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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 257.11
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.419673 - 1.68044i) q^{3} +(1.02449 + 1.77447i) q^{5} +(-2.64365 + 0.105420i) q^{7} +(-2.64775 + 1.41047i) q^{9} +(-5.11564 - 2.95352i) q^{11} +(0.139269 + 0.0804071i) q^{13} +(2.55194 - 2.46630i) q^{15} +(-2.77904 - 4.81345i) q^{17} +(-4.02056 - 2.32127i) q^{19} +(1.28662 + 4.39825i) q^{21} +(0.375194 - 0.216618i) q^{23} +(0.400830 - 0.694257i) q^{25} +(3.48140 + 3.85744i) q^{27} +(-1.95524 + 1.12886i) q^{29} +2.97708i q^{31} +(-2.81630 + 9.83603i) q^{33} +(-2.89547 - 4.58308i) q^{35} +(2.17904 - 3.77422i) q^{37} +(0.0766716 - 0.267778i) q^{39} +(-2.35740 + 4.08313i) q^{41} +(1.82369 + 3.15873i) q^{43} +(-5.21544 - 3.25334i) q^{45} +0.130095 q^{47} +(6.97777 - 0.557389i) q^{49} +(-6.92241 + 6.69009i) q^{51} +(-10.7936 + 6.23170i) q^{53} -12.1034i q^{55} +(-2.21343 + 7.73049i) q^{57} -6.44809 q^{59} -6.90990i q^{61} +(6.85103 - 4.00792i) q^{63} +0.329506i q^{65} +15.2919 q^{67} +(-0.521473 - 0.539582i) q^{69} +1.48027i q^{71} +(-2.60881 + 1.50620i) q^{73} +(-1.33487 - 0.382208i) q^{75} +(13.8353 + 7.26877i) q^{77} +17.3851 q^{79} +(5.02114 - 7.46914i) q^{81} +(-7.62399 - 13.2051i) q^{83} +(5.69422 - 9.86268i) q^{85} +(2.71754 + 2.81191i) q^{87} +(-4.04757 + 7.01059i) q^{89} +(-0.376655 - 0.197886i) q^{91} +(5.00280 - 1.24940i) q^{93} -9.51251i q^{95} +(-2.61123 + 1.50759i) q^{97} +(17.7108 + 0.604706i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75}+ \cdots - 64 q^{99}+O(q^{100}) \) 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
\(3\) −0.419673 − 1.68044i −0.242299 − 0.970202i
\(4\) 0 0
\(5\) 1.02449 + 1.77447i 0.458167 + 0.793569i 0.998864 − 0.0476488i \(-0.0151728\pi\)
−0.540697 + 0.841217i \(0.681839\pi\)
\(6\) 0 0
\(7\) −2.64365 + 0.105420i −0.999206 + 0.0398451i
\(8\) 0 0
\(9\) −2.64775 + 1.41047i −0.882583 + 0.470157i
\(10\) 0 0
\(11\) −5.11564 − 2.95352i −1.54242 − 0.890518i −0.998685 − 0.0512635i \(-0.983675\pi\)
−0.543738 − 0.839255i \(-0.682992\pi\)
\(12\) 0 0
\(13\) 0.139269 + 0.0804071i 0.0386263 + 0.0223009i 0.519189 − 0.854660i \(-0.326234\pi\)
−0.480562 + 0.876960i \(0.659567\pi\)
\(14\) 0 0
\(15\) 2.55194 − 2.46630i 0.658908 − 0.636795i
\(16\) 0 0
\(17\) −2.77904 − 4.81345i −0.674017 − 1.16743i −0.976755 − 0.214358i \(-0.931234\pi\)
0.302738 − 0.953074i \(-0.402099\pi\)
\(18\) 0 0
\(19\) −4.02056 − 2.32127i −0.922381 − 0.532537i −0.0379869 − 0.999278i \(-0.512095\pi\)
−0.884394 + 0.466741i \(0.845428\pi\)
\(20\) 0 0
\(21\) 1.28662 + 4.39825i 0.280764 + 0.959777i
\(22\) 0 0
\(23\) 0.375194 − 0.216618i 0.0782334 − 0.0451681i −0.460373 − 0.887726i \(-0.652284\pi\)
0.538606 + 0.842558i \(0.318951\pi\)
\(24\) 0 0
\(25\) 0.400830 − 0.694257i 0.0801659 − 0.138851i
\(26\) 0 0
\(27\) 3.48140 + 3.85744i 0.669996 + 0.742365i
\(28\) 0 0
\(29\) −1.95524 + 1.12886i −0.363079 + 0.209623i −0.670430 − 0.741972i \(-0.733890\pi\)
0.307352 + 0.951596i \(0.400557\pi\)
\(30\) 0 0
\(31\) 2.97708i 0.534700i 0.963600 + 0.267350i \(0.0861479\pi\)
−0.963600 + 0.267350i \(0.913852\pi\)
\(32\) 0 0
\(33\) −2.81630 + 9.83603i −0.490256 + 1.71223i
\(34\) 0 0
\(35\) −2.89547 − 4.58308i −0.489423 − 0.774683i
\(36\) 0 0
\(37\) 2.17904 − 3.77422i 0.358233 − 0.620477i −0.629433 − 0.777055i \(-0.716713\pi\)
0.987666 + 0.156578i \(0.0500461\pi\)
\(38\) 0 0
\(39\) 0.0766716 − 0.267778i 0.0122773 − 0.0428788i
\(40\) 0 0
\(41\) −2.35740 + 4.08313i −0.368163 + 0.637678i −0.989278 − 0.146042i \(-0.953347\pi\)
0.621115 + 0.783719i \(0.286680\pi\)
\(42\) 0 0
\(43\) 1.82369 + 3.15873i 0.278111 + 0.481702i 0.970915 − 0.239424i \(-0.0769585\pi\)
−0.692805 + 0.721125i \(0.743625\pi\)
\(44\) 0 0
\(45\) −5.21544 − 3.25334i −0.777472 − 0.484980i
\(46\) 0 0
\(47\) 0.130095 0.0189764 0.00948818 − 0.999955i \(-0.496980\pi\)
0.00948818 + 0.999955i \(0.496980\pi\)
\(48\) 0 0
\(49\) 6.97777 − 0.557389i 0.996825 − 0.0796270i
\(50\) 0 0
\(51\) −6.92241 + 6.69009i −0.969331 + 0.936800i
\(52\) 0 0
\(53\) −10.7936 + 6.23170i −1.48262 + 0.855990i −0.999805 − 0.0197331i \(-0.993718\pi\)
−0.482813 + 0.875723i \(0.660385\pi\)
\(54\) 0 0
\(55\) − 12.1034i − 1.63202i
\(56\) 0 0
\(57\) −2.21343 + 7.73049i −0.293177 + 1.02393i
\(58\) 0 0
\(59\) −6.44809 −0.839470 −0.419735 − 0.907647i \(-0.637877\pi\)
−0.419735 + 0.907647i \(0.637877\pi\)
\(60\) 0 0
\(61\) − 6.90990i − 0.884722i −0.896837 − 0.442361i \(-0.854141\pi\)
0.896837 − 0.442361i \(-0.145859\pi\)
\(62\) 0 0
\(63\) 6.85103 − 4.00792i 0.863148 − 0.504950i
\(64\) 0 0
\(65\) 0.329506i 0.0408702i
\(66\) 0 0
\(67\) 15.2919 1.86820 0.934102 − 0.357006i \(-0.116202\pi\)
0.934102 + 0.357006i \(0.116202\pi\)
\(68\) 0 0
\(69\) −0.521473 − 0.539582i −0.0627780 − 0.0649580i
\(70\) 0 0
\(71\) 1.48027i 0.175675i 0.996135 + 0.0878376i \(0.0279957\pi\)
−0.996135 + 0.0878376i \(0.972004\pi\)
\(72\) 0 0
\(73\) −2.60881 + 1.50620i −0.305339 + 0.176287i −0.644839 − 0.764319i \(-0.723075\pi\)
0.339500 + 0.940606i \(0.389742\pi\)
\(74\) 0 0
\(75\) −1.33487 − 0.382208i −0.154138 − 0.0441336i
\(76\) 0 0
\(77\) 13.8353 + 7.26877i 1.57668 + 0.828353i
\(78\) 0 0
\(79\) 17.3851 1.95598 0.977991 − 0.208648i \(-0.0669063\pi\)
0.977991 + 0.208648i \(0.0669063\pi\)
\(80\) 0 0
\(81\) 5.02114 − 7.46914i 0.557905 − 0.829905i
\(82\) 0 0
\(83\) −7.62399 − 13.2051i −0.836841 − 1.44945i −0.892522 − 0.451003i \(-0.851066\pi\)
0.0556811 − 0.998449i \(-0.482267\pi\)
\(84\) 0 0
\(85\) 5.69422 − 9.86268i 0.617625 − 1.06976i
\(86\) 0 0
\(87\) 2.71754 + 2.81191i 0.291350 + 0.301468i
\(88\) 0 0
\(89\) −4.04757 + 7.01059i −0.429041 + 0.743121i −0.996788 − 0.0800819i \(-0.974482\pi\)
0.567747 + 0.823203i \(0.307815\pi\)
\(90\) 0 0
\(91\) −0.376655 − 0.197886i −0.0394842 − 0.0207441i
\(92\) 0 0
\(93\) 5.00280 − 1.24940i 0.518767 − 0.129557i
\(94\) 0 0
\(95\) − 9.51251i − 0.975963i
\(96\) 0 0
\(97\) −2.61123 + 1.50759i −0.265130 + 0.153073i −0.626673 − 0.779283i \(-0.715584\pi\)
0.361542 + 0.932356i \(0.382250\pi\)
\(98\) 0 0
\(99\) 17.7108 + 0.604706i 1.78000 + 0.0607752i
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.257.11 ✓ 48
3.2 odd 2 1512.2.bs.a.1097.8 48
4.3 odd 2 1008.2.ca.e.257.14 48
7.3 odd 6 504.2.cx.a.185.19 yes 48
9.2 odd 6 504.2.cx.a.425.19 yes 48
9.7 even 3 1512.2.cx.a.89.8 48
12.11 even 2 3024.2.ca.e.2609.8 48
21.17 even 6 1512.2.cx.a.17.8 48
28.3 even 6 1008.2.df.e.689.6 48
36.7 odd 6 3024.2.df.e.1601.8 48
36.11 even 6 1008.2.df.e.929.6 48
63.38 even 6 inner 504.2.bs.a.353.11 yes 48
63.52 odd 6 1512.2.bs.a.521.8 48
84.59 odd 6 3024.2.df.e.17.8 48
252.115 even 6 3024.2.ca.e.2033.8 48
252.227 odd 6 1008.2.ca.e.353.14 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.11 ✓ 48 1.1 even 1 trivial
504.2.bs.a.353.11 yes 48 63.38 even 6 inner
504.2.cx.a.185.19 yes 48 7.3 odd 6
504.2.cx.a.425.19 yes 48 9.2 odd 6
1008.2.ca.e.257.14 48 4.3 odd 2
1008.2.ca.e.353.14 48 252.227 odd 6
1008.2.df.e.689.6 48 28.3 even 6
1008.2.df.e.929.6 48 36.11 even 6
1512.2.bs.a.521.8 48 63.52 odd 6
1512.2.bs.a.1097.8 48 3.2 odd 2
1512.2.cx.a.17.8 48 21.17 even 6
1512.2.cx.a.89.8 48 9.7 even 3
3024.2.ca.e.2033.8 48 252.115 even 6
3024.2.ca.e.2609.8 48 12.11 even 2
3024.2.df.e.17.8 48 84.59 odd 6
3024.2.df.e.1601.8 48 36.7 odd 6