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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 = [6,0,0] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) 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 \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 504.169
Dual form 504.2.r.c.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)
 
\(f(q)\) \(=\) \(q+(-1.70574 - 0.300767i) q^{3} +(-0.266044 - 0.460802i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(2.81908 + 1.02606i) q^{9} +(1.11334 - 1.92836i) q^{11} +(-1.03209 - 1.78763i) q^{13} +(0.315207 + 0.866025i) q^{15} +0.815207 q^{17} -7.94356 q^{19} +(1.11334 - 1.32683i) q^{21} +(-3.40033 - 5.88954i) q^{23} +(2.35844 - 4.08494i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(3.73783 - 6.47410i) q^{29} +(-1.14543 - 1.98394i) q^{31} +(-2.47906 + 2.95442i) q^{33} +0.532089 q^{35} -10.2909 q^{37} +(1.22281 + 3.35965i) q^{39} +(1.29813 + 2.24843i) q^{41} +(-0.145430 + 0.251892i) q^{43} +(-0.277189 - 1.57202i) q^{45} +(-0.213011 + 0.368946i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-1.39053 - 0.245188i) q^{51} +1.41147 q^{53} -1.18479 q^{55} +(13.5496 + 2.38917i) q^{57} +(-1.71554 - 2.97140i) q^{59} +(5.23783 - 9.07218i) q^{61} +(-2.29813 + 1.92836i) q^{63} +(-0.549163 + 0.951178i) q^{65} +(2.10220 + 3.64111i) q^{67} +(4.02869 + 11.0687i) q^{69} +12.0865 q^{71} -9.09152 q^{73} +(-5.25150 + 6.25849i) q^{75} +(1.11334 + 1.92836i) q^{77} +(-3.73055 + 6.46151i) q^{79} +(6.89440 + 5.78509i) q^{81} +(8.76264 - 15.1773i) q^{83} +(-0.216881 - 0.375650i) q^{85} +(-8.32295 + 9.91890i) q^{87} -2.09833 q^{89} +2.06418 q^{91} +(1.35710 + 3.72859i) q^{93} +(2.11334 + 3.66041i) q^{95} +(-5.94222 + 10.2922i) q^{97} +(5.11721 - 4.29385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 3 q^{7} + 3 q^{13} + 9 q^{15} + 12 q^{17} - 18 q^{19} - 6 q^{23} + 6 q^{25} - 27 q^{27} + 3 q^{29} + 9 q^{31} - 18 q^{33} - 6 q^{35} - 30 q^{37} + 18 q^{39} - 6 q^{41} + 15 q^{43} + 9 q^{45}+ \cdots - 15 q^{97}+O(q^{100}) \) 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
\(3\) −1.70574 − 0.300767i −0.984808 − 0.173648i
\(4\) 0 0
\(5\) −0.266044 − 0.460802i −0.118979 − 0.206077i 0.800385 − 0.599487i \(-0.204629\pi\)
−0.919363 + 0.393410i \(0.871295\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 0 0
\(11\) 1.11334 − 1.92836i 0.335685 − 0.581423i −0.647931 − 0.761699i \(-0.724366\pi\)
0.983616 + 0.180276i \(0.0576989\pi\)
\(12\) 0 0
\(13\) −1.03209 − 1.78763i −0.286250 − 0.495799i 0.686662 − 0.726977i \(-0.259075\pi\)
−0.972912 + 0.231178i \(0.925742\pi\)
\(14\) 0 0
\(15\) 0.315207 + 0.866025i 0.0813862 + 0.223607i
\(16\) 0 0
\(17\) 0.815207 0.197717 0.0988584 − 0.995102i \(-0.468481\pi\)
0.0988584 + 0.995102i \(0.468481\pi\)
\(18\) 0 0
\(19\) −7.94356 −1.82238 −0.911189 − 0.411988i \(-0.864834\pi\)
−0.911189 + 0.411988i \(0.864834\pi\)
\(20\) 0 0
\(21\) 1.11334 − 1.32683i 0.242951 − 0.289538i
\(22\) 0 0
\(23\) −3.40033 − 5.88954i −0.709018 − 1.22805i −0.965222 − 0.261432i \(-0.915805\pi\)
0.256204 − 0.966623i \(-0.417528\pi\)
\(24\) 0 0
\(25\) 2.35844 − 4.08494i 0.471688 − 0.816988i
\(26\) 0 0
\(27\) −4.50000 − 2.59808i −0.866025 − 0.500000i
\(28\) 0 0
\(29\) 3.73783 − 6.47410i 0.694097 − 1.20221i −0.276387 − 0.961046i \(-0.589137\pi\)
0.970484 − 0.241165i \(-0.0775294\pi\)
\(30\) 0 0
\(31\) −1.14543 − 1.98394i −0.205725 − 0.356327i 0.744638 − 0.667468i \(-0.232622\pi\)
−0.950364 + 0.311142i \(0.899289\pi\)
\(32\) 0 0
\(33\) −2.47906 + 2.95442i −0.431548 + 0.514299i
\(34\) 0 0
\(35\) 0.532089 0.0899394
\(36\) 0 0
\(37\) −10.2909 −1.69181 −0.845903 − 0.533336i \(-0.820938\pi\)
−0.845903 + 0.533336i \(0.820938\pi\)
\(38\) 0 0
\(39\) 1.22281 + 3.35965i 0.195807 + 0.537974i
\(40\) 0 0
\(41\) 1.29813 + 2.24843i 0.202734 + 0.351146i 0.949409 − 0.314044i \(-0.101684\pi\)
−0.746674 + 0.665190i \(0.768351\pi\)
\(42\) 0 0
\(43\) −0.145430 + 0.251892i −0.0221778 + 0.0384131i −0.876901 − 0.480670i \(-0.840393\pi\)
0.854723 + 0.519084i \(0.173727\pi\)
\(44\) 0 0
\(45\) −0.277189 − 1.57202i −0.0413209 − 0.234342i
\(46\) 0 0
\(47\) −0.213011 + 0.368946i −0.0310709 + 0.0538163i −0.881143 − 0.472850i \(-0.843225\pi\)
0.850072 + 0.526667i \(0.176558\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) −1.39053 − 0.245188i −0.194713 − 0.0343332i
\(52\) 0 0
\(53\) 1.41147 0.193881 0.0969404 − 0.995290i \(-0.469094\pi\)
0.0969404 + 0.995290i \(0.469094\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 13.5496 + 2.38917i 1.79469 + 0.316453i
\(58\) 0 0
\(59\) −1.71554 − 2.97140i −0.223344 − 0.386843i 0.732477 − 0.680791i \(-0.238364\pi\)
−0.955821 + 0.293948i \(0.905031\pi\)
\(60\) 0 0
\(61\) 5.23783 − 9.07218i 0.670635 − 1.16157i −0.307089 − 0.951681i \(-0.599355\pi\)
0.977724 − 0.209893i \(-0.0673116\pi\)
\(62\) 0 0
\(63\) −2.29813 + 1.92836i −0.289538 + 0.242951i
\(64\) 0 0
\(65\) −0.549163 + 0.951178i −0.0681153 + 0.117979i
\(66\) 0 0
\(67\) 2.10220 + 3.64111i 0.256824 + 0.444833i 0.965389 − 0.260813i \(-0.0839905\pi\)
−0.708565 + 0.705645i \(0.750657\pi\)
\(68\) 0 0
\(69\) 4.02869 + 11.0687i 0.484997 + 1.33252i
\(70\) 0 0
\(71\) 12.0865 1.43440 0.717200 − 0.696868i \(-0.245423\pi\)
0.717200 + 0.696868i \(0.245423\pi\)
\(72\) 0 0
\(73\) −9.09152 −1.06408 −0.532041 − 0.846719i \(-0.678575\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(74\) 0 0
\(75\) −5.25150 + 6.25849i −0.606391 + 0.722668i
\(76\) 0 0
\(77\) 1.11334 + 1.92836i 0.126877 + 0.219757i
\(78\) 0 0
\(79\) −3.73055 + 6.46151i −0.419720 + 0.726976i −0.995911 − 0.0903388i \(-0.971205\pi\)
0.576191 + 0.817315i \(0.304538\pi\)
\(80\) 0 0
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 8.76264 − 15.1773i 0.961825 − 1.66593i 0.243911 − 0.969798i \(-0.421570\pi\)
0.717914 − 0.696132i \(-0.245097\pi\)
\(84\) 0 0
\(85\) −0.216881 − 0.375650i −0.0235241 − 0.0407449i
\(86\) 0 0
\(87\) −8.32295 + 9.91890i −0.892314 + 1.06342i
\(88\) 0 0
\(89\) −2.09833 −0.222422 −0.111211 − 0.993797i \(-0.535473\pi\)
−0.111211 + 0.993797i \(0.535473\pi\)
\(90\) 0 0
\(91\) 2.06418 0.216385
\(92\) 0 0
\(93\) 1.35710 + 3.72859i 0.140724 + 0.386637i
\(94\) 0 0
\(95\) 2.11334 + 3.66041i 0.216824 + 0.375551i
\(96\) 0 0
\(97\) −5.94222 + 10.2922i −0.603341 + 1.04502i 0.388970 + 0.921250i \(0.372831\pi\)
−0.992311 + 0.123767i \(0.960503\pi\)
\(98\) 0 0
\(99\) 5.11721 − 4.29385i 0.514299 − 0.431548i
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.c.169.1 ✓ 6
3.2 odd 2 1512.2.r.c.505.3 6
4.3 odd 2 1008.2.r.i.673.3 6
9.2 odd 6 4536.2.a.v.1.1 3
9.4 even 3 inner 504.2.r.c.337.1 yes 6
9.5 odd 6 1512.2.r.c.1009.3 6
9.7 even 3 4536.2.a.s.1.3 3
12.11 even 2 3024.2.r.h.2017.3 6
36.7 odd 6 9072.2.a.br.1.3 3
36.11 even 6 9072.2.a.cc.1.1 3
36.23 even 6 3024.2.r.h.1009.3 6
36.31 odd 6 1008.2.r.i.337.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.1 ✓ 6 1.1 even 1 trivial
504.2.r.c.337.1 yes 6 9.4 even 3 inner
1008.2.r.i.337.3 6 36.31 odd 6
1008.2.r.i.673.3 6 4.3 odd 2
1512.2.r.c.505.3 6 3.2 odd 2
1512.2.r.c.1009.3 6 9.5 odd 6
3024.2.r.h.1009.3 6 36.23 even 6
3024.2.r.h.2017.3 6 12.11 even 2
4536.2.a.s.1.3 3 9.7 even 3
4536.2.a.v.1.1 3 9.2 odd 6
9072.2.a.br.1.3 3 36.7 odd 6
9072.2.a.cc.1.1 3 36.11 even 6