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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 257.20
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.29769 - 1.14717i) q^{3} +(-0.643917 - 1.11530i) q^{5} +(-2.63392 - 0.249885i) q^{7} +(0.368015 - 2.97734i) q^{9} +(-3.13384 - 1.80932i) q^{11} +(-3.48808 - 2.01385i) q^{13} +(-2.11504 - 0.708633i) q^{15} +(-0.828246 - 1.43456i) q^{17} +(5.15603 + 2.97683i) q^{19} +(-3.70469 + 2.69728i) q^{21} +(0.372292 - 0.214943i) q^{23} +(1.67074 - 2.89381i) q^{25} +(-2.93794 - 4.28585i) q^{27} +(6.39192 - 3.69038i) q^{29} +0.971739i q^{31} +(-6.14236 + 1.24709i) q^{33} +(1.41733 + 3.09851i) q^{35} +(-5.16236 + 8.94147i) q^{37} +(-6.83668 + 1.38806i) q^{39} +(5.15230 - 8.92404i) q^{41} +(3.67982 + 6.37363i) q^{43} +(-3.55759 + 1.50672i) q^{45} -8.03262 q^{47} +(6.87511 + 1.31636i) q^{49} +(-2.72049 - 0.911487i) q^{51} +(10.4907 - 6.05681i) q^{53} +4.66021i q^{55} +(10.1059 - 2.05181i) q^{57} -1.23719 q^{59} -6.64381i q^{61} +(-1.71332 + 7.75013i) q^{63} +5.18700i q^{65} -2.20562 q^{67} +(0.236545 - 0.706010i) q^{69} -3.66832i q^{71} +(-1.67314 + 0.965987i) q^{73} +(-1.15157 - 5.67190i) q^{75} +(7.80217 + 5.54872i) q^{77} -4.52649 q^{79} +(-8.72913 - 2.19141i) q^{81} +(-0.701322 - 1.21472i) q^{83} +(-1.06664 + 1.84748i) q^{85} +(4.06127 - 12.1216i) q^{87} +(4.81741 - 8.34400i) q^{89} +(8.68412 + 6.17594i) q^{91} +(1.11475 + 1.26102i) q^{93} -7.66734i q^{95} +(8.20853 - 4.73920i) q^{97} +(-6.54027 + 8.66465i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75}+ \cdots - 64 q^{99}+O(q^{100}) \) 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
\(3\) 1.29769 − 1.14717i 0.749223 − 0.662317i
\(4\) 0 0
\(5\) −0.643917 − 1.11530i −0.287968 − 0.498776i 0.685356 − 0.728208i \(-0.259646\pi\)
−0.973325 + 0.229432i \(0.926313\pi\)
\(6\) 0 0
\(7\) −2.63392 − 0.249885i −0.995530 − 0.0944478i
\(8\) 0 0
\(9\) 0.368015 − 2.97734i 0.122672 − 0.992447i
\(10\) 0 0
\(11\) −3.13384 − 1.80932i −0.944888 − 0.545531i −0.0533987 − 0.998573i \(-0.517005\pi\)
−0.891489 + 0.453042i \(0.850339\pi\)
\(12\) 0 0
\(13\) −3.48808 − 2.01385i −0.967420 − 0.558540i −0.0689715 − 0.997619i \(-0.521972\pi\)
−0.898449 + 0.439078i \(0.855305\pi\)
\(14\) 0 0
\(15\) −2.11504 − 0.708633i −0.546101 − 0.182968i
\(16\) 0 0
\(17\) −0.828246 − 1.43456i −0.200879 − 0.347933i 0.747933 − 0.663774i \(-0.231047\pi\)
−0.948812 + 0.315842i \(0.897713\pi\)
\(18\) 0 0
\(19\) 5.15603 + 2.97683i 1.18287 + 0.682933i 0.956678 − 0.291149i \(-0.0940376\pi\)
0.226196 + 0.974082i \(0.427371\pi\)
\(20\) 0 0
\(21\) −3.70469 + 2.69728i −0.808429 + 0.588594i
\(22\) 0 0
\(23\) 0.372292 − 0.214943i 0.0776282 − 0.0448186i −0.460683 − 0.887564i \(-0.652396\pi\)
0.538312 + 0.842746i \(0.319062\pi\)
\(24\) 0 0
\(25\) 1.67074 − 2.89381i 0.334148 − 0.578762i
\(26\) 0 0
\(27\) −2.93794 − 4.28585i −0.565407 − 0.824812i
\(28\) 0 0
\(29\) 6.39192 − 3.69038i 1.18695 − 0.685286i 0.229338 − 0.973347i \(-0.426344\pi\)
0.957612 + 0.288061i \(0.0930105\pi\)
\(30\) 0 0
\(31\) 0.971739i 0.174530i 0.996185 + 0.0872648i \(0.0278126\pi\)
−0.996185 + 0.0872648i \(0.972187\pi\)
\(32\) 0 0
\(33\) −6.14236 + 1.24709i −1.06925 + 0.217091i
\(34\) 0 0
\(35\) 1.41733 + 3.09851i 0.239573 + 0.523744i
\(36\) 0 0
\(37\) −5.16236 + 8.94147i −0.848687 + 1.46997i 0.0336934 + 0.999432i \(0.489273\pi\)
−0.882380 + 0.470537i \(0.844060\pi\)
\(38\) 0 0
\(39\) −6.83668 + 1.38806i −1.09475 + 0.222268i
\(40\) 0 0
\(41\) 5.15230 − 8.92404i 0.804654 − 1.39370i −0.111871 − 0.993723i \(-0.535684\pi\)
0.916524 − 0.399979i \(-0.130982\pi\)
\(42\) 0 0
\(43\) 3.67982 + 6.37363i 0.561167 + 0.971969i 0.997395 + 0.0721330i \(0.0229806\pi\)
−0.436228 + 0.899836i \(0.643686\pi\)
\(44\) 0 0
\(45\) −3.55759 + 1.50672i −0.530334 + 0.224608i
\(46\) 0 0
\(47\) −8.03262 −1.17168 −0.585839 − 0.810427i \(-0.699235\pi\)
−0.585839 + 0.810427i \(0.699235\pi\)
\(48\) 0 0
\(49\) 6.87511 + 1.31636i 0.982159 + 0.188051i
\(50\) 0 0
\(51\) −2.72049 − 0.911487i −0.380945 − 0.127634i
\(52\) 0 0
\(53\) 10.4907 − 6.05681i 1.44101 − 0.831967i 0.443093 − 0.896476i \(-0.353881\pi\)
0.997917 + 0.0645084i \(0.0205479\pi\)
\(54\) 0 0
\(55\) 4.66021i 0.628383i
\(56\) 0 0
\(57\) 10.1059 − 2.05181i 1.33856 − 0.271769i
\(58\) 0 0
\(59\) −1.23719 −0.161069 −0.0805343 − 0.996752i \(-0.525663\pi\)
−0.0805343 + 0.996752i \(0.525663\pi\)
\(60\) 0 0
\(61\) − 6.64381i − 0.850652i −0.905040 − 0.425326i \(-0.860159\pi\)
0.905040 − 0.425326i \(-0.139841\pi\)
\(62\) 0 0
\(63\) −1.71332 + 7.75013i −0.215858 + 0.976425i
\(64\) 0 0
\(65\) 5.18700i 0.643368i
\(66\) 0 0
\(67\) −2.20562 −0.269459 −0.134729 − 0.990882i \(-0.543017\pi\)
−0.134729 + 0.990882i \(0.543017\pi\)
\(68\) 0 0
\(69\) 0.236545 − 0.706010i 0.0284767 − 0.0849936i
\(70\) 0 0
\(71\) − 3.66832i − 0.435349i −0.976021 − 0.217674i \(-0.930153\pi\)
0.976021 − 0.217674i \(-0.0698471\pi\)
\(72\) 0 0
\(73\) −1.67314 + 0.965987i −0.195826 + 0.113060i −0.594707 − 0.803942i \(-0.702732\pi\)
0.398881 + 0.917003i \(0.369399\pi\)
\(74\) 0 0
\(75\) −1.15157 − 5.67190i −0.132972 − 0.654934i
\(76\) 0 0
\(77\) 7.80217 + 5.54872i 0.889140 + 0.632335i
\(78\) 0 0
\(79\) −4.52649 −0.509270 −0.254635 − 0.967037i \(-0.581955\pi\)
−0.254635 + 0.967037i \(0.581955\pi\)
\(80\) 0 0
\(81\) −8.72913 − 2.19141i −0.969903 − 0.243490i
\(82\) 0 0
\(83\) −0.701322 − 1.21472i −0.0769800 − 0.133333i 0.824966 − 0.565183i \(-0.191194\pi\)
−0.901946 + 0.431850i \(0.857861\pi\)
\(84\) 0 0
\(85\) −1.06664 + 1.84748i −0.115694 + 0.200387i
\(86\) 0 0
\(87\) 4.06127 − 12.1216i 0.435414 − 1.29957i
\(88\) 0 0
\(89\) 4.81741 − 8.34400i 0.510644 − 0.884462i −0.489280 − 0.872127i \(-0.662740\pi\)
0.999924 − 0.0123349i \(-0.00392643\pi\)
\(90\) 0 0
\(91\) 8.68412 + 6.17594i 0.910343 + 0.647414i
\(92\) 0 0
\(93\) 1.11475 + 1.26102i 0.115594 + 0.130762i
\(94\) 0 0
\(95\) − 7.66734i − 0.786652i
\(96\) 0 0
\(97\) 8.20853 − 4.73920i 0.833450 − 0.481193i −0.0215824 − 0.999767i \(-0.506870\pi\)
0.855032 + 0.518574i \(0.173537\pi\)
\(98\) 0 0
\(99\) −6.54027 + 8.66465i −0.657322 + 0.870830i
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.257.20 ✓ 48
3.2 odd 2 1512.2.bs.a.1097.17 48
4.3 odd 2 1008.2.ca.e.257.5 48
7.3 odd 6 504.2.cx.a.185.21 yes 48
9.2 odd 6 504.2.cx.a.425.21 yes 48
9.7 even 3 1512.2.cx.a.89.17 48
12.11 even 2 3024.2.ca.e.2609.17 48
21.17 even 6 1512.2.cx.a.17.17 48
28.3 even 6 1008.2.df.e.689.4 48
36.7 odd 6 3024.2.df.e.1601.17 48
36.11 even 6 1008.2.df.e.929.4 48
63.38 even 6 inner 504.2.bs.a.353.20 yes 48
63.52 odd 6 1512.2.bs.a.521.17 48
84.59 odd 6 3024.2.df.e.17.17 48
252.115 even 6 3024.2.ca.e.2033.17 48
252.227 odd 6 1008.2.ca.e.353.5 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.20 ✓ 48 1.1 even 1 trivial
504.2.bs.a.353.20 yes 48 63.38 even 6 inner
504.2.cx.a.185.21 yes 48 7.3 odd 6
504.2.cx.a.425.21 yes 48 9.2 odd 6
1008.2.ca.e.257.5 48 4.3 odd 2
1008.2.ca.e.353.5 48 252.227 odd 6
1008.2.df.e.689.4 48 28.3 even 6
1008.2.df.e.929.4 48 36.11 even 6
1512.2.bs.a.521.17 48 63.52 odd 6
1512.2.bs.a.1097.17 48 3.2 odd 2
1512.2.cx.a.17.17 48 21.17 even 6
1512.2.cx.a.89.17 48 9.7 even 3
3024.2.ca.e.2033.17 48 252.115 even 6
3024.2.ca.e.2609.17 48 12.11 even 2
3024.2.df.e.17.17 48 84.59 odd 6
3024.2.df.e.1601.17 48 36.7 odd 6