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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1512,2,Mod(505,1512)] 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("1512.505"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0733807856\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1009.3
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 1512.1009
Dual form 1512.2.r.c.505.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.266044 - 0.460802i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-1.11334 - 1.92836i) q^{11} +(-1.03209 + 1.78763i) q^{13} -0.815207 q^{17} -7.94356 q^{19} +(3.40033 - 5.88954i) q^{23} +(2.35844 + 4.08494i) q^{25} +(-3.73783 - 6.47410i) q^{29} +(-1.14543 + 1.98394i) q^{31} -0.532089 q^{35} -10.2909 q^{37} +(-1.29813 + 2.24843i) q^{41} +(-0.145430 - 0.251892i) q^{43} +(0.213011 + 0.368946i) q^{47} +(-0.500000 + 0.866025i) q^{49} -1.41147 q^{53} -1.18479 q^{55} +(1.71554 - 2.97140i) q^{59} +(5.23783 + 9.07218i) q^{61} +(0.549163 + 0.951178i) q^{65} +(2.10220 - 3.64111i) q^{67} -12.0865 q^{71} -9.09152 q^{73} +(-1.11334 + 1.92836i) q^{77} +(-3.73055 - 6.46151i) q^{79} +(-8.76264 - 15.1773i) q^{83} +(-0.216881 + 0.375650i) q^{85} +2.09833 q^{89} +2.06418 q^{91} +(-2.11334 + 3.66041i) q^{95} +(-5.94222 - 10.2922i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + 3 q^{13} - 12 q^{17} - 18 q^{19} + 6 q^{23} + 6 q^{25} - 3 q^{29} + 9 q^{31} + 6 q^{35} - 30 q^{37} + 6 q^{41} + 15 q^{43} + 9 q^{47} - 3 q^{49} + 12 q^{53} + 3 q^{59} + 12 q^{61}+ \cdots - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.266044 − 0.460802i 0.118979 − 0.206077i −0.800385 − 0.599487i \(-0.795371\pi\)
0.919363 + 0.393410i \(0.128705\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.11334 − 1.92836i −0.335685 − 0.581423i 0.647931 − 0.761699i \(-0.275634\pi\)
−0.983616 + 0.180276i \(0.942301\pi\)
\(12\) 0 0
\(13\) −1.03209 + 1.78763i −0.286250 + 0.495799i −0.972912 − 0.231178i \(-0.925742\pi\)
0.686662 + 0.726977i \(0.259075\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.815207 −0.197717 −0.0988584 − 0.995102i \(-0.531519\pi\)
−0.0988584 + 0.995102i \(0.531519\pi\)
\(18\) 0 0
\(19\) −7.94356 −1.82238 −0.911189 − 0.411988i \(-0.864834\pi\)
−0.911189 + 0.411988i \(0.864834\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.40033 − 5.88954i 0.709018 − 1.22805i −0.256204 − 0.966623i \(-0.582472\pi\)
0.965222 − 0.261432i \(-0.0841947\pi\)
\(24\) 0 0
\(25\) 2.35844 + 4.08494i 0.471688 + 0.816988i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.73783 − 6.47410i −0.694097 − 1.20221i −0.970484 − 0.241165i \(-0.922471\pi\)
0.276387 − 0.961046i \(-0.410863\pi\)
\(30\) 0 0
\(31\) −1.14543 + 1.98394i −0.205725 + 0.356327i −0.950364 − 0.311142i \(-0.899289\pi\)
0.744638 + 0.667468i \(0.232622\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −1.29813 + 2.24843i −0.202734 + 0.351146i −0.949409 − 0.314044i \(-0.898316\pi\)
0.746674 + 0.665190i \(0.231649\pi\)
\(42\) 0 0
\(43\) −0.145430 − 0.251892i −0.0221778 − 0.0384131i 0.854723 − 0.519084i \(-0.173727\pi\)
−0.876901 + 0.480670i \(0.840393\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.213011 + 0.368946i 0.0310709 + 0.0538163i 0.881143 − 0.472850i \(-0.156775\pi\)
−0.850072 + 0.526667i \(0.823442\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41147 −0.193881 −0.0969404 − 0.995290i \(-0.530906\pi\)
−0.0969404 + 0.995290i \(0.530906\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.71554 − 2.97140i 0.223344 − 0.386843i −0.732477 − 0.680791i \(-0.761636\pi\)
0.955821 + 0.293948i \(0.0949694\pi\)
\(60\) 0 0
\(61\) 5.23783 + 9.07218i 0.670635 + 1.16157i 0.977724 + 0.209893i \(0.0673116\pi\)
−0.307089 + 0.951681i \(0.599355\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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 \(-0.750657\pi\)
0.965389 + 0.260813i \(0.0839905\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 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 \(-0.304538\pi\)
−0.995911 + 0.0903388i \(0.971205\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.76264 − 15.1773i −0.961825 − 1.66593i −0.717914 − 0.696132i \(-0.754903\pi\)
−0.243911 − 0.969798i \(-0.578430\pi\)
\(84\) 0 0
\(85\) −0.216881 + 0.375650i −0.0235241 + 0.0407449i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.09833 0.222422 0.111211 − 0.993797i \(-0.464527\pi\)
0.111211 + 0.993797i \(0.464527\pi\)
\(90\) 0 0
\(91\) 2.06418 0.216385
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.11334 + 3.66041i −0.216824 + 0.375551i
\(96\) 0 0
\(97\) −5.94222 − 10.2922i −0.603341 − 1.04502i −0.992311 − 0.123767i \(-0.960503\pi\)
0.388970 − 0.921250i \(-0.372831\pi\)
\(98\) 0 0
\(99\) 0 0
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 1512.2.r.c.1009.3 6
3.2 odd 2 504.2.r.c.337.1 yes 6
4.3 odd 2 3024.2.r.h.1009.3 6
9.2 odd 6 504.2.r.c.169.1 ✓ 6
9.4 even 3 4536.2.a.v.1.1 3
9.5 odd 6 4536.2.a.s.1.3 3
9.7 even 3 inner 1512.2.r.c.505.3 6
12.11 even 2 1008.2.r.i.337.3 6
36.7 odd 6 3024.2.r.h.2017.3 6
36.11 even 6 1008.2.r.i.673.3 6
36.23 even 6 9072.2.a.br.1.3 3
36.31 odd 6 9072.2.a.cc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.1 ✓ 6 9.2 odd 6
504.2.r.c.337.1 yes 6 3.2 odd 2
1008.2.r.i.337.3 6 12.11 even 2
1008.2.r.i.673.3 6 36.11 even 6
1512.2.r.c.505.3 6 9.7 even 3 inner
1512.2.r.c.1009.3 6 1.1 even 1 trivial
3024.2.r.h.1009.3 6 4.3 odd 2
3024.2.r.h.2017.3 6 36.7 odd 6
4536.2.a.s.1.3 3 9.5 odd 6
4536.2.a.v.1.1 3 9.4 even 3
9072.2.a.br.1.3 3 36.23 even 6
9072.2.a.cc.1.1 3 36.31 odd 6