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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 569.35
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.35

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70701 + 0.293459i) q^{3} +(1.05104 - 3.92255i) q^{5} +(-1.08769 - 1.88394i) q^{7} +(2.82776 + 1.00187i) q^{9} +0.348741 q^{11} +(-4.35295 - 1.16637i) q^{13} +(2.94524 - 6.38739i) q^{15} +(-0.748686 + 0.200610i) q^{17} +(1.26856 + 0.339910i) q^{19} +(-1.30385 - 3.53510i) q^{21} +(3.15159 + 3.15159i) q^{23} +(-9.95154 - 5.74553i) q^{25} +(4.53301 + 2.54004i) q^{27} +(5.26079 - 5.26079i) q^{29} +(-1.77846 - 1.77846i) q^{31} +(0.595305 + 0.102341i) q^{33} +(-8.53306 + 2.28643i) q^{35} +(-5.93010 + 1.35423i) q^{37} +(-7.08825 - 3.26841i) q^{39} +(4.43295 + 7.67810i) q^{41} +(-0.915804 + 0.915804i) q^{43} +(6.90199 - 10.0390i) q^{45} +1.82951i q^{47} +(1.13384 - 1.96387i) q^{49} +(-1.33689 + 0.122735i) q^{51} +(-7.49623 - 4.32795i) q^{53} +(0.366542 - 1.36795i) q^{55} +(2.06570 + 0.952501i) q^{57} +(2.71163 - 0.726580i) q^{59} +(0.345332 - 1.28880i) q^{61} +(-1.18827 - 6.41708i) q^{63} +(-9.15027 + 15.8487i) q^{65} +(3.97901 - 2.29728i) q^{67} +(4.45493 + 6.30465i) q^{69} +(13.0006 - 7.50588i) q^{71} +12.4803i q^{73} +(-15.3013 - 12.7280i) q^{75} +(-0.379324 - 0.657008i) q^{77} +(8.89252 + 2.38274i) q^{79} +(6.99250 + 5.66612i) q^{81} +(14.3609 + 8.29126i) q^{83} +3.14761i q^{85} +(10.5240 - 7.43639i) q^{87} +(-1.26215 - 4.71041i) q^{89} +(2.53731 + 9.46936i) q^{91} +(-2.51394 - 3.55775i) q^{93} +(2.66663 - 4.61874i) q^{95} +(11.9933 - 11.9933i) q^{97} +(0.986158 + 0.349395i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.70701 + 0.293459i 0.985543 + 0.169428i
\(4\) 0 0
\(5\) 1.05104 − 3.92255i 0.470041 − 1.75422i −0.169571 − 0.985518i \(-0.554238\pi\)
0.639612 − 0.768698i \(-0.279095\pi\)
\(6\) 0 0
\(7\) −1.08769 − 1.88394i −0.411110 − 0.712063i 0.583902 − 0.811824i \(-0.301525\pi\)
−0.995011 + 0.0997613i \(0.968192\pi\)
\(8\) 0 0
\(9\) 2.82776 + 1.00187i 0.942588 + 0.333958i
\(10\) 0 0
\(11\) 0.348741 0.105149 0.0525747 − 0.998617i \(-0.483257\pi\)
0.0525747 + 0.998617i \(0.483257\pi\)
\(12\) 0 0
\(13\) −4.35295 − 1.16637i −1.20729 − 0.323493i −0.401592 − 0.915818i \(-0.631543\pi\)
−0.805699 + 0.592326i \(0.798210\pi\)
\(14\) 0 0
\(15\) 2.94524 − 6.38739i 0.760459 − 1.64922i
\(16\) 0 0
\(17\) −0.748686 + 0.200610i −0.181583 + 0.0486550i −0.348465 − 0.937322i \(-0.613297\pi\)
0.166882 + 0.985977i \(0.446630\pi\)
\(18\) 0 0
\(19\) 1.26856 + 0.339910i 0.291028 + 0.0779808i 0.401379 − 0.915912i \(-0.368531\pi\)
−0.110351 + 0.993893i \(0.535198\pi\)
\(20\) 0 0
\(21\) −1.30385 − 3.53510i −0.284523 − 0.771422i
\(22\) 0 0
\(23\) 3.15159 + 3.15159i 0.657151 + 0.657151i 0.954705 − 0.297554i \(-0.0961708\pi\)
−0.297554 + 0.954705i \(0.596171\pi\)
\(24\) 0 0
\(25\) −9.95154 − 5.74553i −1.99031 − 1.14911i
\(26\) 0 0
\(27\) 4.53301 + 2.54004i 0.872379 + 0.488831i
\(28\) 0 0
\(29\) 5.26079 − 5.26079i 0.976904 − 0.976904i −0.0228355 − 0.999739i \(-0.507269\pi\)
0.999739 + 0.0228355i \(0.00726939\pi\)
\(30\) 0 0
\(31\) −1.77846 − 1.77846i −0.319421 − 0.319421i 0.529124 − 0.848545i \(-0.322521\pi\)
−0.848545 + 0.529124i \(0.822521\pi\)
\(32\) 0 0
\(33\) 0.595305 + 0.102341i 0.103629 + 0.0178153i
\(34\) 0 0
\(35\) −8.53306 + 2.28643i −1.44235 + 0.386477i
\(36\) 0 0
\(37\) −5.93010 + 1.35423i −0.974902 + 0.222635i
\(38\) 0 0
\(39\) −7.08825 − 3.26841i −1.13503 − 0.523365i
\(40\) 0 0
\(41\) 4.43295 + 7.67810i 0.692311 + 1.19912i 0.971079 + 0.238760i \(0.0767408\pi\)
−0.278767 + 0.960359i \(0.589926\pi\)
\(42\) 0 0
\(43\) −0.915804 + 0.915804i −0.139659 + 0.139659i −0.773480 − 0.633821i \(-0.781486\pi\)
0.633821 + 0.773480i \(0.281486\pi\)
\(44\) 0 0
\(45\) 6.90199 − 10.0390i 1.02889 − 1.49653i
\(46\) 0 0
\(47\) 1.82951i 0.266861i 0.991058 + 0.133431i \(0.0425993\pi\)
−0.991058 + 0.133431i \(0.957401\pi\)
\(48\) 0 0
\(49\) 1.13384 − 1.96387i 0.161977 − 0.280553i
\(50\) 0 0
\(51\) −1.33689 + 0.122735i −0.187201 + 0.0171863i
\(52\) 0 0
\(53\) −7.49623 − 4.32795i −1.02969 − 0.594490i −0.112792 − 0.993619i \(-0.535979\pi\)
−0.916895 + 0.399129i \(0.869313\pi\)
\(54\) 0 0
\(55\) 0.366542 − 1.36795i 0.0494245 − 0.184455i
\(56\) 0 0
\(57\) 2.06570 + 0.952501i 0.273609 + 0.126162i
\(58\) 0 0
\(59\) 2.71163 − 0.726580i 0.353025 − 0.0945926i −0.0779486 − 0.996957i \(-0.524837\pi\)
0.430973 + 0.902365i \(0.358170\pi\)
\(60\) 0 0
\(61\) 0.345332 − 1.28880i 0.0442153 − 0.165014i −0.940288 − 0.340380i \(-0.889444\pi\)
0.984503 + 0.175366i \(0.0561110\pi\)
\(62\) 0 0
\(63\) −1.18827 − 6.41708i −0.149708 − 0.808476i
\(64\) 0 0
\(65\) −9.15027 + 15.8487i −1.13495 + 1.96579i
\(66\) 0 0
\(67\) 3.97901 − 2.29728i 0.486113 − 0.280658i −0.236847 − 0.971547i \(-0.576114\pi\)
0.722961 + 0.690889i \(0.242781\pi\)
\(68\) 0 0
\(69\) 4.45493 + 6.30465i 0.536310 + 0.758990i
\(70\) 0 0
\(71\) 13.0006 − 7.50588i 1.54288 − 0.890784i 0.544228 − 0.838937i \(-0.316823\pi\)
0.998655 − 0.0518469i \(-0.0165108\pi\)
\(72\) 0 0
\(73\) 12.4803i 1.46071i 0.683067 + 0.730356i \(0.260646\pi\)
−0.683067 + 0.730356i \(0.739354\pi\)
\(74\) 0 0
\(75\) −15.3013 − 12.7280i −1.76684 − 1.46971i
\(76\) 0 0
\(77\) −0.379324 − 0.657008i −0.0432280 − 0.0748731i
\(78\) 0 0
\(79\) 8.89252 + 2.38274i 1.00049 + 0.268080i 0.721649 − 0.692259i \(-0.243384\pi\)
0.278837 + 0.960338i \(0.410051\pi\)
\(80\) 0 0
\(81\) 6.99250 + 5.66612i 0.776944 + 0.629569i
\(82\) 0 0
\(83\) 14.3609 + 8.29126i 1.57631 + 0.910084i 0.995368 + 0.0961403i \(0.0306497\pi\)
0.580944 + 0.813944i \(0.302684\pi\)
\(84\) 0 0
\(85\) 3.14761i 0.341406i
\(86\) 0 0
\(87\) 10.5240 − 7.43639i 1.12830 − 0.797265i
\(88\) 0 0
\(89\) −1.26215 − 4.71041i −0.133788 − 0.499303i 0.866212 − 0.499676i \(-0.166548\pi\)
−1.00000 0.000373562i \(0.999881\pi\)
\(90\) 0 0
\(91\) 2.53731 + 9.46936i 0.265982 + 0.992658i
\(92\) 0 0
\(93\) −2.51394 − 3.55775i −0.260684 − 0.368921i
\(94\) 0 0
\(95\) 2.66663 − 4.61874i 0.273590 − 0.473872i
\(96\) 0 0
\(97\) 11.9933 − 11.9933i 1.21773 − 1.21773i 0.249309 − 0.968424i \(-0.419797\pi\)
0.968424 − 0.249309i \(-0.0802034\pi\)
\(98\) 0 0
\(99\) 0.986158 + 0.349395i 0.0991126 + 0.0351155i
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.569.35 yes 152
3.2 odd 2 inner 888.2.br.a.569.24 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.24 yes 152
111.8 even 12 inner 888.2.br.a.785.35 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.24 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.35 yes 152 1.1 even 1 trivial
888.2.br.a.785.24 yes 152 37.8 odd 12 inner
888.2.br.a.785.35 yes 152 111.8 even 12 inner