Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(169,504)] 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("504.169"); 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, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2091141441.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 169.1
Root \(0.335492 - 1.69925i\) of defining polynomial
Character \(\chi\) \(=\) 504.169
Dual form 504.2.r.e.337.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.30385 + 1.14017i) q^{3} +(-0.164508 - 0.284936i) q^{5} +(0.500000 - 0.866025i) q^{7} +(0.400030 - 2.97321i) q^{9} +(0.664508 - 1.15096i) q^{11} +(-1.53937 - 2.66626i) q^{13} +(0.539368 + 0.183946i) q^{15} +7.35741 q^{17} -2.93671 q^{19} +(0.335492 + 1.69925i) q^{21} +(3.34321 + 5.79062i) q^{23} +(2.44587 - 4.23638i) q^{25} +(2.86838 + 4.33271i) q^{27} +(3.88258 - 6.72483i) q^{29} +(1.63555 + 2.83286i) q^{31} +(0.445874 + 2.25833i) q^{33} -0.329016 q^{35} +0.329016 q^{37} +(5.04709 + 1.72126i) q^{39} +(-0.135552 - 0.234783i) q^{41} +(5.48255 - 9.49606i) q^{43} +(-0.912983 + 0.375134i) q^{45} +(-0.571014 + 0.989025i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-9.59293 + 8.38869i) q^{51} +6.42828 q^{53} -0.437267 q^{55} +(3.82902 - 3.34834i) q^{57} +(-0.372170 - 0.644618i) q^{59} +(-4.42195 + 7.65904i) q^{61} +(-2.37486 - 1.83304i) q^{63} +(-0.506476 + 0.877243i) q^{65} +(-4.28640 - 7.42426i) q^{67} +(-10.9613 - 3.73825i) q^{69} +1.60769 q^{71} -13.4941 q^{73} +(1.64114 + 8.31230i) q^{75} +(-0.664508 - 1.15096i) q^{77} +(0.628926 - 1.08933i) q^{79} +(-8.67995 - 2.37875i) q^{81} +(0.0316459 - 0.0548124i) q^{83} +(-1.21035 - 2.09639i) q^{85} +(2.60515 + 13.1949i) q^{87} +11.3071 q^{89} -3.07874 q^{91} +(-5.36245 - 1.82881i) q^{93} +(0.483112 + 0.836774i) q^{95} +(5.51420 - 9.55087i) q^{97} +(-3.15623 - 2.43614i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 3 q^{5} + 4 q^{7} - q^{9} + 7 q^{11} + 3 q^{13} - 11 q^{15} + 6 q^{17} - 8 q^{19} + q^{21} + 2 q^{23} - 5 q^{25} + 11 q^{27} - 9 q^{29} + 3 q^{31} - 21 q^{33} - 6 q^{35} + 6 q^{37} + 2 q^{39}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −1.30385 + 1.14017i −0.752776 + 0.658277i
\(4\) 0 0
\(5\) −0.164508 − 0.284936i −0.0735702 − 0.127427i 0.826893 − 0.562359i \(-0.190106\pi\)
−0.900464 + 0.434931i \(0.856773\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) 0 0
\(9\) 0.400030 − 2.97321i 0.133343 − 0.991070i
\(10\) 0 0
\(11\) 0.664508 − 1.15096i 0.200357 − 0.347028i −0.748287 − 0.663375i \(-0.769123\pi\)
0.948643 + 0.316348i \(0.102457\pi\)
\(12\) 0 0
\(13\) −1.53937 − 2.66626i −0.426944 − 0.739489i 0.569656 − 0.821883i \(-0.307076\pi\)
−0.996600 + 0.0823948i \(0.973743\pi\)
\(14\) 0 0
\(15\) 0.539368 + 0.183946i 0.139264 + 0.0474946i
\(16\) 0 0
\(17\) 7.35741 1.78443 0.892217 − 0.451606i \(-0.149149\pi\)
0.892217 + 0.451606i \(0.149149\pi\)
\(18\) 0 0
\(19\) −2.93671 −0.673727 −0.336864 − 0.941553i \(-0.609366\pi\)
−0.336864 + 0.941553i \(0.609366\pi\)
\(20\) 0 0
\(21\) 0.335492 + 1.69925i 0.0732104 + 0.370806i
\(22\) 0 0
\(23\) 3.34321 + 5.79062i 0.697108 + 1.20743i 0.969465 + 0.245231i \(0.0788637\pi\)
−0.272356 + 0.962196i \(0.587803\pi\)
\(24\) 0 0
\(25\) 2.44587 − 4.23638i 0.489175 − 0.847276i
\(26\) 0 0
\(27\) 2.86838 + 4.33271i 0.552021 + 0.833830i
\(28\) 0 0
\(29\) 3.88258 − 6.72483i 0.720977 − 1.24877i −0.239631 − 0.970864i \(-0.577026\pi\)
0.960608 − 0.277906i \(-0.0896402\pi\)
\(30\) 0 0
\(31\) 1.63555 + 2.83286i 0.293754 + 0.508796i 0.974694 − 0.223542i \(-0.0717621\pi\)
−0.680940 + 0.732339i \(0.738429\pi\)
\(32\) 0 0
\(33\) 0.445874 + 2.25833i 0.0776168 + 0.393124i
\(34\) 0 0
\(35\) −0.329016 −0.0556138
\(36\) 0 0
\(37\) 0.329016 0.0540899 0.0270449 − 0.999634i \(-0.491390\pi\)
0.0270449 + 0.999634i \(0.491390\pi\)
\(38\) 0 0
\(39\) 5.04709 + 1.72126i 0.808181 + 0.275622i
\(40\) 0 0
\(41\) −0.135552 − 0.234783i −0.0211696 − 0.0366669i 0.855247 − 0.518221i \(-0.173406\pi\)
−0.876416 + 0.481555i \(0.840072\pi\)
\(42\) 0 0
\(43\) 5.48255 − 9.49606i 0.836081 − 1.44814i −0.0570654 − 0.998370i \(-0.518174\pi\)
0.893147 − 0.449765i \(-0.148492\pi\)
\(44\) 0 0
\(45\) −0.912983 + 0.375134i −0.136099 + 0.0559216i
\(46\) 0 0
\(47\) −0.571014 + 0.989025i −0.0832910 + 0.144264i −0.904662 − 0.426131i \(-0.859876\pi\)
0.821371 + 0.570395i \(0.193210\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) −9.59293 + 8.38869i −1.34328 + 1.17465i
\(52\) 0 0
\(53\) 6.42828 0.882992 0.441496 − 0.897263i \(-0.354448\pi\)
0.441496 + 0.897263i \(0.354448\pi\)
\(54\) 0 0
\(55\) −0.437267 −0.0589611
\(56\) 0 0
\(57\) 3.82902 − 3.34834i 0.507166 − 0.443499i
\(58\) 0 0
\(59\) −0.372170 − 0.644618i −0.0484525 − 0.0839221i 0.840782 − 0.541374i \(-0.182096\pi\)
−0.889234 + 0.457452i \(0.848762\pi\)
\(60\) 0 0
\(61\) −4.42195 + 7.65904i −0.566173 + 0.980640i 0.430767 + 0.902463i \(0.358243\pi\)
−0.996940 + 0.0781767i \(0.975090\pi\)
\(62\) 0 0
\(63\) −2.37486 − 1.83304i −0.299204 − 0.230941i
\(64\) 0 0
\(65\) −0.506476 + 0.877243i −0.0628207 + 0.108809i
\(66\) 0 0
\(67\) −4.28640 − 7.42426i −0.523667 − 0.907018i −0.999620 − 0.0275474i \(-0.991230\pi\)
0.475954 − 0.879470i \(-0.342103\pi\)
\(68\) 0 0
\(69\) −10.9613 − 3.73825i −1.31959 − 0.450032i
\(70\) 0 0
\(71\) 1.60769 0.190798 0.0953990 − 0.995439i \(-0.469587\pi\)
0.0953990 + 0.995439i \(0.469587\pi\)
\(72\) 0 0
\(73\) −13.4941 −1.57936 −0.789680 − 0.613519i \(-0.789754\pi\)
−0.789680 + 0.613519i \(0.789754\pi\)
\(74\) 0 0
\(75\) 1.64114 + 8.31230i 0.189503 + 0.959821i
\(76\) 0 0
\(77\) −0.664508 − 1.15096i −0.0757277 − 0.131164i
\(78\) 0 0
\(79\) 0.628926 − 1.08933i 0.0707597 − 0.122559i −0.828475 − 0.560026i \(-0.810791\pi\)
0.899234 + 0.437467i \(0.144124\pi\)
\(80\) 0 0
\(81\) −8.67995 − 2.37875i −0.964439 − 0.264305i
\(82\) 0 0
\(83\) 0.0316459 − 0.0548124i 0.00347359 − 0.00601644i −0.864283 − 0.503005i \(-0.832228\pi\)
0.867757 + 0.496989i \(0.165561\pi\)
\(84\) 0 0
\(85\) −1.21035 − 2.09639i −0.131281 − 0.227386i
\(86\) 0 0
\(87\) 2.60515 + 13.1949i 0.279302 + 1.41465i
\(88\) 0 0
\(89\) 11.3071 1.19855 0.599274 − 0.800544i \(-0.295456\pi\)
0.599274 + 0.800544i \(0.295456\pi\)
\(90\) 0 0
\(91\) −3.07874 −0.322739
\(92\) 0 0
\(93\) −5.36245 − 1.82881i −0.556060 − 0.189638i
\(94\) 0 0
\(95\) 0.483112 + 0.836774i 0.0495662 + 0.0858512i
\(96\) 0 0
\(97\) 5.51420 − 9.55087i 0.559882 − 0.969744i −0.437624 − 0.899158i \(-0.644180\pi\)
0.997506 − 0.0705859i \(-0.0224869\pi\)
\(98\) 0 0
\(99\) −3.15623 − 2.43614i −0.317213 − 0.244841i
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.r.e.169.1 ✓ 8
3.2 odd 2 1512.2.r.e.505.2 8
4.3 odd 2 1008.2.r.l.673.4 8
9.2 odd 6 4536.2.a.y.1.3 4
9.4 even 3 inner 504.2.r.e.337.1 yes 8
9.5 odd 6 1512.2.r.e.1009.2 8
9.7 even 3 4536.2.a.z.1.2 4
12.11 even 2 3024.2.r.m.2017.2 8
36.7 odd 6 9072.2.a.cj.1.2 4
36.11 even 6 9072.2.a.cg.1.3 4
36.23 even 6 3024.2.r.m.1009.2 8
36.31 odd 6 1008.2.r.l.337.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.e.169.1 ✓ 8 1.1 even 1 trivial
504.2.r.e.337.1 yes 8 9.4 even 3 inner
1008.2.r.l.337.4 8 36.31 odd 6
1008.2.r.l.673.4 8 4.3 odd 2
1512.2.r.e.505.2 8 3.2 odd 2
1512.2.r.e.1009.2 8 9.5 odd 6
3024.2.r.m.1009.2 8 36.23 even 6
3024.2.r.m.2017.2 8 12.11 even 2
4536.2.a.y.1.3 4 9.2 odd 6
4536.2.a.z.1.2 4 9.7 even 3
9072.2.a.cg.1.3 4 36.11 even 6
9072.2.a.cj.1.2 4 36.7 odd 6