Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,2,Mod(431,1296)] 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("1296.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0] 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: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 863.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1296.863
Dual form 1296.2.s.h.431.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.67423 - 2.12132i) q^{5} +(-2.00000 + 3.46410i) q^{13} -4.24264i q^{17} +(6.50000 + 11.2583i) q^{25} +(-3.67423 + 2.12132i) q^{29} +2.00000 q^{37} +(11.0227 + 6.36396i) q^{41} +(-3.50000 + 6.06218i) q^{49} +12.7279i q^{53} +(5.00000 + 8.66025i) q^{61} +(14.6969 - 8.48528i) q^{65} +16.0000 q^{73} +(-9.00000 + 15.5885i) q^{85} +4.24264i q^{89} +(4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{13} + 26 q^{25} + 8 q^{37} - 14 q^{49} + 20 q^{61} + 64 q^{73} - 36 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) −3.67423 − 2.12132i −1.64317 − 0.948683i −0.979698 − 0.200480i \(-0.935750\pi\)
−0.663470 − 0.748203i \(-0.730917\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 6.50000 + 11.2583i 1.30000 + 2.25167i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.67423 + 2.12132i −0.682288 + 0.393919i −0.800717 − 0.599043i \(-0.795548\pi\)
0.118428 + 0.992963i \(0.462214\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0227 + 6.36396i 1.72146 + 0.993884i 0.915944 + 0.401305i \(0.131443\pi\)
0.805513 + 0.592578i \(0.201890\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.7279i 1.74831i 0.485643 + 0.874157i \(0.338586\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.6969 − 8.48528i 1.82293 − 1.05247i
\(66\) 0 0
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) −9.00000 + 15.5885i −0.976187 + 1.69081i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 + 6.92820i 0.406138 + 0.703452i 0.994453 − 0.105180i \(-0.0335417\pi\)
−0.588315 + 0.808632i \(0.700208\pi\)
\(98\) 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 1296.2.s.h.863.1 4
3.2 odd 2 inner 1296.2.s.h.863.2 4
4.3 odd 2 CM 1296.2.s.h.863.1 4
9.2 odd 6 inner 1296.2.s.h.431.1 4
9.4 even 3 144.2.c.a.143.2 yes 2
9.5 odd 6 144.2.c.a.143.1 ✓ 2
9.7 even 3 inner 1296.2.s.h.431.2 4
12.11 even 2 inner 1296.2.s.h.863.2 4
36.7 odd 6 inner 1296.2.s.h.431.2 4
36.11 even 6 inner 1296.2.s.h.431.1 4
36.23 even 6 144.2.c.a.143.1 ✓ 2
36.31 odd 6 144.2.c.a.143.2 yes 2
45.4 even 6 3600.2.h.b.1151.2 2
45.13 odd 12 3600.2.o.a.3599.3 4
45.14 odd 6 3600.2.h.b.1151.1 2
45.22 odd 12 3600.2.o.a.3599.2 4
45.23 even 12 3600.2.o.a.3599.4 4
45.32 even 12 3600.2.o.a.3599.1 4
63.13 odd 6 7056.2.h.b.4607.1 2
63.41 even 6 7056.2.h.b.4607.2 2
72.5 odd 6 576.2.c.a.575.2 2
72.13 even 6 576.2.c.a.575.1 2
72.59 even 6 576.2.c.a.575.2 2
72.67 odd 6 576.2.c.a.575.1 2
144.5 odd 12 2304.2.f.f.1151.3 4
144.13 even 12 2304.2.f.f.1151.4 4
144.59 even 12 2304.2.f.f.1151.3 4
144.67 odd 12 2304.2.f.f.1151.4 4
144.77 odd 12 2304.2.f.f.1151.2 4
144.85 even 12 2304.2.f.f.1151.1 4
144.131 even 12 2304.2.f.f.1151.2 4
144.139 odd 12 2304.2.f.f.1151.1 4
180.23 odd 12 3600.2.o.a.3599.4 4
180.59 even 6 3600.2.h.b.1151.1 2
180.67 even 12 3600.2.o.a.3599.2 4
180.103 even 12 3600.2.o.a.3599.3 4
180.139 odd 6 3600.2.h.b.1151.2 2
180.167 odd 12 3600.2.o.a.3599.1 4
252.139 even 6 7056.2.h.b.4607.1 2
252.167 odd 6 7056.2.h.b.4607.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.c.a.143.1 ✓ 2 9.5 odd 6
144.2.c.a.143.1 ✓ 2 36.23 even 6
144.2.c.a.143.2 yes 2 9.4 even 3
144.2.c.a.143.2 yes 2 36.31 odd 6
576.2.c.a.575.1 2 72.13 even 6
576.2.c.a.575.1 2 72.67 odd 6
576.2.c.a.575.2 2 72.5 odd 6
576.2.c.a.575.2 2 72.59 even 6
1296.2.s.h.431.1 4 9.2 odd 6 inner
1296.2.s.h.431.1 4 36.11 even 6 inner
1296.2.s.h.431.2 4 9.7 even 3 inner
1296.2.s.h.431.2 4 36.7 odd 6 inner
1296.2.s.h.863.1 4 1.1 even 1 trivial
1296.2.s.h.863.1 4 4.3 odd 2 CM
1296.2.s.h.863.2 4 3.2 odd 2 inner
1296.2.s.h.863.2 4 12.11 even 2 inner
2304.2.f.f.1151.1 4 144.85 even 12
2304.2.f.f.1151.1 4 144.139 odd 12
2304.2.f.f.1151.2 4 144.77 odd 12
2304.2.f.f.1151.2 4 144.131 even 12
2304.2.f.f.1151.3 4 144.5 odd 12
2304.2.f.f.1151.3 4 144.59 even 12
2304.2.f.f.1151.4 4 144.13 even 12
2304.2.f.f.1151.4 4 144.67 odd 12
3600.2.h.b.1151.1 2 45.14 odd 6
3600.2.h.b.1151.1 2 180.59 even 6
3600.2.h.b.1151.2 2 45.4 even 6
3600.2.h.b.1151.2 2 180.139 odd 6
3600.2.o.a.3599.1 4 45.32 even 12
3600.2.o.a.3599.1 4 180.167 odd 12
3600.2.o.a.3599.2 4 45.22 odd 12
3600.2.o.a.3599.2 4 180.67 even 12
3600.2.o.a.3599.3 4 45.13 odd 12
3600.2.o.a.3599.3 4 180.103 even 12
3600.2.o.a.3599.4 4 45.23 even 12
3600.2.o.a.3599.4 4 180.23 odd 12
7056.2.h.b.4607.1 2 63.13 odd 6
7056.2.h.b.4607.1 2 252.139 even 6
7056.2.h.b.4607.2 2 63.41 even 6
7056.2.h.b.4607.2 2 252.167 odd 6