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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 569.21
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.21

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.169643 + 1.72372i) q^{3} +(0.336068 - 1.25422i) q^{5} +(-1.20899 - 2.09403i) q^{7} +(-2.94244 + 0.584836i) q^{9} -0.238846 q^{11} +(0.935101 + 0.250559i) q^{13} +(2.21894 + 0.366517i) q^{15} +(6.67070 - 1.78741i) q^{17} +(3.10277 + 0.831385i) q^{19} +(3.40444 - 2.43920i) q^{21} +(1.92276 + 1.92276i) q^{23} +(2.87000 + 1.65699i) q^{25} +(-1.50726 - 4.97274i) q^{27} +(3.98251 - 3.98251i) q^{29} +(5.62726 + 5.62726i) q^{31} +(-0.0405187 - 0.411705i) q^{33} +(-3.03268 + 0.812605i) q^{35} +(6.08276 - 0.00225746i) q^{37} +(-0.273262 + 1.65436i) q^{39} +(-5.86662 - 10.1613i) q^{41} +(-5.21255 + 5.21255i) q^{43} +(-0.255346 + 3.88702i) q^{45} -7.53556i q^{47} +(0.576685 - 0.998847i) q^{49} +(4.21264 + 11.1952i) q^{51} +(0.441727 + 0.255031i) q^{53} +(-0.0802686 + 0.299566i) q^{55} +(-0.906713 + 5.48936i) q^{57} +(-11.0903 + 2.97164i) q^{59} +(1.30323 - 4.86372i) q^{61} +(4.78205 + 5.45451i) q^{63} +(0.628514 - 1.08862i) q^{65} +(-3.58431 + 2.06940i) q^{67} +(-2.98813 + 3.64049i) q^{69} +(5.78930 - 3.34245i) q^{71} +10.8670i q^{73} +(-2.36932 + 5.22818i) q^{75} +(0.288763 + 0.500152i) q^{77} +(12.1284 + 3.24980i) q^{79} +(8.31593 - 3.44169i) q^{81} +(4.33751 + 2.50426i) q^{83} -8.96723i q^{85} +(7.54036 + 6.18915i) q^{87} +(-0.463954 - 1.73150i) q^{89} +(-0.605848 - 2.26106i) q^{91} +(-8.74521 + 10.6545i) q^{93} +(2.08548 - 3.61216i) q^{95} +(-4.84959 + 4.84959i) q^{97} +(0.702792 - 0.139686i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 0.169643 + 1.72372i 0.0979436 + 0.995192i
\(4\) 0 0
\(5\) 0.336068 − 1.25422i 0.150294 − 0.560905i −0.849168 − 0.528122i \(-0.822896\pi\)
0.999462 − 0.0327829i \(-0.0104370\pi\)
\(6\) 0 0
\(7\) −1.20899 − 2.09403i −0.456955 − 0.791470i 0.541843 − 0.840480i \(-0.317727\pi\)
−0.998798 + 0.0490098i \(0.984393\pi\)
\(8\) 0 0
\(9\) −2.94244 + 0.584836i −0.980814 + 0.194945i
\(10\) 0 0
\(11\) −0.238846 −0.0720149 −0.0360074 − 0.999352i \(-0.511464\pi\)
−0.0360074 + 0.999352i \(0.511464\pi\)
\(12\) 0 0
\(13\) 0.935101 + 0.250559i 0.259350 + 0.0694927i 0.386151 − 0.922435i \(-0.373804\pi\)
−0.126801 + 0.991928i \(0.540471\pi\)
\(14\) 0 0
\(15\) 2.21894 + 0.366517i 0.572928 + 0.0946344i
\(16\) 0 0
\(17\) 6.67070 − 1.78741i 1.61788 − 0.433510i 0.667504 − 0.744606i \(-0.267363\pi\)
0.950379 + 0.311096i \(0.100696\pi\)
\(18\) 0 0
\(19\) 3.10277 + 0.831385i 0.711824 + 0.190733i 0.596521 − 0.802598i \(-0.296549\pi\)
0.115303 + 0.993330i \(0.463216\pi\)
\(20\) 0 0
\(21\) 3.40444 − 2.43920i 0.742909 − 0.532278i
\(22\) 0 0
\(23\) 1.92276 + 1.92276i 0.400924 + 0.400924i 0.878558 − 0.477635i \(-0.158506\pi\)
−0.477635 + 0.878558i \(0.658506\pi\)
\(24\) 0 0
\(25\) 2.87000 + 1.65699i 0.573999 + 0.331399i
\(26\) 0 0
\(27\) −1.50726 − 4.97274i −0.290072 − 0.957005i
\(28\) 0 0
\(29\) 3.98251 − 3.98251i 0.739534 − 0.739534i −0.232953 − 0.972488i \(-0.574839\pi\)
0.972488 + 0.232953i \(0.0748390\pi\)
\(30\) 0 0
\(31\) 5.62726 + 5.62726i 1.01069 + 1.01069i 0.999942 + 0.0107438i \(0.00341992\pi\)
0.0107438 + 0.999942i \(0.496580\pi\)
\(32\) 0 0
\(33\) −0.0405187 − 0.411705i −0.00705340 − 0.0716686i
\(34\) 0 0
\(35\) −3.03268 + 0.812605i −0.512617 + 0.137355i
\(36\) 0 0
\(37\) 6.08276 − 0.00225746i 1.00000 − 0.000371125i
\(38\) 0 0
\(39\) −0.273262 + 1.65436i −0.0437569 + 0.264910i
\(40\) 0 0
\(41\) −5.86662 − 10.1613i −0.916212 − 1.58693i −0.805117 − 0.593116i \(-0.797897\pi\)
−0.111095 − 0.993810i \(-0.535436\pi\)
\(42\) 0 0
\(43\) −5.21255 + 5.21255i −0.794907 + 0.794907i −0.982287 − 0.187381i \(-0.940000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(44\) 0 0
\(45\) −0.255346 + 3.88702i −0.0380647 + 0.579443i
\(46\) 0 0
\(47\) − 7.53556i − 1.09917i −0.835436 − 0.549587i \(-0.814785\pi\)
0.835436 − 0.549587i \(-0.185215\pi\)
\(48\) 0 0
\(49\) 0.576685 − 0.998847i 0.0823835 − 0.142692i
\(50\) 0 0
\(51\) 4.21264 + 11.1952i 0.589887 + 1.56764i
\(52\) 0 0
\(53\) 0.441727 + 0.255031i 0.0606759 + 0.0350312i 0.530031 − 0.847978i \(-0.322180\pi\)
−0.469355 + 0.883009i \(0.655514\pi\)
\(54\) 0 0
\(55\) −0.0802686 + 0.299566i −0.0108234 + 0.0403935i
\(56\) 0 0
\(57\) −0.906713 + 5.48936i −0.120097 + 0.727083i
\(58\) 0 0
\(59\) −11.0903 + 2.97164i −1.44383 + 0.386874i −0.893875 − 0.448317i \(-0.852024\pi\)
−0.549960 + 0.835191i \(0.685357\pi\)
\(60\) 0 0
\(61\) 1.30323 − 4.86372i 0.166862 − 0.622736i −0.830934 − 0.556371i \(-0.812193\pi\)
0.997795 − 0.0663648i \(-0.0211401\pi\)
\(62\) 0 0
\(63\) 4.78205 + 5.45451i 0.602482 + 0.687204i
\(64\) 0 0
\(65\) 0.628514 − 1.08862i 0.0779576 − 0.135026i
\(66\) 0 0
\(67\) −3.58431 + 2.06940i −0.437893 + 0.252818i −0.702704 − 0.711483i \(-0.748024\pi\)
0.264810 + 0.964301i \(0.414691\pi\)
\(68\) 0 0
\(69\) −2.98813 + 3.64049i −0.359728 + 0.438264i
\(70\) 0 0
\(71\) 5.78930 − 3.34245i 0.687063 − 0.396676i −0.115448 − 0.993314i \(-0.536830\pi\)
0.802511 + 0.596637i \(0.203497\pi\)
\(72\) 0 0
\(73\) 10.8670i 1.27189i 0.771734 + 0.635945i \(0.219390\pi\)
−0.771734 + 0.635945i \(0.780610\pi\)
\(74\) 0 0
\(75\) −2.36932 + 5.22818i −0.273586 + 0.603698i
\(76\) 0 0
\(77\) 0.288763 + 0.500152i 0.0329076 + 0.0569976i
\(78\) 0 0
\(79\) 12.1284 + 3.24980i 1.36455 + 0.365631i 0.865487 − 0.500932i \(-0.167009\pi\)
0.499067 + 0.866563i \(0.333676\pi\)
\(80\) 0 0
\(81\) 8.31593 − 3.44169i 0.923993 − 0.382410i
\(82\) 0 0
\(83\) 4.33751 + 2.50426i 0.476103 + 0.274878i 0.718791 − 0.695226i \(-0.244696\pi\)
−0.242688 + 0.970104i \(0.578029\pi\)
\(84\) 0 0
\(85\) − 8.96723i − 0.972632i
\(86\) 0 0
\(87\) 7.54036 + 6.18915i 0.808411 + 0.663546i
\(88\) 0 0
\(89\) −0.463954 − 1.73150i −0.0491790 − 0.183539i 0.936967 − 0.349418i \(-0.113621\pi\)
−0.986146 + 0.165879i \(0.946954\pi\)
\(90\) 0 0
\(91\) −0.605848 − 2.26106i −0.0635101 − 0.237023i
\(92\) 0 0
\(93\) −8.74521 + 10.6545i −0.906836 + 1.10482i
\(94\) 0 0
\(95\) 2.08548 − 3.61216i 0.213966 − 0.370600i
\(96\) 0 0
\(97\) −4.84959 + 4.84959i −0.492401 + 0.492401i −0.909062 − 0.416661i \(-0.863200\pi\)
0.416661 + 0.909062i \(0.363200\pi\)
\(98\) 0 0
\(99\) 0.702792 − 0.139686i 0.0706332 − 0.0140390i
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.569.21 yes 152
3.2 odd 2 inner 888.2.br.a.569.9 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.9 yes 152
111.8 even 12 inner 888.2.br.a.785.21 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.9 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.21 yes 152 1.1 even 1 trivial
888.2.br.a.785.9 yes 152 37.8 odd 12 inner
888.2.br.a.785.21 yes 152 111.8 even 12 inner