Properties

Label 440.1.bh.a.59.1
Level $440$
Weight $1$
Character 440.59
Analytic conductor $0.220$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -40
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] = [440,1,Mod(59,440)] 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("440.59"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 5, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 440.bh (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.219588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.23425600.1

Embedding invariants

Embedding label 59.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 440.59
Dual form 440.1.bh.a.179.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.809017 - 0.587785i) q^{2} +(0.309017 + 0.951057i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(0.190983 + 0.587785i) q^{7} +(0.309017 - 0.951057i) q^{8} +(-0.809017 - 0.587785i) q^{9} +1.00000 q^{10} +(0.309017 + 0.951057i) q^{11} +(1.30902 + 0.951057i) q^{13} +(0.190983 - 0.587785i) q^{14} +(-0.809017 + 0.587785i) q^{16} +(0.309017 + 0.951057i) q^{18} +(-0.500000 + 1.53884i) q^{19} +(-0.809017 - 0.587785i) q^{20} +(0.309017 - 0.951057i) q^{22} +0.618034 q^{23} +(0.309017 - 0.951057i) q^{25} +(-0.500000 - 1.53884i) q^{26} +(-0.500000 + 0.363271i) q^{28} +1.00000 q^{32} +(-0.500000 - 0.363271i) q^{35} +(0.309017 - 0.951057i) q^{36} +(-0.500000 - 1.53884i) q^{37} +(1.30902 - 0.951057i) q^{38} +(0.309017 + 0.951057i) q^{40} +(0.190983 - 0.587785i) q^{41} +(-0.809017 + 0.587785i) q^{44} +1.00000 q^{45} +(-0.500000 - 0.363271i) q^{46} +(-0.500000 + 1.53884i) q^{47} +(0.500000 - 0.363271i) q^{49} +(-0.809017 + 0.587785i) q^{50} +(-0.500000 + 1.53884i) q^{52} +(-0.500000 - 0.363271i) q^{53} +(-0.809017 - 0.587785i) q^{55} +0.618034 q^{56} +(-0.500000 - 1.53884i) q^{59} +(0.190983 - 0.587785i) q^{63} +(-0.809017 - 0.587785i) q^{64} -1.61803 q^{65} +(0.190983 + 0.587785i) q^{70} +(-0.809017 + 0.587785i) q^{72} +(-0.500000 + 1.53884i) q^{74} -1.61803 q^{76} +(-0.500000 + 0.363271i) q^{77} +(0.309017 - 0.951057i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-0.500000 + 0.363271i) q^{82} +1.00000 q^{88} +0.618034 q^{89} +(-0.809017 - 0.587785i) q^{90} +(-0.309017 + 0.951057i) q^{91} +(0.190983 + 0.587785i) q^{92} +(1.30902 - 0.951057i) q^{94} +(-0.500000 - 1.53884i) q^{95} -0.618034 q^{98} +(0.309017 - 0.951057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{4} - q^{5} + 3 q^{7} - q^{8} - q^{9} + 4 q^{10} - q^{11} + 3 q^{13} + 3 q^{14} - q^{16} - q^{18} - 2 q^{19} - q^{20} - q^{22} - 2 q^{23} - q^{25} - 2 q^{26} - 2 q^{28} + 4 q^{32}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(3\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(4\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(5\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(6\) 0 0
\(7\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(8\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(9\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(10\) 1.00000 1.00000
\(11\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(12\) 0 0
\(13\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(14\) 0.190983 − 0.587785i 0.190983 − 0.587785i
\(15\) 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 \(0.800000\pi\)
\(18\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(19\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(21\) 0 0
\(22\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(23\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(26\) −0.500000 − 1.53884i −0.500000 − 1.53884i
\(27\) 0 0
\(28\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(29\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(36\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(37\) −0.500000 − 1.53884i −0.500000 − 1.53884i −0.809017 − 0.587785i \(-0.800000\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(38\) 1.30902 − 0.951057i 1.30902 − 0.951057i
\(39\) 0 0
\(40\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(41\) 0.190983 − 0.587785i 0.190983 − 0.587785i −0.809017 − 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(45\) 1.00000 1.00000
\(46\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(47\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) 0.500000 − 0.363271i 0.500000 − 0.363271i
\(50\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(51\) 0 0
\(52\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(53\) −0.500000 − 0.363271i −0.500000 − 0.363271i 0.309017 − 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(54\) 0 0
\(55\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(56\) 0.618034 0.618034
\(57\) 0 0
\(58\) 0 0
\(59\) −0.500000 − 1.53884i −0.500000 − 1.53884i −0.809017 − 0.587785i \(-0.800000\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(60\) 0 0
\(61\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) 0.190983 − 0.587785i 0.190983 − 0.587785i
\(64\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(65\) −1.61803 −1.61803
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(71\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(73\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(75\) 0 0
\(76\) −1.61803 −1.61803
\(77\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(78\) 0 0
\(79\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(81\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(82\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(83\) 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
\(88\) 1.00000 1.00000
\(89\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(91\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(92\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(93\) 0 0
\(94\) 1.30902 − 0.951057i 1.30902 − 0.951057i
\(95\) −0.500000 − 1.53884i −0.500000 − 1.53884i
\(96\) 0 0
\(97\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(98\) −0.618034 −0.618034
\(99\) 0.309017 − 0.951057i 0.309017 − 0.951057i
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 440.1.bh.a.59.1 ✓ 4
3.2 odd 2 3960.1.dh.b.1819.1 4
4.3 odd 2 1760.1.cn.a.719.1 4
5.2 odd 4 2200.1.cl.c.851.2 8
5.3 odd 4 2200.1.cl.c.851.1 8
5.4 even 2 440.1.bh.b.59.1 yes 4
8.3 odd 2 440.1.bh.b.59.1 yes 4
8.5 even 2 1760.1.cn.b.719.1 4
11.3 even 5 inner 440.1.bh.a.179.1 yes 4
15.14 odd 2 3960.1.dh.a.1819.1 4
20.19 odd 2 1760.1.cn.b.719.1 4
24.11 even 2 3960.1.dh.a.1819.1 4
33.14 odd 10 3960.1.dh.b.3259.1 4
40.3 even 4 2200.1.cl.c.851.2 8
40.19 odd 2 CM 440.1.bh.a.59.1 ✓ 4
40.27 even 4 2200.1.cl.c.851.1 8
40.29 even 2 1760.1.cn.a.719.1 4
44.3 odd 10 1760.1.cn.a.399.1 4
55.3 odd 20 2200.1.cl.c.1851.2 8
55.14 even 10 440.1.bh.b.179.1 yes 4
55.47 odd 20 2200.1.cl.c.1851.1 8
88.3 odd 10 440.1.bh.b.179.1 yes 4
88.69 even 10 1760.1.cn.b.399.1 4
120.59 even 2 3960.1.dh.b.1819.1 4
165.14 odd 10 3960.1.dh.a.3259.1 4
220.179 odd 10 1760.1.cn.b.399.1 4
264.179 even 10 3960.1.dh.a.3259.1 4
440.3 even 20 2200.1.cl.c.1851.1 8
440.69 even 10 1760.1.cn.a.399.1 4
440.179 odd 10 inner 440.1.bh.a.179.1 yes 4
440.267 even 20 2200.1.cl.c.1851.2 8
1320.179 even 10 3960.1.dh.b.3259.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.1.bh.a.59.1 ✓ 4 1.1 even 1 trivial
440.1.bh.a.59.1 ✓ 4 40.19 odd 2 CM
440.1.bh.a.179.1 yes 4 11.3 even 5 inner
440.1.bh.a.179.1 yes 4 440.179 odd 10 inner
440.1.bh.b.59.1 yes 4 5.4 even 2
440.1.bh.b.59.1 yes 4 8.3 odd 2
440.1.bh.b.179.1 yes 4 55.14 even 10
440.1.bh.b.179.1 yes 4 88.3 odd 10
1760.1.cn.a.399.1 4 44.3 odd 10
1760.1.cn.a.399.1 4 440.69 even 10
1760.1.cn.a.719.1 4 4.3 odd 2
1760.1.cn.a.719.1 4 40.29 even 2
1760.1.cn.b.399.1 4 88.69 even 10
1760.1.cn.b.399.1 4 220.179 odd 10
1760.1.cn.b.719.1 4 8.5 even 2
1760.1.cn.b.719.1 4 20.19 odd 2
2200.1.cl.c.851.1 8 5.3 odd 4
2200.1.cl.c.851.1 8 40.27 even 4
2200.1.cl.c.851.2 8 5.2 odd 4
2200.1.cl.c.851.2 8 40.3 even 4
2200.1.cl.c.1851.1 8 55.47 odd 20
2200.1.cl.c.1851.1 8 440.3 even 20
2200.1.cl.c.1851.2 8 55.3 odd 20
2200.1.cl.c.1851.2 8 440.267 even 20
3960.1.dh.a.1819.1 4 15.14 odd 2
3960.1.dh.a.1819.1 4 24.11 even 2
3960.1.dh.a.3259.1 4 165.14 odd 10
3960.1.dh.a.3259.1 4 264.179 even 10
3960.1.dh.b.1819.1 4 3.2 odd 2
3960.1.dh.b.1819.1 4 120.59 even 2
3960.1.dh.b.3259.1 4 33.14 odd 10
3960.1.dh.b.3259.1 4 1320.179 even 10