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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,3,Mod(593,1296)] 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("1296.593"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,10,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1296.593
Dual form 1296.3.q.j.1025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.59808 + 1.50000i) q^{5} +(2.50000 - 4.33013i) q^{7} +(-12.9904 - 7.50000i) q^{11} +(5.00000 + 8.66025i) q^{13} +18.0000i q^{17} +16.0000 q^{19} +(10.3923 - 6.00000i) q^{23} +(-8.00000 + 13.8564i) q^{25} +(-25.9808 - 15.0000i) q^{29} +(-0.500000 - 0.866025i) q^{31} +15.0000i q^{35} +20.0000 q^{37} +(51.9615 - 30.0000i) q^{41} +(25.0000 - 43.3013i) q^{43} +(-5.19615 - 3.00000i) q^{47} +(12.0000 + 20.7846i) q^{49} -27.0000i q^{53} +45.0000 q^{55} +(25.9808 - 15.0000i) q^{59} +(38.0000 - 65.8179i) q^{61} +(-25.9808 - 15.0000i) q^{65} +(-5.00000 - 8.66025i) q^{67} +90.0000i q^{71} +65.0000 q^{73} +(-64.9519 + 37.5000i) q^{77} +(7.00000 - 12.1244i) q^{79} +(2.59808 + 1.50000i) q^{83} +(-27.0000 - 46.7654i) q^{85} +90.0000i q^{89} +50.0000 q^{91} +(-41.5692 + 24.0000i) q^{95} +(42.5000 - 73.6122i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{7} + 20 q^{13} + 64 q^{19} - 32 q^{25} - 2 q^{31} + 80 q^{37} + 100 q^{43} + 48 q^{49} + 180 q^{55} + 152 q^{61} - 20 q^{67} + 260 q^{73} + 28 q^{79} - 108 q^{85} + 200 q^{91} + 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.59808 + 1.50000i −0.519615 + 0.300000i −0.736777 − 0.676136i \(-0.763653\pi\)
0.217162 + 0.976136i \(0.430320\pi\)
\(6\) 0 0
\(7\) 2.50000 − 4.33013i 0.357143 − 0.618590i −0.630339 − 0.776320i \(-0.717084\pi\)
0.987482 + 0.157730i \(0.0504176\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −12.9904 − 7.50000i −1.18094 − 0.681818i −0.224711 − 0.974425i \(-0.572144\pi\)
−0.956233 + 0.292607i \(0.905477\pi\)
\(12\) 0 0
\(13\) 5.00000 + 8.66025i 0.384615 + 0.666173i 0.991716 − 0.128452i \(-0.0410008\pi\)
−0.607100 + 0.794625i \(0.707667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.0000i 1.05882i 0.848365 + 0.529412i \(0.177587\pi\)
−0.848365 + 0.529412i \(0.822413\pi\)
\(18\) 0 0
\(19\) 16.0000 0.842105 0.421053 − 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 10.3923 − 6.00000i 0.451839 − 0.260870i −0.256767 − 0.966473i \(-0.582657\pi\)
0.708607 + 0.705604i \(0.249324\pi\)
\(24\) 0 0
\(25\) −8.00000 + 13.8564i −0.320000 + 0.554256i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −25.9808 − 15.0000i −0.895888 − 0.517241i −0.0200244 − 0.999799i \(-0.506374\pi\)
−0.875864 + 0.482558i \(0.839708\pi\)
\(30\) 0 0
\(31\) −0.500000 − 0.866025i −0.0161290 − 0.0279363i 0.857848 − 0.513903i \(-0.171801\pi\)
−0.873977 + 0.485967i \(0.838468\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.0000i 0.428571i
\(36\) 0 0
\(37\) 20.0000 0.540541 0.270270 − 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 51.9615 − 30.0000i 1.26735 − 0.731707i 0.292868 − 0.956153i \(-0.405390\pi\)
0.974487 + 0.224446i \(0.0720571\pi\)
\(42\) 0 0
\(43\) 25.0000 − 43.3013i 0.581395 − 1.00701i −0.413919 − 0.910314i \(-0.635840\pi\)
0.995314 − 0.0966925i \(-0.0308264\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.19615 − 3.00000i −0.110556 − 0.0638298i 0.443702 − 0.896174i \(-0.353665\pi\)
−0.554259 + 0.832345i \(0.686998\pi\)
\(48\) 0 0
\(49\) 12.0000 + 20.7846i 0.244898 + 0.424176i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 27.0000i − 0.509434i −0.967016 − 0.254717i \(-0.918018\pi\)
0.967016 − 0.254717i \(-0.0819823\pi\)
\(54\) 0 0
\(55\) 45.0000 0.818182
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 25.9808 − 15.0000i 0.440352 − 0.254237i −0.263395 − 0.964688i \(-0.584842\pi\)
0.703747 + 0.710451i \(0.251509\pi\)
\(60\) 0 0
\(61\) 38.0000 − 65.8179i 0.622951 − 1.07898i −0.365982 − 0.930622i \(-0.619267\pi\)
0.988933 − 0.148361i \(-0.0473997\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25.9808 − 15.0000i −0.399704 − 0.230769i
\(66\) 0 0
\(67\) −5.00000 − 8.66025i −0.0746269 − 0.129258i 0.826297 − 0.563235i \(-0.190443\pi\)
−0.900924 + 0.433977i \(0.857110\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 90.0000i 1.26761i 0.773495 + 0.633803i \(0.218507\pi\)
−0.773495 + 0.633803i \(0.781493\pi\)
\(72\) 0 0
\(73\) 65.0000 0.890411 0.445205 − 0.895428i \(-0.353131\pi\)
0.445205 + 0.895428i \(0.353131\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −64.9519 + 37.5000i −0.843531 + 0.487013i
\(78\) 0 0
\(79\) 7.00000 − 12.1244i 0.0886076 − 0.153473i −0.818315 − 0.574770i \(-0.805092\pi\)
0.906923 + 0.421297i \(0.138425\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.59808 + 1.50000i 0.0313021 + 0.0180723i 0.515569 − 0.856848i \(-0.327580\pi\)
−0.484267 + 0.874920i \(0.660914\pi\)
\(84\) 0 0
\(85\) −27.0000 − 46.7654i −0.317647 − 0.550181i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 0 0
\(91\) 50.0000 0.549451
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −41.5692 + 24.0000i −0.437571 + 0.252632i
\(96\) 0 0
\(97\) 42.5000 − 73.6122i 0.438144 − 0.758888i −0.559402 − 0.828896i \(-0.688969\pi\)
0.997546 + 0.0700082i \(0.0223025\pi\)
\(98\) 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 1296.3.q.j.593.1 4
3.2 odd 2 inner 1296.3.q.j.593.2 4
4.3 odd 2 81.3.d.b.26.1 4
9.2 odd 6 432.3.e.c.161.2 2
9.4 even 3 inner 1296.3.q.j.1025.2 4
9.5 odd 6 inner 1296.3.q.j.1025.1 4
9.7 even 3 432.3.e.c.161.1 2
12.11 even 2 81.3.d.b.26.2 4
36.7 odd 6 27.3.b.b.26.2 yes 2
36.11 even 6 27.3.b.b.26.1 ✓ 2
36.23 even 6 81.3.d.b.53.1 4
36.31 odd 6 81.3.d.b.53.2 4
72.11 even 6 1728.3.e.m.1025.1 2
72.29 odd 6 1728.3.e.g.1025.1 2
72.43 odd 6 1728.3.e.m.1025.2 2
72.61 even 6 1728.3.e.g.1025.2 2
180.7 even 12 675.3.d.a.674.2 2
180.43 even 12 675.3.d.d.674.1 2
180.47 odd 12 675.3.d.d.674.2 2
180.79 odd 6 675.3.c.h.26.1 2
180.83 odd 12 675.3.d.a.674.1 2
180.119 even 6 675.3.c.h.26.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.b.26.1 ✓ 2 36.11 even 6
27.3.b.b.26.2 yes 2 36.7 odd 6
81.3.d.b.26.1 4 4.3 odd 2
81.3.d.b.26.2 4 12.11 even 2
81.3.d.b.53.1 4 36.23 even 6
81.3.d.b.53.2 4 36.31 odd 6
432.3.e.c.161.1 2 9.7 even 3
432.3.e.c.161.2 2 9.2 odd 6
675.3.c.h.26.1 2 180.79 odd 6
675.3.c.h.26.2 2 180.119 even 6
675.3.d.a.674.1 2 180.83 odd 12
675.3.d.a.674.2 2 180.7 even 12
675.3.d.d.674.1 2 180.43 even 12
675.3.d.d.674.2 2 180.47 odd 12
1296.3.q.j.593.1 4 1.1 even 1 trivial
1296.3.q.j.593.2 4 3.2 odd 2 inner
1296.3.q.j.1025.1 4 9.5 odd 6 inner
1296.3.q.j.1025.2 4 9.4 even 3 inner
1728.3.e.g.1025.1 2 72.29 odd 6
1728.3.e.g.1025.2 2 72.61 even 6
1728.3.e.m.1025.1 2 72.11 even 6
1728.3.e.m.1025.2 2 72.43 odd 6