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Properties

Label 5580.2.a.j.1.3
Level $5580$
Weight $2$
Character 5580.1
Self dual yes
Analytic conductor $44.557$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [5580,2,Mod(1,5580)] 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("5580.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5580.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-2,0,0,0,-2,0,2,0,0,0,-10,0,0,0,0,0,-2] 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: yes
Analytic conductor: \(44.5565243279\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 5580.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.34889 q^{7} -4.69779 q^{11} -2.38350 q^{13} +1.21509 q^{17} +1.03461 q^{19} -6.24970 q^{23} +1.00000 q^{25} +5.59859 q^{29} +1.00000 q^{31} +1.34889 q^{35} +5.41811 q^{37} -1.30221 q^{41} +5.73240 q^{43} -3.48270 q^{47} -5.18048 q^{49} -2.51730 q^{53} -4.69779 q^{55} -7.86620 q^{59} -5.46479 q^{61} -2.38350 q^{65} -1.41811 q^{67} -3.79698 q^{71} -1.41811 q^{73} -6.33682 q^{77} -1.81952 q^{79} -13.2151 q^{83} +1.21509 q^{85} +2.20302 q^{89} -3.21509 q^{91} +1.03461 q^{95} +6.69779 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} - 2 q^{23} + 3 q^{25} - 6 q^{29} + 3 q^{31} - 2 q^{35} + 4 q^{37} - 16 q^{41} + 2 q^{43} - 12 q^{47} - 5 q^{49} - 6 q^{53} - 2 q^{55} - 16 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.34889 0.509834 0.254917 − 0.966963i \(-0.417952\pi\)
0.254917 + 0.966963i \(0.417952\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.69779 −1.41644 −0.708218 − 0.705994i \(-0.750501\pi\)
−0.708218 + 0.705994i \(0.750501\pi\)
\(12\) 0 0
\(13\) −2.38350 −0.661065 −0.330532 − 0.943795i \(-0.607228\pi\)
−0.330532 + 0.943795i \(0.607228\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.21509 0.294703 0.147352 − 0.989084i \(-0.452925\pi\)
0.147352 + 0.989084i \(0.452925\pi\)
\(18\) 0 0
\(19\) 1.03461 0.237355 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.24970 −1.30315 −0.651576 − 0.758583i \(-0.725892\pi\)
−0.651576 + 0.758583i \(0.725892\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.59859 1.03963 0.519816 − 0.854278i \(-0.326000\pi\)
0.519816 + 0.854278i \(0.326000\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.34889 0.228005
\(36\) 0 0
\(37\) 5.41811 0.890732 0.445366 − 0.895349i \(-0.353074\pi\)
0.445366 + 0.895349i \(0.353074\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.30221 −0.203371 −0.101686 − 0.994817i \(-0.532424\pi\)
−0.101686 + 0.994817i \(0.532424\pi\)
\(42\) 0 0
\(43\) 5.73240 0.874182 0.437091 − 0.899417i \(-0.356009\pi\)
0.437091 + 0.899417i \(0.356009\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.48270 −0.508003 −0.254002 − 0.967204i \(-0.581747\pi\)
−0.254002 + 0.967204i \(0.581747\pi\)
\(48\) 0 0
\(49\) −5.18048 −0.740069
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.51730 −0.345778 −0.172889 − 0.984941i \(-0.555310\pi\)
−0.172889 + 0.984941i \(0.555310\pi\)
\(54\) 0 0
\(55\) −4.69779 −0.633450
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.86620 −1.02409 −0.512046 − 0.858958i \(-0.671112\pi\)
−0.512046 + 0.858958i \(0.671112\pi\)
\(60\) 0 0
\(61\) −5.46479 −0.699695 −0.349848 − 0.936807i \(-0.613767\pi\)
−0.349848 + 0.936807i \(0.613767\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.38350 −0.295637
\(66\) 0 0
\(67\) −1.41811 −0.173250 −0.0866249 − 0.996241i \(-0.527608\pi\)
−0.0866249 + 0.996241i \(0.527608\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.79698 −0.450619 −0.225309 − 0.974287i \(-0.572339\pi\)
−0.225309 + 0.974287i \(0.572339\pi\)
\(72\) 0 0
\(73\) −1.41811 −0.165977 −0.0829886 − 0.996550i \(-0.526447\pi\)
−0.0829886 + 0.996550i \(0.526447\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.33682 −0.722148
\(78\) 0 0
\(79\) −1.81952 −0.204711 −0.102356 − 0.994748i \(-0.532638\pi\)
−0.102356 + 0.994748i \(0.532638\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −13.2151 −1.45054 −0.725272 − 0.688462i \(-0.758286\pi\)
−0.725272 + 0.688462i \(0.758286\pi\)
\(84\) 0 0
\(85\) 1.21509 0.131795
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.20302 0.233519 0.116760 − 0.993160i \(-0.462749\pi\)
0.116760 + 0.993160i \(0.462749\pi\)
\(90\) 0 0
\(91\) −3.21509 −0.337033
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.03461 0.106149
\(96\) 0 0
\(97\) 6.69779 0.680057 0.340029 − 0.940415i \(-0.389563\pi\)
0.340029 + 0.940415i \(0.389563\pi\)
\(98\) 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 5580.2.a.j.1.3 3
3.2 odd 2 1860.2.a.g.1.3 ✓ 3
12.11 even 2 7440.2.a.bn.1.1 3
15.2 even 4 9300.2.g.q.3349.3 6
15.8 even 4 9300.2.g.q.3349.4 6
15.14 odd 2 9300.2.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.g.1.3 ✓ 3 3.2 odd 2
5580.2.a.j.1.3 3 1.1 even 1 trivial
7440.2.a.bn.1.1 3 12.11 even 2
9300.2.a.u.1.1 3 15.14 odd 2
9300.2.g.q.3349.3 6 15.2 even 4
9300.2.g.q.3349.4 6 15.8 even 4