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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 1009.2
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 1512.1009
Dual form 1512.2.r.c.505.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.326352 + 0.565258i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(1.70574 + 2.95442i) q^{11} +(0.152704 - 0.264490i) q^{13} +0.226682 q^{17} -2.16250 q^{19} +(-3.35117 + 5.80439i) q^{23} +(2.28699 + 3.96118i) q^{25} +(-0.254900 - 0.441500i) q^{29} +(2.85844 - 4.95096i) q^{31} +0.652704 q^{35} -2.28312 q^{37} +(0.479055 - 0.829748i) q^{41} +(3.85844 + 6.68302i) q^{43} +(4.14543 + 7.18009i) q^{47} +(-0.500000 + 0.866025i) q^{49} +3.18479 q^{53} -2.22668 q^{55} +(-6.07785 + 10.5271i) q^{59} +(1.75490 + 3.03958i) q^{61} +(0.0996702 + 0.172634i) q^{65} +(-2.87211 + 4.97464i) q^{67} -14.0273 q^{71} +12.4192 q^{73} +(1.70574 - 2.95442i) q^{77} +(5.98293 + 10.3627i) q^{79} +(2.13563 + 3.69902i) q^{83} +(-0.0739780 + 0.128134i) q^{85} +1.19934 q^{89} -0.305407 q^{91} +(0.705737 - 1.22237i) q^{95} +(7.48932 + 12.9719i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + 3 q^{13} - 12 q^{17} - 18 q^{19} + 6 q^{23} + 6 q^{25} - 3 q^{29} + 9 q^{31} + 6 q^{35} - 30 q^{37} + 6 q^{41} + 15 q^{43} + 9 q^{47} - 3 q^{49} + 12 q^{53} + 3 q^{59} + 12 q^{61}+ \cdots - 15 q^{97}+O(q^{100}) \) 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
\(5\) −0.326352 + 0.565258i −0.145949 + 0.252791i −0.929727 − 0.368251i \(-0.879957\pi\)
0.783778 + 0.621042i \(0.213290\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.70574 + 2.95442i 0.514299 + 0.890792i 0.999862 + 0.0165906i \(0.00528120\pi\)
−0.485563 + 0.874202i \(0.661385\pi\)
\(12\) 0 0
\(13\) 0.152704 − 0.264490i 0.0423524 − 0.0733565i −0.844072 − 0.536230i \(-0.819848\pi\)
0.886425 + 0.462873i \(0.153181\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.226682 0.0549784 0.0274892 − 0.999622i \(-0.491249\pi\)
0.0274892 + 0.999622i \(0.491249\pi\)
\(18\) 0 0
\(19\) −2.16250 −0.496112 −0.248056 − 0.968746i \(-0.579792\pi\)
−0.248056 + 0.968746i \(0.579792\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.35117 + 5.80439i −0.698767 + 1.21030i 0.270128 + 0.962824i \(0.412934\pi\)
−0.968894 + 0.247475i \(0.920399\pi\)
\(24\) 0 0
\(25\) 2.28699 + 3.96118i 0.457398 + 0.792236i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.254900 − 0.441500i −0.0473338 − 0.0819845i 0.841388 − 0.540432i \(-0.181739\pi\)
−0.888722 + 0.458447i \(0.848406\pi\)
\(30\) 0 0
\(31\) 2.85844 − 4.95096i 0.513391 − 0.889219i −0.486488 − 0.873687i \(-0.661722\pi\)
0.999879 − 0.0155324i \(-0.00494430\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.652704 0.110327
\(36\) 0 0
\(37\) −2.28312 −0.375342 −0.187671 − 0.982232i \(-0.560094\pi\)
−0.187671 + 0.982232i \(0.560094\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.479055 − 0.829748i 0.0748159 − 0.129585i −0.826190 − 0.563391i \(-0.809496\pi\)
0.901006 + 0.433806i \(0.142830\pi\)
\(42\) 0 0
\(43\) 3.85844 + 6.68302i 0.588407 + 1.01915i 0.994441 + 0.105293i \(0.0335779\pi\)
−0.406035 + 0.913858i \(0.633089\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.14543 + 7.18009i 0.604673 + 1.04732i 0.992103 + 0.125425i \(0.0400296\pi\)
−0.387430 + 0.921899i \(0.626637\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.18479 0.437465 0.218732 − 0.975785i \(-0.429808\pi\)
0.218732 + 0.975785i \(0.429808\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.07785 + 10.5271i −0.791268 + 1.37052i 0.133913 + 0.990993i \(0.457246\pi\)
−0.925182 + 0.379524i \(0.876088\pi\)
\(60\) 0 0
\(61\) 1.75490 + 3.03958i 0.224692 + 0.389178i 0.956227 − 0.292626i \(-0.0945291\pi\)
−0.731535 + 0.681804i \(0.761196\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.0996702 + 0.172634i 0.0123626 + 0.0214126i
\(66\) 0 0
\(67\) −2.87211 + 4.97464i −0.350884 + 0.607749i −0.986405 − 0.164335i \(-0.947452\pi\)
0.635520 + 0.772084i \(0.280786\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0273 −1.66474 −0.832370 − 0.554221i \(-0.813016\pi\)
−0.832370 + 0.554221i \(0.813016\pi\)
\(72\) 0 0
\(73\) 12.4192 1.45356 0.726780 − 0.686871i \(-0.241016\pi\)
0.726780 + 0.686871i \(0.241016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.70574 − 2.95442i 0.194387 − 0.336688i
\(78\) 0 0
\(79\) 5.98293 + 10.3627i 0.673132 + 1.16590i 0.977011 + 0.213188i \(0.0683847\pi\)
−0.303879 + 0.952710i \(0.598282\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.13563 + 3.69902i 0.234416 + 0.406020i 0.959103 − 0.283058i \(-0.0913490\pi\)
−0.724687 + 0.689078i \(0.758016\pi\)
\(84\) 0 0
\(85\) −0.0739780 + 0.128134i −0.00802404 + 0.0138980i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.19934 0.127130 0.0635649 − 0.997978i \(-0.479753\pi\)
0.0635649 + 0.997978i \(0.479753\pi\)
\(90\) 0 0
\(91\) −0.305407 −0.0320154
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.705737 − 1.22237i 0.0724071 − 0.125413i
\(96\) 0 0
\(97\) 7.48932 + 12.9719i 0.760425 + 1.31710i 0.942631 + 0.333835i \(0.108343\pi\)
−0.182206 + 0.983260i \(0.558324\pi\)
\(98\) 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.1009.2 6
3.2 odd 2 504.2.r.c.337.2 yes 6
4.3 odd 2 3024.2.r.h.1009.2 6
9.2 odd 6 504.2.r.c.169.2 ✓ 6
9.4 even 3 4536.2.a.v.1.2 3
9.5 odd 6 4536.2.a.s.1.2 3
9.7 even 3 inner 1512.2.r.c.505.2 6
12.11 even 2 1008.2.r.i.337.2 6
36.7 odd 6 3024.2.r.h.2017.2 6
36.11 even 6 1008.2.r.i.673.2 6
36.23 even 6 9072.2.a.br.1.2 3
36.31 odd 6 9072.2.a.cc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.2 ✓ 6 9.2 odd 6
504.2.r.c.337.2 yes 6 3.2 odd 2
1008.2.r.i.337.2 6 12.11 even 2
1008.2.r.i.673.2 6 36.11 even 6
1512.2.r.c.505.2 6 9.7 even 3 inner
1512.2.r.c.1009.2 6 1.1 even 1 trivial
3024.2.r.h.1009.2 6 4.3 odd 2
3024.2.r.h.2017.2 6 36.7 odd 6
4536.2.a.s.1.2 3 9.5 odd 6
4536.2.a.v.1.2 3 9.4 even 3
9072.2.a.br.1.2 3 36.23 even 6
9072.2.a.cc.1.2 3 36.31 odd 6