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 = [3,0,0,0,3,0,4,0,0,0,2,0,4,0,0,0,2,0,4,0,0,0,2] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(-2.08613\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +4.43807 q^{7} -4.17226 q^{11} +4.43807 q^{13} +1.35194 q^{17} +2.70388 q^{19} +1.35194 q^{23} +1.00000 q^{25} +7.25839 q^{29} -1.00000 q^{31} +4.43807 q^{35} -10.7826 q^{37} -2.17226 q^{41} +9.46838 q^{43} +4.11644 q^{47} +12.6965 q^{49} -12.8203 q^{53} -4.17226 q^{55} -5.25839 q^{59} +6.00000 q^{61} +4.43807 q^{65} -1.14195 q^{67} +10.7268 q^{71} +10.4381 q^{73} -18.5168 q^{77} -2.05582 q^{79} +6.99258 q^{83} +1.35194 q^{85} +2.55451 q^{89} +19.6965 q^{91} +2.70388 q^{95} -15.9245 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 4 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{41} + 18 q^{43} + 4 q^{47} + 7 q^{49} - 26 q^{53} + 2 q^{55} + 6 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(7\) 4.43807 1.67743 0.838716 − 0.544569i \(-0.183307\pi\)
0.838716 + 0.544569i \(0.183307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.17226 −1.25798 −0.628992 − 0.777412i \(-0.716532\pi\)
−0.628992 + 0.777412i \(0.716532\pi\)
\(12\) 0 0
\(13\) 4.43807 1.23090 0.615449 − 0.788176i \(-0.288975\pi\)
0.615449 + 0.788176i \(0.288975\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.35194 0.327893 0.163947 − 0.986469i \(-0.447577\pi\)
0.163947 + 0.986469i \(0.447577\pi\)
\(18\) 0 0
\(19\) 2.70388 0.620312 0.310156 − 0.950686i \(-0.399619\pi\)
0.310156 + 0.950686i \(0.399619\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.35194 0.281899 0.140949 − 0.990017i \(-0.454985\pi\)
0.140949 + 0.990017i \(0.454985\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.25839 1.34785 0.673925 − 0.738800i \(-0.264607\pi\)
0.673925 + 0.738800i \(0.264607\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.43807 0.750171
\(36\) 0 0
\(37\) −10.7826 −1.77265 −0.886323 − 0.463067i \(-0.846749\pi\)
−0.886323 + 0.463067i \(0.846749\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.17226 −0.339250 −0.169625 − 0.985509i \(-0.554256\pi\)
−0.169625 + 0.985509i \(0.554256\pi\)
\(42\) 0 0
\(43\) 9.46838 1.44391 0.721957 − 0.691938i \(-0.243243\pi\)
0.721957 + 0.691938i \(0.243243\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.11644 0.600445 0.300222 − 0.953869i \(-0.402939\pi\)
0.300222 + 0.953869i \(0.402939\pi\)
\(48\) 0 0
\(49\) 12.6965 1.81378
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.8203 −1.76101 −0.880503 − 0.474040i \(-0.842795\pi\)
−0.880503 + 0.474040i \(0.842795\pi\)
\(54\) 0 0
\(55\) −4.17226 −0.562587
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.25839 −0.684584 −0.342292 − 0.939594i \(-0.611203\pi\)
−0.342292 + 0.939594i \(0.611203\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.43807 0.550475
\(66\) 0 0
\(67\) −1.14195 −0.139511 −0.0697556 − 0.997564i \(-0.522222\pi\)
−0.0697556 + 0.997564i \(0.522222\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.7268 1.27303 0.636517 − 0.771263i \(-0.280375\pi\)
0.636517 + 0.771263i \(0.280375\pi\)
\(72\) 0 0
\(73\) 10.4381 1.22168 0.610842 − 0.791753i \(-0.290831\pi\)
0.610842 + 0.791753i \(0.290831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −18.5168 −2.11018
\(78\) 0 0
\(79\) −2.05582 −0.231298 −0.115649 − 0.993290i \(-0.536895\pi\)
−0.115649 + 0.993290i \(0.536895\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.99258 0.767536 0.383768 − 0.923430i \(-0.374626\pi\)
0.383768 + 0.923430i \(0.374626\pi\)
\(84\) 0 0
\(85\) 1.35194 0.146638
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.55451 0.270778 0.135389 − 0.990793i \(-0.456772\pi\)
0.135389 + 0.990793i \(0.456772\pi\)
\(90\) 0 0
\(91\) 19.6965 2.06475
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.70388 0.277412
\(96\) 0 0
\(97\) −15.9245 −1.61689 −0.808446 − 0.588570i \(-0.799691\pi\)
−0.808446 + 0.588570i \(0.799691\pi\)
\(98\) 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.l.1.3 3
3.2 odd 2 1860.2.a.f.1.3 ✓ 3
12.11 even 2 7440.2.a.bt.1.1 3
15.2 even 4 9300.2.g.p.3349.6 6
15.8 even 4 9300.2.g.p.3349.1 6
15.14 odd 2 9300.2.a.v.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.f.1.3 ✓ 3 3.2 odd 2
5580.2.a.l.1.3 3 1.1 even 1 trivial
7440.2.a.bt.1.1 3 12.11 even 2
9300.2.a.v.1.1 3 15.14 odd 2
9300.2.g.p.3349.1 6 15.8 even 4
9300.2.g.p.3349.6 6 15.2 even 4