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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 257.7
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.14780 - 1.29713i) q^{3} +(-0.527910 - 0.914367i) q^{5} +(0.781227 + 2.52778i) q^{7} +(-0.365108 + 2.97770i) q^{9} +(5.40824 + 3.12245i) q^{11} +(0.872074 + 0.503492i) q^{13} +(-0.580120 + 1.73428i) q^{15} +(-3.26821 - 5.66070i) q^{17} +(1.73329 + 1.00071i) q^{19} +(2.38218 - 3.91475i) q^{21} +(3.81168 - 2.20067i) q^{23} +(1.94262 - 3.36472i) q^{25} +(4.28154 - 2.94421i) q^{27} +(6.12821 - 3.53813i) q^{29} +2.39266i q^{31} +(-2.15735 - 10.5992i) q^{33} +(1.89890 - 2.04877i) q^{35} +(-3.64197 + 6.30808i) q^{37} +(-0.347871 - 1.70910i) q^{39} +(-1.80119 + 3.11974i) q^{41} +(1.60595 + 2.78159i) q^{43} +(2.91545 - 1.23812i) q^{45} -3.74047 q^{47} +(-5.77937 + 3.94954i) q^{49} +(-3.59143 + 10.7367i) q^{51} +(6.02455 - 3.47827i) q^{53} -6.59348i q^{55} +(-0.691409 - 3.39692i) q^{57} +13.3417 q^{59} +8.20683i q^{61} +(-7.81221 + 1.40335i) q^{63} -1.06319i q^{65} +0.122792 q^{67} +(-7.22961 - 2.41832i) q^{69} +5.37678i q^{71} +(14.4429 - 8.33860i) q^{73} +(-6.59423 + 1.34219i) q^{75} +(-3.66781 + 16.1102i) q^{77} -8.86272 q^{79} +(-8.73339 - 2.17436i) q^{81} +(1.07668 + 1.86486i) q^{83} +(-3.45064 + 5.97668i) q^{85} +(-11.6234 - 3.88805i) q^{87} +(2.23201 - 3.86595i) q^{89} +(-0.591431 + 2.59775i) q^{91} +(3.10359 - 2.74629i) q^{93} -2.11315i q^{95} +(-0.960756 + 0.554693i) q^{97} +(-11.2723 + 14.9641i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75}+ \cdots - 64 q^{99}+O(q^{100}) \) 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
\(3\) −1.14780 − 1.29713i −0.662683 − 0.748900i
\(4\) 0 0
\(5\) −0.527910 − 0.914367i −0.236089 − 0.408917i 0.723500 − 0.690324i \(-0.242532\pi\)
−0.959588 + 0.281407i \(0.909199\pi\)
\(6\) 0 0
\(7\) 0.781227 + 2.52778i 0.295276 + 0.955412i
\(8\) 0 0
\(9\) −0.365108 + 2.97770i −0.121703 + 0.992567i
\(10\) 0 0
\(11\) 5.40824 + 3.12245i 1.63065 + 0.941453i 0.983894 + 0.178752i \(0.0572060\pi\)
0.646751 + 0.762701i \(0.276127\pi\)
\(12\) 0 0
\(13\) 0.872074 + 0.503492i 0.241870 + 0.139644i 0.616036 − 0.787718i \(-0.288738\pi\)
−0.374166 + 0.927362i \(0.622071\pi\)
\(14\) 0 0
\(15\) −0.580120 + 1.73428i −0.149786 + 0.447789i
\(16\) 0 0
\(17\) −3.26821 − 5.66070i −0.792656 − 1.37292i −0.924317 − 0.381626i \(-0.875364\pi\)
0.131660 − 0.991295i \(-0.457969\pi\)
\(18\) 0 0
\(19\) 1.73329 + 1.00071i 0.397643 + 0.229579i 0.685466 − 0.728104i \(-0.259598\pi\)
−0.287823 + 0.957683i \(0.592932\pi\)
\(20\) 0 0
\(21\) 2.38218 − 3.91475i 0.519834 − 0.854267i
\(22\) 0 0
\(23\) 3.81168 − 2.20067i 0.794790 − 0.458872i −0.0468562 − 0.998902i \(-0.514920\pi\)
0.841646 + 0.540029i \(0.181587\pi\)
\(24\) 0 0
\(25\) 1.94262 − 3.36472i 0.388524 − 0.672944i
\(26\) 0 0
\(27\) 4.28154 − 2.94421i 0.823983 − 0.566614i
\(28\) 0 0
\(29\) 6.12821 − 3.53813i 1.13798 − 0.657014i 0.192051 − 0.981385i \(-0.438486\pi\)
0.945930 + 0.324371i \(0.105153\pi\)
\(30\) 0 0
\(31\) 2.39266i 0.429734i 0.976643 + 0.214867i \(0.0689317\pi\)
−0.976643 + 0.214867i \(0.931068\pi\)
\(32\) 0 0
\(33\) −2.15735 − 10.5992i −0.375546 − 1.84508i
\(34\) 0 0
\(35\) 1.89890 − 2.04877i 0.320973 − 0.346305i
\(36\) 0 0
\(37\) −3.64197 + 6.30808i −0.598737 + 1.03704i 0.394271 + 0.918994i \(0.370997\pi\)
−0.993008 + 0.118048i \(0.962336\pi\)
\(38\) 0 0
\(39\) −0.347871 − 1.70910i −0.0557039 − 0.273676i
\(40\) 0 0
\(41\) −1.80119 + 3.11974i −0.281298 + 0.487222i −0.971705 − 0.236199i \(-0.924098\pi\)
0.690407 + 0.723421i \(0.257432\pi\)
\(42\) 0 0
\(43\) 1.60595 + 2.78159i 0.244906 + 0.424189i 0.962105 − 0.272679i \(-0.0879097\pi\)
−0.717199 + 0.696868i \(0.754576\pi\)
\(44\) 0 0
\(45\) 2.91545 − 1.23812i 0.434610 − 0.184567i
\(46\) 0 0
\(47\) −3.74047 −0.545604 −0.272802 − 0.962070i \(-0.587950\pi\)
−0.272802 + 0.962070i \(0.587950\pi\)
\(48\) 0 0
\(49\) −5.77937 + 3.94954i −0.825624 + 0.564220i
\(50\) 0 0
\(51\) −3.59143 + 10.7367i −0.502901 + 1.50343i
\(52\) 0 0
\(53\) 6.02455 − 3.47827i 0.827535 − 0.477778i −0.0254729 − 0.999676i \(-0.508109\pi\)
0.853008 + 0.521898i \(0.174776\pi\)
\(54\) 0 0
\(55\) − 6.59348i − 0.889065i
\(56\) 0 0
\(57\) −0.691409 − 3.39692i −0.0915793 − 0.449933i
\(58\) 0 0
\(59\) 13.3417 1.73694 0.868469 − 0.495744i \(-0.165104\pi\)
0.868469 + 0.495744i \(0.165104\pi\)
\(60\) 0 0
\(61\) 8.20683i 1.05078i 0.850862 + 0.525389i \(0.176080\pi\)
−0.850862 + 0.525389i \(0.823920\pi\)
\(62\) 0 0
\(63\) −7.81221 + 1.40335i −0.984246 + 0.176805i
\(64\) 0 0
\(65\) − 1.06319i − 0.131873i
\(66\) 0 0
\(67\) 0.122792 0.0150015 0.00750074 − 0.999972i \(-0.497612\pi\)
0.00750074 + 0.999972i \(0.497612\pi\)
\(68\) 0 0
\(69\) −7.22961 − 2.41832i −0.870343 − 0.291131i
\(70\) 0 0
\(71\) 5.37678i 0.638107i 0.947737 + 0.319053i \(0.103365\pi\)
−0.947737 + 0.319053i \(0.896635\pi\)
\(72\) 0 0
\(73\) 14.4429 − 8.33860i 1.69041 − 0.975959i 0.736228 − 0.676734i \(-0.236605\pi\)
0.954182 − 0.299225i \(-0.0967282\pi\)
\(74\) 0 0
\(75\) −6.59423 + 1.34219i −0.761436 + 0.154983i
\(76\) 0 0
\(77\) −3.66781 + 16.1102i −0.417986 + 1.83593i
\(78\) 0 0
\(79\) −8.86272 −0.997134 −0.498567 − 0.866851i \(-0.666140\pi\)
−0.498567 + 0.866851i \(0.666140\pi\)
\(80\) 0 0
\(81\) −8.73339 − 2.17436i −0.970377 − 0.241596i
\(82\) 0 0
\(83\) 1.07668 + 1.86486i 0.118181 + 0.204695i 0.919047 − 0.394149i \(-0.128960\pi\)
−0.800866 + 0.598844i \(0.795627\pi\)
\(84\) 0 0
\(85\) −3.45064 + 5.97668i −0.374274 + 0.648262i
\(86\) 0 0
\(87\) −11.6234 − 3.88805i −1.24616 − 0.416842i
\(88\) 0 0
\(89\) 2.23201 − 3.86595i 0.236592 − 0.409790i −0.723142 − 0.690699i \(-0.757303\pi\)
0.959734 + 0.280910i \(0.0906361\pi\)
\(90\) 0 0
\(91\) −0.591431 + 2.59775i −0.0619988 + 0.272319i
\(92\) 0 0
\(93\) 3.10359 − 2.74629i 0.321827 − 0.284777i
\(94\) 0 0
\(95\) − 2.11315i − 0.216804i
\(96\) 0 0
\(97\) −0.960756 + 0.554693i −0.0975500 + 0.0563205i −0.547981 − 0.836491i \(-0.684604\pi\)
0.450431 + 0.892811i \(0.351270\pi\)
\(98\) 0 0
\(99\) −11.2723 + 14.9641i −1.13291 + 1.50395i
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.257.7 ✓ 48
3.2 odd 2 1512.2.bs.a.1097.15 48
4.3 odd 2 1008.2.ca.e.257.18 48
7.3 odd 6 504.2.cx.a.185.15 yes 48
9.2 odd 6 504.2.cx.a.425.15 yes 48
9.7 even 3 1512.2.cx.a.89.15 48
12.11 even 2 3024.2.ca.e.2609.15 48
21.17 even 6 1512.2.cx.a.17.15 48
28.3 even 6 1008.2.df.e.689.10 48
36.7 odd 6 3024.2.df.e.1601.15 48
36.11 even 6 1008.2.df.e.929.10 48
63.38 even 6 inner 504.2.bs.a.353.7 yes 48
63.52 odd 6 1512.2.bs.a.521.15 48
84.59 odd 6 3024.2.df.e.17.15 48
252.115 even 6 3024.2.ca.e.2033.15 48
252.227 odd 6 1008.2.ca.e.353.18 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.7 ✓ 48 1.1 even 1 trivial
504.2.bs.a.353.7 yes 48 63.38 even 6 inner
504.2.cx.a.185.15 yes 48 7.3 odd 6
504.2.cx.a.425.15 yes 48 9.2 odd 6
1008.2.ca.e.257.18 48 4.3 odd 2
1008.2.ca.e.353.18 48 252.227 odd 6
1008.2.df.e.689.10 48 28.3 even 6
1008.2.df.e.929.10 48 36.11 even 6
1512.2.bs.a.521.15 48 63.52 odd 6
1512.2.bs.a.1097.15 48 3.2 odd 2
1512.2.cx.a.17.15 48 21.17 even 6
1512.2.cx.a.89.15 48 9.7 even 3
3024.2.ca.e.2033.15 48 252.115 even 6
3024.2.ca.e.2609.15 48 12.11 even 2
3024.2.df.e.17.15 48 84.59 odd 6
3024.2.df.e.1601.15 48 36.7 odd 6