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Properties

Label 5040.2.k.d.1889.1
Level $5040$
Weight $2$
Character 5040.1889
Analytic conductor $40.245$
Analytic rank $0$
Dimension $4$
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] = [5040,2,Mod(1889,5040)] 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("5040.1889"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.1
Root \(-1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1889
Dual form 5040.2.k.d.1889.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.23607i q^{5} +(-1.58114 + 2.12132i) q^{7} -1.41421i q^{11} +3.16228 q^{13} -4.47214i q^{17} +6.00000 q^{23} -5.00000 q^{25} -2.82843i q^{29} +(4.74342 + 3.53553i) q^{35} +4.24264i q^{37} +9.48683 q^{41} +8.48528i q^{43} +4.47214i q^{47} +(-2.00000 - 6.70820i) q^{49} -6.00000 q^{53} -3.16228 q^{55} -9.48683 q^{59} -13.4164i q^{61} -7.07107i q^{65} -5.65685i q^{71} -6.32456 q^{73} +(3.00000 + 2.23607i) q^{77} +4.00000 q^{79} -8.94427i q^{83} -10.0000 q^{85} -9.48683 q^{89} +(-5.00000 + 6.70820i) q^{91} +12.6491 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{23} - 20 q^{25} - 8 q^{49} - 24 q^{53} + 12 q^{77} + 16 q^{79} - 40 q^{85} - 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) − 2.23607i − 1.00000i
\(6\) 0 0
\(7\) −1.58114 + 2.12132i −0.597614 + 0.801784i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.41421i − 0.426401i −0.977008 − 0.213201i \(-0.931611\pi\)
0.977008 − 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 3.16228 0.877058 0.438529 − 0.898717i \(-0.355500\pi\)
0.438529 + 0.898717i \(0.355500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.47214i − 1.08465i −0.840168 − 0.542326i \(-0.817544\pi\)
0.840168 − 0.542326i \(-0.182456\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.82843i − 0.525226i −0.964901 − 0.262613i \(-0.915416\pi\)
0.964901 − 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.74342 + 3.53553i 0.801784 + 0.597614i
\(36\) 0 0
\(37\) 4.24264i 0.697486i 0.937218 + 0.348743i \(0.113391\pi\)
−0.937218 + 0.348743i \(0.886609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.48683 1.48159 0.740797 − 0.671729i \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.47214i 0.652328i 0.945313 + 0.326164i \(0.105756\pi\)
−0.945313 + 0.326164i \(0.894244\pi\)
\(48\) 0 0
\(49\) −2.00000 − 6.70820i −0.285714 − 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −3.16228 −0.426401
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.48683 −1.23508 −0.617540 − 0.786539i \(-0.711871\pi\)
−0.617540 + 0.786539i \(0.711871\pi\)
\(60\) 0 0
\(61\) − 13.4164i − 1.71780i −0.512148 − 0.858898i \(-0.671150\pi\)
0.512148 − 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 7.07107i − 0.877058i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.65685i − 0.671345i −0.941979 − 0.335673i \(-0.891036\pi\)
0.941979 − 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) −6.32456 −0.740233 −0.370117 − 0.928985i \(-0.620682\pi\)
−0.370117 + 0.928985i \(0.620682\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 + 2.23607i 0.341882 + 0.254824i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8.94427i − 0.981761i −0.871227 − 0.490881i \(-0.836675\pi\)
0.871227 − 0.490881i \(-0.163325\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.48683 −1.00560 −0.502801 − 0.864402i \(-0.667697\pi\)
−0.502801 + 0.864402i \(0.667697\pi\)
\(90\) 0 0
\(91\) −5.00000 + 6.70820i −0.524142 + 0.703211i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.6491 1.28432 0.642161 − 0.766570i \(-0.278038\pi\)
0.642161 + 0.766570i \(0.278038\pi\)
\(98\) 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 5040.2.k.d.1889.1 4
3.2 odd 2 5040.2.k.a.1889.3 4
4.3 odd 2 630.2.d.a.629.2 yes 4
5.4 even 2 5040.2.k.a.1889.4 4
7.6 odd 2 inner 5040.2.k.d.1889.4 4
12.11 even 2 630.2.d.d.629.4 yes 4
15.14 odd 2 inner 5040.2.k.d.1889.2 4
20.3 even 4 3150.2.b.c.251.5 8
20.7 even 4 3150.2.b.c.251.4 8
20.19 odd 2 630.2.d.d.629.3 yes 4
21.20 even 2 5040.2.k.a.1889.2 4
28.27 even 2 630.2.d.a.629.3 yes 4
35.34 odd 2 5040.2.k.a.1889.1 4
60.23 odd 4 3150.2.b.c.251.1 8
60.47 odd 4 3150.2.b.c.251.8 8
60.59 even 2 630.2.d.a.629.1 ✓ 4
84.83 odd 2 630.2.d.d.629.1 yes 4
105.104 even 2 inner 5040.2.k.d.1889.3 4
140.27 odd 4 3150.2.b.c.251.3 8
140.83 odd 4 3150.2.b.c.251.6 8
140.139 even 2 630.2.d.d.629.2 yes 4
420.83 even 4 3150.2.b.c.251.2 8
420.167 even 4 3150.2.b.c.251.7 8
420.419 odd 2 630.2.d.a.629.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.d.a.629.1 ✓ 4 60.59 even 2
630.2.d.a.629.2 yes 4 4.3 odd 2
630.2.d.a.629.3 yes 4 28.27 even 2
630.2.d.a.629.4 yes 4 420.419 odd 2
630.2.d.d.629.1 yes 4 84.83 odd 2
630.2.d.d.629.2 yes 4 140.139 even 2
630.2.d.d.629.3 yes 4 20.19 odd 2
630.2.d.d.629.4 yes 4 12.11 even 2
3150.2.b.c.251.1 8 60.23 odd 4
3150.2.b.c.251.2 8 420.83 even 4
3150.2.b.c.251.3 8 140.27 odd 4
3150.2.b.c.251.4 8 20.7 even 4
3150.2.b.c.251.5 8 20.3 even 4
3150.2.b.c.251.6 8 140.83 odd 4
3150.2.b.c.251.7 8 420.167 even 4
3150.2.b.c.251.8 8 60.47 odd 4
5040.2.k.a.1889.1 4 35.34 odd 2
5040.2.k.a.1889.2 4 21.20 even 2
5040.2.k.a.1889.3 4 3.2 odd 2
5040.2.k.a.1889.4 4 5.4 even 2
5040.2.k.d.1889.1 4 1.1 even 1 trivial
5040.2.k.d.1889.2 4 15.14 odd 2 inner
5040.2.k.d.1889.3 4 105.104 even 2 inner
5040.2.k.d.1889.4 4 7.6 odd 2 inner