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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 569.34
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.34

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70098 + 0.326592i) q^{3} +(-0.0585783 + 0.218617i) q^{5} +(1.23175 + 2.13345i) q^{7} +(2.78668 + 1.11105i) q^{9} -4.45126 q^{11} +(1.45800 + 0.390671i) q^{13} +(-0.171039 + 0.352732i) q^{15} +(1.17157 - 0.313921i) q^{17} +(4.74490 + 1.27139i) q^{19} +(1.39841 + 4.03123i) q^{21} +(3.28903 + 3.28903i) q^{23} +(4.28577 + 2.47439i) q^{25} +(4.37722 + 2.79999i) q^{27} +(-5.98185 + 5.98185i) q^{29} +(-7.24947 - 7.24947i) q^{31} +(-7.57150 - 1.45374i) q^{33} +(-0.538561 + 0.144307i) q^{35} +(5.17282 + 3.20030i) q^{37} +(2.35245 + 1.14070i) q^{39} +(-5.48131 - 9.49391i) q^{41} +(-1.65145 + 1.65145i) q^{43} +(-0.406134 + 0.544131i) q^{45} +1.96589i q^{47} +(0.465605 - 0.806452i) q^{49} +(2.09534 - 0.151349i) q^{51} +(9.27137 + 5.35283i) q^{53} +(0.260747 - 0.973120i) q^{55} +(7.65576 + 3.71226i) q^{57} +(6.54296 - 1.75318i) q^{59} +(3.26812 - 12.1968i) q^{61} +(1.06210 + 7.31376i) q^{63} +(-0.170815 + 0.295860i) q^{65} +(-12.3822 + 7.14887i) q^{67} +(4.52041 + 6.66875i) q^{69} +(-3.52135 + 2.03305i) q^{71} -8.68620i q^{73} +(6.48189 + 5.60858i) q^{75} +(-5.48281 - 9.49651i) q^{77} +(-2.07721 - 0.556587i) q^{79} +(6.53112 + 6.19229i) q^{81} +(-1.03984 - 0.600350i) q^{83} +0.274514i q^{85} +(-12.1286 + 8.22139i) q^{87} +(-3.33227 - 12.4362i) q^{89} +(0.962415 + 3.59178i) q^{91} +(-9.96360 - 14.6988i) q^{93} +(-0.555896 + 0.962840i) q^{95} +(7.22624 - 7.22624i) q^{97} +(-12.4042 - 4.94558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.70098 + 0.326592i 0.982062 + 0.188558i
\(4\) 0 0
\(5\) −0.0585783 + 0.218617i −0.0261970 + 0.0977685i −0.977787 − 0.209603i \(-0.932783\pi\)
0.951590 + 0.307372i \(0.0994495\pi\)
\(6\) 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
\(9\) 2.78668 + 1.11105i 0.928892 + 0.370351i
\(10\) 0 0
\(11\) −4.45126 −1.34210 −0.671052 − 0.741410i \(-0.734157\pi\)
−0.671052 + 0.741410i \(0.734157\pi\)
\(12\) 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
\(15\) −0.171039 + 0.352732i −0.0441621 + 0.0910751i
\(16\) 0 0
\(17\) 1.17157 − 0.313921i 0.284147 − 0.0761370i −0.113931 − 0.993489i \(-0.536344\pi\)
0.398078 + 0.917352i \(0.369677\pi\)
\(18\) 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
\(21\) 1.39841 + 4.03123i 0.305158 + 0.879687i
\(22\) 0 0
\(23\) 3.28903 + 3.28903i 0.685810 + 0.685810i 0.961303 − 0.275493i \(-0.0888410\pi\)
−0.275493 + 0.961303i \(0.588841\pi\)
\(24\) 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 \(0.537571\pi\)
−0.993042 + 0.117759i \(0.962429\pi\)
\(30\) 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
\(33\) −7.57150 − 1.45374i −1.31803 − 0.253064i
\(34\) 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
\(39\) 2.35245 + 1.14070i 0.376693 + 0.182658i
\(40\) 0 0
\(41\) −5.48131 − 9.49391i −0.856037 − 1.48270i −0.875680 − 0.482892i \(-0.839586\pi\)
0.0196432 − 0.999807i \(-0.493747\pi\)
\(42\) 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
\(45\) −0.406134 + 0.544131i −0.0605429 + 0.0811143i
\(46\) 0 0
\(47\) 1.96589i 0.286755i 0.989668 + 0.143377i \(0.0457963\pi\)
−0.989668 + 0.143377i \(0.954204\pi\)
\(48\) 0 0
\(49\) 0.465605 − 0.806452i 0.0665150 − 0.115207i
\(50\) 0 0
\(51\) 2.09534 − 0.151349i 0.293406 − 0.0211930i
\(52\) 0 0
\(53\) 9.27137 + 5.35283i 1.27352 + 0.735267i 0.975649 − 0.219339i \(-0.0703899\pi\)
0.297872 + 0.954606i \(0.403723\pi\)
\(54\) 0 0
\(55\) 0.260747 − 0.973120i 0.0351591 − 0.131215i
\(56\) 0 0
\(57\) 7.65576 + 3.71226i 1.01403 + 0.491701i
\(58\) 0 0
\(59\) 6.54296 − 1.75318i 0.851820 − 0.228245i 0.193610 − 0.981079i \(-0.437980\pi\)
0.658210 + 0.752834i \(0.271314\pi\)
\(60\) 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
\(69\) 4.52041 + 6.66875i 0.544193 + 0.802823i
\(70\) 0 0
\(71\) −3.52135 + 2.03305i −0.417907 + 0.241279i −0.694181 − 0.719800i \(-0.744233\pi\)
0.276274 + 0.961079i \(0.410900\pi\)
\(72\) 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
\(81\) 6.53112 + 6.19229i 0.725680 + 0.688032i
\(82\) 0 0
\(83\) −1.03984 − 0.600350i −0.114137 − 0.0658970i 0.441845 − 0.897092i \(-0.354324\pi\)
−0.555982 + 0.831195i \(0.687658\pi\)
\(84\) 0 0
\(85\) 0.274514i 0.0297752i
\(86\) 0 0
\(87\) −12.1286 + 8.22139i −1.30033 + 0.881426i
\(88\) 0 0
\(89\) −3.33227 − 12.4362i −0.353220 − 1.31824i −0.882709 − 0.469919i \(-0.844283\pi\)
0.529489 − 0.848317i \(-0.322384\pi\)
\(90\) 0 0
\(91\) 0.962415 + 3.59178i 0.100889 + 0.376521i
\(92\) 0 0
\(93\) −9.96360 − 14.6988i −1.03318 − 1.52420i
\(94\) 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
\(99\) −12.4042 − 4.94558i −1.24667 − 0.497050i
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.569.34 yes 152
3.2 odd 2 inner 888.2.br.a.569.22 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.22 yes 152
111.8 even 12 inner 888.2.br.a.785.34 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.22 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.34 yes 152 1.1 even 1 trivial
888.2.br.a.785.22 yes 152 37.8 odd 12 inner
888.2.br.a.785.34 yes 152 111.8 even 12 inner