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Properties

Label 1008.1.dd.a.79.2
Level $1008$
Weight $1$
Character 1008.79
Analytic conductor $0.503$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,1,Mod(79,1008)] 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("1008.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.dd (of order \(6\), degree \(2\), 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: \(0.503057532734\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.63504.1

Embedding invariants

Embedding label 79.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1008.79
Dual form 1008.1.dd.a.319.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} -1.00000 q^{5} +(-0.866025 + 0.500000i) q^{7} +(0.500000 + 0.866025i) q^{9} +1.00000i q^{11} +(0.500000 + 0.866025i) q^{13} +(-0.866025 - 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{17} +(0.866025 + 0.500000i) q^{19} -1.00000 q^{21} +1.00000i q^{23} +1.00000i q^{27} +(0.500000 - 0.866025i) q^{29} +(-0.500000 + 0.866025i) q^{33} +(0.866025 - 0.500000i) q^{35} +(0.500000 - 0.866025i) q^{37} +1.00000i q^{39} +(-0.500000 - 0.866025i) q^{41} +(0.866025 + 0.500000i) q^{43} +(-0.500000 - 0.866025i) q^{45} +(0.500000 - 0.866025i) q^{49} -1.00000i q^{51} +(0.500000 + 0.866025i) q^{53} -1.00000i q^{55} +(0.500000 + 0.866025i) q^{57} +(-1.73205 - 1.00000i) q^{59} +(-0.866025 - 0.500000i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(-0.500000 + 0.866025i) q^{69} -2.00000i q^{71} +(-0.500000 - 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{77} +(-0.500000 + 0.866025i) q^{81} +(0.866025 + 0.500000i) q^{83} +(0.500000 + 0.866025i) q^{85} +(0.866025 - 0.500000i) q^{87} +(-0.500000 + 0.866025i) q^{89} +(-0.866025 - 0.500000i) q^{91} +(-0.866025 - 0.500000i) q^{95} +(0.500000 - 0.866025i) q^{97} +(-0.866025 + 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 2 q^{9} + 2 q^{13} - 2 q^{17} - 4 q^{21} + 2 q^{29} - 2 q^{33} + 2 q^{37} - 2 q^{41} - 2 q^{45} + 2 q^{49} + 2 q^{53} + 2 q^{57} - 2 q^{65} - 2 q^{69} - 2 q^{73} - 2 q^{77} - 2 q^{81} + 2 q^{85}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(16\) 0 0
\(17\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(34\) 0 0
\(35\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(36\) 0 0
\(37\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(38\) 0 0
\(39\) 1.00000i 1.00000i
\(40\) 0 0
\(41\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) − 1.00000i − 1.00000i
\(52\) 0 0
\(53\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) − 1.00000i − 1.00000i
\(56\) 0 0
\(57\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(58\) 0 0
\(59\) −1.73205 − 1.00000i −1.73205 − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 − 0.500000i \(-0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(64\) 0 0
\(65\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(66\) 0 0
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(70\) 0 0
\(71\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(72\) 0 0
\(73\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(86\) 0 0
\(87\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(88\) 0 0
\(89\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(96\) 0 0
\(97\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(98\) 0 0
\(99\) −0.866025 + 0.500000i −0.866025 + 0.500000i
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 1008.1.dd.a.79.2 yes 4
3.2 odd 2 3024.1.dd.a.1423.1 4
4.3 odd 2 inner 1008.1.dd.a.79.1 yes 4
7.4 even 3 1008.1.bw.a.655.1 ✓ 4
9.4 even 3 1008.1.bw.a.751.1 yes 4
9.5 odd 6 3024.1.bw.a.415.1 4
12.11 even 2 3024.1.dd.a.1423.2 4
21.11 odd 6 3024.1.bw.a.991.1 4
28.11 odd 6 1008.1.bw.a.655.2 yes 4
36.23 even 6 3024.1.bw.a.415.2 4
36.31 odd 6 1008.1.bw.a.751.2 yes 4
63.4 even 3 inner 1008.1.dd.a.319.1 yes 4
63.32 odd 6 3024.1.dd.a.3007.2 4
84.11 even 6 3024.1.bw.a.991.2 4
252.67 odd 6 inner 1008.1.dd.a.319.2 yes 4
252.95 even 6 3024.1.dd.a.3007.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.1.bw.a.655.1 ✓ 4 7.4 even 3
1008.1.bw.a.655.2 yes 4 28.11 odd 6
1008.1.bw.a.751.1 yes 4 9.4 even 3
1008.1.bw.a.751.2 yes 4 36.31 odd 6
1008.1.dd.a.79.1 yes 4 4.3 odd 2 inner
1008.1.dd.a.79.2 yes 4 1.1 even 1 trivial
1008.1.dd.a.319.1 yes 4 63.4 even 3 inner
1008.1.dd.a.319.2 yes 4 252.67 odd 6 inner
3024.1.bw.a.415.1 4 9.5 odd 6
3024.1.bw.a.415.2 4 36.23 even 6
3024.1.bw.a.991.1 4 21.11 odd 6
3024.1.bw.a.991.2 4 84.11 even 6
3024.1.dd.a.1423.1 4 3.2 odd 2
3024.1.dd.a.1423.2 4 12.11 even 2
3024.1.dd.a.3007.1 4 252.95 even 6
3024.1.dd.a.3007.2 4 63.32 odd 6