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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(257,504)] 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("504.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bs (of order \(6\), degree \(2\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.8
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.09564 + 1.34148i) q^{3} +(-0.271038 - 0.469451i) q^{5} +(1.60655 - 2.10214i) q^{7} +(-0.599164 - 2.93956i) q^{9} +(-0.666300 - 0.384688i) q^{11} +(-2.96386 - 1.71119i) q^{13} +(0.926720 + 0.150754i) q^{15} +(-3.23477 - 5.60278i) q^{17} +(5.60413 + 3.23554i) q^{19} +(1.05981 + 4.45834i) q^{21} +(0.100353 - 0.0579388i) q^{23} +(2.35308 - 4.07565i) q^{25} +(4.59984 + 2.41692i) q^{27} +(4.40174 - 2.54135i) q^{29} -4.63530i q^{31} +(1.24608 - 0.472353i) q^{33} +(-1.42229 - 0.184434i) q^{35} +(5.47518 - 9.48329i) q^{37} +(5.54284 - 2.10114i) q^{39} +(-4.04575 + 7.00745i) q^{41} +(3.32569 + 5.76026i) q^{43} +(-1.21758 + 1.07801i) q^{45} +1.54617 q^{47} +(-1.83802 - 6.75438i) q^{49} +(11.0602 + 1.79922i) q^{51} +(-0.221011 + 0.127601i) q^{53} +0.417060i q^{55} +(-10.4805 + 3.97287i) q^{57} -10.2411 q^{59} +5.57913i q^{61} +(-7.14196 - 3.46301i) q^{63} +1.85518i q^{65} -3.28349 q^{67} +(-0.0322262 + 0.198102i) q^{69} -5.67917i q^{71} +(-5.35354 + 3.09087i) q^{73} +(2.88931 + 7.62205i) q^{75} +(-1.87911 + 0.782639i) q^{77} +4.02459 q^{79} +(-8.28201 + 3.52255i) q^{81} +(-5.80057 - 10.0469i) q^{83} +(-1.75349 + 3.03713i) q^{85} +(-1.41353 + 8.68926i) q^{87} +(-2.00832 + 3.47851i) q^{89} +(-8.35874 + 3.48136i) q^{91} +(6.21818 + 5.07860i) q^{93} -3.50781i q^{95} +(15.0653 - 8.69795i) q^{97} +(-0.731591 + 2.18912i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75}+ \cdots - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.09564 + 1.34148i −0.632566 + 0.774507i
\(4\) 0 0
\(5\) −0.271038 − 0.469451i −0.121212 − 0.209945i 0.799034 − 0.601286i \(-0.205345\pi\)
−0.920246 + 0.391341i \(0.872011\pi\)
\(6\) 0 0
\(7\) 1.60655 − 2.10214i 0.607217 − 0.794536i
\(8\) 0 0
\(9\) −0.599164 − 2.93956i −0.199721 − 0.979853i
\(10\) 0 0
\(11\) −0.666300 − 0.384688i −0.200897 − 0.115988i 0.396177 − 0.918174i \(-0.370337\pi\)
−0.597074 + 0.802186i \(0.703670\pi\)
\(12\) 0 0
\(13\) −2.96386 − 1.71119i −0.822027 − 0.474598i 0.0290877 − 0.999577i \(-0.490740\pi\)
−0.851115 + 0.524979i \(0.824073\pi\)
\(14\) 0 0
\(15\) 0.926720 + 0.150754i 0.239278 + 0.0389246i
\(16\) 0 0
\(17\) −3.23477 − 5.60278i −0.784547 − 1.35887i −0.929269 − 0.369403i \(-0.879562\pi\)
0.144723 − 0.989472i \(-0.453771\pi\)
\(18\) 0 0
\(19\) 5.60413 + 3.23554i 1.28567 + 0.742285i 0.977880 − 0.209168i \(-0.0670756\pi\)
0.307795 + 0.951453i \(0.400409\pi\)
\(20\) 0 0
\(21\) 1.05981 + 4.45834i 0.231268 + 0.972890i
\(22\) 0 0
\(23\) 0.100353 − 0.0579388i 0.0209250 − 0.0120811i −0.489501 − 0.872003i \(-0.662821\pi\)
0.510426 + 0.859922i \(0.329488\pi\)
\(24\) 0 0
\(25\) 2.35308 − 4.07565i 0.470615 − 0.815130i
\(26\) 0 0
\(27\) 4.59984 + 2.41692i 0.885239 + 0.465136i
\(28\) 0 0
\(29\) 4.40174 − 2.54135i 0.817383 − 0.471916i −0.0321304 − 0.999484i \(-0.510229\pi\)
0.849513 + 0.527568i \(0.176896\pi\)
\(30\) 0 0
\(31\) − 4.63530i − 0.832524i −0.909245 − 0.416262i \(-0.863340\pi\)
0.909245 − 0.416262i \(-0.136660\pi\)
\(32\) 0 0
\(33\) 1.24608 − 0.472353i 0.216914 − 0.0822261i
\(34\) 0 0
\(35\) −1.42229 − 0.184434i −0.240410 − 0.0311751i
\(36\) 0 0
\(37\) 5.47518 − 9.48329i 0.900114 − 1.55904i 0.0727692 − 0.997349i \(-0.476816\pi\)
0.827345 − 0.561694i \(-0.189850\pi\)
\(38\) 0 0
\(39\) 5.54284 − 2.10114i 0.887565 − 0.336451i
\(40\) 0 0
\(41\) −4.04575 + 7.00745i −0.631841 + 1.09438i 0.355334 + 0.934739i \(0.384367\pi\)
−0.987175 + 0.159641i \(0.948966\pi\)
\(42\) 0 0
\(43\) 3.32569 + 5.76026i 0.507162 + 0.878431i 0.999966 + 0.00829006i \(0.00263884\pi\)
−0.492803 + 0.870141i \(0.664028\pi\)
\(44\) 0 0
\(45\) −1.21758 + 1.07801i −0.181506 + 0.160700i
\(46\) 0 0
\(47\) 1.54617 0.225532 0.112766 − 0.993622i \(-0.464029\pi\)
0.112766 + 0.993622i \(0.464029\pi\)
\(48\) 0 0
\(49\) −1.83802 − 6.75438i −0.262574 − 0.964912i
\(50\) 0 0
\(51\) 11.0602 + 1.79922i 1.54873 + 0.251941i
\(52\) 0 0
\(53\) −0.221011 + 0.127601i −0.0303582 + 0.0175273i −0.515102 − 0.857129i \(-0.672246\pi\)
0.484744 + 0.874656i \(0.338913\pi\)
\(54\) 0 0
\(55\) 0.417060i 0.0562364i
\(56\) 0 0
\(57\) −10.4805 + 3.97287i −1.38818 + 0.526220i
\(58\) 0 0
\(59\) −10.2411 −1.33328 −0.666640 − 0.745380i \(-0.732268\pi\)
−0.666640 + 0.745380i \(0.732268\pi\)
\(60\) 0 0
\(61\) 5.57913i 0.714335i 0.934040 + 0.357167i \(0.116257\pi\)
−0.934040 + 0.357167i \(0.883743\pi\)
\(62\) 0 0
\(63\) −7.14196 − 3.46301i −0.899802 − 0.436298i
\(64\) 0 0
\(65\) 1.85518i 0.230107i
\(66\) 0 0
\(67\) −3.28349 −0.401143 −0.200571 − 0.979679i \(-0.564280\pi\)
−0.200571 + 0.979679i \(0.564280\pi\)
\(68\) 0 0
\(69\) −0.0322262 + 0.198102i −0.00387958 + 0.0238486i
\(70\) 0 0
\(71\) − 5.67917i − 0.673994i −0.941506 − 0.336997i \(-0.890589\pi\)
0.941506 − 0.336997i \(-0.109411\pi\)
\(72\) 0 0
\(73\) −5.35354 + 3.09087i −0.626585 + 0.361759i −0.779428 − 0.626492i \(-0.784490\pi\)
0.152844 + 0.988250i \(0.451157\pi\)
\(74\) 0 0
\(75\) 2.88931 + 7.62205i 0.333628 + 0.880118i
\(76\) 0 0
\(77\) −1.87911 + 0.782639i −0.214145 + 0.0891900i
\(78\) 0 0
\(79\) 4.02459 0.452802 0.226401 − 0.974034i \(-0.427304\pi\)
0.226401 + 0.974034i \(0.427304\pi\)
\(80\) 0 0
\(81\) −8.28201 + 3.52255i −0.920223 + 0.391395i
\(82\) 0 0
\(83\) −5.80057 − 10.0469i −0.636695 − 1.10279i −0.986153 − 0.165836i \(-0.946968\pi\)
0.349458 − 0.936952i \(-0.386366\pi\)
\(84\) 0 0
\(85\) −1.75349 + 3.03713i −0.190192 + 0.329423i
\(86\) 0 0
\(87\) −1.41353 + 8.68926i −0.151546 + 0.931586i
\(88\) 0 0
\(89\) −2.00832 + 3.47851i −0.212881 + 0.368721i −0.952615 − 0.304178i \(-0.901618\pi\)
0.739734 + 0.672900i \(0.234951\pi\)
\(90\) 0 0
\(91\) −8.35874 + 3.48136i −0.876234 + 0.364946i
\(92\) 0 0
\(93\) 6.21818 + 5.07860i 0.644795 + 0.526626i
\(94\) 0 0
\(95\) − 3.50781i − 0.359894i
\(96\) 0 0
\(97\) 15.0653 − 8.69795i 1.52965 − 0.883143i 0.530272 − 0.847828i \(-0.322090\pi\)
0.999376 − 0.0353150i \(-0.0112435\pi\)
\(98\) 0 0
\(99\) −0.731591 + 2.18912i −0.0735277 + 0.220015i
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 504.2.bs.a.257.8 ✓ 48
3.2 odd 2 1512.2.bs.a.1097.13 48
4.3 odd 2 1008.2.ca.e.257.17 48
7.3 odd 6 504.2.cx.a.185.1 yes 48
9.2 odd 6 504.2.cx.a.425.1 yes 48
9.7 even 3 1512.2.cx.a.89.13 48
12.11 even 2 3024.2.ca.e.2609.13 48
21.17 even 6 1512.2.cx.a.17.13 48
28.3 even 6 1008.2.df.e.689.24 48
36.7 odd 6 3024.2.df.e.1601.13 48
36.11 even 6 1008.2.df.e.929.24 48
63.38 even 6 inner 504.2.bs.a.353.8 yes 48
63.52 odd 6 1512.2.bs.a.521.13 48
84.59 odd 6 3024.2.df.e.17.13 48
252.115 even 6 3024.2.ca.e.2033.13 48
252.227 odd 6 1008.2.ca.e.353.17 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.8 ✓ 48 1.1 even 1 trivial
504.2.bs.a.353.8 yes 48 63.38 even 6 inner
504.2.cx.a.185.1 yes 48 7.3 odd 6
504.2.cx.a.425.1 yes 48 9.2 odd 6
1008.2.ca.e.257.17 48 4.3 odd 2
1008.2.ca.e.353.17 48 252.227 odd 6
1008.2.df.e.689.24 48 28.3 even 6
1008.2.df.e.929.24 48 36.11 even 6
1512.2.bs.a.521.13 48 63.52 odd 6
1512.2.bs.a.1097.13 48 3.2 odd 2
1512.2.cx.a.17.13 48 21.17 even 6
1512.2.cx.a.89.13 48 9.7 even 3
3024.2.ca.e.2033.13 48 252.115 even 6
3024.2.ca.e.2609.13 48 12.11 even 2
3024.2.df.e.17.13 48 84.59 odd 6
3024.2.df.e.1601.13 48 36.7 odd 6