Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(337,1008)] 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("1008.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
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: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 673.3
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1008.673
Dual form 1008.2.r.i.337.3

$q$-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 \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.983616 − 0.180276i \(-0.942301\pi\)
0.647931 + 0.761699i \(0.275634\pi\)
\(12\) 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 \(-0.135166\pi\)
0.911189 + 0.411988i \(0.135166\pi\)
\(20\) 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 \(0.0841947\pi\)
−0.256204 + 0.966623i \(0.582472\pi\)
\(24\) 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 0.950364 − 0.311142i \(-0.100711\pi\)
−0.744638 + 0.667468i \(0.767378\pi\)
\(32\) 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 −0.854723 − 0.519084i \(-0.826273\pi\)
0.876901 + 0.480670i \(0.159607\pi\)
\(44\) 0 0
\(45\) −0.277189 − 1.57202i −0.0413209 − 0.234342i
\(46\) 0 0
\(47\) 0.213011 − 0.368946i 0.0310709 − 0.0538163i −0.850072 − 0.526667i \(-0.823442\pi\)
0.881143 + 0.472850i \(0.156775\pi\)
\(48\) 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 0.955821 − 0.293948i \(-0.0949694\pi\)
−0.732477 + 0.680791i \(0.761636\pi\)
\(60\) 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 0.708565 − 0.705645i \(-0.249343\pi\)
−0.965389 + 0.260813i \(0.916009\pi\)
\(68\) 0 0
\(69\) 4.02869 + 11.0687i 0.484997 + 1.33252i
\(70\) 0 0
\(71\) −12.0865 −1.43440 −0.717200 − 0.696868i \(-0.754577\pi\)
−0.717200 + 0.696868i \(0.754577\pi\)
\(72\) 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 −0.576191 − 0.817315i \(-0.695462\pi\)
0.995911 + 0.0903388i \(0.0287950\pi\)
\(80\) 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 \(0.578430\pi\)
−0.717914 + 0.696132i \(0.754903\pi\)
\(84\) 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 1008.2.r.i.673.3 6
3.2 odd 2 3024.2.r.h.2017.3 6
4.3 odd 2 504.2.r.c.169.1 ✓ 6
9.2 odd 6 9072.2.a.cc.1.1 3
9.4 even 3 inner 1008.2.r.i.337.3 6
9.5 odd 6 3024.2.r.h.1009.3 6
9.7 even 3 9072.2.a.br.1.3 3
12.11 even 2 1512.2.r.c.505.3 6
36.7 odd 6 4536.2.a.s.1.3 3
36.11 even 6 4536.2.a.v.1.1 3
36.23 even 6 1512.2.r.c.1009.3 6
36.31 odd 6 504.2.r.c.337.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.1 ✓ 6 4.3 odd 2
504.2.r.c.337.1 yes 6 36.31 odd 6
1008.2.r.i.337.3 6 9.4 even 3 inner
1008.2.r.i.673.3 6 1.1 even 1 trivial
1512.2.r.c.505.3 6 12.11 even 2
1512.2.r.c.1009.3 6 36.23 even 6
3024.2.r.h.1009.3 6 9.5 odd 6
3024.2.r.h.2017.3 6 3.2 odd 2
4536.2.a.s.1.3 3 36.7 odd 6
4536.2.a.v.1.1 3 36.11 even 6
9072.2.a.br.1.3 3 9.7 even 3
9072.2.a.cc.1.1 3 9.2 odd 6