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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 = [2,0,0,0,0,0,-6] 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: \(1\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 431.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.431
Dual form 1296.2.s.b.863.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{13} -3.46410i q^{19} +(-2.50000 + 4.33013i) q^{25} +(-9.00000 + 5.19615i) q^{31} -10.0000 q^{37} +(-9.00000 - 5.19615i) q^{43} +(2.50000 + 4.33013i) q^{49} +(-7.00000 + 12.1244i) q^{61} +(-3.00000 + 1.73205i) q^{67} +10.0000 q^{73} +(-15.0000 - 8.66025i) q^{79} -6.92820i q^{91} +(7.00000 - 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7} + 2 q^{13} - 5 q^{25} - 18 q^{31} - 20 q^{37} - 18 q^{43} + 5 q^{49} - 14 q^{61} - 6 q^{67} + 20 q^{73} - 30 q^{79} + 14 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −3.00000 − 1.73205i −1.13389 − 0.654654i −0.188982 − 0.981981i \(-0.560519\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 − 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 − 0.397360i \(-0.869927\pi\)
0.917663 − 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −9.00000 + 5.19615i −1.61645 + 0.933257i −0.628619 + 0.777714i \(0.716379\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −9.00000 − 5.19615i −1.37249 − 0.792406i −0.381246 − 0.924473i \(-0.624505\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 2.50000 + 4.33013i 0.357143 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 + 1.73205i −0.366508 + 0.211604i −0.671932 − 0.740613i \(-0.734535\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.0000 − 8.66025i −1.68763 − 0.974355i −0.956325 − 0.292306i \(-0.905577\pi\)
−0.731307 − 0.682048i \(-0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) − 6.92820i − 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 − 12.1244i 0.710742 − 1.23104i −0.253837 − 0.967247i \(-0.581693\pi\)
0.964579 − 0.263795i \(-0.0849741\pi\)
\(98\) 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.b.431.1 2
3.2 odd 2 CM 1296.2.s.b.431.1 2
4.3 odd 2 1296.2.s.e.431.1 2
9.2 odd 6 48.2.c.a.47.1 ✓ 2
9.4 even 3 1296.2.s.e.863.1 2
9.5 odd 6 1296.2.s.e.863.1 2
9.7 even 3 48.2.c.a.47.1 ✓ 2
12.11 even 2 1296.2.s.e.431.1 2
36.7 odd 6 48.2.c.a.47.2 yes 2
36.11 even 6 48.2.c.a.47.2 yes 2
36.23 even 6 inner 1296.2.s.b.863.1 2
36.31 odd 6 inner 1296.2.s.b.863.1 2
45.2 even 12 1200.2.o.i.1199.2 4
45.7 odd 12 1200.2.o.i.1199.2 4
45.29 odd 6 1200.2.h.e.1151.2 2
45.34 even 6 1200.2.h.e.1151.2 2
45.38 even 12 1200.2.o.i.1199.3 4
45.43 odd 12 1200.2.o.i.1199.3 4
63.20 even 6 2352.2.h.c.2255.2 2
63.34 odd 6 2352.2.h.c.2255.2 2
72.11 even 6 192.2.c.a.191.1 2
72.29 odd 6 192.2.c.a.191.2 2
72.43 odd 6 192.2.c.a.191.1 2
72.61 even 6 192.2.c.a.191.2 2
144.11 even 12 768.2.f.d.383.4 4
144.29 odd 12 768.2.f.d.383.3 4
144.43 odd 12 768.2.f.d.383.4 4
144.61 even 12 768.2.f.d.383.3 4
144.83 even 12 768.2.f.d.383.2 4
144.101 odd 12 768.2.f.d.383.1 4
144.115 odd 12 768.2.f.d.383.2 4
144.133 even 12 768.2.f.d.383.1 4
180.7 even 12 1200.2.o.i.1199.4 4
180.43 even 12 1200.2.o.i.1199.1 4
180.47 odd 12 1200.2.o.i.1199.4 4
180.79 odd 6 1200.2.h.e.1151.1 2
180.83 odd 12 1200.2.o.i.1199.1 4
180.119 even 6 1200.2.h.e.1151.1 2
252.83 odd 6 2352.2.h.c.2255.1 2
252.223 even 6 2352.2.h.c.2255.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.c.a.47.1 ✓ 2 9.2 odd 6
48.2.c.a.47.1 ✓ 2 9.7 even 3
48.2.c.a.47.2 yes 2 36.7 odd 6
48.2.c.a.47.2 yes 2 36.11 even 6
192.2.c.a.191.1 2 72.11 even 6
192.2.c.a.191.1 2 72.43 odd 6
192.2.c.a.191.2 2 72.29 odd 6
192.2.c.a.191.2 2 72.61 even 6
768.2.f.d.383.1 4 144.101 odd 12
768.2.f.d.383.1 4 144.133 even 12
768.2.f.d.383.2 4 144.83 even 12
768.2.f.d.383.2 4 144.115 odd 12
768.2.f.d.383.3 4 144.29 odd 12
768.2.f.d.383.3 4 144.61 even 12
768.2.f.d.383.4 4 144.11 even 12
768.2.f.d.383.4 4 144.43 odd 12
1200.2.h.e.1151.1 2 180.79 odd 6
1200.2.h.e.1151.1 2 180.119 even 6
1200.2.h.e.1151.2 2 45.29 odd 6
1200.2.h.e.1151.2 2 45.34 even 6
1200.2.o.i.1199.1 4 180.43 even 12
1200.2.o.i.1199.1 4 180.83 odd 12
1200.2.o.i.1199.2 4 45.2 even 12
1200.2.o.i.1199.2 4 45.7 odd 12
1200.2.o.i.1199.3 4 45.38 even 12
1200.2.o.i.1199.3 4 45.43 odd 12
1200.2.o.i.1199.4 4 180.7 even 12
1200.2.o.i.1199.4 4 180.47 odd 12
1296.2.s.b.431.1 2 1.1 even 1 trivial
1296.2.s.b.431.1 2 3.2 odd 2 CM
1296.2.s.b.863.1 2 36.23 even 6 inner
1296.2.s.b.863.1 2 36.31 odd 6 inner
1296.2.s.e.431.1 2 4.3 odd 2
1296.2.s.e.431.1 2 12.11 even 2
1296.2.s.e.863.1 2 9.4 even 3
1296.2.s.e.863.1 2 9.5 odd 6
2352.2.h.c.2255.1 2 252.83 odd 6
2352.2.h.c.2255.1 2 252.223 even 6
2352.2.h.c.2255.2 2 63.20 even 6
2352.2.h.c.2255.2 2 63.34 odd 6