Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1014,2,Mod(529,1014)] 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("1014.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-1,-1,-4] 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: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 991.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1014.991
Dual form 1014.2.e.e.529.1

$q$-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+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} -2.00000 q^{5} +(0.500000 - 0.866025i) q^{6} +(1.00000 - 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{10} +1.00000 q^{12} +2.00000 q^{14} +(1.00000 + 1.73205i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.00000 + 1.73205i) q^{17} -1.00000 q^{18} +(-3.00000 + 5.19615i) q^{19} +(1.00000 - 1.73205i) q^{20} -2.00000 q^{21} +(2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{24} -1.00000 q^{25} +1.00000 q^{27} +(1.00000 + 1.73205i) q^{28} +(5.00000 + 8.66025i) q^{29} +(-1.00000 + 1.73205i) q^{30} -10.0000 q^{31} +(0.500000 - 0.866025i) q^{32} -2.00000 q^{34} +(-2.00000 + 3.46410i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(-4.00000 - 6.92820i) q^{37} -6.00000 q^{38} +2.00000 q^{40} +(5.00000 + 8.66025i) q^{41} +(-1.00000 - 1.73205i) q^{42} +(2.00000 - 3.46410i) q^{43} +(1.00000 - 1.73205i) q^{45} +(-2.00000 + 3.46410i) q^{46} -12.0000 q^{47} +(-0.500000 + 0.866025i) q^{48} +(1.50000 + 2.59808i) q^{49} +(-0.500000 - 0.866025i) q^{50} +2.00000 q^{51} -6.00000 q^{53} +(0.500000 + 0.866025i) q^{54} +(-1.00000 + 1.73205i) q^{56} +6.00000 q^{57} +(-5.00000 + 8.66025i) q^{58} +(-2.00000 + 3.46410i) q^{59} -2.00000 q^{60} +(-1.00000 + 1.73205i) q^{61} +(-5.00000 - 8.66025i) q^{62} +(1.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{67} +(-1.00000 - 1.73205i) q^{68} +(2.00000 - 3.46410i) q^{69} -4.00000 q^{70} +(0.500000 - 0.866025i) q^{72} -4.00000 q^{73} +(4.00000 - 6.92820i) q^{74} +(0.500000 + 0.866025i) q^{75} +(-3.00000 - 5.19615i) q^{76} +(1.00000 + 1.73205i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-5.00000 + 8.66025i) q^{82} +4.00000 q^{83} +(1.00000 - 1.73205i) q^{84} +(2.00000 - 3.46410i) q^{85} +4.00000 q^{86} +(5.00000 - 8.66025i) q^{87} +(3.00000 + 5.19615i) q^{89} +2.00000 q^{90} -4.00000 q^{92} +(5.00000 + 8.66025i) q^{93} +(-6.00000 - 10.3923i) q^{94} +(6.00000 - 10.3923i) q^{95} -1.00000 q^{96} +(-6.00000 + 10.3923i) q^{97} +(-1.50000 + 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - 4 q^{5} + q^{6} + 2 q^{7} - 2 q^{8} - q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{15} - q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 4 q^{21} + 4 q^{23} + q^{24}+ \cdots - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0.500000 − 0.866025i 0.204124 − 0.353553i
\(7\) 1.00000 − 1.73205i 0.377964 − 0.654654i −0.612801 − 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −1.00000 − 1.73205i −0.316228 − 0.547723i
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000 + 1.73205i 0.258199 + 0.447214i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 − 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 + 5.19615i −0.688247 + 1.19208i 0.284157 + 0.958778i \(0.408286\pi\)
−0.972404 + 0.233301i \(0.925047\pi\)
\(20\) 1.00000 − 1.73205i 0.223607 − 0.387298i
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 − 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0.500000 + 0.866025i 0.102062 + 0.176777i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 + 1.73205i 0.188982 + 0.327327i
\(29\) 5.00000 + 8.66025i 0.928477 + 1.60817i 0.785872 + 0.618389i \(0.212214\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) −1.00000 + 1.73205i −0.182574 + 0.316228i
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −2.00000 + 3.46410i −0.338062 + 0.585540i
\(36\) −0.500000 − 0.866025i −0.0833333 − 0.144338i
\(37\) −4.00000 − 6.92820i −0.657596 − 1.13899i −0.981236 − 0.192809i \(-0.938240\pi\)
0.323640 − 0.946180i \(-0.395093\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 5.00000 + 8.66025i 0.780869 + 1.35250i 0.931436 + 0.363905i \(0.118557\pi\)
−0.150567 + 0.988600i \(0.548110\pi\)
\(42\) −1.00000 − 1.73205i −0.154303 − 0.267261i
\(43\) 2.00000 − 3.46410i 0.304997 − 0.528271i −0.672264 − 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) 0 0
\(45\) 1.00000 − 1.73205i 0.149071 − 0.258199i
\(46\) −2.00000 + 3.46410i −0.294884 + 0.510754i
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) −0.500000 − 0.866025i −0.0707107 − 0.122474i
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) 0 0
\(56\) −1.00000 + 1.73205i −0.133631 + 0.231455i
\(57\) 6.00000 0.794719
\(58\) −5.00000 + 8.66025i −0.656532 + 1.13715i
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 − 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) −2.00000 −0.258199
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 − 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) −5.00000 − 8.66025i −0.635001 − 1.09985i
\(63\) 1.00000 + 1.73205i 0.125988 + 0.218218i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) −1.00000 − 1.73205i −0.121268 − 0.210042i
\(69\) 2.00000 − 3.46410i 0.240772 − 0.417029i
\(70\) −4.00000 −0.478091
\(71\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 4.00000 − 6.92820i 0.464991 − 0.805387i
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) −3.00000 − 5.19615i −0.344124 − 0.596040i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 + 1.73205i 0.111803 + 0.193649i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −5.00000 + 8.66025i −0.552158 + 0.956365i
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 − 1.73205i 0.109109 − 0.188982i
\(85\) 2.00000 − 3.46410i 0.216930 − 0.375735i
\(86\) 4.00000 0.431331
\(87\) 5.00000 − 8.66025i 0.536056 − 0.928477i
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 5.00000 + 8.66025i 0.518476 + 0.898027i
\(94\) −6.00000 − 10.3923i −0.618853 − 1.07188i
\(95\) 6.00000 − 10.3923i 0.615587 − 1.06623i
\(96\) −1.00000 −0.102062
\(97\) −6.00000 + 10.3923i −0.609208 + 1.05518i 0.382164 + 0.924095i \(0.375179\pi\)
−0.991371 + 0.131084i \(0.958154\pi\)
\(98\) −1.50000 + 2.59808i −0.151523 + 0.262445i
\(99\) 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 1014.2.e.e.991.1 2
13.2 odd 12 78.2.b.a.25.2 yes 2
13.3 even 3 1014.2.a.b.1.1 1
13.4 even 6 1014.2.e.b.529.1 2
13.5 odd 4 1014.2.i.c.361.2 4
13.6 odd 12 1014.2.i.c.823.1 4
13.7 odd 12 1014.2.i.c.823.2 4
13.8 odd 4 1014.2.i.c.361.1 4
13.9 even 3 inner 1014.2.e.e.529.1 2
13.10 even 6 1014.2.a.g.1.1 1
13.11 odd 12 78.2.b.a.25.1 ✓ 2
13.12 even 2 1014.2.e.b.991.1 2
39.2 even 12 234.2.b.a.181.1 2
39.11 even 12 234.2.b.a.181.2 2
39.23 odd 6 3042.2.a.c.1.1 1
39.29 odd 6 3042.2.a.n.1.1 1
52.3 odd 6 8112.2.a.g.1.1 1
52.11 even 12 624.2.c.a.337.1 2
52.15 even 12 624.2.c.a.337.2 2
52.23 odd 6 8112.2.a.j.1.1 1
65.2 even 12 1950.2.f.d.649.1 2
65.24 odd 12 1950.2.b.c.1351.2 2
65.28 even 12 1950.2.f.g.649.2 2
65.37 even 12 1950.2.f.g.649.1 2
65.54 odd 12 1950.2.b.c.1351.1 2
65.63 even 12 1950.2.f.d.649.2 2
91.41 even 12 3822.2.c.d.883.2 2
91.76 even 12 3822.2.c.d.883.1 2
104.11 even 12 2496.2.c.m.961.2 2
104.37 odd 12 2496.2.c.f.961.2 2
104.67 even 12 2496.2.c.m.961.1 2
104.93 odd 12 2496.2.c.f.961.1 2
156.11 odd 12 1872.2.c.b.1585.2 2
156.119 odd 12 1872.2.c.b.1585.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 13.11 odd 12
78.2.b.a.25.2 yes 2 13.2 odd 12
234.2.b.a.181.1 2 39.2 even 12
234.2.b.a.181.2 2 39.11 even 12
624.2.c.a.337.1 2 52.11 even 12
624.2.c.a.337.2 2 52.15 even 12
1014.2.a.b.1.1 1 13.3 even 3
1014.2.a.g.1.1 1 13.10 even 6
1014.2.e.b.529.1 2 13.4 even 6
1014.2.e.b.991.1 2 13.12 even 2
1014.2.e.e.529.1 2 13.9 even 3 inner
1014.2.e.e.991.1 2 1.1 even 1 trivial
1014.2.i.c.361.1 4 13.8 odd 4
1014.2.i.c.361.2 4 13.5 odd 4
1014.2.i.c.823.1 4 13.6 odd 12
1014.2.i.c.823.2 4 13.7 odd 12
1872.2.c.b.1585.1 2 156.119 odd 12
1872.2.c.b.1585.2 2 156.11 odd 12
1950.2.b.c.1351.1 2 65.54 odd 12
1950.2.b.c.1351.2 2 65.24 odd 12
1950.2.f.d.649.1 2 65.2 even 12
1950.2.f.d.649.2 2 65.63 even 12
1950.2.f.g.649.1 2 65.37 even 12
1950.2.f.g.649.2 2 65.28 even 12
2496.2.c.f.961.1 2 104.93 odd 12
2496.2.c.f.961.2 2 104.37 odd 12
2496.2.c.m.961.1 2 104.67 even 12
2496.2.c.m.961.2 2 104.11 even 12
3042.2.a.c.1.1 1 39.23 odd 6
3042.2.a.n.1.1 1 39.29 odd 6
3822.2.c.d.883.1 2 91.76 even 12
3822.2.c.d.883.2 2 91.41 even 12
8112.2.a.g.1.1 1 52.3 odd 6
8112.2.a.j.1.1 1 52.23 odd 6