
\(8\) 0 0
\(9\) −2.35554 − 1.85780i −0.785179 − 0.619268i
\(10\) 0 0
\(11\) 4.45126 1.34210 0.671052 − 0.741410i \(-0.265843\pi\)
0.671052 + 0.741410i \(0.265843\pi\)
\(12\) 0 0
\(13\) 1.45800 + 0.390671i 0.404378 + 0.108353i 0.455274 − 0.890351i \(-0.349541\pi\)
−0.0508965 + 0.998704i \(0.516208\pi\)
\(14\) 0 0
\(15\) −0.324490 − 0.219956i −0.0837830 − 0.0567923i
\(16\) 0 0
\(17\) −1.17157 + 0.313921i −0.284147 + 0.0761370i −0.398078 − 0.917352i \(-0.630323\pi\)
0.113931 + 0.993489i \(0.463656\pi\)
\(18\) 0 0
\(19\) 4.74490 + 1.27139i 1.08856 + 0.291677i 0.758096 − 0.652143i \(-0.226130\pi\)
0.330459 + 0.943820i \(0.392796\pi\)
\(20\) 0 0
\(21\) 4.19035 − 0.804556i 0.914410 − 0.175569i
\(22\) 0 0
\(23\) −3.28903 − 3.28903i −0.685810 − 0.685810i 0.275493 − 0.961303i \(-0.411159\pi\)
−0.961303 + 0.275493i \(0.911159\pi\)
\(24\) 0 0
\(25\) 4.28577 + 2.47439i 0.857153 + 0.494878i
\(26\) 0 0
\(27\) −4.37722 + 2.79999i −0.842397 + 0.538858i
\(28\) 0 0
\(29\) 5.98185 − 5.98185i 1.11080 − 1.11080i 0.117759 − 0.993042i \(-0.462429\pi\)
0.993042 − 0.117759i \(-0.0375712\pi\)
\(30\) 0 0
\(31\) −7.24947 − 7.24947i −1.30204 − 1.30204i −0.927012 − 0.375032i \(-0.877632\pi\)
−0.375032 − 0.927012i \(-0.622368\pi\)
\(32\) 0 0
\(33\) 2.52677 − 7.28399i 0.439855 − 1.26798i
\(34\) 0 0
\(35\) 0.538561 − 0.144307i 0.0910334 − 0.0243923i
\(36\) 0 0
\(37\) 5.17282 + 3.20030i 0.850407 + 0.526126i
\(38\) 0 0
\(39\) 1.46693 − 2.16410i 0.234897 − 0.346533i
\(40\) 0 0
\(41\) 5.48131 + 9.49391i 0.856037 + 1.48270i 0.875680 + 0.482892i \(0.160414\pi\)
−0.0196432 + 0.999807i \(0.506253\pi\)
\(42\) 0 0
\(43\) −1.65145 + 1.65145i −0.251843 + 0.251843i −0.821726 − 0.569883i \(-0.806989\pi\)
0.569883 + 0.821726i \(0.306989\pi\)
\(44\) 0 0
\(45\) −0.544131 + 0.406134i −0.0811143 + 0.0605429i
\(46\) 0 0
\(47\) − 1.96589i − 0.286755i −0.989668 − 0.143377i \(-0.954204\pi\)
0.989668 − 0.143377i \(-0.0457963\pi\)
\(48\) 0 0
\(49\) 0.465605 − 0.806452i 0.0665150 − 0.115207i
\(50\) 0 0
\(51\) −0.151349 + 2.09534i −0.0211930 + 0.293406i
\(52\) 0 0
\(53\) −9.27137 − 5.35283i −1.27352 − 0.735267i −0.297872 − 0.954606i \(-0.596277\pi\)
−0.975649 + 0.219339i \(0.929610\pi\)
\(54\) 0 0
\(55\) 0.260747 − 0.973120i 0.0351591 − 0.131215i
\(56\) 0 0
\(57\) 4.77395 − 7.04279i 0.632326 − 0.932841i
\(58\) 0 0
\(59\) −6.54296 + 1.75318i −0.851820 + 0.228245i −0.658210 − 0.752834i \(-0.728686\pi\)
−0.193610 + 0.981079i \(0.562020\pi\)
\(60\) 0 0
\(61\) 3.26812 − 12.1968i 0.418440 − 1.56164i −0.359403 − 0.933182i \(-0.617020\pi\)
0.777844 − 0.628458i \(-0.216314\pi\)
\(62\) 0 0
\(63\) 1.06210 − 7.31376i 0.133812 − 0.921447i
\(64\) 0 0
\(65\) 0.170815 − 0.295860i 0.0211870 − 0.0366969i
\(66\) 0 0
\(67\) −12.3822 + 7.14887i −1.51273 + 0.873374i −0.512838 + 0.858485i \(0.671406\pi\)
−0.999889 + 0.0148885i \(0.995261\pi\)
\(68\) 0 0
\(69\) −7.24917 + 3.51510i −0.872697 + 0.423169i
\(70\) 0 0
\(71\) 3.52135 − 2.03305i 0.417907 − 0.241279i −0.276274 − 0.961079i \(-0.589100\pi\)
0.694181 + 0.719800i \(0.255767\pi\)
\(72\) 0 0
\(73\) − 8.68620i − 1.01664i −0.861168 − 0.508321i \(-0.830266\pi\)
0.861168 − 0.508321i \(-0.169734\pi\)
\(74\) 0 0
\(75\) 6.48189 − 5.60858i 0.748464 − 0.647623i
\(76\) 0 0
\(77\) 5.48281 + 9.49651i 0.624825 + 1.08223i
\(78\) 0 0
\(79\) −2.07721 − 0.556587i −0.233705 − 0.0626210i 0.140066 − 0.990142i \(-0.455269\pi\)
−0.373770 + 0.927521i \(0.621935\pi\)
\(80\) 0 0
\(81\) 2.09712 + 8.75226i 0.233014 + 0.972473i
\(82\) 0 0
\(83\) 1.03984 + 0.600350i 0.114137 + 0.0658970i 0.555982 − 0.831195i \(-0.312342\pi\)
−0.441845 + 0.897092i \(0.645676\pi\)
\(84\) 0 0
\(85\) 0.274514i 0.0297752i
\(86\) 0 0
\(87\) −6.39301 − 13.1843i −0.685403 − 1.41350i
\(88\) 0 0
\(89\) 3.33227 + 12.4362i 0.353220 + 1.31824i 0.882709 + 0.469919i \(0.155717\pi\)
−0.529489 + 0.848317i \(0.677616\pi\)
\(90\) 0 0
\(91\) 0.962415 + 3.59178i 0.100889 + 0.376521i
\(92\) 0 0
\(93\) −15.9781 + 7.74777i −1.65686 + 0.803406i
\(94\) 0 0
\(95\) 0.555896 − 0.962840i 0.0570337 − 0.0987853i
\(96\) 0 0
\(97\) 7.22624 − 7.22624i 0.733713 − 0.733713i −0.237640 − 0.971353i \(-0.576374\pi\)
0.971353 + 0.237640i \(0.0763738\pi\)
\(98\) 0 0
\(99\) −10.4851 − 8.26956i −1.05379 − 0.831122i
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 888.2.br.a.569.22 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.34 yes 152
37.8 odd 12 inner 888.2.br.a.785.34 yes 152
111.8 even 12 inner 888.2.br.a.785.22 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.22 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.34 yes 152 3.2 odd 2 inner
888.2.br.a.785.22 yes 152 111.8 even 12 inner
888.2.br.a.785.34 yes 152 37.8 odd 12 inner