Properties

Label 2000.1.z.a.1151.2
Level $2000$
Weight $1$
Character 2000.1151
Analytic conductor $0.998$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -4
Inner twists $8$

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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] = [2000,1,Mod(351,2000)] 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("2000.351"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 6])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2000.z (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: \(0.998130025266\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 400)
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.195312500000000.4

Embedding invariants

Embedding label 1151.2
Root \(0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 2000.1151
Dual form 2000.1.z.a.351.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.309017 - 0.951057i) q^{9} +(0.363271 - 1.11803i) q^{13} +(-1.53884 - 1.11803i) q^{17} +(0.500000 - 0.363271i) q^{29} +(-0.587785 + 1.80902i) q^{37} +(0.500000 - 1.53884i) q^{41} +1.00000 q^{49} +(0.951057 - 0.690983i) q^{53} +(0.500000 + 1.53884i) q^{61} +(-0.363271 - 1.11803i) q^{73} +(-0.809017 - 0.587785i) q^{81} +(-0.190983 - 0.587785i) q^{89} +(1.53884 - 1.11803i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9} + 4 q^{29} + 4 q^{41} + 8 q^{49} + 4 q^{61} - 2 q^{81} - 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times\).

\(n\) \(501\) \(751\) \(1377\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{5}\right)\)

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 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(10\) 0 0
\(11\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(12\) 0 0
\(13\) 0.363271 − 1.11803i 0.363271 − 1.11803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.53884 − 1.11803i −1.53884 − 1.11803i −0.951057 − 0.309017i \(-0.900000\pi\)
−0.587785 − 0.809017i \(-0.700000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.500000 − 0.363271i 0.500000 − 0.363271i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 − 1.53884i 0.500000 − 1.53884i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.951057 − 0.690983i 0.951057 − 0.690983i − 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(60\) 0 0
\(61\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0 0
\(73\) −0.363271 − 1.11803i −0.363271 − 1.11803i −0.951057 − 0.309017i \(-0.900000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(82\) 0 0
\(83\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.190983 − 0.587785i −0.190983 − 0.587785i 0.809017 − 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.53884 − 1.11803i 1.53884 − 1.11803i 0.587785 − 0.809017i \(-0.300000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(98\) 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 2000.1.z.a.1151.2 8
4.3 odd 2 CM 2000.1.z.a.1151.2 8
5.2 odd 4 400.1.x.a.319.1 yes 4
5.3 odd 4 2000.1.x.a.1599.1 4
5.4 even 2 inner 2000.1.z.a.1151.1 8
15.2 even 4 3600.1.ct.a.2719.1 4
20.3 even 4 2000.1.x.a.1599.1 4
20.7 even 4 400.1.x.a.319.1 yes 4
20.19 odd 2 inner 2000.1.z.a.1151.1 8
25.3 odd 20 400.1.x.a.79.1 ✓ 4
25.4 even 10 inner 2000.1.z.a.351.1 8
25.21 even 5 inner 2000.1.z.a.351.2 8
25.22 odd 20 2000.1.x.a.399.1 4
40.27 even 4 1600.1.bf.a.319.1 4
40.37 odd 4 1600.1.bf.a.319.1 4
60.47 odd 4 3600.1.ct.a.2719.1 4
75.53 even 20 3600.1.ct.a.1279.1 4
100.3 even 20 400.1.x.a.79.1 ✓ 4
100.47 even 20 2000.1.x.a.399.1 4
100.71 odd 10 inner 2000.1.z.a.351.2 8
100.79 odd 10 inner 2000.1.z.a.351.1 8
200.3 even 20 1600.1.bf.a.1279.1 4
200.53 odd 20 1600.1.bf.a.1279.1 4
300.203 odd 20 3600.1.ct.a.1279.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.1.x.a.79.1 ✓ 4 25.3 odd 20
400.1.x.a.79.1 ✓ 4 100.3 even 20
400.1.x.a.319.1 yes 4 5.2 odd 4
400.1.x.a.319.1 yes 4 20.7 even 4
1600.1.bf.a.319.1 4 40.27 even 4
1600.1.bf.a.319.1 4 40.37 odd 4
1600.1.bf.a.1279.1 4 200.3 even 20
1600.1.bf.a.1279.1 4 200.53 odd 20
2000.1.x.a.399.1 4 25.22 odd 20
2000.1.x.a.399.1 4 100.47 even 20
2000.1.x.a.1599.1 4 5.3 odd 4
2000.1.x.a.1599.1 4 20.3 even 4
2000.1.z.a.351.1 8 25.4 even 10 inner
2000.1.z.a.351.1 8 100.79 odd 10 inner
2000.1.z.a.351.2 8 25.21 even 5 inner
2000.1.z.a.351.2 8 100.71 odd 10 inner
2000.1.z.a.1151.1 8 5.4 even 2 inner
2000.1.z.a.1151.1 8 20.19 odd 2 inner
2000.1.z.a.1151.2 8 1.1 even 1 trivial
2000.1.z.a.1151.2 8 4.3 odd 2 CM
3600.1.ct.a.1279.1 4 75.53 even 20
3600.1.ct.a.1279.1 4 300.203 odd 20
3600.1.ct.a.2719.1 4 15.2 even 4
3600.1.ct.a.2719.1 4 60.47 odd 4