Properties

Label 3600.1.dx.a.863.2
Level $3600$
Weight $1$
Character 3600.863
Analytic conductor $1.797$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3600,1,Mod(287,3600)] 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("3600.287"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 0, 10, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.dx (of order \(20\), degree \(8\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

Embedding invariants

Embedding label 863.2
Root \(-0.987688 - 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 3600.863
Dual form 3600.1.dx.a.1727.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.453990 + 0.891007i) q^{5} +(-1.76007 - 0.278768i) q^{13} +(-1.44168 + 0.734572i) q^{17} +(-0.587785 + 0.809017i) q^{25} +(0.610425 + 1.87869i) q^{29} +(-0.309017 + 1.95106i) q^{37} +(1.04744 + 1.44168i) q^{41} -1.00000i q^{49} +(0.550672 + 0.280582i) q^{53} +(-1.53884 - 1.11803i) q^{61} +(-0.550672 - 1.69480i) q^{65} +(-0.142040 - 0.896802i) q^{73} +(-1.30902 - 0.951057i) q^{85} +(0.253116 + 0.183900i) q^{89} +(0.278768 + 0.142040i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{13} + 4 q^{37} + 4 q^{73} - 12 q^{85} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(e\left(\frac{19}{20}\right)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(12\) 0 0
\(13\) −1.76007 − 0.278768i −1.76007 − 0.278768i −0.809017 − 0.587785i \(-0.800000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.44168 + 0.734572i −1.44168 + 0.734572i −0.987688 − 0.156434i \(-0.950000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(24\) 0 0
\(25\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.610425 + 1.87869i 0.610425 + 1.87869i 0.453990 + 0.891007i \(0.350000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(30\) 0 0
\(31\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.309017 + 1.95106i −0.309017 + 1.95106i 1.00000i \(0.5\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.04744 + 1.44168i 1.04744 + 1.44168i 0.891007 + 0.453990i \(0.150000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.453990 − 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.550672 + 0.280582i 0.550672 + 0.280582i 0.707107 − 0.707107i \(-0.250000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) −1.53884 − 1.11803i −1.53884 − 1.11803i −0.951057 − 0.309017i \(-0.900000\pi\)
−0.587785 − 0.809017i \(-0.700000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.550672 − 1.69480i −0.550672 − 1.69480i
\(66\) 0 0
\(67\) 0 0 −0.453990 − 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(72\) 0 0
\(73\) −0.142040 − 0.896802i −0.142040 − 0.896802i −0.951057 − 0.309017i \(-0.900000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.453990 − 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(84\) 0 0
\(85\) −1.30902 − 0.951057i −1.30902 − 0.951057i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.253116 + 0.183900i 0.253116 + 0.183900i 0.707107 − 0.707107i \(-0.250000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.278768 + 0.142040i 0.278768 + 0.142040i 0.587785 − 0.809017i \(-0.300000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(98\) 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 3600.1.dx.a.863.2 yes 16
3.2 odd 2 inner 3600.1.dx.a.863.1 ✓ 16
4.3 odd 2 CM 3600.1.dx.a.863.2 yes 16
12.11 even 2 inner 3600.1.dx.a.863.1 ✓ 16
25.2 odd 20 inner 3600.1.dx.a.1727.1 yes 16
75.2 even 20 inner 3600.1.dx.a.1727.2 yes 16
100.27 even 20 inner 3600.1.dx.a.1727.1 yes 16
300.227 odd 20 inner 3600.1.dx.a.1727.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3600.1.dx.a.863.1 ✓ 16 3.2 odd 2 inner
3600.1.dx.a.863.1 ✓ 16 12.11 even 2 inner
3600.1.dx.a.863.2 yes 16 1.1 even 1 trivial
3600.1.dx.a.863.2 yes 16 4.3 odd 2 CM
3600.1.dx.a.1727.1 yes 16 25.2 odd 20 inner
3600.1.dx.a.1727.1 yes 16 100.27 even 20 inner
3600.1.dx.a.1727.2 yes 16 75.2 even 20 inner
3600.1.dx.a.1727.2 yes 16 300.227 odd 20 inner