↑

Properties

Label 5904.2.a.bh.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 $1$

Related objects

Downloads

Learn more

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,-2,0,-4,0,0,0,-4] 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: 3.3.788.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 328)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.87740\) 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)
 
\(f(q)\) \(=\) \(q+2.27945 q^{5} +0.877404 q^{7} +0.877404 q^{11} +3.75481 q^{13} -6.00000 q^{17} +5.68150 q^{19} +8.55890 q^{23} +0.195903 q^{25} -4.80410 q^{29} +4.55890 q^{31} +2.00000 q^{35} +2.27945 q^{37} -1.00000 q^{41} -6.31371 q^{43} +8.63221 q^{47} -6.23016 q^{49} +8.31371 q^{53} +2.00000 q^{55} +4.55890 q^{59} -3.19590 q^{61} +8.55890 q^{65} -8.24041 q^{67} +11.4363 q^{71} +2.27945 q^{73} +0.769837 q^{77} -2.87740 q^{79} -17.1178 q^{83} -13.6767 q^{85} +10.5589 q^{89} +3.29448 q^{91} +12.9507 q^{95} +13.5096 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 2 q^{13} - 18 q^{17} + 6 q^{19} + 8 q^{23} + 5 q^{25} - 10 q^{29} - 4 q^{31} + 6 q^{35} - 2 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} - q^{49} - 6 q^{53} + 6 q^{55}+ \cdots + 14 q^{97}+O(q^{100}) \) 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
\(5\) 2.27945 1.01940 0.509701 − 0.860352i \(-0.329756\pi\)
0.509701 + 0.860352i \(0.329756\pi\)
\(6\) 0 0
\(7\) 0.877404 0.331627 0.165814 − 0.986157i \(-0.446975\pi\)
0.165814 + 0.986157i \(0.446975\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.877404 0.264547 0.132274 − 0.991213i \(-0.457772\pi\)
0.132274 + 0.991213i \(0.457772\pi\)
\(12\) 0 0
\(13\) 3.75481 1.04140 0.520698 − 0.853741i \(-0.325672\pi\)
0.520698 + 0.853741i \(0.325672\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 5.68150 1.30343 0.651713 − 0.758466i \(-0.274051\pi\)
0.651713 + 0.758466i \(0.274051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.55890 1.78465 0.892327 − 0.451389i \(-0.149071\pi\)
0.892327 + 0.451389i \(0.149071\pi\)
\(24\) 0 0
\(25\) 0.195903 0.0391806
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.80410 −0.892098 −0.446049 − 0.895008i \(-0.647169\pi\)
−0.446049 + 0.895008i \(0.647169\pi\)
\(30\) 0 0
\(31\) 4.55890 0.818803 0.409402 − 0.912354i \(-0.365737\pi\)
0.409402 + 0.912354i \(0.365737\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 2.27945 0.374740 0.187370 − 0.982289i \(-0.440004\pi\)
0.187370 + 0.982289i \(0.440004\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −6.31371 −0.962832 −0.481416 − 0.876492i \(-0.659877\pi\)
−0.481416 + 0.876492i \(0.659877\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.63221 1.25914 0.629569 − 0.776945i \(-0.283232\pi\)
0.629569 + 0.776945i \(0.283232\pi\)
\(48\) 0 0
\(49\) −6.23016 −0.890023
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.31371 1.14198 0.570988 − 0.820958i \(-0.306560\pi\)
0.570988 + 0.820958i \(0.306560\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.55890 0.593519 0.296759 − 0.954952i \(-0.404094\pi\)
0.296759 + 0.954952i \(0.404094\pi\)
\(60\) 0 0
\(61\) −3.19590 −0.409193 −0.204597 − 0.978846i \(-0.565588\pi\)
−0.204597 + 0.978846i \(0.565588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.55890 1.06160
\(66\) 0 0
\(67\) −8.24041 −1.00673 −0.503363 − 0.864075i \(-0.667904\pi\)
−0.503363 + 0.864075i \(0.667904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4363 1.35724 0.678620 − 0.734490i \(-0.262578\pi\)
0.678620 + 0.734490i \(0.262578\pi\)
\(72\) 0 0
\(73\) 2.27945 0.266790 0.133395 − 0.991063i \(-0.457412\pi\)
0.133395 + 0.991063i \(0.457412\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.769837 0.0877311
\(78\) 0 0
\(79\) −2.87740 −0.323733 −0.161867 − 0.986813i \(-0.551751\pi\)
−0.161867 + 0.986813i \(0.551751\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −17.1178 −1.87892 −0.939462 − 0.342654i \(-0.888674\pi\)
−0.939462 + 0.342654i \(0.888674\pi\)
\(84\) 0 0
\(85\) −13.6767 −1.48345
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.5589 1.11924 0.559621 − 0.828749i \(-0.310947\pi\)
0.559621 + 0.828749i \(0.310947\pi\)
\(90\) 0 0
\(91\) 3.29448 0.345356
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.9507 1.32871
\(96\) 0 0
\(97\) 13.5096 1.37169 0.685847 − 0.727746i \(-0.259432\pi\)
0.685847 + 0.727746i \(0.259432\pi\)
\(98\) 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.bh.1.3 3
3.2 odd 2 656.2.a.g.1.3 3
4.3 odd 2 2952.2.a.l.1.3 3
12.11 even 2 328.2.a.e.1.1 ✓ 3
24.5 odd 2 2624.2.a.o.1.1 3
24.11 even 2 2624.2.a.t.1.3 3
60.59 even 2 8200.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
328.2.a.e.1.1 ✓ 3 12.11 even 2
656.2.a.g.1.3 3 3.2 odd 2
2624.2.a.o.1.1 3 24.5 odd 2
2624.2.a.t.1.3 3 24.11 even 2
2952.2.a.l.1.3 3 4.3 odd 2
5904.2.a.bh.1.3 3 1.1 even 1 trivial
8200.2.a.w.1.3 3 60.59 even 2