↑

Properties

Label 3267.1.l.a.2944.3
Level $3267$
Weight $1$
Character 3267.2944
Analytic conductor $1.630$
Analytic rank $0$
Dimension $16$
Projective image $D_{12}$
CM discriminant -3
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3267,1,Mod(838,3267)] 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("3267.838"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

Embedding invariants

Embedding label 2944.3
Root \(-0.304260 + 0.418778i\) of defining polynomial
Character \(\chi\) \(=\) 3267.2944
Dual form 3267.1.l.a.2296.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.309017 + 0.951057i) q^{4} +(0.492303 - 0.159959i) q^{7} +(-1.13551 + 1.56290i) q^{13} +(-0.809017 + 0.587785i) q^{16} +(-1.34500 - 0.437016i) q^{19} +(-0.309017 + 0.951057i) q^{25} +(0.304260 + 0.418778i) q^{28} +(1.40126 + 1.01807i) q^{31} +1.41421i q^{43} +(-0.592242 + 0.430289i) q^{49} +(-1.83730 - 0.596975i) q^{52} +(-0.831254 - 1.14412i) q^{61} +(-0.809017 - 0.587785i) q^{64} +1.73205 q^{67} +(1.83730 - 0.596975i) q^{73} -1.41421i q^{76} +(1.13551 - 1.56290i) q^{79} +(-0.309017 + 0.951057i) q^{91} +(-0.809017 - 0.587785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(1\)

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 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(3\) 0 0
\(4\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(5\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(6\) 0 0
\(7\) 0.492303 − 0.159959i 0.492303 − 0.159959i −0.0523360 − 0.998630i \(-0.516667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.13551 + 1.56290i −1.13551 + 1.56290i −0.358368 + 0.933580i \(0.616667\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(17\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 0 0
\(19\) −1.34500 − 0.437016i −1.34500 − 0.437016i −0.453990 − 0.891007i \(-0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.304260 + 0.418778i 0.304260 + 0.418778i
\(29\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 1.40126 + 1.01807i 1.40126 + 1.01807i 0.994522 + 0.104528i \(0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(42\) 0 0
\(43\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) 0 0
\(49\) −0.592242 + 0.430289i −0.592242 + 0.430289i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.83730 − 0.596975i −1.83730 − 0.596975i
\(53\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(60\) 0 0
\(61\) −0.831254 − 1.14412i −0.831254 − 1.14412i −0.987688 − 0.156434i \(-0.950000\pi\)
0.156434 − 0.987688i \(-0.450000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.73205 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(72\) 0 0
\(73\) 1.83730 − 0.596975i 1.83730 − 0.596975i 0.838671 − 0.544639i \(-0.183333\pi\)
0.998630 − 0.0523360i \(-0.0166667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) − 1.41421i − 1.41421i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.13551 − 1.56290i 1.13551 − 1.56290i 0.358368 − 0.933580i \(-0.383333\pi\)
0.777146 − 0.629320i \(-0.216667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.809017 − 0.587785i −0.809017 − 0.587785i 0.104528 − 0.994522i \(-0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(98\) 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 3267.1.l.a.2944.3 16
3.2 odd 2 CM 3267.1.l.a.2944.3 16
11.2 odd 10 inner 3267.1.l.a.2998.2 16
11.3 even 5 inner 3267.1.l.a.2296.2 16
11.4 even 5 inner 3267.1.l.a.838.2 16
11.5 even 5 3267.1.c.a.2782.2 ✓ 4
11.6 odd 10 3267.1.c.a.2782.3 yes 4
11.7 odd 10 inner 3267.1.l.a.838.3 16
11.8 odd 10 inner 3267.1.l.a.2296.3 16
11.9 even 5 inner 3267.1.l.a.2998.3 16
11.10 odd 2 inner 3267.1.l.a.2944.2 16
33.2 even 10 inner 3267.1.l.a.2998.2 16
33.5 odd 10 3267.1.c.a.2782.2 ✓ 4
33.8 even 10 inner 3267.1.l.a.2296.3 16
33.14 odd 10 inner 3267.1.l.a.2296.2 16
33.17 even 10 3267.1.c.a.2782.3 yes 4
33.20 odd 10 inner 3267.1.l.a.2998.3 16
33.26 odd 10 inner 3267.1.l.a.838.2 16
33.29 even 10 inner 3267.1.l.a.838.3 16
33.32 even 2 inner 3267.1.l.a.2944.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.1.c.a.2782.2 ✓ 4 11.5 even 5
3267.1.c.a.2782.2 ✓ 4 33.5 odd 10
3267.1.c.a.2782.3 yes 4 11.6 odd 10
3267.1.c.a.2782.3 yes 4 33.17 even 10
3267.1.l.a.838.2 16 11.4 even 5 inner
3267.1.l.a.838.2 16 33.26 odd 10 inner
3267.1.l.a.838.3 16 11.7 odd 10 inner
3267.1.l.a.838.3 16 33.29 even 10 inner
3267.1.l.a.2296.2 16 11.3 even 5 inner
3267.1.l.a.2296.2 16 33.14 odd 10 inner
3267.1.l.a.2296.3 16 11.8 odd 10 inner
3267.1.l.a.2296.3 16 33.8 even 10 inner
3267.1.l.a.2944.2 16 11.10 odd 2 inner
3267.1.l.a.2944.2 16 33.32 even 2 inner
3267.1.l.a.2944.3 16 1.1 even 1 trivial
3267.1.l.a.2944.3 16 3.2 odd 2 CM
3267.1.l.a.2998.2 16 11.2 odd 10 inner
3267.1.l.a.2998.2 16 33.2 even 10 inner
3267.1.l.a.2998.3 16 11.9 even 5 inner
3267.1.l.a.2998.3 16 33.20 odd 10 inner