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Properties

Label 5904.2.a.bn.1.3
Level $5904$
Weight $2$
Character 5904.1
Self dual yes
Analytic conductor $47.144$
Analytic rank $0$
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] = [5904,2,Mod(1,5904)] 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("5904.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5904, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5904 = 2^{4} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5904.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,4,0,-4,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.1436773534\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1476)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 5904.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.34889 q^{5} -3.86620 q^{7} -5.38350 q^{11} +0.831590 q^{13} -0.866198 q^{17} -1.86620 q^{19} +5.08129 q^{23} +6.21509 q^{25} +3.65111 q^{29} +9.94749 q^{31} -12.9475 q^{35} -10.9821 q^{37} -1.00000 q^{41} +12.3956 q^{43} -7.56399 q^{47} +7.94749 q^{49} -7.98210 q^{53} -18.0288 q^{55} +10.1626 q^{59} -4.66318 q^{61} +2.78491 q^{65} +10.4302 q^{67} +3.20302 q^{71} +6.03461 q^{73} +20.8137 q^{77} +1.66318 q^{79} -14.1159 q^{83} -2.90081 q^{85} +17.4648 q^{89} -3.21509 q^{91} -6.24970 q^{95} -7.86620 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{5} - 4 q^{7} - 7 q^{11} - 2 q^{13} + 5 q^{17} + 2 q^{19} - 6 q^{23} + 5 q^{25} + 17 q^{29} + q^{31} - 10 q^{35} - q^{37} - 3 q^{41} + 13 q^{43} - 3 q^{47} - 5 q^{49} + 8 q^{53} - 4 q^{55} - 12 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 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
\(5\) 3.34889 1.49767 0.748836 − 0.662756i \(-0.230613\pi\)
0.748836 + 0.662756i \(0.230613\pi\)
\(6\) 0 0
\(7\) −3.86620 −1.46129 −0.730643 − 0.682760i \(-0.760779\pi\)
−0.730643 + 0.682760i \(0.760779\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.38350 −1.62319 −0.811594 − 0.584223i \(-0.801400\pi\)
−0.811594 + 0.584223i \(0.801400\pi\)
\(12\) 0 0
\(13\) 0.831590 0.230642 0.115321 − 0.993328i \(-0.463210\pi\)
0.115321 + 0.993328i \(0.463210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.866198 −0.210084 −0.105042 − 0.994468i \(-0.533498\pi\)
−0.105042 + 0.994468i \(0.533498\pi\)
\(18\) 0 0
\(19\) −1.86620 −0.428135 −0.214068 − 0.976819i \(-0.568671\pi\)
−0.214068 + 0.976819i \(0.568671\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.08129 1.05952 0.529761 − 0.848147i \(-0.322282\pi\)
0.529761 + 0.848147i \(0.322282\pi\)
\(24\) 0 0
\(25\) 6.21509 1.24302
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.65111 0.677993 0.338997 − 0.940788i \(-0.389912\pi\)
0.338997 + 0.940788i \(0.389912\pi\)
\(30\) 0 0
\(31\) 9.94749 1.78662 0.893311 − 0.449439i \(-0.148376\pi\)
0.893311 + 0.449439i \(0.148376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.9475 −2.18853
\(36\) 0 0
\(37\) −10.9821 −1.80545 −0.902723 − 0.430223i \(-0.858435\pi\)
−0.902723 + 0.430223i \(0.858435\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 12.3956 1.89031 0.945154 − 0.326625i \(-0.105912\pi\)
0.945154 + 0.326625i \(0.105912\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.56399 −1.10332 −0.551660 − 0.834069i \(-0.686006\pi\)
−0.551660 + 0.834069i \(0.686006\pi\)
\(48\) 0 0
\(49\) 7.94749 1.13536
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.98210 −1.09643 −0.548213 − 0.836339i \(-0.684692\pi\)
−0.548213 + 0.836339i \(0.684692\pi\)
\(54\) 0 0
\(55\) −18.0288 −2.43100
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.1626 1.32306 0.661528 − 0.749921i \(-0.269908\pi\)
0.661528 + 0.749921i \(0.269908\pi\)
\(60\) 0 0
\(61\) −4.66318 −0.597059 −0.298530 − 0.954400i \(-0.596496\pi\)
−0.298530 + 0.954400i \(0.596496\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.78491 0.345425
\(66\) 0 0
\(67\) 10.4302 1.27425 0.637125 − 0.770761i \(-0.280124\pi\)
0.637125 + 0.770761i \(0.280124\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.20302 0.380128 0.190064 − 0.981772i \(-0.439130\pi\)
0.190064 + 0.981772i \(0.439130\pi\)
\(72\) 0 0
\(73\) 6.03461 0.706297 0.353149 − 0.935567i \(-0.385111\pi\)
0.353149 + 0.935567i \(0.385111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 20.8137 2.37194
\(78\) 0 0
\(79\) 1.66318 0.187122 0.0935612 − 0.995614i \(-0.470175\pi\)
0.0935612 + 0.995614i \(0.470175\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.1159 −1.54942 −0.774711 − 0.632316i \(-0.782104\pi\)
−0.774711 + 0.632316i \(0.782104\pi\)
\(84\) 0 0
\(85\) −2.90081 −0.314637
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.4648 1.85126 0.925632 − 0.378424i \(-0.123534\pi\)
0.925632 + 0.378424i \(0.123534\pi\)
\(90\) 0 0
\(91\) −3.21509 −0.337033
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.24970 −0.641206
\(96\) 0 0
\(97\) −7.86620 −0.798691 −0.399346 − 0.916800i \(-0.630763\pi\)
−0.399346 + 0.916800i \(0.630763\pi\)
\(98\) 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 5904.2.a.bn.1.3 3
3.2 odd 2 5904.2.a.bc.1.1 3
4.3 odd 2 1476.2.a.f.1.3 yes 3
12.11 even 2 1476.2.a.e.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1476.2.a.e.1.1 ✓ 3 12.11 even 2
1476.2.a.f.1.3 yes 3 4.3 odd 2
5904.2.a.bc.1.1 3 3.2 odd 2
5904.2.a.bn.1.3 3 1.1 even 1 trivial