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Properties

Label 2420.1.q.a.1129.1
Level $2420$
Weight $1$
Character 2420.1129
Analytic conductor $1.208$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -11
Inner twists $16$

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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] = [2420,1,Mod(1129,2420)] 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("2420.1129"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 7])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2420.q (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: \(1.20773733057\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.242000.1

Embedding invariants

Embedding label 1129.1
Root \(-0.104528 - 0.994522i\) of defining polynomial
Character \(\chi\) \(=\) 2420.1129
Dual form 2420.1.q.a.1449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.64728 - 0.535233i) q^{3} +(-0.913545 - 0.406737i) q^{5} +(1.61803 + 1.17557i) q^{9} +(1.28716 + 1.15897i) q^{15} +1.73205i q^{23} +(0.669131 + 0.743145i) q^{25} +(-1.01807 - 1.40126i) q^{27} +(-0.809017 - 0.587785i) q^{31} +(1.64728 - 0.535233i) q^{37} +(-1.00000 - 1.73205i) q^{45} +(0.809017 - 0.587785i) q^{49} +(-0.309017 - 0.951057i) q^{59} -1.73205i q^{67} +(0.927051 - 2.85317i) q^{69} +(0.809017 - 0.587785i) q^{71} +(-0.704489 - 1.58231i) q^{75} +(0.309017 + 0.951057i) q^{81} +1.00000 q^{89} +(1.01807 + 1.40126i) q^{93} +(1.01807 - 1.40126i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{5} + 4 q^{9} - 3 q^{15} + q^{25} - 2 q^{31} - 8 q^{45} + 2 q^{49} + 2 q^{59} - 6 q^{69} + 2 q^{71} - 3 q^{75} - 2 q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times\).

\(n\) \(1211\) \(1937\) \(2301\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{7}{10}\right)\)

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
\(3\) −1.64728 − 0.535233i −1.64728 − 0.535233i −0.669131 − 0.743145i \(-0.733333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(4\) 0 0
\(5\) −0.913545 − 0.406737i −0.913545 − 0.406737i
\(6\) 0 0
\(7\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) 0 0
\(9\) 1.61803 + 1.17557i 1.61803 + 1.17557i
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) 1.28716 + 1.15897i 1.28716 + 1.15897i
\(16\) 0 0
\(17\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 0 0
\(19\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(26\) 0 0
\(27\) −1.01807 − 1.40126i −1.01807 − 1.40126i
\(28\) 0 0
\(29\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) −0.809017 − 0.587785i −0.809017 − 0.587785i 0.104528 − 0.994522i \(-0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.64728 − 0.535233i 1.64728 − 0.535233i 0.669131 − 0.743145i \(-0.266667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(38\) 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
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(46\) 0 0
\(47\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(48\) 0 0
\(49\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.309017 − 0.951057i −0.309017 − 0.951057i −0.978148 − 0.207912i \(-0.933333\pi\)
0.669131 − 0.743145i \(-0.266667\pi\)
\(60\) 0 0
\(61\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(68\) 0 0
\(69\) 0.927051 − 2.85317i 0.927051 − 2.85317i
\(70\) 0 0
\(71\) 0.809017 − 0.587785i 0.809017 − 0.587785i −0.104528 − 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(72\) 0 0
\(73\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) 0 0
\(75\) −0.704489 − 1.58231i −0.704489 − 1.58231i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0 0
\(81\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(82\) 0 0
\(83\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.01807 + 1.40126i 1.01807 + 1.40126i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.01807 − 1.40126i 1.01807 − 1.40126i 0.104528 − 0.994522i \(-0.466667\pi\)
0.913545 − 0.406737i \(-0.133333\pi\)
\(98\) 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 2420.1.q.a.1129.1 8
5.4 even 2 inner 2420.1.q.a.1129.2 8
11.2 odd 10 inner 2420.1.q.a.1909.1 8
11.3 even 5 inner 2420.1.q.a.1449.2 8
11.4 even 5 inner 2420.1.q.a.2169.2 8
11.5 even 5 220.1.e.a.109.1 ✓ 2
11.6 odd 10 220.1.e.a.109.1 ✓ 2
11.7 odd 10 inner 2420.1.q.a.2169.2 8
11.8 odd 10 inner 2420.1.q.a.1449.2 8
11.9 even 5 inner 2420.1.q.a.1909.1 8
11.10 odd 2 CM 2420.1.q.a.1129.1 8
33.5 odd 10 1980.1.p.a.109.1 2
33.17 even 10 1980.1.p.a.109.1 2
44.27 odd 10 880.1.i.b.769.2 2
44.39 even 10 880.1.i.b.769.2 2
55.4 even 10 inner 2420.1.q.a.2169.1 8
55.9 even 10 inner 2420.1.q.a.1909.2 8
55.14 even 10 inner 2420.1.q.a.1449.1 8
55.17 even 20 1100.1.f.b.901.1 2
55.19 odd 10 inner 2420.1.q.a.1449.1 8
55.24 odd 10 inner 2420.1.q.a.1909.2 8
55.27 odd 20 1100.1.f.b.901.1 2
55.28 even 20 1100.1.f.b.901.2 2
55.29 odd 10 inner 2420.1.q.a.2169.1 8
55.38 odd 20 1100.1.f.b.901.2 2
55.39 odd 10 220.1.e.a.109.2 yes 2
55.49 even 10 220.1.e.a.109.2 yes 2
55.54 odd 2 inner 2420.1.q.a.1129.2 8
88.5 even 10 3520.1.i.d.769.2 2
88.27 odd 10 3520.1.i.c.769.1 2
88.61 odd 10 3520.1.i.d.769.2 2
88.83 even 10 3520.1.i.c.769.1 2
165.104 odd 10 1980.1.p.a.109.2 2
165.149 even 10 1980.1.p.a.109.2 2
220.39 even 10 880.1.i.b.769.1 2
220.159 odd 10 880.1.i.b.769.1 2
440.149 odd 10 3520.1.i.d.769.1 2
440.259 even 10 3520.1.i.c.769.2 2
440.269 even 10 3520.1.i.d.769.1 2
440.379 odd 10 3520.1.i.c.769.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.1.e.a.109.1 ✓ 2 11.5 even 5
220.1.e.a.109.1 ✓ 2 11.6 odd 10
220.1.e.a.109.2 yes 2 55.39 odd 10
220.1.e.a.109.2 yes 2 55.49 even 10
880.1.i.b.769.1 2 220.39 even 10
880.1.i.b.769.1 2 220.159 odd 10
880.1.i.b.769.2 2 44.27 odd 10
880.1.i.b.769.2 2 44.39 even 10
1100.1.f.b.901.1 2 55.17 even 20
1100.1.f.b.901.1 2 55.27 odd 20
1100.1.f.b.901.2 2 55.28 even 20
1100.1.f.b.901.2 2 55.38 odd 20
1980.1.p.a.109.1 2 33.5 odd 10
1980.1.p.a.109.1 2 33.17 even 10
1980.1.p.a.109.2 2 165.104 odd 10
1980.1.p.a.109.2 2 165.149 even 10
2420.1.q.a.1129.1 8 1.1 even 1 trivial
2420.1.q.a.1129.1 8 11.10 odd 2 CM
2420.1.q.a.1129.2 8 5.4 even 2 inner
2420.1.q.a.1129.2 8 55.54 odd 2 inner
2420.1.q.a.1449.1 8 55.14 even 10 inner
2420.1.q.a.1449.1 8 55.19 odd 10 inner
2420.1.q.a.1449.2 8 11.3 even 5 inner
2420.1.q.a.1449.2 8 11.8 odd 10 inner
2420.1.q.a.1909.1 8 11.2 odd 10 inner
2420.1.q.a.1909.1 8 11.9 even 5 inner
2420.1.q.a.1909.2 8 55.9 even 10 inner
2420.1.q.a.1909.2 8 55.24 odd 10 inner
2420.1.q.a.2169.1 8 55.4 even 10 inner
2420.1.q.a.2169.1 8 55.29 odd 10 inner
2420.1.q.a.2169.2 8 11.4 even 5 inner
2420.1.q.a.2169.2 8 11.7 odd 10 inner
3520.1.i.c.769.1 2 88.27 odd 10
3520.1.i.c.769.1 2 88.83 even 10
3520.1.i.c.769.2 2 440.259 even 10
3520.1.i.c.769.2 2 440.379 odd 10
3520.1.i.d.769.1 2 440.149 odd 10
3520.1.i.d.769.1 2 440.269 even 10
3520.1.i.d.769.2 2 88.5 even 10
3520.1.i.d.769.2 2 88.61 odd 10